Question

The vapour pressure of two miscible liquids $A$ and $B$ are $300$ and $500 \,mm$ of $Hg$ respectively. In a flask $10$ moles of $A$ is mixed with $12$ moles of $B$. However, as soon as $B$ is added, $A$ starts polymerising into a completely insoluble solid. The polymerisation follows first-order kinetics. After $100 \,\min$, $0.525$ mole of a solute is dissolved which arrests the polymerisation completely. The final vapour pressure of the solution is $400 \,mm$ of $Hg$. Estimate the rate constant of the polymerisation reaction. Assume negligible volume change on mixing and polymerisation and ideal behaviour for the final solution.

Answer 1

Let after $100\, \min , x$ moles of $A$ are remaining unpolymerised moles of $B=12$
Moles of non-volatile solute $=0.525$
$\Rightarrow $ Mole fraction of $A=\frac{\chi}{\chi+12+0.525}$
Mole fraction of $B=\frac{12}{\chi+12+0.525}$
$\Rightarrow 400=\left(\frac{\chi}{\chi+12.525}\right) \times 300+\left(\frac{12}{\chi+12.525}\right) \times 500$
$\Rightarrow \chi=9.9$
$\Rightarrow$ Moles of $A$ polymerised in $100\, \min =10-9.9=0.10$
$\Rightarrow k=\frac{1}{t} \ln \frac{10}{9.9}=\frac{1}{100} \ln \frac{10}{9.9} \min ^{-1}$
$=1.005 \times 10^{-4} \min ^{-1}$