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We will be able to generalize a form of this equation to nonideal cases a bit later. Let's first consider some other properties of ideal mixtures. For real solutions, any extensive thermodynamic property such as volume can be considered to be the sum of the volume of the components plus a volume change due to mixing:

(3.27)

The first term on the right reflects the volume resulting from mechanical mixing of the various components. The second term reflects volume changes associated with solution. For example, if we mixed 100 ml of ethanol and 100 ml of water (Figure 3.5), the volume of the resulting solution would be 193 ml. Here, the value of the first term on the right would be 200 ml, and the value of the second term would be −7 ml. We can write similar equations for enthalpy, and so on. But the volume change and enthalpy change due to mixing are both 0 in the ideal case. This is true because both volume and enthalpy changes of mixing arise from intermolecular forces, and, by definition, such intermolecular forces are absent in the ideal case. Thus:

therefore:

and

and therefore:

(where h is the partial molar enthalpy). This, however, is not true of entropy. You can imagine why: if we mix two substances on an atomic level, the number of possible arrangements of our system increases even if they are ideal substances. The entropy of ideal mixing is (compare eqn. 2.110):

(3.28)

(3.29)

Because ΔG_mixing = ΔH_mixing – TΔS_mixing and ΔH_mixing = 0, it follows that:

(3.30)

We stated above that the total expression for an extensive property of a solution is the sum of the partial molar properties of the pure phases (times the mole fractions), plus the mixing term. The partial molar Gibbs free energy is the chemical potential, so the full expression for the Gibbs free energy of an ideal solution is:

(3.31)

Rearranging terms, we can reexpress eqn. 3.31 as:

(3.32)

The term in parentheses is simply the chemical potential of component i, μ_i, as expressed in eqn. 3.26. Substituting eqn. 3.26 into 3.32, we have

(3.33)

Note that for an ideal solution, μ_i is always less than or equal to μ°_i because the term RTln X_i is always negative (because the log of a fraction is always negative).

Let's consider ideal mixing in the simplest case, namely binary mixing. For a two-component (binary) system, X₁ = (1 – X₂), so we can write eqn. 3.30 for the binary case as:

(3.34)

Since X₂ is less than 1, ΔG is negative and becomes increasingly negative with temperature, as illustrated in Figure 3.6. The curve is symmetrical with respect to X, that is, the minimum occurs at X₂ = 0.5.

Now let's see how we can recover information on μ_i from plots such as Figure 3.6, which we will call G-bar–X plots. Substituting X₁ = (1 – X₂) into eqn. 3.33, it becomes:

(3.35)

This is the equation of a straight line on such a plot with slope of (μ₂ – μ₁) and intercept μ₁. This line is illustrated in Figure 3.7. The curved line is described by eqn. 3.31. The dashed line is given by eqn. 3.35. Both eqn. 3.31 and eqn. 3.35 give the same value of for a given value of X₂, such as X′₂. Thus, the straight line and the curved one in Figure 3.7 must touch at X′₂. In fact, the straight line is the tangent to the curved one at X′₂. The intercept of the tangent at X₂ = 0 is μ₁ and the intercept at X₂ = 1 is μ₂. The point is, on a plot of molar free energy vs. mole fraction (a G–X diagram), we can determine the chemical potential of component i in a two-component system by extrapolating a tangent of the free energy curve to X_i = 1. We see that in Figure 3.7, as X₁ approaches 1 (X₂ approaches 0), the intercept of the tangent approaches μ°₁, i.e., μ₁ approaches μ°₁. Looking at eqn. 3.26, this is exactly what we expect. Figure 3.7 illustrates the case of an ideal solution, but the intercept method applies to nonideal solutions as well, as we shall see.

Figure 3.6 Free energy of mixing as a function of temperature in the ideal case. After Nordstrom and Munoz (1986).

Finally, the solid line connecting the μ°'s is the Gibbs free energy of a mechanical mixture of components 1 and 2, which we may express as:

(3.36)

You should satisfy yourself that the ΔG_mixing is the difference between this line and the free energy curve:

(3.37)

Figure 3.7 Molar free energy in an ideal mixture and a graphical illustration of eqn. 3.31. After Nordstrom and Munoz (1986).