2nd order autocatalytic reaction

An autocatalytic reaction is one in which a product catalyses the reaction. An example is the Mn²⁺-catalysed oxidation of oxalic acid by potassium manganate (VII):

$2MnO_4^{\; -}+16H^++5C_2O_4^{\; 2-}\rightarrow 2Mn^{2+}+10CO_2+8H_2O$

The equation can be reduced to A + P → 2P, or simply, A → P, with the rate law:

$\frac{d[A]}{dt}=-k[A][P]\; \; \; \; \; \; \; \; 13a$

Using the same logic described in a previous article, we can rewrite the rate law as:

$\frac{dx}{dt}=k(a-x)(p+x)$

where a = [A₀] and p = [P₀].

Integrating throughout, we have

$\int_{0}^{x}\frac{dx}{(a-x)(p+x)}=k\int_{0}^{t}dt\; \; \; \; \; \; \; \; 14$

Substituting the partial fraction expression $\frac{1}{(a-x)(p+x)}=\frac{1}{a+p}\left ( \frac{1}{a-x}+\frac{1}{p+x} \right )$ in eq14 and after some algebra, we have,