Consider elementary reversible (or irreversible) reactions involving chemical species that can be represented in the general form

(4–1)

The stoichiometric coefficients are integer numbers and is the chemical symbol for the th species. The superscript indicates forward stoichiometric coefficients, while indicates reverse stoichiometric coefficients. Normally, an elementary reaction involves only three or four species; hence the matrix is quite sparse for a large set of reactions. For non-elementary reactions, Equation 4–1 also represents the reaction expression, but the stoichiometric coefficients may be non-integers.

The production rate of the th species can be written as a summation of the rate-of-progress variables for all reactions involving the th species

(4–2)

where

(4–3)

The rate of progress variable for the th reaction is given by the difference of the forward and reverse rates as

(4–4)

where is the molar concentration of the th species and and are the forward and reverse rate constants of the th reaction. As indicated in Equation 4–4, the rate-of-progress of a reaction is evaluated, by default, using the concentration of each reactant or product species raised to the power of its stoichiometric coefficient. Thus, the rate-of-progress of a reaction that includes species with a coefficient of 2 will be second-order with respect to the concentration of . Equation 4–4 is always valid when mass-action kinetics are obeyed, and when the mechanism is written in terms of elementary reactions. The user has the option to define an arbitrary reaction order for a species in a reaction in place of the coefficients used in Equation 4–4. This option is described further below.

The forward rate constants for the reactions are generally assumed to have the following Arrhenius temperature dependence:

(4–5)

where the pre-exponential factor , the temperature exponent , and the activation energy are specified. These three parameters are required input for each reaction.

Two gas constants, and are used throughout this chapter. is used only in conjunction with the activation energy and has compatible units. The reason for the duality is that many users would rather use units of cal/mole for the activation energies even though erg/mole are used elsewhere.

In thermal systems, the reverse rate constants are related to the forward rate constants through the equilibrium constants by

(4–6)

Although is given in concentration units, the equilibrium constants are more easily determined from the thermodynamic properties in pressure units; they are related by

(4–7)

The equilibrium constants are obtained with the relationship

(4–8)