Reaction Quotient

In chemical thermodynamics, the reaction quotient (''Q''_r 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
:\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 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 ''Q''_r, as a function of time ''t'', is defined as

:$Q_\text (t)= \frac ,$

where _''t'' denotes the ''instantaneous'' activity of a species X at time ''t''.
A compact general definition is
:Q_\text(t) = \prod_j _j(t),

where П_''j'' denotes the product across all ''j''-indexed variables, ''a_j''(''t'') is the activity of species ''j'' at time ''t'', and ''ν_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
:$\Delta_G=RT\ln(Q_/K)$,
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 ''Q''_r) when Δ_r''G'' < 0 or in the reverse direction (towards smaller values of ''Q''_r) when Δ_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:
:$Q_\to K$ and $\Delta_G\to 0\quad (t\to\infty)$.

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
:$Q_=1$ and $\Delta_G= \Delta_G^\circ=-RT\ln K\quad (t=0)$.

This quantity, Δ_r''G°'', 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.

Examples

The burning of octane, C₈H₁₈ + ²⁵/₂ O₂ → 8CO₂ + 9H₂O has a Δ_r''G° ~ –''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.

References

External links

Reaction quotient tutorials

tutorial I




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{{Chemical equilibria

Equilibrium chemistry
Physical chemistry