Equilibrium, energy, and rates

A reaction at equilibrium has not stopped. The forward and reverse rates are equal, so concentrations stay constant. The equilibrium constant K expresses that condition. For aA + bB ⇌ cC + dD, K is [C]ᶜ[D]ᵈ / [A]ᵃ[B]ᵇ, with each concentration raised to its coefficient. Pure solids and the solvent in a dilute solution are omitted, because their activities are essentially 1. K depends on temperature and not, directly, on how you mix the starting materials. The reaction quotient Q has the same form using the current concentrations. If Q < K the reaction runs forward; if Q > K, backward.

Enthalpy H is the heat of reaction at constant pressure. Entropy S measures the number of ways the energy can be arranged. The quantity that actually decides spontaneity at constant temperature and pressure is the Gibbs free energy, G = H − TS, and the change ΔG = ΔH − TΔS. Negative ΔG means the forward direction increases the entropy of the universe, which is the second law restated in a convenient potential. The standard free-energy change is tied to the equilibrium constant by ΔG° = −RT ln K. A reaction can be spontaneous and still immeasurably slow. Thermodynamics does not mention time.

Kinetics does. A rate law such as rate = k[A]ᵐ[B]ⁿ is determined by experiment. The exponents are the orders, and their sum is the overall order. For a first-order process, concentration decays as e^{−kt} and the half-life is ln 2 / k, independent of how much you started with. Temperature dependence of k is summarized by the Arrhenius equation: a plot of ln k against 1/T is a straight line whose slope is −Eₐ/R. A catalyst lowers the activation energy and is not consumed. It does not change K.