Physical Chemistry Third Edition

510 11 The Rates of Chemical Reactions

a.Integrate the rate differential equation for the case that

the initial concentrations of both A and B are nonzero.

b. Construct a graph of the concentrations of A and B as

a function of time for the case that

[A][B] 1 .00 mol L−^1 ,kf 10 .0s−^1 and

kr 1 .00 s−^1.

c.For the reaction of part b, find each concentration after

a reaction time of 0.35 s.

11.31Consider the reaction

2A
B+C

The forward reaction is second order, and the reverse

reaction is first order with respect to B and first order with

respect to C. Write a computer program using Euler’s

method to integrate the rate differential equations for the

case that the initial concentration of A is nonzero and

those of B and C are zero.^4

11.5 A Simple Reaction Mechanism:

Two Consecutive Steps

Almost every chemical reaction takes place through a set of steps, called thereaction

mechanism. We now consider the simplest possible mechanism, a two-step mechanism

in which the product of the first step is the reactant of the second step. We assume that

both steps are first order and that the reverse reactions are negligible:

A

k 1

−→B

k 2

−→F (11.5-1a)

wherek 1 is the rate constant for the first step andk 2 is the rate constant for the second

step. The substance B is called areactive intermediate. We will usually number the

steps of a mechanism:

(1) A−→B (11.5-1b)

(2) B−→F (11.5-1c)

We assign a subscript to the rate constant equal to the number of the step.

Since there is no reverse reaction, step (1) has the same rate law as Eq. (11.2-2),

d[A]

dt

−k 1 dt (11.5-2)

Since B is produced by the first step and consumed by the second step,

d[B]

dt

k 1 [A]−k 2 [B] (11.5-3)

Equations (11.5-2) and (11.5-3) are a set of simultaneous differential equations. The

solution to the first equation has already been obtained:

[A]t[A] 0 e−k^1 t (11.5-4)

This solution can be substituted into Eq. (11.5-3) to obtain a single differential equation

for[B]:

d[B]

dt

k 1 [A] 0 e−k^1 t−k 2 [B] (11.5-5)

(^4) R. G. Mortimer,Mathematics for Physical Chemistry, 3rd ed., Elsevier/Academic Press, San Diego, CA, 2005, p. 260. The method is also found in books on
numerical analysis and in some calculus textbooks.