Questions: Compressibility Factor and Reduced Properties

Question 1

At moderate temperature and elevated pressure, nitrogen gas is measured to have Z = 0.87. What does this tell you about nitrogen's behavior compared to an ideal gas at the same conditions?

• A. Nitrogen occupies more volume than an ideal gas — repulsive forces dominate
• B. Nitrogen behaves almost ideally because Z is close to 1, so the ideal gas law is accurate
• C. Nitrogen occupies less volume than an ideal gas — intermolecular attractive forces draw molecules closer than ideal behavior predicts
• D. The measurement indicates nitrogen is in a liquid phase at these conditions

Answer: (C)

Z < 1 means PV < nRT — the actual volume is less than the ideal prediction. This indicates that intermolecular attractive forces are drawing the molecules closer together, compressing the gas beyond what the ideal model predicts. Option B is partially correct that Z ≈ 1 means near-ideal, but misses that Z = 0.87 represents a 13% deviation — significant in engineering calculations. Option A (Z > 1) describes the opposite regime where excluded volume and repulsion dominate at very high pressures.

Question 2

The law of corresponding states allows engineers to estimate compressibility factors for unfamiliar gases using a single generalized chart. What makes this possible?

• A. All gases have the same molecular size and intermolecular forces at high temperatures
• B. The ideal gas law applies to all gases equally, so corrections are universal
• C. When T and P are scaled by critical properties (Tr = T/Tc, Pr = P/Pc), the Z vs. Tr and Pr curves for most simple gases approximately collapse onto a single universal surface
• D. The compressibility factor is defined to equal 1 for all gases, so no gas-specific data is needed

Answer: (C)

The critical point sets the natural energy and length scales for each gas's molecular interactions. By expressing T and P as fractions of these natural scales, you remove the chemical identity of the gas from the problem. Gases 'in corresponding states' (same Tr, same Pr) face the same relative competition between thermal energy and intermolecular forces, producing approximately the same Z. This universality requires only Tc and Pc — tabulated for most industrial gases — to make accurate PVT predictions.

Question 3

At very low pressures, all real gases approach Z = 1 regardless of temperature, because molecules are too far apart for intermolecular interactions to matter.

• T. True
• F. False

Answer: True

As pressure approaches zero, the molar volume becomes very large — molecules are separated by distances where even strong intermolecular forces (van der Waals attractions, etc.) become negligible. In this limit, every gas approaches ideal behavior: PV → nRT and Z → 1. This is why the ideal gas law is accurate for all gases at sufficiently low pressures.

Question 4

A gas with Z > 1 is more compressed than an ideal gas at the same temperature and pressure, indicating strong intermolecular attractive forces.

• T. True
• F. False

Answer: False

Z > 1 means PV > nRT — the gas actually occupies MORE volume than ideal, indicating that repulsive interactions or excluded volume dominate. Strong attractive forces pull molecules together, causing Z < 1 (more compressed than ideal). Z > 1 occurs at high pressures where molecules are forced close enough that hard-core repulsion and finite molecular volume push Z above 1. The two regimes are opposite: Z < 1 is attraction-dominated, Z > 1 is repulsion/excluded-volume-dominated.

Question 5

Explain why reduced properties (Tr = T/Tc, Pr = P/Pc) allow different gases to be compared on the same compressibility chart, rather than needing separate charts for each gas.

Model answer: The critical point reflects the natural energy and length scales for each gas's molecular interactions — the temperature and pressure at which thermal energy and intermolecular attraction are exactly balanced in a characteristic way. By dividing T and P by Tc and Pc respectively, we express conditions in units natural to each gas. Two gases at the same Tr and Pr are experiencing the same relative thermodynamic situation: the same ratio of thermal to intermolecular energy, the same ratio of actual to critical density. In this dimensionless space, their behavior converges, producing approximately the same Z. Only Tc and Pc need to be looked up; the shape of Z(Tr, Pr) is universal.

This is the law of corresponding states, and it works because the critical point is a universal organizing feature of fluid behavior — not just a convenient reference point. Deviations occur for quantum gases (H₂, He) where this classical argument breaks down, requiring modified effective critical constants.