The half life of a radioactive substance is 30 s. Calculate the time taken for the sample to decay to $\frac{3}{4}th$ its initial value.

60 s

The correct answer is Option (4) → 60 s

To calculate the time taken for a radioactive sample to decay $\frac{3}{4}$ of its initial value -

$N(t)=N_0e^{-λt}$ [Radioactive Decay formula]

$\frac{N}{N_0}=\frac{1}{4}=e^{-λt}$ [given]

$⇒\frac{1}{4}=e^{-λt}$

The decay constant $λ$ is related to half-life ($T_{1/2}$)

$λ=\frac{ln(2)}{T_{1/2}}$

$λ=\frac{ln(2)}{30}≃0.0231s^{-1}$

$ln(\frac{1}{4})=-0.0231t$

$=-1.386=-0.0231×t$

$t=60s$