Physical Chemistry Third Edition

502 11 The Rates of Chemical Reactions

When the expressions forGandHare substituted into Eq. (11.3-10) and the resulting

expression is substituted into Eq. (11.3-9), a definite integration gives

1

a[B] 0 −b[A] 0

ln

(

[B]t[A] 0

[A]t[B] 0

)

kft (11.3-11)

This integrated rate law can now be compared with experimental data.

Exercise 11.14

Verify Eq. (11.3-11) by carrying out the integration.

EXAMPLE11.9

Assume that for the reaction

NO+O 3 −→NO 2 +O 2

[NO] 0  1. 00 × 10 −^6 mol L−^1 and [O 3 ] 0  5. 00 × 10 −^7 mol L−^1. Find the concentration

of O 3 after 3.50 s.

Solution

Let O 3 be denoted by A and NO be denoted by B. Use of Eq. (11.3-11) withab1 gives,

with some manipulation:

[B] 0 −x

[A] 0 −x



[B] 0

[A] 0

exp[kt([B] 0 −[A] 0 )]

Using the values given, solution of the equation givesx 4. 73 × 10 −^7 and [O 3 ] 0. 27 ×

10 −^7 mol L−^1.

Exercise 11.15

Carry out the mathematical steps to verify the solution of the preceding example.

The Method of Initial Rates

Consider the hypothetical reaction

aA+bB+fF−→products (11.3-12)

We assume the reaction has definite orders so that the initial rate is

rinitialkf[A]a 0 [B]β 0 [F]φ 0 ≈−

1

a

∆[A]

∆t

(11.3-13)

Several experiments are carried out with the same values of [B] 0 and [F] 0 but

with different values of [A] 0. The initial rate is determined for each experiment. We

write

ln(rinitial)ln(kf[B]β 0 [F]φ 0 )+αln([A] 0 ) (11.3-14)