research summaries for teachers
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The University of Wirconrin Modison, Wkconsin
Transport Phenomena in Gases
The phenomena of diffusion, viscosity, and thermal conductivity are associated with the transport of mass, momentum, and energy, respectively, through a material, and hence are referred to, collectively, as transport phenomena. In the gas phase, the transport is due primarily to the flux of molecules from one region of the gas to another. In more dense systems, the collisional transfer mechanism in which momentum and energy are transferred directly by collision, in bucket brigade fashion, is important. I n t,he liquid phase this is the primary mechanism. I n the gas phase, the transport of the three properties is due to the tendency for the random Brownian motion of the molecules to homogenize the system and establish equilibrium. The "hot" molecules of one region tend to mixwith the "cold" molecules of another. The effect of collisions is to slow down this process by limiting the distance through which a molecule may move without encountering another. If the molecules are idealized as rigid spheres of diameter, c, the mean free path between collisions is 1 = (2'%soz)-', where n is the density of molecules. That is, the larger the molecule the shorter the mean dist,ance between collisions and hence the less easily the three properties may he transported through the gas. Elementary ideas of kinetic theory lead directly to simple expressions for the transport coefficients of a gas.of rigid spheres. For example, the coefficient of viscosity is q = (5/16su2). (?rmkT)'I2. Expressions such as this have long been used to estimate the size of molecules. Of course, real molecules interact according to a soft potential. They attract each other at large separation distances and the molecular cores lead to repulsion a t shorter distances. I t is often assumed that the intermolecular potential depends only on the distance between the molecules, i.e., that the molecules are spherically symmetric. This is true, of course, for the rare gas atoms but for few, if any, real molecules. Nevertheless, this approximation has been highly useful in treating many simple non-polar systems. The kinetic theory treatment of transport phenomena in a gas of spherical molecules, due largely to Chapman and Enskog, is well developed and leads to expressions for the transport coefficients, which are similar to those for the rigid sphere model, except that the cross section is modified 556 / Journal of Chemical Education
theoretical chemistry by temperature dependent factors. These temperature dependent factors are the well known omega integrals. These integrals depend on the form of the intermolecular potential through the dynamics of the collisions. The evaluation of these integrals is a lengthy calculation, but tabulations for a number of realistic potential models, obtained through the use of high speed computers, are now available. With these tabulations, one may use measured values of the transport coefficientsof a gas to obtain information about the intermolecular potential. Much of our quantitative information about the potentials of interactions between simple non-polar molecules has been obtained in this manner. At low temperatures the expressions described above are modified by quantum effects. The analogous quantum mechanical theory, however, is well developed. The resulting expressions are similar to the classical expressions. The essential difference is the manner in which the omega integrals are evaluated. I n the quantum case, these quantities depend upon an additional parameter involving Planck's constant and the mass of a molecule. Quantum effects are interesting, hut are important only for the very light gases such as Hz and DI, and for these only a t very low temperatures. ?!Iost real molecules, of course, are not spherically symmetric. Sometime ago, Wang-Chang, Uhlenbeck and de Boer developed a kinetic theory of molecules with internal degrees of freedom on the basis of a set of modified Boltzmann equations for a set of distribution functions associated with each of the internal states of a molecule. These modified Boltzmann equations involve the differential scattering cross sections for the elastic and inelastic collisions. The development involved a perturbation solution of this set of equations similar to that used by Chapman and Enskog. This perturbation solution leads to expressions for the transport coefficients in terms of the scattering cross sections. The problem is thus reduced to that of evaluating these cross sections. Somewhat later, we developed a classical kinetic theory of molecules which may be idealized as rigid nonspherical bodies. Unfortunately, this model, like the rigid sphere model, is not a very realistic representation of real molecules. Nevertheless, the treatment of rigid ovaloids led to interesting new results. For example, the conservation of angular momentum in the molecular collisions led to the appearance of a new macroscopic
conservation equation. We are only now beginning to understand the physical significance of this equation and we were puzzled by the lack of appearance of a similar equation in the earlier development of WangChang, Uhlenbeck, and de Boer. This anomoly was removed and explained by the later more rigorous independent developments of Snider and Waldmann. I n these quantum mechanical developments, the effect of the degeneracy of the internal quantum states of the molecules is considered. This leads to a macroscopic equation associated with the conservation of angular momentum. I n certain limiting cases, the results of these later developments are similar to those of WangChang, Uhlenbeck, and de Boer, provided their cross sections are reinterpreted as degeneracy averaged cross sections. The study of transport phenomena in systems involving non-spherical molecules is now developing rapidly. On the one hand, experimental studies of the Senftleben effect provide direct evidence of the effect of nonspherical contributions to the potential. On the other hand, the theoretical study of molecular scattering cross sections is developing to a point where it may soon be possible to predict the transport coefficients. The Senftlehen effect is the effect of a magnetic field on the energy flux in a gas or more generally the effect of either a magnetic or electric field on any of the fluxes. I n either case, the field partially aligns the molecules. This partial alignment leads to preferential relative
orientations of the colliding molecules and therefore affects the average collision cross section. The theoretical interpretation of the experimental results will lead directly to information about the non-spherical nature of intermolecular potentials. At moderate densities the collisional transfer mechanism makes significant contributions to the transport coefficients. I n addition, three body collision effects and the effects of correlated two body collisions become important. Both of these effects contribute to the first density corrections to the transport coefficients. Some success has been obtained in the predictions of these. quantities. Our expressions are valid at high temperature where metastable bound pairs of molecules play a minor role. At low temperatures these bound pairs play a primary role and the gas may be considered, approximately, as a mixture of monomers and dimers. Recently the existence of density series expansions of the transport coefficients has been discussed extensively. I t has, apparently, been shown that such expansions are not possible. The first order corrections discussed above exist, but the second order corrections are infinite. The source of this surprising result is a term of the form n2 In n. This term arises from the effect of long range correlated four body collisions. It would appear to be extremely difficult to verify the existence of this term experimentally. I find it difficult to accept its existence and wonder whether its appearance may not indicate fundamental errors in the theory.
Volume 45, Number 9, September 1968
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