From Kerrick and Jacobs (1981).
3.6.3 Activities and activity coefficients
Fugacities are thermodynamic functions that are directly related to chemical potential and can be calculated from measured P–T–V properties of a gas, though we will not discuss how. However, they have meaning for solids and liquids as well as gases, since solids and liquids have finite vapor pressures. Whenever a substance exerts a measurable vapor pressure, a fugacity can be calculated. Fugacities are relevant to the equilibria between species and phase components, because if the vapor phases of the components of some solid or liquid solutions are in equilibrium with each other, and with their respective solid or liquid phases, then the species or phase components in the solid or liquid must be in equilibrium. One important feature of fugacities is that we can use them to define another thermodynamic parameter, the activity, a:
(3.45)
ƒ° is the standard state fugacity. Its value depends on the standard state you choose. You are free to choose a standard state convenient for whatever problem you are addressing.
If we substitute eqn. 3.45 into eqn. 3.43, we obtain the important relationship:
(3.46)
The “catch” on selecting a standard state for ƒ°, and hence for determining ai in eqn. 3.46, is that this state must be the same as the standard state for μ°. Thus, we need to bear in mind that standard states are implicit in the definition of activities, and that those standard states are tied to the standard-state chemical potential. Until the standard state is specified, activities have no meaning.
Comparing eqn. 3.46 with 3.26 leads to:
(3.47)
Thus, in ideal solutions, the activity is equal to the mole fraction.
Chemical potentials can be thought of as driving forces that determine the distribution of components between phases of variable composition in a system. Activities can be thought of as the effective concentration or the availability of components for reaction. In real solutions, it would be convenient to relate all nonideal thermodynamic parameters to the composition of the solution, because composition is generally readily and accurately measured. To relate activities to mole fractions, we define a new parameter, the rational activity coefficient, λ. The relationship is:
(3.48)
The rational activity coefficient differs slightly in definition from the practical activity coefficient, γ, used in aqueous solutions. λ is defined in terms of mole fraction, whereas γ is variously defined in terms of moles of solute per moles of solvent, or more commonly, moles of solute per kg or liter of solution. Consider, for example, the activity of Na in an aqueous sodium chloride solution. For λNa, X is computed as:
whereas for γNa, XNa is:
where n indicates moles of substance. γ is also used for other concentration units that we will introduce in section 3.7.
Example 3.2 Using fugacity to calculate Gibbs free energy
The minerals brucite (Mg(OH)2) and periclase (MgO) are related by the reaction:
Which side of this reaction represents the stable phase assemblage at 600°C and 200 MPa?
Answer: We learned how to solve this sort of problem in Chapter 2: the side with the lowest Gibbs free energy will be the stable assemblage. Hence, we need only to calculate ΔGr at 600°C and 200 MPa. To do so, we use eqn. 2.130:
(2.130)
Our earlier examples dealt with solids, which are incompressible to a good approximation so we could simply treat ΔVr as independent of pressure. In that case, the solution to the first integral on the left was simply ΔVr(P′–Pref). The reaction in this case, like most metamorphic reactions, involves H2O, which is certainly not incompressible: its volume as steam or a supercritical fluid is very much a function of pressure. Let's isolate the difficulty by dividing ΔVr into two parts: the volume change of reaction due to the solids, in this case the difference between molar volumes of periclase and brucite, and the volume change due to H2O. We will denote the former as ΔVS and assume that it is independent of pressure. The second integral in eqn.
