## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 15. Some Special Topics: Surfaces of Liquids and Solutions

### 15.4. Two-Component Liquid Surfaces

For solutions, the preferential adsorption of one of the components at the surface becomes important. For a single-component liquid we described the thermodynamics of the surface first exactly, and thereafter simplified the expression by neglecting the effects resulting from the interphase thickness. This can also be achieved for a two-component solution, but the resulting equations are cumbersome [2], and therefore we neglect this contribution from the start. Rather, we limit ourselves to surfaces – that is, liquid–vapor interfaces. The adsorption behavior is described by the Gibbs adsorption expression __Eq. (15.12)__, and in the indicated approximation reads

(15.58)

Substituting the expression for d*μ*_{1} reading

(15.59)

and the corresponding expression for d*μ*_{2}, we have

__(15.60)__

Employing the Gibbs–Duhem equation at constant *T* and *P*, reading

(15.61)

where 1 − *x* = *x*_{1} and *x* = *x*_{2}, the result is

(15.62)

From this expression it is easy to calculate the temperature derivative

(15.63)

From the chemical potential and Helmholtz energy expressions reading, respectively,

(15.64)

(15.65)

we obtain

(15.66)

From __Eq. (15.60)__ or (15.62) we can calculate similarly the composition derivative as

(15.67)

__(15.68)__

respectively. Since, for an ideal mixture, , __Eq. (15.68)__ reduces to

(15.69)

For solutes with a measurable vapor pressure, we can also replace ln*a*_{2} with ln*P*_{2}. The theory discussed applies to all types of solutes. Nevertheless, it is useful to distinguish between various types of solutes, and we do so in Section 15.6.