Physical Chemistry Third Edition

25.4 The Calculation of Molecular Partition Functions 1067

or by the parameterB ̃e, equal toBe/cwherecis the speed of light. The equilibrium

moment of inertia is denoted byIe, the reduced mass of the nuclei is denoted byμ, and

the equilibrium internuclear distance is denoted byre.

The degeneracy of the rotational level numberJis 2J+1. The rotational partition

function of a dilute diatomic gas is given by the sum over levels:

zrot

∑∞

J 0

gJe−εJ/kBT

∑∞

J 0

(2J+1)e−hBeJ(J+1)/kBT (25.4-9)

Figure 25.4 shows the sum of Eq. (25.4-7) represented by the combined areas of a set of

rectangles, as was done with the translational partition function in Figure 25.2. Because

J0 is the lower limit of the sum, we draw the rectangle for a given value ofJto the

right of that integer value ofJ. Figure 25.4 also shows a curve representing the function

that is obtained by allowingJto take on all real values. The sum is approximately equal

to the area under the curve, which is equal to the integral

zrot≈

∞∫

0

(2J+1)e−hBeJ(J+1)/kBTdJ (25.4-10)

We letuJ(J+1) so thatdu(2J+1)dJ:

zrot≈

∞∫

0

e−hBeu/kBTdu

kBT

hBe



kBT

hcB ̃e



8 π^2 IekBT

h^2



8 π^2 μre 2 kBT

h^2

(25.4-11)

1

1

2

3

4

5

6

7

8

9

10

0 2 3 4 5 6 7 8 9 1011121314151617181920

Value of term

J (quantum number)

Figure 25.4 A Graphical Representation of the Rotational Partition Function (Drawn

forCOat 298.15 K).This figure is analogous to Figure 25.2 for the translational partition

function. The area under the bar graph is equal to the rotational partition function for a

diatomic molecule, and the area under the curve is equal to the integral approximation to

the rotational partition function.