Reaction quotient

In chemical thermodynamics, the reaction quotient (Qr or just Q) is a dimensionless quantity that provides a measurement of the relative amounts 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. In 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

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 Qr, as a function of time t, is defined as


 * $$Q_\text{r} (t)= \frac{\{\mathrm{R}\}^\rho_t\{\mathrm{S}\}^\sigma_t} {\{\mathrm{A}\}^\alpha_t \{\mathrm{B}\}^\beta_t},$$

where {X}t denotes the instantaneous activity of a species X at time t. A compact general definition is
 * $$Q_\text{r}(t) = \prod_j [a_j(t)]^{\nu_j},$$

where Пj denotes the product across all j-indexed variables, aj(t) is the activity of species j at time t, and νj 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
 * $$\Delta_{\mathrm{r}}G=RT\ln(Q_{\mathrm{r}}/K)$$,

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:
 * $$Q_{\mathrm{r}}\to K$$ and $$\Delta_{\mathrm{r}}G\to 0\quad (t\to\infty)$$.

The timescale of this process depends on the rate constants of the forward and reverse reactions. In principle, equilibrium is approached asymptotically at t → ∞; 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
 * $$Q_{\mathrm{r}}=1$$ and $$\Delta_{\mathrm{r}}G= \Delta_{\mathrm{r}}G^\circ=-RT\ln K\quad (t=0)$$.

This quantity, ΔrG°, 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.

In biochemistry
In biochemistry, the reaction quotient is often referred to as the mass-action ratio with the symbol $$ \Gamma $$.

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 a kilogram of octane.

Significance and applications
The reaction quotient plays a crucial role in understanding the direction and extent of a chemical reaction's progress towards equilibrium:


 * 1) Equilibrium condition: At equilibrium, the reaction quotient (Q) is equal to the equilibrium constant (K) for the reaction. This condition is represented as Q = K, indicating that the forward and reverse reaction rates are equal.
 * 2) Predicting reaction direction: If Q < K, the reaction will proceed in the forward direction to establish equilibrium. If Q > K, the reaction will proceed in the reverse direction to reach equilibrium.
 * 3) Extent of reaction: The difference between Q and K provides information about how far the reaction is from equilibrium. A larger difference indicates a greater driving force for the reaction to proceed towards equilibrium.
 * 4) Reaction kinetics: The reaction quotient can be used to study the kinetics of reversible reactions and determine rate laws, as it is related to the concentrations of reactants and products at any given time.
 * 5) Equilibrium constant determination: By measuring the concentrations of reactants and products at equilibrium, the equilibrium constant (K) can be calculated from the reaction quotient (Q = K at equilibrium).

The reaction quotient is a powerful concept in chemical kinetics and thermodynamics, enabling the prediction of reaction directions, the extent of reaction progress, and the determination of equilibrium constants. It finds applications in various fields, including chemical engineering, biochemistry, and environmental chemistry, where understanding the behavior of reversible reactions is crucial.