0. Rice and C. J. G. Raw Soint Louis University st. Louis. Missouri 63156
The Assumption of Elastic Collisions in Elementary Gas Kinetic Theory
Among the assumptions usually made in the elementary kinetic theorv derivation of the ideal eas are: (1) " eauation . ., no intermolecular forces, and (2) elastic collisions between gas molecules and between gas molecules and the walls of the container. The consequences of the first of these assumptions are often discussed in some detail in connection with the corrections which must be made to account for the pressure-volume-temperature relationships of real gases. In most elementary tefibooks, however, nothing appears which might alert the student to the fact that the second assumption concerning elastic molecular collisions must be discarded for an improved model of a gas. As a result, the impression is left that molecular collisons must be elastic. An analogy is often madel between the collisions of gas molecules and the collisions of a rubber ball with the floor. Since the latter are not elastic, the bouncing ball gradually slows down and finally comes to rest. It is then argued that gas molecules would also slow down if molecular collisions were inelastic, and eventually the gas pressure would drop to zero. This very neat explanation of the constancy of the pressure of a gas a t equilibrium is so convincing that the student cannot conceive that any other possibility exists. Thus, it has been our experience that many chemists a t an advanced level of training are shocked to find the assumption of elastic molecular collisions criticized. In an excellent discussion of gas phase energy transfer processes in this Journal, R. J. Campbell2 gives some values of the average number of collisions required for a particular energy transfer process to occur (the "collision number") for various gases. Typical collision numbers for the interchange of translational and rotational energy a t room temperature range from 4-300. Typical collision numbers for the interchange of translational and vibrational energy vary from a few hundred to several million. Mason and Monchick3 and many other workers have considered the effect of inelastic collisions on the properties of gases. While inelastic collisions have an important effect on the study of such problems in gases as heat conductivity, absorption and dispersion of sound waves, and electric discharges, they do not play an explicit role in other problems such as viscosity, diffusion, or the equation of state. The reason for this in the latter case (which is of main interest in this paper) is that, in a gas a t equi-
librium, the principle of microscopic reversihility ensures the persistence of a Maxwell-Boltzmann distrihution of the molecular velocities. Here lies the correct explanation of why the gas pressure remains constant notwithstanding the occurrence of inelastic collisions. For every inelastic collision in which the translational energies of the pair of colliding molecules are decreased an opposite inelastic collision occurs in which the translational energies increase by the same amount. Similarly, in regard to the assumption of elastic collisions of the gas molecules with the walls of the container, a number of authors4 have pointed out that the assumption is not necessary (and is certainly not correct). The number of molecules leaving the wall with a given velocity in a certain direction must he the same as if there were "mirror-like" reflection (elastic collision), simply because the aas is in eauilibrium with no one direction referred over'any other.- here is evidence that only a friction of the molecules striking the wall are reflected elastically, others are temporarily attached to the wall and later emitted with velocities unrelated to those which they had before collision with the wall. But, if the temperature of the wall and gas are the same, then, on the auerage ouer a large number of collisions, gas molecules leave the wall with the same distrihution of speeds they had on approaching it. In summary, while it may not he necessary or wise to present a sophisticated discussion of inelastic collisions in the freshman chemistry course, i t is nevertheless important that the student (and teacher) know that the assumption of elastic collisions between gas.molecules must be abandoned when a more advanced study of gases is undertaken. In the elementary physical chemistry course, however, the ideas presented in this paper should be introduced hecause of the many important processes whose study requires an explicit knowledge of inelastic collisions. See for example, Murphy, D. B., and Rousseau, V., "Foundations of College Chemistry," The Ronald Press Co., New York,
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Campbell, R. J., J. CHEM. EDUC., 45,156 (1968). 3Mason, E. A., and Monchiek, L., 9th International Symposium an Combustion, Academic Press, Inc., New York, 1963, p. . .. 713.
'See for example, Kauzmann, W., "Kinetic Theory of Gases," W.A. Benjamin, Inc., New York, 1966, p. 56.
Volume51. Number 2, February 7974 '/ 139
