Stoichiometric coefficient
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In a chemical reaction system the stoichiometric coefficient of the i–th component is defined as
or
where Ni is the number of molecules of i, and ξ is the progress variable or extent of reaction (Prigogine & Defay, p. 18; Prigogine, pp. 4–7; Guggenheim, p. 37 & 62). The extent of reaction can be regarded as a real (or hypothetical) product, one molecule of which is produced each time the reaction event occurs.
The stoichiometric coefficient νi represents the degree to which a chemical species participates in a reaction. The convention is to assign negative coefficients to "reactants" (which are consumed) and positive ones to "products". However, any reaction may be viewed as "going" in the reverse direction, and all the coefficients then change sign (as does the free energy). Whether a reaction actually will go in the arbitrarily selected forward direction or not depends on the amounts of the substances present at any given time, which determines the kinetics and thermodynamics; i.e. whether equilibrium lies to the "right" or the "left".
If one contemplates actual reaction mechanisms, stoichiometric coefficients will always be integers, since elementary reactions always involve whole molecules. If one uses a composite representation of an "overall" reaction, some may be rational fractions. There are often chemical species present which do not participate in a reaction; their stoichiometric coefficients are therefore zero. Any chemical species which is regenerated, such as a catalyst, also has a stoichiometric coefficient of zero.
The simplest possible case is an isomerism
in which νB = 1 since one molecule of B is produced each time the reaction occurs, while νA = −1 since one molecule of A is necessarily consumed. In any chemical reaction, not only is the total mass conserved, but also the numbers of atoms of each kind, and this imposes a corresponding number of constraints on possible values for the stoichiometric coefficients. Of course, only a small subset of the possible atomic rearrangements will occur.
There are usually multiple reactions proceeding simultaneously in any natural reaction system, including those in biology. Since any chemical component can participate in several reactions simultaneously, the stoichiometric coefficient of the i–th component in the k–th reaction is defined as
so that the total (differential) change in the amount of the i–th component is
.
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 { Ni } (state variables), and the representation in terms of the actual compositional degrees of freedom, as expressed by the extents of reaction { ξk }. The transformation from a vector expressing the extents to a vector expressing the amounts uses a rectangular matrix whose elements are the stoichiometric coefficients [ νi k ].
The maximum and minimum for any ξk 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 kinematic restriction on the reaction simplex, a hyperplane in composition space, or N‑space, whose dimensionality equals the number of 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 of which share 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 point of the simplex. Consequently, 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 molecules or 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 Avogadro's number. While dimensional mass units may be used, the comments about integers are then no longer applicable.
[edit] References
- Ilya Prigogine & R. Defay, translated by D.H. Everett; Chapter IV (1954). Chemical Thermodynamics. Longmans, Green & Co. Exceptionally clear on the logical foundations as applied to chemistry; includes non-equilibrium thermodynamics.
- Ilya Prigogine (1967). Thermodynamics of Irreversible Processes, 3rd ed.. Interscience: John Wiley & Sons. A simple, concise monograph. Library of Congress Catalog No. 67-29540
- E.A. Guggenheim (1967). Thermodynamics: An Advanced Treatment for Chemists and Physicists, 5th ed.. North Holland; John Wiley & Sons (Interscience). A remarkably astute treatise. Library of Congress Catalog No. 67-20003
