Thermochemical equilibrium (${\Delta }_{r}G=0,Q=K$)

At fixed temperature and pressure, an interior reacting equilibrium minimizes Gibbs free energy under the constraints imposed by stoichiometry. Each independent reversible reaction then has zero actual Gibbs energy of reaction,This is not a requirement that the standard Gibbs energy of reaction vanish.

Using dimensionless thermodynamic activities $a_i$, define the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$. The relationtherefore givesThe equilibrium constant $K$ is dimensionless because each activity is measured relative to its declared standard state. In general $K\ne 1$ even though the actual equilibrium Gibbs energy of reaction is zero. ^[ref]

These equations determine an equilibrium composition within the model. They do not supply the reaction rate or guarantee rapid equilibration; that requires the chemical kinetics.

At fixed temperature and pressure, a closed reacting system reaches thermochemical equilibrium by minimizing Gibbs free energy over nonnegative species amounts subject to conserved elemental abundances. For an allowed stoichiometric vector $\nu$, an interior minimum has ${\sum }_i{\nu }_i{\mu }_i=0$. More generally the chemical potential vector lies in the row span of the elemental conservation matrix at positive concentrations, with the appropriate one-sided inequalities for absent species. The first law of thermodynamics gives the thermodynamic differential, while the Second law of thermodynamics supplies the minimization direction.

For an elementary reversible reaction with concentration-based rates obeying the law of mass action, writewhere stoichiometric coefficients ${\nu }_i$ are positive for products. At detailed balance, $r_f=r_r$, soThis $K_c$ carries a power of concentration when $\Delta \nu ={\sum }_i{\nu }_i\ne 0$.

For an ideal solution with activities $a_i=c_i/c^{\circ }$, the dimensionless thermodynamic equilibrium constant is $K={\prod }_i(c_{i,\mathrm{eq}}/c^{\circ }{)}^{{\nu }_i}$. ConsequentlyKeeping the standard concentration factor is what makes the units of both rate coefficients consistent. For $A+B\rightleftharpoons C$, $\Delta \nu =-1$ and $k_r=k_f{c}^{\circ }/K$: a second-order forward coefficient becomes a first-order reverse coefficient.

Thermodynamics fixes this ratio only under the stated elementary, reversible rate-law model. It does not determine either coefficient separately or replace a multistep reaction mechanism by one elementary step.