Arnold diffusion cell

The diffusion coefficient for a gas can be experimentally measured in an Arnold diffusion cell. The device is shown in Figure 3.6 consisting of a narrow tube partially filled with pure liquid A. The system is maintained under constant pressure and gas B flows across the open end of the tube. Component A vaporizes and diffuses into the gas phase, hence the rate of vaporization can be physically measured. Develop a general steady-state expression to describe the diffusion of one gas through a second stagnant gas. Assume that the gas has negligible solubility in liquid A and is also chemically inert in A. 

Whittam, R. (1964). Transport and Diffusion in Red Blood Cells. Edward Arnold, London. 

As it has turned out that consistency in the mean does not hold in general, several people have presented a proof of the fact that the stochastic model of a certain simple special reaction tends to the corresponding deterministic model in the thermodynamic limit. This expression means that the number of particles and the volume of the vessel tend to infinity at the same time and in such a way that the concentration of the individual components (i.e. the ratio of the number and volume) tends to a constant and the two models will be close to each other. In addition to this the fluctuation around the deterministic value is normally distributed as has been shown in a special case by Delbriick (later head of the famous phage group) almost fifty years ago (Delbriick, 1940). To put it into present-day mathematical terms the law of the large numbers, the central limit theorem, and the invariance principle all hold. These statements have been proved for a large class of reactions for those with conservative, reversible mechanisms. Kurtz used the combinatorial model, and the same model was used by L. Arnold (Arnold, 1980) when he generalised the results for the cell model of reactions with diffusion. 

The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, i) of the deterministic model a law of large numbers and a central limit theorem hold (Arnold Theodoso-pulu, 1980 Arnold 1980). 

Mathias M, Roth J, Flemimg J and Lehnert W (2003) Diffusion media materials and characterization. Chapter 46, Handbook of Fuel Cells - Fundamentals, Technology and Applications, edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm. Volume 3 Fuel Cell Technology and Applications, John Wiley Sons, Ltd. 

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