Class 12 NEET Physics Atoms and Nuclei Part 1

Concept:Half-life is the time required for a radioactive sample to decay to half its initial amount, related to decay constant through the exponential decay law.Explanation:The radioactive decay follows the exponential decay equation:$$N(t) = N_0 e^{-\lambda t}$$where $N_0$ is the initial number of nuclei, $\lambda$ is the decay constant, and $t$ is time.At half-life $T_{\frac{1}{2}}$, the number of nuclei remaining is half the initial amount:$$N(T_{\frac{1}{2}}) = \frac{N_0}{2}$$Substituting into the decay equation:$$\frac{N_0}{2} = N_0 e^{-\lambda T_{\frac{1}{2}}}$$Dividing both sides by $N_0$:$$\frac{1}{2} = e^{-\lambda T_{\frac{1}{2}}}$$Taking natural logarithm on both sides:$$\ln\left(\frac{1}{2}\right) = -\lambda T_{\frac{1}{2}}$$$$-\ln 2 = -\lambda T_{\frac{1}{2}}$$$$\ln 2 = \lambda T_{\frac{1}{2}}$$Rearranging:$$T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}$$Answer:Option A: $T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}$