Questions: Van der Waals Equation from Statistical Mechanics

Question 1

A real gas at high density is compressed into a small volume. According to the van der Waals equation, the attractive interaction term (aN²/V²) causes the measured pressure to be which of the following compared to an ideal gas at the same conditions?

• A. Higher than ideal, because attractions pull molecules toward the walls
• B. Lower than ideal, because attractions pull molecules near the walls back toward the interior, reducing momentum delivered to the wall
• C. Equal to ideal, because intermolecular attractions cancel symmetrically
• D. Higher than ideal at low temperature, lower at high temperature

Answer: (B)

Molecules near the container wall are attracted back toward the bulk interior by their neighbors. This inward pull reduces the momentum they transfer to the wall, so the measured pressure is lower than it would be for non-interacting particles. The correction is −aN²/V²: negative, reducing pressure. Option A reverses the direction of the effect — attractions pull molecules *away* from the walls, not toward them.

Question 2

Below the critical temperature, the van der Waals P–V isotherm develops an 'S-shaped' curve with a region where pressure increases as volume increases. What does Maxwell's equal-area construction accomplish?

• A. It provides the exact critical exponents that experiments confirm
• B. It replaces the mechanically unstable S-shaped region with a horizontal tie line representing liquid-gas coexistence
• C. It corrects the excluded-volume term b to account for molecular shape
• D. It extends the mean-field equation to temperatures above the critical point

Answer: (B)

A region with positive slope in P(V) implies negative compressibility — the gas would expand when compressed — which is mechanically unstable and unphysical. Maxwell's construction replaces this unstable portion with a flat coexistence line at the equilibrium vapor pressure, representing the two phases (liquid and gas) coexisting at the same pressure. The equal-area rule determines which pressure: the area under the S-curve on each side of the line must be equal, enforcing thermodynamic equilibrium.

Question 3

The van der Waals equation gives exact predictions for critical exponents (such as how magnetization vanishes at a critical temperature).

• T. True
• F. False

Answer: False

The van der Waals equation is a mean-field approximation that ignores fluctuations. Near the critical point, fluctuations become extremely large (the correlation length diverges), so mean-field theory breaks down. Experimentally, critical exponents differ from mean-field predictions — for example, the coexistence curve (density difference between liquid and gas) near T_c scales as (T_c−T)^β with β≈0.326, not the mean-field value of 0.5. The van der Waals equation gets the qualitative picture right but fails quantitatively near criticality.

Question 4

In the van der Waals equation (P + aN²/V²)(V − Nb) = NkT, replacing V with V − Nb captures the fact that each molecule has less effective space to move in because other molecules physically occupy volume.

• T. True
• F. False

Answer: True

The excluded-volume correction b accounts for the finite size of molecules. The center of one molecule cannot approach the center of another within a distance equal to the molecular diameter, so the free volume available to any molecule is V minus the volume excluded by all other molecules. This term is derived directly from the second virial coefficient for hard-sphere repulsion: it increases the effective pressure because molecules collide more frequently when less free volume is available.

Question 5

Explain why the van der Waals equation predicts a critical point, and why the mean-field approximation that underlies it fails precisely near that critical point.

Model answer: The van der Waals equation predicts a critical point because the attractive term (reducing pressure) and the excluded-volume term (increasing it) compete in a way that — below a critical temperature T_c — creates a region of mechanical instability (the S-curve) representing liquid-gas coexistence. At T_c, this instability just disappears: the liquid and gas densities become equal. The mean-field approximation fails near T_c because it replaces actual molecular interactions with an average field proportional to density, ignoring correlated fluctuations. Near the critical point, fluctuations become large and long-ranged — density fluctuates wildly across large spatial scales — and these correlated fluctuations are exactly what the mean-field approach discards.

Mean-field theory works well when each molecule 'sees' many neighbors that average out, so replacing pairwise interactions with an average is reasonable. This holds at high density or far from the critical point. Near T_c, the system is poised between two phases, so tiny fluctuations can nucleate macroscopic regions of either phase — the correlation length diverges. Mean-field theory predicts wrong critical exponents because it effectively sets the correlation length to zero. More sophisticated renormalization-group theory correctly handles these fluctuations and recovers the observed exponents.