## The Equilibrium Constant

#### Learning Objective

• Duplicate the form of the equation for the thermodynamic equilibrium constant

#### Key Points

• In the equilibrium constant expression, the concentrations of the products go in the numerator and the concentrations of the reactants go the denominator.
• The equilibrium constant is derived from the rate laws for the forward and reverse reactions.
• Only species that exist in the gas or aqueous phases are included in the Keq expression. Reactants and products that exist as solids and liquids are omitted.
• The value of Keq can be used to make qualitative judgments about the thermodynamics of the forward and reverse reactions.

#### Terms

• chemical equilibriumThe state of a reversible reaction in which the rates of the forward and reverse reactions are the same.
• activityReferring to the ideal concentration of a species.

The relationship between forward and reverse reactions in dynamic equilibrium can be expressed mathematically in what is known an equilibrium expression, or Keq expression. Most often, this expression is written in terms of the concentrations of the various reactants and products, and is given by:

$K_{eq}=\frac{[C]^{m}[D]^{n}}{[A]^{a}[B]^{b}}$

Species in brackets represent the concentrations of products, which are always in the numerator, and reactants, which are always in the denominator. Each of the concentrations is raised to a power equal to the stoichiometric coefficient for each species.

## Derivation of the Equilibrium Expression from the Law of Mass Action

Consider the following general, reversible reaction:

$aA+bB\rightleftharpoons mC+nD$

Assuming this reaction is an elementary step, we can write the rate laws for both the forward and reverse reactions:

$\text{rate}_{\text{forward}}=k_1[A]^a[B]^b$

$\text{rate}_{\text{reverse}}=k_2[C]^m[D]^n$

However, we know that the forward and reverse reaction rates are equal in equilibrium:

$k_1[A]^a[B]^b=k_2[C]^m[D]^n$

Rearranging this equation and separating the rate constants from the concentration terms, we get:

$\frac{k_1}{k_2}=\frac{[C]^m[D]^n}{[A]^a[B]^b}$

Notice that the left side of the equation is the quotient of two constants, which is simply another constant. We simplify and write this constant as Keq:

$\frac{k_1}{k_2}=K_{C}=\frac{[C]^{m}[D]^{n}}{[A]^{a}[B]^{b}}$

Keep in mind that the only species that should be included in the Keq expression are reactants and products that exist as gases or are in aqueous solution. Reactants and products in the solid and liquid phases, even if they are involved in the reaction, are not included in the Keq expression, as these species have activities of 1.

“Activity” is a term in physical chemistry used to describe a substance’s ideal concentration. The activity for solids and liquids is 1, so they essentially have a constant concentration of 1, and thereby have no effect on the Keq expression. As such, they are omitted.

## Example

Write the Keq expression for the following reaction:

$H_2O(g)+C(s)\rightleftharpoons H_2(g)+CO(g)$

The expression would be written as:

$K_{eq}=\frac{[H_2][CO]}{[C][H_2O]}=\frac{[H_2][CO]}{1\times[H_2O]}=\frac{[H_2][CO]}{[H_2O]}$

Note that because it is a solid, the activity of C(s) is 1, and it is omitted from the final K expression.

## Predicting the Direction of a Reaction From the Value of Keq

When looking at the Keq expression, we should notice that it is essentially a ratio relating the concentrations of products to the concentrations of reactants at equilibrium. If we know the value of Keq, we can draw some conclusions about the thermodynamics of the forward and reverse reactions. These conclusions are summarized as follows:

• A Keq value of << 1 is indicative that the reverse reaction is highly favored over the forward reaction, and the concentrations of reactants are much higher than those of the products at equilibrium.
• A Keq value $\approx$ 1 is indicative that the forward and reverse reactions are about equally favorable, for the ratio of concentrations of reactants and products is close to unity.
• A Keq >>1 is indicative that the forward reaction is highly favored over the reverse reaction, and at equilibrium, the concentrations of the products are much greater than those of the reactants.