Theory:

The law states that ‘in any radioactive substance, the number of nuclei disintegrating per second is directly proportional to the number of nuclei present’.
 
ΔNΔtαNΔNΔt=λN
 
Where \(λ\) is the decay constant and \(N\) is the number of nuclei in an atom. The negative sign denotes that the number of nuclei decreases with time.
Half-life period:
The half-life period (\(T\)) of a radioactive substance is defined as the time during which half the amount of the substance disintegrates.
A statement 'four atoms with a half-life period of \(2\ seconds\)' means that exactly two atoms will disintegrate approximately after \(2\ seconds\).
 
The half-life period (\(T\)) varies for different substances. The disintegration of a particular atom in a radioactive element is unpredictable. Usually, a smaller value of \(T\) indicates a faster decay.
 
The decay constant is a characteristic property of a radioactive element. The formula is given as
 
Decay constant (λ)=0.693T
 
The higher value of \(λ\) denotes a faster decay.
 
Let the initial number of radioactive nuclei be \(N_0\). Consider the number of nuclei present as \(N_0\)\(\times \frac{1}{2}\) after a half-life period.
 
After two half-lives, the number of nuclei would be
 
12×N0×12=N0122
 
After '\(n\)' half-lives, the number of nuclei would be
 
N=N012n
 
If '\(t\)' is the given time period and '\(T\)' is the half-life period of the radioactive substance, then the value of half-life is
 
n=tT
 
If '\(m_o\)' is the initial mass of the substance and '\(m\)' the mass of un-decayed nuclei, then
\(\)
NN0=mm0
 
The above equation can be re-written as
 
m=m012n
 
This equation helps to find the mass of un-decayed particles.