William M. White - Geochemistry

The excess enthalpy is a measure of the heat released during mixing the pure end-members to form the solution, and the excess entropy is a measure of all the energetic effects resulting from a nonrandom distribution of species in solution. We can express excess enthalpy change in the same way as excess free energy:

(3.53)

But since Δ H ideal mixing= 0, Δ H excess= Δ H real; in other words, the enthalpy change upon mixing is the excess enthalpy change. Similar expressions may, of course, be written for volume and entropy (bearing in mind that unlike volume and enthalpy, Δ S idealis not zero).

Combining eqn. 3.46with eqn. 3.48leads to the following:

(3.54)

We can rewrite this as:

(3.55)

Equation 3.55shows how activity coefficients relate to Henry's and Raoult's laws. Comparing eqn. 3.55with eqn. 3.39, we see that in the region where Henry's law holds, that is, dilute solutions, the activity coefficient is equal to Henry's law constant. In the region where Raoult's law holds, the activity coefficient is 1 and eqn. 3.55reduces to eqn. 3.26since R T ln λ i= 0.

Since we know that

comparing equations 3.51and 3.55, we find that:

(3.56)

which is the same as:

(3.56a)

so that the molar excess free energy associated with component i is simply R T times the log of the activity coefficient. The total molar excess free energy of the solution is then:

(3.57)

We will see the usefulness of the concept of excess free energy shortly when we consider activities in electrolyte solutions. It will also prove important in our treatment of nonideal solid solutions and exsolution phenomena in the next chapter.

In northern climates, salting roads and sidewalks to melt snow and ice is a common practice in winter. We have now acquired the thermodynamic tools to show why salt melts ice, and that this effect does not depend on any special properties of salt or water. Depression of the melting point by addition of a second component to a pure substance is a general phenomenon. Suppose that we have an aqueous solution containing sodium chloride coexisting with pure ice. If the two phases are at equilibrium, then the chemical potential of water in ice must equal that of water in the solution:

(3.58)

(we are using subscripts to denote the component, and superscripts to denote the phase; aq denotes the liquid aqueous solution). We define our standard state as that of the pure substance. According to eqn. 3.48, the chemical potential of water in the solution can be expressed as:

(3.59)

where denotes the chemical potential of pure liquid water. Substituting eqn. 3.59into 3.58 and rearranging, we have:

(3.60)

Ice will incorporate very little salt; if we assume it is a pure phase, we may write eqn. 3.60as:

(3.60a)

or

(3.61)

(The order is important: eqn. 3.60a describes the freezing process, 3.61the melting process. These processes will have equal and opposite entropies, enthalpies, and free energies.) The left-hand side of eqn. 3.61is the Gibbs free energy of melting for pure water, which we denote as ( is 0 at the melting temperature of pure water, which we denote T o m, but nonzero at any other temperature).

We may rewrite eqn. 3.61as:

(3.62)

If we assume that Δ H and Δ S are independent of temperature (which is not unreasonable over a limited temperature range) and we assume pressure is constant as well, the left-hand side of the equation may also be written as:

(3.63)

Substituting eqn. 3.63into 3.62:

(3.64)

At the melting temperature of pure water, is zero, so that:

Substituting this into eqn. 3.64and rearranging:

(3.65)

Further rearrangement yields: