Question

If $f$ represents the intermolecular force between molecules and $V$ is the volume occupied by the gas molecules, then at high pressure and at low temperatures, the gas shows large deviation from the ideal behaviour because the fact that

• A. $f$ is negligible
• B. both $f$ and $V$ are negligible
• C. $V$ is negligible
• D. $f$ is appreciable and $V$ is negligible
• E. $f$ is appreciable and $V$ is not negligible

Hint:

Logic Tip: The van der Waals equation $(P + \frac{an^2}{V^2})(V - nb) = nRT$ mathematically corrects for these exact two non-ideal behaviors. The term $\frac{an^2}{V^2}$ corrects for appreciable intermolecular forces ($f$), and $nb$ corrects for non-negligible molecular volume ($V$).

Answer 1

Answer: (E)

Concept:
The Kinetic Theory of Gases postulates two major assumptions for an \textit{ideal} gas: 1. The volume occupied by the gas molecules themselves is negligible compared to the total volume of the container. 2. There are no attractive or repulsive intermolecular forces between the gas molecules. Real gases obey ideal gas laws only at high temperatures and low pressures. Under extreme conditions, these ideal assumptions fail.
Step 1: Analyze the effect of high pressure.
At high pressures, the gas is severely compressed, decreasing the overall volume of the container. In this compressed state, the actual physical volume occupied by the individual gas molecules ($V$) becomes a significant fraction of the total volume. Therefore, the volume of the molecules is not negligible.
Step 2: Analyze the effect of low temperature.
At low temperatures, the kinetic energy of the gas molecules decreases significantly. Because the molecules are moving much slower, they spend more time in close proximity to one another. This allows the intermolecular attractive forces ($f$) to take effect and pull the molecules together. Therefore, the intermolecular forces become appreciable (significant).
Step 3: Combine the real gas conditions.
Combining both physical realities under these extreme conditions means neither ideal assumption holds true: $f$ is appreciable, and $V$ is not negligible. This leads to the derivation of the van der Waals equation for real gases, which includes correction factors for both $f$ and $V$.