| context | $R,J\in \mathbb N$ |
| context | $ \nu^-,\nu^+\in\mathrm{Matrix}(R,J,\mathbb Q) $ |
| context | $ k\in \mathbb R^R $ |
| definiendum | $ [A] \in \mathrm{it} $ |
| $j\in \text{range}(J)$ | |
| postulate | $ [A]:C(\mathbb R,\mathbb R^J) $ |
| range | $ ::[A](t) $ |
| postulate | $ \frac{\partial}{\partial t}[A]_j=\sum_{r=1}^R k_r\cdot(\nu_{rj}^+-\nu_{rj}^-)\cdot\prod_{i=1}^J [A]_i^{\nu_{ri}^-} $ |
The quantities $R$ and $J$ denote the number of reactions and the number of different species. Then $\nu_{rj}^-$ and $\nu_{rj}^+$ are stochastic coefficients of the reactants and products and $k_r$ is the reaction rate coefficient of the $r$'s reaction.
Physically speaking, for each microscopic particle collision the reaction rate coefficient $k_r$ gives the probability that $\nu_{rj}^-$ of the reactants transform into $\nu_{rj}^+$ of the products. The rate is moreover proportional to the probability of encounter and hence the product to the momentary concentrations themselves.
Non-time resolved, this reads for all $r$
$\sum_{j=1}^J \nu_{rj}^{(e)} A_j \overset{k_r}{\longrightarrow} \sum_{j=1}^J \nu_{rj}^{(p)} A_j.$
For example, the simplest carbon combustion process: $\mathrm{C}\mathrm{H}_4 + 2\ \mathrm{O}_2 \longrightarrow \mathrm{C}\mathrm{O}_2 + 2\ \mathrm{H}_2\mathrm{O}.$
(Or more explicitly: $1\ \mathrm{C}\mathrm{H}_4 + 2\ \mathrm{O}_2 + 0\ \mathrm{C}\mathrm{O}_2 + 0\ \mathrm{H}_2\mathrm{O} \longrightarrow 0\ \mathrm{C}\mathrm{H}_4 + 0\ \mathrm{O}_2 + 1\ \mathrm{C}\mathrm{O}_2 + 2\ \mathrm{H}_2\mathrm{O}$.)
In practice, $k$ depends on the temperature, which, through the equation of state, can again be a nonlinear function of the concentrations.
Wikipedia: Rate equation