Kinetics of the reaction 2A → B

Chemical reaction kinetics: Second-order kinetics

Kinetics of the reaction 2A → B

The chemical reaction is of the type \[2\text{A }{\mathop{\longrightarrow}\limits_{}^{k}} \text{ B}\] with reaction rate constant \(k\). When we assume that this is an elementary reaction, then we can write down the following differential equation: \[-\frac{1}{2}\frac{\dd[\text{A}]}{\dd t}=k\, [\text{A}]^2\] In other words \[\frac{\dd[\text{A}]}{\dd t}=-2\,k\, [\text{A}]^2\] This differential equation has the solution \[[\text{A}]=\frac{[\text{A}]_0}{1+2\,k\,t\,[\text{A}]_0}\] where \([\text{A}]_0\) is the concentration of A at time \(t=0\).

For math enthusiast we give a proof.

Let us denote, for convenience, the concentration \([\text{A}]\) as the function \(x(t)\). Then we have the following differential equation: \[x'(t)=-2\,k\,x(t)^2\] We can rewrite it as \[-\frac{x'(t)}{x(t)^2}=2k\] but with the calculation rules for differentiating even as \[\left(\frac{1}{x(t)}\right)'=2k\] But then: \[\frac{1}{x(t)}=2\,k\,t+C\] for some constant \(C\). Substitution of \(t=0\) with \(x(0)=x_0\) gives: \[\frac{1}{x_0}=C\] So: \[\frac{1}{x(t)}=2\,k\,t+\frac{1}{x_0}=\frac{k\,t\,x_0+1}{x_0}\] Thus: \[x(t)=\frac{x_0}{1+2\,k\,t\,x_0}\]

The following simulation shows the time course of the concentration of A.

A concrete example of a chemical reaction with this type of kinetics is the gas reaction in which nitrogen dioxide is converted into nitrogen monoxide and oxygen: \[2\text{NO}_2(g)\longrightarrow 2\text{NO}(g)+\text{O}_2(g)\]