Questions: Virial Equation and Intermolecular Forces

Question 1

A gas at moderate pressure has a compressibility factor Z < 1. What does this tell you about its intermolecular forces?

• A. Repulsive forces dominate, so the gas occupies more volume than ideal
• B. Attractive forces dominate, so the gas is more compressed than the ideal law predicts
• C. The gas is behaving ideally, since Z is close to 1
• D. The virial coefficients are all zero at this temperature

Answer: (B)

Z = PV/nRT < 1 means the actual volume is smaller than the ideal gas law predicts. This happens when attractive intermolecular forces pull molecules closer together, reducing the pressure (and thus the volume) below the ideal value. The second virial coefficient B is negative when attractions dominate. Repulsions would push Z above 1.

Question 2

As temperature is raised well above room temperature for a gas whose second virial coefficient B is initially negative, what happens to Z at moderate pressures?

• A. Z remains below 1 because the intermolecular attraction potential doesn't change with temperature
• B. Z approaches 1 and may exceed it as B becomes less negative and eventually positive
• C. Z drops further below 1 because faster-moving molecules collide more often
• D. Z stays at exactly 1 because the ideal gas law always applies at high temperatures

Answer: (B)

As temperature rises, thermal kinetic energy increasingly overcomes intermolecular attraction. B becomes less negative, passes through zero at the Boyle temperature, and eventually becomes positive — at which point repulsions dominate and Z > 1. This is why the sign of B (and of Z−1) is a direct readout of which intermolecular force regime dominates at a given temperature.

Question 3

The virial equation predicts that most real gases have Z < 1 at most temperatures and pressures.

• T. True
• F. False

Answer: False

This is false. At high pressures or elevated temperatures where repulsive forces dominate, Z > 1 — the gas is less compressed than ideal behavior predicts. Z < 1 occurs when attractive forces dominate (typically at moderate pressures and temperatures for common gases). The sign of Z−1 is determined by the balance between attraction and repulsion encoded in the virial coefficient B.

Question 4

The ideal gas law is a special case of the virial equation, valid when the correction terms B/V, C/V², etc. are negligibly small.

• T. True
• F. False

Answer: True

The virial equation is PV = nRT(1 + B/V + C/V² + ...). When molar volume V is large — i.e., at low pressure and high temperature where molecules are far apart and interact rarely — all correction terms vanish and the equation reduces exactly to PV = nRT. The ideal gas law is not a separate model; it is the leading-order approximation of the virial expansion.

Question 5

Why does the sign of the second virial coefficient B change from negative to positive as temperature increases, and what physical transition does this mark?

Model answer: At low temperatures, the kinetic energy of molecules is small relative to the depth of the intermolecular attraction well, so attractive forces dominate pairwise interactions and B is negative. As temperature rises, molecules move faster and the thermal energy exceeds the attractive well depth; repulsive forces (from electron shell overlap at short distances) then dominate, making B positive. The temperature at which B = 0 is the Boyle temperature, where the gas behaves ideally at moderate pressures because attractive and repulsive corrections exactly cancel.

This connects the macroscopic observable (the sign of B, measurable from PVT data) directly to the shape of the intermolecular potential: the Lennard-Jones-like well with a negative (attractive) region at intermediate separations and a steep repulsive wall at short separations. The Boyle temperature marks the crossover between these two regimes.