Questions: Deviations of Real Gases from Ideal Behavior

Question 1

At moderate pressures, the compressibility factor Z for CO₂ is measured to be 0.88. This means:

• A. CO₂ occupies more volume than an ideal gas would at the same temperature and pressure
• B. CO₂ molecules repel each other strongly, making the gas harder to compress
• C. CO₂ exerts less pressure than the ideal gas law predicts, because intermolecular attractive forces reduce the molecules' momentum at the walls
• D. CO₂ has a higher molar mass than predicted by its formula weight

Answer: (C)

Z = PV/nRT < 1 means the gas behaves as if it exerts less pressure than the ideal model predicts (at fixed T and V). The physical reason is intermolecular attractive forces: as a molecule approaches the container wall, nearby molecules pull it back, so it arrives with slightly less momentum. The result is lower wall-collision force — lower pressure — than a gas of non-interacting particles would show. Option A is incorrect: Z < 1 at fixed T and P means V < nRT/P (less volume), not more.

Question 2

Helium at room temperature shows Z > 1 even at moderate pressures. Carbon dioxide at room temperature shows Z < 1 at moderate pressures before rising above 1 at very high pressures. Why the difference?

• A. Helium has a higher boiling point, making its molecules harder to compress
• B. Helium atoms are much larger than CO₂ molecules, so excluded volume effects dominate immediately
• C. CO₂ has stronger intermolecular attractive forces relative to its kinetic energy at room temperature, so attraction (Z < 1) dominates at moderate pressures; helium's weak attractions mean repulsive excluded-volume effects (Z > 1) dominate first
• D. CO₂ is a linear molecule and behaves differently from monatomic helium because of rotational degrees of freedom

Answer: (C)

The two effects competing are: (1) intermolecular attraction → Z < 1, and (2) excluded volume (finite molecular size) → Z > 1. CO₂ has significant van der Waals attraction, so at moderate pressures when molecules are moderately close, attraction dominates and Z dips below 1. Helium has extremely weak attractive forces (it barely liquefies at all), so excluded-volume repulsion dominates from the start and Z > 1. Room temperature is 'high' for helium relative to its critical temperature (5.2 K) but 'low' for CO₂ relative to its critical temperature (304 K), explaining why CO₂ still shows significant attractive effects at room temperature.

Question 3

A gas in a regime where intermolecular attractive forces dominate will exert lower pressure on its container walls than an ideal gas at the same temperature and volume.

• T. True
• F. False

Answer: True

True. Attractive forces create a net inward pull on molecules approaching the wall — they are slowed slightly before impact and thus transfer less momentum to the wall. Pressure is force per unit area, arising from molecular collisions, so reduced collision momentum means reduced pressure. This effect is captured by Z < 1: the actual PV product is less than nRT, indicating the gas is 'underperforming' relative to ideal predictions.

Question 4

When the compressibility factor Z < 1, the real gas occupies a larger volume than an ideal gas would at the same temperature and pressure.

• T. True
• F. False

Answer: False

False. Z = PV/nRT < 1 means PV < nRT. At fixed temperature T and pressure P, this rearranges to V < nRT/P — the real gas occupies a smaller volume than an ideal gas. The attractive forces pulling molecules together cause the gas to be more compact. (At fixed T and V, Z < 1 instead means P < nRT/V — lower pressure.) Either way, Z < 1 corresponds to the gas being 'pulled together' by attraction, not expanded.

Question 5

Explain the two physical mechanisms that cause real gases to deviate from ideal behavior, and describe the conditions under which each mechanism dominates.

Model answer: Two mechanisms cause deviations: (1) Intermolecular attractive forces — at moderate pressures and low temperatures, molecules are close enough to attract each other. A molecule approaching the wall is pulled inward, reducing its impact momentum and lowering pressure below ideal predictions, so Z < 1. (2) Finite molecular volume (excluded volume) — at high pressures, molecules cannot overlap. The volume available to any molecule is less than the total container volume, making the gas harder to compress than ideal, so Z > 1. Attractive effects dominate when molecules are moderately close (moderate P, low T); repulsive/volume effects dominate when molecules are very close (high P). The Boyle temperature is where the two effects cancel and Z ≈ 1.

Both effects appear in the van der Waals equation: the a/V² term corrects for attraction (reduces effective pressure), and the b term corrects for excluded volume (reduces effective volume available). These corrections predict the characteristic dip-then-rise shape of Z vs. P curves observed experimentally for most gases.