(f) Autocatalytic Reactions

\[A + R \stackrel{k}{\rightarrow} R + R\] Here, one of the products catalyzes the reaction. \[-r_A = -\frac{dC_A}{dt} = kC_AC_R \tag*{(10)}\] Since the total number of moles are balanced, we have, at any time \[C_{A0} + C_{R0} = C_A + C_R = C_0 \quad (\text{constant}) \tag*{(11)}\] Integrating the rate expression, we get \[\begin{align*} \ln\left[\frac{C_{A0}(C_0-C_A)}{C_A(C_0-C_{A0})}\right] = \ln\left[\frac{C_R/C_{R0}}{C_A/C_{A0}}\right] &= C_0kt = (C_{A0}+C_{R0})kt\end{align*}\] If \(M=C_{R0}/C_{A0}\), then the above equation reduces to (in terms of conversion), as: \[\ln\left[\frac{M+X_A}{M(1-X_A)}\right] = (M+1)C_{A0}kt \tag*{(12)} \] Sigmoidal shaped curve for conversion and parabolic shaped rate-concentration curve, as shown in Fig.(10) are typical of autocatalytic reactions.

At the maximum rate, \[\begin{align*} \frac{d(-r_A)}{dC_A} &= 0 \\ \text{Using Eqns.(10), and (11), we get} \\ \frac{d[kC_AC_0-kC_A^2]}{dC_A} &= 0 \\ 2kC_A &= kC_0 \\ 2C_A &= C_A + C_R \\ \Longrightarrow \quad C_A &= C_R \end{align*}\]