Two radioactive substances X and Y originally have N _ 1 and N _ 2...

Two radioactive substances

X

and

Y

originally have

{N}_{1}

and

{N}_{2}

nuclei respectively. Half life of

X

is half of the half life of

Y

. After three half lives of

Y

, number of nuclei of both are equal. The ratio

\dfrac{{N}_{1}}{{N}_{2}}

will be equal to :

Options

A $\dfrac{1}{3}$
B $\dfrac{8}{1}$
C $\dfrac{3}{1}$
D $\dfrac{1}{8}$

Correct answer

B. $\dfrac{8}{1}$

Step-by-step solution

${T}_{x} = t; {T}_{y} = 2t$

$3{T}_{y} = 6t,$

${N}_{1}^{'} = {N}_{2}^{'}$

${N}_{1}{e}^{- {\lambda}_{1}6t} = {N}_{2}{e}^{- {\lambda}_{2}6t}$

$\dfrac{{N}_{1}}{{N}_{2}} = {e}^{\left({\lambda}_{1} - {\lambda}_{2}\right)6t} = {e}^{\left[\left(\dfrac{1}{t} - \dfrac{1}{2t}\right) \times 6t\right]\ln 2} = {e}^{\ln \left({2}^{3}\right)} = {e}^{\ln 8} = 8$

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