6

# Reaction Rates

The rate of a reaction can be expressed either in terms of the decrease in the amount of a reactant or the increase in the amount of a product per unit time. Relations between different rate expressions for a given reaction are derived directly from the stoichiometric coefficients of the equation representing the reaction. The rate of a chemical reaction is affected by several parameters. Reactions involving two phases proceed more rapidly when there is greater surface area contact. If temperature or reactant concentration is increased, the rate of a given reaction generally increases as well. A catalyst can increase the rate of a reaction by providing an alternative pathway with a lower activation energy.

## 6.1 Chemical Reaction Rates

### Learning Objectives

By the end of this section, you will be able to:

• Define chemical reaction rate
• Derive rate expressions from the balanced equation for a given chemical reaction
• Calculate reaction rates from experimental data

Figure 6.1

An agama lizard basks in the sun. As its body warms the chemical reactions of its metabolism speed up.

The lizard in the photograph is not simply enjoying the sunshine or working on its tan. The heat from the sun’s rays is critical to the lizard’s survival. A warm lizard can move faster than a cold one because the chemical reactions that allow its muscles to move occur more rapidly at higher temperatures. A cold lizard is a slower lizard and an easier meal for predators.

From baking a cake to determining the useful lifespan of a bridge, rates of chemical reactions play important roles in our understanding of processes that involve chemical changes. Two questions are typically posed when planning to carry out a chemical reaction. The first is: “Will the reaction produce the desired products in useful quantities?” The second question is: “How rapidly will the reaction occur?” A third question is often asked when investigating reactions in greater detail: “What specific molecular-level processes take place as the reaction occurs?” Knowing the answer to this question is of practical importance when the yield or rate of a reaction needs to be controlled.

The study of chemical kinetics concerns the second and third questions—that is, the rate at which a reaction yields products and the molecular-scale means by which a reaction occurs. This chapter examines the factors that influence the rates of chemical reactions, the mechanisms by which reactions proceed and the quantitative techniques used to describe the rates at which reactions occur.

A rate is a measure of how some property varies with time. Speed is a familiar rate that expresses the distance traveled by an object in a given amount of time. Wage is a rate that represents the amount of money earned by a person working for a given amount of time. Likewise, the rate of a chemical reaction is a measure of how much reactant is consumed, or how much product is produced by the reaction in a given amount of time.

The rate of reaction is the change in the amount of a reactant or product per unit time. Reaction rates are therefore determined by measuring the time dependence of some property that can be related to reactant or product amounts. Rates of reactions that consume or produce gaseous substances, for example, are conveniently determined by measuring changes in volume or pressure. For reactions involving one or more colored substances, rates may be monitored via measurements of light absorption. For reactions involving aqueous electrolytes, rates may be measured via changes in a solution’s conductivity.

For reactants and products in solution, their relative amounts (concentrations) are conveniently used for purposes of expressing reaction rates. For example, the concentration of hydrogen peroxide, H2O2, in an aqueous solution changes slowly over time as it decomposes according to the equation:

${\text{2H}}_{2}{\text{O}}_{2}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{2H}}_{2}\text{O}\left(l\right)+{\text{O}}_{2}\left(g\right)$

The rate at which the hydrogen peroxide decomposes can be expressed in terms of the rate of change of its concentration as shown here:

$\begin{array}{cc}\hfill \text{rate of decomposition of}\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}{\text{O}}_{2}& =-\phantom{\rule{0.2em}{0ex}}\frac{\text{change in concentration of reactant}}{\text{time interval}}\hfill \\ & =-\phantom{\rule{0.2em}{0ex}}\frac{{\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{1}}}{{t}_{2}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{t}_{1}}\hfill \\ & =-\phantom{\rule{0.2em}{0ex}}\frac{\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\hfill \end{array}$

This mathematical representation of the change in species concentration over time is the rate expression for the reaction. The brackets indicate molar concentrations, and the symbol delta (Δ) indicates “change in.” Thus, ${\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{1}}$ represents the molar concentration of hydrogen peroxide at some time t1; likewise, ${\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{2}}$ represents the molar concentration of hydrogen peroxide at a later time t2; and Δ[H2O2] represents the change in molar concentration of hydrogen peroxide during the time interval Δt (that is, t2t1). Since the reactant concentration decreases as the reaction proceeds, Δ[H2O2] is a negative quantity. Reaction rates are, by convention, positive quantities, and so this negative change in concentration is multiplied by −1. Figure 6.2 provides an example of data collected during the decomposition of H2O2.

Figure 6.2

The rate of decomposition of H2O2 in an aqueous solution decreases as the concentration of H2O2 decreases.

To obtain the tabulated results for this decomposition, the concentration of hydrogen peroxide was measured every 6 hours over the course of a day at a constant temperature of 40 °C. Reaction rates were computed for each time interval by dividing the change in concentration by the corresponding time increment, as shown here for the first 6-hour period:

$\frac{-\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{-\left(\text{0.500 mol/L}-\text{1.000 mol/L}\right)}{\left(\text{6.00 h}-\text{0.00 h}\right)}\phantom{\rule{0.1em}{0ex}}=0.0833 mol\phantom{\rule{0.2em}{0ex}}{\text{L}}^{-1}\phantom{\rule{0.2em}{0ex}}{\text{h}}^{-1}$

Notice that the reaction rates vary with time, decreasing as the reaction proceeds. Results for the last 6-hour period yield a reaction rate of:

$\frac{-\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{-\left(0.0625\phantom{\rule{0.2em}{0ex}}\text{mol/L}-0.125\phantom{\rule{0.2em}{0ex}}\text{mol/L}\right)}{\left(24.00\phantom{\rule{0.2em}{0ex}}\text{h}-18.00\phantom{\rule{0.2em}{0ex}}\text{h}\right)}\phantom{\rule{0.1em}{0ex}}=0.010\phantom{\rule{0.2em}{0ex}}\text{mol}\phantom{\rule{0.2em}{0ex}}{\text{L}}^{-1}\phantom{\rule{0.2em}{0ex}}{\text{h}}^{-1}$

This behavior indicates the reaction continually slows with time. Using the concentrations at the beginning and end of a time period over which the reaction rate is changing results in the calculation of an average rate for the reaction over this time interval. At any specific time, the rate at which a reaction is proceeding is known as its instantaneous rate. The instantaneous rate of a reaction at “time zero,” when the reaction commences, is its initial rate. Consider the analogy of a car slowing down as it approaches a stop sign. The vehicle’s initial rate—analogous to the beginning of a chemical reaction—would be the speedometer reading at the moment the driver begins pressing the brakes (t0). A few moments later, the instantaneous rate at a specific moment—call it t1—would be somewhat slower, as indicated by the speedometer reading at that point in time. As time passes, the instantaneous rate will continue to fall until it reaches zero, when the car (or reaction) stops. Unlike instantaneous speed, the car’s average speed is not indicated by the speedometer; but it can be calculated as the ratio of the distance traveled to the time required to bring the vehicle to a complete stop (Δt). Like the decelerating car, the average rate of a chemical reaction will fall somewhere between its initial and final rates.

The instantaneous rate of a reaction may be determined one of two ways. If experimental conditions permit the measurement of concentration changes over very short time intervals, then average rates computed as described earlier provide reasonably good approximations of instantaneous rates. Alternatively, a graphical procedure may be used that, in effect, yields the results that would be obtained if short time interval measurements were possible. In a plot of the concentration of hydrogen peroxide against time, the instantaneous rate of decomposition of H2O2 at any time t is given by the slope of a straight line that is tangent to the curve at that time (Figure 6.3). These tangent line slopes may be evaluated using calculus, but the procedure for doing so is beyond the scope of this chapter.

Figure 6.3

This graph shows a plot of concentration versus time for a 1.000 M solution of H2O2. The rate at any time is equal to the negative of the slope of a line tangent to the curve at that time. Tangents are shown at t = 0 h (“initial rate”) and at t = 12 h (“instantaneous rate” at 12 h).

### 6.1.1 CHEMISTRY IN EVERYDAY LIFE

#### Reaction Rates in Analysis: Test Strips for Urinalysis

Physicians often use disposable test strips to measure the amounts of various substances in a patient’s urine (Figure 6.4). These test strips contain various chemical reagents, embedded in small pads at various locations along the strip, which undergo changes in color upon exposure to sufficient concentrations of specific substances. The usage instructions for test strips often stress that proper read time is critical for optimal results. This emphasis on read time suggests that kinetic aspects of the chemical reactions occurring on the test strip are important considerations.

The test for urinary glucose relies on a two-step process represented by the chemical equations shown here:

${\text{C}}_{6}{\text{H}}_{12}{\text{O}}_{6}+{\text{O}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\phantom{\rule{0.2em}{0ex}}\text{catalyst}\phantom{\rule{0.2em}{0ex}}}{\to }\phantom{\rule{0.2em}{0ex}}{\text{C}}_{6}{\text{H}}_{10}{\text{O}}_{6}+{\text{H}}_{2}{\text{O}}_{2}$
${\text{2H}}_{2}{\text{O}}_{2}+2{\text{I}}^{\text{−}}\phantom{\rule{0.2em}{0ex}}\stackrel{\phantom{\rule{0.2em}{0ex}}\text{catalyst}\phantom{\rule{0.2em}{0ex}}}{\to }\phantom{\rule{0.2em}{0ex}}{\text{I}}_{\text{2}}+{\text{2H}}_{2}\text{O}+{\text{O}}_{2}$

The first equation depicts the oxidation of glucose in the urine to yield glucolactone and hydrogen peroxide. The hydrogen peroxide produced subsequently oxidizes colorless iodide ion to yield brown iodine, which may be visually detected. Some strips include an additional substance that reacts with iodine to produce a more distinct color change.

The two test reactions shown above are inherently very slow, but their rates are increased by special enzymes embedded in the test strip pad. This is an example of catalysis, a topic discussed later in this chapter. A typical glucose test strip for use with urine requires approximately 30 seconds for completion of the color-forming reactions. Reading the result too soon might lead one to conclude that the glucose concentration of the urine sample is lower than it actually is (a false-negative result). Waiting too long to assess the color change can lead to a false positive due to the slower (not catalyzed) oxidation of iodide ion by other substances found in urine.

Figure 6.4

Test strips are commonly used to detect the presence of specific substances in a person’s urine. Many test strips have several pads containing various reagents to permit the detection of multiple substances on a single strip. (credit: Iqbal Osman)

### 6.1.2 Relative Rates of Reaction

The rate of a reaction may be expressed as the change in concentration of any reactant or product. For any given reaction, these rate expressions are all related simply to one another according to the reaction stoichiometry. The rate of the general reaction

$\text{aA}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{bB}$

can be expressed in terms of the decrease in the concentration of A or the increase in the concentration of B. These two rate expressions are related by the stoichiometry of the reaction:

$\text{rate}=-\left(\frac{1}{\text{a}}\right)\left(\frac{\text{ΔA}}{\text{Δ}t}\right)=\left(\frac{1}{\text{b}}\right)\left(\frac{\text{ΔB}}{\text{Δ}t}\right)$

Consider the reaction represented by the following equation:

${\text{2NH}}_{3}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}\left(g\right)+{\text{3H}}_{2}\left(g\right)$

The relation between the reaction rates expressed in terms of nitrogen production and ammonia consumption, for example, is:

$-\phantom{\rule{0.2em}{0ex}}\frac{{\text{Δmol NH}}_{3}}{\text{Δ}t}\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.4em}{0ex}}\frac{\text{1 mol}\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}}{\text{2 mol}\phantom{\rule{0.2em}{0ex}}{\text{NH}}_{3}}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{\text{Δmol}\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}}{\text{Δ}t}$

This may be represented in an abbreviated format by omitting the units of the stoichiometric factor:

$-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\frac{\text{Δmol}\phantom{\rule{0.2em}{0ex}}{\text{NH}}_{3}}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{\text{Δmol}\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}}{\text{Δ}t}$

Note that a negative sign has been included as a factor to account for the opposite signs of the two amount changes (the reactant amount is decreasing while the product amount is increasing). For homogeneous reactions, both the reactants and products are present in the same solution and thus occupy the same volume, so the molar amounts may be replaced with molar concentrations:

$-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\frac{\text{Δ}\left[{\text{NH}}_{3}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{\text{Δ}\left[{\text{N}}_{\text{2}}\right]}{\text{Δ}t}$

Similarly, the rate of formation of H2 is three times the rate of formation of N2 because three moles of H2 are produced for each mole of N2 produced.

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\frac{\text{Δ}\left[{\text{H}}_{2}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\frac{\text{Δ}\left[{\text{N}}_{\text{2}}\right]}{\text{Δ}t}$

Figure 6.5 illustrates the change in concentrations over time for the decomposition of ammonia into nitrogen and hydrogen at 1100 °C. Slopes of the tangent lines at t = 500 s show that the instantaneous rates derived from all three species involved in the reaction are related by their stoichiometric factors. The rate of hydrogen production, for example, is observed to be three times greater than that for nitrogen production:

$\frac{2.91\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}M\text{/s}}{9.70\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\phantom{\rule{0.2em}{0ex}}M\text{/s}}\phantom{\rule{0.2em}{0ex}}\approx 3$

Figure 6.5

Changes in concentrations of the reactant and products for the reaction $$$2{\text{NH}}_{3}\text{}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{N}}_{2}+{\text{3H}}_{2}.$$$ The rates of change of the three concentrations are related by the reaction stoichiometry, as shown by the different slopes of the tangents at t = 500 s. ## 6.2 Factors Affecting Reaction Rates

### Learning Objectives

By the end of this section, you will be able to:

• Describe the effects of chemical nature, physical state, temperature, concentration, and catalysis on reaction rates

The rates at which reactants are consumed and products are formed during chemical reactions vary greatly. Five factors typically affecting the rates of chemical reactions will be explored in this section: the chemical nature of the reacting substances, the state of subdivision (one large lump versus many small particles) of the reactants, the temperature of the reactants, the concentration of the reactants, and the presence of a catalyst.

### 6.2.1 The Chemical Nature of the Reacting Substances

The rate of a reaction depends on the nature of the participating substances. Reactions that appear similar may have different rates under the same conditions, depending on the identity of the reactants. For example, when small pieces of the metals iron and sodium are exposed to air, the sodium reacts completely with air overnight, whereas the iron is barely affected. The active metals calcium and sodium both react with water to form hydrogen gas and a base. Yet calcium reacts at a moderate rate, whereas sodium reacts so rapidly that the reaction is almost explosive.

### 6.2.2 The Physical States of the Reactants

A chemical reaction between two or more substances requires intimate contact between the reactants. When reactants are in different physical states, or phases (solid, liquid, gaseous, dissolved), the reaction takes place only at the interface between the phases. Consider the heterogeneous reaction between a solid phase and either a liquid or gaseous phase. Compared with the reaction rate for large solid particles, the rate for smaller particles will be greater because the surface area in contact with the other reactant phase is greater. For example, large pieces of iron react more slowly with acids than they do with finely divided iron powder (Figure 6.6). Large pieces of wood smolder, smaller pieces burn rapidly, and saw dust burns explosively.

Figure 6.6 (a) Iron powder reacts rapidly with dilute hydrochloric acid and produces bubbles of hydrogen gas: 2Fe(s) + 6HCl(aq 2FeCl3(aq) + 3H2(g). (b) An iron nail reacts more slowly because the surface area exposed to the acid is much less.

Watch this video to see the reaction of cesium with water in slow motion and a discussion of how the state of reactants and particle size affect reaction rates.

### 6.2.3 Temperature of the Reactants

Chemical reactions typically occur faster at higher temperatures. Food can spoil quickly when left on the kitchen counter. However, the lower temperature inside of a refrigerator slows that process so that the same food remains fresh for days. Gas burners, hot plates, and ovens are often used in the laboratory to increase the speed of reactions that proceed slowly at ordinary temperatures. For many chemical processes, reaction rates are approximately doubled when the temperature is raised by 10 °C.

### 6.2.4 Concentrations of the Reactants

The rates of many reactions depend on the concentrations of the reactants. Rates usually increase when the concentration of one or more of the reactants increases. For example, calcium carbonate (CaCO3) deteriorates as a result of its reaction with the pollutant sulfur dioxide. The rate of this reaction depends on the amount of sulfur dioxide in the air (Figure 6.7). An acidic oxide, sulfur dioxide combines with water vapor in the air to produce sulfurous acid in the following reaction:

${\text{SO}}_{\text{2}}\left(g\right)+{\text{H}}_{2}\text{O(}g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}{\text{SO}}_{\text{3}}\left(aq\right)$

Calcium carbonate reacts with sulfurous acid as follows:

${\text{CaCO}}_{\text{3}}\left(s\right)+{\text{H}}_{2}{\text{SO}}_{\text{3}}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{CaSO}}_{\text{3}}\left(aq\right)+{\text{CO}}_{\text{2}}\left(g\right)+{\text{H}}_{2}\text{O(}l\right)$

In a polluted atmosphere where the concentration of sulfur dioxide is high, calcium carbonate deteriorates more rapidly than in less polluted air. Similarly, phosphorus burns much more rapidly in an atmosphere of pure oxygen than in air, which is only about 20% oxygen.

Figure 6.6

Statues made from carbonate compounds such as limestone and marble typically weather slowly over time due to the actions of water, and thermal expansion and contraction. However, pollutants like sulfur dioxide can accelerate weathering. As the concentration of air pollutants increases, deterioration of limestone occurs more rapidly. (credit: James P Fisher III)

Phosphorus burns rapidly in air, but it will burn even more rapidly if the concentration of oxygen is higher. Watch this video to see an example.

### 6.2.5 The Presence of a Catalyst

Relatively dilute aqueous solutions of hydrogen peroxide, H2O2, are commonly used as topical antiseptics. Hydrogen peroxide decomposes to yield water and oxygen gas according to the equation:

$2{\text{H}}_{2}{\text{O}}_{2}\left(\text{aq}\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}2{\text{H}}_{2}\text{O}\left(l\right)+{\text{O}}_{2}\left(\text{g}\right)$

Under typical conditions, this decomposition occurs very slowly. When dilute H2O2(aq) is poured onto an open wound, however, the reaction occurs rapidly and the solution foams because of the vigorous production of oxygen gas. This dramatic difference is caused by the presence of substances within the wound’s exposed tissues that accelerate the decomposition process. Substances that function to increase the rate of a reaction are called catalysts, a topic treated in greater detail later in this chapter.