two equations, the atomic wt. cation terms cancel and we have:
Next, we want to calculate the number of moles of each cation per formula unit. A general formula for feldspar is: XY4O8, where X is Na, K, or Ca in the A site and Y is Al or Si in the tetrahedral site. So to calculate formula units in the A site, we divide the number of moles of Na, K, and Ca by the sum of moles of Na, K, and Ca. To calculate formula units in the tetrahedral site, we divide the number of moles of Al and Si by the sum of moles of Al and Si and multiply by 4, since there are four ions in this site. Since the number of oxygens is constant, we can refer to these quantities as the moles per eight oxygens. The following table shows the results of these calculations.
Cation formula units
| Mol. wt. oxide | Moles cation | Moles per 8 oxygens | |
| Si | 60.06 | 0.7385 | 2.077 |
| Al | 101.96 | 0.6836 | 1.923 |
| Ca | 56.08 | 0.3322 | 0.926 |
| Na | 61.98 | 0.0255 | 0.071 |
| K | 94.2 | 0.0011 | 0.003 |
The activity of albite is equal to the mole fraction of Na, 0.07; the activity of anorthite is 0.93.
3.9 EQUILIBRIUM CONSTANTS
Now that we have introduced the concepts of activity and activity coefficients, we are ready for one of the most useful parameters in physical chemistry: the equilibrium constant. Though we can predict the equilibrium state of a system, and therefore the final result of a chemical reaction, from the Gibbs free energy alone, the equilibrium constant is a convenient and succinct way express this. As we shall see, it is closely related to, and readily derived from, the Gibbs free energy.
3.9.1 Derivation and definition
Consider a chemical reaction such as:
carried out under isobaric and isothermal conditions. The Gibbs free energy change of this reaction can be expressed as:
(3.81)
At equilibrium, ΔG must be zero. A general expression then is:
(3.82)
where νi is the stoichiometric coefficient of species i. Equilibrium in such situations need not mean that all the reactants (i.e., those phases on the left side of the equation) are consumed to leave only products. Indeed, this is generally not so. Substituting eqn. 3.46 into 3.82 we obtain:
(3.83)
or:
(3.84)
The first term is simply the standard state Gibbs free energy change, ΔG°, for the reaction. There can be only one fixed value of ΔG° for a fixed standard state pressure and temperature, and therefore of the activity products. The activity products are therefore called the equilibrium constant K, familiar from elementary chemistry:
(3.85)
Substituting eqn. 3.85 into 3.84 and rearranging, we see that the equilibrium constant is related to the Gibbs free energy change of the reaction by the equation:
(3.86)
At this point, it is worth saying some more about standard states. We mentioned that one is free to choose a standard state, but there are pitfalls. In general, there are two kinds of standard states, fixed pressure–temperature standard states and variable P–T standard states. If you chose a fixed temperature standard state, then eqn. 3.86 is only valid at that standard-state temperature. If you chose a variable-temperature standard state, then eqn. 3.86 is valid for all temperatures, but ΔG° is then a function of temperature. The same goes for pressure. Whereas most thermodynamic quantities we have dealt with thus far are additive, equilibrium constants are multiplicative (see Example 3.6).
3.9.2 Law of mass action
Let's attempt to understand the implications of eqn. 3.85. Consider the dissociation of carbonic acid, an important geologic reaction:
For this particular case, eqn. 3.85 is expressed as:
The right side of the equation is a quotient, the product of the activities of the products divided by the product of the activities of the reactants and is called the reaction quotient. At equilibrium, the reaction quotient is equal to the equilibrium constant. The equilibrium constant therefore allows us to predict the relative amounts of products and reactants that will be present when a system reaches equilibrium.
Suppose now that we prepare a beaker of carbonic acid solution; it is not hard to prepare: we just allow pure water to equilibrate with the atmosphere. Let's simplify things by assuming that this is an ideal solution. This allows us to replace activities with concentrations (the concentration units will dictate how we define the equilibrium constant; see below). When the solution has reached equilibrium, just enough carbonic acid will have dissociated so that the reaction quotient will be equal to the equilibrium constant. Now let's add some H+ ions, perhaps by adding a little HCl.
