States of Matter 1 Question 92
92. The pressure in a bulb dropped from 2000 to $1500 \mathrm{~mm}$ of mercury in $47 \mathrm{~min}$ when the contained oxygen leaked through a small hole. The bulb was then evacuated. A mixture of oxygen and another gas of molecular weight 79 in the molar ratio of $1: 1$ at a total pressure of $4000 \mathrm{~mm}$ of mercury was introduced. Find the molar ratio of the two gases remaining in the bulb after a period of $74 \mathrm{~min}$.
$(1981,3 \mathrm{M})$
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Solution:
- Rate of effusion is expressed as $-\frac{d p}{d t}=\frac{k p}{\sqrt{M}}$
$k=$ constant,$p=$ instantaneous pressure
$\Rightarrow \quad-\frac{d p}{p}=\frac{k d t}{\sqrt{M}}$
Integration of above equation gives $\ln \left(\frac{p_{0}}{p}\right)=\frac{k t}{\sqrt{M}}$
Using first information : $\ln \left(\frac{2000}{1500}\right)=\frac{k 47}{\sqrt{32}}$
$$ \Rightarrow \quad k=\frac{\sqrt{32}}{47} \ln \left(\frac{4}{3}\right) $$
Now in mixture, initially gases are taken in equal mole ratio, hence they have same initial partial pressure of $2000 \mathrm{~mm}$ of $\mathrm{Hg}$ each.
After $74 \mathrm{~min}$ :
For $\mathrm{O}{2} \quad \ln \left(\frac{2000}{p{\mathrm{O}_{2}}}\right)=\frac{74 k}{\sqrt{32}}$
Substituting $k$ from Eq. (i) gives
$$ \ln \left(\frac{2000}{p_{\mathrm{O}_{2}}}\right)=\frac{74}{\sqrt{32}} \times \frac{\sqrt{32}}{47} \ln \left(\frac{4}{3}\right) $$
$$ \ln \left(\frac{2000}{p_{\mathrm{O}_{2}}}\right)=\frac{74}{47} \ln \left(\frac{4}{3}\right) $$
Solving gives $p\left(\mathrm{O}_{2}\right)$ at $74 \mathrm{~min}=1271.5 \mathrm{~mm}$
For unknown gas : $\ln \left(\frac{2000}{p_{g}}\right)=\frac{74 k}{\sqrt{79}}$
Substituting $k$ from (i) gives
$$ \ln \left(\frac{2000}{p_{g}}\right)=\frac{74}{\sqrt{79}} \times \frac{\sqrt{32}}{47} \ln \left(\frac{4}{3}\right) $$
Solving gives : $\quad p_{g}=1500 \mathrm{~mm}$
$\Rightarrow$ After 74 min, $p\left(\mathrm{O}_{2}\right): p(g)=1271.5: 1500$
Also, in a mixture, partial pressure $\propto$ number of moles
$$ \Rightarrow \quad n\left(\mathrm{O}_{2}\right): n(g)=1: 1.18 $$
