Editing Stoichiometry (section)

== Stoichiometric coefficient and stoichiometric number ==
In lay terms, the ''stoichiometric coefficient'' of any given component is the number of molecules and/or [[formula units]] that participate in the reaction as written. A related concept is the ''stoichiometric number'' (using IUPAC nomenclature), wherein the stoichiometric coefficient is multiplied by +1 for all products and by −1 for all reactants.

For example, in the reaction {{chem2|CH4 + 2 O2 → [[Carbon dioxide|CO2]] + 2 H2O}}, the stoichiometric number of {{chem2|CH4}} is −1, the stoichiometric number of {{chem2|O2}} is −2, for {{CO2}} it would be +1 and for {{chem2|H2O}} it is +2.

In more technically precise terms, the stoichiometric number in a [[chemical reaction]] [[system]] of the ''i''-th component is defined as
: <math>\nu_i = \frac{\Delta N_i}{\Delta \xi} \,</math>
or
: <math> \Delta N_i = \nu_i \, \Delta \xi  \,</math>
where <math>N_i</math> is the number of [[molecule]]s of ''i'', and <math>\xi</math> is the progress variable or [[extent of reaction]].<ref>Prigogine & Defay, p.&nbsp;18; Prigogine, pp.&nbsp;4–7; Guggenheim, p.&nbsp;37&nbsp;&&nbsp;62</ref><ref>{{GoldBookRef |title=extent of reaction, ''ξ'' |file=E02283 |accessdate=4 May 2015 }}</ref>

The stoichiometric number&nbsp;<math>\nu_i</math> represents the degree to which a chemical species participates in a reaction. The convention is to assign negative numbers to ''reactants'' (which are consumed) and positive ones to ''products'', consistent with the convention that increasing the extent of reaction will correspond to shifting the composition from reactants towards products. However, any reaction may be viewed as going in the reverse direction, and in that point of view, would change in the negative direction in order to lower the system's Gibbs free energy. Whether a reaction actually ''will'' go in the arbitrarily selected forward direction or not depends on the amounts of the [[chemical substance|substances]] present at any given time, which determines the [[chemical kinetics|kinetics]] and [[thermodynamic equilibrium|thermodynamics]], i.e., whether [[chemical equilibrium|equilibrium]] lies to the ''right'' or the ''left'' of the initial state,

In [[reaction mechanism]]s, stoichiometric coefficients for each step are always [[integer]]s, since elementary reactions always involve whole molecules. If one uses a composite representation of an overall reaction, some may be [[rational number|rational]] [[fraction (mathematics)|fractions]]. There are often chemical species present that do not participate in a reaction; their stoichiometric coefficients are therefore zero. Any chemical species that is regenerated, such as a [[catalyst]], also has a stoichiometric coefficient of zero.

The simplest possible case is an [[isomerization]]
: A → B

in which {{math|1=''ν''<sub>B</sub>&nbsp;=&nbsp;1}} since one molecule of B is produced each time the reaction occurs, while {{math|1=''ν''<sub>A</sub>&nbsp;=&nbsp;−1}} since one molecule of A is necessarily consumed. In any chemical reaction, not only is the total [[conservation of mass|mass conserved]] but also the numbers of [[atom]]s of each [[periodic table|kind]] are conserved, and this imposes corresponding constraints on possible values for the stoichiometric coefficients.

There are usually multiple reactions proceeding simultaneously in any [[nature|natural]] reaction system, including those in [[biology]]. Since any chemical component can participate in several reactions simultaneously, the stoichiometric number of the ''i''-th component in the ''k''-th reaction is defined as
: <math>\nu_{ik} = \frac{\partial N_i}{\partial \xi_k} \,</math>
so that the total (differential) change in the amount of the ''i''-th component is
: <math> dN_i = \sum_k \nu_{ik} \, d\xi_k. \,</math>

Extents of reaction provide the clearest and most explicit way of representing compositional change, although they are not yet widely used.

With complex reaction systems, it is often useful to consider both the representation of a reaction system in terms of the amounts of the chemicals present {{math|1={{mset|&nbsp;''N<sub>i</sub>''&nbsp;}}}} ([[thermodynamic variable|state variables]]), and the representation in terms of the actual compositional [[Degrees of freedom (physics and chemistry)|degrees of freedom]], as expressed by the extents of reaction {{math|1={{mset|&nbsp;''ξ<sub>k</sub>''&nbsp;}}}}. The transformation from a [[vector space|vector]] expressing the extents to a vector expressing the amounts uses a rectangular [[matrix (mathematics)|matrix]] whose elements are the stoichiometric numbers {{math|1=[&nbsp;''ν<sub>i&thinsp;k</sub>''&nbsp;]}}.

The [[extreme value|maximum and minimum]] for any ''ξ<sub>k</sub>'' occur whenever the first of the reactants is depleted for the forward reaction; or the first of the "products" is depleted if the reaction as viewed as being pushed in the reverse direction. This is a purely [[kinematics|kinematic]] restriction on the reaction [[simplex]], a [[hyperplane]] in composition space, or ''N''‑space, whose [[dimension]]ality equals the number of ''[[linear independence|linearly-independent]]'' chemical reactions. This is necessarily less than the number of chemical components, since each reaction manifests a relation between at least two chemicals. The accessible region of the hyperplane depends on the amounts of each chemical species actually present, a contingent fact. Different such amounts can even generate different hyperplanes, all sharing the same algebraic stoichiometry.

In accord with the principles of [[chemical kinetics]] and [[thermodynamic equilibrium]], every chemical reaction is ''reversible'', at least to some degree, so that each equilibrium point must be an [[interior (topology)|interior point]] of the simplex. As a consequence, extrema for the ''ξ''s will not occur unless an experimental system is prepared with zero initial amounts of some products.

The number of ''physically''-independent reactions can be even greater than the number of chemical components, and depends on the various reaction mechanisms. For example, there may be two (or more) reaction ''paths'' for the isomerism above. The reaction may occur by itself, but faster and with different intermediates, in the presence of a catalyst.

The (dimensionless) "units" may be taken to be [[molecule]]s or [[mole (unit)|moles]]. Moles are most commonly used, but it is more suggestive to picture incremental chemical reactions in terms of molecules. The ''N''s and ''ξ''s are reduced to molar units by dividing by the [[Avogadro constant]]. While dimensional [[mass]] units may be used, the comments about integers are then no longer applicable.