A system has two microstates: one with energy E and one with energy 2E. As temperature T increases toward infinity, what happens to the ratio of their probabilities?
AThe high-energy state becomes even less likely
BThe ratio approaches 1 — both states become equally probable
CThe high-energy state becomes impossible
DThe ratio depends only on the volume, not temperature
The probability of a microstate is proportional to exp(−E/kT). The ratio of probabilities is exp(−E/kT) / exp(−2E/kT) = exp(E/kT). As T → ∞, E/kT → 0, so exp(E/kT) → 1 — the two states become equally probable. The Boltzmann factor suppresses high-energy states at low T, but thermal fluctuations wash out energy differences at very high T.
Question 2 True / False
In the canonical ensemble, the total energy of the system is exactly fixed at most times.
TTrue
FFalse
Answer: False
Only N (particle number), V (volume), and T (temperature) are fixed in the canonical ensemble. Because the system is in thermal contact with a heat bath, energy can fluctuate — individual microstates have different energies. The average energy is well-defined, but instantaneous energy is not fixed. This contrasts with the microcanonical ensemble, where energy is fixed exactly.
Question 3 Short Answer
Why is the canonical ensemble more practically useful than the microcanonical ensemble for most calculations?
Think about your answer, then reveal below.
Model answer: The canonical ensemble holds temperature constant by coupling to a heat bath, which is how real laboratory experiments are typically conducted. The microcanonical ensemble fixes energy exactly, which is mathematically convenient but physically harder to realize. Fixing T is easier in practice than isolating a system to prevent any energy exchange.
Most real experiments are performed in contact with an environment at a known temperature, not in perfect thermal isolation. The canonical ensemble captures this situation directly. The partition function Z = Σ exp(−E_i/kT) also provides a convenient route to all thermodynamic quantities (free energy, entropy, heat capacity) via derivatives.