CBSE · Class 12 · Physics · NucleiWhat is radioactivity? State the law of radioactive decay and derive the mathematical expression $N(t) = N0 e^{-\lambda t}$. Define half-life and mean life of a radioactive substance, and write down the relation between them.

Definition of Radioactivity\nRadioactivity is a spontaneous phenomenon in which unstable atomic nuclei disintegrate by emitting radiation (such as alpha particles, beta particles, and gamma rays) to attain a more stable configuration. This is a purely random and nuclear process unaffected by external physical conditions such as temperature, pressure, or chemical combinations.

Law of Radioactive Decay\nThe law of radioactive decay states that the rate of disintegration of radioactive nuclei at any given instant is directly proportional to the number of undecayed radioactive nuclei present at that instant.

Mathematical Derivation of Decay Law\nLet $N$ be the number of undecayed radioactive nuclei present at any time $t$, and $dN$ be the number of nuclei that decay in a small time interval $dt$.\nAccording to the law of radioactive decay:

$$-\frac{dN}{dt} \propto N$$ $$\frac{dN}{dt} = -\lambda N$$\nwhere $\lambda$ is a positive constant called the radioactive decay constant or disintegration constant. The negative sign indicates that the number of nuclei decreases with time. \nRearranging the terms to separate variables: $$\frac{dN}{N} = -\lambda dt$$ \nIntegrating both sides on the limits from time $t = 0$ (where $N = N_0$, the initial number of nuclei) to time $t$ (where $N = N(t)$): $$\int_{N_0}^{N(t)} \frac{dN}{N} = -\lambda \int_{0}^{t} dt$$ $$\ln\left(\frac{N(t)}{N_0}\right) = -\lambda t$$ \nTaking the exponential on both sides: $$\frac{N(t)}{N_0} = e^{-\lambda t}$$ $$N(t) = N_0 e^{-\lambda t}$|\nThis is the exponential law of radioactive decay.

Half-Life ($T_{1/2}$)\nHalf-life is defined as the time interval required for half of the radioactive nuclei initially present in a sample to disintegrate or decay.\nFormula: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

Mean Life ($\tau$)\nMean life is defined as the ratio of the total lifetime of all radioactive nuclei to the total initial number of nuclei present in the sample. It is also equal to the reciprocal of the decay constant.\nFormula: $\tau = \frac{1}{\lambda}$

Relation Between Half-Life and Mean Life\nFrom the definitions above, the relation connecting half-life and mean life is:

$$T_{1/2} = \tau \ln 2 \approx 0.693 \tau$$