An autocatalytic reaction is represented as $A + B ⟶ D$ , where the initial concentrations of A and B are each equal to $a_{0}$ , and the initial concentration of D is zero.
The rate of reaction at any time, $t$ , is $r = k [A] [B] [D]$ , except at $t = 0$ . Then,

Question

An autocatalytic reaction is represented as $A + B ⟶ D$ , where the initial concentrations of A and B are each equal to $a_{0}$ , and the initial concentration of D is zero.

The rate of reaction at any time, $t$ , is $r = k [A] [B] [D]$ , except at $t = 0$ . Then,

TestHub · Accepted Answer

A + B ⟶ D a 0 ​ ​ a 0 ​ ​ 0 ​ ( a 0 ​ − x ) ( a 0 ​ − x ) x r = d t d x ​ = k ( a 0 ​ − x ) 2 ( x ) ⇒ r = k ( a 0 2 ​ x + x 3 − 2 a 0 ​ x 2 ) At maximum rate, dx dr ​ = 0 . 3 x 2 − 4 a 0 ​ x + a 0 2 ​ = 0 Solving, x = a 0 ​ and x = a 0 ​ /3 . ​ ∴ x = 3 a 0 ​ ​ ( ∵ x  = a 0 ​ ) ​ r m a x ​ = 27 4 k ⋅ a 0 3 ​ ​ ​

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JEE Advanced (2028)