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,
$\text{N}_{0} = \text{5000, N} = 1 2 5 0 , t = \text{5 min}$
As $\text{ N} = \text{N}_{0} \text{e}^{- \lambda \text{t}}$
$\therefore \text{ln} \frac{N }{\left(N ⁡\right)_{0}} = - \lambda t \text{ ; } \text{ln} \left(\frac{1 2 5 0}{5 0 0 0}\right) = - \lambda t \text{ ; } \text{ln} \left(\frac{1}{4}\right) = - \lambda \times 5 ; - 2 \text{ln} 2 = - \lambda \times 5 \text{ ; } \lambda = 0 \text{.} 4 \text{ln} 2$