(a) Decay constant: Let the number of nuclei of a radioactive substance present at any time be $N$. Decay of number of nuclei $(\Delta N)$ in a small interval of time $(\Delta t)$ is proportional to $N$ and $\Delta t$. $\;\Delta N \propto N \Delta t$ $\;\text{ Or, }\;\;\Delta N = -\lambda N \Delta t$ The proportionality constant $\lambda$ is known as the decay constant. (b) Half-life $=4.5 \times 10^9$ years Mass number $=238$ $\;\lambda = \frac{0.693}{T} = \frac{0.693}{4.5 \times 10^9} \;\text{ per year }\;$ $\; N = \frac{10 \times 6.02 \times 10^{23}}{238}$ Hence, the activity, $A = \lambda N$ $\;\;\text{ Or, }\;\; A = \frac{0.693}{4.5 \times 10^9} \times \frac{10 \times 6.02 \times 10^{23}}{238}$ $\; \therefore A = 0.0039 \times 10^{15} = 3.9 \times 10^{12} \;\text{ per year }\;$