Physical Chemistry Third Edition

11.7 The Experimental Study of Fast Reactions 519

Exercise 11.23

a.Verify the steps of algebra leading to Eq. (11.7-7).

b.Verify that Eq. (11.7-8), with Eq. (11.7-9), is a solution to Eq. (11.7-7).

c.Write the expressions for∆[A] and∆[B].

If we assume that hydrogen ions in water primarily occur as hydronium ions, the

reaction of hydrogen ions and hydroxide ions can be written

H 3 O++OH−

k 1




k′ 1

2H 2 O (11.7-10)

which has the general form

A+B

k 1




k′ 1

2C (11.7-11)

Assume that the reaction is second order in both directions. We write

[C][C]eq+∆[C] (11.7-12a)

[A][A]eq−

1

2

∆[C] (11.7-12b)

[B][B]eq−

1

2

∆[C] (11.7-12c)

When Eqs. (11.7-12) are substituted into the differential equation for the rate of the

reaction and the necessary steps of algebra are carried out with neglect of terms pro-

portional to (∆[C])^2 , we obtain

∆[C]∆[C] 0 e−t/τ (11.7-13)

where

1

τ

k 1

[A]eq+[B]eq

2

+ 2 k′ 1 [C]eq (11.7-14)

EXAMPLE11.12

At 25◦C, the forward rate constant of Eq. (11.7-10) is equal to 1. 4 × 1011 L mol−^1 s−^1 .At

this temperature, the dissociation equilibrium constantKwequals 1. 008 × 10 −^14 mol^2 L−^2.

a.Using Eq. (11.5-3), find the value ofk′ 1.

b.For pure water, find the relaxation time if a T-jump experiment has a final temperature of

25 ◦C.

Solution

a.We convertKw, for which the H 2 O activity is expressed in terms of the mole fraction, to

the equilibrium constant for the reaction of Eq. (11.7-10) with all concentrations in terms

of molarities: