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Answer: \(N = N_0e^{-\lambda t}\); SI unit of activity is the becquerel (Bq).

The rate of radioactive decay at any instant is directly proportional to the number of undecayed nuclei \(N\) present at that instant:

\[\frac{dN}{dt} \propto N \Rightarrow \frac{dN}{dt} = -\lambda N\]

where \(\lambda\) is the decay constant; the negative sign shows that \(N\) decreases with time. Rearranging, \(\dfrac{dN}{N} = -\lambda\,dt\). Let \(N = N_0\) at \(t = 0\) and \(N\) at time \(t\). Integrating,

\[\int_{N_0}^{N}\frac{dN}{N} = -\lambda\int_0^t dt \Rightarrow \ln\frac{N}{N_0} = -\lambda t\]\[\boxed{N = N_0e^{-\lambda t}}\]

This is the law of radioactive decay: the number of undecayed nuclei decreases exponentially with time.

Activity \(A = \left|\dfrac{dN}{dt}\right| = \lambda N\). Its SI unit is the becquerel (1 Bq = 1 decay per second). (A practical unit is the curie, 1 Ci = \(3.7\times10^{10}\) Bq.)

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report
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