Test of the Onsager relation for ideal gas transport in membranes

Abstract

The mathematical description of gas transport across inert membranes permeable to all components is considered from two points of view: a phenomenological approach through non-equilibrium thermodynamics, and a kinetic-theory approach through the dusty-gas model. The results are shown to be equivalent in local (differential) forms. Extension of the equations to overall or finite-difference linear forms is made by use of arithmetic means for concentrations and pressure inside the membrane. Comparison with the accurately integrated kinetic-theory equations shows that the results are valid only over a relatively narrow regime that becomes rapidly constricted as either or both of the driving forces departs significantly from zero. Use of logarithmic mean concentrations that preserve the overall entropy production yields results of even more restricted validity. Data for the system He + Ar in a graphite membrane are compared with the theoretical results, and used to test the overall Onsager relation for the arithmetic-mean equations. The results suggest that “second-order” coefficients in flux-force equations may be artifacts of the integration of linear differential equation over finite differences. It is concluded that no completely accurate integration of local transport equations is possible without detailed information on the dependence of the local coefficients on state variables, and that the attempt to bypass the difficulty by assuming an overall form that correctly represents the entropy production gives flux equations inferior to those obtained using simple arithmetic means.