Velocity and Energy Distributions in Gases B. A. Morrow and D. F. Tessier University of Ottawa, Onawa, Ontario, Canada KIN 984 Physical Chemistry textbooks generally show plots of the velocity1 distribution function [F(u)] which is derived from the kinetic theory of ideal gases (see also Fig. A). The velocity a t the maximum of this curve is easily evaluated by solving d[F(u)]ldu = 0 to yield the most probable velocity (up) as
where k is the Boltzmann constant, T the temperature, and m the molecular mass. Many books also derive the kinetic energy distribution function (Fig. B), F ( E ) , and one can similarly show that the most probable energy is
We have found that students are frequently puzzled to find that if E,, is equated to the translational kinetic energy (l/2mu2),then the velocity at the most probable energy is
. .
a result which differs by 4 from the most probable velocity itself. The purpose of the present paper is to show how this apparent inconsistency arises. The answer lies in the mathematical nature of the problem. Although we may be considering -loz3 molecules, it is not reasonable to ask how many molecules will have an exact velocity since exactness implies knowledge to an infinite number of significant figures. Instead, one seeks an expression which will give the fraction of molecules (WJN, where N is the total number of molecules) which have a velocity within a specific range, say from u to u du. The desired relationship is W J N = F(u)du whereF(u), the velocity distribution function, (see any Physical Chemistry text for the derivation) is given by
+
Distribution function olots for H" at 300 K. A. 8v)versus v-B. Am versus E: C.
"7".
By a similar argument, the fraction of molecules ( W E I N ) which have kinetic energies from E to E + dE is F(E)dE, where F ( E ) is the energy distribution function. Because E = mu2/2, the latter can be derived from eqn. (4) by making the substitution u = (ZElm)ll%nd du = (2mE)-ll2dE to yield ,
,
Plots of F(u) versus u and of F(E) versus E for Hz at 300 K are shown in Figures A and B respectively. The area under these curves at a given u or E and of width du or dE of course gives dN,,/N or W E I N , respectively. However, because d E = mudu, a volume element in energy space is not proportional to a volume element in velocity space. Therefore, whereas an area at u of width du under curve A correctly gives the fraction of molecules which have velocities in this interval, the fraction of molecules which have energies in this same interval is a function of u and depends on the location of the interval. By analogy, the volume of water collected at various locations during a rainstorm in a pail of fixed dimensions will reflect the fraction of the total number of drops falling at those locations. But, if the radius of the pail also varies as afunction of location in the storm, a modified interpretation must be ascribed to the data. The reason why u,, and u at Ep differ by 4 can be seen more clearly if we express F ( E ) as a function of velocity by
substituting mu212 for E in (5) to yield;
(6)
Although the numerical value of F E ( u )is obviously the same as that of F(E), FE(v)is pot a distribution function2because dEldu is not a constant. That is, if PE(U)is plotted against velocity (Fig. C ) , then an area bounded by an interval du corresponds to
However, as long as du and dl3 are infinitesimal, we can say that such an area re~resentsthe fraction of molecules ( d N r / N ) having energies i n the interval dElmu = mudulmu= du. The fraction of molecules which have uelocities in this same interval du and at the same velocity u differs by a factor of mu because m u F ~ ( u ) d u= F(u)du. Therefore, whereas the maximum in curve C corresponds to the velocity at the most probable energy (eqn. (311, the most probable velocity itself is necessarily greater when FE(u)is multiplied by the linearly increasing function mu.
' Strictly we mean speed, not velocity, smce we are not concerned
with direction. The FE(v)function is also not normalized.
Volume 59
Number 3
March 1982
193
