Entropies of Ideal Mixing

The entropy of mixing ideal gases was treated already by Gibbs. The result is known as the Gibbs paradox. Several scientists were engaged with this paradox, among them Rayleigh, Lorentz, Einstein, and von Neumann [1,2]. 

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

We express the calculated entropies of mixing in units of R. For ideal solutions the values of are evaluated directly from Eq. (8.28) ... 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

A plot of these values is shown in Fig. 8.1. Note the increase in the entropy of mixing over the ideal value with increasing n value. Also note that the maximum occurs at decreasing mole fractions of polymer with increasing degree of polymerization. 

The drermodynamic data for CuaS-FeS (Krivsky and Schuhmann, 1957) show that drese sulphides mix to form approximately ideal ionic liquids. These are molten salts in which the heat of mixing is essentially zero, and die entropy of mixing is related to the ionic fractions of die cations and anions. In the present case die ionic fractions yield values for the activities of the two sulphides... 

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4>, and 4>2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda. ... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

This result is nearly equal to 4.87 J K hmoT. the value that would be calculated for the entropy of mixing to form an ideal solution. We will show in Chapter 7 that the equation to calculate AmixSm for the ideal mixing process is the same as the one to calculate the entropy of mixing of two ideal gases. That is. AmixSm = -R. Vj ln.Yj. 

Ideal solution behavior over extended ranges in both composition and temperature requires that the following conditions be fulfilled (i) the entropy of mixing must be given by ... 

A particular type of nonideal solution is the regular solution which is characterized by a nonzero enthalpy of mixing but an ideal entropy of mixing. Thus, for a regular solution,... 

The Gibbs energy of mixing of an ideal solution is negative due to the positive entropy of mixing obtained by differentiation of Ald.xGm with respect to temperature ... 

Figure 3.3 Thermodynamic properties of an arbitrary ideal solution A-B at 1000 K. (a) The Gibbs energy, enthalpy and entropy, (b) The entropy of mixing and the partial entropy of mixing of component A. (c) The Gibbs energy of mixing and the partial Gibbs energy of mixing of component A.

The excess molar entropy of mixing is the real entropy of mixing minus the ideal entropy of mixing. Using a binary A-B solution as an example, Ae xSm s... 

For a large number of the more commonly used microscopic solution models it is assumed, as we will see in Chapter 9, that the entropy of mixing is ideal. The different atoms are assumed to be randomly distributed in the solution. This means that the excess Gibbs energy is most often assumed to be purely enthalpic in nature. However, in systems with large interactions, the excess entropy may be large and negative. 

The entropy of mixing of many real solutions will deviate considerably from the ideal entropy of mixing. However, accurate data are available only in a few cases. The simplest model to account for a non-ideal entropy of mixing is the quasi-regular model, where the excess Gibbs energy of mixing is expressed as... 

The molar entropy of mixing for the ideal solution follows as... 

The ideal solution approximation is well suited for systems where the A and B atoms are of similar size and in general have similar properties. In such systems a given atom has nearly the same interaction with its neighbours, whether in a mixture or in the pure state. If the size and/or chemical nature of the atoms or molecules deviate sufficiently from each other, the deviation from the ideal model may be considerable and other models are needed which allow excess enthalpies and possibly excess entropies of mixing. 

The entropy of mixing is the same as for the ideal solution model. 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

So far we have considered mixtures of atoms or species of similar size and shape. Now we will consider a mixture of a polymeric solute and a solvent of monomers [7, 8], The ideal entropy of mixing used until now cannot possibly hold for this polymer solution, in which the solute molecule may be thousands or more times the size of the solvent. The long chain polymer may be considered to consist of r chain segments, each of which is equal in size to the solvent molecule. Therefore r is also equal to the ratio of the molar volumes of the solute and the solvent. The solute and the solvent can be distributed in a lattice where each lattice site can contain one solvent molecule. The coordination number of a lattice site is z. 

Figure 9.6 (a) Molar entropy of mixing of ideal polymer solutions for r = 10, 100 and 1000 plotted as a function of the mole fraction of polymer compared with the entropy of mixing of two atoms of similar size, r = 1. (b) Activity of the two components for the same conditions. 

The first term is the ideal entropy of mixing while the second term is the enthalpy of mixing in the regular solution approximation ... 

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