Chemistry - Real Gas Question with Solution | TestHub

$\mathrm{PV}=\mathrm{RT}-\frac{\mathrm{A}}{\mathrm{V}}+\frac{2\mathrm{B}}{{\mathrm{V}}^2}$

$\mathrm{P}=\frac{\mathrm{RT}}{\mathrm{V}}-\frac{\mathrm{A}}{{\mathrm{V}}^2}+\frac{2\mathrm{B}}{{\mathrm{V}}^3}$

At critical point $\frac{\mathrm{dP}}{\mathrm{dV}}=0$

$\frac{dP}{dV}=-\frac{RT}{V^2}+\frac{2A}{V^3}-\frac{6B}{V^4}$

$0=\frac{1}{{\mathrm{V}}^2}\left[-\mathrm{RT}+\frac{2\mathrm{A}}{\mathrm{V}}-\frac{6\mathrm{B}}{{\mathrm{V}}^2}\right]$

${\mathrm{RTV}}^2-2\mathrm{AV}+6\mathrm{B}=0$

$\mathrm{D}=4{\mathrm{A}}^2-4\times \mathrm{RT}\times 6\mathrm{B}=0$

${\mathrm{T}}_{\mathrm{C}}=\frac{{\mathrm{A}}^2}{6\mathrm{RB}};{\mathrm{V}}_{\mathrm{C}}=\frac{6\mathrm{B}}{\mathrm{A}}$

${\mathrm{P}}_{\mathrm{C}}=\frac{{\mathrm{A}}^3}{108{\mathrm{B}}^2};\mathrm{Z}=\frac{{\mathrm{P}}_{\mathrm{C}}{\mathrm{V}}_{\mathrm{C}}}{{\mathrm{RT}}_{\mathrm{C}}}\Rightarrow \frac{1}{\mathrm{Z}}=3$