Some nuclei of a radioactive material are undergoing radioactive...

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as: (where $\lambda$ is the decay constant)

Options

A $\dfrac{1}{2}\dfrac{\ln 2}{\lambda}$
B $\dfrac{\ln 2}{\lambda}$
C $\dfrac{2\ln 2}{\lambda}$
D $\dfrac{ln\dfrac{3}{2}}{\lambda}$

Correct answer

D. $\dfrac{ln\dfrac{3}{2}}{\lambda}$

Step-by-step solution

$\dfrac{3{N}_{0}}{4} = {N}_{0}{e}^{- \lambda{t}_{1}}$

$\dfrac{{N}_{0}}{2} = {N}_{0}{e}^{- {\lambda}_{2}}$

$\ln \left(3 / 4\right) = - \lambda{t}_{1} \ldots ( 1 )$

$\ln \left(1 / 2\right) = - \lambda{t}_{2} \ldots ( 2 )$

$\ln \left(3 / 4\right) - \ln \left(1 / 2\right) = \lambda\left({t}_{2} - {t}_{1}\right) \ldots ( 3 )$

$\Deltat = \dfrac{\ln \left(3 / 2\right)}{\lambda}$

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