Question

Radioactive nuclei P and Q disintegrate into R with half lives 1 month and 2 months respectively . At time t= 0 , number of nuclei of each P and Q is x. time at which rate of disintegration of P and Q are equal, number of nuclei of R is.

Answer 1

When the activity (disintegration rate) of both are equal,at time $t$

$A={\lambda }_1{N}_1={\lambda }_2{N}_2$

$={\lambda }_1{N}_0{e}^{{\lambda }_{1}t}={\lambda }_2{N}_0{e}^{e-{\lambda }_{2}t}$

$=\frac{{\lambda }_1}{{\lambda }_2}=e^{{\lambda }_{1}t-{\lambda }_{2}t}$

$=t=\frac{ln\left(\frac{{\lambda }_1}{{\lambda }_2}\right)}{({\lambda }_1-{\lambda }_2)}=\frac{ln\left(\frac{T_2}{T_1}\right)}{ln2((\frac{1}{T_1}-\frac{1}{T_2})}$

$=\frac{ln\left(2\right)}{ln\left(2\right)(1-\frac{1}{2})}=2$ months

In 2 months, P undergoes 2 half lives, so 75% = $\frac{3x}{4}$ disintegrates to R

and Q undergoes 1 half life, so 50% = $\frac{x}{2}$ disintegrates to K.

$\therefore$ Amount of R formed = $\frac{3x}{4}+\frac{x}{2}=\frac{5x}{4}$