Which of the following is incorrect for real gas at high pressure and room temperature

Question

TestHub · Accepted Answer

The compressibility factor Z for a real gas is given by $Z = \frac{P V _{m}}{RT}$ .

At high pressure and room temperature, the volume of the gas molecules becomes significant, and repulsive forces dominate. The van der Waals equation for real gases is $(P + \frac{a}{V _{m}^{2}}) (V_{m} - b) = RT$ .

At high pressure, the term $\frac{a}{V _{m}^{2}}$ becomes negligible compared to P, so the equation simplifies to $P (V_{m} - b) = RT$ .

Dividing by RT, we get $\frac{P V _{m}}{RT} - \frac{P b}{RT} = 1$ .

Therefore, $Z = 1 + \frac{P b}{RT}$ .

This shows that Z > 1, indicating positive deviation from ideal behavior due to dominant repulsive forces.

Option B, $Z = 1 + \frac{a}{RT V _{m}}$ , is incorrect. This expression is derived under conditions of low pressure, where attractive forces are more significant, and the volume term $b$ is negligible. In that case, the van der Waals equation becomes $(P + \frac{a}{V _{m}^{2}}) V_{m} = RT$ , leading to $P V_{m} + \frac{a}{V _{m}} = RT$ , and thus $Z = 1 - \frac{a}{RT V _{m}}$ .

The final answer is $B$

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