William M. White - Geochemistry

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.

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, K D, as:

(3.88)

K Dis related to the equilibrium constant K as:

(3.89)

where K λis 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 K Dvalues 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 K γdefined analogously to K λ. 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 constant is the solubility product . Consider the dissolution 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 K spis 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).

The apparent (molar) solubility product of fluorite (CaF 2) at 25°C is 3.9 × 10 −11. What is the concentration of Ca 2+ion in groundwater containing 0.1 mM of F –in equilibrium with fluorite?

Answer: Expressing eqn. 3.93for this case we have:

We take the activity of CaF 2as 1. Rearranging and substituting in values, we have:

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.

Since Δ G° = Δ H° – T Δ S° and Δ G° r= −R T 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 3.95allows us to calculate an equilibrium constant from fundamental thermodynamic data (see Example 3.8). Conversely, we can estimate values for Δ S ° and Δ H ° from the equilibrium constant, which is readily calculated if we know the activities of reactants and products. Equation 3.95has the form: