Reaction Quotient

The reaction quotient, usually denoted $Q$, is a quantity calculated from the current composition of a mixture undergoing a chemical reaction. It has the same mathematical form as the equilibrium constant, but its value can be evaluated whether or not the mixture has reached chemical equilibrium. Comparing $Q$ with the appropriate equilibrium constant indicates whether formation of products or reactants is thermodynamically favored under the specified conditions. (openstax.org)

Definition and mathematical form

For a balanced chemical equation

the thermodynamic reaction quotient is

where $a_i$ denotes the activity of species $i$, not its stoichiometric coefficient. Activities are dimensionless quantities defined relative to a chosen standard state. Products appear in the numerator, reactants in the denominator, and the exponents come from the reaction’s stoichiometry. More generally,

with positive stoichiometric numbers $\nu_i$ for products and negative numbers for reactants. (faculty.washington.edu)

The expression depends on how the reaction is written. Reversing the equation replaces $Q$ by $1/Q$; multiplying every coefficient by a factor $m$ replaces it by $Q^m$. The corresponding equilibrium constant transforms identically. Thus, numerical values cannot be compared meaningfully unless they refer to the same reaction equation and conventions. (openstax.org)

Concentration and pressure forms

Introductory treatments commonly use a concentration quotient,

where brackets denote molar concentration. For gaseous mixtures, a pressure quotient is often written

where each $p_i$ is a partial pressure. These expressions use the mixture’s current concentrations or pressures, rather than necessarily their equilibrium values. (openstax.org)

These convenient forms require care about units and nonideal behavior. On a concentration-based standard-state convention,

where $\gamma_i$ is an activity coefficient. For a gas, activity can be expressed as $f_i/p^\circ$, where $f_i$ is fugacity. In the ideal-gas limit, fugacity approaches partial pressure. Standard-state normalization makes the thermodynamic quotient dimensionless; an unnormalized textbook $Q_c$ or $Q_p$ may carry apparent units. (goldbook.iupac.org)

Pure solids and pure liquids are normally assigned unit activity in elementary equilibrium calculations and therefore disappear from the written expression. A solvent such as water can also be approximated as having unit activity in sufficiently dilute aqueous solutions. This approximation need not hold in concentrated mixtures. (openstax.org)

Comparison with equilibrium

At equilibrium,

For a specified reaction and consistent standard states, the thermodynamic equilibrium constant is determined by temperature, whereas $Q$ changes as the mixture’s composition changes. The usual direction criteria are:

• $Q<K$: forward reaction is favored, consuming reactants and forming products.
• $Q>K$: reverse reaction is favored, consuming products and forming reactants.
• $Q=K$: there is no net thermodynamic driving force for that reaction. (openstax.org)

For example, consider

Its concentration quotient is

As an illustrative calculation, concentrations of $0.40$, $0.20$, and $0.10\ \mathrm{mol\,L^{-1}}$ for HI, hydrogen, and iodine give $Q_c=8$. If the applicable $K_c$ were 50, forward reaction would be favored; if it were 2, reverse reaction would be favored. These assumed values illustrate the comparison rather than specify a measured equilibrium. (openstax.org)

Thermodynamic basis

In thermodynamics, activity is related to chemical potential by

Summing these chemical potentials with their stoichiometric numbers gives the reaction Gibbs energy,

where $R$ is the gas constant and $T$ is absolute temperature. At equilibrium, $\Delta_rG=0$, so

These relations explain the direction criteria: $Q<K$ gives a negative forward reaction Gibbs energy, while $Q>K$ gives a positive one. (goldbook.iupac.org)

At constant temperature and pressure, $\Delta_rG$ is the derivative of the system’s Gibbs energy with respect to reaction extent. It describes the local driving force at the current composition, not generally the total Gibbs-energy difference between two arbitrary mixtures. The logarithmic expression assumes positive activities; absent species are treated through appropriate limiting behavior. (faculty.washington.edu)

Perturbations, rates, and electrochemistry

The quotient provides a quantitative interpretation of Le Châtelier’s principle. Adding a reactant generally lowers $Q$, while removing a product does likewise, favoring forward reaction when $K$ is unchanged. For an ideal gaseous mixture, multiplying all partial pressures by a factor $\lambda$ changes $Q_p$ by $\lambda^{\Delta\nu_g}$, where $\Delta\nu_g$ is the total gaseous product coefficient minus the total gaseous reactant coefficient. (openstax.org)

Reaction quotients do not determine reaction speeds. Chemical kinetics and activation barriers govern how rapidly a mixture changes. Catalysis can accelerate approach to equilibrium without changing the equilibrium constant or equilibrium composition. (openstax.org)

In electrochemistry, the Nernst equation relates the quotient for a redox reaction to the reversible cell potential:

where $n$ is the stoichiometric number of transferred electrons and $F$ is the Faraday constant. Consequently, changes in participating species’ activities alter cell potential even when the standard potential remains unchanged. (openstax.org)