Question

The mean lives of a radioactive substance for α and β emissions are 1620 years and 405 years respectively. After how much time will the activity be reduced to one fourth

• A. 405 year
• B. 1620 year
• C. 449 year
• D. None

Answer 1

Answer: (C)

449 year

Answer 2

$\lambda_{\alpha} = \frac{1}{1620}$ per year and $\lambda_{\beta} = \frac{1}{405}$ per year and it is given that the fraction of the remained activity $\frac{A}{A_{0}} = \frac{1}{4}$

Total decay constant .

$\lambda = \lambda_{\alpha} + \lambda_{\beta} = \frac{1}{1620} + \frac{1}{405} = \frac{1}{324}peryear$

We know that $A = A_{0}e^{- \lambda t}$⇒ $t = \frac{1}{\lambda}\log_{e}\frac{A_{0}}{A}$

⇒ $t = \frac{1}{\lambda}\log_{e}4 = \frac{2}{\lambda}\log_{e}2$ = 324 × 2 × 0.693 = 449 years