Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

THE EQUILIBRIUM CONSTANT: A STATISTICAL APPROACH 117

This leads to

Q=QtrQrotQvibQel

where

Qtr=

∑

i

e−(εtr)/kBT, etc.

Thus, if one knows the energy absorption for each possible mode of motion or

excitation, one can write down the partition function and obtain all the thermodynamic

information for that system. In practice this may be a difficult task because systems

may not beweaklyinteracting; instead, they may interact in such a way that the

vibrational energy spacing for each bond in the molecule depends on its neighbors.

Other interactions may occur. Also, real chemical bonds are not truly harmonic and

their energy level spacings are equal only as an approximation.

8.8 THE EQUILIBRIUM CONSTANT: A STATISTICAL APPROACH

First taking into account the energy spacing between reactants and products and then

considering their partition functions, we can writeKeqin terms of the productof

molecular partition functions Q

lnKeq=−

ε 0

RT

+ln

(

Qi

NA

)ξ

whereξis the appropriate stoichiometric coefficient andNAis Avogadro’s num-

ber. Note the correspondence between the second term in this equation and the

entropy change of reactionrS◦as it appears in the classical van’t Hoff and related

equations.

Sodium metal can be vaporized at moderately high temperatures. The vapor

exists as diatomic sodium molecules Na 2 (g) in equilibrium with atomic sodium

vapor Na(g).

Na 2 (g)→←2Na(g)

Maczek (1998) has applied the statistical thermodynamic equation forKeqto several

reactions, including the dissociation of Na 2 molecules in the gas phase at 1274 K. Not

all modes of motion apply in this case. For the product state, there is no partition func-

tionQvib(Na) orQrot(Na) because there can be no internuclear vibration or rotation

of atoms, which are essentially point masses. The reactant state consists of connected

masses separated in space for which vibration and rotationarequantum mechanically