the reaction quotient increases above that of the equilibrium constant and the system is no longer in equilibrium. Systems will always respond to disturbances by moving toward equilibrium (how fast they respond is another matter, and one that we will address in Chapter 5). The system will respond by adjusting the concentrations of the three species until equilibrium is again achieved; in this case, hydrogen and bicarbonate ions will combine to form carbonic acid until the reaction quotient again equals the equilibrium constant. We can also see that had we reduced the number of hydrogen ions in the solution (perhaps by adding a base), the reaction would have been driven the other way (i.e., hydrogen ions would be produced by dissociation). Equation 3.85 is known as the law of mass action, which we can state more generally: Changing the concentration of one species in a system at equilibrium will cause a reaction in a direction that minimizes that change.
Example 3.6 Manipulating reactions and equilibrium constant expressions
Often we encounter a reaction for which we have no value of the equilibrium constant. In many cases, however, we can derive an equilibrium constant by considering the reaction of interest to be the algebraic sum of several reactions for which we do have equilibrium constant values. For example, the concentration of carbonate ion is often much lower than that of the bicarbonate ion. In such cases, it is more convenient to write the reaction for the dissolution of calcite as:
(3.87)
Given the following equilibrium constants, what is the equilibrium constant expression for the above reaction?
Answer: Reaction 3.87 can be written as the algebraic sum of three reactions:
The initial inclination might be to think that if we can sum the reactions, the equilibrium constant of the resulting reaction is the sum of the equilibrium constants of the components. However, this is not the case. Whereas we sum the reactions, we take the product of the equilibrium constants. Thus, our new equilibrium constant is:
For several reasons (chief among them that equilibrium constants can be very large or very small numbers), it is often more convenient to work with the log of the equilibrium constant. A commonly used notation is pK. pK is the negative logarithm (base 10) of the corresponding equilibrium constant (note this notation is analogous to that used for pH). The pK's sum and our equilibrium constant expression is:
3.9.2.1 Le Chatelier's principle
We can generalize this principle to the effects of temperature and pressure as well. Recall that:
(2.128)
and
(2.129)
and that systems respond to changes imposed on them by minimizing G. Thus, a system undergoing reaction will respond to an increase in pressure by minimizing volume. Similarly, it will respond to an increase in temperature by maximizing entropy. The reaction ice → water illustrates this. If the pressure is increased on a system containing water and ice, the equilibrium will shift to favor the phase with the least volume, which is water (recall that water is unusual in that the liquid has a smaller molar volume than the solid). If the temperature of that system is increased, the phase with the greatest molar entropy is favored, which is also water.
Another way of looking at the effect of temperature is to recall that:
Combining this with eqn. 2.129, we can see that if a reaction A + B → C + D generates heat, then increasing the temperature will retard formation of the products, that is, the reactants will be favored.
A general statement that encompasses both the law of mass action and the effects we have just discussed is then:
When perturbed, a system reacts to minimize the effect of the perturbation.
This is known as Le Chatelier's principle.
3.9.3 KD values, apparent equilibrium constants, and the solubility product
It is often difficult to determine activities for phase components or species, and therefore it is more convenient to work with concentrations. We can define a new “constant,” the distribution coefficient, KD, as:
(3.88)
KD is related to the equilibrium constant K as:
(3.89)
where Kλ is simply the ratio of activity coefficients:
(3.90)
Distribution coefficients are functions of temperature and pressure, as are the equilibrium constants, though the dependence of the two may differ. The difference is that KD values are also functions of composition.
An alternative to the distribution coefficient is the apparent equilibrium constant, which we define as:
(3.91)
(3.92)
with Kγ defined analogously to Kλ. The difference between the apparent equilibrium constant and the distribution coefficient is that we have defined the former in terms of molality and the latter in terms of mole fraction. Igneous geochemists tend to use the distribution coefficient, aqueous geochemists the apparent equilibrium constant.
Another special form of the equilibrium
