Describe the laws of radioactive decay. Derive equation of radioactive decay using these laws.

Explanation

Radioactivity is the spontaneous transformation of an element into another with the emission of some particles.
Laws of Radioactive disintegration
1. Radioactivity is spontaneous process which does not depend upon external factors.
During disintegration either `alpha` or `beta`-particle is emitted. Both are never emitted simultaneously.
3. Emission of `alpha`-particle decreases atomic number by two and mass number by 4.
4. Emission of `beta`-particle increases atomic number by one but mass number remains the same.
5. Emission of `gamma`-ray does not change atomic or mass number.
6. The number of atoms disintegrated per second is directly proportional to the number of radioactive atoms actually present at that instant. This law is called radioactive decay law.
i.e. `-(dN)/(dt) prop N`
or `-(dN)/(dt)= lambda N`, ...(i)
where `lambda` is a constant called disintegration constant and depends upon the nature of the radioactive substance.
Now from (i) , we have
`(1)/(N) dN = - lambda dt`
or `int 1/N dN = lambda int dt`
or `log_(e)N=lambda t +C`, ...(ii)
where C is constant of integration. To determine its value.
Let `N=N_(0)` initially,
i.e. when `t=0, N=N_(0)`
`log_(e)=N=0+C`
Substituting the value of C in (ii) , we have
`log_(e)N=-lambda t + log_(e)N_(0)`
or `log_(e)N-log_(e)N_(0)=-lambda t`
or `log_(e). (N_(0))/(N)= -lambdat`
or `(N)/(N_(0))=e^(-lambda t)`
or `N=N_(0)e^(-lambda t)`
which is the required equation.
Disintegration constant `(lambda)` is defined as the time after which the number of radioactive atoms reduce to 1/e times the original number of atoms.
Half life period (T) is the time during which the number of atoms of a radioactive material reduces to half of the original number .