Extent of reaction

In physical chemistry, extent of reaction is a quantity that measures the extent in which the reaction proceeds. It is usually denoted by the Greek letter ξ. The extent of a reaction has units of amount (moles). It was introduced by the Belgian scientist Théophile de Donder.

Definition

Consider the reaction

A ⇌ B

Suppose an infinitesimal amount dξ of the reactant A changes into B. The change of the amount of A can be represented by the equation dn_A = – dξ and the change of B is dn_B = dξ.[1]

The extent of reaction is then defined as[2][3]

${\displaystyle d\xi ={\frac {dn_{i}}{\nu _{i}}}}$

where ${\displaystyle n_{i}}$ denotes the amount of the i-th reactant and ${\displaystyle \nu _{i}}$is the stoichiometric coefficient (or stoichiometric number using IUPAC nomenclature[4]) of the i-th reactant. In other words, it is the amount of substance that is being changed in an equilibrium reaction. Considering finite changes instead of infinitesimal changes, one can write the equation for the extent of a reaction as

${\displaystyle \Delta \xi ={\frac {\Delta n_{i}}{\nu _{i}}}}$

The extent of a reaction is defined as zero at the beginning of the reaction. Thus the change of ξ is the extent itself.

${\displaystyle \xi ={\frac {\Delta n_{i}}{\nu _{i}}}={\frac {n_{equilibrium}-n_{initial}}{\nu _{i}}}}$

Relations

The relation between the change in Gibbs reaction energy and Gibbs energy can be defined as the slope of the Gibbs energy plotted against the extent of reaction at constant pressure and temperature.[1]

${\displaystyle \Delta _{r}G=\left({\frac {\partial G}{\partial \xi }}\right)_{p,T}}$

Analogously, the relation between the change in reaction enthalpy and enthalpy can be defined.[5]

${\displaystyle \Delta _{r}H=\left({\frac {\partial H}{\partial \xi }}\right)_{p,T}}$

Example

The extent of reaction is a useful quantity in computations with equilibrium reactions. Consider the reaction

2A ⇌ B + 3 C

where the initial amounts are ${\displaystyle n_{A}=2\ {\text{mol}},n_{B}=1\ {\text{mol}},n_{C}=0\ {\text{mol}}}$, and the equilibrium amount of A is 0.5 mol. We can calculate the extent of reaction from its definition

${\displaystyle \xi ={\frac {\Delta n_{A}}{\nu _{A}}}={\frac {0.5\ {\text{mol}}-2\ {\text{mol}}}{-2}}=0.75\ {\text{mol}}}$

Do not forget that the stoichiometric number of reactants is negative. Now when we know the extent, we can rearrange the equation and calculate the equilibrium amounts of B and C.

${\displaystyle n_{equilibrium}=\xi \nu _{i}+n_{initial}}$

${\displaystyle n_{B}=0.75\ {\text{mol}}\times 1+1\ {\text{mol}}=1.75\ {\text{mol}}}$

${\displaystyle n_{C}=0.75\ {\text{mol}}\times 3+0\ {\text{mol}}=2.25\ {\text{mol}}}$

References

1. Atkins, Peter; de Paula, Julio (2006). Physical chemistry (8 ed.). p. 201. ISBN 978-0-7167-8759-4.
2. Lisý, Ján Mikuláš; Valko, Ladislav (1979). Príklady a úlohy z fyzikálnej chémie. p. 593.
3. Ulický, Ladislav (1983). Chemický náučný slovník. p. 313.
4. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "stoichiometric number, ν". doi:10.1351/goldbook.S06025
5. Lisý, Ján Mikuláš; Valko, Ladislav (1979). Príklady a úlohy z fyzikálnej chémie. p. 593.