Julian H. Gibbs Brown University Providence, Rhode Island 02912
I
I
Elementary Derivation of the Boltztnann Distribution Law
Consider an elementary collision process in which one or more "varticles" of a svstem undereo an upward transition from quantum state i, of energy El, to quantum state j, of energy E,, while an appropriate number of other "particles" simultaneously undergo a downward transition from state m , of energy E,, to state 1, of energy El.
-
At equilibrium the rates of the forward and baclcward Drocesses must be eoual
According to the principle of microscopic reversibility
Therefore, solving eqn. (4) for Nj/Nr, we have Let the number of particles undergoing the upward transition in this elementary process be A and the number undergoing the downward transition be B, so that the process may be written in a fashion analogous to that used for a chemical reaction as
N ~ / N= ~( N , / N , ) ~ ~ ~ ~ ~ (6) ~ ~
That this is a form of the Boltzmann distribution law is easily seen. Define a real number B by
Ai+Bm-Aj+B1
For this elementary process to conserve energy, it is necessary that A(Ej
- Ei) B
=
=
-B(Ez - E,)
as is obviously always possible for one pair of levels. The content of the Boltzmann distribution law is that, if this choice of p is made for one pair of levels (here 1 and m), then, at equilibrium, the ratio of populations of any other pair of levels can be written in the same form with the same value of p i f the appropriate AE i s used. To see that this has been proved here for levels i and j, one need only insert eqn. (7) for (Nm/NI)into eqn. (6) for NJN,
AAEcf/AAEt,
where
a.. - E . - E, -3
9
and AE,,
=
Em - E,
Thus, by the same sorts of arguments as are used in chemical kinetics and with use of eqn. (I), the rate of this elementary process is given by
where Nt is the concentration, or density, of particles in level i, N, is the corresponding concentration in level m , and W +j is a proportionality constant characm-1
teristic of the transition. The rate of the inverse process is correspondingly rote. . = W . . u j A N , A ( A E ~ A E Z - )
,1-*-7"
542
/
I-,
1-m
Journal of Chemical Educafion
(3)
Since the choice of levels i and j (and for that matter, 1 and m ) was completely arbitarary, the proof obviously can be extended to any other pair of levels, r and s, since r and s can be related back to 1 and m either directly through processes involving r and s and 1 and m or indirectly through processes relating r and s to levels i and j which have already been related to I and m. The identification of p with l/kT, where k is Boltzmann's constant and T is the thermodynamic temperature, can be effected by the usual arguments.
