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A first order decomposition reaction is given as A 3B + C . At time t , the concentration of product B is found to be 45 16 times the initial concentration of A . If the time t is x times the half-life ( t_ 1/2 ) of the reaction, the value of x is ________.

Correct answer

4

Step-by-step solution

Let the initial concentration of A be [A]₀ . The reaction is: A 3B + C At time t=0 : [A]₀ 0 0 At time t : [A]₀ - y 3y y Given that the concentration of B at time t is 45 16 [A]₀ . 3y = 45 16 [A]₀ y = 15 16 [A]₀ The remaining concentration of A at time t is: [A]_t = [A]₀ - y = [A]₀ - 15 16 [A]₀ = 1 16 [A]₀ For a first order reaction, the concentration of reactant remaining after n half-lives is given by: [A]_t = [A]₀ 2^n Comparing the two expressions: 1 2^n = 1 16 2^n = 16 n = 4 Therefore, t = 4 t_ 1/2 , which means

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