The rate of radioactive material disintegration is independent of physical and chemical conditions. The number of atom that break up at any instant is proportional to the number of atoms present at that instant. Let $N$ be the number of atoms present in a particular radioactive element at a given instant $t$. Then rate of decrease $-dN/dt$ is proportional to $N$.

$-\dfrac{dN}{dt} = \lambda N$ (1)

here $\lambda$ is constant known as disintegration constant or decay constant of the radioactive element. It is defined as the ratio of the substance which disintegrates in a unit time to the amount of substance present

$\lambda = \dfrac{-dN/dt}{N}$

Equation (1) can be written as $\dfrac{dN}{N} = - \lambda dt$

Integrating we get

$log_e N = - \lambda t + C$ (2) here C is constant of integration

Let $N_0$ be number of radioactive atoms present initially.

Then when $t = 0$ , $N = N_0$

Therefore, $log_e N_0 = C$

Substituting for C in (2) we get

$log N = - \lambda t + log N_0$

$log_e \dfrac{N}{N_0} = - \lambda t$

$N = N_0 e^{-\lambda t}$ (3)

This equation shows that number of atoms on taken radioactive element decreases exponentially with time. Theoretically, infinite time is required for radioactivity to disappear completely and this is same for all elements. Hence to compare two element’s radioactive properties, a quantity called half-life period is used.

Half life period: The half life period of a radioactive substance is defined as the time required for one half of the radioactive substance to disintegrate. For any given radioactive element, at the end of time $T_{1/2}$ only 50% of the radioactive atoms remain unchanged and at the end of $2T_{1/2}$ only 25% remain and at the end of $3T_{1/2}$ 12.5% remain and after $4T_{1/2}$ 6.25% remain and so on.

Value of half-life period

We have the relation $N = N_0 e^{-\lambda t}$

If $T_{1/2}$ be the half life period, then at $t = T_{1/2}$ and $N = N_0 / 2$

$\dfrac{N_0}{2} = N_0 e^{- \lambda T_{1/2}}$

$e^{\lambda T_{1/2}} = 2$

$\lambda T_{1/2} = log_e 2$

$T_{1/2} = \dfrac{log_e 2}{\lambda}$

$T_{1/2} = \dfrac{0.6931}{\lambda}$

Thus we can conclude that half life is inversely proportional to disintegration constant lambda. Half life is different for different radioactive substances. Uranium has a half life period of $4.5 \times 10^9 years$ and Radium has a half life of 1622 years while element Radon has half life of 3.8 days.