An
unstable nucleus emits
energetic particles or waves and the unstable nucleus transforms to some other
nucleus or energy level. This is referred to as radioactive
decay. Three types of radioactive decay occur in nature:
(i) Alpha
decay in which a helium nucleus is emitted;
(ii) Beta
decay in which electrons or positrons (particles with the same mass as
electrons, but with a charge exactly opposite to that of electron) are emitted;
(iii) Gama
decay in which high energy (hundreds of keV or more) photons are emitted.
Law of radioactive decay
The
number of nuclei undergoing the decay per unit time is proportional to the
total number of nuclei in the sample. If N is the
number of nuclei in the sample and ΔN undergo decay in time Δt
then
\[\frac{\Delta N }{\Delta t}= \lambda N\]
Where λ is called the radioactive decay constant or disintegration constant. The change in the number of nuclei in the sample is dN = – ΔN in time Δt. Thus the rate of change of N is (in the limit Δt → 0)
\[\frac{dN }{dt}=- \lambda N\]
\[\Rightarrow \frac{dN}{N}=-\lambda dt\]
Integrating we get,
\[\Rightarrow\int_{N_{0}}^{N} \frac{dN}{N}=-\lambda \int_{0}^{t}dt\]
\[\Rightarrow ln\frac{N}{N_{0}}=-\lambda t\]
\[\therefore N(t)=N_{0}e^{-\lambda t}\]
The above equation represents the law of radioactivity.The SI unit for radioactive
decay is Becquerel (Bq). Where 1Bq=1 decay per second. Old unit for radioactive
decay was Curie. 1 Curie = 3.7 x 1010 Bq
\[\therefore \frac{N_{0}}{2}=N_{0}e^{-\lambda T_{\frac{1}{2}}}\]
\[\Rightarrow T_{\frac{1}{2}}=\frac{ln2}{\lambda }=\frac{0.693}{\lambda }\]
