8.5: Equilibria

Now that we have a good idea about the factors that affect how fast a reaction goes, let us return to a discussion of what factors affect how far a reaction goes. As previously discussed, a reaction reaches equilibrium when the rate of the forward reaction equals the rate of the reverse reaction, so the concentrations of reactants and products do not change over time. The equilibrium state of a particular reaction is characterized by what is known as the equilibrium constant, K_eq. We can generalize this relationship for a general reaction:

nA+mB⇄ oC+pD.

Note that each concentration is raised to the power of its coefficient in the balanced reaction. By convention, the constant is always written with the products on the numerator, and the reactants in the denominator. So large values of K_eq indicate that, at equilibrium, the reaction mixture has more products than reactants. Conversely, a small value of K_eq (typically <1, depending on the form of K_eq) indicates that there are fewer products than reactants in the mixture at equilibrium. The expression for K_eq depends on how you write the direction of the reaction. You can work out for yourself that K_eq(forward) = 1/K_eq(reverse). One other thing to note is that if a pure liquid or solid participates in the reaction, it is omitted from the equilibrium expression for K_eq. This makes sense because the concentration of a pure solid or liquid is constant (at constant temperature). The equilibrium constant for any reaction at a particular temperature is a constant. This means that you can add reactants or products and the constant does not change.¹⁶⁵ You cannot , however, change the temperature, because that will change the equilibrium constant as we will see shortly. The implications of this are quite profound. For example, if you add or take away products or reactants from a reaction, the amounts of reactants or products will change so that the reaction reaches equilibrium again—with the same value of K_eq. And because we know (or can look up and calculate) what the equilibrium constant is, we are able to figure out exactly what the system will do to reassert the equilibrium condition.

Let us return to the reaction of acetic acid and water: AcOH + H₂O ⇄ H₃O⁺ + AcO^–,

we can figure out that the equilibrium constant would be written as: K_eq = [H₃O⁺][AcO^–]/[AcOH].

The H2O term in the reactants can be omitted even though it participates in the reaction, because it is a pure liquid and its concentration does not change appreciably during the reaction. (Can you calculate the concentration of pure water?) We already know that a 0.10-M solution of AcOH has a pH of 2.9, so we can use this experimentally-determined data to calculate the equilibrium constant for a solution of acetic acid. A helpful way to think about this is to set up a table in which you note the concentrations of all species before and after equilibrium.

	[AcOH] M	[H3O^+]	[AcO^–] M
Initial Concentration	0.10	1 x 10^–7 (from water)	0
Change in Concentration (this is equal to the amount of AcOH that ionized - and can be calculated from the pH)	– 1.3 x 10^–3 M (because the AcOH must reduce by the same amount that the H+ increases)	10–pH = 1.3 x 10^–3 M	1.3 x 10^–3 M (because the same amount of acetate must be produced as H^+)
Final (equilibrium concentration)	0.10 – 1.3 x 10^–3 ~ 0.10	(1.3 x 10^–3) + (1 x 10^–7) ~ 1.3 x 10^–3	1.3 x 10^–3

You can also include the change in concentration as the system moves to the equilibrium state: AcOH + H₂O ⇄ H₃O⁺ + AcO^–. Using the data from this type of analysis, we can calculate the equilibrium constant: K_eq = (1.3 x 10^–3)²/0.1, which indicates that K_eq for this reactions equals 1.8 x 10^–5. Note that we do not use a large number of significant figures to calculate K_eq because they are not particularly useful, since we are making approximations that make a more accurate calculation not justifiable. In addition, note that K_eq itself does not have units associated with it.

References

¹⁶⁵ Strictly speaking, it is not concentrations that appear in the expression for K. Rather, it is another property called the activity (a)—often called the effective concentration. The activity takes into account the interactions between molecules and ions and solvents, but for our purposes it is acceptable to use concentrations in the expressions for Keq. One outcome of this is that activity is a dimensionless quantity, so equilibrium constants are one of the few places where we don’t have to worry about getting the right units!