Radioactivity is a spontaneous and continuous phenomenon. By the continuous emission of radioactive radiations, a radioactive element is converted into another radioactive element which is also convened or disintegrated into a new element. This process continues unit a new non-radioactive element is produced. During disintegration α or β-rays are emitted. In 1902, Rutherford and Soddy, after extensive studies formulated a law, known as the Law of radioactive disintegration or decay. The law is stated below:

*At any moment the number of radioactive atoms that disintegrate in unit time is directly proportional to the number of unchanged radioactive atoms remaining.*

If the rate of radioactive disintegration of atoms is dN/dt and if N is the number of unchanged atoms at time t, then – – dN/dt ∞ N or, – dN/dt = constant x N.

or, dN/dt = – λN

or, DN/N = – λ.dt. … … … (1)

Here λ is the radioactive decay constant. Since the number of atoms of radioactive substance decreases, hence a negative sign has been used in the equation.

Integrating the equation (1) we get,

log_{e} N = – λt + C … … … (2)

Here C is the constant of integration. Its value is to be determined.

Suppose at the initial time, t = 0, then N = N_{0}.

From equation (2) we get,

log_{e} N_{0} = C … … … (3)

Now from equations (2) and (3) we get,

log_{e}N = – λt + log_{e}N_{0}.

or, log_{e}N – log_{e}N_{0} = – λt

or, log_{e }N/N_{0} = log_{e}e^{-λt}

or, N/N_{0} = e^{-λt}

So, N = N_{0} e^{-λt} … … …. (4)

This is the law of radioactive decay or disintegration.

This law obeys the exponential law [Figure). Its graph i.e., the graph of time (t) versus unchanged atoms N is shown in the adjoining figure.

From the figure, it is seen that at the start i.e., t = 0, if there is a fixed amount of radioactive atoms then after T its amount decreases by half and after 2T time that residue amount becomes half again. i.e., then it will be ¼ part of the original amount.

From N – t curve it is proved that for total disintegration of a radioactive material it takes infinite time.

*Decay constant or disintegration constant:*

From equation (4) we get, dN/dt = – λN

or, λ = – (dN/dt) / N.

Now, if N = 1, then from the above equation we get,

λ = – dN/dt.

That means decay constant indicates the probability of decay of an atom in unit time.

**Definition:** The probability of disintegration or decay of an atom of a radioactive substance in unit time is called the decay constant or disintegration constant.

Its unit is s^{-1} or day^{-1} or yr^{-1}.