is generally given units of molality in this case (it is dimensionless, as we defined it in eqn. 3.45), so that in this hypothetical standard state, activity equals molality. The standard state is hypothetical because, for most electrolytes, the activity will be less than 1 in a 1 m (molal) solution. Because the standard state generally is unattainable in reality, we must also define an attainable reference state, from which experimental measurements can be extrapolated. By convention, the reference state is that of an infinitely dilute solution – the Henry's law state. For multicomponent solutions, we also specify that the concentrations of all other components be held constant. Hence the reference state is:
Figure 3.11 Relationship of activity and molality, reference state, and standard state for aqueous solutions. After Nordstrom and Munoz (1986).
(3.70)
This convention is illustrated in Figure 3.11. In such solutions, the activity coefficient can be shown to depend on the charge of the ion, its concentration, and the concentration of other ions in the solution as well as temperature and other parameters of the solute. Comparing eqn. 3.70 with 3.46 and 3.48, we see that under these conditions, the activity coefficient is 1. By referring to infinite dilution, we are removing the effect of solute–solute interactions. The standard state properties of an electrolyte solution therefore only take account of solvent–solute interactions.
Clearly, it is impossible to measure the properties of the solute, such as chemical potential or molar volume, at infinite dilution. In practice, this problem is overcome by measuring properties at some finite dilution and extrapolating the result to infinite dilution. Indeed, even at finite concentrations, it is not possible to measure directly many properties of electrolytes. Volume is a good example. One cannot measure the volume of the solute, but one can measure the volume change of the solution as a function of concentration of the solute. Then by assuming that the partial molar volume of water does not change, a partial molar volume of the solute can be calculated. This is called the apparent molar volume,
Figure 3.12 Apparent molar volume of NaCl in aqueous solution as a function of molality. The standard molar volume, V°, is the apparent molar volume at infinite dilution.
(3.71)
This convention leads to some interesting effects. For example, the apparent molar volume of magnesium sulfate increases with pressure, and many other salts, including NaCl (Figure 3.13), exhibit the same behavior. Just as curiously, the apparent molar volume of sodium chloride in saturated aqueous solution becomes negative above ∼200°C (Figure 3.13). Many other salts show the same effect. These examples emphasize the “apparent” nature of molar volume when defined in this way. Of course, the molar volume of NaCl does not actually become negative; rather, this is the result of the interaction between Na+ and Cl– and H2O (electrostriction) and the convention of assigning all nonideality to sodium chloride.
Figure 3.13 Standard molar volume of NaCl in aqueous solution as a function of temperature and pressure. Based on the data of Helgeson and Kirkham (1976).
The concentration of a salt consisting of νA moles of cation A and νB moles of cation B is related to the concentration of its constituent ionic species as:
(3.72)
By convention, the thermodynamic properties of ionic species A and B are related to those of the salt AB by:
(3.73)
where Ψ is some thermodynamic property. Thus the chemical potential of MgCl2 is related to that of Mg2+ and Cl– as:
The same holds for enthalpy of formation, entropy, molar volume, and so on.
A final important convention is that the partial molar properties and energies of formation for the proton (H+) are taken to be zero under all conditions.
3.7.3 Activities in electrolytes
The assumption we made for ideal solution behavior was that interactions between molecules (species might be a better term in the case of electrolyte solutions) of solute and molecules of solvent were not different from those interactions between solvent ions only. In light of the discussion of aqueous solutions earlier, we can see this is clearly not going to be the case for an electrolyte solution. We have seen significant deviations from ideality even where the components have no net charge (e.g., water–ethanol); we can expect greater deviations due to electrostatic interactions between charged species.
The nature of these interactions suggests that a purely macroscopic viewpoint, which takes no account of molecular and ionic interactions, may have severe limitations in predicting equilibria involving electrolyte solutions. Thus, chemists and geochemists concerned with the behavior of electrolytes have had to incorporate a microscopic viewpoint into electrolyte theory. On the other hand, they did not want to abandon entirely the useful description of equilibria based on thermodynamics. We have already introduced concepts, the activity and the activity coefficient, which allow us to treat nonideal behavior within a thermodynamic framework. The additional task imposed by electrolyte solutions, and indeed all real solutions, therefore, is not to rebuild the framework, but simply to determine activities from readily measurable properties of the solution. The dependence of all partial molar properties of a solute on concentration can be determined once the activity coefficient and its temperature and pressure dependence are known.
3.7.3.1 The Debye–Hückel and Davies equations
