As you work through this section, if you find that you need a bit more background material to help you understand the topics at hand, you can consult “Chemistry: The Molecular Science” (5th ed. Moore and Stanitski) Chapter 11-1 through 11-2, and/or Chapter 9.1-9.5 in the Additional Reading Materials section.

D18.1 Reaction Rate

The rate of a chemical reaction is usually defined as the change in concentration of a reactant or a product per unit time. Rates can be determined by measuring time from the start of reaction and either concentration or some property that can be related to concentration. For example, for reactions involving one or more colored substances, light absorption could be measured at different times after the start of the reaction.

Consider the decomposition reaction of cyclobutane to ethene in the gas phase:

The rate at which C₄H₈ decomposes can be expressed in terms of the rate of change of its concentration:

In this equation, square brackets represent concentration in mol/L  so [C₄H₈]_t1 represents the concentration of cyclobutane at time t₁ and Δ[C₄H₈] represents the change in concentration of cyclobutane during the time interval Δt (that is, t₂ − t₁). The minus sign is in the equation because reaction rate is defined to be positive. The reactant concentration decreases as the reaction proceeds, making Δ[C₄H₈] a negative quantity, so a negative sign is needed to make it positive.

Table 1 provides an example of data collected during the decomposition of C₄H₈. (M represents the units mol/L.)

Notice that the reaction rate varies with time, decreasing as the reaction proceeds and the concentration of C₄H₈ decreases. Results for the first and last 20-second period are:

An average rate over a given time period can be calculated using the concentrations at the beginning and end of the period. The rate of reaction at any specific time is known as the instantaneous rate. The instantaneous rate at “time zero,” when the reaction starts, is the initial rate.

The instantaneous rate of a reaction may be determined one of two ways:

• If concentration changes can be measured over very short time intervals, then average rates over these very short time intervals provide reasonably good approximations of instantaneous rates.
• If we plot concentration vs. time, the instantaneous rate at any time t is given by the slope of a straight line that is tangent to the curve at that time (Figure 1).

D18.2 Relative Rates of Reaction

The rate of a reaction may be expressed in terms of the change in concentration of any reactant or product, and but depends on the stoichiometry of the reaction. Consider the ammonia decomposition reaction:

Because one nitrogen molecule is produced for every two ammonia molecules that react, the rate of formation of nitrogen is half the rate of disappearance of ammonia. Thus,

The negative sign accounts for the fact that the reactant concentration is decreasing while the product concentration is increasing. The fraction 1/2 accounts for the stoichiometry.

Similarly, because 3 mol H₂ forms during the time required for formation of 1 mol N₂,

Figure 2 plots concentrations vs. time for the reaction. At any time, the instantaneous rates for reactants and products are related by the reciprocals of their stoichiometric coefficients. For example, at 500 s the rate of hydrogen production three times greater than that for nitrogen production.

Thus, the rate of reaction is defined by taking the change in concentration of a reactant or product and multiplying by the reciprocal of the stoichiometric coefficient for each reactant or product. For a reactant the stoichiometric coefficient is given a negative sign (because the concentration is decreasing and rates are defined to be positive). For the reaction aA + bB → cC + dD, where lower-case letters are coefficients and upper-case letters represent chemical formulas,

D18.3 Factors Affecting Reaction Rates

When a chemical reaction occurs, one or more reactant substances are changed into one or more product substances. This involves changes in bonding, which requires that atoms, molecules, or ions be in close contact; that is, atomic-scale particles must collide to react. Anything that affects the number and/or effectiveness of those collisions will affect the rate of reaction.

Chemical Nature of the Reacting Substances

Some substances react faster than others. Different substances have different atomic-scale structures and different bonding. For example, calcium and sodium both react with water to form hydrogen gas and a basic solution. Yet calcium reacts at a moderate rate, whereas sodium reacts so rapidly that the reaction is almost explosive. The reaction involves loss of electrons from sodium or calcium atoms, and sodium has a smaller first ionization energy.

Temperature of the Reaction

Chemical reactions typically occur faster at higher temperatures. At higher temperatures atomic-scale particles move faster, so they collide harder and more often, both of which increase the probability that they will react. For example, natural gas (CH₄) does not react rapidly with air at room temperature, but strike a match and BOOM!

Concentrations

Rates usually increase when the concentration of one or more of the reactants increases. In some cases, rates depend on the concentrations of products as well. 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 concentration of sulfur dioxide in the air. Sulfur dioxide reacts with water vapor to produce sulfurous acid,

Sulfurous acid then reacts with calcium carbonate,

In a more polluted atmosphere where the concentration of sulfur dioxide is higher, calcium carbonate deteriorates more rapidly (Figure 4).

Presence and Concentration of a Catalyst

A catalyst is a substance that increases the rate of a chemical reaction by providing an alternative reaction pathway but is not consumed by the reaction. The greater the concentration of a catalyst the more the catalyst can speed up a reaction. How catalysts work will be discussed in detail later.

Surface Area

The factors discussed so far apply to homogeneous reactions, reactions that occur in a single phase (solid, liquid, or gas). If a reaction occurs at a surface, the intersection of two phases (such as the surface of a solid in contact with a gas), an increase in surface area can increase the rate. Reactions that take place at a surface are called heterogeneous reactions.  A finely divided solid has more surface area available for reaction than does one large piece of the same substance. For example, large pieces of iron can be held in a burner flame for a long time and hardly react whereas iron powder blown into the flame sparkles as the tiny particles burn. (Click here for a video.) Large pieces of wood smolder, smaller pieces burn rapidly, and sawdust burns explosively.

D18.4 Effect of Concentration: Rate Laws

Rate laws or rate equations are mathematical expressions that relate the rate of a chemical reaction to the concentrations of reactants (and sometimes products or catalysts). Often the rate of reaction is proportional to concentration, or to a power of a concentration, of a substance involved in the reaction:

Here, [A], [B], and [C] represent concentrations, k is the rate constant, and each exponent, m, n, and p, defines the reaction order with respect to each reactant. The rate constant is a proportionality constant independent of reactant concentrations and specific for a particular reaction at a particular temperature. The order of a reaction is the power to which a concentration must be raised to correctly calculate the rate. For example, if m = 1, the reaction is first-order with respect to A; if n = 2, the reaction is second-order with respect to B; if p = 0, the reaction is zeroth-order with respect to C, which means that the rate of the reaction is not affected by the concentration of C because [C]⁰ = 1. The overall reaction order is the sum of the individual orders, m + n + p + … Reaction orders are usually positive integers, although they can be fractions or negative numbers.

Method of Initial Rates

The rate constant and the reaction orders must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. One way to do this is the method of initial rates. To use this method, select two sets of rate data where all concentrations but one are the same and set up a ratio of the two rates and the two rate laws. This will yield an equation that contains only one unknown: the reaction order of the substance whose concentration varies.

Example 4

Determining Rate Law from Initial Rates

Ozone in the upper atmosphere is depleted when it reacts with nitrogen oxides. The rates of the reactions of nitrogen oxides with ozone are important factors in deciding how significant these reactions are in the formation of the ozone hole over Antarctica.

One such reaction is the combination of nitric oxide, NO, with ozone, O₃:

NO(g) + O₃(g) → NO₂(g) + O₂(g)

This reaction has been studied in the laboratory, and these rate data were determined at 25 °C.

Trial	[NO] (M)	[O3] (M)	[latex]\frac{{\Delta}[\text{NO}_2]}{{\Delta}t}\;(\text{M/s})[/latex]
1	1.00 × 10−6	3.00 × 10−6	6.60 × 10−5
2	1.00 × 10−6	6.00 × 10−6	1.32 × 10−4
3	1.00 × 10−6	9.00 × 10−6	1.98 × 10−4
4	2.00 × 10−6	9.00 × 10−6	3.96 × 10−4
5	3.00 × 10−6	9.00 × 10−6	5.94 × 10−4

Determine the rate law and the rate constant for the reaction at 25 °C.

Solution

The rate law will have the form:

rate = k[NO]^m[O₃]ⁿ

We can determine the values of m, n, and k from the experimental data using the following three-part process:

1. Determine the value of m from the data in which [NO] varies and [O₃] is constant (Trials 3, 4, and 5).
   • When [NO] doubles from trial 3 to 4, the rate doubles, and when [NO] triples from trial 3 to 5, the rate also triples. Thus, the rate is directly proportional to [NO], and m = 1.
2. Determine the value of n from data in which [O₃] varies and [NO] is constant (Trials 1, 2, and 3).
   •  When [O₃] doubles from trial 1 to 2, the rate doubles; when [O₃] triples from trial 1 to 3, the rate increases also triples. Thus, the rate is directly proportional to [O₃], and n = 1.

• The rate law is thus:  rate = k[NO]¹[O₃]¹

1. Determine the value of k from one set of concentrations and the corresponding rate.
   [latex]\begin{array}{r @{{}={}} l} k &= \dfrac{\text{rate}}{[\text{NO}][\text{O}_3]} \\[1.2em] &= \dfrac{6.60\;\times\;10^{-5}\;\rule[0.5ex]{0.7em}{0.1ex}\hspace{-0.7em}\text{M}\text{s}^{-1}}{(1.00\;\times\;10^{-6}\;\rule[0.5ex]{0.7em}{0.1ex}\hspace{-0.7em}\text{M})(3.00\;\times\;10^{-6}\;\text{M})} \\[1.0em] &= 2.20\;\times\;10^{7}\;\text{M}^{-1}\text{s}^{-1} \end{array}[/latex]

   The large value of k tells us that, at 25 °C, this is a fast reaction that could play an important role in ozone depletion if [NO] is large enough.

Check Your Learning

Acetaldehyde decomposes when heated to yield methane and carbon monoxide according to the equation:

CH₃CHO(g) → CH₄(g) + CO(g)

Determine the rate law and the rate constant for the reaction from the following experimental data:

Trial	[CH3CHO] (M)	[latex]-\frac{{\Delta}[\text{CH}_3\text{CHO}]}{{\Delta}t}\;(\text{M/s})[/latex]
1	1.75 × 10−3	2.06 × 10−11
2	3.50 × 10−3	8.24 × 10−11
3	7.00 × 10−3	3.30 × 10−10

Answer:

rate = k[CH₃CHO]², with k = 6.73 × 10⁻⁶ [latex]\frac{1}{M \cdot s}[/latex]

D18.5 Reaction Order and Rate Constant Units

It is often true that, as in the last “Check Your Learning” example, the reaction orders in the rate law are different from the coefficients in the chemical equation for the reaction. It is important to note that rate laws must be determined by experiment and are not reliably predicted by reaction stoichiometry.

Reaction orders play a role in determining the units for the rate constant—the units for k are whatever is needed so that substituting into the rate law expression affords the appropriate units for the rate.

The units for the rate constant for common reaction orders are summarized below.