Читать книгу Geochemistry - William M. White - Страница 95

Free energy and other thermodynamic properties are dependent on composition. We need a way of expressing this dependence. For any extensive property of the system, such as volume, entropy, energy, or free energy, we can define a partial molar value, which expresses how that property will depend on changes in amount of one component. For example, we define the partial molar volume of component i in phase φ as:

(3.11)

(we will use small letters to denote partial molar quantities; the superscript refers to the phase and the subscript refers to the component). The plain language interpretation of eqn. 3.11 is that the partial molar volume of component i in phase φ tells us how the volume of phase φ will vary with an infinitesimal addition of component i, if all other variables are held constant. For example, the partial molar volume of Na in an aqueous solution such as seawater would tell us how the volume of that solution would change for an infinitesimal addition of Na. In this case i would refer to the Na component and φ would refer to the aqueous solution phase. In Table 2.2, we see that the molar volumes of the albite and anorthite end-members of the plagioclase solid solution are different. We could define as the partial molar volume of albite in plagioclase, which would tell us how the volume of plagioclase would vary for an infinitesimal addition of albite. (In this example, we have chosen our component as albite rather than Na. While we could have chosen Na, the choice of albite simplifies matters because the replacement of Na with Ca is accompanied by the replacement of Si by Al.)

Figure 3.5 Variation of the partial molar volumes of water and ethanol as a function of the mole fraction of ethanol in a binary solution. This figure also illustrates the behavior of a very nonideal solution. After Nordstrom and Munoz (1986).

The second expression in eqn. 3.11 says that the volume of a phase is the sum of the partial molar volumes of the components times the number of moles of each component present. Thus, the volume of plagioclase would be the sum of the partial molar volumes of the albite and anorthite components weighted by the number of moles of each.

Another example might be a solution of water and ethanol. The variation of the partial molar volumes of water and ethanol in a binary solution is illustrated in Figure 3.5. This system illustrates very clearly why the qualification “for an infinitesimal addition” is always added: the value of a partial molar quantity of a component may vary with the amount of that component present.

Equation 3.11 can be generalized to all partial molar quantities and also expresses an important property of partial molar quantities: an extensive variable of a system or phase is the sum of its partial molar quantities for each component in the system. In our example above, this means that the volume of plagioclase is the sum of the partial molar volume of the albite and anorthite components.

Generally, we find it more convenient to convert extensive properties to intensive properties by dividing by the total number of moles in the system, Σn. Dividing both sides of eqn. 3.11 by Σn we have:

(3.12)

This equation says that the molar volume of a substance is the sum of the partial molar volumes of its components times their mole fractions. For a pure phase, the partial molar volume equals the molar volume since X = 1.