If the molecules of A and B are kept in separate containers, each container will be in a state of equilibrium with the populations of A and B given by their separate Boltzmann distributions. This is shown in Fig 33.2(a). The partition functions of the systems A and B are given by qA and qB. Fig 33.2(b) shows the combined system and the population of the levels of the combined system. The population of the i th level of the combined system is given by
i = (N / q) e
(33.13)
i = NPi = N e
(33.14)
Where q is the partition function of the combined system and Ei is the energy of the i th level of the combined system. In the combined system, the total population of A molecules at equilibrium is given by summing ni for A molecules. Similarly for B. We thus have,
NA =
i = N / q
e
= (N /q) qA
(33.15)
NB =
= N /q
e
(33.16)
The prime in eq (33.16) indicates that the levels (Ei )'B are now measured from the lowest level of the combined states of A and B, which is (E0) A. Expressing E' i in terms of Ei, we have,
i = (N / q) e
e
= (N /q) q
= N /q
e 
E
e
= N (q 
