Two distinct reactants X and Y decompose in a mixture via independent first-order kinetics. The initial concentrations of the two reactants are equal ( [ X ]₀ = [ Y ]₀ ). The half-life of X is 30 min and the half-life of Y is 60 min . The time after which the concentration of Y will be 8 times the concentration of X is ________ min.

Step-by-step solution

For a first order reaction, the concentration after time t can be expressed in terms of half-life as: [ A ] = [ A ]₀ ( 1 2 )^ t t_ 1/2 For reactant X: [ X ] = [ X ]₀ ( 1 2 )^ t 30 For reactant Y: [ Y ] = [ Y ]₀ ( 1 2 )^ t 60 We are given that [ X ]₀ = [ Y ]₀ and we need to find the time t when [ Y ] = 8[ X ] . Taking the ratio of the concentrations: [ Y ] [ X ] = ( 1 2 )^ t 60 ( 1 2 )^ t 30 8 = ( 1 2 )^ t 60 - t 30 = ( 1 2 )^ - t 60 = 2^ t 60 Since 8 = 2^3 , we have: 2^ t 60 = 2^3 Equating the exponents: t 60 = 3 t