Questions: Compressibility Factor and Generalized Correlations

Question 1

Carbon dioxide at 300 K and 10 MPa has a compressibility factor Z ≈ 0.2. What does this value indicate about CO₂'s behavior at these conditions?

• A. CO₂ behaves nearly ideally — Z = 0.2 means only a 20% correction is needed
• B. Intermolecular attractions dominate strongly, causing CO₂ to occupy about 20% of the volume that an ideal gas would at these conditions
• C. Intermolecular repulsions dominate, forcing CO₂ to occupy more volume than the ideal gas law predicts
• D. Z = 0.2 means the ideal gas law overestimates the pressure by 80%

Answer: (B)

Z = PV/(nRT). When Z < 1, the actual volume V is less than the ideal-gas prediction nRT/P — the molecules are 'pulled together' by attractive intermolecular forces. Z ≈ 0.2 means the real volume is only 20% of the ideal prediction — an extreme deviation. This occurs for CO₂ near its critical point (T_c = 304 K, P_c = 7.4 MPa) where attractions are very strong. Option A misreads Z as a correction factor rather than a ratio of real to ideal volume.

Question 2

An engineer needs to estimate the molar volume of propane (C₃H₈) at 400 K and 5 MPa but has no propane-specific equation of state. She knows T_c = 370 K, P_c = 4.25 MPa, and ω = 0.152 for propane. What is her best approach?

• A. Assume ideal gas behavior — propane is a small molecule and deviates little from ideality
• B. Use generalized correlations with T_r = 400/370 ≈ 1.08 and P_r = 5/4.25 ≈ 1.18 and the acentric factor to look up Z, then compute V = ZnRT/P
• C. Use Z = 1.0 since T_r > 1 means the gas is above its critical temperature and must behave ideally
• D. The problem cannot be solved without a substance-specific equation of state for propane

Answer: (B)

At T_r ≈ 1.08 and P_r ≈ 1.18 — near-critical conditions — propane deviates significantly from ideal behavior (Z << 1). The generalized correlation approach uses the reduced properties and acentric factor with the Pitzer or Lee-Kesler correlation to read Z, then compute V = ZnRT/P. This gives accurate results without a substance-specific EOS. Option C is wrong: above T_c does not mean ideal — at high pressures near T_c, Z can still be well below 1.

Question 3

A gas with Z = 1.15 at some temperature and pressure is exhibiting behavior closer to ideal than a gas with Z = 0.85 at different conditions.

• T. True
• F. False

Answer: False

Z = 1 is the ideal reference. Both Z = 1.15 and Z = 0.85 represent deviations from ideality — the magnitudes of deviation are |1.15 − 1| = 0.15 and |0.85 − 1| = 0.15, identical. Z > 1 indicates repulsive interactions dominate (molecules effectively occupy more volume than ideal); Z < 1 indicates attractive interactions dominate. Neither direction is 'more ideal' than the other — only Z = 1 is ideal.

Question 4

At sufficiently low pressures (P << P_c), real gases approach Z = 1 and behave approximately as ideal gases, regardless of molecular polarity or size.

• T. True
• F. False

Answer: True

At low pressures, average intermolecular distances are large, so both attractive and repulsive interactions become negligible relative to thermal kinetic energy. The ratio PV/(nRT) → 1 for all real gases as P → 0. This is why the ideal gas law is a universal low-pressure limiting behavior — not an approximation that works only for simple molecules. Polarity and size matter more near critical conditions or at high pressures.

Question 5

Explain why the corresponding states principle is so useful in engineering practice, and what role the acentric factor plays in extending its accuracy.

Model answer: The corresponding states principle shows that most gases follow the same Z surface when plotted against reduced temperature T_r = T/T_c and reduced pressure P_r = P/P_c. This means a single universal set of tables or equations (like Lee-Kesler) predicts real-gas behavior for any substance using only its critical temperature and pressure — both tabulated for hundreds of compounds. Without this principle, each substance would require its own experimentally fitted equation of state. The acentric factor ω extends the principle to non-spherical molecules: simple spherical molecules (noble gases, methane) follow the two-parameter T_r/P_r correlation almost exactly, but elongated or polar molecules deviate because their shape affects intermolecular orientation and packing. The Pitzer correlation Z = Z⁰(T_r, P_r) + ω·Z¹(T_r, P_r) adds ω as a third parameter that captures molecular shape effects with a linear correction, recovering accuracy for hydrocarbons, refrigerants, and other industrially important fluids.

The practical consequence is that engineers can estimate Z, and then departure functions for enthalpy and entropy, for any compound in a process — even novel or rare ones — using three tabulated numbers (T_c, P_c, ω). This is essential for process simulation software like Aspen or HYSYS, where hundreds of compounds must be handled without individual EOS fits.