Question

A certain gas A polymerizes to a small extent at a given temperature & pressure,$nA\rightleftharpoons A_n.$ Determine the approx. value of $\frac{PV}{RT}$. Assume that initially one mole of A was taken in the container.

• A. $\frac{PV}{RT}=[1-\frac{(n-1)K_c}{V^{n-1}}]$
• B. $\frac{PV}{RT}=[1+\frac{(n+1)K_c}{V^{n-1}}]$
• C. $\frac{PV}{RT}=[1-\frac{(n+1)K_c}{V^{n-1}}]$
• D. None of these

Answer 1

The correct option is A $\frac{PV}{RT}=[1-\frac{(n-1)K_c}{V^{n-1}}]$

For $nA\rightleftharpoons {\left[A\right]}_n$
At $t=010$
At equilibrium $1-\alpha \frac{\alpha }{n}$ ($\alpha$ is very small)
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,
$\frac{PV}{RT}=[1-\frac{(n-1)K_c}{V^{n-1}}]$