Editing Equilibrium constant (section)

== 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.<ref>{{cite book|last=Denbigh|first=K.|title=The principles of chemical equilibrium|publisher=Cambridge University Press |location=Cambridge|year=1981|edition=4th|isbn=978-0-521-28150-8|chapter= Chapter 4}}</ref> 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. 
:<math>\left(\frac{\partial G}{\partial \xi }\right)_{T,P}=0</math>

The free energy change, d''G''<sub>r</sub>, 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, ''μ<sub>i</sub>'', of the ''i''th species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, ''N''<sub>i</sub>
:<math>\mu_i=\left(\frac{\partial G}{\partial N_i}\right)_{T,P}</math>

A general chemical equilibrium can be written as

:<math>\sum_j n_j \mathrm{Reactant}_j \rightleftharpoons \sum_k m_k \mathrm{Product}_k</math> {{spaces|5}}

where ''n<sub>j</sub>'' are the [[stoichiometric coefficient]]s of the reactants in the equilibrium equation, and ''m<sub>j</sub>'' are the coefficients of the products. At equilibrium
:<math>\sum_k m_k \mu_k = \sum_j n_j \mu_j </math>

The chemical potential, ''μ<sub>i</sub>'', of the ''i''th species can be calculated in terms of its [[activity (chemistry)|activity]], ''a<sub>i</sub>''.
:<math>\mu_i = \mu_i^\ominus + RT \ln a_i</math>
''μ''{{su|b=''i''|p=<s>o</s>}} 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 ''δG''<sub>r</sub>(Eq)&nbsp;=&nbsp;0
:<math>\sum_j n_j(\mu_j^\ominus +RT\ln a_j)=\sum_k m_k(\mu_k^\ominus +RT\ln a_k) </math>
Rearranging the terms,
:<math>\sum_k m_k\mu_k^\ominus-\sum_j n_j\mu_j^\ominus =-RT \left(\sum_k \ln {a_k}^{m_k}-\sum_j \ln {a_j}^{n_j}\right)</math>
:<math>\Delta G^\ominus = -RT \ln K.</math>
This relates the [[standard state|standard]] Gibbs free energy change, Δ''G''<sup><s>o</s></sup> to an equilibrium constant, ''K'', the [[reaction quotient]] of activity values at equilibrium.
:<math>\Delta G^\ominus = \sum_k m_k\mu_k^\ominus-\sum_j n_j\mu_j^\ominus</math>
:<math>\ln K= \sum_k \ln {a_k}^{m_k}-\sum_j \ln {a_j}^{n_j};
K=\frac{\prod_k {a_k}^{m_k}}{\prod_j {a_j}^{n_j}} \equiv \frac{{\{\mathrm{R}\}} ^\rho {\{\mathrm{S}\}}^\sigma ... } {{\{\mathrm{A}\}}^\alpha {\{\mathrm{B}\}}^\beta ...}
</math>

===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]]
:<chem>AB + H2O <=> AH + B(OH)</chem>
has reaction rates given by expressions
:<math chem>\text{forward rate} = k_f\ce{[AB][H2O]}</math>
:<math chem>\text{backward rate} = k_b\ce{[AH][B(OH)]}
</math>
According to [[Cato Maximilian Guldberg|Guldberg]] and [[Peter Waage|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  
:<math chem>K=\frac{k_f}{k_b}=\frac\ce{[AH][B(OH)]}\ce{[AB][H2O]}</math>.
The concentration of water may be taken to be constant, resulting in the simpler expression 
:<math chem>K^c=\frac\ce{[AH][B(OH)]}\ce{[AB]}</math>.
This particular concentration quotient, <math>K^c</math>, has the dimension of concentration, but the thermodynamic equilibrium constant, {{mvar|K}}, is always dimensionless.