Question

Identify the wrong relation for real gases:

• A. \( Z = \frac{V_{\text{ideal}}}{V_{\text{real}}} \)
• B. \( p_{\text{ideal}} = p_{\text{real}} + \frac{an^2}{V^2} \)
• C. \( V_{\text{real}} = V_{\text{ideal}} - nb \)
• D. \( \left( p + \frac{a}{V^2} \right) (V - b) = RT \)

Hint:

The compressibility factor Z measures deviations from ideal gas behavior, involving pressure, volume, and temperature, not the ratio of ideal to real volumes.

Answer 1

The Correct Option is A

To determine the wrong relation for real gases among the given options, let's analyze each equation based on the principles of real gases and the Van der Waals equation. 

1. \(Z = \frac{V_{\text{ideal}}}{V_{\text{real}}}\): This equation appears incorrect because the compressibility factor \(Z\) is defined as the ratio of the volume of a real gas to the volume of an ideal gas under the same conditions, i.e., \(Z = \frac{V_{\text{real}}}{V_{\text{ideal}}}\). Therefore, this relation is incorrect.
2. \(p_{\text{ideal}} = p_{\text{real}} + \frac{an^2}{V^2}\): This equation reflects the correction applied to the pressure of a real gas. Due to intermolecular forces, the actual pressure is higher than the ideal pressure. Hence, this relation represents how the actual pressure of a real gas is augmented over the ideal pressure.
3. \(V_{\text{real}} = V_{\text{ideal}} - nb\): This is a part of the volume correction in the Van der Waals equation, where \(b\) is the volume occupied by the gas particles. This equation correctly depicts how the volume is adjusted for the finite size of gas molecules.
4. \(\left( p + \frac{a}{V^2} \right) (V - b) = RT\): This is the Van der Waals equation, which accounts for the pressure and volume corrections for real gases. It represents how real gases deviate from ideal gas behavior.

Based on the above analysis, the relation \(Z = \frac{V_{\text{ideal}}}{V_{\text{real}}}\) is incorrect, as it contradicts the standard definition of the compressibility factor. The compressibility factor should be \(Z = \frac{V_{\text{real}}}{V_{\text{ideal}}}\), reflecting that real gases occupy more volume than predicted by the ideal gas law.

Hence, the correct answer is: \(Z = \frac{V_{\text{ideal}}}{V_{\text{real}}}\).

Answer 2

The compressibility factor \( Z \) is a measure of deviation from ideal gas behavior and is defined as:
\[ Z = \frac{p_{\text{real}}V_{\text{real}}}{RT} \]
Option (A), which states \( Z = \frac{V_{\text{ideal}}}{V_{\text{real}}} \), is incorrect because \( Z \) is not defined in terms of the ratio of ideal to real volume. Instead, it involves pressure, volume, and temperature of the real gas relative to the ideal gas law.

Options (B), (C), and (D) are correct:

• (B) Adjusts pressure due to intermolecular forces (\( \frac{an^2}{V^2} \)).
• (C) Accounts for excluded volume (\( nb \)) in real gases.
• (D) Represents the van der Waals equation, which adjusts both pressure and volume.

Thus, the wrong relation is Option (A).