Question

In the following compressibility factor $(Z)$ versus pressure graph at $300\, K$, the compressibility of $CH_4$ at pressure $< 200$ bar deviates from ideal behaviour because

• A. the molar volume of $CH_4$ is less than its molar volume in the ideal state
• B. the molar volume of $CH_4$ is same as that in its ideal state
• C. inrermolecular interactions between $CH_4$ molecules decreases
• D. the molar volume of $CH_4$ is more than its molar volume in the ideal state

Answer 1

Answer: (A)

Compressibility factor,$ Z = \frac{pV}{nRT}$
For ideal gas $Z = 1$ (at all temperature and pressures)
For real gas,$Z > 1 $(At high pressures) or $Z < 1$(At low pressures )
For real gas, $Z = \frac{pV_{\text{real}}}{nRT} \,\,...(i)$
If the gas shows ideal behaviour, then
$V_{\text{ideal}} = \frac{nRT}{p} $
or $\frac{p}{nRT} = \frac{1}{V_{\text{ideal}}}\,\,...(ii)$
Putting $(ii)$ in $(i)$
$Z = \frac{V_\text{real}}{V_\text{ideal}}$
If we consider molar volume, then
$Z = \frac{(V_m)_\text{real}}{(V_m)_\text{ideal}}$
As in given case
$Z = \frac{(V_m)_\text{real}}{(V_m)_\text{ideal}} < 1$ at $ p < 200 $ bar
$\therefore (V_m)_{CH_4} $ real $ < (V_m)_{CH_4}$ ideal