Question

The compressibility factor for one mole of Vander Waal's constant gas at 0C and 100 atm pressures is to be
0.5. Assuming that the volume of a gas molecule is negligible, calculate the Vander Waals constant a.

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}, \\ \mathrm{Z}=\frac{\mathrm{pV}}{\mathrm{nRT}} \\ \mathrm{or},\mathrm{V}=\frac{\mathrm{ZnRT}}{\mathrm{p}} \\ =\frac{0.5\times 1\times 0.0821\times 273}{100} \\ =0.1119\mathrm{L} \\ \mathrm{Now}, \\ (\mathrm{p}+\frac{\mathrm{a}}{{\mathrm{V}}^2})(\mathrm{V}-\mathrm{b})=\mathrm{RT}(\mathrm{since}\mathrm{n}=1) \\ \mathrm{Neglecting}\mathrm{b}, \\ (\mathrm{p}+\frac{\mathrm{a}}{{\mathrm{V}}^2})\mathrm{V}=\mathrm{RT} \\ \mathrm{or},\mathrm{pV}+\frac{\mathrm{a}}{\mathrm{V}}=\mathrm{RT} \\ \mathrm{or},\frac{\mathrm{pV}}{\mathrm{RT}}+\frac{\mathrm{a}}{\mathrm{VRT}}=1 \\ \mathrm{or},\mathrm{a}=(1-\frac{\mathrm{pV}}{\mathrm{RT}})\mathrm{VRT} \\ =(1-0.5)0.1119\times 0.0821\times 273 \\ =1.252{\mathrm{atmL}}^2{\mathrm{mol}}^{-2} \end{gathered}$$