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.

0,0

0.005

$k$

$[A]$

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:

$2\text{NO}_2(g)\longrightarrow 2\text{NO}(g)+\text{O}_2(g)$