of NaCl in water. The equilibrium constant is:
where aq denotes the dissolved ion and s denotes solid. Because the activity of NaCl in pure sodium chloride solid is 1, this reduces to:
(3.93)
where Ksp is called the solubility product. You should note that it is generally the case in dissolution reactions such as this that we take the denominator (i.e., the activity of the solid) to be 1 (see Example 3.7).
Example 3.7 Using the solubility product
The apparent (molar) solubility product of fluorite (CaF2) at 25°C is 3.9 × 10−11. What is the concentration of Ca2+ ion in groundwater containing 0.1 mM of F– in equilibrium with fluorite?
Answer: Expressing eqn. 3.93 for this case we have:
We take the activity of CaF2 as 1. Rearranging and substituting in values, we have:
3.9.4 Henry's law and gas solubilities
Consider a liquid, water for example, in equilibrium with a gas, the atmosphere for example. Earlier in this chapter, we found that the partial pressure of component i in the gas could be related to the concentration of a component i in the liquid by Henry's law:
(3.10)
where h is Henry's law constant. We can rearrange this as:
(3.94)
Notice that this equation is analogous in form to the equilibrium constant expression (3.88), except that we have used a partial pressure in place of one of the concentrations. A Henry's law constant is thus a form of equilibrium constant used for gas solubility: it relates the equilibrium concentration of a substance in a liquid solution to that component's partial pressure in a gas.
3.9.5 Temperature dependence of equilibrium constant
Since ΔG° = ΔH° – TΔS° and ΔG°r = −RT ln K, it follows that in the standard state, the equilibrium constant is related to enthalpy and entropy change of reaction as:
(3.95)
Equation 3.95 allows us to calculate an equilibrium constant from fundamental thermodynamic data (see Example 3.8). Conversely, we can estimate values for ΔS° and ΔH° from the equilibrium constant, which is readily calculated if we know the activities of reactants and products. Equation 3.95 has the form:
where a and b are ΔH°/R and ΔS°/R, respectively. If we can assume that ΔH and ΔS are constant over some temperature range (this is likely to be the case provided the temperature interval is small), then a plot of ln K vs. 1/T will have a slope of ΔH°/R and an intercept of ΔS°/R. Thus, measurements of ln K made over a range of temperatures and plotted vs. 1/T provide estimates of ΔH° and ΔS°. Even if ΔH and ΔS are not constant, they can be estimated from the instantaneous slope and intercept of a curve of ln K plotted against 1/T. This is illustrated in Figure 3.17, which shows measurements of the solubility constant for barite (BaSO4) plotted in this fashion (though in this case the log10 rather than natural logarithm is used). From changes of ΔH and ΔS with changing temperature and knowing the heat capacity of barite, we can also estimate heat capacities of the Ba2+ and SO42– ions, which would obviously be difficult to measure directly. We can, of course, also calculate ΔG directly from eqn. 3.86. Thus, a series of measurements of the equilibrium constant for simple systems allows us to deduce the fundamental thermodynamic data needed to predict equilibrium in more complex systems.
Figure 3.17 Log of the solubility constant of barite plotted against the inverse of temperature. The slope of a tangent to the curve is equal to −ΔH/R. The intercept of the tangent (which occurs at 1/T = 0 and is off the plot) is equal to ΔS/R. After Blount (1977).
Example 3.8 Calculating equilibrium constants and equilibrium concentrations
The hydration of olivine to form chrysotile (a serpentine mineral) may be represented in a pure Mg system as:
If this reaction controlled the concentration of Mg2+ of the metamorphic fluid, what would the activity of Mg2+ be in that fluid if it had a pH of 4.0 at 300° C?
Answer: Helgeson (1967) gives the thermodynamic data shown in the table below for the reactants at 300° C. From these data, we use Hess's law to calculate
| Species | ΔH° kJ | S° J/K |
| Mg3Si2O5(OH)4 | −4272.87 | 434.84 |
| Mg2+ | −366.46 |
|
