Question

The activity of a radioactive sample falls from $700s^{-1}$ to $500s^{-1}$ in 30 minutes. Its half life is close to :

• A. $72$ min
• B. $62$ min
• C. $66$ min
• D. $52$ min

Answer 1

Answer: (B)

From equation of radioactivity we know
$<br/>A_t=A_0{e}^{-\lambda t}<br/>$
Where $A_t=$ Amount of material at time ' $t$ '
$A_0=$ Amount of substance at $t=0$
$\lambda =$ decay constant
$t=$ time.
Taking log we have
$\ln \left[\frac{A_0}{A_t}\right]=\lambda t$
For half life $A_t=\frac{A_0}{2}$.
$\Rightarrow \ln 2=\lambda t_{1/2}$ (A) $t_{1/2}=$ half life
From given condition.
$\ln \left[\frac{700}{500}\right]=\lambda (30min)$
(A) $-(B)$
$\frac{\ln 2}{\ln [7/5]}=\frac{t_{1/2}}{30}$
$\Rightarrow t_{1/2}=\frac{\ln 2\times 30}{\ln [7/5]}\Rightarrow t_{1/2}=\frac{0.693\times 30}{0.336}$
$\Rightarrow t_{1/2}=61.8min$