Equilibrium constant ($K$)

For a specified chemical reaction and standard state convention, the dimensionless equilibrium constant is the equilibrium value of the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$, where $a_i$ denotes thermodynamic activity and ${\nu }_i$ a signed stoichiometric vector entry. Thermodynamic activities incorporate the reference state and departures from ideal behavior. See IUPAC Gold Book, equilibrium constant.

In the ideal concentration model for A converting to B, use $a_{\mathrm{A}}=[\mathrm{A}]/c^{\circ }$ and $a_{\mathrm{B}}=[\mathrm{B}]/c^{\circ }$ with the same reference $c^{\circ }=1\mathrm{M}$. Their ratio gives $Q=[\mathrm{B}]/[\mathrm{A}]$ and $K=0.75/0.25=3$. Equality $K=k_f/k_r$ applies to this elementary first-order pair; it is not a universal ratio of two measured rate constants for an arbitrary multistep chemical reaction.

At a specified constant temperature, with consistent standard states, the standard molar reaction Gibbs free energy isHere ${\Delta }_{r}G$ is the local slope per mole of extent of reaction at fixed temperature and pressure, rather than the finite total Gibbs free energy change between two mixtures. At equilibrium ${\Delta }_{r}G=0$, while ${\Delta }_r{G}^{\circ }$ generally remains nonzero. For $K=3$ at $298\mathrm{K}$, it is about $-2.72kJ/mol$. For positive activities, $Q<K$ favors forward conversion and $Q>K$ favors reverse conversion; neither sign supplies a reaction rate. See MIT 5.111, Lecture 18, reaction free energy and equilibrium.

For ideal concentration activities and signed stoichiometric coefficients ${\nu }_i$, $K_c={\prod }_i(c_i/c^{\circ }{)}^{{\nu }_i}$ at equilibrium. Its numerical value depends on the stated standard concentration when ${\sum }_i{\nu }_i\ne 0$. Number concentrations require a number-concentration reference; molar concentrations require the corresponding molar reference.

For $A+B\rightleftharpoons AB$, a concentration association constant is $K_a=[AB]/([A][B])$. Comparisons require matched solvent, temperature and standard-state conventions.

A dissociation constant is an equilibrium constant for a reaction in which a combined species separates into components. For $AB\rightleftharpoons A+B$, a concentration convention writesIt has units of concentration. The dimensionless thermodynamic form instead uses thermodynamic activities, $K_d^{\circ }=a_A{a}_B/a_{AB}$. For ideal concentration activities, $K_d^{\circ }=K_d^{(c)}/c^{\circ }$, with specified standard concentration $c^{\circ }$. The choice of quantity and reference is therefore part of quoting the constant.^[ref]

In the simple dilute $1:1$ model with free $c_A$ held fixed, the fraction of B present as AB is $c_A/(K_d^{(c)}+c_A)$. A smaller $K_d^{(c)}$ therefore means more association at the same free concentration. The inverse association constant describes the reversed reaction only when conventions, solvent conditions and temperature match. A dissociation rate is a kinetic quantity and is not specified by this equilibrium ratio alone.