## 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 A_{1−x}B* _{x}* and the surface composition A

_{1−y}B

*. 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*

_{y}*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

(15.70)

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

__(15.71)__

The Helmholtz energies are then given by

__(15.72)__

For a one-component system the surface tension is given by , so that

__(15.73)__

where *a _{i}* =

*A*/

*N*denotes the area per molecule. If we subtract

_{i}__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

__(15.74)__

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 *μ _{i}*,

*N*and

*V*constant (see

__Chapter 5__). The configurational partition function of the surface layer

*Q*

^{(σ)}is given by

(15.75)

where

(15.76)

The energy *U*^{(σ)} is evaluated as for a regular solution (in the zeroth approximation) and reads, using as before the interaction energy 2*w* = 2*w*_{AB} − *w*_{AA} − *w*_{BB},

(15.77)

Here, is the coordination number for “bonds” in the surface layer with composition A_{1−y}B_{y}, 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

(15.78)

We now use the maximum-term method (see Justification 5.4), replacing the sum by its largest term and obtain

(15.79)

The equilibrium condition is

(15.80)

and leads to

__(15.81)__

From and __Eq. (15.71)__ we obtain

__(15.82)__

so that combining __Eq. (15.81)__ and __(15.82)__ gives

(15.83)

This is the equivalent of __Eq. (15.74)__ in the zeroth approximation. We see that the surface tension term −(*a*_{B}*γ*_{B} − *a*_{A}*γ*_{A}) is replaced by ½(*z *− *z*^{(σ)})(*w*_{BB} − *w*_{AA}), while the interaction term with *w* is obviously absent for the ideal solution.

Estimating *w _{ii}* from the heat of vaporization

*L*in the nearest-neighbor approximation

_{i}*L*= −½

_{i}*zN*

_{A}

*w*(with

_{ii}*N*

_{A}= Avogadro's constant) and

*w*from the heat of solution

*L*

_{AB}of component A in component B (or

*vice versa*from

*L*

_{BA}), we have

(15.84)

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 Δ*L*_{AB} = Δ*L*_{BA}. 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 *N*_{A} as Avogadro's number,

(15.85)

Using this approximation we obviously neglect entropy terms and have identified ½*zw _{ii}* with and ½

*z*

^{(σ)}

*w*with . The discussion in Section 15.2 showed that, numerically, the difference is far from negligible. The fraction of missing “bonds” (

_{ii}*z*−

*z*

^{(σ)})/

*z*, as also indicated in Section 15.2, is ∼1/4.