Question

Two radio isotopes $P$ and $Q$ of atomic mass 10 and 20 respectively are mixed in equal amount by mass. After 20days, their weight ratio $\frac{W_P}{W_Q}$ is found to be 1 : 4. Isotope $Q$ has a half-life of 10 days. The half-life of isotope $P$ is :

• A. zero
• B. 5 days
• C. 20 days
• D. infinite

Answer 1

The correct option is B 5 days

Let w g of each was taken, then
$InitialmoleofP=\frac{w}{10}$
$moleofQ=\frac{w}{20}$
$FinalmoleofP=\frac{w_1}{5\times 10}$;
$FinalmoleofQ=\frac{4w_1}{20\times 5}$
For $P,\frac{N_{0_P}}{N_P}=e^{{\lambda }_{P\times t}}$
For $Q,\frac{N_{0_Q}}{N_Q}=e^{{\lambda }_{Q\times t}}$
$\therefore ForP,\frac{w\times 5\times 10}{10\times w_1}=e^{{\lambda }_{P\times 20}}$ ...( i)
$ForQ,\frac{w\times 20\times 5}{20\times w_1\times 4}=e^{{\lambda }_{Q\times 20}}$ ...( ii)
From Eqs. (i) and (ii), $4=e^{({\lambda }_P-{\lambda }_Q)\times 20}$
$\therefore 20({\lambda }_P-{\lambda }_Q)={\log }_{e}4$
$20[\frac{0.693}{t_{1/2}}-\frac{0.693}{10}]={\log }_{e}4$
$\therefore t_{\left(1/2\right)P}=5$