The Foundations of Chemistry

One method of assessing reaction order is based on comparing successive half-lives. As

we have seen, t1/2for a first-order reaction does not depend on initial concentration. We

can measure the time required for different concentrations of a reactant to fall to half of

their original values. If this time remains constant, it is an indication that the reaction is

first order for that reactant and first order overall (see Figure 16-4a). By contrast, for other

orders of reaction, t1/2would change depending on initial concentration. For a second-

order reaction, successively measured t1/2values would increase by a factor of 2 as [A] 0

decreases by a factor of 2 (see Figure 16-4b). [A] 0 is measured at the beginning of each

particular measurement period.

You should test this method using the
concentration-versus-time data of
Example 16-10, plotted in Figure
16-8b.

670 CHAPTER 16: Chemical Kinetics

Problem-Solving Tip:Which Equation Should Be Used?

How can you tell which equation to use to solve a particular problem?

1.You must decide whether to use the rate-law expression or the integrated rate equa-

tion. Remember that

the rate-law expressionrelates rate and concentration

whereas

the integrated rate equationrelates time and concentration.

When you need to find the ratethat corresponds to particular concentrations, or

the concentrations needed to give a desired rate, you should use the rate-law

expression. When timeis involved in the problem, you should use the integrated

rate equation.

2.You must choose the form of the rate-law expression or the integrated rate equation

—zero, first, or second order—that is appropriate to the order of the reaction. These

are summarized in Table 16-2. One of the following usually helps you decide.

a. The statement of the problem may state explicitly what the order of the reaction

is.

b. The rate-law expression may be given, so that you can tell the order of the reac-

tion from the exponents in that expression.

c. The units of the specific rate constant, k,may be given; you can interpret these

stated units to tell you the order of the reaction.

Order Units of k

0 Mtime^1

1 time^1

2 M^1 time^1

E

nrichment

Calculus Derivation of Integrated Rate Equations

The derivation of the integrated rate equation is an example of the use of calculus in chem-

istry. The following derivation is for a reaction that is assumed to be first order in a reactant

A and first order overall. If you do not know calculus, you can still use the results of this

derivation, as we have already shown in this section. For the reaction