Reaction quotient

A general chemical reaction in which ${\displaystyle \alpha }$ moles of a reactant A and ${\displaystyle \beta }$ moles of a reactant B react to give ${\displaystyle \rho }$ moles of a product R and ${\displaystyle \sigma }$ moles of a product S can be written as

${\displaystyle \alpha \text{A}+\beta \text{B}\stackrel{-\rightharpoonup }{\leftharpoondown -}\rho \text{R}+\sigma \text{S}}$.

The reaction is written as an equilibrium even though, in many cases, it may appear that all of the reactants on one side have been converted to the other side. When any initial mixture of A, B, R, and S is made, and the reaction is allowed to proceed (either in the forward or reverse direction), the reaction quotient ${\displaystyle Q_{\mathrm{r}}}$, as a function of time ${\displaystyle t}$, is defined as

${\displaystyle Q_{\mathrm{r}}(t)=\frac{\{\mathrm{R}{\}}_t^{\rho }\{\mathrm{S}{\}}_t^{\sigma }}{\{\mathrm{A}{\}}_t^{\alpha }\{\mathrm{B}{\}}_t^{\beta }},}$

where ${\displaystyle \{\mathrm{X}{\}}_t}$ denotes the instantaneous activity of a species ${\displaystyle \mathrm{X}}$ at time ${\displaystyle t}$. A compact general definition is

${\displaystyle Q_{\mathrm{r}}(t)=\prod _j[a_j(t){]}^{{\nu }_j},}$

where ${\displaystyle \prod _j}$ denotes the product across all ${\displaystyle j}$-indexed variables, ${\displaystyle a_j(t)}$ is the activity of species ${\displaystyle j}$ at time ${\displaystyle t}$, and ${\displaystyle {\nu }_j}$ is the stoichiometric number (the stoichiometric coefficient multiplied by +1 for products and −1 for starting materials).

Relationship to K (the equilibrium constant)

As the reaction proceeds with the passage of time, the species' activities, and hence the reaction quotient, change in a way that reduces the free energy of the chemical system. The direction of the change is governed by the Gibbs free energy of reaction by the relation

${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G=RT\ln \frac{Q_{\mathrm{r}}}{K}}$,

where K is a constant independent of initial composition, known as the equilibrium constant. The reaction proceeds in the forward direction (towards larger values of ${\displaystyle Q_{\mathrm{r}}}$) when ${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G<0}$ or in the reverse direction (towards smaller values of ${\displaystyle Q_r}$) when ${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}G>0}$. Eventually, as the reaction mixture reaches chemical equilibrium, the activities of the components (and thus the reaction quotient) approach constant values. The equilibrium constant is defined to be the asymptotic value approached by the reaction quotient:

${\displaystyle Q_{\mathrm{r}}\to K\text{and}{\mathrm{\Delta }}_{\mathrm{r}}G\to 0(t\to \mathrm{\infty })}$.

The timescale of this process depends on the rate constants of the forward and reverse reactions. In principle, equilibrium is approached asymptotically at ${\displaystyle t\to \mathrm{\infty }}$; in practice, equilibrium is considered to be reached, in a practical sense, when concentrations of the equilibrating species no longer change perceptibly with respect to the analytical instruments and methods used.

If a reaction mixture is initialized with all components having an activity of unity, that is, in their standard states, then

${\displaystyle Q_{\mathrm{r}}=1\text{and}{\mathrm{\Delta }}_{\mathrm{r}}G={\mathrm{\Delta }}_{\mathrm{r}}{G}^{\circ }=-RT\ln K(t=0)}$.

This quantity, ${\displaystyle {\mathrm{\Delta }}_{\mathrm{r}}{G}^{\circ }}$, is called the standard Gibbs free energy of reaction.

All reactions, regardless of how favorable, are equilibrium processes, though practically speaking, if no starting material is detected after a certain point by a particular analytical technique in question, the reaction is said to go to completion.