Concentration profiles for the reactions in series

A reaction in which one of the products of reaction acts as a catalyst is called an autocatal 4 ic reaction. In a n-th order in a batch reactor, the rate of product formation or disappearance of reactant is high initially as concentration of the reactant is high, and reaction slows down as the reactant disappears. However, in an autocatalytic reaction, the rate is low initially as little product is present, the rate increases as more and more product gets formed and then drops again because reactant is consumed. 

The simplest case of such a reaction is given below. 

For this reaction, a reaction rate based on first-order kinetics can be written 

We know from the reaction Equation 3.43 that the total number of moles remains constant, therefore, 

Substituting value of Cr from Equation 3.45 into Equation 3.44 and integrating results in 

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