Question

The half life of a radioactive nucleus is 50 days. find the time interval (t2 – t1) where the time t2 whenof it has decayed and the time t1 whenof it had decayed is

Answer 1

Question is ----->
the half life of a radioactive nucleus is 50 days. The time interval (t2-t1) between the time t2 when 2/3 of it has decayed and the time t1 when 1/3 of it had decayed as?

Solution: Half life of radioactive nucleus is 50 days ,
$\bold{T_{1/2}=50\:days}$

We know, radioactive nucleus decayed at time t is given by
$\bold{N=N_0e^{-\lambda.t}}$
A/C to question,
at t₁ time ,1/3 of it had decayed .
e.g., $\bold{\frac{N_0}{3}=N_0e^{-\lambda.t_1}} \\\\\bold{\frac{1}{3}=e^{-\lambda.t_1}}------------------(1)$

similarly, at t₂ time, 2/3 of it had decayed ,
e.g., $\bold{\frac{2N_0}{3}=N_0e^{-\lambda.t_2}} \\\\\bold{\frac{2}{3}=e^{-\lambda.t_2}}------------------(2)$

Now, divide equation (1) and (2)
$\bold{\frac{1}{2}=e^{-\lambda(t_1-t_2)}}\\\\\bold{ln2=\lambda(t_1-t_2)}$------(3)
We also know,
$\bold{T_{1/2}=\frac{ln2}{\lambda}}$

so, $\bold{\lambda=\frac{ln2}{50}}------(4)$

Put equation (4) in equation (3)
Then, (t₂ - t₁) = $\bold{T_{1/2}=50\:days}$