JEE Main & Advanced Physics Nuclear Physics And Radioactivity Successive Disintegration and Radioactive Equilibrium

Suppose a radioactive element A disintegrates to form another radioactive element B which intern disintegrates to still another element C; such decays are called successive disintegration.

Rate of disintegration of \[A=\frac{d{{N}_{1}}}{dt}=-{{\lambda }_{1}}{{N}_{1}}\]  (which is also the rate of formation of B)

Rate of disintegration of \[B=\frac{d{{N}_{2}}}{dt}=-{{\lambda }_{2}}{{N}_{2}}\]

\[\therefore \] Net rate of formation of B = Rate of disintegration of A - Rate of disintegration of B

\[={{\lambda }_{1}}{{N}_{1}}-{{\lambda }_{2}}{{N}_{2}}\]

Equilibrium

In radioactive equilibrium, the rate of decay of any radioactive product is just equal to it's rate of production from the previous member.

i.e.\[{{\lambda }_{1}}{{N}_{1}}={{\lambda }_{2}}{{N}_{2}}\Rightarrow \]\[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}=\frac{{{N}_{2}}}{{{N}_{2}}}=\frac{{{\tau }_{2}}}{{{\tau }_{1}}}=\frac{({{T}_{1/2}})}{{{({{T}_{1/2}})}_{1}}}\]