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Equilibrium Constant

Represents the ratio of product and reactant concentrations at equilibrium, derived from the partition function and temperature.
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The statement of the theorem

Let S\mathcal{S} be a system undergoing a reaction ∑i=1NνiXi=0\sum_{i=1}^{N} \nu_i X_i = 0, where XiX_i are chemical species and νi\nu_i are stoichiometric coefficients. Define the grand canonical partition function Z(μ,T,V)\mathcal{Z}(\mu, T, V) such that the chemical potential of species ii is μi=−(∂∂Niln⁡Z)T,V,Nj≠i\mu_i = -\left(\frac{\partial}{\partial N_i} \ln \mathcal{Z}\right)_{T, V, N_{j\neq i}}. The equilibrium constant KK for the reaction is defined by the ratio of activities (or concentrations) at equilibrium, K=∏i=1Naiνi/νirefK = \prod_{i=1}^{N} a_i^{\nu_i/\nu_i^\text{ref}}, where aia_i is the activity of species ii. Furthermore, KK is rigorously related to the standard Gibbs free energy change ΔG∘\Delta G^\circ by the fundamental thermodynamic relation derived from the partition function: $$ \Delta G^\circ = -RT \ln K = -k_B T \ln \left(\frac{\mathcal{Z}_{products}}{\mathcal{Z}_{reactants}}\right) \quad \text{or equivalently,} \quad \Delta G^\circ = -RT \ln \left(\frac{\prod_{i=1}^{N} a_i^{\nu_i}}{\prod_{j=1}^{N} a_j^{\nu_j}}\right) \text{ for } \sum \nu_i X_i = 0 \text{.
Source: Wikipedia