Question

The mean lives of a radioactive material for $\alpha$ and $\beta$ radiations are 1620 years and 520 years respectively. The material decays simultaneously for $\alpha$ and $\beta$ radiations.The time after which one fourth of the material remains undecayed is

• A. 546 years
• B. 324 years
• C. 720 years
• D. 840 years

Answer 1

The correct option is A 546 years

Let, the mean life of a radioactive material for $\alpha$ radiations is ${\tau }_{\alpha }=1620years$,
and the mean life of a radioactive material for $\beta$ radiations is ${\tau }_{\beta }=520years$,
$\frac{N}{N_0}=\frac{1}{4}$ ..........(given)

The mean life of a radioactive material for both radiations will be:
$\tau =\frac{{\tau }_{\alpha }.{\tau }_{\beta }}{{\tau }_{\alpha }+{\tau }_{\beta }}$

$\tau =\frac{1620\times 520}{1620+520}$

$\tau =393.64\approx 394years$

The time of decay of a radioactive material is:
$T=\tau 2.303{\log }_{10}\frac{N_0}{N}$
$T=394\times 2.303{\log }_{10}\left(4\right)$
$T=394\times 2.303\times 0.602$
$T=546.243\approx 546years$