Wilke–Chang Liquid Diffusivity

Liquid diffusivities are troublesome to measure and there are far more solute-solvent pairs in the world than anyone has measured. Charles Wilke and Pin Chang published their correlation in 1955 to fill the gap, and seventy years later it remains the default estimate when nothing better exists. It rests on the Stokes–Einstein picture — a solute molecule dragged through a continuum by Brownian motion, with diffusion proportional to temperature and inversely proportional to viscosity — modified by two empirical adjustments where that picture fails.

The first adjustment is the association parameter $\phi$, and it is the interesting one. Stokes–Einstein treats the solvent as a structureless continuum, but water is nothing of the kind: hydrogen bonding makes it behave as though its molecules were larger and more organised than their formula weight suggests. Wilke and Chang absorbed that into a fitted factor, 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for unassociated solvents such as benzene and hexane. It is frankly a fudge factor, but it is a physically motivated one and it works. The second adjustment is the exponent 0.6 on the solute's molar volume, where the hydrodynamic derivation would predict one third; the difference is what the correlation had to do to fit real data across a range of molecular shapes.

Almost every error made with this equation comes from mixing up which molecule owns which property. The molar mass, the viscosity and the association parameter all belong to the SOLVENT. The molar volume belongs to the SOLUTE. Getting that backwards produces a plausible number with no meaning, and nothing in the arithmetic complains. The molar volume is also a specific quantity — the Le Bas additive volume at the normal boiling point, built up from tabulated atomic contributions — and it is neither the molar volume at room temperature nor the critical volume. Those differ by tens of percent, and the 0.6 power passes most of that straight through.

Hold the result loosely and inside its limits. Wilke and Chang reported an average error near 10 percent against their own data set, and individual pairs can do considerably worse. The correlation applies at INFINITE DILUTION; at any real concentration the diffusivity depends on composition through the thermodynamic activity, and in strongly non-ideal mixtures it can differ severalfold from this value. It should not be used for very large solutes, where a hydrodynamic treatment is better, nor when water is the solute rather than the solvent. And when carrying a diffusivity from one temperature to another, do not use the linear $T$ on top — the viscosity underneath moves far more steeply. The grouping that stays roughly constant is $D\mu/T$, and that is the right bridge between temperatures.