#### 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 K
_{eq}expression. Reactants and products that exist as solids and liquids are omitted. - The value of K
_{eq}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 K_{eq} expression. Most often, this expression is written in terms of the concentrations of the various reactants and products, and is given by:

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

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:

[latex]aA+bB\rightleftharpoons mC+nD[/latex]

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

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

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

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

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

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

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

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 K_{eq}:

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

Keep in mind that the only species that should be included in the K_{eq} 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 K_{eq} 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 K_{eq} expression. As such, they are omitted.

## Example

Write the K_{eq} expression for the following reaction:

[latex]H_2O(g)+C(s)\rightleftharpoons H_2(g)+CO(g)[/latex]

The expression would be written as:

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

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 K_{eq}

When looking at the K_{eq} 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 K_{eq}, we can draw some conclusions about the thermodynamics of the forward and reverse reactions. These conclusions are summarized as follows:

- A K
_{eq}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 K
_{eq}value [latex]\approx[/latex] 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 K
_{eq}>>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.