A Thermodynamic Interpretation of the “Excluded-Volume Effect” in Coupled Diffusion

Abstract

Strongly coupled diffusion has been reported for fluxes of solutes that differ significantly in molecular size. According to the excluded-volume model, a solute increases the effective concentrations of other solutes by reducing the volume of solution they can occupy. The flux of solute 2 produced by the gradient ∇c1 in the concentration of solute 1 is interpreted as the ordinary diffusion of solute 2 down its effective concentration gradient. Cross-diffusion coefficient D21 = D22c2V1,eff/(1 − c1V1,eff)2 is predicted, where V1,effis the effective molar volume of solute 1. This model does not account for countercurrent coupled diffusion (D21 < 0) and is found to be inconsistent with the Onsager reciprocal relation (ORR). A thermodynamic model of coupled transport is developed by approximating the flux of solute i as the product of its concentration, mobility, and chemical potential gradient driving force (−∇μi), which gives D21 = D22c2(V1 − V0)/[1 − c1(V1 − V0)], where Vi is the partial molar volume of component i and the solvent is component 0. For dilute solutions with V1 ≫ V0, the thermodynamic prediction D21 ≈ c2V1D22 and the excluded-volume prediction D21 ≈ c2V1,effD22 are qualitatively similar, but the thermodynamic model does not require the assumption of effective concentrations or effective volumes, provides a physical explanation for coupled diffusion, and is consistent with the ORR. Moreover, because ∂μ2/∂c1 is proportional to V1 − V0, the thermodynamic model suggests that a concentration gradient in solute 1 can drive co-current or counter-current flows of solute 2, depending on the relative volumes of solute 1 and solvent 0. These features are illustrated by comparing measured and predicted Dik coefficients for solutions of n-octane(1) + n-hexadecane(2) in n-dodecane(0) at nine different compositions at 25 °C.