Q.Derive an expression for law of radioactive decay. Define one becquerel (Bq).

Radioactive decay is a random process in which the decay rate at any instant is proportional to the number of undecayed nuclei present, giving an exponential decay law.

Derivation: Let $N$ be the number of undecayed radioactive nuclei present at time $t$. The rate of decay $\left(-\dfrac{dN}{dt}\right)$ is directly proportional to the number of nuclei present at that instant (each nucleus decays independently and randomly, with a fixed probability per unit time):

$-\dfrac{dN}{dt} \propto N \;\Rightarrow\; -\dfrac{dN}{dt} = \lambda N$

where $\lambda$ is the decay constant, characteristic of the radioactive nuclide. Separating variables and integrating,

$\int_{N_0}^{N}\dfrac{dN}{N} = -\lambda\int_0^t dt \;\Rightarrow\; \ln\dfrac{N}{N_0} = -\lambda t$

$N = N_0 e^{-\lambda t}$

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