E. A. M a s o n
Brown Universitv Providence, Rhode Island 02912 and R. B. Evans III Ook Ridge Notional Loborotory ~ a k ~ ~ i Tennessee d ~ e , 37830
I
I1
Graham's Laws:
of Gases in Motion Part I, Theory
The kinetic-moIecular theory of matter is central to much of chemistry, but there are remarkably few gas-kinetic experiments available for undergraduate instruction. Thus, most expositions are based strongly on the results of modern physics, in which the theoretical (and experimental) apparatus is often so elaborate that the instructor's presentation seems more akin to divine revelation than to a straightforward experimental science. Moreover, many of the textbook discussions of kinetic theory are seriously in error, especially where diffusion is involved. Those that are correct are usually rather superficial and never mention the interesting parts, those with real kinetic-theory content. I n a previous paper (1) some simple equilibrium experiments based on the work of Thomas Graham were presented as suitable for lecture demonstrations or elementary exercises for freshmen. I n this and a subsequent paper we present a simple exposition of kinetic theory and a description of two, simple, gaskinetic experiments (rate measurements) that are recommended for more advanced students and that illustrate interactions among several phenomena. We hope that the theoretical analysis and the experiments presented will encourage instructors to incorporate more of the kinetic theory of gases in undergraduate courses, where apt-but simple-illustrations of the molecular nature of matter are distinguished by their absence. Naturally, we take the subject appreciably beyond where Graham left it, a t least in the sense of quantitative mathematical description, but it was Graham who had the genius to actually recognize and point out the major underlying physical pbenomena. This paper presents the theoretical background, with particular reference to the following points (1) Three different aspects of ges transport (motion) as ernbodled in Graham's studies of diffusion in 1833 (8),and of effusionand transpiration in 1846 (3). (2) The proper way to describe situations where two or more of the types of transport occur together. This does not seem to be generally known. (3) The generalization to multicomponent mixtures. We believe that some of these results are new.
The emphasis is on the simplest possible explanations that are essentially correct without resort to the heavy mathematical artillery of the Chapman-Enskog theory of gases. I t is remarkable how far one can go in this way. Modes of G a s Motion
In the absence of conditions giving rise to turbulent flow, there are three main types of isothermal gas 358
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Journal o f Chemical Education
Simple Demonstrations
transport through tubes or porous media, as was clearly discussed by Graham in a final paper (4) in which he summarized and extended his earlier work ( I ) %fusion, now more commonly called free-molecule or Knudsen flow. Here the pressure is so low that collisions b e tween molecules can be ignored compared to collisions of molecules with the walls of the tube ar porous medium. (2) Transpi~alion,or laminar viseous flow, in which the gas acts as a continuum fluid driven by a pressure gradient. This is sometimes called convective or bulk flow. Here the pressure is high enough so that molecule-molecule collisions dominate over moleculewall collisions. ( 3 ) Diffusion, in which the different species of s. mixture move under the influence of composition gradients. This is still a continuum phenomenon in the sense that molecule-molecule collisions dominate over moleculewall collisions.
Corresponding to these three transport mechanisms are three transport coefficients: the Ihudsen diffusion coefficient, D,K (for species i), the viscosity coefficient, v , and the continuum or normal diffusion coefficient, p,, (for a binary mixture of species i and j). Similarly, there are three corresponding parameters characteristic of the solid medium through which the gas moves: the Ihudsen flow parameter, KO,the viscous flow parameter, Bo, and the porosity-tortuosity ratio, s/q., for continuum diffusion. For a medium of simple geometry, such as a long, circular capillary, these parameters are all simply related. But for a complex case, such as an actual porous medium, the relation is complicated, and the three parameters are usuaLly found from experiment rather than by calculation from an assumed geometry. Other types of transport also exist, such as thermal transpiration and thermal diffusion in a temperature gradient or surface diffusion of adsorbed molecules. We ignore these in our discussions, and we shall consider only isothermal phenomena as did Graham. The foregoing types of gas transport can also occur in combination. For instance, the combination of viscous flow and free-molecule flow leads to the phenomenon known as "viscous slip." This slipping of a gas over a solid surface was discovered experimentally in 1875 by Kundt and Warburg (5); it turns out to be of the nature of a special diffusion process. Another combination is that between viscous flow and continuum diffusion, which gives rise to the phenomenon known as "diffusive slip," first discussed in 1943 by ICramers and Kistemaker (6). Still another combination is that between free-molecule and continuum diffusion, which was studied intensively during World War I1 in connection with isotope separation (7). Finally, all three mechanisms are present in a flowing,
diffusiug gas mixt,ure in the t,ransition regime. Incidentally, t,hc coupling between flow and diffusion is much more commou than is generally realized, for almost all experimental arrangements involving diffusion also causc flow to occur. The above classification of gas transport mechanisms makes an importaut distinction between viscous (or bulk flow) transport and diffusive transport. This distinction holds only in the continuum regime; in the free-molecule regime, the molecules act independently, and all flow can be regarded simply as diffusion. The imp~rt~ance of the distinction lies in the fact that the two mechanisms contribute independently to the t,ot.al transport, which is t,hus simply a sum of diffusive and viscous flow terms with 110 extra terms due to coupling between the two mechanisms. For gases, this additivity can be shown to follow from the rigorous Chapman-Enskog kinetic theory to a very high degree of approximation (8-11), but the conclusion is quite a general one, valid for any isotropic system. I t depends only on the fact that the various flows are proportional to first derivatives of gradients (linear laws) and that quantities of different tensorial character do not couple in t,he linear approximation. This last result is sometimes known as Curie's theorem (12-14). The coupling between free-molecule and coutinuum diffusion is slightly more subtle, and requires some explicit use of kinetic-theory arguments. We now describe each of these simple transport. mechanisms in turn and then obtain the general result by coupling t.he three together.
two ends. I t is customary to write this in differential form as (for one species, i, of a mixture)
Effusion (Free-Molecule or Knudsen Flow)
where n = n, n,. Two limiting cases are particularly simple. In the first case, there is gas on both sides of the hole, so that the total density (or pressure) is uniform and dnldz = 0. Then eqn. (6) yields
=
umu
D
