Maxwell Maxwell-Stefan Diffusion Maxwell Legacy Concepts
In a nutshell

Diffusion is a hugely important process in many practical situations. Examples include the rate of spread of polutants in air or in water, the mixing of gases in the atmosphere, the rate solids dissolve in unstirred liquids, leakage from a gas container, passage through membrane walls, the spread of scents, and so on. Diffusion is of particular interest to the applied chemist, physicist or engineer but is a phenomenon that is not immediately appealing to theoreticians due to the complex situations that quickly arise. Looking at the problems Maxwell tackled, many of them have strong practical applications. It's not surprising that in 1866 when Maxwell developed his ideas on the molecular basis of the behaviour of gases, he should include a detailed discussion of diffusion in multi-component systems. When Joseph Stefan (better known for his radiation work) came to look at much the same problem five years later, he commented that 'the study of Maxwell's treatise is not easy'.

Maxwell tailored his approach bearing in mind the experimental work of the Glaswegian chemist Thomas Graham. Graham had investigated diffusion in different circumstances. In one, a heavy gas was introduced at the base of a vertical column and a light gas above it at the same temperature and pressure. Everyone knows that if less dense oil is floated on water it simply remains there, forming two distinct layers. Not so in the case of gases. The diffusion of the heavy gas into the top layer of light gas at various time intervals was measured by Graham. In another experiment, different gases diffused into air through a narrow hole. This has morphed into the 'classic' two-bulb experiment, where bulbs containing different gases are connected by a thin tube that can be opened for a given amount of time before the contents of either bulb are determined. This highlights that diffusion is always a 2-way process. Maxwell examined how gas would diffuse in different circumstances, including such detail as the viscosity of the gases involved. He derived values for the diffusion constants he introduced, based on Graham's data.

So what is Maxwell-Stefan diffusion? Surprisingly to the modern mind, one of Maxwell's missions in the 1860s was to promote the reality of molecules. The concept of atoms at least had filtered down centuries earlier from metaphysics and philosophy. Many people accepted the concept of the finite divisibility of matter but found it much harder to accept that molecules were not simply an idea but were really there in front of them, occupying a small but definite amount of space, moving around, colliding and rebounding all below the level of matter revealed by the microscope. That was hard to swallow. To digress slightly, the chemists building on the earlier 19th century work of Dalton and the late 1860s publication of Dmitri Mendeleev's periodic table were working on a similar premise that atoms were the fundamental units in forming compounds. One can well argue that the coming together of the chemists' convictions and the physicists' approach to molecular dynamics gave rise to the modern discipline of physical chemistry in the 1880s.

Returning to diffusion and focusing on the molecules, what were they? Centres of force or impenetrable balls? Spherical or asymmetric? Smooth or rough? Maxwell's approach to gaseous properties including diffusion was to model the molecules of gaseous mixtures and work out from the mechanics of their interactions the properties of the model. As he put it: 'One class of phenomena are those which are due to the motion of agitation by which the molecules of a liquid or gas are continually working their way from one place to another, and continually changing their course, like people hustled in a crowd.' He compared his results with experiments, which were in fact thin on the ground in the 1860s. A good agreement was taken as another piece of evidence that molecules were a reality. By 1873, Maxwell could say to the British Association that diffusion experiments form 'one of the most convincing proofs of the motion of molecules'. Later in the 1870s, Maxwell reworked his ideas, starting from more general concepts in an attempt to make his results less dependent on particular assumptions.

Maxwell's approach was therefore a 'first principles' approach that resulted in expressions for diffusion that depended on the assumptions made. As he pointed out, molecules in a gas at room temperature travel between collisions faster than a speeding bullet. Yet we know that diffusion takes place slower than walking pace, so getting the reduction factor correct is not straightforward. For example in his earlier work in his famous paper of 1860 Illustrations of the Dynamical Theory of Gases he assumed hard spherical molecules and found that the diffusion constant varied with temperature T as T3/2 whereas in his 1866 paper considering molecules whose repulsion varied as the inverse 5th power from their centre, the diffusion varied as T2. Maxwell himself wrote a summary of his approach to diffusion in the article of that title running to about 6000 words in the 9th edition of the Encyclopaedia Britannica (vol. VII, 1877). In it he discusses his assumptions, some of the formulae reached and the supporting evidence of experiments, notably those of Loschmidt in the 1870s.

Fishy smellDiffusion - not a fishy story but an important process in the spread of compounds in many circumstances.
Technical detail

One of the reasons that Stefan had difficulty with Maxwell's 1866 paper is that Maxwell only gets to the topic of diffusion once he has discussed several other properties of gases. By then he is well into his stride in developing his notation and his approach. An easier place to start is his later Encyclopaedia Britannica article mentioned above. In it he begins by noting that the diffusion of a particular gas across a small volume depends on the difference in pressure of the particular gas on the opposite sides of the volume. Motion is resisted by the other gases present that are moving with a different speed. This resistance is proportional to the velocity of the first gas relative to the second, to the product of their densities, to a coefficient that depends on the nature of the gases and to temperature. This leads Maxwell to his general equation of motion for the diffusing component and after some time to the definition of a diffusion constant D.

Maxwell mentions in passing in this article some experimental diffusion results of Adolf Fick with liquids but does not refer to Fick's formulation of the diffusion problem. Fick formulated an empirical rule that diffusion is driven by the concentration gradient of each species in a system. Fick's approach has been widely applied but does not, like Maxwell's, analyse the detailed motion of the constituents. As a result, Fick's approach does not include the full complexity of diffusion processes. In particular, Fick's diffusion constant D is not the same as Maxwell's, which is a warning that tabulations of diffusion data must be interpreted within the framework in which they were analysed. A review article by Krishna and Wesselingh emphasises the advantages of the Maxwell-Stefan approach in a wide range of circumstances. It also, in passing, confirms that the subject is one where attention to fine detail is necessary if a theoretical approach is going to be useful in practical circumstances. I'll leave those who want to take up the technicalities to peruse the literature.

References: Maxwell, J. C. 1866, On the dynamical theory of gases. Phil. Trans. Roy. Soc. 157, 49-88.

R. Krishna and J. A. Wesselingh, 1997, The Maxwell-Stefan approach to mass transfer, Chemical Engineering Science, Vol. 52, No. 6, pp. 861-911.

JSR 2016