Question

) Two radioactive nuclei having half life periods
20 years and 15 years respectively have initial
activities in the ratio 1 : 2. After how many years,
both posses equal activity?​

Answer 1

The number of years after which the 2 nuclei will have the same activity is 60

• Number of Nuclei at a time is given by N =N0.$e^{-Lt}$

• at t = T, half life period, N = N0/2

• N/N0  = $e^{-LT}$ = 1/2

• natural log, ln(1/2) = -LT

• ln 2 = LT ==> L = 0.693/T

Let L be the decay constant and T be the half life

If a and b are the 2 nuclei,

• La = 0.693/Ta and Lb = 0.693/Tb

• La = 0.03465 , Lb = 0.0462

Activity of the nuclei is given by

• R =  $\frac{dN}{dt}$ = -LN0$e^{-Lt}$ = -LN.

Therefore given ratio of initial activities:

• $\frac{Ra(0)}{Rb(0)}$ = $\frac{1}{2}$

• $\frac{1}{2} = \frac{LaNa(0)}{LbNb(0)}$

To find the time when both have equal activity,

• Ra = Rb
• -LaNa(0)$e^{-Lat}$ = -LbNb(0)$e^{-Lbt}$

rearranging ==>

• $\frac{1}{2} =$ $e^{-(Lb - La)t}$
• ln 2 = (Lb - La)t

==>

• t = ln 2/ (Lb - La) = ln 2/(0.0462 - 0.03465) = 60 years