Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
11. Mixing Liquids: Molecular Solutions
11.6. A Slightly Different Approach
So far the model has been based on mole fractions, but here we will iterate the arguments for the regular solution model in slightly different terms, that is, using the pair-correlation function . From that set of arguments we will see that volume fractions are to be preferred. Other arguments in favor of using volume fractions are given in Section 11.8. The outcome leads to the solubility parameter approach, based on the Berthelot rule, Eq. (11.81), and therefore we will assess the validity of that approximation once more. Finally, it leads to the so-called one-fluid and two-fluid models of mixtures.
The argumentation starts with the energy expression
in which the first term represents the kinetic energy and the second term the potential energy (denoting the integral by U0 for later reference). Furthermore, u(r) and g(r) represent the pair energy8) and pair-correlation function, respectively. We extend this expression to a binary mixture, considering the potential energy only. In a solution around a central molecule of component 1, molecules of type 1 and 2 are found. The probability of finding a molecule of component 1 is given by (N1/V)g11(r), while the probability of finding a molecule of component 2 is given by (N2/V)g12(r). Similar relationships hold for a central molecule of component 2. Since the probabilities that the central molecule is 1 or 2 are x1 and x2, respectively, we may write for the total volume V, using Vα and Vm for the partial and molar volume9) of component α and the solution, respectively,
Remembering that u12 = u21 and g12 = g21, we may further simplify, meanwhile using volume fractions ϕi = (xiVi)/(x1V1 + x2V2), to obtain
Subtracting the energy of the separate components, given by
we obtain for the mixing energy
So far everything is exact, apart from the pair potential approximation. The next argument used is scaling. We use, based on the principle of corresponding states,
Moreover, we use the experimental similarity of g for simple molecules and write
in combination with σ12 = ½(σ11 + σ22). Defining y = r/σ, we obtain for 1 mole
We now make the identification
which leads to
Assuming that the Berthelot approximation c12 = −(c11c22)1/2 is valid, we get
or, defining δα ≡ |cαα|1/2 = |wαα/Vα|1/2,
This expression is essentially the same as Eq. (11.56) combined with Eq. (11.81), except that mole fractions have been replaced by volume fractions. The derivation shows that the regular solution model is not necessarily based on a lattice model, but can be based on the structural analogy between solvent and solution.
11.6.1 The Solubility Parameter Approach
Hildebrand  introduced the solubility parameter δα = (ΔUα/Vα)1/2 so that
The term solubility parameter is based on the original application of the model in the solubility of compounds. Consider that miscibility occurs when ΔmixGm ≤ 0. Since in this model ΔmixSm ≥ 0 and ΔmixHm ≥ 0 always, ΔmixHmmust be not too large to result in miscibility. In other words, miscibility is predicted if w is small. Two ways to estimate δ-values exist. One way is to use the experimental enthalpy of vaporization ΔvapHα for the pure compounds α to calculate ΔvapUα = ΔvapHα − RT which, when combined with Vα, yields δα. Another approach employs group contribution methods which have been devised to estimate δ (see Chapter 13).
In order to estimate how well the geometric mean approximation for w is, we resort to the dispersion energy. Recall that in Chapter 3 an approximate expression for the dispersion interaction was given reading αβ
with α the polarizability and I the characteristic energy. It was indicated that estimating I as the ionization potential is doubtful, while an estimate for α is much more reliable.
Assuming , a reasonable assumption for spherical molecules, we may write from Eq. (11.95)
Using Eq. (11.100) for εαβ at r = σαβ and the assumption σ12 = (σ11 + σ22)/2, we obtain
Because the geometric average is invariably less than the arithmetic average, the approximation c12 = −(c11c22)1/2 is only valid if I1 = I2 and σ11 = σ22 (as discussed in Section 3.6); otherwise, |c12| < (c11c22)1/2. Writing
Reed  estimated that for hydrocarbon + fluorocarbon systems fI ≅ 0.97 and fσ ≅ 0.995, rendering large deviations from the Berthelot rule using typical δ-values (this is consistent with our discussion in Section 3.6). Finally, we note that although the solubility parameter approach is empirically rather successful, its theoretical basis is flimsy.
11.6.2 The One- and Two-Fluid Model
The approach discussed provides a basis for the one-fluid and two-fluid model of mixtures. We saw that using gαβ = g (r/σαβ) for all pairs of components α and β, ρ = N/V, and returning to mole fractions xα, the configurational energy is given by
Using the symbols ε and σ for the mixture we can define
In this way a mixture can be described by a hypothetical pure, model fluid with properties defined by ε and σ given the scaled interaction and correlation functions f(y) and g(y). In the literature this model is usually referred to as the one-fluid van der Waals (or vdW1) model.
Another “Ansatz” is to use gαβ = ½[gαα(r/σαα) + gββ(r/σββ)]. This leads to
Here, gαα represents the correlation function for a pure, pseudo-component α with εα and σα and the mixture is described as an ideal mixture of two pseudo-components. It is referred to as the two-fluid van der Waals (or vdW2) model. Although it is difficult to discriminate between the vdW1 and vdW2 models on an experimental basis, simulations using the Lennard-Jones potential clearly show that the vdW1 model is better  that the vdW2 model (Figure 11.6). The expressions for F and G in the vdW1 model read
respectively, where F0 and G0 correspond to U0. For the vdW2 model, these expressions must be summed over the components using the appropriate values for ε and σ. In contrast to the vdW2 model, the vdW1 model introduces no ambiguities between F and G, which makes it also preferable. Since for mixtures, P and T usually are the normal variables, the use of G is more convenient.
Figure 11.6 (a) Excess Gibbs energy of mixing and (b) excess enthalpy of mixing for an equimolar mixture of 6 : 12 molecules for which σ22/σ12 = 1.06 at 97 K and P = 0 (ε12 = 133.50 K, σ12 = 3.596 Å). The points give the simulation results, and the curves marked 1 and 2 give the results of the vdWl and vdW2 theories, respectively.