William M. White - Geochemistry

where a and b are Δ H° /R and Δ S° /R, respectively. If we can assume that Δ H and Δ S 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 Δ H° /R and an intercept of Δ S° /R. Thus, measurements of ln K made over a range of temperatures and plotted vs. 1/ T provide estimates of Δ H° and Δ S° . Even if Δ H and Δ S 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 (BaSO 4) plotted in this fashion (though in this case the log 10rather than natural logarithm is used). From changes of Δ H and Δ S with changing temperature and knowing the heat capacity of barite, we can also estimate heat capacities of the Ba 2+and SO 4 2–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 −Δ H / R . The intercept of the tangent (which occurs at 1/ T = 0 and is off the plot) is equal to Δ S / R . After Blount (1977).

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 Mg 2+of the metamorphic fluid, what would the activity of Mg 2+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 and as −231.38 kJ and −253.01 J/K respectively. The equilibrium constant for the reaction may be calculated as:

The equilibrium constant for this reaction can be written as:

which reduces to if we take the activities of water, chrysotile, and forsterite as 1. Since pH = – log aH +, we may rearrange and obtain the activity of the magnesium ion as:

Taking the derivative with respect to temperature of both sides of eqn. 3.95(while holding pressure constant), we have:

(3.96)

This equation is known as the van't Hoff equation .

Since

and

then

(3.97)

If Δ V rdoes not depend on pressure, this equation can be integrated to obtain:

(3.97a)

This assumption will be pretty good for solids because their compressibilities are very low, but slightly less satisfactory for reactions involving liquids (such as dissolution), because they are more compressible. This assumption will be essentially totally invalid for reactions involving gases, because their volumes are highly pressure-dependent.

With the equilibrium constant now in our geochemical toolbox, we have the tools necessary to roll up our sleeves and get to work on some real geochemical problems. Even setting aside nonideal behavior, electrolyte solutions (geologic ones in particular) often have many components and can be extremely complex. Predicting their equilibrium state can therefore be difficult. There are, however, a few rules for approaching problems of electrolyte solutions that, when properly employed, make the task much more tractable.

We emphasized at the beginning of the chapter the importance of choosing the components in a system. How well we choose components will make a difference to how easily we can solve a given problem. Morel and Hering (1993) suggested these rules for choosing components and species in aqueous systems:

1 All species should be expressible as stoichiometric functions of the components, the stoichiometry being defined by chemical reactions.

2 Each species has a unique stoichiometric expression as a function of the components.

3 H2O should always be chosen as a component.

4 H+ should always be chosen as a component.

H +activity, or pH, is very often the critical variable, also called the master variable , in problems in natural waters. In addition, recall that we define the free energy of formation of H +as 0. For these reasons, it is both convenient and important that H +be chosen as a component.