In a sample of radioactive material, the probability that a particular nucleus will decay in next 2 hour is 8 x 10-4. The half-life (in hour) of the radioactive sample is  [Take ln⁡2=0.693,ln⁡(0.9992)=−8×10−4

Probability that a particular nucleus will decay in next 2 hr is P=N0−NN0=8×10−4[N= surviving population after 2hr]⇒N01−8×10−4=N⇒ 1−8×10−4=e−λt ∵NN0=e−λt⇒ e−λt=0.9992⇒−λt=ℓn(0.9992)⇒−ln2t1/2×2=ln(0.9992) [∵t=2hr]⇒t1/2=2ln2ln(0.9992)hr=2×0.6938×10−4=1732.5hr