# Sales variable

The turnover variable (also outdated extent of reaction , English extent of reaction ) ξ with the symbols is a measure of the progress of a given chemical reaction . The conversion variable has the SI unit of mole and is an extensive quantity . It is defined in such a way that it assumes the same value for all substances involved in the reaction for every possible conversion and changes by the same value with increasing conversion.

The variable is used to integrate the stoichiometry of a reaction into mathematical equations of physical chemistry . It was introduced as a degré d'avancement by Théophile de Donder .

## definition

A reaction equation of a chemical reaction consists of the chemical formulas for the reactants and reaction products as well as the stoichiometric factors that indicate the number ratio of the particles involved. In physical chemistry, these factors are extended to the stoichiometric numbers ν i of the particles i , with the reactants having negative and the products positive signs. In a reaction equation, on the other hand, there are only the amounts of the stoichiometric numbers:

${\ displaystyle | \ nu _ {\ mathrm {A}} | \ mathrm {A} + | \ nu _ {\ mathrm {B}} | \ mathrm {B} + ... \ longrightarrow | \ nu _ {\ mathrm {K}} | \ mathrm {K} + | \ nu _ {\ mathrm {L}} | \ mathrm {L} + ...}$

The amount of substance n i (unit mol ) of the particles i , which are formed or consumed with a differential conversion d n i , take on the same value when divided by the respective stoichiometric number:

${\ displaystyle {\ frac {\ mathrm {d} n _ {\ mathrm {A}}} {\ nu _ {\ mathrm {A}}}} = {\ frac {\ mathrm {d} n _ {\ mathrm {B }}} {\ nu _ {\ mathrm {B}}}} = {\ frac {\ mathrm {d} n _ {\ mathrm {K}}} {\ nu _ {\ mathrm {K}}}} = { \ frac {\ mathrm {d} n _ {\ mathrm {L}}} {\ nu _ {\ mathrm {L}}}} = ...}$

The following applies to the general definition of the turnover variables ξ according to IUPAC and DIN 32642:

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

In the more specific case, the sales variable is called

${\ displaystyle \ xi = {\ frac {n_ {i} -n_ {i, 0}} {\ nu _ {i}}}}$

considered, where n i, 0 is the amount of substance of the particle i before the start of the reaction (ξ = 0) and n i is the amount of substance of the substance i at a certain point of conversion. For n i the following applies:

${\ displaystyle n_ {i} = n_ {i, 0} + \ nu _ {i} \ xi}$

For a turnover from state I with the turnover variable ξ I to a turnover II with ξ II, the following applies:

${\ displaystyle \ Delta \ xi = \ xi ^ {\ text {II}} - \ xi ^ {\ text {I}} = {\ frac {\ mathrm {1}} {\ nu _ {i}}} ( n_ {i} ^ {\ text {II}} - n_ {i} ^ {\ text {I}}) = {\ frac {\ mathrm {\ Delta} n_ {i}} {\ nu _ {i}} }}$

The conversion variable of a reaction depends on the formulation of the reaction equation, since the stoichiometric numbers are taken from the reaction equation. It can only be used if the stoichiometry of the reaction under consideration is known.

### Examples

For a chemical reaction, the same conversion variable applies to a certain conversion for each particle (here molecule), which should be illustrated by the following reaction equation:

${\ displaystyle \ mathrm {3 \, H_ {2} + N_ {2} \, \ longrightarrow \, 2 \, NH_ {3}}}$

The turnover variable results here as follows:

${\ displaystyle \ xi \ = {\ frac {\ Delta n {\ mathrm {(H_ {2})}}} {- 3}} = {\ frac {\ Delta n {\ mathrm {(N_ {2}) }}} {- 1}} = {\ frac {\ Delta n {\ mathrm {(NH_ {3})}}} {2}}}$

For physico-chemical considerations, such as the molar enthalpy of reaction , a formula conversion is considered. The reaction runs from ξ = 0 to formula conversion ξ = 1:

${\ displaystyle \ xi \ = {\ frac {-3 \ \ mathrm {mol \ H_ {2}}} {- 3}} = {\ frac {-1 \ \ mathrm {mol \ N_ {2}}} { -1}} = {\ frac {+2 \ \ mathrm {mol \ NH_ {3}}} {2}} = 1 {\ text {mol}}}$

If the reaction equation is not formulated as a cardinal equation, but based on the amount of ammonia formed and formulated with fractional numbers, the result is from the equation

${\ displaystyle \ mathrm {{\ frac {3} {2}} \, H_ {2} + {\ frac {1} {2}} N_ {2} \, \ longrightarrow \ NH_ {3}}}$

for the turnover variant

${\ displaystyle \ xi \ = {\ frac {\ Delta n {\ mathrm {(H_ {2})}}} {- {\ frac {3} {2}}}} = {\ frac {\ Delta n { \ mathrm {(N_ {2})}}} {- {\ frac {1} {2}}}} = {\ frac {\ Delta n {\ mathrm {(NH_ {3})}}} {1} }}$

and for formula sales

${\ displaystyle \ xi \ = {\ frac {- {\ frac {3} {2}} \ \ mathrm {mol \ H_ {2}}} {- {\ frac {3} {2}}}} = { \ frac {- {\ frac {1} {2}} \ \ mathrm {mol \ N_ {2}}} {- {\ frac {1} {2}}}} = {\ frac {+1 \ \ mathrm {mol \ NH_ {3}}} {1}} = 1 {\ text {mol}}}$

The molar enthalpy of reaction based on the cardinal equation is based on the formation of 2 mol, the broken equation on the formation of 1 mol of NH 3 . The numerical value of the enthalpy of reaction is half as large in the broken equation, so it is important to state a reaction equation for the respective value.

## Degree of reaction

The degree of reaction α is the ratio of the conversion variable ξ to the complete conversion ξ max :

${\ displaystyle \ alpha = {\ frac {\ xi} {\ xi _ {max}}}}$

The degree of reaction is dimensionless and takes values ​​between 0 and 1. The term “degree of reaction” is rarely used in German literature; words such as degree of dissociation or degree of ionization are used for special reactions .

## Remarks

1. The word reaction sequence number is no longer recommended by the IUPAC because ξ is not a pure number, but a quantity with the dimension amount of substance .

## Individual evidence

1. a b c d Klaus H. Homann (Ed.): Sizes, units and symbols in der Physikalischen Chemie / International Union of Pure and Applied Chemistry (IUPAC), German version, VCH, Weinheim, 1995, ISBN 3-527-29326 -4 .
2. a b Entry on extent of reaction . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.E02283 Version: 2.3.3 ..
3. a b DIN 32642: Symbolic description of chemical reactions , January 1992.
4. Entry on degree of reaction . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.D01570 Version: 2.3.2.

## literature

• Gerd Wedler, Hans-Joachim Freund: Textbook of Physical Chemistry , 6th edition, Wiley, Weinheim, 2012, p. 34 ff.
• Quantities, units and symbols in physical chemistry / International Union of Pure and Applied Chemistry (IUPAC), Blackwell Science, Oxford, 1993. ISBN 0-632-03583-8 PDF