NOVEMBER, 1953
A PROBLEM ON THE ENERGIES OF MOLECULES EMIL J . SLOWINSKI, JR. University of Connecticut, Storm, Connecticut
INSTUDIES in statistical mechanics and in the kinetic theory of gases much interest is centered on the energies possessed by the molecules in a gas, since a knowledge of such energies and the laws governing them is of irnportance not only of itself but in deriving information as t o other properties of the gas. One quantity which one finds can be determined from the relations given is the most probable energy of the molecules-that is, that energy a t which the largest fraction of the molecules will be found. The approach to the problem can be made in several slightly diierent ways, which lead, strangely enough, t o different results and thus to an interesting and rather anomalous s i t u e tion. Let us consider three possible attacks. The first might well be through the use of t.he simple Boltzmann function, which states that
where N, is the number of molecules in a state of energy
E, No the number in a state of energy zero, and kT is the familiar product of the Boltzmann constant and the absolute temperature. This expression quickly shows, since N,/No is a fraction less than one for all energies greater than zero, that the energy a t which there will be the most molecules, that is, the most probable energy, will be zero. Here then, E, = 0. However, one might take another approach and use the Maxwell-Boltzmann function for molecular speeds, which is given by the relation: f(u)dv = 4 d A%-~""='= dv
wheref(u)dv is the fraction of the molecules having speed in the small interval between u and u av, rn is the mass of a molecule, and A is a constant equal to (rn/Z~kT)"~. Determining the most probable speed from this expression merely involves maximizing f(u) with respect to v , and this, of course, is most easily done by diierentiating f(u) with respect to v, and setting the result equal t o zero. Doing this, we get:
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Clearing the expression of the common factors and multiplying through by (2E/m)'" we see that: which, on removing the common factor, yields uz = 2kT/m, where v is the most probable speed. This value for the speed we might use to determine the most probable energy, since the energy of motion is simply equal to '/2mvz. From the expression for v2 we see then that in this case E, = kT. There is still another way that the problen might be approached by the enterprising. Using the relation E = '/2muZand its differential, we can readily convert f(u)dv to f(E)dE, where the latter expression equals the fraction of molecules having energies in the small interval between E and & dE. This turns out to be:
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f(E)dE = 4rA3 (2E/m)'l. ( l / m ) e - E l k T d ~
Maximizing f ( E ) as we did f(v) we get:
l / m = 2E/mkT,
or that E, = '/*kT
So, by three somewhat different methods, we find that there are three somewhat different values for the most probable energy of a molecule. One suspects that somewhere in a t least two of the methods there is some sort of error. The question is, which of the results is the one that is acceptable, and where do the other methods fail to treat properly of the problem?' (It is obvious that here we consider only energy of translational motion. This sacrifices a bit of generality, but is of course necessary in view of the fact that the speed function gives information only about such energy.) A sequel to this paper will be published lrtter.
