Questions: The Compressibility Factor Z

Question 1

A gas is measured at conditions where Z = 0.85. What does this tell you about the gas relative to an ideal gas at the same temperature and pressure?

• A. The gas occupies 15% more volume than an ideal gas — repulsive forces dominate
• B. The gas occupies less volume than an ideal gas — intermolecular attractions are pulling molecules together, making it easier to compress
• C. The gas exerts 15% less pressure than an ideal gas, but its volume is unchanged
• D. Z < 1 indicates the gas has lost 15% of its molecules to condensation

Answer: (B)

Z = PV/nRT < 1 means PV < nRT, so the actual volume is less than the ideal prediction. Intermolecular attractions pull molecules together, reducing volume and making the gas easier to compress than ideal. Z > 1 would indicate the opposite: repulsive (finite-volume) effects dominate, and the gas resists compression. Option A has the direction backwards.

Question 2

Nitrogen (N₂) and methane (CH₄) are placed at the same reduced temperature T_r and reduced pressure P_r. According to the law of corresponding states, what should be true?

• A. Their compressibility factors Z are approximately equal, because the reduced variables already account for each substance's critical properties
• B. Their Z values differ because they have different molecular weights
• C. Their Z values differ because they have different critical temperatures — the law only works within a single chemical family
• D. The law of corresponding states only applies to monatomic noble gases

Answer: (A)

The law of corresponding states says Z is approximately the same function of T_r and P_r for all simple nonpolar gases. The critical properties T_c and P_c are already embedded in the reduced variables — they encode the energy scale and density scale of molecular interactions. Normalizing by them makes the dimensionless behavior approximately universal. Molecular weight is irrelevant once you're working in reduced units.

Question 3

When Z > 1, a gas is easier to compress than an ideal gas at the same temperature and pressure.

• T. True
• F. False

Answer: False

Z > 1 means PV > nRT — the gas occupies MORE volume than ideal. This happens when finite molecular volume (repulsive interactions) dominates: molecules resist being packed together. Such a gas is HARDER to compress than ideal, not easier. Z < 1 corresponds to easier-than-ideal compression (attractions dominant). The direction of deviation determines whether the gas is compressed or expanded relative to ideal.

Question 4

If you know a gas's reduced temperature T_r and reduced pressure P_r, you can estimate its real molar volume as V = ZRT/P, where Z is read from a generalized compressibility chart.

• T. True
• F. False

Answer: True

This is the direct engineering application of Z. The ideal gas law gives V_ideal = RT/P. The compressibility factor Z is a multiplicative correction: V_real = Z × (RT/P). A generalized chart maps T_r and P_r to Z for many substances on a single curve set, so you need only the critical constants (T_c, P_c) of your specific gas to enter the chart. This correction can account for 5–30% deviations at high pressures.

Question 5

Why does the law of corresponding states allow a single generalized compressibility chart to describe many chemically different gases, rather than requiring a separate chart for each substance?

Model answer: Simple nonpolar molecules all have intermolecular potentials of similar shape — differing mainly in the depth and range of the potential well. The critical temperature T_c captures the energy scale of molecular attraction (well depth) and the critical pressure P_c captures the density scale (range). When temperature and pressure are normalized by these critical values into reduced variables T_r and P_r, the dimensionless molecular physics becomes approximately universal. Two gases at the same T_r and P_r are, in reduced units, experiencing equivalent molecular conditions — so their Z values are approximately the same.

The acentric factor ω extends this principle to non-spherical polar molecules (like water or ammonia), where the simple two-parameter correlation breaks down. But the underlying logic remains: critical properties encode the key features of molecular interaction, and normalizing by them collapses substance-specific behavior into an approximately universal curve.