# Reaction quotient

In chemical thermodynamics, the reaction quotient (Qr or just Q) is a quantity that provides a measurement of the relative quantities of products and reactants present in a reaction mixture for a reaction with well-defined overall stoichiometry, at a particular point in time. Mathematically, it is defined as the ratio of the activities (or molar concentrations) of the product species over those of the reactant species involved in the chemical reaction, taking stoichiometric coefficients of the reaction into account as exponents of the concentrations. This functional form obeys the law of mass action, and, in the special case that the reaction is at equilibrium, the reaction quotient is constant over time and is equal to the equilibrium constant.

A general chemical reaction in which α moles of a reactant A and β moles of a reactant B react to give ρ moles of a product R and σ moles of a product S can be written as

${\displaystyle {\ce {\it {\alpha \,{\rm {A{}+{\it {\beta \,{\rm {B{}<=>{\it {\rho \,{\rm {R{}+{\it {\sigma \,{\rm {S{}}}}}}}}}}}}}}}}}}}$.

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 a mixture of A and B is made up and the reaction is allowed to occur, the reaction quotient Qr, as a function of time t, is defined as[1]

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

where {X}t denotes the instantaneous activity[2] of a species X at time t. A compact general definition is (where Пj is the product across all j-indexed variables, and same for Пi):

${\displaystyle Q_{\text{r}}(t)={\frac {\prod _{j}^{\mathrm {pdt.} }a_{j}^{\nu _{j}}(t)}{\prod _{i}^{\mathrm {rct.} }a_{i}^{\nu _{i}}(t)}},}$

where the numerator is a product of reaction product activities aj, each raised to the power of a stoichiometric coefficient νj, and the denominator is a similar product of reactant activities. All activities refer to a time t.

## Relationship to K (the equilibrium constant)

As the reaction proceeds with the passage of time, assuming the activation energy does not make the reaction prohibitively slow for a given timescale, 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

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

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 Qr) when ΔrG < 0 or in the reverse direction (towards smaller values of Qr) when ΔrG > 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:

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

In principle, reactions take an infinite amount of time to reach equilibrium; 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 used).

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

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

This quantity, Δr, is called the standard Gibbs free energy of reaction.[3]

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. For example, the burning of octane, C8H18 + 25/2 O2 → 8CO2 + 9H2O has a ΔrG° ~ –240 kcal/mol, corresponding to an equilibrium constant of 10175, a number so large that it is of no practical significance, since there are only ~5 × 1024 molecules in kilogram of octane. Nevertheless, the process is an equilibrium, in principle.

## References

1. Zumdahl, Steven; Zumdahl, Susan (2003). Chemistry (6th ed.). Houghton Mifflin. ISBN 0-618-22158-1.
2. Under certain circumstances (see chemical equilibrium) each activity term such as {A} may be replaced by a concentration term, [A]. Both the reaction quotient and the equilibrium constant are then concentration quotients.
3. The standard free energy of reaction can be determined using the difference between the sum of the standard free energies of formation of products and the sum of the standard free energies of formation of reactants, accounting for stoichiometries: ${\textstyle \Delta _{\mathrm {r} }G^{\circ }=\sum _{\mathrm {prod.} }^{i}\nu _{i}\Delta _{\mathrm {f} }G^{\circ }-\sum _{\mathrm {react.} }^{j}\nu _{j}\Delta _{\mathrm {f} }G^{\circ }}$.