Question

A radioactive sample at any instant has its disintegration rate $5000$ disintegrations per minute. After $5$ $minutes$ , the rate is $1250$ disintegrations per minute. Then the decay constant per minute is

• A. $\text{0.4}ln\left(2\right)$
• B. $\text{0.2}ln\left(2\right)$
• C. $\text{0.1}ln\left(\text{2}\right)$
• D. $\text{0.8}ln\left(2\right)$

Answer 1

Answer: (A)

As rate of disintegration is proportional to the number of atoms actually present, therefore,
${\text{N}}_0=\text{5000, N}=1250,t=\text{5 min}$
As $\text{ N}={\text{N}}_0{\text{e}}^{-\lambda \text{t}}$
$\therefore \text{ln}\frac{N}{{\left(N\right)}_0}=-\lambda t\text{ ; }\text{ln}\left(\frac{1250}{5000}\right)=-\lambda t\text{ ; }\text{ln}\left(\frac{1}{4}\right)=-\lambda \times 5;-2\text{ln}2=-\lambda \times 5\text{ ; }\lambda =0\text{.}4\text{ln}2$