The surface chemical potential from a surface equation of state versus Butler's equation

• Two models are shown to be equivalent to Sprow and Prausnitz equation.
• Surface chemical potential can be deduced from a surface equation of state.
• Surface pressure is a measure of solute's surface activity.
• Approximations for surface activity coefficients and surface composition are avoided.
• Butler's bulk–surface equilibrium approach is inconsistent with Gibbs adsorption equation.

In the last years, our research group has been committed to the development of bulk–surface equilibrium models that relate surface tension, bulk composition, and bulk activity coefficients; in this paper, we focus on two of them. The starting premise is the equality of bulk and surface chemical potentials. Instead of the common practice of expressing the surface chemical potential analogously to the bulk, in the first model, surface chemical potential is obtained from the integration of Gibbs adsorption equation coupled to Volmer's equation of state, and written as a function of surface pressure shown to be a measure of surface activity. In the second model, Gibbs adsorption equation was evaluated in the surface saturated region and then coupled to the bulk chemical potential avoiding the use of an explicit expression for the surface. Both models are equivalent in meaning and structure to the widely accepted Sprow and Prausnitz equation (Trans. Faraday Soc. 62 (1966) 1105), originally derived by Butler (Proc. R. Soc. A 138 (1932) 348), where it is necessary to introduce non-thermodynamic approximations for the calculation of partial molar areas, surface activity coefficients and surface composition. On the contrary, the parameters of the two proposed models are attained from the fitting to the experimental data. Obeying a monolayer model, Butler arrived to the conclusion that bulk and surface chemical potentials are not equal, but it is demonstrated that in Gibbs adsorption equation, the equality of chemical potentials is valid and that the models are thermodynamically consistent.

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