Two substances X and Y undergo first-order decay. Their initial concentrations are in the ratio [X]₀ : [Y]₀ = 4:1 . The half-life of X is 30 minutes . After 120 minutes , the concentrations of X and Y become exactly equal. The half-life of Y is ____ minutes .

Step-by-step solution

For a first-order reaction, the concentration at time t is given by [A]_t = [A]₀ ( 1 2 )^ t t_ 1/2 . For substance X at t = 120 min : [X]₁₂₀ = [X]₀ ( 1 2 )^ 120 30 = [X]₀ ( 1 2 )^4 = [X]₀ 16 For substance Y at t = 120 min : [Y]₁₂₀ = [Y]₀ ( 1 2 )^ 120 t_ 1/2, Y Given that [X]₁₂₀ = [Y]₁₂₀ : [X]₀ 16 = [Y]₀ ( 1 2 )^ 120 t_ 1/2, Y Substitute [X]₀ = 4[Y]₀ into the equation: 4[Y]₀ 16 = [Y]₀ ( 1 2 )^ 120 t_ 1/2, Y 1 4 = ( 1 2 )^ 120 t_ 1/2, Y ( 1 2 )^2 = ( 1 2 )^ 120 t_ 1/2, Y Equating the exponents: 2 = 120 t_ 1/2, Y t_ 1