Question

A radioactive isotope decays as ${}_z{A}^m{\to }_{z-2}{B}^{m-4}{\to }_{z-1}{C}^{m-4}$. The half lives of $A$ and $B$ are $6$ months and $10$ months respectively. Assuming that initially only $A$ was present, will it be possible to achieve radioactive equilibrium for $B$. If so what would be the ratios of $A$ and $B$ at equilibrium. What would happen if the half lives $A$ and $B$ were $10$ months and $6$ months respectively.

Answer 1

${}_Z^m\text{A}\stackrel{-\alpha }{\to }{}_{Z-2}^{m-4}\text{B}\stackrel{-\beta }{\to }{}_{Z-1}^{m-4}\text{C}$
$(t_{1/2}{)}_A=10$ month

$(t_{1/2}{)}_B=6$ month
The radioactive equilibrium is attached then, at equilibrium the ratio of atoms of $A$ and $B$ left is:
$\frac{N_A}{N_B}=\frac{t_{1/2}A}{t_{1/2}B}=\frac{10}{6}=1.66$

Therefore the answer is 2.
If half-life of $A=6$ month and $B$ is 10 month, then since $t_{1/2}A<t_{1/2}B$ or ${\lambda }_A>{\lambda }_A$ and thus no equilibrium will be set.
$Note:$ Radioactive process are first order in nature and any radioactive species completely decays only at infinite time.