William M. White - Geochemistry

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 Δ G mixing= Δ H mixing– T Δ S mixingand Δ H 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. 3.31as:

(3.32)

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

(3.33)

Note that for an ideal solution, μ iis always less than or equal to μ° ibecause the term R T ln X iis 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, X 1= (1 – X 2), so we can write eqn. 3.30for the binary case as:

(3.34)

Since X 2is less than 1, Δ G 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 X 2= 0.5.

Now let's see how we can recover information on μ ifrom plots such as Figure 3.6, which we will call G-bar–X plots. Substituting X 1= (1 – X 2) into eqn. 3.33, it becomes:

(3.35)

This is the equation of a straight line on such a plot with slope of ( μ 2– μ 1) and intercept μ 1. 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. 3.31and eqn. 3.35give the same value of for a given value of X 2, such as X′ 2. Thus, the straight line and the curved one in Figure 3.7must 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 X 2= 0 is μ 1and the intercept at X 2= 1 is μ 2. 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 X 1approaches 1 ( X 2approaches 0), the intercept of the tangent approaches μ° 1, i.e., μ 1approaches μ° 1. Looking at eqn. 3.26, this is exactly what we expect. Figure 3.7illustrates 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 Δ G mixingis 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).

We now turn our attention to real solutions, which are somewhat more complex than ideal ones, as you might imagine. We will need to introduce a few new tools to help us deal with these complexities.

Let's consider the behavior of a real solution in view of the two solution models we have already introduced: Raoult's law and Henry's law. Figure 3.8illustrates the variation of chemical potential as a function of composition in a hypothetical real solution. We can identify three regions where the behavior of the chemical potential is distinct:

1 The first is where the mole fraction of component Xi is close to 1 and Raoult's law holds. In this case, the amount of solute dissolved in i is trivially small, so molecular interactions involving solute molecules do not significantly affect the thermodynamic properties of the solution, and the behavior of μi is close to that in an ideal solution:(3.26)