Editing Activity coefficient (section)

== Application to chemical equilibrium ==
At equilibrium, the sum of the chemical potentials of the reactants is equal to the sum of the chemical potentials of the products. The [[Gibbs free energy]] change for the reactions, Δ<sub>r</sub>''G'', is equal to the difference between these sums and therefore, at equilibrium, is equal to zero. Thus, for an equilibrium such as

:<math> \alpha_\mathrm{A} + \beta_\mathrm{B} = \sigma_\mathrm{S} + \tau_\mathrm{T},</math>
:<math> \Delta_\mathrm{r} G =  \sigma \mu_\mathrm{S} + \tau \mu_\mathrm{T} - (\alpha \mu_\mathrm{A} + \beta \mu_\mathrm{B}) = 0\,</math>
Substitute in the expressions for the chemical potential of each reactant:

:<math> \Delta_\mathrm{r} G = \sigma \mu_S^\ominus + \sigma RT \ln a_\mathrm{S} + \tau \mu_\mathrm{T}^\ominus + \tau RT \ln a_\mathrm{T} -(\alpha \mu_\mathrm{A}^\ominus + \alpha RT \ln a_\mathrm{A} + \beta \mu_\mathrm{B}^\ominus + \beta RT \ln a_\mathrm{B})=0</math>
Upon rearrangement this expression becomes

:<math> \Delta_\mathrm{r} G =\left(\sigma \mu_\mathrm{S}^\ominus+\tau \mu_\mathrm{T}^\ominus -\alpha \mu_\mathrm{A}^\ominus- \beta \mu_\mathrm{B}^\ominus \right) + RT \ln \frac{a_\mathrm{S}^\sigma a_\mathrm{T}^\tau} {a_\mathrm{A}^\alpha a_\mathrm{B}^\beta} =0</math>

The sum 
{{nowrap|''σμ''{{su|b=S|p=<s>o</s>}} + ''τμ''{{su|b=T|p=<s>o</s>}} − ''αμ''{{su|b=A|p=<s>o</s>}} − ''βμ''{{su|b=B|p=<s>o</s>}}}} is the standard free energy change for the reaction, <math>\Delta_\mathrm{r} G^\ominus</math>. 

Therefore,
:<math> \Delta_r G^\ominus = -RT \ln K </math>

where {{mvar|K}} is the [[equilibrium constant]]. Note that activities and equilibrium constants are dimensionless numbers.

This derivation serves two purposes. It shows the relationship between standard free energy change and equilibrium constant. It also shows that an equilibrium constant is defined as a quotient of activities. In practical terms this is inconvenient. When each activity is replaced by the product of a concentration and an activity coefficient, the equilibrium constant is defined as

:<math>K= \frac{[\mathrm{S}]^\sigma[\mathrm{T}]^\tau}{[\mathrm{A}]^\alpha[\mathrm{B}]^\beta} \times \frac{\gamma_\mathrm{S}^\sigma \gamma_\mathrm{T}^\tau}{\gamma_\mathrm{A}^\alpha \gamma_\mathrm{B}^\beta}</math>
where [S] denotes the [[concentration]] of S, etc. In practice equilibrium constants are [[Determination of equilibrium constants|determined]] in a medium such that the quotient of activity coefficients is constant and can be ignored, leading to the usual expression
:<math>K= \frac{[\mathrm{S}]^\sigma[\mathrm{T}]^\tau}{[\mathrm{A}]^\alpha[\mathrm{B}]^\beta}</math>
which applies under the conditions that the activity quotient has a particular (constant) value.