Question

A radioactive nucleus $A$ with a half-life $T$, decays into a nucleus $B$. At $t = 0$, there is no nucleus $B$. At sometime t, the ratio of the number of $B$ to that of $A$ is $0.3$. Then, $t$ is given by :

• A. $t = \frac{T}{2} \frac{\log \, 2}{\log \, (1.3)}$
• B. $t = T \frac{\log \, (1.3)}{\log \, 2}$
• C. $t = T \, \log (1.3)$
• D. $t = \frac{T }{\log \, (1.3)}$

Answer 1

Answer: (B)

$t = T \frac{\log \, (1.3)}{\log \, 2}$

Answer 2

$\frac{N_{0} - N_{0 }e^{-\lambda t}}{N_{0} e^{-\lambda t}} = 0.3$
$\Rightarrow e^{\lambda t} = 1.3$
$\therefore \lambda t =$ In $1.3$
$\left(\frac{In\, 2}{T}\right) t =$ In $1.3$
$t = T. \frac{In\, \left(1.3\right)}{In\, 2}$
$t = T \frac{\log\left(1.3\right)}{\log2 }$