Question

Two radioactive nuclei x and y initially contain equal number of atoms the half life is one hour and two hours respectively calculate the ratio of the rates of disintegration after 2 hours

Answer 1

Answer: Ratio of Thus ratio of rates of disintegration of X and y is 2:1

Explanation: Radioactive decay follows first order kinetics.

Half-life of nuclei X = 1 hour

Half-life of nuclei Y = 2 hour

$\lambda_X =\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{1}= 0.693hour^{-1}$

$\lambda_Y=\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{2}= 0.346hour^{-1}$

Rate of disintegration=$\frac{N}{N_o}=e^{-\lambda t}$

N = amount left after time t

$N_0$ = initial amount

$\lambda$ = rate constant

t= time  = 2 hours

Given: $N_0{X}=N_0{Y}=N_0$

Thus $\frac{N_X}{N_o}=e^{-0.693\times 1}=0.5$

$\frac{N_Y}{N_o}=e^{-0.693\times 2}=0.25$

Thus ratio of rates of disintegration of X and Y is $\frac{0.5}{0.25}=\frac{2}{1}$