A sealed container of gas at room temperature spontaneously contracts so that all molecules collect in one half. A student says this is impossible because the Second Law forbids entropy decrease. What is the more precise statement?
AThe student is correct — the Second Law absolutely forbids this event
BThe event is possible but astronomically improbable — for N ≈ 10²³ molecules, the probability is roughly 2^(−N)
CThe event is possible because entropy can decrease locally as long as the surroundings compensate
DThe event would violate conservation of energy, not the Second Law
The Second Law is a statistical statement, not an absolute prohibition. For N molecules, the probability of spontaneous contraction to half the volume is 2^(−N) — a number indistinguishable from zero for macroscopic systems, but not mathematically zero. Option A overstates the law: no physical law assigns a probability of exactly zero to this event; it is just so improbable as to be effectively impossible. Option C describes a different point (local entropy decrease in open systems) and does not address spontaneous contraction of an isolated gas.
Question 2 Multiple Choice
Why does Boltzmann's formula use ln Ω rather than Ω itself to define entropy?
ABecause Ω grows too fast for practical calculation and the logarithm makes the numbers manageable
BBecause the logarithm ensures entropy is additive: when two independent systems combine, their entropies add rather than multiply
CBecause ln Ω is always larger than Ω, giving entropy its characteristically large values
DBecause Boltzmann wanted to match the classical thermodynamic definition dS = dQ/T
The key reason is additivity. If system A has Ω_A microstates and system B has Ω_B microstates, the combined system has Ω_A × Ω_B microstates (independent possibilities multiply). For entropy to be an extensive quantity (scaling with system size, as thermodynamics requires), we need S_total = S_A + S_B. Taking the logarithm converts the product to a sum: k ln(Ω_A × Ω_B) = k ln Ω_A + k ln Ω_B. Option A is true but secondary — the deeper reason is extensivity, not computational convenience.
Question 3 True / False
The Second Law of Thermodynamics is a probabilistic statement: for macroscopic systems, spontaneous entropy decrease is so improbable as to be practically impossible, but it is not logically forbidden.
TTrue
FFalse
Answer: True
This is precisely the statistical interpretation Boltzmann provided. The 'impossibility' of entropy decrease is a probability statement about counting microstates, not an absolute logical law. For macroscopic N, the relevant probabilities (like 2^(−N)) are so small that we never observe violations — but this is a feature of large numbers, not a fundamental prohibition.
Question 4 True / False
A gas expands to fill a vacuum because gas molecules are repelled from high-density regions toward low-density regions.
TTrue
FFalse
Answer: False
This confuses macroscopic diffusion with microscopic probability. Individual gas molecules are not 'repelled' from crowded regions — they move randomly. Expansion occurs because the macrostate with gas spread throughout the full volume has overwhelmingly more microstates (Ω_final/Ω_initial = 2^N for a doubling of volume) than the initial state. There is no force pushing molecules outward; the vast numerical superiority of the expanded macrostate makes expansion the overwhelmingly probable outcome of random molecular motion.
Question 5 Short Answer
Explain why the number of microstates Ω peaks so sharply at the equal-distribution macrostate as the number of particles N grows large.
Think about your answer, then reveal below.
Model answer: For N particles split between two halves of a box, the number of microstates with k particles on the left is (N choose k). The binomial coefficient peaks at k = N/2 and falls off sharply away from this peak. For large N, the peak becomes extremely narrow relative to N — the ratio of the peak value to off-peak values grows exponentially. By Stirling's approximation, the entropy at the peak is S_max = Nk ln 2, and deviations from equal distribution represent exponentially fewer microstates.
This sharpening of the peak is what makes the Second Law so reliable for macroscopic systems. Even at N = 100, the distribution around N/2 is noticeably tight. At N = 10²³, deviations from equal partition are unobservable. The sharpness comes from the combinatorial structure: the ratio of microstates at the peak to microstates slightly off-peak grows as e^(N × something), an exponentially growing advantage.