Physical Chemistry Third Edition

26.3 Chemical Equilibrium in Dilute Gases 1103

or

Ke−∆ε^0 /kBT

∏s

a 1

(

z′◦a

NAv

)νa

(26.3-15)

wherez′◦az′′aVm◦is the molecular partition function of substance numberawith the

volume set equal to the standard molar volume at pressureP◦and with the zero of

energy set equal to the ground-state energy of the molecule.

The result in Eq. (26.3-15) has a physical interpretation that helps in remembering

the formula. The partition function is a measure of the availability of states for the

molecules of a given substance. If the partition functions of the products are larger than

those of the reactants, the equilibrium constant will be larger than unity, corresponding

to a larger number of accessible states for the products and a larger probability for

the products than for the reactants. Similarly, if there are fewer states available for the

products than for the reactants, the equilibrium constant will be smaller than unity.

EXAMPLE26.11

Calculate the equilibrium constant for the reaction

H 2 
2H

at 500.0 K. Assume that the vibrational and electronic partition functions of the hydrogen

molecule are both approximately equal to 1.000. Assume that the H 2 is an equilibrium mixture

ofortho- andpara-hydrogen. If this is the case, the nuclear spin degeneracy will cancel (there

is a degeneracy of 4 for the molecule and a degeneracy of 2 for each atom).

Solution

From Eq. (26.3-8),

zrot,H 2 

1

σ

8 π^2 IekBT

h^2



1

2

8 π^2 μr^2 ekBT

h^2



1

2

8 π^2 (8. 37 × 10 −^28 kg)(0. 741 × 10 −^10 m)^2 (1. 3807 × 10 −^23 JK−^1 )(500 K)

(6. 6261 × 10 −^34 Js)^2

 2. 85

A better approximation to the rotational partition function could be obtained by explicit

summation, but we will use this approximation. The translational partition function of atomic

hydrogen is

z′tr,H◦

NAv



(

2 πmkBT

h^2

) 3 / 2

Vm◦

NAv



(

2 π(1. 674 × 10 −^27 kg)(1. 3807 × 10 −^23 JK−^1 )(500 K)

(6. 6261 × 10 −^34 Js)^2

) 3 / 2

×

0 .04157 m^3 mol−^1

6. 022 × 1023 mol−^1

 1. 468 × 105

The electronic partition function of atomic hydrogen is approximately equal to 2.00 since

the ground level is a doublet. The electronic partition function of molecular hydrogen is