Question

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

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

Answer 1

The correct option is B $5$ days

Let $wg$ of each we take, then
Initial mole of $P=\frac{w}{10}$
mole of $Q=\frac{w}{20}$
Final mole of $P=\frac{w_1}{5\times 10}$
Final mole of $Q=\frac{4w_1}{20\times 5}$
For $P,\frac{N_{0p}}{N_P}=e^{{\lambda }_{P\times t}}$
For $Q,\frac{N_{0Q}}{N_Q}=e^{{\lambda }_{Q\times t}}$
$\therefore$ For $P,\frac{w\times 5\times 10}{10\times w_1}=e^{{\lambda }_{P\times 20}}...\left(i\right)$
For $QP,\frac{w\times 20\times 5}{20\times w_1\times 4}=e^{{\lambda }_{Q\times 20}}...\left(ii\right)$
From Eqn, $\left(i\right)$ and $\left(ii\right)$, $4=e^{({\lambda }_P-{\lambda }_q)\times 20}$
$\therefore 20({\lambda }_P-{\lambda }_q)={\log }_{e}4$
$20[\frac{0.693}{t_{1/2}}-\frac{0.693}{10}]={\log }_{e}4$
$\therefore t_{\left(t/2\right)}=5$