Concise Physical Chemistry

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

DEGENERACY AND EQUILIBRIUM 111

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A B

FIGURE 8.5 A two-level equilibrium with many A and many B levels.

and the total number in state B is given by a similar summation

nB=

∑

i

nBi=

∑

i

nB 0 e−(εBi−εB^0 )/kBT

The equilibrium constant can be written as the quotient of the number of molecules

in B over the number of molecules in A:

Keq=

nB

nA

=

∑

i

nB 0 e−(εBi−εB^0 )/kBT

∑

i

nA 0 e−(εAi−εA^0 )/kT

Factoring outnB 0 /nA 0 =e−(εB^0 −εA^0 )/kBT, which is common to each term in the sums,

we obtain

Keq=

nB 0

nA 0

∑

i

e−(εBi−εB^0 )/kBT

∑

i

e−(εAi−εA^0 )/kBT

=e−(εB^0 −εA^0 )/kBT

∑

i

e−(εBi−εB^0 )/kBT

∑

i

e−(εAi−εA^0 )/kBT

which is to say

Keq=e−(εB^0 −εA^0 )/kBT

QB

QA

whereQAandQBare called thepartition functionsfor the A and B states. Please do

not confuse the partition functionQwith the concentration quotientsQorQ.