Physical Chemistry Third Edition

1058 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

wheremis the molecular mass. We write the translational partition function as a sum

over states:

ztr

∑∞

nx 1

∑∞

ny 1

∑∞

nz 1

exp

(

−βh^2 (n^2 x/a^2 +n^2 y/b^2 +n^2 z/c^2 )

8 m

)

(25.3-13)

Each value ofnxcan occur with every value ofny, and so on, so the sum factors into

three separate sums:

ztrzxzyzz (25.3-14)

where

zx

∑∞

nx 1

exp

(

−βh^2 n^2 x

8 ma^2

)

(25.3-15)

and wherezyandzzare given by analogous formulas.

Exercise 25.12

By writing all the terms on both sides of the equation, show that

∑^3

i 1

∑^3

j 1

aibj

⎛

⎝

∑^3

i 1

ai

⎞

⎠

⎛

⎝

∑^3

j 1

bj

⎞

⎠

whereaiandbjrepresent arbitrary quantities.

Figure 25.2 shows a representation of the sum of Eq. (25.3-15) in which each term

is equal to the area of a rectangle with unit width and with height equal to the value

of the term. The rectangle for a given value ofnxis drawn to the left of that value on

the horizontal axis, so the areas begin atnx0 while the sum begins atnx1. Also

shown in the figure is a curve representing the function

f(nx)exp

(

−βh^2 n^2 x

8 ma^2

)

(25.3-16)

wherenxnow takes on any positive real value, not just integral values. This function

is equal to the terms in the sum shown in Eq. (25.3-15) for integral values ofnxand

interpolates between these values for nonintegral values ofnx.

We approximate the sum by the area under the curve, which is equal to the integral

zx≈

∞∫

0

exp

(

−βh^2 n^2 x

8 ma^2

)

dnx

(

2 πm

h^2 β

) 1 / 2

a (25.3-17)

The approximation of Eq. (25.3-17) will be a good approximation if nearly all of the

areas between the rectangles and the curve in Figure 25.2 are small, as will be shown