Questions: Real Gas Thermodynamics and Equations of State

Question 1

Engineers designing a high-pressure gas system find that the ideal gas law predicts a molar volume 12% higher than experimentally measured at operating conditions. What is the most likely physical explanation?

• A. The gas is behaving more ideally at high pressure than at low pressure due to increased molecular collisions
• B. Intermolecular attractive forces pull molecules slightly inward, reducing the pressure they exert on container walls and resulting in a smaller actual volume than the ideal prediction
• C. The molecular volume correction overestimates the space occupied by molecules at high pressure
• D. Temperature dominates at high pressure, making the ideal gas a better approximation than at low pressure

Answer: (B)

At high pressure, molecules are close together and intermolecular attractions become significant. These attractions pull molecules away from the container walls, reducing the pressure the gas exerts. Since PV = nRT assumes ideal pressure, the actual pressure is lower than ideal at the same V and T — meaning you need a smaller V to reach the same measured pressure. The compressibility factor Z = PV/nRT < 1 in this regime, reflecting that the gas occupies less volume than the ideal law predicts. The van der Waals correction −a/V² captures this effect.

Question 2

The principle of corresponding states allows a single generalized compressibility chart to estimate Z for many different gases. What makes this possible?

• A. All gases have the same molecular size and interaction strength at room temperature and pressure
• B. The van der Waals equation has identical mathematical form for every gas
• C. When expressed in reduced variables (T_r = T/T_c, P_r = P/P_c), all gases exhibit approximately the same compressibility factor Z
• D. Gases with the same molecular weight behave identically at any given temperature and pressure

Answer: (C)

The principle of corresponding states emerges from the observation that cubic equations of state, when written in reduced variables (normalized by critical point values), take a universal form with no gas-specific parameters. This means that all gases at the same fraction of their critical temperature and pressure should have approximately the same Z. The principle is not exact — acentric factor corrections improve it for polar and non-spherical molecules — but it is accurate enough for engineering estimates when precise EOS data is unavailable, and underlies all generalized compressibility charts.

Question 3

The van der Waals correction term −a/V² acts to reduce the pressure below that predicted by the ideal gas law because attractive forces pull molecules away from the container walls.

• T. True
• F. False

Answer: True

In the interior of a dense gas, attractive forces on a molecule are roughly isotropic — it is pulled equally in all directions by its neighbors. But a molecule near the container wall is pulled backward (inward) by the bulk of the gas with no compensating pull from the wall side. This net inward pull reduces the molecule's effective velocity when it strikes the wall, lowering the pressure it exerts. The van der Waals equation accounts for this by subtracting a/V² from the pressure term, where a reflects the strength of intermolecular attractions and V² the square of molar volume (density effect).

Question 4

Real gas behavior deviates most strongly from the ideal gas law at high temperatures and low pressures, where molecules move fastest.

• T. True
• F. False

Answer: False

The opposite is true: ideal gas assumptions hold best at high temperature and low pressure, where molecules are fast (kinetic energy dominates over intermolecular potential energy) and widely spaced (molecular volume is negligible compared to total volume). Deviations are largest near and below the critical point, where high pressure brings molecules into close proximity (making molecular volume significant) and relatively low temperature allows intermolecular attractions to become comparable to kinetic energy. This is precisely the regime where van der Waals and Peng-Robinson equations are most necessary.

Question 5

Why do the two corrections in the van der Waals equation — molecular volume (b) and intermolecular attraction (a) — affect the compressibility factor Z in opposite directions?

Model answer: The molecular volume correction (replacing V with V − nb) accounts for the fact that molecules occupy space and cannot overlap. This makes the effective free volume smaller than the total volume, which means the gas behaves as if it is in a more compressed space than it actually is — the pressure it exerts is higher than the ideal prediction for the same total volume. This drives Z > 1. The intermolecular attraction correction (−a/V²) reduces the pressure below ideal because attractive forces pull molecules back from the container walls. This drives Z < 1. At moderate pressures, attractions typically dominate and Z < 1; at very high pressures, molecular volume dominates and Z > 1. The interplay of these two effects produces the characteristic dip-and-rise shape of Z vs. pressure curves.

This competition explains why Z is not monotonically greater or less than 1 for real gases — it depends on which effect dominates at a given temperature and pressure. At the Boyle temperature, the two effects exactly cancel and Z ≈ 1 over a wide pressure range even for a real gas.