Physical Chemistry Third Edition

538 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

EXAMPLE12.8

Assuming the activation energy and the value of the rate constant at 373 K from Example

12.6, find the effective collision diameter of a hydrogen molecule and an iodine molecule.

Solution

Using Eq. (12.3-17)

πd^2 

(

πμ

8 kBT

) 1 / 2

k

NAv

eEa/RT

k

〈vrel〉NAv

eEa/RT

The value of〈vrel〉is 1988 m s−^1 , so that

d^2 

8. 74 × 10 −^18 m^3 mol−^1 s−^1

π

(

1988 m−^1 s−^1

)(

6. 022 × 1023 mol−^1

)e^54.^794  1. 45 × 10 −^21 m^2

d 3. 8 × 10 −^11 m38 pm 0. 38 A ̊

The value of the collision diameter in Example 12.8 is too small by a factor of about

1/10, which is typical of the collision theory. The explanation for the smallness of the

apparent collision diameter is that the molecules not only have to collide with at least

a minimum relative energy, but in many reactions must also be oriented properly in

order to react. Even two small molecules like H 2 and I 2 would presumably have to be

oriented correctly for the reaction to occur.

To account for the orientation dependence, a factor called thesteric factoris intro-

duced into the collision theory. The steric factor is defined to be the fraction of collisions

in which the orientation of the molecules is appropriate for reaction. If this factor is

denoted byφ, Eq. (12.3-17) becomes

kφNAvπd^212

(

8 kBT

πμ

) 1 / 2

e−Ec/RT (12.3-18)

Exercise 12.7

Find the value of the steric factor for the reaction of Example 12.8 that will give a value for the

collision diameter equal to the mean of the hard-sphere diameters of H 2 and I 2 in Table A.15 of

Appendix A.

Liquid-State Reactions

The temperature dependence of rate constants for both gaseous and liquid-state reactions

is usually well described by the Arrhenius formula, Eq. (12.3-2). For activation-limited

reactions, the activation energies are roughly equal to those for gas-phase reactions.

This is as expected, since the collisional activation is very similar to that of gaseous

reactions. In the case of diffusion-limited reactions, the temperature dependence of the

rate constant is governed by the diffusion coefficients. Diffusion coefficients in liquids

commonly have a temperature dependence given by Eq. (10.4-5), which is also of the

same form as the Arrhenius formula:

DD 0 e−Ea,d/RT (12.3-19)