Question

A and B are two radioactive substances whose half-lives are 1 and 2 years respectively. Initially 10 g of A and, 1 g of B is taken. The time (approximate) after which they will have same quantity remaining is

• A. 6.62 year
• B. 5 year
• C. 3.2 year
• D. 7 year

Answer 1

The correct option is A 6.62 year

$N=N_0{\frac{1}{2}}^{t/T_{1/2}}$
$\Rightarrow N_A=10{\frac{1}{2}}^{t/2}$ and $N_B=1{\frac{1}{2}}^{t/2}$
Given:
$\Rightarrow 10{\frac{1}{2}}^t={\frac{1}{2}}^{t/2}\Rightarrow 10={\frac{1}{2}}^{-t/2}$
$\Rightarrow 10=2^{t/2}$

Taking log on both the sides:
$lo{g}_{10}=\frac{t}{2}lo{g}_{10}2$

$\Rightarrow 1=\frac{t}{2}\times 0.3010$
$\Rightarrow t=$ 6.62 years