## 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 [4]. 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

(11.86)

in which the first term represents the kinetic energy and the second term the potential energy (denoting the integral by *U*_{0} for later reference). Furthermore, *u*(*r*) and *g*(*r*) represent the pair energy__ ^{8)}__ 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 (

*N*

_{1}/

*V*)

*g*

_{11}(

*r*), while the probability of finding a molecule of component 2 is given by (

*N*

_{2}/

*V*)

*g*

_{12}(

*r*). Similar relationships hold for a central molecule of component 2. Since the probabilities that the central molecule is 1 or 2 are

*x*

_{1}and

*x*

_{2}, respectively, we may write for the total volume

*V*, using

*V*

_{α}and

*V*

_{m}for the partial and molar volume

__of component α and the solution, respectively,__

^{9)}(11.87)

(11.88)

Remembering that *u*_{12} = *u*_{21} and *g*_{12} = *g*_{21}, we may further simplify, meanwhile using volume fractions *ϕ _{i}* = (

*x*)/(

_{i}V_{i}*x*

_{1}

*V*

_{1}+

*x*

_{2}

*V*

_{2}), to obtain

(11.89)

Subtracting the energy of the separate components, given by

(11.90)

we obtain for the mixing energy

(11.91)

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,

(11.92)

Moreover, we use the experimental similarity of *g* for simple molecules and write

(11.93)

in combination with *σ*_{12} = ½(*σ*_{11} + *σ*_{22}). Defining *y* = *r*/*σ*, we obtain for 1 mole

(11.94)

We now make the identification

__(11.95)__

which leads to

(11.96)

Assuming that the Berthelot approximation *c*_{12} = −(*c*_{11}*c*_{22})^{1/2} is valid, we get

(11.97)

or, defining *δ*_{α} ≡ |*c*_{αα}|^{1/2} = |*w*_{αα}/*V*_{α}|^{1/2},

(11.98)

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 [5] introduced the *solubility parameter* *δ*_{α} = (Δ*U*_{α}/*V*_{α})^{1/2} so that

(11.99)

The term solubility parameter is based on the original application of the model in the solubility of compounds. Consider that miscibility occurs when Δ_{mix}*G*_{m} ≤ 0. Since in this model Δ_{mix}*S*_{m} ≥ 0 and Δ_{mix}*H*_{m} ≥ 0 always, Δ_{mix}*H*_{m}must 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 Δ_{vap}*H*_{α} for the pure compounds α to calculate Δ_{vap}*U*_{α} = Δ_{vap}*H*_{α} − *RT* which, when combined with *V*_{α}, yields *δ*_{α}. 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*

__(11.100)__

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)__

(11.101)

Using __Eq. (11.100)__ for *ε*_{αβ} at *r* = *σ*_{αβ} and the assumption *σ*_{12} = (*σ*_{11} + *σ*_{22})/2, we obtain

(11.102)

Because the geometric average is invariably less than the arithmetic average, the approximation *c*_{12} = −(*c*_{11}*c*_{22})^{1/2} is only valid if *I*_{1} = *I*_{2} and *σ*_{11} = *σ*_{22} (as discussed in Section 3.6); otherwise, |*c*_{12}| < (*c*_{11}*c*_{22})^{1/2}. Writing

(11.103)

(11.104)

Reed [6] estimated that for hydrocarbon + fluorocarbon systems *f _{I}* ≅ 0.97 and

*f*≅ 0.995, rendering large deviations from the Berthelot rule using typical

_{σ}*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 *g*_{αβ} = *g* (*r*/*σ*_{αβ}) for all pairs of components α and β, *ρ* = *N*/*V*, and returning to mole fractions *x*_{α}, the configurational energy is given by

(11.105)

Using the symbols *ε* and *σ* for the mixture we can define

(11.106)

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 *g*_{αβ} = ½[*g*_{αα}(*r*/*σ*_{αα}) + *g*_{ββ}(*r*/*σ*_{ββ})]. This leads to

(11.107)

(11.108)

Here, *g*_{αα} represents the correlation function for a pure, pseudo-component α with *ε*_{α} and *σ*_{α} 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 [7] that the vdW2 model (__Figure 11.6__). The expressions for *F* and *G* in the vdW1 model read

(11.109)

(11.110)

respectively, where *F*_{0} and *G*_{0} correspond to *U*_{0}. 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

*σ*

_{22}/

*σ*

_{12}= 1.06 at 97 K and

*P*= 0 (

*ε*

_{12}= 133.50 K,

*σ*

_{12}= 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.