For example, for the equilibrium 2NO2 N2O4, Fugacity is related to partial pressure, , by a dimensionless fugacity coefficient ϕ: . Thus, for the example, Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas phase then resemble the expression for solution equilibria with fugacity coefficient in place of activity coefficient and partial pressure in place of concentration. Thermodynamic basis for equilibrium constant expressions Thermodynamic equilibrium is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure the Gibbs free energy is minimum. The slope of the reaction free energy with respect to the extent of reaction, ξ, is zero when the free energy is at its minimum value. The free energy change, dGr, can be expressed as a weighted sum of change in amount times the chemical potential, the partial molar free energy of the species. The chemical potential, μi, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, Ni A general chemical equilibrium can be written as where nj are the stoichiometric coefficients of the reactants in the equilibrium equation, and mj are the coefficients of the products. At equilibrium The chemical potential, μi, of the ith species can be calculated in terms of its activity, ai. μ is the standard chemical potential of the species, R is the gas constant and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δGr(Eq) = 0 Rearranging the terms, This relates the standard Gibbs free energy change, ΔGo to an equilibrium constant, K, the reaction quotient of activity values at equilibrium. Equivalence of thermodynamic and kinetic expressions for equilibrium constants At equilibrium the rate of the forward reaction is equal to the backward reaction rate. A simple reaction, such as ester hydrolysis AB + H2O <=> AH + B(OH) has reaction rates given by expressions According to Guldberg and Waage, equilibrium is attained when the forward and backward reaction rates are equal to each other. In these circumstances, an equilibrium constant is defined to be equal to the ratio of the forward and backward reaction rate constants . The concentration of water may be taken to be constant, resulting in the simpler expression