4.1 Equilibrium Constant

The equilibrium constant, K, denotes a ratio of products to reactants for a system that is at equilibrium and, for this class, will be treated as dimensionless. We construct our equilibrium constant for the reversible reaction

\[2\mathrm{NO_2}(g) \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} \mathrm{N_2O_4}(g)\]

from the ratio of the reverse and forward rate laws.

\[\begin{align*} \mathrm{rate}_{\mathrm{f}} &= k_1[\mathrm{NO_2}]^2 \\[1.5ex] \mathrm{rate}_{\mathrm{r}} &= k_{-1}[\mathrm{N_2O_4}] \end{align*}\]

At equilibrium, these two rates are equal.

\[\begin{align*} \mathrm{rate}_{\mathrm{f}} &= \mathrm{rate}_{\mathrm{r}} \quad \therefore \\[1.5ex] k_1[\mathrm{NO_2}]^2 &=k_{-1}[\mathrm{N_2O_4}] \end{align*}\]

Since the rates are equal, they cancel. We can rearrange the expression by placing the rate constants on one side.

\[\begin{align*} k_1[\mathrm{NO_2}]^2 &=k_{-1}[\mathrm{N_2O_4}] \\[1.5ex] \dfrac{k_1}{k_{-1}} &= \dfrac{[\mathrm{N_2O_4}]}{[\mathrm{NO_2}]^2}\\[1.5ex] K &= \dfrac{[\mathrm{N_2O_4}]}{[\mathrm{NO_2}]^2} \end{align*}\]

The ratio of rate constants becomes a numerical value referred to as K, the equilibrium constant for the reaction. The final equation is called the equilibrium expression and, as seen here, is expressed as the ratio of products to reactants. Notice the exponent “2” in the denominator comes from the stoichiometric coefficient from the balanced chemical equation.