The meaning of heterogeneous equilibrium

So far, the equilibria we have discussed have involved a single phase of matter, either gas or solution. There we could express the equilibrium constant in terms of partial pressures of gases or concentrations of dissolved species in a solvent. These are known as homogeneous equilibrium situations.

Chemical reactions and chemical equilibria that involve more than one phase are called heterogeneous. Examples are phase equilibria, e.g.

$${\rm H}_2{\rm O}(l) \rightleftharpoons {\rm H}_2{\rm O}(g)$$

or solvation of solids in water, e.g.

$${\rm I}_2(s) \rightleftharpoons {\rm I}_2(aq)$$

Acid-base reactions are also examples of heterogeneous equilibria as they involve both pure liquids (the solvent) and the dissolved species. Thus, before we can consider acid-base equilibria, we need to discuss the rules for writing equilibrium constant expressions for heterogeneous equilibria.

How can we generalize the law of mass action (equilibrium constant expressions) to deal with heterogeneous equilibria? Since the equilibrium constant is derived from the Gibbs free energy, we need some quantity that can generally characterize a thermodynamic state for a solid, liquid or gas. Recall that for $n$ moles of an ideal gas at temperature $T$, the change in the Gibbs free energy from a reference state, where its pressure is $P_{\rm ref}$, to an arbitrary state where its pressure is $P$, the change in the Gibbs free energy is

$$\Delta G = nRT\ln \left({P \over P_{\rm ref}}\right)$$

That is, at fixed $n$ and $T$, the pressure alone can be used to characterize the thermodynamic state (the volume can always be determined from the equation of state known the pressure, $P$, e.g., for an ideal gas, $V=nRT/P$).

Likewise, for $n$ moles of a solute in an ideal, dilute solution at a temperature $T$, the change in the Gibbs free energy from a reference state where the concentration is $c_{\rm ref}$ to an arbitrary state where the concentration is $c$ (created by adding or removing solvent only), the change in the Gibbs free energy is

$$\Delta G = nRT\ln \left({c \over c_{\rm ref}}\right)$$

That is, at fixed $n$ of solute and fixed $T$, the concentration alone can be used to characterize the thermodynamic state.

The pressure of a gas or concentration of a solution are examples of a more general thermodynamic quantity called the activity, $a$. The specific definition of an activity depends on the particular phase of matter being described. It is generally derived by considering the number of microscopic states available to a system. For a gas, the general definition of activity is

$$a = {\gamma P \over P_{\rm ref}}$$

where $\gamma$ is called the activity coefficient, and is a measure of non-ideal behavior. For a solution, the activity is defined to be

$$a = {\gamma c \over c_{\rm ref}}$$

For solids and liquids, the general convention is that the activity of a pure substance is taken to be 1.

The change in Gibbs free energy when $n$ moles of an arbitrary substance is changed from a reference state to an arbitrary thermodynamic state at constant temperature $T$ can then be defined in terms of the activity as

$$\Delta G = nRT\ln a$$

since $a$ is a dimensionless quantity. Note that for an ideal gas or an ideal solution, the activity coefficient $\gamma=1$.

Starting with this definition, it is straightforward to apply the reasoning in Chapter 9 and derive the equilibrium constant expression for the generic reaction (possibly involving heterogeneous phases)

$$a{\rm A} + b{\rm B}\rightleftharpoons c{\rm C} + d{\rm D}$$

$$K = {a_{\rm C}^c a_{\rm D}^d \over a_{\rm A}^a a_{\rm B}^b}$$

Note that this definition reduces to the definitions we had before when the reaction involves either only gaseous species or dissolved species.

With this general definition, and the convention that activities of pure substances are 1, we can mix the kinds of quantities that determine an equilibrium constant expression (e.g. partial pressures and concentrations can both appear in the expression). The general rules are:

1.

Gases enter equilibrium constant expressions as partial pressures in atm.

2.

Dissolved species enter as concentrations in moles/L.

3.

Pure solids and pure liquids, having activities equal to 1, do not enter into equilibrium constant expressions, nor does a solvent taking part in a chemical reaction, provided the solution is dilute.

Example: Consider the reaction:

$${\rm C}(s,gr) + {\rm CO}_2(g)\rightleftharpoons 2{\rm CO}(g)$$

Assuming that CO$_2$ and CO can be treated as ideal gases, what is the equilibrium constant expression? If the initial partial pressure of CO$_2$ is 1.4 atm, what is the partial pressure of each gas at equilibrium? Take the value of $K$ to be 4.2.

Solution: The equilibrium constant expression is generally:

$$K = {(a_{\rm CO})^2 \over (a_{{\rm CO}_2}) (a_{\rm C(s,gr)})}$$

The activities of the gaseous species are just their partial pressures. The activity of graphite is 1. Thus, $K$ reduces to

$$K = {(P_{\rm CO})^2 \over (P_{{\rm CO}_2})}$$

Let $2x$ be the partial pressure of CO gas at equilibrium. Then the partial pressure of CO$_2$ gas is 1.4-$x$, and we can use the equilibrium constant expression to solve for $x$:

$$4.2 = {(2x)^2 \over (1.4-x)}$$

$$x = 0.8\;{\rm atm}$$

leading to partial pressures of 1.6 atm for CO and 0.6 atm for CO$_2$.