6.2.4: Chemical Kinetics

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THERE ARE MANY PROBLEMS IN THE CHEMISTRY of chemical reactions which lead to systems of differential equations. The simplest reaction is when a chemical \(A\) turns into chemical \(B\). This happens at a certain rate, \(k>0 .\) This reaction can be represented by the chemical formula

\[A \stackrel{\longrightarrow}{k} B \nonumber \]

In this case we have that the rates of change of the concentrations of \(A,[A]\), and \(B,[B]\), are given by

The chemical reactions used in these examples are first order reactions. Second

\[ \begin{aligned} \dfrac{d[A]}{d t} &=-k[A] \\ \dfrac{d[B]}{d t} &=k[A] \end{aligned} \label{6.52} \]

order reactions have rates proportional to the square of the concentration.

Think about this as it is a key to understanding the next reactions.

A more complicated reaction is given by

\[A \underset{k_{1}}{\longrightarrow} B \underset{k_{2}}{\longrightarrow} C. \nonumber \]

Here there are three concentrations and two rates of change. The system of equations governing the reaction is

\[ \begin{aligned} \dfrac{d[A]}{d t} &=-k_{1}[A] \\ \dfrac{d[B]}{d t} &=k_{1}[A]-k_{2}[B] \\ \dfrac{d[C]}{d t} &=k_{2}[B] \end{aligned} \label{6.53} \]

The more complication rate of change is when [B] increases from [A] changing to [B] and decrease when [B] changes to [C]. Thus, there are two terms in the rate of change equation for concentration [B].

One can further consider reactions in which a reverse reaction is possible. Thus, a further generalization occurs for the reaction

\[A \ll{k_{1}}^{k_{3}} \longrightarrow B \underset{k_{2}}{\longrightarrow} C . \nonumber \]

The reverse reaction rates contribute to the reaction equations for [A] and [B]. The resulting system of equations is

\[ \begin{aligned} \dfrac{d[A]}{d t} &=-k_{1}[A]+k_{3}[B] \\ \dfrac{d[B]}{d t} &=k_{1}[A]-k_{2}[B]-k_{3}[B] \\ \dfrac{d[C]}{d t} &=k_{2}[B] \end{aligned} \label{6.54} \]

Nonlinear chemical reactions will be discussed in the next chapter.