Questions: Statistical Entropy and Molecular Disorder

Question 1

A perfectly ordered crystal of salt at 800 K has many accessible vibrational energy levels. A gas of the same substance at 10 K is spatially disordered but has very few thermally accessible states. Which has higher entropy, and why?

• A. The cold gas, because entropy measures spatial disorder and a gas is always more disordered than a crystal
• B. The hot crystal, because it has more accessible microstates due to the large number of thermally populated vibrational levels
• C. They are equal because both contain the same number of molecules
• D. The hot crystal has lower entropy because spatial order always reduces entropy

Answer: (B)

Entropy counts accessible microstates (S = k_B ln Ω), not spatial arrangement. The hot crystal has an enormous number of vibrational energy microstates available at 800 K, giving it high entropy despite its spatial regularity. The cold gas, though spatially disordered, has very few thermally accessible states at 10 K. 'Disorder' in the everyday visual sense is a poor proxy for entropy — microstate count is what matters.

Question 2

Two gas molecules start confined to the left half of a sealed box. When a partition is removed, they spontaneously spread throughout the full volume. The best explanation for this behavior is:

• A. The molecules repel each other and spread to maximize their mutual distances
• B. There are vastly more microstates available when the molecules can be anywhere in the full volume, making the spread configuration overwhelmingly more probable
• C. Expansion is driven by a decrease in the molecules' internal energy as they move to regions of lower potential energy
• D. The molecules move toward regions of lower pressure until equilibrium is reached

Answer: (B)

No force drives the expansion — the molecules are not repelling each other or decreasing their energy. The full volume simply offers far more microstates (positions and momenta configurations) for the two molecules than the half-volume does. With 10²³ particles, the probability of all molecules spontaneously returning to one half is so astronomically small that it is effectively impossible. The second law emerges from this probabilistic argument, not from any directed force.

Question 3

According to S = k_B ln(Ω), combining two identical, independent systems doubles the total entropy because the total microstate count doubles.

• T. True
• F. False

Answer: False

Combining independent systems multiplies their microstate counts: Ω_total = Ω₁ × Ω₂. But because of the logarithm in Boltzmann's formula, S_total = k_B ln(Ω₁ × Ω₂) = k_B ln Ω₁ + k_B ln Ω₂ = S₁ + S₂. So entropy is additive (extensive), not multiplicative. The logarithm is precisely what makes entropy a sensible thermodynamic quantity — it converts the multiplicative nature of microstates into additive entropy.

Question 4

The statistical interpretation of the second law treats the spontaneous increase of entropy not as a fundamental constraint imposed on nature, but as the inevitable outcome of a system evolving toward its most probable macrostate.

• T. True
• F. False

Answer: True

Classical thermodynamics states the second law as a postulate. Statistical mechanics explains why: for any macroscopic system, the equilibrium macrostate corresponds to overwhelmingly more microstates than any ordered configuration. Systems 'evolve toward higher entropy' simply because they evolve toward more probable states. A spontaneous decrease in entropy is statistically possible but so improbable for ~10²³ particles that it never occurs on observable timescales.

Question 5

Why is describing entropy as 'molecular disorder' or 'messiness' potentially misleading? Provide a concrete example that illustrates the limitation of this description.

Model answer: The 'disorder' metaphor implies spatial or visual messiness, but entropy precisely counts accessible microstates — which need not correlate with visual arrangement. For example, a crystalline solid at high temperature has high entropy because its atoms have many accessible vibrational energy levels, even though the spatial arrangement is highly regular. Conversely, a very cold gas may appear spatially 'disordered' but have low entropy because few energy microstates are thermally accessible. The correct definition is always microstate count: S = k_B ln Ω.

This distinction matters particularly when comparing substances across different phases or temperatures. Entropy tracks how many microscopic configurations are compatible with the observed macroscopic state, whether those configurations involve spatial arrangement, energy distribution, or both.