c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come
110 A STATISTICAL APPROACH TO THERMODYNAMICS· · · · · ·· · · · ·A B
FIGURE 8.3 A degenerate two-level equilibrium.Even if the two levels of state B are not exactly at the same energy, their capacity
to accommodate B molecules is greater than it would be if there were only one state
as in the original equilibrium. Suppose now that both state A and state B consist
of many levels. The distribution of molecules in state A will be controlled by one
Boltzmann factor and the distribution of molecules in state B will be controlled by
another Boltzmann factor, but the distribution between A and B will also be controlled
by a Boltzmann factor. We now have three factors controlling the equilibrium: the
distribution within A, the distribution within B, and the distribution between A and
B. The distribution within the levels of state A isnAi
nA 0=e−(εAi−εA^0 )/kBTThe distribution within state B isnBi
nB 0=e−(εBi−εB^0 )/kBTThe distribution between the lowest levels in states A and B isnB 0
nA 0=e−(εB^0 −εA^0 )/kBTThe total number of molecules in state A is the summation over all the states in AnA=∑
inAi=∑
inA 0 e−(εAi−εA^0 )/kBT· · · · · ·· · · · ·A B
FIGURE 8.4 A degenerate two-level equilibrium. The two energies at the B level are not
exactly the same.