2nd order reaction of the type (A + 2B → P)

The rate law for the 2nd order reaction of the type A + 2B → P is:

$\frac{d[A]}{dt}=-k[A][B]$

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

$\frac{dx}{dt}=k(a-x)(b-2x)$

where a = [A₀] and b = [B₀].

Rearranging the above equation and integrating throughout, we have

$\int_{0}^{x}\frac{dx}{(a-x)(\frac{b}{2}-x)}=2k\int_{0}^{t}dt\; \; \; \; \; \; \; \; 13$

Substituting the partial fraction expression $\frac{1}{(a-x)(\frac{b}{2}-x)}=\frac{1}{a-\frac{b}{2}}\left ( \frac{1}{\frac{b}{2}-x}-\frac{1}{a-x} \right )$ in eq13 and after some algebra, we have,

$kt=\frac{1}{b-2a}ln\frac{a(b-2x)}{b(a-x)}$

which is equivalent to

$kt=\frac{1}{[B_0]-2[A_0]}ln\frac{[A_o]([B])}{[B_0]([A])}$

2nd order reaction of the type (A + 2B → P)

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