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

### 11. Mixing Liquids: Molecular Solutions

### 11.8. Empirical Improvements*

In this section we indicate briefly some empirical improvements that can be made on the regular solution model, largely based on an expansion of the excess Gibbs function. Theoretical improvements deal with a better description for the local configuration than being based on random mixing with consequences for entropy and energy, and we discuss these in the next section.

The excess Gibbs energy as used in the regular solution model can be systematically but largely empirically extended. As a first approximation in the regular solution model, we wrote , where *w*_{m} is a (temperature-dependent) parameter (we omit the subscript m from now on). Corrections on this expression may be modeled as a power law in *x*_{1} and *x*_{2}. Hence, we can write the *Redlich–Kister expansion* [8]

(11.115)

The inclusion of these terms generates systematic improvements in the description.

When *A* = *B* = *C* = ··· = 0, *G*^{E} = 0, *RT*ln*γ*_{1} = *RT*ln*γ*_{2} = 0 and *γ*_{1} = *γ*_{2} = 1. Accordingly, we regain the ideal solution.

When *B* = *C* = ··· = 0, *G*^{E} = *Ax*_{1}*x*_{2} and and . Obviously, these results represent the regular solution and the expressions are symmetric in nature. For infinite dilution one obtains .

When *C* = ··· = 0, *G*^{E}/*x*_{1}*x*_{2} = *A* + *B* (*x*_{1} − *x*_{2}), implying a deviation from the regular solution model. Equivalently, since *x*_{1} + *x*_{2} = 1, we may write *G*^{E}/*x*_{1}*x*_{2} = *A* (*x*_{1} + *x*_{2}) + *B* (*x*_{1} − *x*_{2}) or defining *A*_{21} = *A* + *B* and *A*_{12} = *A* − *B*,

(11.116)

a result known as the *Margules equation*. Obviously, this approach can be extended although, according to Guggenheim [9], an expansion further than third order is hardly ever required in view of the experimental accuracy. In __Figure 11.7__ and __Table 11.5__ some examples are presented of *G*^{E} functions, which shows that this approach is flexible and can describe rather different type of behaviors, that is, continuously increasing *G*^{E}, continuously decreasing *G*^{E}, and a *G*^{E} showing a maximum or minimum.

** Figure 11.7** Excess Gibbs energies for the systems of

__Table 11.5__.

** Table 11.5** Excess Gibbs function parameters for various solutions.

One may also expand the reciprocal *x*_{1}*x*_{2}/*G*^{E} in the mole fractions to obtain the *van Laar expansion* [10]

(11.117)

Equivalently, we may write, since (*x*_{1} + *x*_{2}) = 1,

(11.118)

Defining and , we obtain

__(11.119)__

The expressions for the activity coefficients, known as the *van Laar equations*, are

(11.120)

In this case for *x*_{1} = 0, and for *x*_{2} = 0, .

Some algebraic manipulation, meanwhile, identifying and , where *V*_{α} is the partial volume of component α and *C* is a constant, shows that __Eq. (11.119)__ is equivalent to an expansion of *G*^{E} in the volume fraction *ϕ*_{1} = *V*_{1}*x*_{1}/(*V*_{1}*x*_{1} + *V*_{2}*x*_{2}) = *V*_{1}*x*_{1}/*V*_{m}, where *V*_{m} ≡ *V*_{1}*x*_{1} + *V*_{2}*x*_{2} is the molar volume of the mixture, that is, . This relation fits data well for many binary systems of normal liquids. In the van Laar version the constant *C* was originally given in terms of the *a-* and *b*-parameters of the vdW equation, that is

(11.121)

To be preferred is the expression, rationalized by Scatchard [11],

(11.122)

where Δ*U*_{α} = Δ_{vap}*H*_{α} − *RT* is the molar energy of vaporization, and the solubility parameter *δ*_{α} = (Δ*U*_{α}*/V*_{α})^{1/2} is used. The latter expression is also given by the approximate statistical mechanical calculation in Section 11.6. Using and as parameters, the ratio corresponds reasonably well to *V*_{1}/*V*_{2}, for example, for the mixture benzene and iso-octane [12] at 45 °C , while experimentally *V*_{1}/*V*_{2} = 0.536.

Although these expressions show a relatively large flexibility in fitting the experimental data, their theoretical basis is flimsy and the equations are difficult to extend to multicomponent systems. Finally, the temperature dependence of the parameters involved is not explicitly included. Developments based on the concept of local composition have led to various models which remedy these defects, and of which the *Wilson model* [13] is the archetype. For a solution of components 1 and 2, the number of 1–1 and 2–2 interactions relative to the number of 1–2 interactions is given by the ratio of component 1 to 2 weighted by a Boltzmann factor. Hence

(11.123)

The volume fractions *ϕ*_{α}, using as pure component molar volume *V*_{α}, are then defined by

(11.124)

(11.125)

The Gibbs energy of mixing is assumed to be

(11.126)

(11.127)

Defining

(11.128)

we can write

(11.129)

After some calculation the result for the activity coefficients *γ*_{1} and *γ*_{2} reads

(11.130)

(11.131)

The quantities *r*_{12} and *r*_{21} are considered as empirical parameters and have been extensively tabulated [14]. For a fixed *T*, only the two parameters *r*_{12} and *r*_{21} are required to describe the system, but for a range of temperatures the parameters ∂*r*_{12}/∂*T* and ∂*r*_{21}/∂*T* are also required. The Wilson expressions can be easily generalized to a multicomponent system since, in principle, no new parameters are required. There are, however, several drawbacks. For example, when *r*_{12} = *r*_{21} = 1, or Δ_{mix}*G*_{m} = 0, the model cannot describe phase separation, nor for other values of *r*_{12} and *r*_{21}. It also appears that the model cannot describe a minimum or maximum in *γ*_{α}(*x*).

In summary, Wilson-type models can be used effectively for binary and multicomponent systems, and contain an explicit temperature dependence. Consequently, the approach is rather extensively used. For further information, the reader is referred to the literature (e.g., Ref. [15]).