5 Steps to a 5 AP Chemistry

Integrated Rate Laws

Thus far, only cases in which instantaneous data are used in the rate expression have been shown.

These expressions allow us to answer questions concerning the speed of the reaction at a partic-

ular moment, but not questions about how long it might take to use up a certain reactant, etc.

If changes in the concentration of reactants or products over time are taken into account, as in

the integrated rate laws, these questions can be answered. Consider the following reaction:

A →B

Assuming that this reaction is first order, then the rate of reaction can be expressed as

the change in concentration of reactant Awith time:

and also as the rate law:

Rate =k[A]

Setting these terms equal to each other gives:

and integrating over time gives:

ln[A]t− ln[A] 0 = −kt

where ln is the natural logarithm, [A] 0 is the concentration of reactant A at time =0,

and [A]tis the concentration of reactant Aat some time t.

If the reaction is second order in A, then the following equation can be derived using

the same procedure:

Consider the following problem: Hydrogen iodide, HI, decomposes through a second-

order process to the elements. The rate constant is 2.40 × 10 −^21 /M s at 25°C. How long

will it take for the concentration of HI to drop from 0.200 M to 0.190 M at 25°C?

Answer:

1.10 × 1020 s. In this problem, k=2.40 × 10 −^21 /M s, [A] 0 =0.200 M, and [A] 1 =0.190 M.

You can simply insert the values and solve for t, or you first can rearrange the equation to

give t=[1/[A]t– 1/[A] 0 ]/k. You will get the same answer in either case. If you get a nega-

tive answer, you interchanged [A]tand [A] 0. A common mistake is to use the first-order

equation instead of the second-order equation. The problem will always give you the infor-

mation needed to determine whether the first-order or second-order equation is required.

The order of reaction can be determined graphically through the use of the integrated

rate law. If a plot of the ln[A] versus time yields a straight line, then the reaction is first order

with respect to reactant A. If a plot of versus time yields a straight line, then the reaction

is second order with respect to reactant A.

The reaction half-life, t1/2, is the amount of time that it takes for a reactant concentra-

tion to decrease to one-half its initial concentration. For a first-order reaction, the half-life

1

[A]

11

[]AAt0]

−=kt

[

−

Δ

Δ

=

[

]

A

t

kA

]

[

Rate

]

=−

Δ

Δ

[A

t

Kinetics  201