A comprehensive thermodynamic theory of the cross-effect in isothermal diffusion (“Hertz effect”) in a multicomponent gas, liquid, or solid

Abstract

The cross-effect between diffusion fluxes in a nonuniform, multicomponent mixture is the cause of diffusion of a component against its own concentration gradient. Gustav Hertz was the first to use this cross-effect for the separation of isotopes. The calculation of the magnitude of the cross-effect, which we call Hertz effect, is possible if all coefficients of the diffusion tensor of the mixture are known. For dilute gases, kinetic gas theory enables the calculation of the diffusion tensor, but calculation methods for dense gases, liquids, and solids require input of measured binary or tracer diffusion coefficients. This paper presents for the first time a calculation method for the magnitude of the Hertz effect without using input of any measured diffusion coefficient. Input is required from the equation-of-state of the mixture and from kinetic gas theory. The theory is based on the principle of a maximum number of possible microstates in the stationary, nonequilibrium state. The theory has been compared with all measured diffusion data of mixtures that can be represented by our phase-behavior package. There is fair to good agreement between measurement and theory, but the Hertz effect in these mixtures is small. A test against diffusion data of highly nonideal mixtures is therefore recommended.