Question:
A radioactive sample is undergoing $\alpha$ decay.
At any time $\mathrm{t}_{1}$, its activity is A and another time
$t_{2}$, the activity is $\frac{\mathrm{A}}{5}$. What is the average life
time for the sample ?
Correct Option: , 3
Solution:
Let initial activity be $\mathrm{A}_{0}$
$\mathrm{A}=\mathrm{A}_{0} \mathrm{e}^{-\lambda \mathrm{t}_{1}}$ .........(i)
$\frac{A}{5}=A_{0} e^{-\lambda_{2}}$ ...........(II)
(i) $\div$ (ii)
$5=\mathrm{e}^{\lambda\left(\mathrm{t}_{2}-\mathrm{t}_{1}\right)}$
$\lambda=\frac{\ln 5}{\mathrm{t}_{2}-\mathrm{t}_{1}}=\frac{1}{\tau}$
$\tau=\frac{\mathrm{t}_{2}-\mathrm{t}_{1}}{\ell \ln 5}$
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