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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<div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_JIuf" name="h3_JIuf">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_LLVp" class="" name="li_LLVp">Describe the fundamental properties of solutions</li><li id="li_fscX" class="" name="li_fscX">Calculate solution concentrations using molarity</li><li id="li_Hijf" class="" name="li_Hijf">Perform dilution calculations using the dilution equation</li></ul></div><p id="fs-idm232512" class="" name="fs-idm232512">Preceding sections of this chapter focused on the composition of substances: samples of matter that contain only one type of element or compound. However, mixtures—samples of matter containing two or more substances physically combined—are more commonly encountered in nature than are pure substances. Similar to a pure substance, the relative composition of a mixture plays an important role in determining its properties. The relative amount of oxygen in a planet’s atmosphere determines its ability to sustain aerobic life. The relative amounts of iron, carbon, nickel, and other elements in steel (a mixture known as an “alloy”) determine its physical strength and resistance to corrosion. The relative amount of the active ingredient in a medicine determines its effectiveness in achieving the desired pharmacological effect. The relative amount of sugar in a beverage determines its sweetness (see <a class="autogenerated-content" href="#Figure_2.1" data-mce-href="#Figure_2.1">Figure 2.1</a>). This section will describe one of the most common ways in which the relative compositions of mixtures may be quantified.</p><p id="p_PjRn" class="os-caption-container table_figure_number" name="p_PjRn">Figure <span data-etb-numbered-element="1">2.1</span></p><p id="p_nkhk" class="os-caption table_figure_title" name="p_nkhk">Sugar is one of many components in the complex mixture known as coffee. The amount of sugar in a given amount of coffee is an important determinant of the beverage’s sweetness. (credit: Jane Whitney)</p><figure class="scaled-down text-center" data-id="CNX_Chem_03_03_espresso"><span id="fs-idm63360640" class="" data-type="media" data-alt="A picture is shown of sugar being poured from a spoon into a cup."><img id="img_inGL" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_1.jpg" alt="A picture is shown of sugar being poured from a spoon into a cup." name="img_inGL" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_1.jpg"></span></figure><h2 id="h3_PBTu" class="" data-type="title" name="h3_PBTu"><span data-etb-numbered-element="1">2.1</span> Solutions</h2><p id="fs-idm60561504" class="" name="fs-idm60561504">Solutions have previously been defined as homogeneous mixtures, meaning that the composition of the mixture (and therefore its properties) is uniform throughout its entire volume. Solutions occur frequently in nature and have also been implemented in many forms of manmade technology. A more thorough treatment of solution properties is provided in the chapter on solutions and colloids, but provided here is an introduction to some of the basic properties of solutions.</p><p id="fs-idm635664" class="" name="fs-idm635664">The relative amount of a given solution component is known as its <span id="term278" class="" data-type="term">concentration</span>. Often, though not always, a solution contains one component with a concentration that is significantly greater than that of all other components. This component is called the <span id="term279" class="" data-type="term">solvent</span> and may be viewed as the medium in which the other components are dispersed, or <span id="term280" class="" data-type="term">dissolved</span>. Solutions in which water is the solvent are, of course very common on our planet. A solution in which water is the solvent is called an <span id="term281" class="" data-type="term">aqueous solution</span>.</p><p id="fs-idm64279024" class="" name="fs-idm64279024">A <span id="term282" class="" data-type="term">solute</span> is a component of a solution that is typically present at a much lower concentration than the solvent. Solute concentrations are often described with qualitative terms such as <span id="term283" class="" data-type="term">dilute</span> (of relatively low concentration) and <span id="term284" class="" data-type="term">concentrated</span> (of relatively high concentration).</p><p id="fs-idm62212352" class="" name="fs-idm62212352">Concentrations may be quantitatively assessed using a wide variety of measurement units, each convenient for particular applications. <span id="term285" class="" data-type="term">Molarity (<em class="" data-effect="italics">M</em>)</span> is a useful concentration unit for many applications in chemistry. Molarity is defined as the number of moles of solute in exactly 1 liter (1 L) of the solution:</p><div id="fs-idm57520096" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi>M</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mol solute</mtext></mrow><mrow><mtext>L solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">M</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mol solute</mtext></mrow><mrow class=""><mtext class="">L solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mrow></annotation-xml></semantics></math></div></div><div id="fs-idm98982768" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_iccP" name="h3_iccP">EXAMPLE <span data-etb-numbered-element="1">2.1.1</span></h3></header><div class="body"><h4 id="h4_EBVm" class="" data-type="title" name="h4_EBVm">Calculating Molar Concentrations</h4>A 355-mL soft drink sample contains 0.133 mol of sucrose (table sugar). What is the molar concentration of sucrose in the beverage?<h4 id="h4_ZgtH" class="" data-type="title" name="h4_ZgtH">Solution</h4>Since the molar amount of solute and the volume of solution are both given, the molarity can be calculated using the definition of molarity. Per this definition, the solution volume must be converted from mL to L:<div id="fs-idm77939616" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd></mtd></mtr><mtr><mtd><mi>M</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mol solute</mtext></mrow><mrow><mtext>L solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr><mtr><mtd><mtext>mol solute</mtext><mo>=</mo><mi>M</mi><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>L solution</mtext></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>mol solute</mtext><mspace width="0.2em"></mspace><mo>=</mo><mspace width="0.2em"></mspace><mn>0.375</mn><mspace width="0.4em"></mspace><mfrac><mrow><mtext>mol sugar</mtext></mrow><mtext>L</mtext></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mn>10</mn><mspace width="0.2em"></mspace><mtext>mL</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>L</mtext></mrow><mrow><mn>1000</mn><mspace width="0.2em"></mspace><mtext>mL</mtext></mrow></mfrac></mrow><mo>)</mo><mspace width="0.2em"></mspace></mrow><mo>=</mo><mspace width="0.2em"></mspace><mn>0.004</mn><mspace width="0.2em"></mspace><mtext>mol sugar</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mi class="">M</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mol solute</mtext></mrow><mrow class=""><mtext class="">L solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mol solute</mtext><mo class="">=</mo><mi class="">M</mi><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mtext class="">L solution</mtext></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mol solute</mtext><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mn class="">0.375</mn><mspace width="0.4em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mol sugar</mtext></mrow><mtext class="">L</mtext></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">10</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext></mrow><mrow class=""><mn class="">1000</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL</mtext></mrow></mfrac></mrow><mo class="">)</mo><mspace width="0.2em" class=""></mspace></mrow><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mn class="">0.004</mn><mspace width="0.2em" class=""></mspace><mtext class="">mol sugar</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><h4 id="h4_qVav" class="" data-type="title" name="h4_qVav">Check Your Learning</h4>A teaspoon of table sugar contains about 0.01 mol sucrose. What is the molarity of sucrose if a teaspoon of sugar has been dissolved in a cup of tea with a volume of 200 mL?<div id="fs-idm62768528" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_4SP4gpqG" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_4SP4gpqG" data-bs-toggle="collapse" id="h3_fnns" name="h3_fnns" data-mce-href="#answer_4SP4gpqG" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_4SP4gpqG" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm62230608" class="" name="fs-idm62230608">0.05 M</p></div></section></div></div></div><div id="fs-idm64107376" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_wwTc" name="h3_wwTc">EXAMPLE <span data-etb-numbered-element="1">2.1.2</span></h3></header><div class="body"><h4 id="h4_zoBy" class="" data-type="title" name="h4_zoBy">Deriving Moles and Volumes from Molar Concentrations</h4>How much sugar (mol) is contained in a modest sip (~10 mL) of the soft drink from <a class="autogenerated-content" href="#Example_2.1" data-shortened-url="https://edtechbooks.org/-PFe" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/6-3-molarity#fs-idm98982768" data-mce-href="#Example_2.1">Example 2.1</a>?<h4 id="h4_oaGE" class="" data-type="title" name="h4_oaGE">Solution</h4>Rearrange the definition of molarity to isolate the quantity sought, moles of sugar, then substitute the value for molarity derived in <a class="autogenerated-content" href="#Example_2.1" data-shortened-url="https://edtechbooks.org/-PFe" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/6-3-molarity#fs-idm98982768" data-mce-href="#Example_2.1">Example 2.1</a>, 0.375 <em class="" data-effect="italics">M</em>:<div id="fs-idm72962928" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd></mtd></mtr><mtr><mtd><mi>M</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mol solute</mtext></mrow><mrow><mtext>L solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>0.839</mn><mspace width="0.2em"></mspace><mi>M</mi></mtd></mtr><mtr><mtd><mi>M</mi><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>0.839</mn><mspace width="0.2em"></mspace><mtext>mol solute</mtext></mrow><mrow><mn>1.00</mn><mspace width="0.2em"></mspace><mtext>L solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace></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=""></mtd></mtr><mtr class=""><mtd class=""><mi class="">M</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mol solute</mtext></mrow><mrow class=""><mtext class="">L solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">0.839</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mtd></mtr><mtr class=""><mtd class=""><mi class="">M</mi><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">0.839</mn><mspace width="0.2em" class=""></mspace><mtext class="">mol solute</mtext></mrow><mrow class=""><mn class="">1.00</mn><mspace width="0.2em" class=""></mspace><mtext class="">L solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><h4 id="h4_rLfX" class="" data-type="title" name="h4_rLfX">Check Your Learning</h4>What volume (mL) of the sweetened tea described in <a class="autogenerated-content" href="#Example_2.1" data-shortened-url="https://edtechbooks.org/-PFe" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/6-3-molarity#fs-idm98982768" data-mce-href="#Example_2.1">Example 2.1</a> contains the same amount of sugar (mol) as 10 mL of the soft drink in this example?<div id="fs-idm58199648" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_Mll3JF0m" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_Mll3JF0m" data-bs-toggle="collapse" id="h3_kwie" name="h3_kwie" data-mce-href="#answer_Mll3JF0m" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_Mll3JF0m" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm89023504" class="" name="fs-idm89023504">80 mL</p></div></section></div></div></div><div id="fs-idm81897840" class="ui-has-child-title callout" data-type="example"><h3 class="os-title" id="h3_aqna" name="h3_aqna">EXAMPLE <span data-etb-numbered-element="1">2.1.3</span></h3><h4 id="h4_SMNs" class="" data-type="title" name="h4_SMNs">Calculating Molar Concentrations from the Mass of Solute</h4>Distilled white vinegar (<a href="#Figure_2.2" data-mce-href="#Figure_2.2">Figure 2.2</a>) is a solution of acetic acid, CH<sub class="">3</sub>CO<sub class="">2</sub>H, in water. A 0.500-L vinegar solution contains 25.2 g of acetic acid. What is the concentration of the acetic acid solution in units of molarity?<p id="p_QLod" class="os-caption-container table_figure_number" name="p_QLod">Figure <span data-etb-numbered-element="1">2.2</span></p><p id="p_tpPW" class="os-caption table_figure_title" name="p_tpPW">Distilled white vinegar is a solution of acetic acid in water.</p><figure class="scaled-down text-center" data-id="CNX_Chem_03_04_vinegar"><span id="fs-idm69003664" class="" data-type="media" data-alt="A label on a container is shown. The label has a picture of a salad with the words “Distilled White Vinegar,” and, “Reduced with water to 5% acidity,” written above it."><img id="img_ByaJ" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_2.jpg" alt="A label on a container is shown. The label has a picture of a salad with the words “Distilled White Vinegar,” and, “Reduced with water to 5% acidity,” written above it." name="img_ByaJ" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_2.jpg"></span></figure><h4 id="h4_jvEw" class="" data-type="title" name="h4_jvEw">Solution</h4>As in previous examples, the definition of molarity is the primary equation used to calculate the quantity sought. Since the mass of solute is provided instead of its molar amount, use the solute’s molar mass to obtain the amount of solute in moles:<div id="fs-idm81742128" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>1.325 mol NaCl</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>58.44</mn><mspace width="0.2em"></mspace><mtext>g NaCl</mtext></mrow><mrow><mtext>mol NaCl</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>77.4</mn><mspace width="0.2em"></mspace><mtext>g NaCl</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">1.325 mol NaCl</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">58.44</mn><mspace width="0.2em" class=""></mspace><mtext class="">g NaCl</mtext></mrow><mrow class=""><mtext class="">mol NaCl</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">77.4</mn><mspace width="0.2em" class=""></mspace><mtext class="">g NaCl</mtext></mrow></annotation-xml></semantics></math></div></div><div id="fs-idm8804720" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>mol solute</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>L solution</mtext></mrow><mrow><mtext>mol solute</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mtext>L solution</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">mol solute</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">L solution</mtext></mrow><mrow class=""><mtext class="">mol solute</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mtext class="">L solution</mtext></mrow></annotation-xml></semantics></math></div></div><h4 id="h4_RpzD" class="" data-type="title" name="h4_RpzD">Check Your Learning</h4><div id="DIV_bauz2" name="DIV_bauz2">Calculate the molarity of 6.52 g of CoCl<sub>2</sub> (128.9 g/mol) dissolved in an aqueous solution with a total volume of 75.0 mL.</div><div id="fs-idm2856432" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_v1pJMC0y" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_v1pJMC0y" data-bs-toggle="collapse" id="h3_SgeC" name="h3_SgeC" data-mce-href="#answer_v1pJMC0y" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_v1pJMC0y" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm103606496" class="" name="fs-idm103606496">0.674 <em class="" data-effect="italics">M</em></p></div></section></div></div><div id="fs-idm104693104" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_PmTr" name="h3_PmTr">EXAMPLE <span data-etb-numbered-element="1">2.1.4</span></h3></header><h4 id="h4_GSas" class="" data-type="title" name="h4_GSas">Determining the Mass of Solute in a Given Volume of Solution</h4><p id="p_roFt" name="p_roFt">How many grams of NaCl are contained in 0.250 L of a 5.30-<em class="" data-effect="italics" style="font-size: 19.2px;" data-mce-style="font-size: 19.2px;">M</em> solution?</p><h4 id="h4_UQZV" class="" data-type="title" name="h4_UQZV">Solution</h4><p id="p_MpCs" name="p_MpCs">The volume and molarity of the solution are specified, so the amount (mol) of solute is easily computed as demonstrated in <a class="autogenerated-content" href="https://edtechbooks.org/general_college_chemistry_2/chapter_2?book_nav=true&opens=10866,9477,9595,9596,10867,10902,9597,9607,9631&nav_pos=0#Example_2.2" data-shortened-url="https://edtechbooks.org/-CUad" data-unshortened-url="https://edtechbooks.org/general_college_chemistry_2/chapter_2?book_nav=true&opens=10866,9477,9595,9596,10867,10902,9597,9607,9631&nav_pos=0#Example_2.2" style="font-size: 19.2px;" data-mce-href="https://edtechbooks.org/general_college_chemistry_2/chapter_2?book_nav=true&opens=10866,9477,9595,9596,10867,10902,9597,9607,9631&nav_pos=0#Example_2.2" data-mce-style="font-size: 19.2px;">Example 2.2</a>:</p><div id="fs-idm85599968" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mn>75.6</mn><mspace width="0.2em"></mspace><mtext>g</mtext><mspace width="0.2em"></mspace><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mtext>H</mtext><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mtext>mol</mtext><mspace width="0.2em"></mspace><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mtext>H</mtext><mspace width="0.2em"></mspace></mrow><mrow><mn>60.05</mn><mspace width="0.2em"></mspace><mtext>g</mtext></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mtext>L solution</mtext></mrow><mrow><mn>0.839</mn><mspace width="0.2em"></mspace><mtext>mol</mtext><mspace width="0.2em"></mspace><msub><mrow><mtext>CH</mtext></mrow><mn>3</mn></msub><msub><mrow><mtext>CO</mtext></mrow><mn>2</mn></msub><mtext>H</mtext></mrow></mfrac></mrow><mo>)</mo><mspace width="0.2em"></mspace></mrow><mo>=</mo><mspace width="0.2em"></mspace><mn>1.50</mn><mspace width="0.2em"></mspace><mtext>L solution</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mn class="">75.6</mn><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">CH</mtext></mrow><mn class="">3</mn></msub><msub class=""><mrow class=""><mtext class="">CO</mtext></mrow><mn class="">2</mn></msub><mtext class="">H</mtext><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mtext class="">mol</mtext><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">CH</mtext></mrow><mn class="">3</mn></msub><msub class=""><mrow class=""><mtext class="">CO</mtext></mrow><mn class="">2</mn></msub><mtext class="">H</mtext><mspace width="0.2em" class=""></mspace></mrow><mrow class=""><mn class="">60.05</mn><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mrow></mfrac></mrow><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mtext class="">L solution</mtext></mrow><mrow class=""><mn class="">0.839</mn><mspace width="0.2em" class=""></mspace><mtext class="">mol</mtext><mspace width="0.2em" class=""></mspace><msub class=""><mrow class=""><mtext class="">CH</mtext></mrow><mn class="">3</mn></msub><msub class=""><mrow class=""><mtext class="">CO</mtext></mrow><mn class="">2</mn></msub><mtext class="">H</mtext></mrow></mfrac></mrow><mo class="">)</mo><mspace width="0.2em" class=""></mspace></mrow><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mn class="">1.50</mn><mspace width="0.2em" class=""></mspace><mtext class="">L solution</mtext></mrow></annotation-xml></semantics></math></div></div><p id="fs-idm108465360" class="" name="fs-idm108465360">Finally, this molar amount is used to derive the mass of NaCl:</p><div id="fs-idm1818176" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>M</mi><mn>1</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mi class="">n</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">M</mi><mn class="">1</mn></msub><msub class=""><mi class="">L</mi><mn class="">1</mn></msub></mrow></annotation-xml></semantics></math></div></div><h4 id="h4_kKYy" class="" data-type="title" name="h4_kKYy">Check Your Learning</h4>How many grams of CaCl<sub class="">2</sub> (110.98 g/mol) are contained in 250.0 mL of a 0.200-<em class="" data-effect="italics">M</em> solution of calcium chloride?<div id="fs-idm80099776" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_I83YdVDH" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_I83YdVDH" data-bs-toggle="collapse" id="h3_fNuq" name="h3_fNuq" data-mce-href="#answer_I83YdVDH" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_I83YdVDH" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm73628272" class="" name="fs-idm73628272">5.55 g CaCl<sub class="">2</sub></p></div></section></div></div><p id="fs-idm75726960" class="" name="fs-idm75726960">When performing calculations stepwise, as in <a class="autogenerated-content" href="#Example_2.4" data-shortened-url="https://edtechbooks.org/-vGR" data-unshortened-url="https://openstax.org/books/chemistry-atoms-first-2e/pages/6-3-molarity#fs-idm104693104" data-mce-href="#Example_2.4">Example 2.4</a>, it is important to refrain from rounding any intermediate calculation results, which can lead to rounding errors in the final result. In <a href="#Example_2.4" data-mce-href="#Example_2.4">Example 2.4</a>, the molar amount of NaCl computed in the first step, 1.325 mol, would be properly rounded to 1.32 mol if it were to be reported; however, although the last digit (5) is not significant, it must be retained as a guard digit in the intermediate calculation. If the guard digit had not been retained, the final calculation for the mass of NaCl would have been 77.1 g, a difference of 0.3 g.</p><p id="fs-idm105202592" class="" name="fs-idm105202592">In addition to retaining a guard digit for intermediate calculations, rounding errors may also be avoided by performing computations in a single step (see <a href="#Example_2.5" data-mce-href="#Example_2.5">Example 2.5</a>). This eliminates intermediate steps so that only the final result is rounded.</p><div id="fs-idm88061984" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_ztMv" name="h3_ztMv">EXAMPLE <span data-etb-numbered-element="1">2.1.5</span></h3></header><div class="body"><h4 id="h4_iwBr" class="" data-type="title" name="h4_iwBr">Determining the Volume of Solution Containing a Given Mass of Solute</h4>In <a href="#Example_2.3" data-mce-href="#Example_2.3">Example 2.3</a>, the concentration of acetic acid in white vinegar was determined to be 0.839 <em class="" data-effect="italics">M</em>. What volume of vinegar contains 75.6 g of acetic acid?<h4 id="h4_Adke" class="" data-type="title" name="h4_Adke">Solution</h4>First, use the molar mass to calculate moles of acetic acid from the given mass:<div id="fs-idm112801328" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>L</mi><mn>1</mn></msub><mo>=</mo><msub><mi>M</mi><mn>2</mn></msub><msub><mi>L</mi><mn>2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mi class="">M</mi><mn class="">1</mn></msub><msub class=""><mi class="">L</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">M</mi><mn class="">2</mn></msub><msub class=""><mi class="">L</mi><mn class="">2</mn></msub></mrow></annotation-xml></semantics></math></div></div><p id="fs-idm67563520" class="" name="fs-idm67563520">Then, use the molarity of the solution to calculate the volume of solution containing this molar amount of solute:</p><div id="fs-idm79868224" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub></mrow><mrow><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub></mrow><mrow class=""><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><p id="fs-idm141610240" class="" name="fs-idm141610240">Combining these two steps into one yields:</p><div id="fs-idm104886656" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub></mrow><mrow><msub><mi>C</mi><mn>2</mn></msub></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">2</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub></mrow><mrow class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><div id="fs-idm79860368" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mrow><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mrow><mrow class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><h4 id="h4_fmBr" class="" data-type="title" name="h4_fmBr">Check Your Learning</h4>What volume of a 1.50-<em class="" data-effect="italics">M</em> KBr solution contains 66.0 g KBr?<div id="fs-idm87238064" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_7u72nymM" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_7u72nymM" data-bs-toggle="collapse" id="h3_IjRL" name="h3_IjRL" data-mce-href="#answer_7u72nymM" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_7u72nymM" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm80069312" class="" name="fs-idm80069312">0.370 L</p></div></section></div></div></div><h2 id="h3_wnTz" class="" data-type="title" name="h3_wnTz"><span data-etb-numbered-element="1">2.2</span> Dilution of Solutions</h2><p id="fs-idm134700400" class="" name="fs-idm134700400"><span id="term286" class="" data-type="term">Dilution</span> is the process whereby the concentration of a solution is lessened by the addition of solvent. For example, a glass of iced tea becomes increasingly diluted as the ice melts. The water from the melting ice increases the volume of the solvent (water) and the overall volume of the solution (iced tea), thereby reducing the relative concentrations of the solutes that give the beverage its taste (<a class="autogenerated-content" href="#Figure_2.3" data-mce-href="#Figure_2.3">Figure 2.3</a>).</p><div id="CNX_Chem_03_04_dilution" class="os-figure"><p id="p_FHVN" class="os-caption-container table_figure_number" name="p_FHVN">Figure <span data-etb-numbered-element="1">2.3</span></p><p id="p_fwvT" class="os-caption table_figure_title" name="p_fwvT">Both solutions contain the same mass of copper nitrate. The solution on the right is more dilute because the copper nitrate is dissolved in more solvent. (credit: Mark Ott)</p><figure class="scaled-down text-center" data-id="CNX_Chem_03_04_dilution"><span id="fs-idp133169776" class="" data-type="media" data-alt="This figure shows two graduated cylinders side-by-side. The first has about half as much blue liquid as the second. The blue liquid is darker in the first cylinder than in the second."><img id="img_dPMN" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_3.jpg" alt="This figure shows two graduated cylinders side-by-side. The first has about half as much blue liquid as the second. The blue liquid is darker in the first cylinder than in the second." name="img_dPMN" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_3.jpg"></span></figure></div><p id="fs-idm77995840" class="" name="fs-idm77995840">Dilution is also a common means of preparing solutions of a desired concentration. By adding solvent to a measured portion of a more concentrated <em class="" data-effect="italics">stock solution</em>, a solution of lesser concentration may be prepared. For example, commercial pesticides are typically sold as solutions in which the active ingredients are far more concentrated than is appropriate for their application. Before they can be used on crops the pesticides must be diluted. This is also a very common practice for the preparation of a number of common laboratory reagents.</p><p id="fs-idm77996832" class="" name="fs-idm77996832">A simple mathematical relationship can be used to relate the volumes and concentrations of a solution before and after the dilution process. According to the definition of molarity, the number of moles of solute in a solution (<em class="" data-effect="italics">n</em>) is equal to the product of the solution’s molarity (<em class="" data-effect="italics">M</em>) and its volume in liters (<em class="" data-effect="italics">L</em>):</p><div id="fs-idm32193760" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>mass percentage</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass of component</mtext></mrow><mrow><mtext>mass of solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>100</mn><mi>%</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">mass percentage</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass of component</mtext></mrow><mrow class=""><mtext class="">mass of solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">100</mn><mi class="">%</mi></mrow></annotation-xml></semantics></math></div></div><p id="fs-idm104760288" class="" name="fs-idm104760288">Expressions like these may be written for a solution before and after it is diluted:</p><div id="fs-idm18261728" class="unnumbered" data-type="equation"><div class="MathJax_Display"><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></div></div><div id="fs-idm61402048" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtext>volume percentage</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>volume solute</mtext></mrow><mrow><mtext>volume solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>100</mn><mi>%</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">volume percentage</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">volume solute</mtext></mrow><mrow class=""><mtext class="">volume solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">100</mn><mi class="">%</mi></mrow></annotation-xml></semantics></math></div></div><p id="fs-idm123215808" class="" name="fs-idm123215808">where the subscripts “1” and “2” refer to the solution before and after the dilution, respectively. Since the dilution process <em class="" data-effect="italics">does not change the amount of solute in the solution, </em><em class="" data-effect="italics">n</em><sub class="">1</sub> = <em class="" data-effect="italics">n</em><sub class="">2</sub>. Thus, these two equations may be set equal to one another:</p><div id="fs-idp188032944" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd></mtd></mtr><mtr><mtd><mtext>ppm</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>6</mn></msup><mspace width="0.2em"></mspace><mtext>ppm</mtext></mtd></mtr><mtr><mtd><mtext>ppb</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></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=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppm</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">6</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppm</mtext></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppb</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><p id="fs-idm103311152" class="" name="fs-idm103311152">This relation is commonly referred to as the dilution equation. Although this equation uses molarity as the unit of concentration and liters as the unit of volume, other units of concentration and volume may be used as long as the units properly cancel per the factor-label method. Reflecting this versatility, the dilution equation is often written in the more general form:</p><div id="fs-idm69146864" class="unnumbered" data-type="equation"><div class="MathJax_Display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable><mtr><mtd></mtd></mtr><mtr><mtd><mtext>ppb</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>mass solute</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>ppb</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>mass solution</mtext></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>mass solute</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>15</mn><mspace width="0.2em"></mspace><mtext>ppb</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>300</mn><mspace width="0.2em"></mspace><mtext>mL</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1.00</mn><mspace width="0.2em"></mspace><mtext>g</mtext></mrow><mrow><mtext>mL</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>4.5</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mrow><mn>−6</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>g</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppb</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mass solute</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">ppb</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mtext class="">mass solution</mtext></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mass solute</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">15</mn><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">300</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1.00</mn><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mrow><mrow class=""><mtext class="">mL</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">4.5</mn><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mrow class=""><mn class="">−6</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></div></div><p id="fs-idm81143040" class="" name="fs-idm81143040">where <em class="" data-effect="italics">C</em> and <em class="" data-effect="italics">V</em> are concentration and volume, respectively.</p><div id="fs-idm98324208" 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_huIw" name="h3_huIw"><span class="os-title-label">LINK TO LEARNING</span></h3></header><div class="os-note-body"><p id="fs-idm105278864" class="" name="fs-idm105278864">Use the <a class="" href="http://openstax.org/l/16Phetsolvents" target="_blank" rel="noopener nofollow" data-shortened-url="https://edtechbooks.org/-xtHA" data-unshortened-url="http://openstax.org/l/16Phetsolvents" data-mce-href="http://openstax.org/l/16Phetsolvents">simulation</a> to explore the relations between solute amount, solution volume, and concentration and to confirm the dilution equation.</p></div></div><div id="fs-idm81737600" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_cZwN" name="h3_cZwN">EXAMPLE <span data-etb-numbered-element="1">2.2.1</span></h3></header><div class="body"><h4 id="h4_iWJQ" class="" data-type="title" name="h4_iWJQ">Determining the Concentration of a Diluted Solution</h4>If 0.850 L of a 5.00-<em class="" data-effect="italics">M</em> solution of copper nitrate, Cu(NO<sub class="">3</sub>)<sub class="">2</sub>, is diluted to a volume of 1.80 L by the addition of water, what is the molarity of the diluted solution?<h4 id="h4_ktkZ" class="" data-type="title" name="h4_ktkZ">Solution</h4>The stock concentration, <em class="" data-effect="italics">C</em><sub class="">1</sub>, and volume, <em class="" data-effect="italics">V</em><sub class="">1</sub>, are provided as well as the volume of the diluted solution, <em class="" data-effect="italics">V</em><sub class="">2</sub>. Rearrange the dilution equation to isolate the unknown property, the concentration of the diluted solution, <em class="" data-effect="italics">C</em><sub class="">2</sub>:<div id="fs-idm82973104" class="unnumbered" data-type="equation"><div class="MathJax_Display"><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><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>+</mo><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><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>3</mn></msub><msup><mtext>O</mtext><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mtext>OH</mtext><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 xmlns="http://www.w3.org/1998/Math/MathML" 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><mo class="">+</mo><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><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">3</mn></msub><msup class=""><mtext class="">O</mtext><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=""><mtext class="">OH</mtext><mtext class="">−</mtext></msup><mo stretchy="false" class="">(</mo><mi class="">a</mi><mi class="">q</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></div></div><p id="fs-idm81759792" class="" name="fs-idm81759792">Since the stock solution is being diluted by more than two-fold (volume is increased from 0.85 L to 1.80 L), the diluted solution’s concentration is expected to be less than one-half 5 <em class="" data-effect="italics">M</em>. This ballpark estimate will be compared to the calculated result to check for any gross errors in computation (for example, such as an improper substitution of the given quantities). Substituting the given values for the terms on the right side of this equation yields:</p><div id="fs-idm61572480" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="17" 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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><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" 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-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c30"></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-c4C"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 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class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><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" 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-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c30"></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-c4C"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c 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type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c4C"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 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><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>0.850</mn><mspace width="0.2em"></mspace><mtext>L</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>5.00</mn><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mol</mtext></mrow><mtext>L</mtext></mfrac><mspace width="0.2em"></mspace></mrow><mrow><mn>1.80 L</mn></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>2.36</mn><mspace width="0.2em"></mspace><mi>M</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">0.850</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">5.00</mn><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mol</mtext></mrow><mtext class="">L</mtext></mfrac><mspace width="0.2em" class=""></mspace></mrow><mrow class=""><mn class="">1.80 L</mn></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">2.36</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm125598048" class="" name="fs-idm125598048">This result compares well to our ballpark estimate (it’s a bit less than one-half the stock concentration, 5 <em class="" data-effect="italics">M</em>).</p><h4 id="h4_qrbA" class="" data-type="title" name="h4_qrbA">Check Your Learning</h4>What is the concentration of the solution that results from diluting 25.0 mL of a 2.04-<em class="" data-effect="italics">M</em> solution of CH<sub class="">3</sub>OH to 500.0 mL?<div id="fs-idm112503952" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_hYNeCqPy" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_hYNeCqPy" data-bs-toggle="collapse" id="h3_pDrw" name="h3_pDrw" data-mce-href="#answer_hYNeCqPy" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_hYNeCqPy" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm1594608" class="" name="fs-idm1594608">0.102 <em class="" data-effect="italics">M</em> CH<sub class="">3</sub>OH</p></div></section></div></div></div><div id="fs-idm81033056" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_bnWx" name="h3_bnWx">EXAMPLE <span data-etb-numbered-element="1">2.2.2</span></h3></header><div class="body"><h4 id="h4_Xsaz" class="" data-type="title" name="h4_Xsaz">Volume of a Diluted Solution</h4>What volume of 0.12 <em class="" data-effect="italics">M</em> HBr can be prepared from 11 mL (0.011 L) of 0.45 <em class="" data-effect="italics">M</em> HBr?<h4 id="h4_aQxf" class="" data-type="title" name="h4_aQxf">Solution</h4>Provided are the volume and concentration of a stock solution, <em class="" data-effect="italics">V</em><sub class="">1</sub> and <em class="" data-effect="italics">C</em><sub class="">1</sub>, and the concentration of the resultant diluted solution, <em class="" data-effect="italics">C</em><sub class="">2</sub>. Find the volume of the diluted solution, <em class="" data-effect="italics">V</em><sub class="">2</sub> by rearranging the dilution equation to isolate <em class="" data-effect="italics">V</em><sub class="">2</sub>:<div id="fs-idm80850240" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="18" 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-mtable style="min-width: 5.676em;" data-mce-style="min-width: 5.676em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; 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margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><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" 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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub></mrow><mrow><msub><mi>C</mi><mn>2</mn></msub></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">2</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub></mrow><mrow class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm61209504" class="" name="fs-idm61209504">Since the diluted concentration (0.12 <em class="" data-effect="italics">M</em>) is slightly more than one-fourth the original concentration (0.45 <em class="" data-effect="italics">M</em>), the volume of the diluted solution is expected to be roughly four times the original volume, or around 44 mL. Substituting the given values and solving for the unknown volume yields:</p><div id="fs-idm82787888" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="19" 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-mtable style="min-width: 8.629em;" data-mce-style="min-width: 8.629em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><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" 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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><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" 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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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="19" 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-mtable style="min-width: 8.629em;" data-mce-style="min-width: 8.629em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><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" 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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><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" 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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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><mtable><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mrow><mo>(</mo><mrow><mn>0.45</mn><mspace width="0.2em"></mspace><mi>M</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>0.011</mn><mspace width="0.2em"></mspace><mtext>L</mtext></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>0.12</mn><mspace width="0.2em"></mspace><mi>M</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mn>0.041</mn><mspace width="0.2em"></mspace><mtext>L</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">2</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">0.45</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mrow><mo class="">)</mo></mrow><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">0.011</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext></mrow><mo class="">)</mo></mrow></mrow><mrow class=""><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">0.12</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mrow><mo class="">)</mo></mrow></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">2</mn></msub><mo class="">=</mo><mn class="">0.041</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm80652096" class="" name="fs-idm80652096">The volume of the 0.12-<em class="" data-effect="italics">M</em> solution is 0.041 L (41 mL). The result is reasonable and compares well with the rough estimate.</p><h4 id="h4_driX" class="" data-type="title" name="h4_driX">Check Your Learning</h4>A laboratory experiment calls for 0.125 <em class="" data-effect="italics">M</em> HNO<sub class="">3</sub>. What volume of 0.125 <em class="" data-effect="italics">M</em> HNO<sub class="">3</sub> can be prepared from 0.250 L of 1.88 <em class="" data-effect="italics">M</em> HNO<sub class="">3</sub>?<div id="fs-idm58675056" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_Nrzejq3L" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_Nrzejq3L" data-bs-toggle="collapse" id="h3_ZbnY" name="h3_ZbnY" data-mce-href="#answer_Nrzejq3L" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_Nrzejq3L" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm590752" class="" name="fs-idm590752">3.76 L</p></div></section></div></div></div><div id="fs-idm60422464" class="ui-has-child-title callout" data-type="example"><h3 class="os-title" id="h3_HGhy" name="h3_HGhy">EXAMPLE <span data-etb-numbered-element="1">2.2.3</span></h3><h4 id="h4_DUEi" class="" data-type="title" name="h4_DUEi">Volume of a Concentrated Solution Needed for Dilution</h4>What volume of 1.59 <em class="" data-effect="italics">M</em> KOH is required to prepare 5.00 L of 0.100 <em class="" data-effect="italics">M</em> KOH?<h4 id="h4_EeXY" class="" data-type="title" name="h4_EeXY">Solution</h4>Given are the concentration of a stock solution, <em class="" data-effect="italics">C</em><sub class="">1</sub>, and the volume and concentration of the resultant diluted solution, <em class="" data-effect="italics">V</em><sub class="">2</sub> and <em class="" data-effect="italics">C</em><sub class="">2</sub>. Find the volume of the stock solution, <em class="" data-effect="italics">V</em><sub class="">1</sub> by rearranging the dilution equation to isolate <em class="" data-effect="italics">V</em><sub class="">1</sub>:<div id="fs-idm108330752" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="20" 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-mtable style="min-width: 5.676em;" data-mce-style="min-width: 5.676em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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="20" 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-mtable style="min-width: 5.676em;" data-mce-style="min-width: 5.676em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.045em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.045em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><msub><mi>C</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mrow><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><msub class=""><mi class="">C</mi><mn class="">2</mn></msub><msub class=""><mi class="">V</mi><mn class="">2</mn></msub></mrow><mrow class=""><msub class=""><mi class="">C</mi><mn class="">1</mn></msub></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm108418736" class="" name="fs-idm108418736">Since the concentration of the diluted solution 0.100 <em class="" data-effect="italics">M</em> is roughly one-sixteenth that of the stock solution (1.59 <em class="" data-effect="italics">M</em>), the volume of the stock solution is expected to be about one-sixteenth that of the diluted solution or around 0.3 liters. Substituting the given values and solving for the unknown volume yields:</p><div id="fs-idm129889216" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="21" 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-mtable style="min-width: 8.629em;" data-mce-style="min-width: 8.629em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></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-c4C"></mjx-c></mjx-mtext></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></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-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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="21" 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-mtable style="min-width: 8.629em;" data-mce-style="min-width: 8.629em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></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-c4C"></mjx-c></mjx-mtext></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></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-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D449 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.186em;" data-mce-style="vertical-align: -0.15em; margin-left: -0.186em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></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-c4C"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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><mtable><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mrow><mo>(</mo><mrow><mn>0.100</mn><mspace width="0.2em"></mspace><mi>M</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5.00</mn><mspace width="0.2em"></mspace><mtext>L</mtext></mrow><mo>)</mo></mrow></mrow><mrow><mn>1.59</mn><mspace width="0.2em"></mspace><mi>M</mi></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr><mtr><mtd><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mn>0.314</mn><mspace width="0.2em"></mspace><mtext>L</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">0.100</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mrow><mo class="">)</mo></mrow><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">5.00</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext></mrow><mo class="">)</mo></mrow></mrow><mrow class=""><mn class="">1.59</mn><mspace width="0.2em" class=""></mspace><mi class="">M</mi></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr><mtr class=""><mtd class=""><msub class=""><mi class="">V</mi><mn class="">1</mn></msub><mo class="">=</mo><mn class="">0.314</mn><mspace width="0.2em" class=""></mspace><mtext class="">L</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm85673088" class="" name="fs-idm85673088">Thus, 0.314 L of the 1.59-<em class="" data-effect="italics">M</em> solution is needed to prepare the desired solution. This result is consistent with the rough estimate.</p><h4 id="h4_Nnvc" class="" data-type="title" name="h4_Nnvc">Check Your Learning</h4>What volume of a 0.575-<em class="" data-effect="italics">M</em> solution of glucose, C<sub class="">6</sub>H<sub class="">12</sub>O<sub class="">6</sub>, can be prepared from 50.00 mL of a 3.00-<em class="" data-effect="italics">M</em> glucose solution?<div id="fs-idm150115024" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_nV91xDld" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_nV91xDld" data-bs-toggle="collapse" id="h3_dzQG" name="h3_dzQG" data-mce-href="#answer_nV91xDld" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_nV91xDld" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm68276304" class="" name="fs-idm68276304">0.261 L</p></div></section></div></div><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><h3 class="" data-type="title" id="h3_aHcZ" name="h3_aHcZ">Learning Objectives</h3><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_tdIN" class="" name="li_tdIN">Define the concentration units of mass percentage, volume percentage, mass-volume percentage, parts-per-million (ppm), and parts-per-billion (ppb)</li><li id="li_xaLd" class="" name="li_xaLd">Perform computations relating a solution’s concentration and its components’ volumes and/or masses using these units</li></ul></div><p id="fs-idm32304496" class="" name="fs-idm32304496">The previous section introduced molarity, a very useful measurement unit for evaluating the concentration of solutions. However, molarity is only one measure of concentration. This section will describe some other units of concentration that are commonly used in various applications, either for convenience or by convention.</p><h2 id="h3_kSvE" class="" data-type="title" name="h3_kSvE"><span data-etb-numbered-element="1">2.3</span> Mass Percentage</h2><p id="fs-idm26991728" class="" name="fs-idm26991728">Earlier in this chapter, percent composition was introduced as a measure of the relative amount of a given element in a compound. Percentages are also commonly used to express the composition of mixtures including solutions. The <span id="term287" class="" data-type="term">mass percentage</span> of a solution component is defined as the ratio of the component’s mass to the solution’s mass, expressed as a percentage:</p><div id="fs-idm5911152" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="22" 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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></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-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mtext></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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></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-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c25 TEX-MI"></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="22" 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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></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-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mtext></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-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></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-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c25 TEX-MI"></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>mass percentage</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass of component</mtext></mrow><mrow><mtext>mass of solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>100</mn><mi>%</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">mass percentage</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass of component</mtext></mrow><mrow class=""><mtext class="">mass of solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">100</mn><mi class="">%</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm3484528" class="" name="fs-idm3484528">Mass percentage is also referred to by similar names such as <em class="" data-effect="italics">percent mass, percent weight, weight/weight percent</em>, and other variations on this theme. The most common symbol for mass percentage is simply the percent sign, %, although more detailed symbols are often used including %mass, %weight, and (w/w)%. Use of these more detailed symbols can prevent confusion of mass percentages with other types of percentages, such as volume percentages (to be discussed later in this section).</p><p id="fs-idm21823696" class="" name="fs-idm21823696">Mass percentages are popular concentration units for consumer products. The label of a typical liquid bleach bottle (<a class="autogenerated-content" href="#Figure_2.4" data-mce-href="#Figure_2.4">Figure 2.4</a>) cites the concentration of its active ingredient, sodium hypochlorite (NaOCl), as being 7.4%. A 100.0-g sample of bleach would therefore contain 7.4 g of NaOCl.</p><div id="CNX_Chem_03_05_bleach" class="os-figure"><p id="p_GLXS" class="os-caption-container table_figure_number" name="p_GLXS">Figure <span data-etb-numbered-element="1">2.4</span></p><p id="p_XmtW" class="os-caption table_figure_title" name="p_XmtW">Liquid bleach is an aqueous solution of sodium hypochlorite (NaOCl). This brand has a concentration of 7.4% NaOCl by mass.</p><figure class="scaled-down text-center" data-id="CNX_Chem_03_05_bleach"><span id="fs-idm21370256" class="" data-type="media" data-alt="The sides of two cylindrical containers are shown. Each container’s label is partially visible. The left container’s label reads “Bleach.” The right label contains more information about the product including the phrase, “Contains: Sodium hypochlorite 7.4 %.”"><img id="img_gDvR" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_4.jpg" alt="The sides of two cylindrical containers are shown. Each container’s label is partially visible. The left container’s label reads “Bleach.” The right label contains more information about the product including the phrase, “Contains: Sodium hypochlorite 7.4 %.”" name="img_gDvR" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_4.jpg"></span></figure></div><div id="fs-idm23566960" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_WKib" name="h3_WKib">EXAMPLE <span data-etb-numbered-element="1">2.3.1</span></h3></header><div class="body"><h4 id="h4_hoRk" class="" data-type="title" name="h4_hoRk">Calculation of Percent by Mass</h4>A 5.0-g sample of spinal fluid contains 3.75 mg (0.00375 g) of glucose. What is the percent by mass of glucose in spinal fluid?<h4 id="h4_ooKG" class="" data-type="title" name="h4_ooKG">Solution</h4>The spinal fluid sample contains roughly 4 mg of glucose in 5000 mg of fluid, so the mass fraction of glucose should be a bit less than one part in 1000, or about 0.1%. Substituting the given masses into the equation defining mass percentage yields:<div id="fs-idm17789648" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="23" 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-c25 TEX-MI"></mjx-c></mjx-mi><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c 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class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace></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-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></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-c67"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c66"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c25 TEX-MI"></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>%</mi><mspace width="0.2em"></mspace><mtext>glucose</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>3.75</mn><mspace width="0.2em"></mspace><mtext>mg glucose</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>g</mtext></mrow><mrow><mn>1000</mn><mspace width="0.2em"></mspace><mtext>mg</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mrow><mrow><mn>5.0</mn><mspace width="0.2em"></mspace><mtext>g spinal fluid</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>0.075</mn><mi>%</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mi class="">%</mi><mspace width="0.2em" class=""></mspace><mtext class="">glucose</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">3.75</mn><mspace width="0.2em" class=""></mspace><mtext class="">mg glucose</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mrow><mrow class=""><mn class="">1000</mn><mspace width="0.2em" class=""></mspace><mtext class="">mg</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mrow><mrow class=""><mn class="">5.0</mn><mspace width="0.2em" class=""></mspace><mtext class="">g spinal fluid</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">0.075</mn><mi class="">%</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm36099936" class="" name="fs-idm36099936">The computed mass percentage agrees with our rough estimate (it’s a bit less than 0.1%).</p><p id="fs-idm6973776" class="" name="fs-idm6973776">Note that while any mass unit may be used to compute a mass percentage (mg, g, kg, oz, and so on), the same unit must be used for both the solute and the solution so that the mass units cancel, yielding a dimensionless ratio. In this case, the solute mass unit in the numerator was converted from mg to g to match the units in the denominator. Alternatively, the spinal fluid mass unit in the denominator could have been converted from g to mg instead. As long as identical mass units are used for both solute and solution, the computed mass percentage will be correct.</p><h4 id="h4_DZrL" class="" data-type="title" name="h4_DZrL">Check Your Learning</h4>A bottle of a tile cleanser contains 135 g of HCl and 775 g of water. What is the percent by mass of HCl in this cleanser?<div id="fs-idp47230144" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_wfbKQw8u" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_wfbKQw8u" data-bs-toggle="collapse" id="h3_wzZi" name="h3_wzZi" data-mce-href="#answer_wfbKQw8u" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_wfbKQw8u" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm9307808" class="" name="fs-idm9307808">14.8%</p></div></section></div></div></div><div id="fs-idm27131760" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_EWaq" name="h3_EWaq">EXAMPLE <span data-etb-numbered-element="1">2.3.2</span></h3></header><div class="body"><h4 id="h4_GZCA" class="" data-type="title" name="h4_GZCA">Calculations using Mass Percentage</h4>“Concentrated” hydrochloric acid is an aqueous solution of 37.2% HCl that is commonly used as a laboratory reagent. The density of this solution is 1.19 g/mL. What mass of HCl is contained in 0.500 L of this solution?<h4 id="h4_NMLB" class="" data-type="title" name="h4_NMLB">Solution</h4>The HCl concentration is near 40%, so a 100-g portion of this solution would contain about 40 g of HCl. Since the solution density isn’t greatly different from that of water (1 g/mL), a reasonable estimate of the HCl mass in 500 g (0.5 L) of the solution is about five times greater than that in a 100 g portion, or 5 <span class="os-math-in-para"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" ctxtmenu_counter="24" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-semantics><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" ctxtmenu_counter="24" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; position: relative;" data-mce-tabindex="0"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-semantics><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><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></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></span> 40 = 200 g. In order to derive the mass of solute in a solution from its mass percentage, the mass of the solution must be known. Using the solution density given, convert the solution’s volume to mass, and then use the given mass percentage to calculate the solute mass. This mathematical approach is outlined in this flowchart:<span id="fs-idm3435008" class="" data-type="media" data-alt="A diagram of three boxes connected by a right-facing arrow in between each is shown. The box on the left contains the phrase, “Volume of solution ( m L ),” the middle box reads, “Mass of solution ( g ),” while the one on the right contains the phrase, “Mass of H C l ( g ).” There is a phrase under the left arrow that says, “Multiply by density ( g / m L )” and under the right arrow it states, “Multiply by mass percent as ratio ( g H C l / g solution ).”"><img id="img_LJiw" class="img-fluid col-12" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_5.jpg" alt="A diagram of three boxes connected by a right-facing arrow in between each is shown. The box on the left contains the phrase, “Volume of solution ( m L ),” the middle box reads, “Mass of solution ( g ),” while the one on the right contains the phrase, “Mass of H C l ( g ).” There is a phrase under the left arrow that says, “Multiply by density ( g / m L )” and under the right arrow it states, “Multiply by mass percent as ratio ( g H C l / g solution ).”" name="img_LJiw" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_5.jpg"></span><p id="fs-idm3493296" class="" name="fs-idm3493296">For proper unit cancellation, the 0.500-L volume is converted into 500 mL, and the mass percentage is expressed as a ratio, 37.2 g HCl/g solution:</p><div id="fs-idm17430336" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="25" style="font-size: 119.5%; position: relative;" data-mce-style="font-size: 119.5%; 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space="4"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></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-c67"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c48"></mjx-c><mjx-c class="mjx-c43"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mtext></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>500 mL solution</mtext><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1.19</mn><mspace width="0.2em"></mspace><mtext>g solution</mtext></mrow><mrow><mtext>mL solution</mtext></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mn>37.2</mn><mspace width="0.2em"></mspace><mtext>g HCl</mtext></mrow><mrow><mn>100</mn><mspace width="0.2em"></mspace><mtext>g solution</mtext></mrow></mfrac></mrow><mo>)</mo><mspace width="0.2em"></mspace></mrow><mo>=</mo><mspace width="0.2em"></mspace><mn>221</mn><mspace width="0.2em"></mspace><mtext>g HCl</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">500 mL solution</mtext><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mn class="">1.19</mn><mspace width="0.2em" class=""></mspace><mtext class="">g solution</mtext></mrow><mrow class=""><mtext class="">mL solution</mtext></mrow></mfrac></mrow><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mn class="">37.2</mn><mspace width="0.2em" class=""></mspace><mtext class="">g HCl</mtext></mrow><mrow class=""><mn class="">100</mn><mspace width="0.2em" class=""></mspace><mtext class="">g solution</mtext></mrow></mfrac></mrow><mo class="">)</mo><mspace width="0.2em" class=""></mspace></mrow><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mn class="">221</mn><mspace width="0.2em" class=""></mspace><mtext class="">g HCl</mtext></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm2814416" class="" name="fs-idm2814416">This mass of HCl is consistent with our rough estimate of approximately 200 g.</p><h4 id="h4_ATqj" class="" data-type="title" name="h4_ATqj">Check Your Learning</h4>What volume of concentrated HCl solution contains 125 g of HCl?<div id="fs-idm31069568" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_Z0T0lGJQ" 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_Z0T0lGJQ" data-bs-toggle="collapse" id="h3_qIat" name="h3_qIat" data-mce-href="#answer_Z0T0lGJQ" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_Z0T0lGJQ" class="list-group-item collapse" style=""><div class="os-note-body"><p id="fs-idm33390192" class="" name="fs-idm33390192">282 mL</p></div></section></div></div></div><h2 id="h3_DPjI" class="" data-type="title" name="h3_DPjI"><span data-etb-numbered-element="1">2.4</span> Volume Percentage</h2><p id="fs-idp14105408" class="" name="fs-idp14105408">Liquid volumes over a wide range of magnitudes are conveniently measured using common and relatively inexpensive laboratory equipment. The concentration of a solution formed by dissolving a liquid solute in a liquid solvent is therefore often expressed as a <span id="term288" class="" data-type="term">volume percentage</span>, %vol or (v/v)%:</p><div id="fs-idm2717488" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="26" 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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext></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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c25 TEX-MI"></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="26" 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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c6E"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c67"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext></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-c76"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c25 TEX-MI"></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>volume percentage</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>volume solute</mtext></mrow><mrow><mtext>volume solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>100</mn><mi>%</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtext class="">volume percentage</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">volume solute</mtext></mrow><mrow class=""><mtext class="">volume solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">100</mn><mi class="">%</mi></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><div id="fs-idm1441136" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_khzC" name="h3_khzC">EXAMPLE <span data-etb-numbered-element="1">2.4.1</span></h3></header><div class="body"><h4 id="h4_HnTU" class="" data-type="title" name="h4_HnTU">Calculations using Volume Percentage</h4>Rubbing alcohol (isopropanol) is usually sold as a 70%vol aqueous solution. If the density of isopropyl alcohol is 0.785 g/mL, how many grams of isopropyl alcohol are present in a 355 mL bottle of rubbing alcohol?<h4 id="h4_fWzt" class="" data-type="title" name="h4_fWzt">Solution</h4>Per the definition of volume percentage, the isopropanol volume is 70% of the total solution volume. 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class="mjx-c6C"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c29 TEX-S3"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace></mjx-mrow><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-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c39"></mjx-c><mjx-c class="mjx-c35"></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-c67"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c79"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c></mjx-mtext></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><mrow><mo>(</mo><mrow><mn>355</mn><mspace width="0.2em"></mspace><mtext>mL solution</mtext></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mn>70</mn><mspace width="0.2em"></mspace><mtext>mL isopropyl alcohol</mtext></mrow><mrow><mn>100</mn><mspace width="0.2em"></mspace><mtext>mL solution</mtext></mrow></mfrac></mrow><mo>)</mo></mrow><mspace width="0.2em"></mspace><mrow><mo>(</mo><mrow><mfrac><mrow><mn>0.785</mn><mspace width="0.2em"></mspace><mtext>g isopropyl alcohol</mtext></mrow><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>mL isopropyl alcohol</mtext></mrow></mfrac></mrow><mo>)</mo><mspace width="0.2em"></mspace></mrow><mo>=</mo><mspace width="0.2em"></mspace><mn>195</mn><mspace width="0.2em"></mspace><mtext>g isopropyl alchol</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mrow class=""><mo class="">(</mo><mrow class=""><mn class="">355</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL solution</mtext></mrow><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mn class="">70</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL isopropyl alcohol</mtext></mrow><mrow class=""><mn class="">100</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL solution</mtext></mrow></mfrac></mrow><mo class="">)</mo></mrow><mspace width="0.2em" class=""></mspace><mrow class=""><mo class="">(</mo><mrow class=""><mfrac class=""><mrow class=""><mn class="">0.785</mn><mspace width="0.2em" class=""></mspace><mtext class="">g isopropyl alcohol</mtext></mrow><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL isopropyl alcohol</mtext></mrow></mfrac></mrow><mo class="">)</mo><mspace width="0.2em" class=""></mspace></mrow><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mn class="">195</mn><mspace width="0.2em" class=""></mspace><mtext class="">g isopropyl alchol</mtext></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><h4 id="h4_wMfm" class="" data-type="title" name="h4_wMfm">Check Your Learning</h4>Wine is approximately 12% ethanol (CH<sub class="">3</sub>CH<sub class="">2</sub>OH) by volume. Ethanol has a molar mass of 46.06 g/mol and a density 0.789 g/mL. How many moles of ethanol are present in a 750-mL bottle of wine?<div id="fs-idp41917088" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_sp9VhTGT" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_sp9VhTGT" data-bs-toggle="collapse" id="h3_ogqB" name="h3_ogqB" data-mce-href="#answer_sp9VhTGT" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_sp9VhTGT" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm15159936" class="" name="fs-idm15159936">1.5 mol ethanol</p></div></section></div></div></div><h2 id="h3_spJA" class="" data-type="title" name="h3_spJA"><span data-etb-numbered-element="1">2.5</span> Mass-Volume Percentage</h2><p id="fs-idm16796176" class="" name="fs-idm16796176">“Mixed” percentage units, derived from the mass of solute and the volume of solution, are popular for certain biochemical and medical applications. A <span id="term289" class="" data-type="term">mass-volume percent</span> is a ratio of a solute’s mass to the solution’s volume expressed as a percentage. The specific units used for solute mass and solution volume may vary, depending on the solution. For example, physiological saline solution, used to prepare intravenous fluids, has a concentration of 0.9% mass/volume (m/v), indicating that the composition is 0.9 g of solute per 100 mL of solution. The concentration of glucose in blood (commonly referred to as “blood sugar”) is also typically expressed in terms of a mass-volume ratio. Though not expressed explicitly as a percentage, its concentration is usually given in milligrams of glucose per deciliter (100 mL) of blood (<a class="autogenerated-content" href="#Figure_2.5" data-mce-href="#Figure_2.5">Figure 2.5</a>).</p><p class="os-caption-container table_figure_number" id="p_zEII" name="p_zEII">Figure <span data-etb-numbered-element="1">2.5</span></p><p id="p_cdta" class="os-caption table_figure_title" name="p_cdta">“Mixed” mass-volume units are commonly encountered in medical settings. (a) The NaCl concentration of physiological saline is 0.9% (m/v). (b) This device measures glucose levels in a sample of blood. The normal range for glucose concentration in blood (fasting) is around 70–100 mg/dL. (credit a: modification of work by “The National Guard”/Flickr; credit b: modification of work by Biswarup Ganguly)</p><figure class="text-center" data-id="CNX_Chem_03_05_saline"><span id="fs-idp12420736" class="" data-type="media" data-alt="Two pictures are shown labeled a and b. Picture a depicts a clear, colorless solution in a plastic bag being held in a person’s hand. Picture b shows a person’s hand holding a detection meter with a digital readout screen while another hand holds someone’s finger up to the end of the meter. The meter is pressed to the drop of blood that is at the end of the person’s finger."><img id="img_BbSx" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_6.jpg" alt="Two pictures are shown labeled a and b. Picture a depicts a clear, colorless solution in a plastic bag being held in a person’s hand. Picture b shows a person’s hand holding a detection meter with a digital readout screen while another hand holds someone’s finger up to the end of the meter. The meter is pressed to the drop of blood that is at the end of the person’s finger." name="img_BbSx" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_6.jpg"></span></figure><h2 id="h3_ACAB" class="" data-type="title" name="h3_ACAB"><span data-etb-numbered-element="1">2.6</span> Parts per Million and Parts per Billion</h2><p id="fs-idm39037584" class="" name="fs-idm39037584">Very low solute concentrations are often expressed using appropriately small units such as <span id="term290" class="" data-type="term">parts per million (ppm)</span> or <span id="term291" class="" data-type="term">parts per billion (ppb)</span>. Like percentage (“part per hundred”) units, ppm and ppb may be defined in terms of masses, volumes, or mixed mass-volume units. There are also ppm and ppb units defined with respect to numbers of atoms and molecules.</p><p id="fs-idm18786352" class="" name="fs-idm18786352">The mass-based definitions of ppm and ppb are given here:</p><div id="fs-idp17696176" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="28" 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-mtable style="min-width: 13.239em;" data-mce-style="min-width: 13.239em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: left;" data-mce-style="text-align: left;"><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: left;" 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unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable columnalign="left"><mtr><mtd></mtd></mtr><mtr><mtd><mtext>ppm</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>6</mn></msup><mspace width="0.2em"></mspace><mtext>ppm</mtext></mtd></mtr><mtr><mtd><mtext>ppb</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></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=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppm</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">6</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppm</mtext></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppb</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm37568416" class="" name="fs-idm37568416">Both ppm and ppb are convenient units for reporting the concentrations of pollutants and other trace contaminants in water. Concentrations of these contaminants are typically very low in treated and natural waters, and their levels cannot exceed relatively low concentration thresholds without causing adverse effects on health and wildlife. For example, the EPA has identified the maximum safe level of fluoride ion in tap water to be 4 ppm. Inline water filters are designed to reduce the concentration of fluoride and several other trace-level contaminants in tap water (<a class="autogenerated-content" href="#Figure_2.6" data-mce-href="#Figure_2.6">Figure 2.6</a>).</p><p class="os-caption-container table_figure_number" id="p_SMYU" name="p_SMYU">Figure <span data-etb-numbered-element="1">2.6</span></p><p id="p_xLMx" class="os-caption table_figure_title" name="p_xLMx">(a) In some areas, trace-level concentrations of contaminants can render unfiltered tap water unsafe for drinking and cooking. (b) Inline water filters reduce the concentration of solutes in tap water. (credit a: modification of work by Jenn Durfey; credit b: modification of work by “vastateparkstaff”/Wikimedia commons)</p><figure class="text-center" data-id="CNX_Chem_03_05_faucet"><span id="fs-idm10006048" class="" data-type="media" data-alt="Two pictures are shown labeled a and b. Picture a is a close-up shot of water coming out of a faucet. Picture b shows a machine with the words, “Filtered Water Dispenser.” This machine appears to be inside a refrigerator."><img id="img_UGfw" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_7.jpg" alt="Two pictures are shown labeled a and b. Picture a is a close-up shot of water coming out of a faucet. Picture b shows a machine with the words, “Filtered Water Dispenser.” This machine appears to be inside a refrigerator." name="img_UGfw" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_7.jpg"></span></figure><div id="fs-idm22950992" class="ui-has-child-title callout" data-type="example"><header class=""><h3 class="os-title" id="h3_BvfG" name="h3_BvfG">EXAMPLE <span data-etb-numbered-element="1">2.6.1</span></h3></header><div class="body"><h4 id="h4_CNyR" class="" data-type="title" name="h4_CNyR">Calculation of Parts per Million and Parts per Billion Concentrations</h4>According to the EPA, when the concentration of lead in tap water reaches 15 ppb, certain remedial actions must be taken. What is this concentration in ppm? At this concentration, what mass of lead (μg) would be contained in a typical glass of water (300 mL)?<h4 id="h4_phTo" class="" data-type="title" name="h4_phTo">Solution</h4>The definitions of the ppm and ppb units may be used to convert the given concentration from ppb to ppm. Comparing these two unit definitions shows that ppm is 1000 times greater than ppb (1 ppm = 10<sup class="">3</sup> ppb). Thus:<div id="fs-idm1779776" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="29" 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-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mtext></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-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mtext></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="29" 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-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mtext></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-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mtext></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><mn>15</mn><mspace width="0.2em"></mspace><mtext>ppb</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>ppm</mtext></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>0.015</mn><mspace width="0.2em"></mspace><mtext>ppm</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mn class="">15</mn><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">ppm</mtext></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">3</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">0.015</mn><mspace width="0.2em" class=""></mspace><mtext class="">ppm</mtext></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idm35694560" class="" name="fs-idm35694560">The definition of the ppb unit may be used to calculate the requested mass if the mass of the solution is provided. Since the volume of solution (300 mL) is given, its density must be used to derive the corresponding mass. Assume the density of tap water to be roughly the same as that of pure water (~1.00 g/mL), since the concentrations of any dissolved substances should not be very large. 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display="true" ctxtmenu_counter="30" 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-mtable style="min-width: 21.645em;" data-mce-style="min-width: 21.645em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-msup space="3"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></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-mtext></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c20"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c75"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext><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><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></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-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></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-c6D"></mjx-c><mjx-c class="mjx-c4C"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c30"></mjx-c><mjx-c class="mjx-c30"></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-c67"></mjx-c></mjx-mtext></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c4C"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c62"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c34"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-msup space="3"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mtext><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></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><mtable><mtr><mtd></mtd></mtr><mtr><mtd><mtext>ppb</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>mass solute</mtext></mrow><mrow><mtext>mass solution</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>mass solute</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mtext>ppb</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mtext>mass solution</mtext></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>mass solute</mtext><mo>=</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>15</mn><mspace width="0.2em"></mspace><mtext>ppb</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mn>300</mn><mspace width="0.2em"></mspace><mtext>mL</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1.00</mn><mspace width="0.2em"></mspace><mtext>g</mtext></mrow><mrow><mtext>mL</mtext></mrow></mfrac><mspace width="0.2em"></mspace></mrow><mrow><msup><mrow><mn>10</mn></mrow><mn>9</mn></msup><mspace width="0.2em"></mspace><mtext>ppb</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>4.5</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mn>10</mn><mrow><mn>−6</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>g</mtext></mtd></mtr></mtable></mrow><annotation-xml encoding="MathML-Content"><mtable xmlns="http://www.w3.org/1998/Math/MathML" class=""><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">ppb</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">mass solute</mtext></mrow><mrow class=""><mtext class="">mass solution</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mass solute</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mtext class="">ppb</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mtext class="">mass solution</mtext></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mtd></mtr><mtr class=""><mtd class=""></mtd></mtr><mtr class=""><mtd class=""><mtext class="">mass solute</mtext><mo class="">=</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">15</mn><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mn class="">300</mn><mspace width="0.2em" class=""></mspace><mtext class="">mL</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1.00</mn><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mrow><mrow class=""><mtext class="">mL</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mn class="">9</mn></msup><mspace width="0.2em" class=""></mspace><mtext class="">ppb</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">4.5</mn><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><msup class=""><mn class="">10</mn><mrow class=""><mn class="">−6</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mtd></mtr></mtable></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><p id="fs-idp31649040" class="" name="fs-idp31649040">Finally, convert this mass to the requested unit of micrograms:</p><div id="fs-idm4570368" class="unnumbered" data-type="equation"><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="31" 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-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-msup space="3"><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.413em;" data-mce-style="vertical-align: 0.413em;"><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-utext variant="normal" style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;" data-mce-style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;">μ</mjx-utext><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></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-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c34"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-utext variant="normal" style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;" data-mce-style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;">μ</mjx-utext><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" 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class="mjx-c2212"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mtext><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-utext variant="normal" style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;" data-mce-style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;">μ</mjx-utext><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></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-msup><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow><mjx-script style="vertical-align: 0.393em;" data-mce-style="vertical-align: 0.393em;"><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><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-c34"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mspace style="width: 0.2em;" data-mce-style="width: 0.2em;"></mjx-mspace><mjx-mtext class="mjx-n"><mjx-utext variant="normal" style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;" data-mce-style="font-size: 83.7%; padding: 0.896em 0px 0.239em; font-family: MJXZERO, serif;">μ</mjx-utext><mjx-c class="mjx-c67"></mjx-c></mjx-mtext></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><mn>4.5</mn><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><msup><mrow><mn>10</mn></mrow><mrow><mn>−6</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>g</mtext><mspace width="0.2em"></mspace><mo>×</mo><mspace width="0.2em"></mspace><mfrac><mrow><mn>1</mn><mspace width="0.2em"></mspace><mtext>μg</mtext></mrow><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>−6</mn></mrow></msup><mspace width="0.2em"></mspace><mtext>g</mtext></mrow></mfrac><mspace width="0.2em"></mspace><mo>=</mo><mn>4.5</mn><mspace width="0.2em"></mspace><mtext>μg</mtext></mrow></mrow><annotation-xml encoding="MathML-Content"><mrow xmlns="http://www.w3.org/1998/Math/MathML" class=""><mn class="">4.5</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="">−6</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext><mspace width="0.2em" class=""></mspace><mo class="">×</mo><mspace width="0.2em" class=""></mspace><mfrac class=""><mrow class=""><mn class="">1</mn><mspace width="0.2em" class=""></mspace><mtext class="">μg</mtext></mrow><mrow class=""><msup class=""><mrow class=""><mn class="">10</mn></mrow><mrow class=""><mn class="">−6</mn></mrow></msup><mspace width="0.2em" class=""></mspace><mtext class="">g</mtext></mrow></mfrac><mspace width="0.2em" class=""></mspace><mo class="">=</mo><mn class="">4.5</mn><mspace width="0.2em" class=""></mspace><mtext class="">μg</mtext></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div></div><h4 id="h4_NwEN" class="" data-type="title" name="h4_NwEN">Check Your Learning</h4>A 50.0-g sample of industrial wastewater was determined to contain 0.48 mg of mercury. Express the mercury concentration of the wastewater in ppm and ppb units.<div id="fs-idm45688384" class="ui-has-child-title list-group my-4" data-type="note"><header class="mx-0 my-0 p-0"><h3 href="#answer_BGUDHiBO" style="cursor: pointer;" class="os-title my-0 py-2 px-0 px-3 no-border-radius" data-type="title" aria-expanded="false" aria-controls="answer_BGUDHiBO" data-bs-toggle="collapse" id="h3_JALZ" name="h3_JALZ" data-mce-href="#answer_BGUDHiBO" data-mce-style="cursor: pointer;">Answer</h3></header><section id="answer_BGUDHiBO" class="collapse list-group-item"><div class="os-note-body"><p id="fs-idm2529904" class="" name="fs-idm2529904">9.6 ppm, 9600 ppb</p></div></section></div></div></div><h2 id="h2_XIwP" name="h2_XIwP"><span data-etb-numbered-element="1">2.7</span> Electrolytes</h2><div id="1" class="ui-has-child-title callout objectives" data-type="abstract"><header class=""><h3 class="" data-type="title" id="h3_jjAo" name="h3_jjAo">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_LQQQ" class="" name="li_LQQQ">Define and give examples of electrolytes</li><li id="li_EwLQ" class="" name="li_EwLQ">Distinguish between the physical and chemical changes that accompany dissolution of ionic and covalent electrolytes</li><li id="li_mwkH" class="" name="li_mwkH">Relate electrolyte strength to solute-solvent attractive forces</li></ul></div><p id="fs-idp18381520" class="" name="fs-idp18381520">When some substances are dissolved in water, they undergo either a physical or a chemical change that yields ions in solution. These substances constitute an important class of compounds called <span id="term487" class="" data-type="term">electrolytes</span>. Substances that do not yield ions when dissolved are called <span id="term488" class="" data-type="term">nonelectrolytes</span>. If the physical or chemical process that generates the ions is essentially 100% efficient (all of the dissolved compound yields ions), then the substance is known as a <span id="term489" class="" data-type="term">strong electrolyte</span>. If only a relatively small fraction of the dissolved substance undergoes the ion-producing process, it is called a <span id="term490" class="" data-type="term">weak electrolyte</span>.</p><p id="fs-idm8509568" class="" name="fs-idm8509568">Substances may be identified as strong, weak, or nonelectrolytes by measuring the electrical conductance of an aqueous solution containing the substance. To conduct electricity, a substance must contain freely mobile, charged species. Most familiar is the conduction of electricity through metallic wires, in which case the mobile, charged entities are electrons. Solutions may also conduct electricity if they contain dissolved ions, with conductivity increasing as ion concentration increases. Applying a voltage to electrodes immersed in a solution permits assessment of the relative concentration of dissolved ions, either quantitatively, by measuring the electrical current flow, or qualitatively, by observing the brightness of a light bulb included in the circuit (<a class="autogenerated-content anchor-link" href="#p_SWGb" data-mce-href="#p_SWGb">Figure 2.7</a>).</p><p class="os-caption-container table_figure_number" id="p_SWGb" name="p_SWGb">Figure <span data-etb-numbered-element="1">2.7</span></p><p id="p_PpdG" class="os-caption table_figure_title" name="p_PpdG">Solutions of nonelectrolytes such as ethanol do not contain dissolved ions and cannot conduct electricity. Solutions of electrolytes contain ions that permit the passage of electricity. The conductivity of an electrolyte solution is related to the strength of the electrolyte.</p><figure class="text-center" data-id="CNX_Chem_11_02_electrolyt"><img id="img_haZt" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_8.jpg" alt="This diagram shows three separate beakers. Each has a wire plugged into a wall outlet. In each case, the wire leads from the wall to the beaker and is split resulting in two ends. One end leads to a light bulb and continues on to a rectangle labeled with a plus sign. The other end leads to a rectangle labeled with a minus sign. The rectangles are in a solution. In the first beaker, labeled “Ethanol No Conductivity,” four pairs of linked small green spheres suspended in the solution between the rectangles. In the second beaker, labeled “K C l Strong Conductivity,” six individual green spheres, three labeled plus and three labeled minus are suspended in the solution. Each of the six spheres has an arrow extending from it pointing to the rectangle labeled with the opposite sign. In the third beaker, labeled “Acetic acid solution Weak conductivity,” two pairs of joined green spheres and two individual spheres, one labeled plus and one labeled minus are shown suspended between the two rectangles. The plus labeled sphere has an arrow pointing to the rectangle labeled minus and the minus labeled sphere has an arrow pointing to the rectangle labeled plus." name="img_haZt" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_8.jpg"></figure><h2 id="h3_JVSR" class="" data-type="title" name="h3_JVSR"><span data-etb-numbered-element="1">2.8</span> Ionic Electrolytes</h2><p id="fs-idp2501472" class="" name="fs-idp2501472">Water and other polar molecules are attracted to ions, as shown in <a href="#p_gSTP" class="anchor-link" data-mce-href="#p_gSTP">Figure 2.8</a>. The electrostatic attraction between an ion and a molecule with a dipole is called an <span id="term491" class="" data-type="term">ion-dipole attraction</span>. These attractions play an important role in the dissolution of ionic compounds in water.</p><p class="os-caption-container table_figure_number" id="p_gSTP" name="p_gSTP">Figure <span data-etb-numbered-element="1">2.8</span></p><p id="p_LCVr" class="os-caption table_figure_title" name="p_LCVr">As potassium chloride (KCl) dissolves in water, the ions are hydrated. The polar water molecules are attracted by the charges on the K<sup class="">+</sup> and Cl<sup class="">−</sup> ions. Water molecules in front of and behind the ions are not shown.</p><figure class="scaled-down text-center" data-id="CNX_Chem_11_02_waterions"><img id="img_xbEf" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_9.jpg" alt="The diagram shows eight purple spheres labeled K superscript plus and eight green spheres labeled C l superscript minus mixed and touching near the center of the diagram. Outside of this cluster of spheres are seventeen clusters of three spheres, which include one red and two white spheres. A red sphere in one of these clusters is labeled O. A white sphere is labeled H. Two of the green C l superscript minus spheres are surrounded by three of the red and white clusters, with the red spheres closer to the green spheres than the white spheres. One of the K superscript plus purple spheres is surrounded by four of the red and white clusters. The white spheres of these clusters are closest to the purple spheres." name="img_xbEf" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_9.jpg"></figure><p id="fs-idp43584496" class="" name="fs-idp43584496">When ionic compounds dissolve in water, the ions in the solid separate and disperse uniformly throughout the solution because water molecules surround and solvate the ions, reducing the strong electrostatic forces between them. This process represents a physical change known as <span id="term492" class="" data-type="term">dissociation</span>. Under most conditions, ionic compounds will dissociate nearly completely when dissolved, and so they are classified as strong electrolytes. Even sparingly, soluble ionic compounds are strong electrolytes, since the small amount that does dissolve will dissociate completely.</p><p id="fs-idp39918704" class="" name="fs-idp39918704">Consider what happens at the microscopic level when solid KCl is added to water. Ion-dipole forces attract the positive (hydrogen) end of the polar water molecules to the negative chloride ions at the surface of the solid, and they attract the negative (oxygen) ends to the positive potassium ions. The water molecules surround individual K<sup class="">+</sup> and Cl<sup class="">−</sup> ions, reducing the strong interionic forces that bind the ions together and letting them move off into solution as solvated ions, as <a class="autogenerated-content" href="#Figure_2.8" data-mce-href="#Figure_2.8">Figure 2.8</a> shows. Overcoming the electrostatic attraction permits the independent motion of each hydrated ion in a dilute solution as the ions transition from fixed positions in the undissolved compound to widely dispersed, solvated ions in solution.</p><h2 id="h3_qRjG" class="" data-type="title" name="h3_qRjG"><span data-etb-numbered-element="1">2.9</span> Covalent Electrolytes</h2><p id="fs-idp617792" class="" name="fs-idp617792">Pure water is an extremely poor conductor of electricity because it is only very slightly ionized—only about two out of every 1 billion molecules ionize at 25 °C. Water ionizes when one molecule of water gives up a proton (H<sup class="">+</sup> ion) to another molecule of water, yielding hydronium and hydroxide ions.</p><div class="MathJax_Display"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" ctxtmenu_counter="32" 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-msub><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-c4F"></mjx-c></mjx-mtext><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-c 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class="mjx-c2B"></mjx-c></mjx-mtext></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E 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-msup space="3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c4F"></mjx-c><mjx-c class="mjx-c48"></mjx-c></mjx-mtext><mjx-script style="vertical-align: 0.432em;" data-mce-style="vertical-align: 0.432em;"><mjx-mtext class="mjx-n" size="s"><mjx-c class="mjx-c2212"></mjx-c></mjx-mtext></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45E 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><msub><mtext>H</mtext><mn>2</mn></msub><mtext>O</mtext><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>+</mo><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><mspace width="0.2em"></mspace><msub><mtext>H</mtext><mn>3</mn></msub><msup><mtext>O</mtext><mtext>+</mtext></msup><mo stretchy="false">(</mo><mi>a</mi><mi>q</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mtext>OH</mtext><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 xmlns="http://www.w3.org/1998/Math/MathML" 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><mo class="">+</mo><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><mspace width="0.2em" class=""></mspace><msub class=""><mtext class="">H</mtext><mn class="">3</mn></msub><msup class=""><mtext class="">O</mtext><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=""><mtext class="">OH</mtext><mtext class="">−</mtext></msup><mo stretchy="false" class="">(</mo><mi class="">a</mi><mi class="">q</mi><mo stretchy="false" class="">)</mo></mrow></annotation-xml></semantics></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container></div><p id="fs-idm10993712" class="" name="fs-idm10993712">In some cases, solutions prepared from covalent compounds conduct electricity because the solute molecules react chemically with the solvent to produce ions. For example, pure hydrogen chloride is a gas consisting of covalent HCl molecules. This gas contains no ions. However, an aqueous solution of HCl is a very good conductor, indicating that an appreciable concentration of ions exists within the solution.</p><p id="fs-idm25981680" class="" name="fs-idm25981680">Because HCl is an <em class="" data-effect="italics">acid</em>, its molecules react with water, transferring H<sup class="">+</sup> ions to form hydronium ions (H<sub class="">3</sub>O<sup class="">+</sup>) and chloride ions (Cl<sup class="">−</sup>):</p><p id="p_Bwxb" class="" name="p_Bwxb"><span id="fs-idp29456368" class="scaled-down" data-type="media" data-alt="A chemical equation is shown. To the left, two hydrogen atoms are linked, each with a single dash to a central oxygen atom to the left and below the oxygen symbol, which has two pairs of dots, above and to the right of the atom. A plus sign is shown to the right, then a hydrogen atom linked to the left side of chlorine atom by a single dash with three pairs of dots, above, to the right, and below the element symbol. An arrow points to the products which are three hydrogen atoms linked by single dashes to a central oxygen atom shown in brackets with superscript plus. The oxygen atom has a single pair of dots above the element symbol. This is followed by a plus and C l superscript minus. This symbol is surrounded by four pairs of dots, above and below and to the left and right of the element symbol."><img id="img_JNcr" class="img-fluid" src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_10.jpg" alt="A chemical equation is shown. To the left, two hydrogen atoms are linked, each with a single dash to a central oxygen atom to the left and below the oxygen symbol, which has two pairs of dots, above and to the right of the atom. A plus sign is shown to the right, then a hydrogen atom linked to the left side of chlorine atom by a single dash with three pairs of dots, above, to the right, and below the element symbol. An arrow points to the products which are three hydrogen atoms linked by single dashes to a central oxygen atom shown in brackets with superscript plus. The oxygen atom has a single pair of dots above the element symbol. This is followed by a plus and C l superscript minus. This symbol is surrounded by four pairs of dots, above and below and to the left and right of the element symbol." name="img_JNcr" data-media-type="image/jpeg" data-mce-src="https://edtechbooks.org//content_images/general_college_chemistry_2/IgFb_10.jpg"></span></p><p id="fs-idm4855136" class="" name="fs-idm4855136">This reaction is essentially 100% complete for HCl (i.e., it is a <em class="" data-effect="italics">strong acid</em> and, consequently, a strong electrolyte). Likewise, weak acids and bases that only react partially generate relatively low concentrations of ions when dissolved in water and are classified as weak electrolytes. The reader may wish to review the discussion of strong and weak acids provided in the earlier chapter of this text on reaction classes and stoichiometry.</p>
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