Parallel reactions

6.6 Parallel reactions

As a last example we will discuss parallel reactions of reactions of different order.

\begin{equation*} \begin{split} A &\xrightarrow{k_1} B \quad \mbox{first order}\\ A &\xrightarrow{k_2} C \quad \mbox{second order}\\ \end{split} \end{equation*}

(6.44)

So the set of differential equations is

\begin{equation*} \begin{split} \frac{dA}{dt}&= - k_1\, A - k_2\, A^2 = - A\,(k_1+k_2\,A)\\ \frac{dB}{dt}&= k_1\, A \\ \frac{dC}{dt}&= k_2\, A^2 \\ \end{split} \end{equation*}

(6.45)

Having solved the differential equation for \(A(t)\), the functions \(B(t)\) and \(C(t)\) are just found by integration. Separating the variables we get for the first differential equation

\begin{equation*} -t=\int_{A_0}^A \frac{dA}{A\;(k_1-k_2\,A)}=\int_{A_0}^A \left( \frac{1}{A}-\frac{k_2}{k_1+k_2\,A}\right)\frac{1}{k_1} dA = \frac{1}{k_1} \left(\ln \frac{A}{k_1+k_2\,A} - \ln \frac{A_0}{k_1+k_2\,A_0}\right) \label{eq:A_par_1_2_ode} \end{equation*}

(6.46)

Solving for \(A(t)\) we get

\begin{equation*} \frac{1}{A(t)}=\frac{1}{A_0}\left[\exp(k_1\,t)\left(1+\frac{k_2\,A_0}{k_1}\right)-\frac{k_2\,A_0}{k_1}\right] \label{eq:A_par_1_2_sol} \end{equation*}

(6.47)

Analyzing the limiting cases we get

for \(k_1 \gg k_2\,A_0\)

\begin{equation*} \frac{1}{A(t)}=\frac{1}{A_0}\exp(k_1\,t) \quad \mbox{i.e.} \quad A(t) = A_0 \, \exp(-k_1\,t) \label{eq:A_par_1_2_sol_lim1} \end{equation*} (6.48)

which is the solution for a simple first order reaction.

for \(k_1 \ll k_2\,A_0\) we simplify

\begin{equation*} \exp(k_1\,t) \approx 1 + k_1\,t \quad . \end{equation*}

(6.49)

Including this into Eq. (6.47) we find

\begin{equation*} \frac{1}{A(t)}\approx \frac{1}{A_0}\left(1+k_1\,t+\frac{k_2\,A_0}{k_1} + \frac{k_2\,A_0\,k_1\,t}{k_1}-\frac{k_2\,A_0}{k_1}\right) \approx \frac{1}{A_0} \left(1 + A_0\, k_2 \, t\right) = \frac{1}{A_0} + k_2\,t \label{eq:A_par_1_2_sol_lim2} \end{equation*}

(6.50)

which is the solution for a simple second order reaction as shown in the table in section 6.2.