The equilibrium constant for this reaction can be written as:
which reduces to
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.
3.9.6 Pressure dependence of equilibrium constant
Since
and
then
(3.97)
If ΔVr does 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.
3.10 PRACTICAL APPROACH TO ELECTROLYTE EQUILIBRIUM
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.
3.10.1 Choosing components and species
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.
3.10.2 Mass balance
This constraint, also sometimes called mole balance, is a very simple one, and as such it is easily overlooked. When a salt is dissolved in water, the anion and cation are added in stoichiometric proportions. If the dissolution of the salt is the only source of these ions in the solution, then for a salt of composition Cν+Aν– we may write:
(3.98)
Thus, for example, for a solution formed by dissolution of CaCl2 in water, the concentration of Cl– ion will be twice that of the Ca2+ ion. Even if CaCl2 is not the only source of these ions in solution, its congruent dissolution allows us to write the mass balance constraint in the form of a differential equation:
which just says that CaCl2 dissolution adds two Cl– ions to solution for every Ca2+ ion added.
By carefully choosing components and boundaries of our system, we can often write conservation equations for components. For example, suppose we have a liter of water containing dissolved CO2 in equilibrium with calcite (for example, groundwater in limestone). In some circumstances, we may want to choose our system as the water plus the limestone, in which case we may consider Ca conserved and write:
where CaCO3s is calcite (limestone) and
Choosing carbon as a component has the disadvantage that some carbon will be present as organic compounds, which we may not wish to consider. A wiser choice is to define CO2 as a component. Total CO2 would then include all carbonate species as well as CO2 (very often, total CO2 is expressed instead as total carbonate). The conservation equation for total CO2 for our system would be:
Here we see the importance of the distinction we made between components and species earlier in the chapter. Example 3.9 illustrates the use of mass balance.
