Questions: Real Gases and the van der Waals Equation

Question 1

A real gas is compressed to very high pressure at room temperature. Which statement best describes its compressibility factor Z?

• A. Z = 1, because high pressure forces ideal behavior
• B. Z < 1, because intermolecular attractions dominate at high pressure
• C. Z > 1, because molecular excluded volume dominates at very high pressure
• D. Z < 1, because molecular volume shrinks when molecules are crowded

Answer: (C)

At very high pressures, molecules are so tightly packed that their own finite volume becomes the dominant effect. The excluded volume (b correction) means the gas occupies more volume than the ideal law predicts, pushing Z above 1. The attraction-dominated regime (Z < 1) occurs at moderate pressures where molecules are close enough to attract but not yet so crowded that their volumes dominate. Many students pick option B because they associate 'compressed' with 'attractions matter,' but at extreme compression the repulsive volume exclusion wins.

Question 2

Gas A has a large van der Waals constant 'a' and Gas B has a large constant 'b'. Which statement correctly describes their molecular properties?

• A. Gas A has large molecules; Gas B has strong intermolecular attractions
• B. Gas A has strong intermolecular attractions; Gas B has large molecules
• C. Both gases deviate from ideality in the same way at all conditions
• D. Gas A is more ideal than Gas B at low temperatures

Answer: (B)

The 'a' constant corrects for intermolecular attractive forces — gases like water and ammonia with strong hydrogen bonding have large 'a' values. The 'b' constant corrects for the physical space molecules occupy — large, bulky molecules like SF₆ or noble gases like Xe have large 'b' values. The two constants are independent and reflect different molecular properties. Gas A (large 'a') will show Z < 1 at moderate pressures; Gas B (large 'b') will show Z > 1 at high pressures.

Question 3

A gas is most ideal (closest to ideal gas behavior) at high temperatures and high pressures.

• T. True
• F. False

Answer: False

High temperature promotes ideal behavior because kinetic energy overwhelms intermolecular attractions. But high pressure pushes molecules together, making both the excluded volume and intermolecular attraction corrections more significant — this moves the gas further from ideal behavior. Ideal conditions are low pressure (molecules far apart, volume correction negligible) combined with high temperature (kinetic energy dominates attractions). The Z vs pressure plot for any real gas shows the greatest deviation from Z = 1 at high pressures.

Question 4

A gas with Z < 1 at a given temperature and pressure is producing more pressure on its container walls than an ideal gas would under identical conditions.

• T. True
• F. False

Answer: False

Z = PV/nRT < 1 means the actual pressure P is less than the ideal prediction nRT/V. Physically, this occurs because intermolecular attractions pull molecules back as they approach the container wall, reducing the force of impact. The van der Waals 'a' correction adds a(n/V)² to the measured pressure to account for this deficit — the measured pressure is lower than ideal, not higher. This is the opposite of what many students expect from the intuition that 'crowded molecules = more collisions = more pressure.'

Question 5

Explain why the van der Waals equation collapses to the ideal gas law at low pressure and high temperature.

Model answer: At low pressure, molecules are far apart so intermolecular attractions are negligible (the 'a' term → 0) and molecular volume is tiny compared to the container volume (nb ≪ V, so V − nb ≈ V). At high temperature, kinetic energy dominates any residual attractions. When both corrections vanish, (P + a(n/V)²)(V − nb) = nRT reduces to PV = nRT.

This question tests whether students understand the corrections as physical effects that grow or shrink with conditions, rather than as fixed mathematical additions. The 'a' correction depends on molecular density (n/V)² — at low pressure this density is low and the term is negligible. The 'b' correction matters only when nb is comparable to V — at low pressure, V is large and nb/V → 0. Real understanding means being able to explain when and why the corrections switch off.