Write the definition of half-life of a radioactive substance and obtain its relation to the decay constant.

(N/A) The half-life of a radioactive substance is defined as the time interval during which the number of radioactive nuclei reduces to half of its initial value.
Let $N_0$ be the initial number of nuclei at $t = 0$. After one half-life $T_{1/2}$,the number of nuclei $N$ becomes $N_0 / 2$.
According to the radioactive decay law:
$N = N_0 e^{-\lambda t}$
Substituting $N = N_0 / 2$ and $t = T_{1/2}$:
$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$
$\frac{1}{2} = e^{-\lambda T_{1/2}}$
Taking the natural logarithm $(\ln)$ on both sides:
$\ln(1/2) = -\lambda T_{1/2}$
$-\ln(2) = -\lambda T_{1/2}$
$\ln(2) = \lambda T_{1/2}$
Since $\ln(2) \approx 0.693$:
$0.693 = \lambda T_{1/2}$
Therefore,the relation is:
$T_{1/2} = \frac{0.693}{\lambda}$
Hence,the half-life of a radioactive element is inversely proportional to the decay constant $\lambda$ and is independent of the number of nuclei present in the sample.