Reino W. Hakala Howard University
Washington, D. C.
A Quanta1 Evaluation of the Exponent in B O ~ Z ~ U N NDistribution 'S Law
Various ways of evaluating the parameter bin Boltzmann's distribution law, N i = Ag, exp (-h%)
have been proposed. The usual method is to find the average energy of an ideal gas with the aid of Boltzmann's distribution law and then to compare the result with that obtained in the elementary kinetic theory of gases. In this approach, it is necessary to assume that the velocities (or momenta) of the molecules vary practically continuously so that the sums that are involved may be approximated by integrals. It is also customary to provide an elementary quantal proof of the near-continuum of translational energy (for example, in E. A. Guggenheim's masterful little treatise, "Boltzmann's Distribution Law," Interscience Publishers, Inc., New York, 1955, pp. 3 4 ) . As will be shown presently, it is possible to make use of this proof to evaluate b. The derivation of the value of b that results is somewhat simpler than, although equivalent to, the usual purely classical one based on velocities. The quantal proof has the pedagogical advantage of being a nat.ura1 appendage of, rather than an artificial accretion to, the quantal proof that translational energy is practically continuous. This quanta! evaluation of b now follows: According to the quantal "particle-in-a-box" model (which is discussed in a very clear and succinct manner by Guggenheim, loc. cit.), the translational kinetic energy of a single gaseous molecule is given by
where h is:Planck's constant, m is the mass of the molecule, nz, n,, and n, are translational quantum numbers referring to the x-, y-, and z-components of translational motion, and l,, l,, and 1, are the lengths of the edges of the,box parallel to the x-, y-, and z-axes. The translational kinetic energy is more conveniently expressed in the abbreviated form Let us now consider a monatomic ideal gas. Since it is ideal, it possesses only kinetic energy. Since it is monatomic, the only kind of kinetic energy it possesses is translational. Consequently, the average energy, per molecule, of a monatomic ideal gas is given by where em is given as above. If, in a gas consisting of a total of N molecules, No of the molecules possess the "w-component" of energy eo, N1 possess el, etc., then the average w-component of the energy is
Making use of the Boltzmann distribution law, we then find that b =
C Aexp (-he
