Extension of the Gibbs-Duhem Equation to the Partial Molar Surface Thermodynamic Properties of Solutions

In this paper, the Gibbs-Duhem equation is extended to the partial molar surface thermodynamic properties of solutions. According to the surface Gibbs-Duhem equations, the sum of the mole fractions of the components in the surface region of a bulk solution multiplied by different partial molar surface quantities should equal zero if summation is taken by all components of the solution. There are four different partial molar surface quantities identified in this paper for which the surface Gibbs-Duhem equation is proven to be valid: (i) the reduced surface chemical potential, (ii) the surface chemical potential, (iii) the partial molar surface area, and (iv) the partial molar excess surface Gibbs energy = the product of partial molar surface area and the partial surface tension. The first one is known since Guggenheim (1940), but the other three are presented here for the first time. It is also demonstrated here how to apply the surface Gibbs-Duhem equations: (i) it is proven that the model equation applied by us recently for the reduced chemical potential [Adv Coll Interf Sci 2020, 283, 102212] obeys one of the surface Gibbs-Duhem equations, (ii) in contrary, it is proven that the model equation suggested by us recently for the partial molar surface area contradicts one of the surface Gibbs-Duhem equations; therefore, a new (and simpler) model equation for the partial molar surface areas of the components is suggested here that obeys the surface Gibbs-Duhem equation. It is also shown that the Butler equation obeys one of the surface Gibbs-Duhem equations. It is also concluded that surface composition in equilibrium should be one that ensures minimum surface tension.