Physical Chemistry Third Edition

9.3 The Velocity Probability Distribution 395

Ifh(u) is a function ofu, its mean value is given by

〈h(u)〉

∫u 2

u 1

h(u)f(u)du (9.3-4)

EXAMPLE 9.2

If all values ofubetweenu 1 andu 2 are equally probable, then

f(u)

1

u 2 −u 1

Find the value of<u^2 >ifu 1 0 andu 2 2.

Solution

<u^2 >

1

u 2 −u 1

∫ 2

0

u^2 du

1

2

1

3

(2^3 −0)

8

6



4

3

 1. 333 ···

Derivation of the Probability Distribution for Our

Model System

We now seek a formula that represents the probability distribution for the molecular

states in our model system. Because there are no forces on the molecules inside the box

we assert that all positions inside the box are equally probable and focus on the proba-

bility distribution for molecular velocities. We begin with a reasonable (but unproved)

assumption:The probability distribution of each velocity component is independent of

the other velocity components. The validity of this assumption must be tested by com-

paring our results with experiment. Consider a velocity with componentsvx,vy, and

vz. Let the probability thatvxlies betweenv′x(a particular value ofvx) andv′x+dvx

be given by

(probability)f(v′x)dvx (9.3-5)

Because the effects of gravity are assumed to be negligible there is no physical dif-

ference between any two directions. The probability distributions forvyandvzmust

be given by the same function as that forvx. Letv′y, andv′zbe particular values ofvy

andvz. The probability thatvxlies betweenv′xandv′x+dvx, thatvyis in the range

betweenv′yandv′y+dvy, and thatvzlies betweenv′zandv′z+dvzis given by

(probability)g(v′x,v′y,v′z)dvxdvydvzg(v′)d^3 v (9.3-6)

wherev′is the vector with componentsv′x,v′y, andv′zand where we abbreviate

dvxdvydvzby the symbold^3 v. The probability distribution (probability density)g(v)

is a probability per unit volume in velocity space. The volume elementdvxdvydvzis

infinitesimal, but Figure 9.6 shows the velocity space with a finite rectangular region

crudely representingdvxdvydvz.

vz

vx

vy

dvy

dvx

dvz

Volume element

v

0

Figure 9.6 A Velocity Vector in
Velocity Space, with a Cartesian Vol-
ume Element.