Law of mass action

Remarks

In an ideal gas mixture $\mu_i = \mu_i^\circ(T) + RT \ln(p_i/p^\circ)$. The equilibrium condition $\sum_i \nu_i \mu_i = 0$ becomes $\Delta_r G^\circ(T) + RT \sum_i \nu_i \ln(p_i/p^\circ) = 0$, and taking exponentials gives the law. For $\mathrm{N_2} + 3\,\mathrm{H_2} \rightleftharpoons 2\,\mathrm{NH_3}$ it reads $\dfrac{(p_{\mathrm{NH_3}}/p^\circ)^2}{(p_{\mathrm{N_2}}/p^\circ)(p_{\mathrm{H_2}}/p^\circ)^3} = K(T)$; raising the total pressure therefore favours ammonia, since the left side has total degree $-2$ in the pressures. For non-ideal mixtures, the same law holds with activities in place of $p_i/p^\circ$. The Gibbs free energy change $\Delta_r G^\circ$ is negative exactly when $K > 1$.

Going deeper

For a student

This is the thermodynamic, equilibrium form of the law; it is how one computes the composition of a reacting mixture from tabulated standard Gibbs energies. There is also a kinetic form, due to Guldberg and Waage, about the rate of an elementary reaction. It says that the rate is proportional to the product of the reactant concentrations, each raised to its stoichiometric coefficient. For an elementary reversible reaction, setting the forward and reverse rates equal recovers the equilibrium form. The equilibrium constant, written in concentrations, is then the ratio of the forward and reverse rate constants. The kinetic form is the basis of the differential equations of chemical reaction networks. The temperature dependence of $K$ is given by the van 't Hoff equation.