Читать книгу Geochemistry - William M. White - Страница 109

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. 3.59 into 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. 3.60 as:

(3.60a)

or

(3.61)

(The order is important: eqn. 3.60a describes the freezing process, 3.61 the melting process. These processes will have equal and opposite entropies, enthalpies, and free energies.) The left-hand side of eqn. 3.61 is 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. 3.61 as:

(3.62)

If we assume that ΔH and ΔS 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. 3.63 into 3.62:

(3.64)

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

Substituting this into eqn. 3.64 and rearranging:

(3.65)

Further rearrangement yields:

For a reasonably dilute solution, the activity of water will approximately equal its mole fraction, so that:

(3.66)

The entropy of melting is always positive, and since X is always less than 1, the left-hand side of eqn. 3.66 must always be positive. Thus, the ratio must always be greater than 1. So the temperature at which an aqueous solution will freeze will always be less than the melting point of pure water. Salting of roads is not a question of geochemical interest, but there are many examples of depression of the freezing point of geological interest. For example, the freezing point of the ocean is about –2°C, and this phenomenon is important in igneous petrology, as we shall see in the next chapter. A related phenomenon of geological interest is elevation of the boiling point of a liquid: for example, hydrothermal solutions boil at temperatures significantly above that of pure water. Can you demonstrate that elevation of the boiling point of an ideal solution depends only on the mole fraction of the solute?