# Laws of Radioactivity & Half Life

The emission of radiation (α, β, γ) from the nucleus accompanied by its spontaneous disintegration is called radioactivity.

It is observed to occur in Uranium, Radium and other heavy elements and their compounds.
Radioactive decay is observed to be a random phenomenon. It is impossible to predict when a particular nucleus will decay.

The rate of radioactive radiation from an element does not depend on temperature, chemical combination, pressure, etc. since this is a nuclear phenomenon.

The Rutherford-Soddy displacement laws were one of the first quantitative laws of radioactivity to be proposed:
(i) In α decay, the daughter element (product) is observed to be displaced by two positions below (to the Left) the parent in the periodic table and the mass number reduces by 4 units.

(ii) In β decay, the daughter element is observed to be displaced by one position above (i.e. to the right) w.r.t. the parent, while its mass number does not change.

The activity of a sample is observed to be directly proportional to number of parent nuclei present in the sample, since the contribution of each nucleus is constant and independent of all other nuclei.

If the sample contains only the element A which decays into a daughter B,

A → B, then,

$\displaystyle \frac{dN}{dt} = -\lambda N$

Here λ is a constant (called decay constant) and N is the instantaneous number of active nuclei emitting the radiation.

$\displaystyle \frac{dN}{N} = -\lambda dt$

If at t = 0 number of active nuclei be No and at any time it is N then

$\displaystyle \int_{N_0}^{N} \frac{dN}{N} = -\lambda \int_{0}^{t}dt$

$\displaystyle ln\frac{N}{N_0} = -\lambda t$

$\displaystyle N = N_0e^{-\lambda t}$

This expression gives the relation between instantaneous number of active nuclei and initial number of active nuclei.

The quantity λN gives the number of decays per unit time and is known as activity of the sample.

$\displaystyle A = \lambda N$

$\displaystyle A = \lambda N_0e^{-\lambda t}$

$\displaystyle A = A_0 e^{-\lambda t}$

Unit of Activity :
SI unit of activity is Becquerel (Bq)
1 Becquerel = 1 disintegration per second. Popular unit of activity is curie (Ci)
1 Curie = 3.7 x 1010 disintegration per second.

### Half life :

The time interval in which the number of active nuclei reduces to half of its initial value is called half-life

$\displaystyle N = N_0e^{-\lambda t}$

If half life is denoted by T1/2 then at

$\displaystyle T=T_{1/2} , N = \frac{N_0}{2}$

$\displaystyle \frac{N_0}{2} = N_0e^{-\lambda T_{1/2}}$

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

$\displaystyle T_{1/2} = \frac{ln 2}{\lambda}$

Since , ln 2 = 0.693

$\displaystyle T_{1/2} = \frac{0.693}{\lambda}$