The law of mass action is a fundamental principle for translating elementary chemical reactions into sets of differential equations, which are called rate equations. > [!NOTE] **Definition** (Law of Mass Action) > Let $M,N\in \mathbb{N}^{+}$ Consider the following [[Reaction|reaction]] of $M$ reactants $R_{1},\dots,R_{M}$ to $N$ products $P_{1},\dots,P_{N}$,$m_{1}R_{1}+\dots m_{M}R_{M} \stackrel{k}{\to} n_{1}P_{1} +\dots n_{N}P_{N}$where $m_{k} \in \mathbb{N}$ and $n_{l} \in \mathbb{N}$ are [[Stochiometry|stoichiometries]] (indicating how many molecules are consumed or generated in one reaction) and $k>0$ is a rate factor. > > Let $r_{1}(t), \dots, r_{M}(t)$ denote the time-dependent concentrations of the reactants > and $p_{1}(t), \dots,p_{N}(t)$ denote the time-dependent concentrations of the products. > > The reaction rate $r(t)$ is defined by $r(t)= k \prod_{k=1}^{M} r_{k}(t)^{m_{k}} $ > The rate of change of the concentrations of the reactants is $\frac{d}{dt} r_{k}(t)= -m_{k} r(t), \quad k=1,\dots,M.$and the rate of the change of the concentrations of the products is $\frac{d}{dt} p_{l}(t) = n_{l} r(t), \quad l=1,\dots,N.$ > For any system of reactions, the rates add. # Applications **Examples**: ...