Question

A certain gas $A$ polymerizes to a small extent at a given temperature and pressure, $nA\rightleftharpoons A_n$. Show that the gas obeys the approx. equation $\frac{PV}{RT}=[1-\frac{(n-1)K_C}{V^{n-1}}]$ where $K_C=\frac{\left[A_n\right]}{[A{]}^n}$ and $V$ is the volume of the container. Assume that initially one mole of $A$ was taken in the container.

Answer 1

For $\underset{1(1-\alpha )}{nA}\rightleftharpoons \underset{0\frac{\alpha }{n}}{{\left[A\right]}_n}$

Total mole at equilibrium $=1-\alpha +\frac{\alpha }{n}$

$\therefore$ Gas equation is :

$\frac{PV}{RT}=1-\alpha +\frac{\alpha }{n}$ or $\frac{P}{RT}=\frac{1-\alpha +\frac{\alpha }{n}}{V}$ ...(i)

Given, $\frac{PV}{RT}=1-\frac{(n-1)K_c}{V^{n-1}}$

$\frac{P}{RT}=\frac{1}{V}-\frac{(n-1)K_c}{V^n}$

$=\frac{1}{V}-\frac{(n-1)\times \frac{\alpha }{n\cdot V}}{V^n\times {\left(\frac{1-\alpha }{V}\right)}^n}$ $(\because K_c=\frac{\frac{\alpha }{nV}}{{\left(\frac{1-\alpha }{V}\right)}^n})$

$=[1-\frac{(n-1)\alpha }{n{(1-\alpha )}^n}]\frac{1}{V}$

or $\frac{P}{RT}=\frac{1}{V}[1-\frac{(n-1)\alpha }{n}]$

$[\because {(1-\alpha )}^n=1]$

or $\frac{P}{RT}=\frac{1}{V}[1-\alpha +\frac{\alpha }{n}]$

$\frac{P}{RT}=\frac{1-\alpha +\frac{\alpha }{n}}{V}$ ...(ii)

by these equations it can be seen that,

$\frac{PV}{RT}=[1-\frac{(n-1)K_c}{V^{n-1}}]$