Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
15. Some Special Topics: Surfaces of Liquids and Solutions
15.5. Statistics of Adsorption
Apart from a thermodynamic description of liquid surfaces, another route is to use a statistical mechanical approach. We do so briefly in this section, limiting ourselves again to surfaces, and to that purpose consider a binary system for which we denote the bulk composition with A1−xBx and the surface composition A1−yBy. We again use a lattice-type model, and assume that only the first molecular layer has a composition different from the bulk. As before, the components are indicated by subscript i, while the surface and bulk phase are indicated by superscripts (σ) and (α), respectively. We start with an ideal solution and thereafter consider the influence of nonideality.
We recall that the chemical potential of component i is given by
where and are the standard chemical potentials of the pure components in the bulk and the surface, respectively, while x and y denote the mole fractions. Further, at equilibrium we have . From these equations we easily obtain
The Helmholtz energies are then given by
For a one-component system the surface tension is given by , so that
where ai = A/Ni denotes the area per molecule. If we subtract Eq. (15.71-2) from Eq. (15.71-1) and Eq. (15.72-2) from Eq. (15.72-1) and insert Eq. (15.73) for both components, we obtain
This is the adsorption equation for ideal solutions.
We now introduce interactions and “change gear” to statistics, employing the grand (canonical) partition function Ξ in connection with the lattice model. Note that Ξ is the proper partition function to use, since we keep μi, N and Vconstant (see Chapter 5). The configurational partition function of the surface layer Q(σ) is given by
The energy U(σ) is evaluated as for a regular solution (in the zeroth approximation) and reads, using as before the interaction energy 2w = 2wAB − wAA − wBB,
Here, is the coordination number for “bonds” in the surface layer with composition A1−yBy, and is the coordination number for “bonds” between the surface layer and the first bulk layer, so that the total surface coordination number reads (e.g., for a FCC (111) plane and ). The grand partition function becomes
We now use the maximum-term method (see Justification 5.4), replacing the sum by its largest term and obtain
The equilibrium condition is
and leads to
From and Eq. (15.71) we obtain
so that combining Eq. (15.81) and (15.82) gives
This is the equivalent of Eq. (15.74) in the zeroth approximation. We see that the surface tension term −(aBγB − aAγA) is replaced by ½(z − z(σ))(wBB − wAA), while the interaction term with w is obviously absent for the ideal solution.
Estimating wii from the heat of vaporization Li in the nearest-neighbor approximation Li = −½zNAwii (with NA = Avogadro's constant) and w from the heat of solution LAB of component A in component B (or vice versa from LBA), we have
Note that the heat of vaporization is counted positive, while the bond energy is counted negative. In the zeroth approximation of the regular solution theory ΔLAB = ΔLBA. Experimentally, this condition is only exceptionally fulfilled, which indicates a need for further improvement. However, the approach does provide a clear basis for a picture in which the surface tension – that is, the surface Helmholtz energy – is estimated as, with again NA as Avogadro's number,
Using this approximation we obviously neglect entropy terms and have identified ½zwii with and ½z(σ)wii with . The discussion in Section 15.2 showed that, numerically, the difference is far from negligible. The fraction of missing “bonds” (z−z(σ))/z, as also indicated in Section 15.2, is ∼1/4.