# Reaction quotient

In chemistry, a reaction quotient (Qr or just Q) is a function of the activities or concentrations of the chemical species involved in a chemical reaction. In the special case that the reaction is at equilibrium the reaction quotient is constant and equal to the equilibrium constant that appears in the expression of the law of mass action.

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):

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

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 energy barrier does not make the reaction prohibitively slow for a given timescale, the species' activities and hence the reaction quotient change. 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) if ΔrG < 0 or in the reverse direction (towards smaller values of Qr) if 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:

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

In principle, reactions take an infinite amount of time to reach equilibrium; in practice, equilibrium is effectively reached in a finite time for most systems studied in the laboratory. If a reaction mixture is initialized with only reactants present, then

${\displaystyle Q_{\mathrm {r} }=0}$ and ${\displaystyle \Delta _{\mathrm {r} }G=-\infty \quad (t=0)}$,

If a reaction mixture is initialized with all components having an activity of unity, then

${\displaystyle Q_{\mathrm {r} }=1}$ and ${\displaystyle \Delta _{\mathrm {r} }G=\Delta _{\mathrm {r} }G^{\circ }\quad (t=0)}$,

where Δr is the standard Gibbs free energy of reaction,[3] a quantity directly related to the equilibrium constant by the formula

${\displaystyle \Delta _{\mathrm {r} }G^{\circ }=-RT\ln K}$.

As a result, all reactions, regardless of how favorable, are equilibrium processes. 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.

## 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 is defined to be 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: ${\textstyle \Delta _{\mathrm {r} }G^{\circ }=\sum _{\mathrm {prod.} }^{i}\nu _{i}\Delta _{\mathrm {f} }G^{\circ }-\sum _{\mathrm {react.} }^{j}\nu _{j}\Delta _{\mathrm {f} }G^{\circ }}$.