Two different first-order reactions, X and Y, have rate constants 0.0693 min ⁻¹ and 0.0231 min ⁻¹ respectively. If both reactions start with the same initial concentration, what will be the ratio of the remaining concentrations of the reactants, [ X ] [ Y ] , after 30 minutes ?

Step-by-step solution

For reaction X: t_ 1/2, X = 0.693 k_ X = 0.693 0.0693 = 10 min Number of half-lives in 30 min for X, n_ X = 30 10 = 3 Remaining concentration of X, [ X ] = [ X ]₀ 2^3 = [ X ]₀ 8 For reaction Y: t_ 1/2, Y = 0.693 k_ Y = 0.693 0.0231 = 30 min Number of half-lives in 30 min for Y, n_ Y = 30 30 = 1 Remaining concentration of Y, [ Y ] = [ Y ]₀ 2^1 = [ Y ]₀ 2 Since the initial concentrations are equal ( [ X ]₀ = [ Y ]₀ ), the ratio of their remaining concentrations is: [ X ] [ Y ] = [ X ]₀ / 8 [ Y ]₀ / 2 = 2 8 = 1 4 Ther