General College Chemistry
Foreword
Unit 1. Solutions and Colligative Properties
1. Introduction and Review of Thermodynamics
2. Solutions
3. Osmotic Pressure
4. Vapor Pressure
5. Melting/Boiling
Unit 2. Kinetics
6. Reaction Rates
7. Experimental Rate Laws
8. Integrated Rate Laws
9. Temperature and Rate
10. Reaction Mechanisms and Catalysis
Unit 3. Nuclear Chemistry
11. Radioactive Decay
12. Rates of Decay, Radiodating
13. Fission and Fusion
14. Biological Effects
Unit 4. Equilibrium and Thermodynamics
15. Equilibrium Constants
16. Reaction Quotients and Le Chatelier
17. Equilibrium Calculations
18. Review of Thermodynamics Calculations
19. ΔG and Q, ΔG° and K
20. Temperature Dependence of K
Unit 5. Acids, Bases, and Buffers
21. Acids and Bases
22. Autoionization and pH Calculations
23. Weak Acids and Bases
24. Polyprotic Acids
25. pH of Salt Solutions
26. Buffers
27. DBL Acid Base
28. Titrations I
29. Titrations II
Unit 6. Solubility and Complex Ion Equilibria
30. Solubility Equilibria
31. Complex Ion Equilibria
32. Chelation, Biological Effects
33. Crystal Field Theory
34. Isomerism, Equilibria and Biological Systems
Unit 7. Electrochemistry and Solids
35. Redox, Electrochemical Cells
36. Cell Potentials
37. Nernst Equation
38. Batteries and Fuel Cells
39. Metals, Alloys and Semiconductors
40. Covalent Network Solids, Salts, Polymers
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Introduction and Review of Thermodynamics
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<div class="callout default" data-callout-custom-class="default"><h3 class="" id="h3_cNmu" name="h3_cNmu">How to Study Chem <span data-etb-numbered-element="1">1.0.1</span> More Effectively</h3><p id="p_GQmb" class="" name="p_GQmb"></p><figure class="col-12"><a href="https://www.youtube.com/watch?v=nWxgeENGbns&autoplay=1" data-mce-href="https://www.youtube.com/watch?v=nWxgeENGbns&autoplay=1" class="position-relative text-center d-block"><img class="img-fluid" id="YT_nWxgeENGbns" data-youtube-thumbnail-style="max" src="https://i3.ytimg.com/vi/nWxgeENGbns/maxresdefault.jpg" data-mce-src="https://i3.ytimg.com/vi/nWxgeENGbns/maxresdefault.jpg"></a><figcaption class="text-center"><a target="_blank" href="https://www.youtube.com/watch?v=nWxgeENGbns&autoplay=1" rel="noopener" data-mce-href="https://www.youtube.com/watch?v=nWxgeENGbns&autoplay=1">Watch on YouTube</a></figcaption></figure></div><h2 id="h2_ChJr" class="" name="h2_ChJr"><span data-etb-numbered-element="1">1.1</span> Spontaneity</h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_GgGG" name="h3_GgGG">Learning Objectives</h3></header><p id="para-00001" class="" name="para-00001">By the end of this section, you will be able to:</p><ul id="list-00001" class="" name="list-00001"><li id="li_aqYX" class="" name="li_aqYX">Distinguish between spontaneous and nonspontaneous processes</li><li id="li_Jxna" class="" name="li_Jxna">Describe the dispersal of matter and energy that accompanies certain spontaneous processes</li></ul></div><p id="fs-idp116177632" class="" name="fs-idp116177632">Processes have a natural tendency to occur in one direction under a given set of conditions. Water will naturally flow downhill but uphill flow requires outside intervention such as the use of a pump. Iron exposed to the earth’s atmosphere will corrode but rust is not converted to iron without intentional chemical treatment. A <span id="term532" class="" data-type="term">spontaneous process</span> is one that occurs naturally under certain conditions. A <span id="term533" class="" data-type="term">nonspontaneous process </span>on the other hand, will not take place unless it is “driven” by the continual input of energy from an external source. A process that is spontaneous in one direction under a particular set of conditions is nonspontaneous in the reverse direction. At room temperature and typical atmospheric pressure, for example, ice will spontaneously melt but water will not spontaneously freeze.</p><p id="fs-idp3956976" class="" name="fs-idp3956976">The spontaneity of a process is <em class="" data-effect="italics">not</em> correlated to the speed of the process. A spontaneous change may be so rapid that it is essentially instantaneous or so slow that it cannot be observed over any practical period of time. To illustrate this concept, consider the decay of radioactive isotopes, a topic more thoroughly treated in the chapter on nuclear chemistry. Radioactive decay is by definition a spontaneous process in which the nuclei of unstable isotopes emit radiation as they are converted to more stable nuclei. All the decay processes occur spontaneously but the rates at which different isotopes decay vary widely. Technetium-99m is a popular radioisotope for medical imaging studies that undergoes relatively rapid decay and exhibits a half-life of about six hours. Uranium-238 is the most abundant isotope of uranium and its decay occurs much more slowly exhibiting a half-life of more than four billion years (<a class="autogenerated-content anchor-link" href="#p_MLFR" data-mce-href="#p_MLFR">Figure 1.1</a>).</p><p id="p_MLFR" class="os-caption-container table_figure_number" name="p_MLFR">Figure <span data-etb-numbered-element="1">1.1</span></p><p id="p_CAPo" class="os-caption table_figure_title" name="p_CAPo">Both U-238 and Tc-99m undergo spontaneous radioactive decay, but at drastically different rates. Over the course of one week, essentially all of a Tc-99m sample and none of a U-238 sample will have decayed.</p><figure class="scaled-down" data-id="CNX_Chem_16_01_decayrates"><span id="fs-idp34729504" class="" data-type="media" data-alt="A graph of two lines is shown where the y-axis is labeled, “amount of isotope remaining ( percent sign ),” and has values zero through one hundred, in increments of ten, written along the axis. The x-axis is labeled, “time ( days )” and has values zero through seven, in increments of one, written along the axis. The first graph, drawn with a blue line, begins at the top left value of one hundred on the y-axis and zero on the x-axis and falls steeply over the first three minutes, then the graphed line becomes almost horizontal until it reaches seven minutes on the x-axis. The second graph, drawn in red, begins at the same point as the first, but remains perfectly horizontal with no change along the y-axis. A legend labels the red line as, “U dash 238,” and the blue line as,"><img id="img_nagR" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_3.jpg" alt="A graph of two lines is shown where the y-axis is labeled, “amount of isotope remaining ( percent sign ),” and has values zero through one hundred, in increments of ten, written along the axis. The x-axis is labeled, “time ( days )” and has values zero through seven, in increments of one, written along the axis. The first graph, drawn with a blue line, begins at the top left value of one hundred on the y-axis and zero on the x-axis and falls steeply over the first three minutes, then the graphed line becomes almost horizontal until it reaches seven minutes on the x-axis. The second graph, drawn in red, begins at the same point as the first, but remains perfectly horizontal with no change along the y-axis. A legend labels the red line as, “U dash 238,” and the blue line as," name="img_nagR" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_3.jpg" style="display: block; margin-left: auto; margin-right: auto;" data-mce-style="display: block; margin-left: auto; margin-right: auto;"></span></figure><p id="fs-idm7164864" class="" name="fs-idm7164864">As another example, consider the conversion of diamond into graphite (<a class="autogenerated-content anchor-link" href="#p_IpbV" data-mce-href="#p_IpbV">Figure 1.2</a>).</p><div id="fs-idm44290000" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-270-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>diamond</mtext><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>graphite</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>diamond</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>graphite</mtext><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>diamond</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>graphite</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">C</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">diamond</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">C</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">graphite</mtext><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp124196272" class="" name="fs-idp124196272">The phase diagram for carbon indicates that graphite is the stable form of this element under ambient atmospheric pressure while diamond is the stable allotrope at very high pressures such as those present during its geologic formation. Thermodynamic calculations of the sort described in the last section of this chapter indicate that the conversion of diamond to graphite at ambient pressure occurs spontaneously yet diamonds are observed to exist, and persist, under these conditions. Though the process is spontaneous under typical ambient conditions, its rate is extremely slow; so, for all practical purposes diamonds are indeed “forever.” Situations such as these emphasize the important distinction between the thermodynamic and the kinetic aspects of a process. In this particular case, diamonds are said to be <em class="" data-effect="italics">thermodynamically unstable</em> but <em class="" data-effect="italics">kinetically stable</em> under ambient conditions.</p><p id="p_IpbV" class="os-caption-container table_figure_number" name="p_IpbV">Figure <span data-etb-numbered-element="1">1.2</span></p><p id="p_qoKc" class="os-caption table_figure_title" name="p_qoKc">The conversion of carbon from the diamond allotrope to the graphite allotrope is spontaneous at ambient pressure, but its rate is immeasurably slow at low to moderate temperatures. This process is known as <em class="" data-effect="italics">graphitization </em>and its rate can be increased to easily measurable values at temperatures in the 1000–2000 K range. (credit "diamond" photo: modification of work by "Fancy Diamonds"/Flickr; credit "graphite" photo: modification of work by images-of-elements.com/carbon.php)</p><figure class="col-md-12 mx-auto" data-id="CNX_Chem_16_01_carbon"><img id="img_Hfip" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_4.jpg" alt="Two pairs of images are shown. The left pair, labeled, “C, ( diamond ),” has a picture of a diamond held by a pair of plyers and a diagram of the molecular arrangement. The second pair, labeled, “C ( graphite ),” has a picture of a large, black, slightly shiny rock and a diagram of four sheets composed of many atoms arranged in large squares in a stacked arrangement with space between each." name="img_Hfip" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_4.jpg"></figure><h3 class="" data-type="title" id="h3_ykpb" name="h3_ykpb"><span data-etb-numbered-element="1">1.1.1</span> Dispersal of Matter and Energy</h3><p id="fs-idp148339312" class="" name="fs-idp148339312">Extending the discussion of thermodynamic concepts toward the objective of predicting spontaneity, consider now an isolated system consisting of two flasks connected with a closed valve. Initially there is an ideal gas in one flask and the other flask is empty (<em class="" data-effect="italics">P</em> = 0). (<a class="autogenerated-content anchor-link" href="#p_kdZU" data-mce-href="#p_kdZU">Figure 1.3</a>). When the valve is opened, the gas spontaneously expands to fill both flasks equally. Recalling the definition of pressure-volume work from the chapter on thermochemistry note that no work has been done because the pressure in a vacuum is zero.</p><div id="fs-idm38278672" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-271-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>w</mi><mo>=</mo><mtext>−</mtext><mi>P</mi><mtext>Δ</mtext><mi>V</mi><mo>=</mo><mn>0</mn><mspace width="5em" ></mspace><mo stretchy="false">(</mo><mi>P</mi><mo>=</mo><mtext>0 in a vacuum</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>w</mi><mo>=</mo><mtext>−</mtext><mi>P</mi><mtext>Δ</mtext><mi>V</mi><mo>=</mo><mn>0</mn><mspace width="5em"></mspace><mo stretchy="false">(</mo><mi>P</mi><mo>=</mo><mtext>0 in a vacuum</mtext><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>U</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>w</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">U</mi><mo class="">=</mo><mi class="">q</mi><mo class="">+</mo><mi class="">w</mi><mo class="">=</mo><mn class="">0</mn><mo class="">+</mo><mn class="">0</mn><mo class="">=</mo><mn class="">0</mn></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm42330576" class="" name="fs-idm42330576">Note as well that since the system is isolated, no heat has been exchanged with the surroundings (<em class="" data-effect="italics">q</em> = 0). The <em class="" data-effect="italics">first law of thermodynamics</em> confirms that there has been no change in the system’s internal energy as a result of this process.</p><div id="fs-idp27062144" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-272-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>U</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>w</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>U</mi><mo>=</mo><mi>q</mi><mo>+</mo><mi>w</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">q</mi><mrow class=""><mtext class="">rev</mtext></mrow></msub></mrow><mi class="">T</mi></mfrac></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm47467840" class="" name="fs-idm47467840">The spontaneity of this process is therefore not a consequence of any change in energy that accompanies the process. Instead, the driving force appears to be related to the <em class="" data-effect="italics">greater, more uniform dispersal of matter</em> that results when the gas is allowed to expand. Initially, the system was comprised of one flask containing matter and another flask containing nothing. After the spontaneous expansion took place, the matter was distributed both more widely (occupying twice its original volume) and more uniformly (present in equal amounts in each flask).</p><div id="CNX_Chem_16_02_Gas" class="os-figure"><p id="p_kdZU" class="os-caption-container table_figure_number" name="p_kdZU">Figure <span data-etb-numbered-element="1">1.3</span></p><p id="p_fgiE" class="os-caption table_figure_title" name="p_fgiE">An isolated system consists of an ideal gas in one flask that is connected by a closed valve to a second flask containing a vacuum. Once the valve is opened, the gas spontaneously becomes evenly distributed between the flasks.</p><figure class="" data-id="CNX_Chem_16_02_Gas"><span id="fs-idp117381408" class="" data-type="media" data-alt="A diagram shows two two-sided flasks connected by a right-facing arrow labeled “Spontaneous” and a left-facing arrow labeled “Nonspontaneous.” Each pair of flasks are connected to one another by a tube with a stopcock. In the left pair of flasks, the left flask contains thirty particles evenly dispersed while the right flask contains nothing and the stopcock is closed. The right pair of flasks has an open stopcock and equal numbers of particles in both flasks."><img id="img_jhBi" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_5.jpg" alt="A diagram shows two two-sided flasks connected by a right-facing arrow labeled “Spontaneous” and a left-facing arrow labeled “Nonspontaneous.” Each pair of flasks are connected to one another by a tube with a stopcock. In the left pair of flasks, the left flask contains thirty particles evenly dispersed while the right flask contains nothing and the stopcock is closed. The right pair of flasks has an open stopcock and equal numbers of particles in both flasks." name="img_jhBi" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_5.jpg"></span></figure></div><p id="fs-idp202853936" class="" name="fs-idp202853936">Now consider two objects at different temperatures: object X at temperature <em class="" data-effect="italics">T</em><sub class="">X</sub> and object Y at temperature <em class="" data-effect="italics">T</em><sub class="">Y</sub>, with <em class="" data-effect="italics">T</em><sub class="">X</sub> > <em class="" data-effect="italics">T</em><sub class="">Y</sub> (<a class="autogenerated-content anchor-link" href="#p_CTqe" data-mce-href="#p_CTqe">Figure 1.4</a>). When these objects come into contact, heat spontaneously flows from the hotter object (X) to the colder one (Y). This corresponds to a loss of thermal energy by X and a gain of thermal energy by Y.</p><div id="fs-idp36377040" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-273-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mi>q</mi><mtext>X</mtext></msub><mo><</mo><mn>0</mn><mspace width="2em" ></mspace><mtext>and</mtext><mspace width="2em" ></mspace><msub><mi>q</mi><mtext>Y</mtext></msub><mo>=</mo><mtext>−</mtext><msub><mi>q</mi><mtext>X</mtext></msub><mo>></mo><mn>0</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mi>q</mi><mtext>X</mtext></msub><mo><</mo><mn>0</mn><mspace width="2em"></mspace><mtext>and</mtext><mspace width="2em"></mspace><msub><mi>q</mi><mtext>Y</mtext></msub><mo>=</mo><mtext>−</mtext><msub><mi>q</mi><mtext>X</mtext></msub><mo>></mo><mn>0</mn></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>×</mo></mrow><annotation-xml encoding="MathML-Content"><mo xmlns="http://www.w3.org/1998/Math/MathML" class="">×</mo></annotation-xml></semantics></math></span></div></div><p id="fs-idp120750768" class="" name="fs-idp120750768">From the perspective of this two-object system, there was no net gain or loss of thermal energy, rather the available thermal energy was redistributed among the two objects. This spontaneous process resulted in a <em class="" data-effect="italics">more uniform dispersal of energy</em>.</p><p id="p_CTqe" class="os-caption-container table_figure_number" name="p_CTqe">Figure <span data-etb-numbered-element="1">1.4</span></p><p id="p_nTZw" class="os-caption table_figure_title" name="p_nTZw">When two objects at different temperatures come in contact, heat spontaneously flows from the hotter to the colder object.</p><figure class="scaled-down" data-id="CNX_Chem_16_02_Temperature"><span id="fs-idp106848832" class="" data-type="media" data-alt="Two diagrams are shown. The left diagram is comprised of two separated squares; the left is red and labeled “X” and the right is blue and labeled “Y.” Below this diagram is the label “T subscript X, a greater than sign, T subscript Y.” The right diagram shows the boxes next to one another, shaded red on the left, blue on the right, and blended red and blue together in the middle. The left box is red and labeled “X,” the right is blue and labeled “Y” and a right-facing arrow labeled “Heat” is written above them. Below this diagram is the label “X and Y in contact."><img id="img_iQmr" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_6.jpg" alt="Two diagrams are shown. The left diagram is comprised of two separated squares; the left is red and labeled “X” and the right is blue and labeled “Y.” Below this diagram is the label “T subscript X, a greater than sign, T subscript Y.” The right diagram shows the boxes next to one another, shaded red on the left, blue on the right, and blended red and blue together in the middle. The left box is red and labeled “X,” the right is blue and labeled “Y” and a right-facing arrow labeled “Heat” is written above them. Below this diagram is the label “X and Y in contact." name="img_iQmr" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_6.jpg"></span></figure><p id="fs-idp55398608" class="" name="fs-idp55398608">As illustrated by the two processes described, an important factor in determining the spontaneity of a process is the extent to which it changes the dispersal or distribution of matter and/or energy. In each case, a spontaneous process took place that resulted in a more uniform distribution of matter or energy.</p><div id="fs-idp106049744" class="ui-has-child-title callout" data-type="example"><h3 class="os-title" id="h3_Xnks" name="h3_Xnks">EXAMPLE <span data-etb-numbered-element="1">1.1.2</span></h3><h4 id="h4_MYaT" class="" data-type="title" name="h4_MYaT">Redistribution of Matter during a Spontaneous Process</h4><p id="p_HaCw" name="p_HaCw">Describe how matter is redistributed when the following spontaneous processes take place:</p><p id="fs-idm25379552" class="" name="fs-idm25379552">(a) A solid sublimes.</p><p id="fs-idm37056304" class="" name="fs-idm37056304">(b) A gas condenses.</p><p id="fs-idp198218368" class="" name="fs-idp198218368">(c) A drop of food coloring added to a glass of water forms a solution with uniform color.</p><h4 id="h4_AWUs" class="" data-type="title" name="h4_AWUs"><a id="Figure_1.5" class="" href="https://edtechbooks.org/" data-mce-href="https://edtechbooks.org/"></a>Solution</h4><p id="p_gcND" class="os-caption-container table_figure_number" name="p_gcND">Figure <span data-etb-numbered-element="1">1.5</span></p><p id="p_MVdY" class="os-caption table_figure_title" name="p_MVdY">(credit a: modification of work by Jenny Downing; credit b: modification of work by “Fuzzy Gerdes”/Flickr; credit c: modification of work by Paul A. Flowers)</p><figure class="scaled-down" data-id="CNX_Chem_16_02_Process"><span id="fs-idp205626528" class="" data-type="media" data-alt="This figure has three photos labeled, “a,” “b,” and “c.” Photo a shows a glass with a solid in water. There is steam or smoke coming from the top of the glass. Photo b shows the bottom half of a glass with water sticking to its outside surface. Photo c shows three images of the same container. The first shows a clear liquid in the container. The second shows a red liquid mixing with the clear liquid in the container. The third shows a red liquid."><img id="img_MrTq" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_7.jpg" alt="This figure has three photos labeled, “a,” “b,” and “c.” Photo a shows a glass with a solid in water. There is steam or smoke coming from the top of the glass. Photo b shows the bottom half of a glass with water sticking to its outside surface. Photo c shows three images of the same container. The first shows a clear liquid in the container. The second shows a red liquid mixing with the clear liquid in the container. The third shows a red liquid." name="img_MrTq" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_7.jpg"></span></figure><p id="fs-idm48323968" class="" name="fs-idm48323968">(a) Sublimation is the conversion of a solid (relatively high density) to a gas (much lesser density). This process yields a much greater dispersal of matter, since the molecules will occupy a much greater volume after the solid-to-gas transition.</p><p id="fs-idm4737472" class="" name="fs-idm4737472">(b) Condensation is the conversion of a gas (relatively low density) to a liquid (much greater density). This process yields a much lesser dispersal of matter, since the molecules will occupy a much lesser volume after the gas-to-liquid transition.</p><p id="fs-idp54655312" class="" name="fs-idp54655312">(c) The process in question is diffusion. This process yields a more uniform dispersal of matter since the initial state of the system involves two regions of different dye concentrations (high in the drop of dye, zero in the water), and the final state of the system contains a single dye concentration throughout.</p><h4 id="h4_eXFo" class="" data-type="title" name="h4_eXFo">Check Your Learning</h4><p id="p_PZst" name="p_PZst">Describe how energy is redistributed when a spoon at room temperature is placed in a cup of hot coffee.</p><div id="fs-idp37406832" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_a7mZPPgo" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_a7mZPPgo" data-bs-toggle="collapse" id="h3_FAKK" name="h3_FAKK" data-mce-href="#answer_a7mZPPgo" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_a7mZPPgo" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm37784928" class="" name="fs-idm37784928">Heat will spontaneously flow from the hotter object (coffee) to the colder object (spoon), resulting in a more uniform distribution of thermal energy as the spoon warms and the coffee cools.</p></div></section></div></div><h2 id="h2_JJTq" class="" name="h2_JJTq"><span data-etb-numbered-element="1">1.2</span> Entropy </h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_gJcC" name="h3_gJcC">Learning Objectives</h3></header><p id="para-00001" class="" name="para-00001">By the end of this section, you will be able to:</p><ul id="list-00001" class="" name="list-00001"><li id="li_gwqn" class="" name="li_gwqn">Define entropy</li><li id="li_RutE" class="" name="li_RutE">Explain the relationship between entropy and the number of microstates</li><li id="li_NxwF" class="" name="li_NxwF">Predict the sign of the entropy change for chemical and physical processes</li></ul></div><p id="fs-idm1359840" class="" name="fs-idm1359840">In 1824, at the age of 28, Nicolas Léonard Sadi <span id="term534" class="no-emphasis" data-type="term">Carnot</span> (<a class="autogenerated-content" href="#Figure_1.6" data-mce-href="#Figure_1.6">Figure 1.6</a>) published the results of an extensive study regarding the efficiency of steam heat engines. A later review of Carnot’s findings by Rudolf <span id="term535" class="no-emphasis" data-type="term">Clausius</span> introduced a new thermodynamic property that relates the spontaneous heat flow accompanying a process to the temperature at which the process takes place. This new property was expressed as the ratio of the <em class="" data-effect="italics">reversible</em> heat (<em class="" data-effect="italics">q</em><sub class="">rev</sub>) and the kelvin temperature (<em class="" data-effect="italics">T</em>). In thermodynamics, a <span id="term536" class="" data-type="term">reversible process</span> is one that takes place at such a slow rate that it is always at equilibrium and its direction can be changed (it can be “reversed”) by an infinitesimally small change in some condition. Note that the idea of a reversible process is a formalism required to support the development of various thermodynamic concepts; no real processes are truly reversible rather they are classified as <em class="" data-effect="italics">irreversible</em>.</p><p id="p_SESV" class="os-caption-container table_figure_number" name="p_SESV">Figure <span data-etb-numbered-element="1">1.6</span></p><p id="p_hHBD" class="os-caption table_figure_title" name="p_hHBD">(a) Nicholas Léonard Sadi Carnot’s research into steam-powered machinery and (b) Rudolf Clausius’s later study of those findings led to groundbreaking discoveries about spontaneous heat flow processes.</p><figure class="scaled-down" data-id="CNX_Chem_16_03_Carnot"><span id="fs-idm106556080" class="" data-type="media" data-alt=" A portrait of Rudolf Clasius is shown."><img id="img_gefs" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_8.jpg" alt=" A portrait of Rudolf Clasius is shown." name="img_gefs" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_8.jpg"></span></figure><p id="fs-idm194920048" class="" name="fs-idm194920048">Similar to other thermodynamic properties, this new quantity is a state function so its change depends only upon the initial and final states of a system. In 1865, Clausius named this property <span id="term537" class="" data-type="term">entropy (<em class="" data-effect="italics">S</em>)</span> and defined its change for any process as the following:</p><div id="fs-idm146915824" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-274-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>6</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mfrac class=""><mn class="">6</mn><mrow class=""><mn class="">16</mn></mrow></mfrac></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm100781680" class="" name="fs-idm100781680">The entropy change for a real, irreversible process is then equal to that for the theoretical reversible process that involves the same initial and final states.</p><h3 class="" data-type="title" id="h3_JNqJ" name="h3_JNqJ"><span data-etb-numbered-element="1">1.2.1</span> Entropy and Microstates</h3><p id="fs-idp173054032" class="" name="fs-idp173054032">Following the work of Carnot and Clausius, Ludwig <span id="term538" class="no-emphasis" data-type="term">Boltzmann</span> developed a molecular-scale statistical model that related the entropy of a system to the <em class="" data-effect="italics">number of microstates</em> (<em class="" data-effect="italics">W</em>) possible for the system. A <span id="term539" class="" data-type="term">microstate</span> is a specific configuration of all the locations and energies of the atoms or molecules that make up a system. The relation between a system’s entropy and the number of possible microstates is</p><div id="fs-idp33531488" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-275-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><mi>W</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><mi>W</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mfrac class=""><mn class="">1</mn><mrow class=""><mn class="">16</mn></mrow></mfrac><mo class="">.</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm109445936" class="" name="fs-idm109445936">where <em class="" data-effect="italics">k</em> is the Boltzmann constant, 1.38 <span class="os-math-in-para"><span id="MathJax-Element-276-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>×</mo></mrow><annotation-xml encoding="MathML-Content"><mo>×</mo></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mfrac class=""><mn class="">1</mn><mn class="">8</mn></mfrac><mo class="">.</mo></mrow></annotation-xml></semantics></math></span></span> 10<sup class="">−23</sup> J/K.</p><p id="fs-idm172030064" class="" name="fs-idm172030064">As for other state functions, the change in entropy for a process is the difference between its final (<em class="" data-effect="italics">S</em><sub class="">f</sub>) and initial (<em class="" data-effect="italics">S</em><sub class="">i</sub>) values:</p><div id="fs-idm76606496" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-277-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><msub><mi>S</mi><mtext>f</mtext></msub><mo>−</mo><msub><mi>S</mi><mtext>i</mtext></msub><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><msub><mi>W</mi><mtext>f</mtext></msub><mo>−</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><msub><mi>W</mi><mtext>i</mtext></msub><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>W</mi><mtext>f</mtext></msub></mrow><mrow><msub><mi>W</mi><mtext>i</mtext></msub></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><msub><mi>S</mi><mtext>f</mtext></msub><mo>−</mo><msub><mi>S</mi><mtext>i</mtext></msub><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><msub><mi>W</mi><mtext>f</mtext></msub><mo>−</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><msub><mi>W</mi><mtext>i</mtext></msub><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>W</mi><mtext>f</mtext></msub></mrow><mrow><msub><mi>W</mi><mtext>i</mtext></msub></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mfrac class=""><mn class="">7</mn><mrow class=""><mn class="">10</mn></mrow></mfrac><mo class="">.</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm69202464" class="" name="fs-idm69202464">For processes involving an increase in the number of microstates, <em class="" data-effect="italics">W</em><sub class="">f</sub> > <em class="" data-effect="italics">W</em><sub class="">i</sub>, the entropy of the system increases and Δ<em class="" data-effect="italics">S</em> > 0. Conversely, processes that reduce the number of microstates, <em class="" data-effect="italics">W</em><sub class="">f</sub> < <em class="" data-effect="italics">W</em><sub class="">i</sub>, yield a decrease in system entropy, Δ<em class="" data-effect="italics">S</em> < 0. This molecular-scale interpretation of entropy provides a link to the probability that a process will occur as illustrated in the next paragraphs.</p><p id="fs-idm119038640" class="" name="fs-idm119038640">Consider the general case of a system comprised of <em class="" data-effect="italics">N</em> particles distributed among <em class="" data-effect="italics">n</em> boxes. The number of microstates possible for such a system is <em class="" data-effect="italics">n<sup class="">N</sup></em>. For example, distributing four particles among two boxes will result in 2<sup class="">4</sup> = 16 different microstates as illustrated in <a class="autogenerated-content" href="#Figure_1.7" data-mce-href="#Figure_1.7">Figure 1.7</a>. Microstates with equivalent particle arrangements (not considering individual particle identities) are grouped together and are called <em class="" data-effect="italics">distributions</em>. The probability that a system will exist with its components in a given distribution is proportional to the number of microstates within the distribution. Since entropy increases logarithmically with the number of microstates, <em class="" data-effect="italics">the most probable distribution is therefore the one of greatest entropy</em>.</p><div id="CNX_Chem_16_02_Microstates" class="os-figure"><p id="p_iGdT" class="os-caption-container table_figure_number" name="p_iGdT">Figure <span data-etb-numbered-element="1">1.7</span></p><p id="p_eLXR" class="os-caption table_figure_title" name="p_eLXR">The sixteen microstates associated with placing four particles in two boxes are shown. The microstates are collected into five distributions—(a), (b), (c), (d), and (e)—based on the numbers of particles in each box.</p><figure class="" data-id="CNX_Chem_16_02_Microstates"><span id="fs-idm219643568" class="" data-type="media" data-alt="Five rows of diagrams that look like dominoes are shown and labeled a, b, c, d, and e. Row a has one “domino” that has four dots on the left side, red, green, blue and yellow in a clockwise pattern from the top left, and no dots on the right. Row b has four “dominos,” each with three dots on the left and one dot on the right. The first shows a “domino” with green, yellow and blue on the left and red on the right. The second “domino” has yellow, blue and red on the left and green on the right. The third “domino” has red, green and yellow on the left and blue on the right while the fourth has red, green and blue on the left and yellow on the right. Row c has six “dominos”, each with two dots on either side. The first has a red and green on the left and a blue and yellow on the right. The second has a red and blue on the left and a green and yellow on the right while the third has a yellow and red on the left and a green and blue on the right. The fourth has a green and blue on the left and a red and yellow on the right. The fifth has a green and yellow on the left and a red and blue on the right. The sixth has a blue and yellow on the left and a green and red on the right. Row d has four “dominos,” each with one dot on the left and three on the right. The first “domino” has red on the left and a blue, green and yellow on the right. The second has a green on the left and a red, yellow and blue on the right. The third has a blue on the left and a red, green and yellow on the right. The fourth has a yellow on the left and a red, green and blue on the right. Row e has 1 “domino” with no dots on the left and four dots on the right that are red, green, blue and yellow."><img id="img_MJTD" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_9.jpg" alt="Five rows of diagrams that look like dominoes are shown and labeled a, b, c, d, and e. Row a has one “domino” that has four dots on the left side, red, green, blue and yellow in a clockwise pattern from the top left, and no dots on the right. Row b has four “dominos,” each with three dots on the left and one dot on the right. The first shows a “domino” with green, yellow and blue on the left and red on the right. The second “domino” has yellow, blue and red on the left and green on the right. The third “domino” has red, green and yellow on the left and blue on the right while the fourth has red, green and blue on the left and yellow on the right. Row c has six “dominos”, each with two dots on either side. The first has a red and green on the left and a blue and yellow on the right. The second has a red and blue on the left and a green and yellow on the right while the third has a yellow and red on the left and a green and blue on the right. The fourth has a green and blue on the left and a red and yellow on the right. The fifth has a green and yellow on the left and a red and blue on the right. The sixth has a blue and yellow on the left and a green and red on the right. Row d has four “dominos,” each with one dot on the left and three on the right. The first “domino” has red on the left and a blue, green and yellow on the right. The second has a green on the left and a red, yellow and blue on the right. The third has a blue on the left and a red, green and yellow on the right. The fourth has a yellow on the left and a red, green and blue on the right. Row e has 1 “domino” with no dots on the left and four dots on the right that are red, green, blue and yellow." name="img_MJTD" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_9.jpg"></span></figure></div><p id="fs-idm104062656" class="" name="fs-idm104062656">For this system, the most probable configuration is one of the six microstates associated with distribution (c) where the particles are evenly distributed between the boxes, that is, a configuration of two particles in each box. The probability of finding the system in this configuration is <span class="os-math-in-para"><span id="MathJax-Element-278-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>6</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>6</mn><mrow><mn>16</mn></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>W</mi><mtext>c</mtext></msub></mrow><mrow><msub><mi>W</mi><mtext>a</mtext></msub></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>1.38</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>J/K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.4em"></mspace><mfrac><mn>6</mn><mn>1</mn></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>2.47</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>J/K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mo class="">=</mo><mi class="">k</mi><mspace width="0.2em" class=""></mspace><mtext class="">ln</mtext><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">W</mi><mtext class="">c</mtext></msub></mrow><mrow class=""><msub class=""><mi class="">W</mi><mtext class="">a</mtext></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">1.38</mn><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mrow class=""><mn class="">10</mn></mrow><mrow class=""><mn class="">−23</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">J/K</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mtext class="">ln</mtext><mspace width="0.4em" class=""></mspace><mfrac class=""><mn class="">6</mn><mn class="">1</mn></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">2.47</mn><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mrow class=""><mn class="">10</mn></mrow><mrow class=""><mn class="">−23</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">J/K</mtext></mrow></annotation-xml></semantics></math></span></span> or <span class="os-math-in-para"><span id="MathJax-Element-279-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>3</mn><mrow><mn>8</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn><mrow><mn>8</mn></mrow></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msup><mrow><mtext>Ag</mtext></mrow><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mtext>Cl</mtext></mrow><mtext>−</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>AgCl</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msup class=""><mrow class=""><mtext class="">Ag</mtext></mrow><mtext class="">+</mtext></msup><mo stretchy="false" class="">(</mo><mi class="">a</mi><mi class="">q</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msup class=""><mrow class=""><mtext class="">Cl</mtext></mrow><mtext class="">−</mtext></msup><mo stretchy="false" class="">(</mo><mi class="">a</mi><mi class="">q</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">AgCl</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> The least probable configuration of the system is one in which all four particles are in one box, corresponding to distributions (a) and (e), each with a probability of <span class="os-math-in-para"><span id="MathJax-Element-280-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mrow><mtext>NH</mtext></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>NH</mtext></mrow><mtext>3</mtext></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mrow class=""><mtext class="">NH</mtext></mrow><mn class="">3</mn></msub><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">NH</mtext></mrow><mtext class="">3</mtext></msub><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> The probability of finding all particles in only one box (either the left box or right box) is then <span class="os-math-in-para"><span id="MathJax-Element-281-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mn>2</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mn>2</mn><mrow><mn>16</mn></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mrow class=""><mtext class="">CO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">CO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> or <span class="os-math-in-para"><span id="MathJax-Element-282-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub><mo class="">=</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub><mo class="">+</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub></mrow></annotation-xml></semantics></math></span></span></p><p id="fs-idm150227312" class="" name="fs-idm150227312">As you add more particles to the system, the number of possible microstates increases exponentially (2<sup class=""><em class="" data-effect="italics">N</em></sup>). A macroscopic (laboratory-sized) system would typically consist of moles of particles (<em class="" data-effect="italics">N</em> ~ 10<sup class="">23</sup>), and the corresponding number of microstates would be staggeringly huge. Regardless of the number of particles in the system, however, the distributions in which roughly equal numbers of particles are found in each box are always the most probable configurations.</p><p id="fs-idm249639376" class="" name="fs-idm249639376">This matter dispersal model of entropy is often described qualitatively in terms of the <em class="" data-effect="italics">disorder</em> of the system. By this description, microstates in which all the particles are in a single box are the most ordered thus possessing the least entropy. Microstates in which the particles are more evenly distributed among the boxes are more disordered possessing greater entropy.</p><p id="fs-idm158412992" class="" name="fs-idm158412992">The previous description of an ideal gas expanding into a vacuum (<a class="" href="#Figure_1.3" data-mce-href="#Figure_1.3">Figure 1.3</a>) is a macroscopic example of this particle-in-a-box model. For this system the most probable distribution is confirmed to be the one in which the matter is most uniformly dispersed or distributed between the two flasks. Initially, the gas molecules are confined to just one of the two flasks. Opening the valve between the flasks increases the volume available to the gas molecules and, correspondingly, the number of microstates possible for the system. Since <em class="" data-effect="italics">W</em><sub class="">f</sub> > <em class="" data-effect="italics">W</em><sub class="">i</sub>, the expansion process involves an increase in entropy (Δ<em class="" data-effect="italics">S</em> > 0) and is spontaneous.</p><p id="fs-idm167918400" class="" name="fs-idm167918400">A similar approach may be used to describe the spontaneous flow of heat. Consider a system consisting of two objects, each containing two particles, and two units of thermal energy (represented as “*”) in <a class="autogenerated-content" href="#Figure_1.8" data-mce-href="#Figure_1.8">Figure 1.8</a>. The hot object is comprised of particles <strong class="">A</strong> and <strong class="">B</strong> and initially contains both energy units. The cold object is comprised of particles <strong class="">C</strong> and <strong class="">D</strong>, which initially has no energy units. Distribution (a) shows the three microstates possible for the initial state of the system, with both units of energy contained within the hot object. If one of the two energy units is transferred, the result is distribution (b) consisting of four microstates. If both energy units are transferred, the result is distribution (c) consisting of three microstates. Thus, we may describe this system by a total of ten microstates. The probability that the heat does not flow when the two objects are brought into contact, that is, that the system remains in distribution (a), is <span class="os-math-in-para"><span id="MathJax-Element-283-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>3</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow><mo>|</mo></mrow><mo><</mo><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" columnalign="left" class=""><mtr class=""><mtd class=""><mrow class=""><mo class="">|</mo> <mrow class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub></mrow> <mo class="">|</mo></mrow><mo class=""><</mo><mrow class=""><mo class="">|</mo> <mrow class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub></mrow> <mo class="">|</mo></mrow></mtd></mtr><mtr class=""><mtd class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub><mo class="">=</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub><mo class="">+</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub><mo class="">></mo><mn class="">0</mn></mtd></mtr></mtable></annotation-xml></semantics></math></span></span> More likely is the flow of heat to yield one of the other two distribution, the combined probability being <span class="os-math-in-para"><span id="MathJax-Element-284-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow><mo>|</mo></mrow><mo>≈</mo><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" columnalign="left" class=""><mtr class=""><mtd class=""><mrow class=""><mo class="">|</mo> <mrow class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub></mrow> <mo class="">|</mo></mrow><mo class="">≈</mo><mrow class=""><mo class="">|</mo> <mrow class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub></mrow> <mo class="">|</mo></mrow></mtd></mtr><mtr class=""><mtd class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub><mo class="">=</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub><mo class="">+</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub><mo class="">=</mo><mn class="">0</mn></mtd></mtr></mtable></annotation-xml></semantics></math></span></span> The most likely result is the flow of heat to yield the uniform dispersal of energy represented by distribution (b), the probability of this configuration being <span class="os-math-in-para"><span id="MathJax-Element-285-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mfrac><mn>4</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>4</mn><mrow><mn>10</mn></mrow></mfrac><mo>.</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> This supports the common observation that placing hot and cold objects in contact results in spontaneous heat flow that ultimately equalizes the objects’ temperatures. And, again, this spontaneous process is also characterized by an increase in system entropy.</p><div class="os-caption-container table_figure_number">Figure <span data-etb-numbered-element="1">1.8</span></div><p id="p_fXkw" class="os-caption table_figure_title" name="p_fXkw">This shows a microstate model describing the flow of heat from a hot object to a cold object. (a) Before the heat flow occurs, the object comprised of particles <strong class="">A</strong> and <strong class="">B</strong> contains both units of energy and as represented by a distribution of three microstates. (b) If the heat flow results in an even dispersal of energy (one energy unit transferred), a distribution of four microstates results. (c) If both energy units are transferred, the resulting distribution has three microstates.</p><figure class="scaled-down" data-id="CNX_Chem_16_03_Energy"><span id="fs-idm215022256" class="" data-type="media" data-alt="Three rows labeled a, b, and c are shown and each contains rectangles with two sides where the left side is labeled, “A,” and “B,” and the right is labeled, “C,” and “D.” Row a has three rectangles where the first has a dot above and below the letter A, the second has a dot above the A and B, and the third which has a dot above and below the letter B. Row b has four rectangles; the first has a dot above A and C, the second has a dot above A and D, the third has a dot above B and C and the fourth has a dot above B and D. Row c has three rectangles; the first has a dot above and below the letter C, the second has a dot above C and D and the third has a dot above and below the letter D."><img id="img_ihkG" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_10.jpg" alt="Three rows labeled a, b, and c are shown and each contains rectangles with two sides where the left side is labeled, “A,” and “B,” and the right is labeled, “C,” and “D.” Row a has three rectangles where the first has a dot above and below the letter A, the second has a dot above the A and B, and the third which has a dot above and below the letter B. Row b has four rectangles; the first has a dot above A and C, the second has a dot above A and D, the third has a dot above B and C and the fourth has a dot above B and D. Row c has three rectangles; the first has a dot above and below the letter C, the second has a dot above C and D and the third has a dot above and below the letter D." name="img_ihkG" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_10.jpg"></span></figure><div id="fs-idm141128112" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_YEtb" name="h3_YEtb">EXAMPLE <span data-etb-numbered-element="1">1.2.2</span></h3></header><div class="body"><h4 id="h4_qdBk" class="" data-type="title" name="h4_qdBk">Determination of Δ<em class="" data-effect="italics">S</em></h4>Calculate the change in entropy for the process depicted below.<span id="CNX_Chem_16_03_Matter_img" class="" data-type="media" data-alt="A diagram shows one rectangle with two sides that has four dots, red, green, yellow and blue written on the left side. A right-facing arrow leads to six more two-sided rectangles, each with two dots on the left and right sides. The first rectangle has a red and green dot on the left and a blue and yellow on the right, while the second shows a red and blue on the left and a green and yellow on the right. The third rectangle has a red and yellow dot on the left and a blue and green on the right, while the fourth shows a green and blue on the left and a red and yellow on the right. The fifth rectangle has a yellow and green dot on the left and a blue and red on the right, while the sixth shows a yellow and blue on the left and a green and red on the right."><img id="img_qCxp" class="img-fluid col-12" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_11.jpg" alt="A diagram shows one rectangle with two sides that has four dots, red, green, yellow and blue written on the left side. A right-facing arrow leads to six more two-sided rectangles, each with two dots on the left and right sides. The first rectangle has a red and green dot on the left and a blue and yellow on the right, while the second shows a red and blue on the left and a green and yellow on the right. The third rectangle has a red and yellow dot on the left and a blue and green on the right, while the fourth shows a green and blue on the left and a red and yellow on the right. The fifth rectangle has a yellow and green dot on the left and a blue and red on the right, while the sixth shows a yellow and blue on the left and a green and red on the right." name="img_qCxp" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_11.jpg"></span><h4 id="h4_LApf" class="" data-type="title" name="h4_LApf">Solution</h4><p id="fs-idm97076112" class="" name="fs-idm97076112">The initial number of microstates is one, the final six:</p><div id="fs-idm221426432" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-286-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>W</mi><mtext>c</mtext></msub></mrow><mrow><msub><mi>W</mi><mtext>a</mtext></msub></mrow></mfrac><mspace width="0.2em" ></mspace><mo>=</mo><mn>1.38</mn><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em" ></mspace><mtext>J/K</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.4em" ></mspace><mfrac><mn>6</mn><mn>1</mn></mfrac><mspace width="0.2em" ></mspace><mo>=</mo><mn>2.47</mn><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em" ></mspace><mtext>J/K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>W</mi><mtext>c</mtext></msub></mrow><mrow><msub><mi>W</mi><mtext>a</mtext></msub></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>1.38</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>J/K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.4em"></mspace><mfrac><mn>6</mn><mn>1</mn></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>2.47</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−23</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>J/K</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mn>22.1</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext><mo>+</mo><mfrac><mrow><mn>−6.00</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em"></mspace><mtext>J</mtext></mrow><mrow><mtext>283.15 K</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mtext>+0.9 J/K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""></mtr><mtr class=""><mtd class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub><mo class="">=</mo><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">sys</mtext></mrow></msub><mo class="">+</mo><mfrac class=""><mrow class=""><msub class=""><mi class="">q</mi><mrow class=""><mtext class="">surr</mtext></mrow></msub></mrow><mi class="">T</mi></mfrac></mtd></mtr><mtr class=""></mtr><mtr class=""><mtd class=""><mo class="">=</mo><mn class="">22.1</mn><mspace width="0.2em" class=""></mspace><mtext class="">J/K</mtext><mo class="">+</mo><mfrac class=""><mrow class=""><mn class="">−6.00</mn><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">3</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">J</mtext></mrow><mrow class=""><mtext class="">283.15 K</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mtext class="">+0.9 J/K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></span></div></div><p id="fs-idm190514064" class="" name="fs-idm190514064">The sign of this result is consistent with expectation; since there are more microstates possible for the final state than for the initial state, the change in entropy should be positive.</p><h4 id="h4_xeJL" class="" data-type="title" name="h4_xeJL">Check Your Learning</h4>Consider the system shown in <a class="autogenerated-content" href="#Figure_1.8" data-mce-href="#Figure_1.8">Figure 1.8</a>. What is the change in entropy for the process where <em class="" data-effect="italics">all</em> the energy is transferred from the hot object (<strong class="">AB</strong>) to the cold object (<strong class="">CD</strong>)?<div id="fs-idp48055552" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_MMrU6ksO" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_MMrU6ksO" data-bs-toggle="collapse" id="h3_IIJn" name="h3_IIJn" data-mce-href="#answer_MMrU6ksO" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_MMrU6ksO" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm145487536" class="" name="fs-idm145487536">0 J/K</p></div></section></div></div></div><h3 class="" data-type="title" id="h3_xYEi" name="h3_xYEi"><span data-etb-numbered-element="1">1.2.3</span> Predicting the Sign of ΔS</h3><p id="fs-idm169703584" class="" name="fs-idm169703584">The relationships between entropy, microstates, and matter/energy dispersal described previously allow us to make generalizations regarding the relative entropies of substances and to predict the sign of entropy changes for chemical and physical processes. Consider the phase changes illustrated in <a class="autogenerated-content" href="#Figure_1.9" data-mce-href="#Figure_1.9">Figure 1.9</a>. In the solid phase, the atoms or molecules are restricted to nearly fixed positions with respect to each other and are capable of only modest oscillations about these positions. With essentially fixed locations for the system’s component particles, the number of microstates is relatively small. In the liquid phase, the atoms or molecules are free to move over and around each other, though they remain in relatively close proximity to one another. This increased freedom of motion results in a greater variation in possible particle locations, so the number of microstates is correspondingly greater than for the solid. As a result, <em class="" data-effect="italics">S</em><sub class="">liquid</sub> > <em class="" data-effect="italics">S</em><sub class="">solid</sub> and the process of converting a substance from solid to liquid (melting) is characterized by an increase in entropy, Δ<em class="" data-effect="italics">S</em> > 0. By the same logic, the reciprocal process (freezing) exhibits a decrease in entropy, Δ<em class="" data-effect="italics">S</em> < 0.</p><p class="os-caption-container table_figure_number" id="p_ZszU" name="p_ZszU">Figure <span data-etb-numbered-element="1">1.9</span></p><p id="p_VMNN" class="os-caption table_figure_title" name="p_VMNN">The entropy of a substance increases (Δ<em class="" data-effect="italics">S</em> > 0) as it transforms from a relatively ordered solid, to a less-ordered liquid, and then to a still less-ordered gas. The entropy decreases (Δ<em class="" data-effect="italics">S</em> < 0) as the substance transforms from a gas to a liquid and then to a solid.</p><figure class="" data-id="CNX_Chem_16_03_Entropies"><span id="fs-idm187243744" class="" data-type="media" data-alt="Three stoppered flasks are shown with right and left-facing arrows in between each; the first is labeled above as, “delta S greater than 0,” and below as, “delta S less than 0,” while the second is labeled above as, “delta S greater than 0,” and below as, “delta S less than 0.” A long, right-facing arrow is drawn above all the flasks and labeled, “Increasing entropy.” The left flask contains twenty-seven particles arranged in a cube in the bottom of the flask and is labeled, “Crystalline solid,” below. The middle flask contains twenty-seven particles dispersed randomly in the bottom of the flask and is labeled, “Liquid,” below. The right flask contains twenty-seven particles dispersed inside of the flask and moving rapidly and is labeled, “Gas,” below."><img id="img_nnvt" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_12.jpg" alt="Three stoppered flasks are shown with right and left-facing arrows in between each; the first is labeled above as, “delta S greater than 0,” and below as, “delta S less than 0,” while the second is labeled above as, “delta S greater than 0,” and below as, “delta S less than 0.” A long, right-facing arrow is drawn above all the flasks and labeled, “Increasing entropy.” The left flask contains twenty-seven particles arranged in a cube in the bottom of the flask and is labeled, “Crystalline solid,” below. The middle flask contains twenty-seven particles dispersed randomly in the bottom of the flask and is labeled, “Liquid,” below. The right flask contains twenty-seven particles dispersed inside of the flask and moving rapidly and is labeled, “Gas,” below." name="img_nnvt" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_12.jpg"></span></figure><p id="fs-idm123213200" class="" name="fs-idm123213200">Now consider the gaseous phase in which a given number of atoms or molecules occupy a <em class="" data-effect="italics">much</em> greater volume than in the liquid phase. Each atom or molecule can be found in many more locations, corresponding to a much greater number of microstates. Consequently, for any substance, <em class="" data-effect="italics">S</em><sub class="">gas</sub> > <em class="" data-effect="italics">S</em><sub class="">liquid</sub> > <em class="" data-effect="italics">S</em><sub class="">solid</sub>, and the processes of vaporization and sublimation likewise involve increases in entropy, Δ<em class="" data-effect="italics">S</em> > 0. Likewise, the reciprocal phase transitions, condensation and deposition, involve decreases in entropy, Δ<em class="" data-effect="italics">S</em> < 0.</p><p id="fs-idp32863824" class="" name="fs-idp32863824">According to kinetic-molecular theory, the temperature of a substance is proportional to the average kinetic energy of its particles. Raising the temperature of a substance will result in more extensive vibrations of the particles in solids and more rapid translations of the particles in liquids and gases. At higher temperatures the distribution of kinetic energies among the atoms or molecules of the substance is also broader (more dispersed) than at lower temperatures. Thus, the entropy for any substance increases with temperature (<a class="autogenerated-content" href="#Figure_1.10" data-mce-href="#Figure_1.10">Figure 1.10</a>).</p><p id="p_KbCz" class="os-caption-container table_figure_number" name="p_KbCz">Figure <span data-etb-numbered-element="1">1.10</span></p><p id="p_LxTN" class="os-caption table_figure_title" name="p_LxTN">Entropy increases as the temperature of a substance is raised which corresponds to the greater spread of kinetic energies. When a substance undergoes a phase transition its entropy changes significantly.</p><figure class="" data-id="CNX_Chem_16_02_EntGraph"><span id="fs-idm190770944" class="" data-type="media" data-alt="Two graphs are shown. The y-axis of the left graph is labeled, “Fraction of molecules,” while the x-axis is labeled, “Velocity, v ( m / s ),” and has values of 0 through 1,500 along the axis with increments of 500. Four lines are plotted on this graph. The first, labeled, “100 K,” peaks around 200 m / s while the second, labeled, “200 K,” peaks near 300 m / s and is slightly lower on the y-axis than the first. The third line, labeled, “500 K,” peaks around 550 m / s and is lower than the first two on the y-axis. The fourth line, labeled, “1000 K,” peaks around 750 m / s and is the lowest of the four on the y-axis. Each line get increasingly broad. The second graph has a y-axis labeled, “Entropy, S,” with an upward-facing arrow and an x-axis labeled, “Temperature ( K ),” and a right-facing arrow. The graph has three equally spaced columns in the background, labeled, “Solid,” “Liquid,” and, “Gas,” from left to right. A line extends slightly upward through the first column in a slight upward direction, then goes straight up in the transition between the first two columns. In then progresses in a slight upward direction through the second column, then goes up dramatically between the second and third columns, then continues in a slight upward direction once more. The first vertical region of this line is labeled, “Melting,” and the second is labeled, “Boiling.”"><img id="img_BaMP" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_13.jpg" alt="Two graphs are shown. The y-axis of the left graph is labeled, “Fraction of molecules,” while the x-axis is labeled, “Velocity, v ( m / s ),” and has values of 0 through 1,500 along the axis with increments of 500. Four lines are plotted on this graph. The first, labeled, “100 K,” peaks around 200 m / s while the second, labeled, “200 K,” peaks near 300 m / s and is slightly lower on the y-axis than the first. The third line, labeled, “500 K,” peaks around 550 m / s and is lower than the first two on the y-axis. The fourth line, labeled, “1000 K,” peaks around 750 m / s and is the lowest of the four on the y-axis. Each line get increasingly broad. The second graph has a y-axis labeled, “Entropy, S,” with an upward-facing arrow and an x-axis labeled, “Temperature ( K ),” and a right-facing arrow. The graph has three equally spaced columns in the background, labeled, “Solid,” “Liquid,” and, “Gas,” from left to right. A line extends slightly upward through the first column in a slight upward direction, then goes straight up in the transition between the first two columns. In then progresses in a slight upward direction through the second column, then goes up dramatically between the second and third columns, then continues in a slight upward direction once more. The first vertical region of this line is labeled, “Melting,” and the second is labeled, “Boiling.”" name="img_BaMP" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_13.jpg"></span></figure><div id="fs-idm189737360" class="chemistry link-to-learning ui-has-child-title callout info" data-type="note"><header class=""><h3 class="os-title" data-type="title" id="h3_PSzq" name="h3_PSzq"><span class="os-title-label">LINK TO LEARNING</span></h3></header><div class="os-note-body"><p id="fs-idm305735232" class="" name="fs-idm305735232">Try this <a class="" href="http://openstax.org/l/16freemotion" target="_blank" rel="noopener nofollow" data-shortened-url="https://edtechbooks.org/-FxKB" data-unshortened-url="http://openstax.org/l/16freemotion" data-mce-href="http://openstax.org/l/16freemotion">simulator</a> with interactive visualization of the dependence of particle location and freedom of motion on physical state and temperature.</p></div></div><p id="fs-idm135226880" class="" name="fs-idm135226880">The entropy of a substance is influenced by the structure of the particles (atoms or molecules) that comprise the substance. With regard to atomic substances, heavier atoms possess greater entropy at a given temperature than lighter atoms, which is a consequence of the relation between a particle’s mass and the spacing of quantized translational energy levels (a topic beyond the scope of this text). For molecules, greater numbers of atoms increase the number of ways in which the molecules can vibrate and thus the number of possible microstates and the entropy of the system.</p><p id="fs-idm96833856" class="" name="fs-idm96833856">Finally, variations in the types of particles affects the entropy of a system. Compared to a pure substance in which all particles are identical, the entropy of a mixture of two or more different particle types is greater. This is because of the additional orientations and interactions that are possible in a system comprised of nonidentical components. For example, when a solid dissolves in a liquid the particles of the solid experience both a greater freedom of motion and additional interactions with the solvent particles. This corresponds to a more uniform dispersal of matter and energy and a greater number of microstates. The process of dissolution therefore involves an increase in entropy, Δ<em class="" data-effect="italics">S</em> > 0.</p><p id="fs-idm224301776" class="" name="fs-idm224301776">Considering the various factors that affect entropy allows us to make informed predictions of the sign of Δ<em class="" data-effect="italics">S</em> for various chemical and physical processes as illustrated in <a class="autogenerated-content" href="#Example_1.3" data-shortened-url="https://edtechbooks.org/-bes" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/12-2-entropy#fs-idm151499136" data-mce-href="#Example_1.3">Example 1.2.4</a>.</p><div id="fs-idm151499136" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_iNUZ" name="h3_iNUZ">Example <span data-etb-numbered-element="1">1.2.4</span></h3></header><div class="body"><h4 id="h4_EnWD" class="" data-type="title" name="h4_EnWD">Predicting the Sign of ∆<em class="" data-effect="italics">S</em></h4>Predict the sign of the entropy change for the following processes. Indicate the reason for each of your predictions.<p id="fs-idm216333072" class="" name="fs-idm216333072">(a) One mole liquid water at room temperature <span class="os-math-in-para"><span id="MathJax-Element-287-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">⟶</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">⟶</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mo class="">∑</mo><mi class="">ν</mi><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(products)</mtext><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mo class="">∑</mo><mi class="">ν</mi><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(reactants)</mtext></mrow></annotation-xml></semantics></math></span></span> one mole liquid water at 50 °C</p><p id="fs-idm120870816" class="" name="fs-idm120870816">(b) <span class="os-math-in-para"><span id="MathJax-Element-288-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msup><mrow><mtext>Ag</mtext></mrow><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mtext>Cl</mtext></mrow><mtext>−</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mtext>AgCl</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mtext>Ag</mtext></mrow><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mtext>Cl</mtext></mrow><mtext>−</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>AgCl</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mi>x</mi><mi>S</mi><mo data-mjx-pseudoscript="true">°</mo><mo stretchy="false">(</mo><mtext>C</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>D</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mrow><mo>[</mo><mrow><mi>m</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>A</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>B</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mrow class=""><mo class="">[</mo><mrow class=""><mi class="">x</mi><mi class="">S</mi><mo class="">°</mo><mo stretchy="false" class="">(</mo><mtext class="">C</mtext><mo stretchy="false" class="">)</mo><mo class="">+</mo><mi class="">y</mi><mi class="">S</mi><mtext class="">°</mtext><mo stretchy="false" class="">(</mo><mtext class="">D</mtext><mo stretchy="false" class="">)</mo></mrow><mo class="">]</mo></mrow><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">[</mo><mrow class=""><mi class="">m</mi><mi class="">S</mi><mtext class="">°</mtext><mo stretchy="false" class="">(</mo><mtext class="">A</mtext><mo stretchy="false" class="">)</mo><mo class="">+</mo><mi class="">n</mi><mi class="">S</mi><mtext class="">°</mtext><mo stretchy="false" class="">(</mo><mtext class="">B</mtext><mo stretchy="false" class="">)</mo></mrow><mo class="">]</mo></mrow></mrow></annotation-xml></semantics></math></span></span></p><p id="fs-idm126368960" class="" name="fs-idm126368960">(c) <span class="os-math-in-para"><span id="MathJax-Element-289-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>6</mn></msub><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn>15</mn></mrow><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mn>6</mn><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>C</mtext><mn>6</mn></msub><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn>15</mn></mrow><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mn>6</mn><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span></p><p id="fs-idm146695616" class="" name="fs-idm146695616">(d) <span class="os-math-in-para"><span id="MathJax-Element-290-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mrow><mtext>NH</mtext></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mrow><mtext>NH</mtext></mrow><mtext>3</mtext></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mrow><mtext>NH</mtext></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>NH</mtext></mrow><mtext>3</mtext></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><msub class=""><mtext class="">H</mtext><mn class="">4</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><msub class=""><mtext class="">H</mtext><mn class="">6</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span></p><h4 id="h4_BPtc" class="" data-type="title" name="h4_BPtc">Solution</h4>(a) positive, temperature increases<p id="fs-idm136327104" class="" name="fs-idm136327104">(b) negative, reduction in the number of ions (particles) in solution, decreased dispersal of matter</p><p id="fs-idm137858960" class="" name="fs-idm137858960">(c) negative, net decrease in the amount of gaseous species</p><p id="fs-idm119107312" class="" name="fs-idm119107312">(d) positive, phase transition from solid to liquid, net increase in dispersal of matter</p><h4 id="h4_qjAK" class="" data-type="title" name="h4_qjAK">Check Your Learning</h4>Predict the sign of the entropy change for the following processes. Give a reason for your prediction.<p id="fs-idm40731056" class="" name="fs-idm40731056">(a) <span class="os-math-in-para"><span id="MathJax-Element-291-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mrow><mtext>NaNO</mtext></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msup><mrow><mtext>Na</mtext></mrow><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>NO</mtext></mrow><mn>3</mn></msub><msup><mrow ></mrow><mtext>−</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mrow><mtext>NaNO</mtext></mrow><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msup><mrow><mtext>Na</mtext></mrow><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>NO</mtext></mrow><mn>3</mn></msub><msup><mrow></mrow><mtext>−</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mo class="">∑</mo><mi class="">ν</mi><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(products)</mtext><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mo class="">∑</mo><mi class="">ν</mi><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(reactants)</mtext></mrow></annotation-xml></semantics></math></span></span></p><p id="fs-idp86372496" class="" name="fs-idp86372496">(b) the freezing of liquid water</p><p id="fs-idm69138640" class="" name="fs-idm69138640">(c) <span class="os-math-in-para"><span id="MathJax-Element-292-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Ca</mtext><msub><mrow><mo stretchy="false">(</mo><mtext>OH</mtext><mo stretchy="false">)</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mtext>s</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>CaO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Ca</mtext><msub class=""><mrow class=""><mo stretchy="false" class="">(</mo><mtext class="">OH</mtext><mo stretchy="false" class="">)</mo></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mtext class="">s</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">CaO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span></p><p id="fs-idm47554640" class="" name="fs-idm47554640">(d) <span class="os-math-in-para"><span id="MathJax-Element-293-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mtext>CaCO</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mtext>CaO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>CaCO</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>CaO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></span></p><div id="fs-idm8656368" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_zb6M530t" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_zb6M530t" data-bs-toggle="collapse" id="h3_EHSa" name="h3_EHSa" data-mce-href="#answer_zb6M530t" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_zb6M530t" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm140480768" class="" name="fs-idm140480768">(a) Positive; The solid dissolves to give an increase of mobile ions in solution. (b) Negative; The liquid becomes a more ordered solid. (c) Positive; The relatively ordered solid becomes a gas. (d) Positive; There is a net increase in the amount of gaseous species.</p></div></section></div></div></div><h2 id="h2_JLKx" class="" data-type="title" name="h2_JLKx"><span data-etb-numbered-element="1">1.3</span> The Second and Third Laws of Thermodynamics </h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_AEvR" name="h3_AEvR">Learning Objectives</h3></header><p id="para-00001" class="" name="para-00001">By the end of this section, you will be able to:</p><ul id="list-00001" class="" name="list-00001"><li id="li_WRKT" class="" name="li_WRKT">State and explain the second and third laws of thermodynamics</li><li id="li_kRVA" class="" name="li_kRVA">Calculate entropy changes for phase transitions and chemical reactions under standard conditions</li></ul></div><h3 class="" data-type="title" id="h3_xVyX" name="h3_xVyX"><span data-etb-numbered-element="1">1.3.1</span> The Second Law of Thermodynamics</h3><p id="fs-idp3609760" class="" name="fs-idp3609760">In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy <em class="" data-effect="italics">of the system</em> (Δ<em class="" data-effect="italics">S</em> > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include <em class="" data-effect="italics">the surroundings</em>, we may reach a significant conclusion regarding the relation between this property and spontaneity. In thermodynamic models, the system and surroundings comprise everything, that is, the universe, and so the following is true:</p><div id="fs-idm7595760" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-294-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>S</mi><mo>−</mo><mspace width="0.1em"></mspace><mfrac><mrow><mtext>Δ</mtext><mi>H</mi></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">S</mi><mo class="">−</mo><mspace width="0.1em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">H</mi></mrow><mi class="">T</mi></mfrac></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp115428960" class="" name="fs-idp115428960">To illustrate this relation, consider again the process of heat flow between two objects, one identified as the system and the other as the surroundings. There are three possibilities for such a process:</p><ol id="fs-idm90348816" class="" type="1" name="fs-idm90348816"><li id="li_KMre" class="" name="li_KMre">The objects are at different temperatures, and heat flows from the hotter to the cooler object. <em class="" data-effect="italics">This is always observed to occur spontaneously.</em> Designating the hotter object as the system and invoking the definition of entropy yields the following:<div id="fs-idp67516128" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-295-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mrow><mtext>−</mtext><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>sys</mtext></mrow></msub></mrow></mfrac><mspace width="2em" ></mspace><mtext>and</mtext><mspace width="2em" ></mspace><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>−</mtext><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>sys</mtext></mrow></msub></mrow></mfrac><mspace width="2em"></mspace><mtext>and</mtext><mspace width="2em"></mspace><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>−</mtext><mi>T</mi><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mtext class="">−</mtext><mi class="">T</mi><mtext class="">Δ</mtext><msub class=""><mi class="">S</mi><mrow class=""><mtext class="">univ</mtext></mrow></msub></mrow></annotation-xml></semantics></math></span></div></div>The magnitudes of −<em class="" data-effect="italics">q</em><sub class="">rev</sub> and <em class="" data-effect="italics">q</em><sub class="">rev</sub> are equal, their opposite arithmetic signs denoting loss of heat by the system and gain of heat by the surroundings. Since <em class="" data-effect="italics">T</em><sub class="">sys</sub> > <em class="" data-effect="italics">T</em><sub class="">surr</sub> in this scenario, the entropy <em class="" data-effect="italics">decrease</em> of the system will be less than the entropy <em class="" data-effect="italics">increase</em> of the surroundings, and so <em class="" data-effect="italics">the entropy of the universe will increase</em>:<div id="fs-idm241116384" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-296-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow><mo>|</mo></mrow><mo><</mo><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo> <mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow> <mo>|</mo></mrow><mo><</mo><mrow><mo>|</mo> <mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow> <mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>></mo><mn>0</mn></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">Δ</mi><mi class="">G</mi><mo class="">=</mo><msub class=""><mi class="">w</mi><mrow class=""><mi class="">max</mi></mrow></msub></mrow></annotation-xml></semantics></math></span></div></div></li><li id="li_Yuev" class="" name="li_Yuev">The objects are at different temperatures, and heat flows from the cooler to the hotter object. <em class="" data-effect="italics">This is never observed to occur spontaneously.</em> Again designating the hotter object as the system and invoking the definition of entropy yields the following:<div id="fs-idp14273600" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-297-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>sys</mtext></mrow></msub></mrow></mfrac><mspace width="2em" ></mspace><mtext>and</mtext><mspace width="2em" ></mspace><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mspace width="0.2em" ></mspace><mfrac><mrow><mtext>−</mtext><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>sys</mtext></mrow></msub></mrow></mfrac><mspace width="2em"></mspace><mtext>and</mtext><mspace width="2em"></mspace><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>−</mtext><msub><mi>q</mi><mrow><mtext>rev</mtext></mrow></msub></mrow><mrow><msub><mi>T</mi><mrow><mtext>surr</mtext></mrow></msub></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext></mrow></annotation-xml></semantics></math></span></div></div>The arithmetic signs of <em class="" data-effect="italics">q</em><sub class="">rev</sub> denote the gain of heat by the system and the loss of heat by the surroundings. The magnitude of the entropy change for the surroundings will again be greater than that for the system, but in this case, the signs of the heat changes (that is, <em class="" data-effect="italics">the direction of the heat flow</em>) will yield a negative value for Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub>. <em class="" data-effect="italics">This process involves a decrease in the entropy of the universe.</em></li><li id="li_NNHb" class="" name="li_NNHb">The objects are at essentially the same temperature, <em class="" data-effect="italics">T</em><sub class="">sys</sub> ≈ <em class="" data-effect="italics">T</em><sub class="">surr</sub>, and so the magnitudes of the entropy changes are essentially the same for both the system and the surroundings. In this case, the entropy change of the universe is zero, and the system is <em class="" data-effect="italics">at equilibrium</em>.<div id="fs-idm250691808" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-298-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow><mo>|</mo></mrow><mo>≈</mo><mrow><mo>|</mo><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo> <mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub></mrow> <mo>|</mo></mrow><mo>≈</mo><mrow><mo>|</mo> <mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub></mrow> <mo>|</mo></mrow></mtd></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext></mrow></annotation-xml></semantics></math></span></div></div></li></ol><p id="fs-idp26583536" class="" name="fs-idp26583536">These results lead to a profound statement regarding the relation between entropy and spontaneity known as the <span id="term540" class="" data-type="term">second law of thermodynamics</span>: <em class="" data-effect="italics">all spontaneous changes cause an increase in the entropy of the universe.</em> A summary of these three relations is provided in <a class="autogenerated-content" href="#Table_1.1" data-mce-href="#Table_1.1">Table 1.1</a>.</p><p id="p_hMVo" class="os-caption-container table_figure_number" name="p_hMVo">Table <span data-etb-numbered-element="1">1.11</span></p><p id="p_RjpQ" class="os-table-title table_figure_title" name="p_RjpQ">The Second Law of Thermodynamics</p><table id="fs-idp41455824" class="table table-sm table-striped table-responsive mce-item-table"><thead class="thead-dark"></thead><tbody class=""><tr class="" valign="top"><td class="">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> > 0</td><td class="">spontaneous</td></tr><tr class="" valign="top"><td class="">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> < 0</td><td class="">nonspontaneous (spontaneous in opposite direction)</td></tr><tr class="" valign="top"><td class="">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> = 0</td><td class="">at equilibrium</td></tr></tbody></table><p id="fs-idp172517200" class="" name="fs-idp172517200">For many realistic applications, the surroundings are vast in comparison to the system. In such cases, the heat gained or lost by the surroundings as a result of some process represents a very small, nearly infinitesimal, fraction of its total thermal energy. For example, combustion of a fuel in air involves transfer of heat from a system (the fuel and oxygen molecules undergoing reaction) to surroundings that are infinitely more massive (the earth’s atmosphere). As a result, <em class="" data-effect="italics">q</em><sub class="">surr</sub> is a good approximation of <em class="" data-effect="italics">q</em><sub class="">rev</sub>, and the second law may be stated as the following:</p><div id="fs-idp179481536" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-299-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mrow><mtext>(J/K·mol)</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">S</mi><mtext class="">°</mtext><mrow class=""><mtext class="">(J/K·mol)</mtext></mrow></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp5341920" class="" name="fs-idp5341920">We may use this equation to predict the spontaneity of a process as illustrated in <a class="autogenerated-content" href="#Example_1.3.2" data-shortened-url="https://edtechbooks.org/-rsta" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/12-3-the-second-and-third-laws-of-thermodynamics#fs-idp33042160" data-mce-href="#Example_1.3.2">Example 1.3.2</a>.</p><div id="fs-idp33042160" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_qUQL" name="h3_qUQL">EXAMPLE <span data-etb-numbered-element="1">1.3.2</span></h3></header><div class="body"><h4 id="h4_hXTF" class="" data-type="title" name="h4_hXTF">Will Ice Spontaneously Melt?</h4>The entropy change for the process<div id="fs-idp179791152" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-300-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mn>(1 mol)188.8</mn><mspace width="0.2em"></mspace><mtext>J/mol·K</mtext><mo>−</mo><mn>(1</mn><mspace width="0.2em"></mspace><mtext>mol)</mtext><mn>70.0</mn><mspace width="0.2em"></mspace><mtext>J/mol K</mtext><mo>=</mo><mn>118.8</mn><mspace width="0.2em"></mspace><mtext>J/mol·K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mrow class=""><mtext class="">1 mol</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace></mrow><mi class="">S</mi><mtext class="">°</mtext><mrow class=""><mo class="">(</mo><mrow class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mrow class=""><mtext class="">1 mol</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace></mrow><mi class="">S</mi><mtext class="">°</mtext><mrow class=""><mo class="">(</mo><mrow class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo></mrow><mo class="">)</mo></mrow></mtd></mtr><mtr class=""><mtd class=""><mo class="">=</mo><mn class="">(1 mol)188.8</mn><mspace width="0.2em" class=""></mspace><mtext class="">J/mol·K</mtext><mo class="">−</mo><mn class="">(1</mn><mspace width="0.2em" class=""></mspace><mtext class="">mol)</mtext><mn class="">70.0</mn><mspace width="0.2em" class=""></mspace><mtext class="">J/mol K</mtext><mo class="">=</mo><mn class="">118.8</mn><mspace width="0.2em" class=""></mspace><mtext class="">J/mol·K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></span></div></div><p id="fs-idp175389120" class="" name="fs-idp175389120">is 22.1 J/K and requires that the surroundings transfer 6.00 kJ of heat to the system. Is the process spontaneous at −10.00 °C? Is it spontaneous at +10.00 °C?</p><h4 id="h4_LKKt" class="" data-type="title" name="h4_LKKt">Solution</h4>We can assess the spontaneity of the process by calculating the entropy change of the universe. If Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> is positive, then the process is spontaneous. At both temperatures, Δ<em class="" data-effect="italics">S</em><sub class="">sys</sub> = 22.1 J/K and <em class="" data-effect="italics">q</em><sub class="">surr</sub> = −6.00 kJ.<p id="fs-idp9805936" class="" name="fs-idp9805936">At −10.00 °C (263.15 K), the following is true:</p><div id="fs-idp170559696" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-301-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd columnalign="right"><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub></mtd><mtd columnalign="left"><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mtd></mtr><mtr><mtd ></mtd><mtd columnalign="left"><mo>=</mo><mtext>22.1 J/K</mtext><mo>+</mo><mfrac><mrow><mn>−6.00</mn><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em" ></mspace><mtext>J</mtext></mrow><mrow><mtext>263.15 K</mtext></mrow></mfrac><mspace width="0.2em" ></mspace><mo>=</mo><mn>−0.7</mn><mspace width="0.2em" ></mspace><mtext>J/K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd columnalign="right"><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub></mtd><mtd columnalign="left"><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>surr</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mtd></mtr><mtr><mtd></mtd><mtd columnalign="left"><mo>=</mo><mtext>22.1 J/K</mtext><mo>+</mo><mfrac><mrow><mn>−6.00</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em"></mspace><mtext>J</mtext></mrow><mrow><mtext>263.15 K</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>−0.7</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mtext>45.01 kJ</mtext><mo>−</mo><mrow><mo>(</mo><mrow><mtext>298 K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>118.8</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext></mrow><mo>)</mo></mrow><mspace width="0.4em"></mspace><mo>×</mo><mspace width="0.4em"></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext></mtd></mtr><mtr class=""></mtr><mtr class=""><mtd class=""><mo class="">=</mo><mtext class="">45.01 kJ</mtext><mo class="">−</mo><mrow class=""><mo class="">(</mo><mrow class=""><mtext class="">298 K</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">118.8</mn><mspace width="0.2em" class=""></mspace><mtext class="">J/K</mtext></mrow><mo class="">)</mo></mrow><mspace width="0.4em" class=""></mspace><mo class="">×</mo><mspace width="0.4em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">1 kJ</mtext></mrow><mrow class=""><mtext class="">1000 J</mtext></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math></span></div></div><p id="fs-idp43894736" class="" name="fs-idp43894736"><em class="" data-effect="italics">S</em><sub class="">univ</sub> < 0, so melting is nonspontaneous (<em class="" data-effect="italics">not</em> spontaneous) at −10.0 °C.</p><p id="fs-idp156182416" class="" name="fs-idp156182416">At 10.00 °C (283.15 K), the following is true:</p><div id="fs-idm42720272" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-302-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr ></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mtd></mtr><mtr ></mtr><mtr><mtd><mo>=</mo><mn>22.1</mn><mspace width="0.2em" ></mspace><mtext>J/K</mtext><mo>+</mo><mfrac><mrow><mn>−6.00</mn><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em" ></mspace><mtext>J</mtext></mrow><mrow><mtext>283.15 K</mtext></mrow></mfrac><mspace width="0.2em" ></mspace><mo>=</mo><mtext>+0.9 J/K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr></mtr><mtr><mtd><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>sys</mtext></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mn>22.1</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext><mo>+</mo><mfrac><mrow><mn>−6.00</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em"></mspace><mtext>J</mtext></mrow><mrow><mtext>283.15 K</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mtext>+0.9 J/K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>></mo><mn>0</mn><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">></mo><mn class="">0</mn><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp34700864" class="" name="fs-idp34700864"><em class="" data-effect="italics">S</em><sub class="">univ</sub> > 0, so melting <em class="" data-effect="italics">is</em> spontaneous at 10.00 °C.</p><h4 id="h4_IHTk" class="" data-type="title" name="h4_IHTk">Check Your Learning</h4>Using this information, determine if liquid water will spontaneously freeze at the same temperatures. What can you say about the values of <em class="" data-effect="italics">S</em><sub class="">univ</sub>?<div id="fs-idp106254880" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_TiDqtiiV" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_TiDqtiiV" data-bs-toggle="collapse" id="h3_FfvG" name="h3_FfvG" data-mce-href="#answer_TiDqtiiV" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_TiDqtiiV" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idp105321744" class="" name="fs-idp105321744">Entropy is a state function, so Δ<em class="" data-effect="italics">S</em><sub class="">freezing</sub> = −Δ<em class="" data-effect="italics">S</em><sub class="">melting</sub> = −22.1 J/K and <em class="" data-effect="italics">q</em><sub class="">surr</sub> = +6.00 kJ. At −10.00 °C spontaneous, +0.7 J/K; at +10.00 °C nonspontaneous, −0.9 J/K.</p></div></section></div></div></div><h3 class="" data-type="title" id="h3_bCKS" name="h3_bCKS"><span data-etb-numbered-element="1">1.3.3</span> The Third Law of Thermodynamics</h3><p id="fs-idm3499504" class="" name="fs-idm3499504">The previous section described the various contributions of matter and energy dispersal that contribute to the entropy of a system. With these contributions in mind, consider the entropy of a pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K). This system may be described by a single microstate, as its purity, perfect crystallinity and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (<em class="" data-effect="italics">W</em> = 1). According to the Boltzmann equation, the entropy of this system is zero.</p><div id="fs-idp179521936" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-303-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.2em" ></mspace><mi>W</mi><mo>=</mo><mi>k</mi><mspace width="0.2em" ></mspace><mtext>ln</mtext><mspace width="0.1em" ></mspace><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.2em"></mspace><mi>W</mi><mo>=</mo><mi>k</mi><mspace width="0.2em"></mspace><mtext>ln</mtext><mspace width="0.1em"></mspace><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>102.0 kJ/mol</mtext><mo>;</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">102.0 kJ/mol</mtext><mo class="">;</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm15206896" class="" name="fs-idm15206896">This limiting condition for a system’s entropy represents the <span id="term541" class="" data-type="term">third law of thermodynamics</span>: <em class="" data-effect="italics">the entropy of a pure, perfect crystalline substance at 0 K is zero.</em></p><p id="fs-idp12220368" class="" name="fs-idp12220368">Careful calorimetric measurements can be made to determine the temperature dependence of a substance’s entropy and to derive absolute entropy values under specific conditions. <span id="term542" class="" data-type="term">Standard entropies (<em class="" data-effect="italics">S</em>°)</span> are for one mole of substance under standard conditions (a pressure of 1 bar and a temperature of 298.15 K; see details regarding standard conditions in the thermochemistry chapter of this text). The <span id="term543" class="" data-type="term">standard entropy change (Δ<em class="" data-effect="italics">S</em>°)</span> for a reaction may be computed using standard entropies as shown below:</p><div id="fs-idp35409536" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-304-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em" ></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msup><mi>G</mi><mo>°</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msup class=""><mi class="">G</mi><mo class="">°</mo></msup></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm44012816" class="" name="fs-idm44012816">where ν represents stoichiometric coefficients in the balanced equation representing the process. For example, Δ<em class="" data-effect="italics">S</em>° for the following reaction at room temperature</p><div id="fs-idm48571920" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-305-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D,</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D,</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">m</mi><mtext class="">A</mtext><mo class="">+</mo><mi class="">n</mi><mtext class="">B</mtext><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mi class="">x</mi><mtext class="">C</mtext><mo class="">+</mo><mi class="">y</mi><mtext class="">D</mtext><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp156085024" class="" name="fs-idp156085024">is computed as:</p><div id="fs-idp34261696" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-306-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mi>x</mi><mi>S</mi><mo>°</mo><mo stretchy="false">(</mo><mtext>C</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>D</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mspace width="0.2em" ></mspace><mrow><mo>[</mo><mrow><mi>m</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>A</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>B</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mi>x</mi><mi>S</mi><mo>°</mo><mo stretchy="false">(</mo><mtext>C</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>D</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mrow><mo>[</mo><mrow><mi>m</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>A</mtext><mo stretchy="false">)</mo><mo>+</mo><mi>n</mi><mi>S</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>B</mtext><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">HgO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">yellow</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">Hg</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp43524064" class="" name="fs-idp43524064">A partial listing of standard entropies is provided in <a class="autogenerated-content" href="#Table1.2" data-mce-href="#Table1.2">Table 1.2</a>, and additional values are provided in <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a>. The example exercises that follow demonstrate the use of <em class="" data-effect="italics">S</em>° values in calculating standard entropy changes for physical and chemical processes.</p><p id="p_gLLH" class="os-caption-container table_figure_number" name="p_gLLH">Table <span data-etb-numbered-element="1">1.12</span></p><p id="p_PJAE" class="os-caption table_figure_title" name="p_PJAE">Standard entropies for selected substances measured at 1 atm and 298.15 K. (Values are approximately equal to those measured at 1 bar, the currently accepted standard state pressure.)</p><table class="table table-sm table-striped table-responsive mce-item-table" data-id="fs-idm78597984"><thead class="thead-dark"><tr class="" valign="top"><th class="" data-align="left"><strong class="">Substance</strong></th><th class="" data-align="left"><strong class=""><span id="MathJax-Element-307-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mtext>°</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mspace width="0.2em"></mspace><mtext>(kJ/mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mtext class="">f</mtext><mo class="">°</mo></msubsup><mspace width="0.2em" class=""></mspace><mtext class="">(kJ/mol)</mtext></mrow></annotation-xml></semantics></math></span> (J mol<sup class="">−1</sup> K<sup class="">−1</sup>)</strong></th></tr></thead><tbody class=""><tr class="" valign="top"><td class="" colspan="2">carbon</td></tr><tr class="" valign="top"><td class="" data-align="left">C(<em class="" data-effect="italics">s</em>, graphite)</td><td class="" data-align="center">5.740</td></tr><tr class="" valign="top"><td class="" data-align="left">C(<em class="" data-effect="italics">s</em>, diamond)</td><td class="" data-align="center">2.38</td></tr><tr class="" valign="top"><td class="" data-align="left">CO(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">197.7</td></tr><tr class="" valign="top"><td class="" data-align="left">CO<sub class="">2</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">213.8</td></tr><tr class="" valign="top"><td class="" data-align="left">CH<sub class="">4</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">186.3</td></tr><tr class="" valign="top"><td class="" data-align="left">C<sub class="">2</sub>H<sub class="">4</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">219.5</td></tr><tr class="" valign="top"><td class="" data-align="left">C<sub class="">2</sub>H<sub class="">6</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">229.5</td></tr><tr class="" valign="top"><td class="" data-align="left">CH<sub class="">3</sub>OH(<em class="" data-effect="italics">l</em>)</td><td class="" data-align="center">126.8</td></tr><tr class="" valign="top"><td class="" data-align="left">C<sub class="">2</sub>H<sub class="">5</sub>OH(<em class="" data-effect="italics">l</em>)</td><td class="" data-align="center">160.7</td></tr><tr class="" valign="top"><td class="" colspan="2" data-align="left">hydrogen</td></tr><tr class="" valign="top"><td class="" data-align="left">H<sub class="">2</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">130.57</td></tr><tr class="" valign="top"><td class="" data-align="left">H(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">114.6</td></tr><tr class="" valign="top"><td class="" data-align="left">H<sub class="">2</sub>O(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">188.71</td></tr><tr class="" valign="top"><td class="" data-align="left">H<sub class="">2</sub>O(<em class="" data-effect="italics">l</em>)</td><td class="" data-align="center">69.91</td></tr><tr class="" valign="top"><td class="" data-align="left">HCI(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">186.8</td></tr><tr class="" valign="top"><td class="" data-align="left">H<sub class="">2</sub>S(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">205.7</td></tr><tr class="" valign="top"><td class="" colspan="2" data-align="left">oxygen</td></tr><tr class="" valign="top"><td class="" data-align="left">O<sub class="">2</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">205.03</td></tr></tbody></table><div id="fs-idp173892208" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_fBnq" name="h3_fBnq">EXAMPLE <span data-etb-numbered-element="1">1.3.4</span></h3></header><div class="body"><h4 id="h4_mIYa" class="" data-type="title" name="h4_mIYa">Determination of Δ<em class="" data-effect="italics">S</em>°</h4>Calculate the standard entropy change for the following process:<div id="fs-idp582800" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-308-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mspace width="0.2em"></mspace><mtext>(J/K·mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">S</mi><mtext class="">°</mtext><mspace width="0.2em" class=""></mspace><mtext class="">(J/K·mol)</mtext></mrow></annotation-xml></semantics></math></span></div></div><h4 id="h4_Hxpa" class="" data-type="title" name="h4_Hxpa">Solution</h4>Calculate the entropy change using standard entropies as shown above:<div id="fs-idm7244112" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-309-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>Δ</mi><mi>S</mi><mo>°</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>70.0</mn><mspace width="0.2em" ></mspace><mtext>J</mtext><mspace width="0.2em" ></mspace><msup><mrow><mtext>mol</mtext></mrow><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mspace width="0.2em" ></mspace><msup><mtext>K</mtext><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>188.8</mn><mspace width="0.2em" ></mspace><mtext>J</mtext><mspace width="0.2em" ></mspace><msup><mrow><mtext>mol</mtext></mrow><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mspace width="0.2em" ></mspace><msup><mtext>K</mtext><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mtext>−</mtext><mn>118.8</mn><mspace width="0.2em" ></mspace><mtext>J/K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>Δ</mi><mi>S</mi><mo>°</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>70.0</mn><mspace width="0.2em"></mspace><mtext>J</mtext><mspace width="0.2em"></mspace><msup><mrow><mtext>mol</mtext></mrow><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mspace width="0.2em"></mspace><msup><mtext>K</mtext><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>188.8</mn><mspace width="0.2em"></mspace><mtext>J</mtext><mspace width="0.2em"></mspace><msup><mrow><mtext>mol</mtext></mrow><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mspace width="0.2em"></mspace><msup><mtext>K</mtext><mrow><mtext>−</mtext><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mtext>−</mtext><mn>118.8</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mtext>2</mtext></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mrow class=""><mo class="">[</mo><mrow class=""><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">Hg</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><msub class=""><mtext class="">O</mtext><mtext class="">2</mtext></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow><mo class="">]</mo></mrow><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mspace width="0.2em" class=""></mspace><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">HgO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">yellow</mtext><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm47317216" class="" name="fs-idm47317216">The value for Δ<em class="" data-effect="italics">S</em>° is negative, as expected for this phase transition (condensation), which the previous section discussed.</p><h4 id="h4_vQeY" class="" data-type="title" name="h4_vQeY">Check Your Learning</h4>Calculate the standard entropy change for the following process:<div id="fs-idp43728512" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-310-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>C</mtext><mn>2</mn></msub><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">=</mo><mstyle displaystyle="true" class=""><mo class="">∑</mo><mrow class=""><mi class="">ν</mi><mtext class="">Δ</mtext><msubsup class=""><mi class="">H</mi><mrow class=""><mtext class="">f</mtext></mrow><mo class="">°</mo></msubsup><mtext class="">(products)</mtext><mspace width="0.2em" class=""></mspace><mo class="">−</mo></mrow></mstyle><mstyle displaystyle="true" class=""><mo class="">∑</mo><mrow class=""><mi class="">ν</mi><mtext class="">Δ</mtext><msubsup class=""><mi class="">H</mi><mrow class=""><mtext class="">f</mtext></mrow><mo class="">°</mo></msubsup><mtext class="">(reactants)</mtext></mrow></mstyle></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idp34672944" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_5cGdNT0U" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_5cGdNT0U" data-bs-toggle="collapse" id="h3_ZpUq" name="h3_ZpUq" data-mce-href="#answer_5cGdNT0U" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_5cGdNT0U" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm48282704" class="" name="fs-idm48282704">−120.6 J K<sup class="">–1</sup> mol<sup class="">–1</sup></p></div></section></div></div></div><div id="fs-idp34274768" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_cuos" name="h3_cuos">Example <span data-etb-numbered-element="1">1.3.5</span></h3></header><div class="body"><h4 id="h4_rxud" class="" data-type="title" name="h4_rxud">Determination of Δ<em class="" data-effect="italics">S</em>°</h4>Calculate the standard entropy change for the combustion of methanol, CH<sub class="">3</sub>OH:<div id="fs-idp22586368" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-311-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mn>2</mn><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><mtext>OH</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mn>2</mn><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>4</mn><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><mtext>OH</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mn>2</mn><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><mn>4</mn><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>−90.46 kJ/mol</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>90.46 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">0 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">0 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">−90.46 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mo class="">=</mo><mtext class="">90.46 kJ/mol</mtext></mrow></annotation-xml></semantics></math></span></div></div><h4 id="h4_SQER" class="" data-type="title" name="h4_SQER">Solution</h4>Calculate the entropy change using standard entropies as shown above:<div id="fs-idp26460064" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-312-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em" ></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo><mo>∑</mo><mi>ν</mi><mi>S</mi><mtext>°</mtext><mtext>(reactants)</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mn class="">1</mn><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mtext class="">Hg</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mn class="">1</mn><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mtext class="">HgO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">yellow</mtext><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm2438144" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-313-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><mrow><mo>[</mo><mrow><mn>2</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mn>4</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mspace width="0.2em" ></mspace><mrow><mo>[</mo><mrow><mn>2</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><mtext>OH</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo>+</mo><mn>3</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr ></mtr><mtr><mtd><mo>=</mo><mrow><mo mathvariant="bold">{</mo><mrow><mspace width="0.1em" ></mspace><mrow><mo>[</mo><mrow><mn>2</mn><mo>(</mo><mn>213.8</mn><mo>)</mo><mo>+</mo><mn>4</mn><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mn>70.0</mn></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mspace width="0.2em" ></mspace><mrow><mo stretchy="false">[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>126.8</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>205.03</mn></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow><mspace width="0.2em" ></mspace></mrow><mo mathvariant="bold">}</mo></mrow><mo>=</mo><mn>−161.1</mn><mspace width="0.2em" ></mspace><mtext>J/K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mrow><mo>[</mo><mrow><mn>2</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mn>4</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mrow><mo>[</mo><mrow><mn>2</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><mtext>OH</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>+</mo><mn>3</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mrow><mo mathvariant="bold">{</mo><mrow><mspace width="0.1em"></mspace><mrow><mo>[</mo><mrow><mn>2</mn><mo>(</mo><mn>213.8</mn><mo>)</mo><mo>+</mo><mn>4</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>70.0</mn></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mrow><mo stretchy="false">[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>126.8</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>205.03</mn></mrow><mo>)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow><mspace width="0.2em"></mspace></mrow><mo mathvariant="bold">}</mo></mrow><mo>=</mo><mn>−161.1</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mtext>90.46 kJ</mtext><mo>−</mo><mtext>298.15 K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>107.4 J/K·mol</mtext><mspace width="0.4em"></mspace><mo>×</mo><mspace width="0.4em"></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">90.46 kJ</mtext><mo class="">−</mo><mtext class="">298.15 K</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mtext class="">107.4 J/K·mol</mtext><mspace width="0.4em" class=""></mspace><mo class="">×</mo><mspace width="0.4em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">1 kJ</mtext></mrow><mrow class=""><mtext class="">1000 J</mtext></mrow></mfrac></mrow></annotation-xml></semantics></math></span></div></div><h4 id="h4_kVqL" class="" data-type="title" name="h4_kVqL">Check Your Learning</h4>Calculate the standard entropy change for the following reaction:<div id="fs-idp53679664" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-314-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Ca</mtext><msub><mrow><mo stretchy="false">(</mo><mtext>OH</mtext><mo stretchy="false">)</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mtext>s</mtext><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mtext>CaO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Ca</mtext><msub><mrow><mo stretchy="false">(</mo><mtext>OH</mtext><mo stretchy="false">)</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mtext>s</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>CaO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">4</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idp174350576" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_c8SsXJaD" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius collapsed" data-type="title" aria-expanded="false" aria-controls="answer_c8SsXJaD" data-bs-toggle="collapse" id="h3_EeNJ" name="h3_EeNJ" data-mce-href="#answer_c8SsXJaD" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_c8SsXJaD" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idp170733152" class="" name="fs-idp170733152">24.7 J/K</p></div></section></div></div></div><h2 id="h2_KVBh" class="" name="h2_KVBh"><span data-etb-numbered-element="1">1.4</span> Free Energy</h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_jKZJ" name="h3_jKZJ">Learning Objectives</h3></header>By the end of this section, you will be able to:<ul id="list-00001" class="" name="list-00001"><li id="li_fJna" class="" name="li_fJna">Define Gibbs free energy, and describe its relation to spontaneity</li><li id="li_ogcB" class="" name="li_ogcB">Calculate free energy change for a process using free energies of formation for its reactants and products</li><li id="li_ZoNR" class="" name="li_ZoNR">Calculate free energy change for a process using enthalpies of formation and the entropies for its reactants and products</li></ul></div><p id="fs-idm21370272" class="" name="fs-idm21370272">One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system <em class="" data-effect="italics">and</em> the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard <span id="term544" class="no-emphasis" data-type="term">Gibbs</span>. This new property is called the <span id="term545" class="" data-type="term">Gibbs free energy (<em class="" data-effect="italics">G</em>)</span> (or simply the <em class="" data-effect="italics">free energy</em>), and it is defined in terms of a system’s enthalpy and entropy as the following:</p><div id="fs-idm129849344" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-315-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>G</mi><mo>=</mo><mi>H</mi><mo>−</mo><mi>T</mi><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>G</mi><mo>=</mo><mi>H</mi><mo>−</mo><mi>T</mi><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd columnalign="left"><munder accentunder="true"><mtable><mtr><mtd columnalign="left"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mtext>Δ</mtext><msubsup><mrow><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mspace width="5em"></mspace></mtd></mtr></mtable><mo stretchy="true">¯</mo></munder></mtd></mtr><mtr><mtd columnalign="left"><mtable><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo>=</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd columnalign="left" class=""><munder accentunder="true" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mo class="">+</mo><mfrac bevelled="true" class=""><mn class="">1</mn><mn class="">2</mn></mfrac><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="1em" class=""></mspace><mtext class="">Δ</mtext><msubsup class=""><mrow class=""><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">gas</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">l</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mo class="">+</mo><mfrac bevelled="true" class=""><mn class="">1</mn><mn class="">2</mn></mfrac><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="1em" class=""></mspace><mo class="">−</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">liquid</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mspace width="5em" class=""></mspace></mtd></mtr></mtable><mo stretchy="true" class="">¯</mo></munder></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">l</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="5em" class=""></mspace><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">gas</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">−</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">liquid</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math></span></div></div><p id="fs-idm247387040" class="" name="fs-idm247387040">Free energy is a state function, and at constant temperature and pressure, the <span id="term546" class="" data-type="term">free energy change (Δ<em class="" data-effect="italics">G</em>)</span> may be expressed as the following:</p><div id="fs-idm279983248" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-316-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mrow class=""><mtext class="">SO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="5em" class=""></mspace><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mtext class="">−</mtext><mn class="">300.1</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm111872960" class="" name="fs-idm111872960">(For simplicity’s sake, the subscript “sys” will be omitted henceforth.)</p><p id="fs-idm275842432" class="" name="fs-idm275842432">The relationship between this system property and the spontaneity of a process may be understood by recalling the previously derived second law expression:</p><div id="fs-idm33912192" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-317-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>S</mi><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>S</mi><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>q</mi><mrow><mtext>surr</mtext></mrow></msub></mrow><mi>T</mi></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">1</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mn class="">167.9</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm170940880" class="" name="fs-idm170940880">The first law requires that <em class="" data-effect="italics">q</em><sub class="">surr</sub> = −<em class="" data-effect="italics">q</em><sub class="">sys</sub>, and at constant pressure <em class="" data-effect="italics">q</em><sub class="">sys</sub> = Δ<em class="" data-effect="italics">H</em>, so this expression may be rewritten as:</p><div id="fs-idm195954976" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-318-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>S</mi><mo>−</mo><mspace width="0.1em" ></mspace><mfrac><mrow><mtext>Δ</mtext><mi>H</mi></mrow><mi>T</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>S</mi><mo>−</mo><mspace width="0.1em"></mspace><mfrac><mrow><mtext>Δ</mtext><mi>H</mi></mrow><mi>T</mi></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm169794560" class="" name="fs-idm169794560">Multiplying both sides of this equation by −<em class="" data-effect="italics">T</em>, and rearranging yields the following:</p><div id="fs-idm248567088" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-319-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>−</mtext><mi>T</mi><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>−</mtext><mi>T</mi><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>4Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>3O</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>2Fe</mtext></mrow><mn>2</mn></msub><mrow></mrow><msub><mtext>O</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">4Fe</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mrow class=""><mtext class="">3O</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">2Fe</mtext></mrow><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">O</mtext><mn class="">3</mn></msub><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm229909296" class="" name="fs-idm229909296">Comparing this equation to the previous one for free energy change shows the following relation:</p><div id="fs-idm117423648" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-320-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>−</mtext><mi>T</mi><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>−</mtext><mi>T</mi><mtext>Δ</mtext><msub><mi>S</mi><mrow><mtext>univ</mtext></mrow></msub></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>y</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>m</mi><mi>x</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">y</mi><mo class="">=</mo><mi class="">b</mi><mo class="">+</mo><mi class="">m</mi><mi class="">x</mi></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm228637552" class="" name="fs-idm228637552">The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub>. <a class="autogenerated-content" href="#Table_1.3" data-mce-href="#Table_1.3">Table 1.3</a> summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.</p><p id="p_xxPB" class="os-caption-container table_figure_number" name="p_xxPB">Table <span data-etb-numbered-element="1">1.13</span></p><p id="p_SugG" class="os-table-title table_figure_title" name="p_SugG">Relation between Process Spontaneity and Signs of Thermodynamic Properties</p><table id="fs-idm211518768" class="table table-sm table-striped table-responsive mce-item-table"><thead class="thead-dark"></thead><tbody class=""><tr class="" valign="top"><td class="" data-align="center">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> > 0</td><td class="" data-align="center">Δ<em class="" data-effect="italics">G</em> < 0</td><td class="" data-align="center">spontaneous</td></tr><tr class="" valign="top"><td class="" data-align="center">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> < 0</td><td class="" data-align="center">Δ<em class="" data-effect="italics">G</em> > 0</td><td class="" data-align="center">nonspontaneous</td></tr><tr class="" valign="top"><td class="" data-align="center">Δ<em class="" data-effect="italics">S</em><sub class="">univ</sub> = 0</td><td class="" data-align="center">Δ<em class="" data-effect="italics">G</em> = 0</td><td class="" data-align="center">at equilibrium</td></tr></tbody></table><h3 class="" data-type="title" id="h3_uaLh" name="h3_uaLh"><span data-etb-numbered-element="1">1.4.1</span> What’s “Free” about ΔG?</h3><p id="fs-idm229806576" class="" name="fs-idm229806576">In addition to indicating spontaneity, the free energy change also provides information regarding the amount of useful work (<em class="" data-effect="italics">w</em>) that may be accomplished by a spontaneous process. Although a rigorous treatment of this subject is beyond the scope of an introductory chemistry text, a brief discussion is helpful for gaining a better perspective on this important thermodynamic property.</p><p id="fs-idm215836800" class="" name="fs-idm215836800">For this purpose, consider a spontaneous, exothermic process that involves a decrease in entropy. The free energy, as defined by</p><div id="fs-idm222474000" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-321-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><mi>Δ</mi><mi>H</mi><mo>−</mo><mi>T</mi><mi>Δ</mi><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><mi>Δ</mi><mi>H</mi><mo>−</mo><mi>T</mi><mi>Δ</mi><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>T</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>Δ</mtext><mi>H</mi></mrow><mrow><mtext>Δ</mtext><mi>S</mi></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">T</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">H</mi></mrow><mrow class=""><mtext class="">Δ</mtext><mi class="">S</mi></mrow></mfrac></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm229204848" class="" name="fs-idm229204848">may be interpreted as representing the difference between the energy produced by the process, Δ<em class="" data-effect="italics">H</em>, and the energy lost to the surroundings, <em class="" data-effect="italics">T</em>Δ<em class="" data-effect="italics">S</em>. The difference between the energy produced and the energy lost is the energy available (or “free”) to do useful work by the process, Δ<em class="" data-effect="italics">G</em>. If the process somehow could be made to take place under conditions of thermodynamic reversibility, the amount of work that could be done would be maximal:</p><div id="fs-idm240532528" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-322-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>Δ</mi><mi>G</mi><mo>=</mo><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">Δ</mi><mi class="">G</mi><mo class="">=</mo><msub class=""><mi class="">w</mi><mrow class=""><mi class="">max</mi></mrow></msub></mrow></annotation-xml></semantics></math></span></div></div><p id="para-00001" class="" name="para-00001">where <span class="os-math-in-para"><span id="MathJax-Element-323-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mi>w</mi><mrow><mi>max</mi></mrow></msub></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> refers to all types of work except expansion (pressure-volume) work.</p><p id="fs-idm236757664" class="" name="fs-idm236757664">However, as noted previously in this chapter, such conditions are not realistic. In addition, the technologies used to extract work from a spontaneous process (e.g., batteries) are never 100% efficient, and so the work done by these processes is always less than the theoretical maximum. Similar reasoning may be applied to a nonspontaneous process, for which the free energy change represents the <em class="" data-effect="italics">minimum</em> amount of work that must be done on the system to carry out the process.</p><h3 class="" data-type="title" id="h3_tGAC" name="h3_tGAC"><span data-etb-numbered-element="1">1.4.2</span> Calculating Free Energy Change</h3><p id="fs-idm233417152" class="" name="fs-idm233417152">Free energy is a state function, so its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the use of standard enthalpies and entropies to compute <span id="term547" class="" data-type="term">standard free energy changes, Δ<em class="" data-effect="italics">G</em>°</span>, according to the following relation:</p><div id="fs-idm182227520" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-324-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mrow><mtext>(J/K·mol)</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">S</mi><mtext class="">°</mtext><mrow class=""><mtext class="">(J/K·mol)</mtext></mrow></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm117444432" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_KIvn" name="h3_KIvn">Example <span data-etb-numbered-element="1">1.4.3</span></h3></header><div class="body"><h4 id="h4_kqxF" class="" data-type="title" name="h4_kqxF">Using Standard Enthalpy and Entropy Changes to Calculate Δ<em class="" data-effect="italics">G</em>°</h4>Use standard enthalpy and entropy data from <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a> to calculate the standard free energy change for the vaporization of water at room temperature (298 K). What does the computed value for Δ<em class="" data-effect="italics">G</em>° say about the spontaneity of this process?<h4 id="h4_Nkbt" class="" data-type="title" name="h4_Nkbt">Solution</h4>The process of interest is the following:<div id="fs-idm59589120" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-325-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mtext class="">°</mtext><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm23409248" class="" name="fs-idm23409248">The standard change in free energy may be calculated using the following equation:</p><div id="fs-idm165858560" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-326-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>></mo><mn>0</mn><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">></mo><mn class="">0</mn><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm272232416" class="" name="fs-idm272232416">From <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a>:</p><div id="fs-idm230329280" class="os-table"><div class="table-responsive"><table class="table table-sm table-striped table-responsive mce-item-table" data-id="fs-idm230329280" data-label=""><thead class="thead-dark"><tr class="" valign="top"><th class="" scope="col" data-align="center">Substance</th><th class="" scope="col" data-align="center"><span id="MathJax-Element-327-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mtext>(kJ/mol)</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mtext>(kJ/mol)</mtext></mrow></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mtext class="">f</mtext><mo class="">°</mo></msubsup></mrow></annotation-xml></semantics></math></span></th><th class="" scope="col" data-align="center"><span id="MathJax-Element-328-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mrow><mtext>(J/K·mol)</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mtext>°</mtext><mrow><mtext>(J/K·mol)</mtext></mrow></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">m</mi><mtext class="">A</mtext><mo class="">+</mo><mi class="">n</mi><mtext class="">B</mtext><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mi class="">x</mi><mtext class="">C</mtext><mo class="">+</mo><mi class="">y</mi><mtext class="">D</mtext><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></th></tr></thead><tbody class=""><tr class="" valign="top"><td class="" data-align="center">H<sub class="">2</sub>O(<em class="" data-effect="italics">l</em>)</td><td class="" data-align="center">−286.83</td><td class="" data-align="center">70.0</td></tr><tr class="" valign="top"><td class="" data-align="center">H<sub class="">2</sub>O(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">−241.82</td><td class="" data-align="center">188.8</td></tr></tbody></table></div></div><p id="fs-idm123711584" class="" name="fs-idm123711584">Using the appendix data to calculate the standard enthalpy and entropy changes yields:</p><div id="fs-idm164142400" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-329-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>[</mo><mrow><mtext>−241.82 kJ/mol</mtext><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−286.83</mtext></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mtext>kJ/mol</mtext><mo>=</mo><mtext>45.01 kJ</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mrow><mo>[</mo><mrow><mtext>−241.82 kJ/mol</mtext><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−286.83</mtext></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mtext>kJ/mol</mtext><mo>=</mo><mtext>45.01 kJ</mtext></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm171591584" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-330-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mn>(1 mol)188.8</mn><mspace width="0.2em" ></mspace><mtext>J/mol·K</mtext><mo>−</mo><mn>(1</mn><mspace width="0.2em" ></mspace><mtext>mol)</mtext><mn>70.0</mn><mspace width="0.2em" ></mspace><mtext>J/mol K</mtext><mo>=</mo><mn>118.8</mn><mspace width="0.2em" ></mspace><mtext>J/mol·K</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mrow><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace></mrow><mi>S</mi><mtext>°</mtext><mrow><mo>(</mo><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mn>(1 mol)188.8</mn><mspace width="0.2em"></mspace><mtext>J/mol·K</mtext><mo>−</mo><mn>(1</mn><mspace width="0.2em"></mspace><mtext>mol)</mtext><mn>70.0</mn><mspace width="0.2em"></mspace><mtext>J/mol K</mtext><mo>=</mo><mn>118.8</mn><mspace width="0.2em"></mspace><mtext>J/mol·K</mtext></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mspace width="0.2em"></mspace><mtext>(J/K·mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">S</mi><mtext class="">°</mtext><mspace width="0.2em" class=""></mspace><mtext class="">(J/K·mol)</mtext></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm129668064" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-331-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mtext>(0 kJ/mol)</mtext><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol(0 kJ/mol)</mtext></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol(−58.43 kJ/mol)</mtext><mo>=</mo><mtext>58.43 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mrow class=""><mo class="">[</mo><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mtext class="">(0 kJ/mol)</mtext><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mspace width="0.2em" class=""></mspace><mtext class="">mol(0 kJ/mol)</mtext></mrow><mo class="">]</mo></mrow><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mtext class="">1 mol(−58.43 kJ/mol)</mtext><mo class="">=</mo><mtext class="">58.43 kJ/mol</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idp11689536" class="" name="fs-idp11689536">Substitution into the standard free energy equation yields:</p><div id="fs-idm246654432" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-332-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mtd></mtr><mtr ></mtr><mtr><mtd><mo>=</mo><mtext>45.01 kJ</mtext><mo>−</mo><mrow><mo>(</mo><mrow><mtext>298 K</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mn>118.8</mn><mspace width="0.2em" ></mspace><mtext>J/K</mtext></mrow><mo>)</mo></mrow><mspace width="0.4em" ></mspace><mo>×</mo><mspace width="0.4em" ></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mtext>45.01 kJ</mtext><mo>−</mo><mrow><mo>(</mo><mrow><mtext>298 K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>118.8</mn><mspace width="0.2em"></mspace><mtext>J/K</mtext></mrow><mo>)</mo></mrow><mspace width="0.4em"></mspace><mo>×</mo><mspace width="0.4em"></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>−90.46 kJ/mol</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>90.46 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">0 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">0 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">−90.46 kJ/mol</mtext><mo stretchy="false" class="">)</mo><mo class="">=</mo><mtext class="">90.46 kJ/mol</mtext></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm135070448" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-333-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>45.01 kJ</mtext><mo>−</mo><mtext>35.4 kJ</mtext><mo>=</mo><mtext>9.6 kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>45.01 kJ</mtext><mo>−</mo><mtext>35.4 kJ</mtext><mo>=</mo><mtext>9.6 kJ</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo stretchy="false">(</mo><mtext>75.9 J/mol K</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>205.2 J/mol K</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>71.13 J/mol K</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>107.4 J/mol K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mtext class="">1 mol</mtext><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">(</mo><mtext class="">75.9 J/mol K</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">205.2 J/mol K</mtext><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">71.13 J/mol K</mtext><mo stretchy="false" class="">)</mo><mo class="">=</mo><mtext class="">107.4 J/mol K</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm276144320" class="" name="fs-idm276144320">At 298 K (25 °C) <span class="os-math-in-para"><span id="MathJax-Element-334-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>></mo><mn>0</mn><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>></mo><mn>0</mn><mo>,</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">4</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> so boiling is nonspontaneous (<em class="" data-effect="italics">not</em> spontaneous).</p><h4 id="h4_vRZf" class="" data-type="title" name="h4_vRZf">Check Your Learning</h4>Use standard enthalpy and entropy data from <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a> to calculate the standard free energy change for the reaction shown here (298 K). What does the computed value for Δ<em class="" data-effect="italics">G</em>° say about the spontaneity of this process?<div id="fs-idm276750416" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-335-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow ></mrow><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow ></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>6</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mspace width="5em" class=""></mspace><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">1</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm97664544" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_Dm3263IW" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="true" aria-controls="answer_Dm3263IW" data-bs-toggle="collapse" id="h3_QVAx" name="h3_QVAx" data-mce-href="#answer_Dm3263IW" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_Dm3263IW" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm157144416" class="" name="fs-idm157144416"><span class="os-math-in-para"><span id="MathJax-Element-336-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>102.0 kJ/mol</mtext><mo>;</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>102.0 kJ/mol</mtext><mo>;</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">1</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mn class="">167.9</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math></span></span> the reaction is nonspontaneous (<em class="" data-effect="italics">not</em> spontaneous) at 25 °C.</p></div></section></div></div></div><p id="fs-idm43672192" class="" name="fs-idm43672192">The standard free energy change for a reaction may also be calculated from <span id="term548" class="" data-type="term">standard free energy of formation ΔG°<sub class="">f</sub></span> values of the reactants and products involved in the reaction. The standard free energy of formation is the free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. Similar to the standard enthalpy of formation, <span class="os-math-in-para"><span id="MathJax-Element-337-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>2C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>2CO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">2C</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">2CO</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></span> is by definition zero for elemental substances in their standard states. The approach used to calculate <span class="os-math-in-para"><span id="MathJax-Element-338-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msup><mi>G</mi><mo>°</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msup><mi>G</mi><mo>°</mo></msup></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>y</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>m</mi><mi>x</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">y</mi><mo class="">=</mo><mi class="">b</mi><mo class="">+</mo><mi class="">m</mi><mi class="">x</mi></mrow></annotation-xml></semantics></math></span></span> for a reaction from <span class="os-math-in-para"><span id="MathJax-Element-339-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mtext class="">f</mtext><mo class="">°</mo></msubsup></mrow></annotation-xml></semantics></math></span></span> values is the same as that demonstrated previously for enthalpy and entropy changes. For the reaction</p><div id="fs-idm250621408" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-340-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mtext>A</mtext><mo>+</mo><mi>n</mi><mtext>B</mtext><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mi>x</mi><mtext>C</mtext><mo>+</mo><mi>y</mi><mtext>D</mtext><mo>,</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">,</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm124979680" class="" name="fs-idm124979680">the standard free energy change at room temperature may be calculated as</p><div id="fs-idm193663184" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-341-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr ></mtr><mtr ></mtr><mtr><mtd><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mo>ν</mo><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>products</mtext><mo stretchy="false">)</mo></mrow></mstyle><mspace width="0.2em" ></mspace><mo>−</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mo>ν</mo><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>reactants</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr ></mtr><mtr><mtd><mo>=</mo><mrow><mo>[</mo><mrow><mi>x</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>C</mtext><mo>)</mo></mrow><mo>+</mo><mi>y</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>D</mtext><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mspace width="0.2em" ></mspace><mrow><mo>[</mo><mrow><mi>m</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>A</mtext><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>B</mtext><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr></mtr><mtr></mtr><mtr><mtd><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mo>ν</mo><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>products</mtext><mo stretchy="false">)</mo></mrow></mstyle><mspace width="0.2em"></mspace><mo>−</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mo>ν</mo><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo stretchy="false">(</mo><mtext>reactants</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr></mtr><mtr><mtd><mo>=</mo><mrow><mo>[</mo><mrow><mi>x</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>C</mtext><mo>)</mo></mrow><mo>+</mo><mi>y</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>D</mtext><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mrow><mo>[</mo><mrow><mi>m</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>A</mtext><mo>)</mo></mrow><mo>+</mo><mi>n</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mrow><mo>(</mo><mtext>B</mtext><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mstyle displaystyle="true" class=""><mo class="">∑</mo><mrow class=""><mi class="">ν</mi><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">(products)</mtext><mspace width="0.2em" class=""></mspace><mo class="">−</mo></mrow></mstyle><mstyle displaystyle="true" class=""><mo class="">∑</mo><mrow class=""><mi class="">ν</mi><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">(reactants)</mtext></mrow></mstyle></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm202485584" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_voPK" name="h3_voPK">Example <span data-etb-numbered-element="1">1.4.4</span></h3></header><div class="body"><h4 id="h4_EgiB" class="" data-type="title" name="h4_EgiB">Using Standard Free Energies of Formation to Calculate Δ<em class="" data-effect="italics">G</em>°</h4>Consider the decomposition of yellow mercury(II) oxide.<div id="fs-idm217690608" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-342-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">H</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">Hg</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mtext class="">Δ</mtext><msubsup class=""><mi class="">H</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">H</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">HgO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">yellow</mtext><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm274288288" class="" name="fs-idm274288288">Calculate the standard free energy change at room temperature, <span class="os-math-in-para"><span id="MathJax-Element-343-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>,</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>,</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo stretchy="false">(</mo><mtext>75.9 J/mol K</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>205.2 J/mol K</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>71.13 J/mol K</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>107.4 J/mol K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mo class="">[</mo><mtext class="">1 mol</mtext><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">(</mo><mtext class="">75.9 J/mol K</mtext><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">205.2 J/mol K</mtext><mo stretchy="false" class="">)</mo><mo class="">]</mo><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mtext class="">1 mol</mtext><mo stretchy="false" class="">(</mo><mtext class="">71.13 J/mol K</mtext><mo stretchy="false" class="">)</mo><mo class="">=</mo><mtext class="">107.4 J/mol K</mtext></mrow></annotation-xml></semantics></math></span></span> using (a) standard free energies of formation and (b) standard enthalpies of formation and standard entropies. Do the results indicate the reaction to be spontaneous or nonspontaneous under standard conditions?</p><h4 id="h4_WAcF" class="" data-type="title" name="h4_WAcF">Solution</h4>The required data are available in <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a> and are shown here.<div id="fs-idm232730384" class="os-table"><div class="table-responsive"><table class="table table-sm table-striped table-responsive mce-item-table" data-id="fs-idm232730384" data-label=""><thead class="thead-dark"><tr class="" valign="top"><th class="" scope="col" data-align="center">Compound</th><th class="" scope="col" data-align="center"><span id="MathJax-Element-344-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mspace width="0.2em" ></mspace><mtext>(kJ/mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msubsup><mi>G</mi><mtext>f</mtext><mo>°</mo></msubsup><mspace width="0.2em"></mspace><mtext>(kJ/mol)</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></th><th class="" scope="col" data-align="center"><span id="MathJax-Element-345-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mspace width="0.2em" ></mspace><mtext>(kJ/mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><msubsup><mi>H</mi><mtext>f</mtext><mo>°</mo></msubsup><mspace width="0.2em"></mspace><mtext>(kJ/mol)</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>SO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mtext>−</mtext><mn>98.8</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">SO</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="5em" class=""></mspace><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mtext class="">−</mtext><mn class="">300.1</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mtext class="">−</mtext><mn class="">98.8</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></annotation-xml></semantics></math></span></th><th class="" scope="col" data-align="center"><span id="MathJax-Element-346-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>S</mi><mtext>°</mtext><mspace width="0.2em" ></mspace><mtext>(J/K·mol)</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mtext>°</mtext><mspace width="0.2em"></mspace><mtext>(J/K·mol)</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>2C</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><mtext>2CO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">2C</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><mtext class="">2CO</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></th></tr></thead><tbody class=""><tr class="" valign="top"><td class="" data-align="center">HgO (<em class="" data-effect="italics">s</em>, yellow)</td><td class="" data-align="center">−58.43</td><td class="" data-align="center">−90.46</td><td class="" data-align="center">71.13</td></tr><tr class="" valign="top"><td class="" data-align="center">Hg(<em class="" data-effect="italics">l</em>)</td><td class="" data-align="center">0</td><td class="" data-align="center">0</td><td class="" data-align="center">75.9</td></tr><tr class="" valign="top"><td class="" data-align="center">O<sub class="">2</sub>(<em class="" data-effect="italics">g</em>)</td><td class="" data-align="center">0</td><td class="" data-align="center">0</td><td class="" data-align="center">205.2</td></tr></tbody></table></div></div><p id="fs-idm192751264" class="" name="fs-idm192751264">(a) Using free energies of formation:</p><div id="fs-idm211589632" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-347-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em" ></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mn class="">0</mn><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm178475312" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-348-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mtext>2</mtext></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mspace width="0.2em" ></mspace><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mtext>2</mtext></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mtext>2</mtext></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mspace width="0.2em"></mspace><mn>1</mn><mtext>Δ</mtext><msubsup><mi>G</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mo class="">=</mo><mrow class=""><mo class="">[</mo><mrow class=""><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">Hg</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mn class="">1</mn><mn class="">2</mn></mfrac><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><msub class=""><mtext class="">O</mtext><mtext class="">2</mtext></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow><mo class="">]</mo></mrow><mspace width="0.2em" class=""></mspace><mo class="">−</mo><mspace width="0.2em" class=""></mspace><mn class="">1</mn><mtext class="">Δ</mtext><msubsup class=""><mi class="">G</mi><mrow class=""><mn class="">f</mn></mrow><mo class="">°</mo></msubsup><mtext class="">HgO</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo class="">,</mo><mspace width="0.2em" class=""></mspace><mtext class="">yellow</mtext><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm164697728" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-349-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mspace width="0.2em" ></mspace><mtext>mol</mtext><mtext>(0 kJ/mol)</mtext><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em" ></mspace><mtext>mol(0 kJ/mol)</mtext></mrow><mo>]</mo></mrow><mspace width="0.2em" ></mspace><mo>−</mo><mtext>1 mol(−58.43 kJ/mol)</mtext><mo>=</mo><mtext>58.43 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mrow><mo>[</mo><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mtext>(0 kJ/mol)</mtext><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol(0 kJ/mol)</mtext></mrow><mo>]</mo></mrow><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol(−58.43 kJ/mol)</mtext><mo>=</mo><mtext>58.43 kJ/mol</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mo>−</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi></mrow><mtext>°</mtext><mtext>(reactants)</mtext></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mo class="">∑</mo><mtext class="">ν</mtext><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(products)</mtext><mo class="">−</mo><mo class="">∑</mo><mtext class="">ν</mtext><mtext class="">Δ</mtext><mi class="">S</mi></mrow><mtext xmlns="http://www.w3.org/1998/Math/MathML" class="">°</mtext><mtext xmlns="http://www.w3.org/1998/Math/MathML" class="">(reactants)</mtext></annotation-xml></semantics></math></span></div></div><p id="fs-idm112946608" class="" name="fs-idm112946608">(b) Using enthalpies and entropies of formation:</p><div id="fs-idm147522592" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-350-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em" ></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(products)</mtext><mspace width="0.2em"></mspace><mo>−</mo></mrow></mstyle><mstyle displaystyle="true"><mo>∑</mo><mrow><mi>ν</mi><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mtext>f</mtext></mrow><mo>°</mo></msubsup><mtext>(reactants)</mtext></mrow></mstyle></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">4</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">C</mtext><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm39030688" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-351-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em" ></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><msubsup><mi>H</mi><mrow><mn>f</mn></mrow><mo>°</mo></msubsup><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>SO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mtext>−</mtext><mn>98.8</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">SO</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="5em" class=""></mspace><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mtext class="">−</mtext><mn class="">300.1</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mtext class="">−</mtext><mn class="">98.8</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm261408512" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-352-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em" ></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em" ></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>−90.46 kJ/mol</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>90.46 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>0 kJ/mol</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>−90.46 kJ/mol</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>90.46 kJ/mol</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>4Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>3O</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>2Fe</mtext></mrow><mn>2</mn></msub><mrow></mrow><msub><mtext>O</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">4Fe</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mrow class=""><mtext class="">3O</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">2Fe</mtext></mrow><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">O</mtext><mn class="">3</mn></msub><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idp6944752" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-353-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mo>−</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi></mrow><mtext>°</mtext><mtext>(reactants)</mtext></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mo>−</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi></mrow><mtext>°</mtext><mtext>(reactants)</mtext></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>(products)</mtext><mo>−</mo><mo>∑</mo><mtext>ν</mtext><mtext>Δ</mtext><mi>S</mi></mrow><mtext>°</mtext><mtext>(reactants)</mtext></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mo class="">=</mo><mo class="">∑</mo><mtext class="">ν</mtext><mtext class="">Δ</mtext><mi class="">S</mi><mtext class="">°</mtext><mtext class="">(products)</mtext><mo class="">−</mo><mo class="">∑</mo><mtext class="">ν</mtext><mtext class="">Δ</mtext><mi class="">S</mi></mrow><mtext xmlns="http://www.w3.org/1998/Math/MathML" class="">°</mtext><mtext xmlns="http://www.w3.org/1998/Math/MathML" class="">(reactants)</mtext></annotation-xml></semantics></math></span></div></div><div id="fs-idm61090368" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-354-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em" ></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em" ></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>Hg</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mn>1</mn><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mtext>HgO</mtext><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mspace width="0.2em"></mspace><mtext>yellow</mtext><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">l</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm131288608" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-355-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mspace width="0.2em" ></mspace><mo stretchy="false">(</mo><mtext>75.9 J/mol K</mtext><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo>+</mo><mspace width="0.2em" ></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em" ></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>205.2 J/mol K</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em" ></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>71.13 J/mol K</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>107.4 J/mol K</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mo>[</mo><mtext>1 mol</mtext><mspace width="0.2em"></mspace><mo stretchy="false">(</mo><mtext>75.9 J/mol K</mtext><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo>+</mo><mspace width="0.2em"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mspace width="0.2em"></mspace><mtext>mol</mtext><mo stretchy="false">(</mo><mtext>205.2 J/mol K</mtext><mo stretchy="false">)</mo><mo>]</mo><mspace width="0.2em"></mspace><mo>−</mo><mtext>1 mol</mtext><mo stretchy="false">(</mo><mtext>71.13 J/mol K</mtext><mo stretchy="false">)</mo><mo>=</mo><mtext>107.4 J/mol K</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>FeS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">FeS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Fe</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mrow class=""><mtext class="">SO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm153411216" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-356-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mtext>90.46 kJ</mtext><mo>−</mo><mtext>298.15 K</mtext><mspace width="0.2em" ></mspace><mo>×</mo><mspace width="0.2em" ></mspace><mtext>107.4 J/K·mol</mtext><mspace width="0.4em" ></mspace><mo>×</mo><mspace width="0.4em" ></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mtext>°</mtext><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi><mtext>°</mtext><mo>=</mo><mtext>90.46 kJ</mtext><mo>−</mo><mtext>298.15 K</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>107.4 J/K·mol</mtext><mspace width="0.4em"></mspace><mo>×</mo><mspace width="0.4em"></mspace><mfrac><mrow><mtext>1 kJ</mtext></mrow><mrow><mtext>1000 J</mtext></mrow></mfrac></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mn class="">0</mn><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm190648624" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-357-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mo stretchy="false">(</mo><mn>90.46</mn><mo>−</mo><mn>32.01</mn><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mtext>kJ/mol</mtext><mo>=</mo><mtext>58.45 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mo stretchy="false">(</mo><mn>90.46</mn><mo>−</mo><mn>32.01</mn><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mtext>kJ/mol</mtext><mo>=</mo><mtext>58.45 kJ/mol</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mtext>°</mtext><mo>=</mo><mo stretchy="false">(</mo><mn>90.46</mn><mo>−</mo><mn>32.01</mn><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mtext>kJ/mol</mtext><mo>=</mo><mtext>58.45 kJ/mol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mtext class="">°</mtext><mo class="">=</mo><mo stretchy="false" class="">(</mo><mn class="">90.46</mn><mo class="">−</mo><mn class="">32.01</mn><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mtext class="">kJ/mol</mtext><mo class="">=</mo><mtext class="">58.45 kJ/mol</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm269603872" class="" name="fs-idm269603872">Both ways to calculate the standard free energy change at 25 °C give the same numerical value (to three significant figures), and both predict that the process is nonspontaneous (<em class="" data-effect="italics">not</em> spontaneous) at room temperature.</p><h4 id="h4_kDck" class="" data-type="title" name="h4_kDck">Check Your Learning</h4>Calculate Δ<em class="" data-effect="italics">G</em>° using (a) free energies of formation and (b) enthalpies of formation and entropies (<a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a>). Do the results indicate the reaction to be spontaneous or nonspontaneous at 25 °C?<div id="fs-idm213619408" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-358-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow ></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow ></mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>4</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>C</mtext><mn>2</mn></msub><mrow></mrow><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">1</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mn class="">167.9</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math></span></div></div><div id="fs-idm254989456" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_MASEtC8H" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="true" aria-controls="answer_MASEtC8H" data-bs-toggle="collapse" id="h3_BMeh" name="h3_BMeh" data-mce-href="#answer_MASEtC8H" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_MASEtC8H" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idp2244016" class="" name="fs-idp2244016">(a) 140.8 kJ/mol, nonspontaneous<span class="" data-type="newline"><br class=""></span>(b) 141.5 kJ/mol, nonspontaneous</p></div></section></div></div></div><h3 class="" data-type="title" id="h3_MuSY" name="h3_MuSY"><span data-etb-numbered-element="1">1.4.5</span> Free Energy Changes for Coupled Reactions</h3><p id="fs-idm243247344" class="" name="fs-idm243247344">The use of free energies of formation to compute free energy changes for reactions as described above is possible because ΔG is a state function, and the approach is analogous to the use of Hess’ Law in computing enthalpy changes (see the chapter on thermochemistry). Consider the vaporization of water as an example:</p><div id="fs-idm237314080" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-359-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mn class="">0</mn><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm244158208" class="" name="fs-idm244158208">An equation representing this process may be derived by adding the formation reactions for the two phases of water (necessarily reversing the reaction for the liquid phase). The free energy change for the sum reaction is the sum of free energy changes for the two added reactions:</p><div id="fs-idm245138624" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-360-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd columnalign="left"><munder accentunder="true"><mtable><mtr><mtd columnalign="left"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo stretchy="false">→</mo><mspace width="0.2em" ></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></mtd><mtd ></mtd><mtd ></mtd><mtd columnalign="left"><mspace width="1em" ></mspace><mtext>Δ</mtext><msubsup><mrow><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd ></mtd><mtd ></mtd><mtd columnalign="left"><mspace width="1em" ></mspace><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mspace width="5em" ></mspace></mtd></mtr></mtable><mo stretchy="true">¯</mo></munder></mtd></mtr><mtr><mtd columnalign="left"><mtable><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em" ></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd ></mtd><mtd ></mtd><mtd columnalign="left"><mspace width="5em" ></mspace><mtext>Δ</mtext><mi>G</mi><mo>=</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable><mtr><mtd columnalign="left"><munder accentunder="true"><mtable><mtr><mtd columnalign="left"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mtext>Δ</mtext><msubsup><mrow><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mspace width="5em"></mspace></mtd></mtr></mtable><mo stretchy="true">¯</mo></munder></mtd></mtr><mtr><mtd columnalign="left"><mtable><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo>=</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd columnalign="left"><munder accentunder="true"><mtable><mtr><mtd columnalign="left"><mrow><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mtext>Δ</mtext><msubsup><mrow><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>+</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><msub><mtext>O</mtext><mn>2</mn></msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mspace width="5em"></mspace></mtd></mtr></mtable><mo stretchy="true">¯</mo></munder></mtd></mtr><mtr><mtd columnalign="left"><mtable><mtr><mtd columnalign="left"><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mspace width="0.2em"></mspace><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mtd><mtd></mtd><mtd></mtd><mtd columnalign="left"><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo>=</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>gas</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>−</mo><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>f</mtext><mo>,</mo><mtext>liquid</mtext></mrow><mrow><mo>°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd columnalign="left" class=""><munder accentunder="true" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mo class="">+</mo><mfrac bevelled="true" class=""><mn class="">1</mn><mn class="">2</mn></mfrac><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="1em" class=""></mspace><mtext class="">Δ</mtext><msubsup class=""><mrow class=""><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">gas</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">l</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow><mo class="">+</mo><mfrac bevelled="true" class=""><mn class="">1</mn><mn class="">2</mn></mfrac><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="1em" class=""></mspace><mo class="">−</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">liquid</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mspace width="5em" class=""></mspace></mtd></mtr></mtable><mo stretchy="true" class="">¯</mo></munder></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">l</mi><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">O</mtext><mrow class=""><mo class="">(</mo><mi class="">g</mi><mo class="">)</mo></mrow></mtd><mtd class=""></mtd><mtd class=""></mtd><mtd columnalign="left" class=""><mspace width="5em" class=""></mspace><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">gas</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">−</mo><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">f</mtext><mo class="">,</mo><mtext class="">liquid</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math></span></div></div><p id="fs-idm243360800" class="" name="fs-idm243360800">This approach may also be used in cases where a nonspontaneous reaction is enabled by coupling it to a spontaneous reaction. For example, the production of elemental zinc from zinc sulfide is thermodynamically unfavorable, as indicated by a positive value for ΔG°:</p><div id="fs-idm244734352" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-361-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="5em" ></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>4Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>3O</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>2Fe</mtext></mrow><mn>2</mn></msub><mrow></mrow><msub><mtext>O</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">4Fe</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mrow class=""><mtext class="">3O</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="0.2em" class=""></mspace><mo stretchy="false" class="">⟶</mo><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">2Fe</mtext></mrow><mn class="">2</mn></msub><mrow class=""></mrow><msub class=""><mtext class="">O</mtext><mn class="">3</mn></msub><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm243301424" class="" name="fs-idm243301424">The industrial process for production of zinc from sulfidic ores involves coupling this decomposition reaction to the thermodynamically favorable oxidation of sulfur:</p><div id="fs-idm242211664" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-362-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em" ></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mrow class=""><mtext class="">SO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mspace width="5em" class=""></mspace><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mtext class="">−</mtext><mn class="">300.1</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm244876016" class="" name="fs-idm244876016">The coupled reaction exhibits a negative free energy change and is spontaneous:</p><div id="fs-idm243579984" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-363-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>SO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em" ></mspace><mtext>Δ</mtext><mi>G</mi><mo>°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext><mo>+</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext><mo>=</mo><mtext>−</mtext><mn>98.8</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>SO</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="5em"></mspace><mtext>Δ</mtext><mi>G</mi><mo>°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mtext>−</mtext><mn>300.1</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mtext>−</mtext><mn>98.8</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" data-mce-tabindex="0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mn class="">0</mn><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></span></div></div><p id="fs-idm225059872" class="" name="fs-idm225059872">This process is typically carried out at elevated temperatures, so this result obtained using standard free energy values is just an estimate. The gist of the calculation, however, holds true.</p><div id="fs-idm243536208" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_fihY" name="h3_fihY">Example <span data-etb-numbered-element="1">1.4.6</span></h3></header><div class="body"><h4 id="h4_ucRL" class="" data-type="title" name="h4_ucRL">Calculating Free Energy Change for a Coupled Reaction</h4>Is a reaction coupling the decomposition of ZnS to the formation of H<sub class="">2</sub>S expected to be spontaneous under standard conditions?<h4 id="h4_mXhh" class="" data-type="title" name="h4_mXhh">Solution</h4>Following the approach outlined above and using free energy values from <a class="" href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-page-slug="g-standard-thermodynamic-properties-for-selected-substances" data-page-uuid="667adccf-f900-4d86-a13d-409c014086ea" data-page-fragment="" data-shortened-url="https://edtechbooks.org/-oWT" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances" data-mce-href="https://openstax.org/books/chemistry-atoms-first-2e/pages/g-standard-thermodynamic-properties-for-selected-substances">Appendix G</a>:<div id="fs-idm195100848" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-364-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo>°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em" ></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo>°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; 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class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-msubsup><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-script style="vertical-align: -0.247em; margin-left: 0px;" data-mce-style="vertical-align: -0.247em; margin-left: 0px;"><mjx-mrow size="s"><mjx-mo class="mjx-var"><mjx-c class="mjx-cB0"></mjx-c></mjx-mo></mjx-mrow><mjx-spacer style="margin-top: 0.602em;" data-mce-style="margin-top: 0.602em;"></mjx-spacer><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mtext></mjx-mrow></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6B"></mjx-c><mjx-c class="mjx-c4A"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c46"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c79"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c3A"></mjx-c></mjx-mtext></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c53"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c48"></mjx-c></mjx-mtext><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c48"></mjx-c></mjx-mtext><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-c53"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-msubsup><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-script style="vertical-align: -0.247em; margin-left: 0px;" data-mce-style="vertical-align: -0.247em; margin-left: 0px;"><mjx-mrow size="s"><mjx-mo class="mjx-var"><mjx-c class="mjx-cB0"></mjx-c></mjx-mo></mjx-mrow><mjx-spacer style="margin-top: 0.602em;" data-mce-style="margin-top: 0.602em;"></mjx-spacer><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mtext></mjx-mrow></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6B"></mjx-c><mjx-c class="mjx-c4A"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c43"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c3A"></mjx-c></mjx-mtext></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c5A"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c53"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c48"></mjx-c></mjx-mtext><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c5A"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c48"></mjx-c></mjx-mtext><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mtext class="mjx-n"><mjx-c class="mjx-c53"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-cB0"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6B"></mjx-c><mjx-c class="mjx-c4A"></mjx-c></mjx-mtext><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6B"></mjx-c><mjx-c class="mjx-c4A"></mjx-c></mjx-mtext><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6B"></mjx-c><mjx-c class="mjx-c4A"></mjx-c></mjx-mtext></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd columnalign="left"><mrow><mtext>Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>1</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>S</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><msubsup><mrow><mtext>Δ</mtext><mi>G</mi></mrow><mrow><mtext>2</mtext></mrow><mrow><mo>°</mo></mrow></msubsup><mo>=</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mtd></mtr><mtr><mtd columnalign="left"><mrow><mtext>Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left"><mrow><mtext>ZnS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Zn</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>H</mtext><mn>2</mn></msub><mtext>S</mtext><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mtext>Δ</mtext><mi>G</mi><mo data-mjx-pseudoscript="true">°</mo><mo>=</mo><mn>201.3</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>+</mo><mn>−33.4</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext><mo>=</mo><mn>167.9</mn><mspace width="0.2em"></mspace><mtext>kJ</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtable class=""><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Decomposition of zinc sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">1</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Formation of hydrogen sulfide:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><msubsup class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi></mrow><mrow class=""><mtext class="">2</mtext></mrow><mrow class=""><mo class="">°</mo></mrow></msubsup><mo class="">=</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mtd></mtr><mtr class=""><mtd columnalign="left" class=""><mrow class=""><mtext class="">Coupled reaction:</mtext></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">ZnS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Zn</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">H</mtext><mn class="">2</mn></msub><mtext class="">S</mtext><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></mtd><mtd columnalign="left" class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">°</mo><mo class="">=</mo><mn class="">201.3</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">+</mo><mn class="">−33.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext><mo class="">=</mo><mn class="">167.9</mn><mspace width="0.2em" class=""></mspace><mtext class="">kJ</mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><p id="fs-idm213857504" class="" name="fs-idm213857504">The coupled reaction exhibits a positive free energy change and is thus nonspontaneous.</p><h4 id="h4_kuqw" class="" data-type="title" name="h4_kuqw">Check Your Learning</h4>What is the standard free energy change for the reaction below? Is the reaction expected to be spontaneous under standard conditions?<div id="fs-idm212071344" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-365-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>FeS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>FeS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c46"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c53"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c4F"></mjx-c></mjx-mtext><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c46"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c53"></mjx-c><mjx-c class="mjx-c4F"></mjx-c></mjx-mtext></mjx-mrow><mjx-script style="vertical-align: -0.15em;" data-mce-style="vertical-align: -0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>FeS</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>O</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mtext>Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>SO</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">FeS</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mtext class="">O</mtext><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo><mo stretchy="false" class="">→</mo><mtext class="">Fe</mtext><mo stretchy="false" class="">(</mo><mi class="">s</mi><mo stretchy="false" class="">)</mo><mo class="">+</mo><msub class=""><mrow class=""><mtext class="">SO</mtext></mrow><mn class="">2</mn></msub><mo stretchy="false" class="">(</mo><mi class="">g</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><div id="fs-idm224787152" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_tmIBvqPs" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="true" aria-controls="answer_tmIBvqPs" data-bs-toggle="collapse" id="h3_eZLV" name="h3_eZLV" data-mce-href="#answer_tmIBvqPs" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_tmIBvqPs" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm206530384" class="" name="fs-idm206530384">–199.7 kJ; spontaneous</p></div></section></div></div></div><div class="callout default" data-callout-custom-class="default"><h3 class="" id="h3_WVYp" name="h3_WVYp">Summary of Enthalpy, Entropy, and Free Energy</h3><p id="p_LiEH" class="" name="p_LiEH"></p><small class="text-muted"></small><p id="p_CTXr" class="" name="p_CTXr"></p><figure class="col-12"><a href="https://www.youtube.com/watch?v=7jSwtKF-wyA&autoplay=1" data-mce-href="https://www.youtube.com/watch?v=7jSwtKF-wyA&autoplay=1" class="position-relative text-center d-block"><img class="img-fluid" id="YT_7jSwtKF-wyA" data-youtube-thumbnail-style="hq" src="https://i3.ytimg.com/vi/7jSwtKF-wyA/hqdefault.jpg" data-mce-src="https://i3.ytimg.com/vi/7jSwtKF-wyA/hqdefault.jpg" style="display: block; margin-left: auto; margin-right: auto;" data-mce-style="display: block; margin-left: auto; margin-right: auto;"></a><figcaption class="text-center"><a target="_blank" href="https://www.youtube.com/watch?v=7jSwtKF-wyA&autoplay=1" rel="noopener" data-mce-href="https://www.youtube.com/watch?v=7jSwtKF-wyA&autoplay=1">Watch on YouTube</a></figcaption></figure> <p id="p_ciSQ" name="p_ciSQ"></p></div><h2 id="p_yvJk" class="os-caption table_figure_title" name="p_yvJk"><span data-etb-numbered-element="1">1.5</span> Equilibrium</h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_zcMj" name="h3_zcMj">Learning Objectives</h3></header><p id="para-00001" class="" name="para-00001">By the end of this section, you will be able to:</p><ul id="list-00001" class="" name="list-00001"><li id="li_gUwt" class="" name="li_gUwt">Explain how temperature affects the spontaneity of some processes</li></ul></div><h3 class="" data-type="title" id="h3_iLge" name="h3_iLge">Temperature Dependence of Spontaneity</h3><p id="fs-idm158748720" class="" name="fs-idm158748720">As was previously demonstrated in the section on entropy in an earlier chapter, the spontaneity of a process may depend upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or the other depending upon the temperature of the substance in question. Likewise, some chemical reactions can also exhibit temperature dependent spontaneities. 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="96" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><p id="fs-idm176739120" class="" name="fs-idm176739120">The spontaneity of a process, as reflected in the arithmetic sign of its free energy change, is then determined by the signs of the enthalpy and entropy changes and, in some cases, the absolute temperature. Since <em class="" data-effect="italics">T</em> is the absolute (kelvin) temperature, it can only have positive values. Four possibilities therefore exist with regard to the signs of the enthalpy and entropy changes:</p><ol id="fs-idm124567280" class="" type="1" name="fs-idm124567280"><li id="li_vWpE" class="" name="li_vWpE"><strong class="">Both Δ<em class="" data-effect="italics">H</em> and Δ<em class="" data-effect="italics">S</em> are positive.</strong> This condition describes an endothermic process that involves an increase in system entropy. In this case, Δ<em class="" data-effect="italics">G</em> will be negative if the magnitude of the <em class="" data-effect="italics">T</em>Δ<em class="" data-effect="italics">S</em> term is greater than Δ<em class="" data-effect="italics">H</em>. If the <em class="" data-effect="italics">T</em>Δ<em class="" data-effect="italics">S</em> term is less than Δ<em class="" data-effect="italics">H</em>, the free energy change will be positive. Such a process is <em class="" data-effect="italics">spontaneous at high temperatures and nonspontaneous at low temperatures.</em></li><li id="li_VFTX" class="" name="li_VFTX"><strong class="">Both Δ<em class="" data-effect="italics">H</em> and Δ<em class="" data-effect="italics">S</em> are negative.</strong> This condition describes an exothermic process that involves a decrease in system entropy. In this case, Δ<em class="" data-effect="italics">G</em> will be negative if the magnitude of the <em class="" data-effect="italics">T</em>Δ<em class="" data-effect="italics">S</em> term is less than Δ<em class="" data-effect="italics">H</em>. If the <em class="" data-effect="italics">T</em>Δ<em class="" data-effect="italics">S</em> term’s magnitude is greater than Δ<em class="" data-effect="italics">H</em>, the free energy change will be positive. Such a process is <em class="" data-effect="italics">spontaneous at low temperatures and nonspontaneous at high temperatures.</em></li><li id="li_zjif" class="" name="li_zjif"><strong class="">Δ<em class="" data-effect="italics">H</em> is positive and Δ<em class="" data-effect="italics">S</em> is negative.</strong> This condition describes an endothermic process that involves a decrease in system entropy. In this case, Δ<em class="" data-effect="italics">G</em> will be positive regardless of the temperature. Such a process is <em class="" data-effect="italics">nonspontaneous at all temperatures.</em></li><li id="li_pDYr" class="" name="li_pDYr"><strong class="">Δ<em class="" data-effect="italics">H</em> is negative and Δ<em class="" data-effect="italics">S</em> is positive.</strong> This condition describes an exothermic process that involves an increase in system entropy. In this case, Δ<em class="" data-effect="italics">G</em> will be negative regardless of the temperature. Such a process is <em class="" data-effect="italics">spontaneous at all temperatures.</em></li></ol><p id="fs-idm39683376" class="" name="fs-idm39683376">These four scenarios are summarized in <a class="autogenerated-content" href="#Figure_1.11" data-mce-href="#Figure_1.11">Figure 1.11</a>.</p><p id="p_jrWL" class="os-caption-container table_figure_number" name="p_jrWL">Figure <span data-etb-numbered-element="1">1.14</span></p><p id="p_swdT" class="os-caption table_figure_title" name="p_swdT">There are four possibilities regarding the signs of enthalpy and entropy changes.</p><figure class="" data-id="CNX_Chem_16_04_Scenarios"><span id="fs-idm123583264" class="" data-type="media" data-alt="A table with three columns and four rows is shown. The first column has the phrase, “Delta S greater than zero ( increase in entropy ),” in the third row and the phrase, “Delta S less than zero ( decrease in entropy),” in the fourth row. The second and third columns have the phrase, “Summary of the Four Scenarios for Enthalpy and Entropy Changes,” written above them. The second column has, “delta H greater than zero ( endothermic ),” in the second row, “delta G less than zero at high temperature, delta G greater than zero at low temperature, Process is spontaneous at high temperature,” in the third row, and “delta G greater than zero at any temperature, Process is nonspontaneous at any temperature,” in the fourth row. The third column has, “delta H less than zero ( exothermic ),” in the second row, “delta G less than zero at any temperature, Process is spontaneous at any temperature,” in the third row, and “delta G less than zero at low temperature, delta G greater than zero at high temperature, Process is spontaneous at low temperature.”"><img id="img_Cuod" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_16.jpg" alt="A table with three columns and four rows is shown. The first column has the phrase, “Delta S greater than zero ( increase in entropy ),” in the third row and the phrase, “Delta S less than zero ( decrease in entropy),” in the fourth row. The second and third columns have the phrase, “Summary of the Four Scenarios for Enthalpy and Entropy Changes,” written above them. The second column has, “delta H greater than zero ( endothermic ),” in the second row, “delta G less than zero at high temperature, delta G greater than zero at low temperature, Process is spontaneous at high temperature,” in the third row, and “delta G greater than zero at any temperature, Process is nonspontaneous at any temperature,” in the fourth row. The third column has, “delta H less than zero ( exothermic ),” in the second row, “delta G less than zero at any temperature, Process is spontaneous at any temperature,” in the third row, and “delta G less than zero at low temperature, delta G greater than zero at high temperature, Process is spontaneous at low temperature.”" name="img_Cuod" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_16.jpg"></span></figure><div id="fs-idm299243200" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_pxjs" name="h3_pxjs">EXAMPLE <span data-etb-numbered-element="1">1.5.1</span></h3></header><div class="body"><h4 id="h4_FcSP" class="" data-type="title" name="h4_FcSP">Predicting the Temperature Dependence of Spontaneity</h4>The incomplete combustion of carbon is described by the following equation:<div id="fs-idm124549040" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-433-Frame" class="MathJax" style="box-sizing: border-box; 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This particular reaction involves an increase in entropy due to the accompanying increase in the amount of gaseous species (net gain of one mole of gas, Δ<em class="" data-effect="italics">S</em> > 0). The reaction is therefore spontaneous (Δ<em class="" data-effect="italics">G</em> < 0) at all temperatures.<h4 id="h4_svgK" class="" data-type="title" name="h4_svgK">Check Your Learning</h4>Popular chemical hand warmers generate heat by the air-oxidation of iron:<div id="fs-idm247358864" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-434-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>4Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>3O</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em" ></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em" ></mspace><msub><mrow><mtext>2Fe</mtext></mrow><mn>2</mn></msub><mrow ></mrow><msub><mtext>O</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>4Fe</mtext><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mtext>3O</mtext></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="0.2em"></mspace><mo stretchy="false">⟶</mo><mspace width="0.2em"></mspace><msub><mrow><mtext>2Fe</mtext></mrow><mn>2</mn></msub><mrow></mrow><msub><mtext>O</mtext><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; 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the reaction is spontaneous at low temperatures.</p></div></section></div></div></div><p id="fs-idp2328928" class="" name="fs-idp2328928">When considering the conclusions drawn regarding the temperature dependence of spontaneity, it is important to keep in mind what the terms “high” and “low” mean. Since these terms are adjectives, the temperatures in question are deemed high or low relative to some reference temperature. A process that is nonspontaneous at one temperature but spontaneous at another will necessarily undergo a change in “spontaneity” (as reflected by its Δ<em class="" data-effect="italics">G</em>) as temperature varies. This is clearly illustrated by a graphical presentation of the free energy change equation, in which Δ<em class="" data-effect="italics">G</em> is plotted on the <em class="" data-effect="italics">y</em> axis versus <em class="" data-effect="italics">T</em> on the <em class="" data-effect="italics">x</em> axis:</p><div id="fs-idm161001120" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-435-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><div id="fs-idm285695760" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-436-Frame" class="MathJax" style="box-sizing: border-box; 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="100" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="100" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>y</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>m</mi><mi>x</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">y</mi><mo class="">=</mo><mi class="">b</mi><mo class="">+</mo><mi class="">m</mi><mi class="">x</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><p id="fs-idm206759776" class="" name="fs-idm206759776">Such a plot is shown in <a class="autogenerated-content" href="#Figure_1.12" data-mce-href="#Figure_1.12">Figure 1.12</a>. A process whose enthalpy and entropy changes are of the same arithmetic sign will exhibit a temperature-dependent spontaneity as depicted by the two yellow lines in the plot. Each line crosses from one spontaneity domain (positive or negative Δ<em class="" data-effect="italics">G</em>) to the other at a temperature that is characteristic of the process in question. This temperature is represented by the <em class="" data-effect="italics">x</em>-intercept of the line, that is, the value of <em class="" data-effect="italics">T</em> for which Δ<em class="" data-effect="italics">G</em> is zero:</p><div id="fs-idm187292464" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-437-Frame" class="MathJax" style="box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></annotation-xml></semantics></math>" data-mce-style="box-sizing: border-box; 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mtext class="mjx-n" space="4"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>Δ</mtext><mi>G</mi><mo>=</mo><mn>0</mn><mo>=</mo><mtext>Δ</mtext><mi>H</mi><mo>−</mo><mi>T</mi><mtext>Δ</mtext><mi>S</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">Δ</mtext><mi class="">G</mi><mo class="">=</mo><mn class="">0</mn><mo class="">=</mo><mtext class="">Δ</mtext><mi class="">H</mi><mo class="">−</mo><mi class="">T</mi><mtext class="">Δ</mtext><mi class="">S</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><div id="fs-idm280698832" class="unnumbered" data-type="equation"><div class="MathJax_Display"><span id="MathJax-Element-438-Frame" class="MathJax" style="box-sizing: border-box; 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position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;" data-mce-style="margin-left: 0px; margin-right: 0px;"><mjx-semantics><mjx-mrow><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mtext class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mtext><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>T</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>Δ</mtext><mi>H</mi></mrow><mrow><mtext>Δ</mtext><mi>S</mi></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">T</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">Δ</mtext><mi class="">H</mi></mrow><mrow class=""><mtext class="">Δ</mtext><mi class="">S</mi></mrow></mfrac></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span></div></div><p id="fs-idm44045216" class="" name="fs-idm44045216">So, saying a process is spontaneous at “high” or “low” temperatures means the temperature is above or below, respectively, that temperature at which Δ<em class="" data-effect="italics">G</em> for the process is zero. As noted earlier, the condition of ΔG = 0 describes a system at equilibrium.</p><p class="os-caption-container table_figure_number" id="p_XVEq" name="p_XVEq">Figure <span data-etb-numbered-element="1">1.15</span></p><p id="p_kPJk" class="os-caption table_figure_title" name="p_kPJk">These plots show the variation in Δ<em class="" data-effect="italics">G</em> with temperature for the four possible combinations of arithmetic sign for Δ<em class="" data-effect="italics">H</em> and Δ<em class="" data-effect="italics">S</em>.</p><figure class="col-12 mx-auto col-md-10"><img id="img_QniG" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_17.jpg" alt="A graph is shown where the y-axis is labeled, “Free energy,” and the x-axis is labeled, “Increasing temperature ( K ).” The value of zero is written midway up the y-axis with the label, “delta G greater than 0,” written above this line and, “delta G less than 0,” written below it. The bottom half of the graph is labeled on the right as, “Spontaneous,” and the top half is labeled on the right as, “Nonspontaneous.” A green line labeled, “delta H less than 0, delta S greater than 0,” extends from a quarter of the way up the y-axis to the bottom right of the graph. A yellow line labeled, “delta H less than 0, delta S less than 0,” extends from a quarter of the way up the y-axis to the middle right of the graph. A second yellow line labeled, “delta H greater than 0, delta S greater than 0,” extends from three quarters of the way up the y-axis to the middle right of the graph. A red line labeled, “delta H greater than 0, delta S less than 0,” extends from three quarters of the way up the y-axis to the top right of the graph." name="img_QniG" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/yXqJ_17.jpg"></figure><p><span id="fs-idm275434144" class="" data-type="media" data-alt="A graph is shown where the y-axis is labeled, “Free energy,” and the x-axis is labeled, “Increasing temperature ( K ).” The value of zero is written midway up the y-axis with the label, “delta G greater than 0,” written above this line and, “delta G less than 0,” written below it. The bottom half of the graph is labeled on the right as, “Spontaneous,” and the top half is labeled on the right as, “Nonspontaneous.” A green line labeled, “delta H less than 0, delta S greater than 0,” extends from a quarter of the way up the y-axis to the bottom right of the graph. A yellow line labeled, “delta H less than 0, delta S less than 0,” extends from a quarter of the way up the y-axis to the middle right of the graph. A second yellow line labeled, “delta H greater than 0, delta S greater than 0,” extends from three quarters of the way up the y-axis to the middle right of the graph. A red line labeled, “delta H greater than 0, delta S less than 0,” extends from three quarters of the way up the y-axis to the top right of the graph."></span></p>
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