Question

A radioactive element has a rate of disintegration $10,000$ disintegrations per minute at a particular instant. After four minutes it becomes $2500$ disintegrations per minute. The decay constant per minute is

• A. $0.2log_{e}^{} 2 \, $
• B. $ \, 0.5log_{e}^{} 2 \, $
• C. $0.6log_{e}^{} 2 \, $
• D. $0.8log_{e}^{} 2 \, $

Answer 1

Answer: (B)

$\frac{N}{N_{0}}=e^{- \lambda t}$
$\therefore \, \frac{2500}{10000}=e^{- \lambda \times 4}$
$\therefore \, \, \, \frac{1}{4}=e^{- 4 \lambda }$
$\therefore \, e^{4 \lambda }=4$
$\therefore \, \, \, 4\lambda =log_{e} 4 \, \, $
$\therefore \, \, \, 4\lambda =log_{e} 2^{2} \, \, $
$\therefore \, \, \, 4\lambda =2log_{e} 2^{ \, } \, \, $
$\therefore \, \, \, \lambda =\frac{2}{4}log_{e} 2$
$\therefore \, \lambda =0.5log_{e} 2^{ \, }$