A and B are two radioactive substances whose half lives are 1 and… - Vidyadip

MCQ

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

✓
$6.62$
B
$5$
C
$3.2$
D
$7$

✓

Answer

Correct option: A.

$6.62$

a
(a) $N = {N_0}{\left( {\frac{1}{2}} \right)^{t/{T_{1/2}}}} \Rightarrow {N_A} = 10{\left( {\frac{1}{2}} \right)^{t/1}}$

and ${N_B} = 1{\left( {\frac{1}{2}} \right)^{t/2}}$

Given ${N_A} = {N_B} \Rightarrow 10{\left( {\frac{1}{2}} \right)^t} = {\left( {\frac{1}{2}} \right)^{t/2}}$

$ \Rightarrow 10 = {\left( {\frac{1}{2}} \right)^{ - t/2}} \Rightarrow 10 = {2^{t/2}}.$

Taking log both the sides.${\log _{10}}10 = \frac{t}{2}{\log _{10}}2$

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

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