Physical Chemistry Third Edition

528 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

The rapidity of some liquid-state reactions at first seems surprising since ordinary

diffusion processes in liquids take hours or days to occur. The difference is that in an

ordinary diffusion process the root-mean-square distance traveled by molecules might

be several centimeters, whereas the mean distance between reacting molecules in a

solution reaction might be a few nanometers.

EXAMPLE12.2

a.Estimate the mean distance between a molecule of substance 2 and the nearest-neighbor

molecule of substance 3 in a well-mixed solution with a concentration of each substance

equal to 0.10 mol L−^1.

b.For a substance with a diffusion coefficient equal to 1. 0 × 10 −^9 m^2 s−^1 , find the time

for which the root-mean-square distance diffused in one direction is equal to the mean

distance in part a.

Solution

a.We estimate the mean distance as the side of a cube in the solution that contains on the

average one molecule of each solute.

Vcube

1L

0 .10 mol

1 mol

6. 0 × 1023

1m^3

1000 L

 1. 7 × 10 −^26 m^3

Side of cubed

(

1. 7 × 10 −^26 m^3

) 1 / 3

 2. 6 × 10 −^9 m 2 .6nm

b.From Eq. (10.2-19),

〈

x^2

〉 1 / 2



√

2 Dt

t

d^2

2 D



(2. 6 × 10 −^9 m)^2

2

(

1. 0 × 10 −^9 m^2 s−^1

) 3. 4 × 10 −^9 s 3 .4ns

Bimolecular Liquid-Phase Reactions

As in Chapter 9, we picture a molecule in a liquid as being confined in a “cage” of

other molecules, but occasionally moving past its nearest neighbors into an adjacent

cage. Assume that substances 2 and 3 are dissolved in a solvent, substance 1. In a dilute

solution, most cages surrounding a solute molecule are made up entirely of solvent

molecules, but a few will also have one molecule of type 2 or type 3. The motion of

a type-2 molecule into a cage with a type-3 molecule is called anencounterof the

two molecules. In adiffusion-limitedordiffusion-controlledbimolecular elementary

process, the chemical part of the process is so rapid that every encounter of the reacting

molecules immediately leads to reaction. The rate is therefore controlled (limited) by

the rate at which the reactant molecules diffuse together.

A theory for the rate of a bimolecular elementary diffusion-limited process was

developed by Smoluchowski.^2 The first version of the theory was based on the assump-

tion that molecules of type 2 are diffusing toward stationary molecules of type 3.

On the average, the motion of type 2 molecules toward the fixed molecules of type 3

(^2) M. V. Smoluchowski,Z. Phys. Chem., 92 , 129 (1917). See K. J. Laidler,Chemical Kinetics, Harper and
Row, New York, 1987, p. 212ff.