Question

Two radioactive substances $A$ and $B$ have half lives $T$ and $2 T$ respectively. Samples of $A$ and $B$ contain equal number of nuclei initially. After a time $4 T$, the ratio of the number of undecayed nuclei of $A$ to that of $B$ is

• A. $1: 4$
• B. $1: 2$
• C. $2: 1$
• D. $4: 1$

Answer 1

Answer: (A)

Using $N=\frac{N_{0}}{2^{n}}$
$\Rightarrow \frac{N_{A}}{N_{B}}=\frac{2^{n_{B}}}{2^{n_{A}}}=2^{n_{B}-n_{A}}$
From $t=n T$
$\Rightarrow 4 T=n_{A} T, \Rightarrow n_{A}=4 $
$4 T=n_{B}(2 T) \Rightarrow n_{B}=2$
sub in (1)
$\frac{N_{A}}{N_{B}}=2^{2-4}=2^{-2}=\frac{1}{4}=1: 4$