Thermodynamics and statistical mechanics
| English | 中文 | Pinyin |
|---|---|---|
| entropy/ˈentrəpi/ | 熵 | shāng |
| partition function/pɑːˈtɪʃn ˈfʌŋkʃn/ | 配分函数 | pèi fēn hán shù |
A decision before an answer
- Two engines can obey energy conservation, yet only one obeys the entropy limit on efficiency.
- Your goal: Apply the first and second laws with a sign convention.
Read the relationship
- State the first-law sign convention: ΔU=Q−W when W is work done by the system.
- Relate ensembles, Boltzmann factors and entropy.
A Carnot engine between 400 K and 300 K has efficiency:
1−300/400=0.25.
Use the defining rule
- For a reversible engine between reservoirs, efficiency is bounded by 1−Tc/Th using absolute temperatures.
- Analyse ideal gases, heat engines and quantum statistics.
A higher-energy state’s Boltzmann weight at fixed T is:
The negative exponential decreases with energy.
Check the conditions
- Boltzmann weights are exp(−E/kT). Probabilities require division by the partition function.
- Analyse ideal gases, heat engines and quantum statistics.
Reservoirs at 600 K and 300 K give a maximum reversible efficiency 1−300/600=0.5. An engine taking 1000 J can produce at most 500 J of work under these conditions. Conservation alone would not impose that bound.
A system receives 100 J heat and does 40 J work. ΔU=____ J.
ΔU=Q−W=100−40.
Apply the task format
- Distinguish Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac assumptions. Fermions cannot share an identical single-particle state.
- Analyse ideal gases, heat engines and quantum statistics.
Celsius temperature ratios cannot be used in the Carnot efficiency formula.
Which answer fits this case?
Apply the first and second laws with a sign convention
An engine may convert all heat from one reservoir into work in a complete cycle with no other effect.
That violates the second law.
Keep the distinctions
- entropy 熵 — A state quantity connected to multiplicity and reversible heat transfer.
- partition function 配分函数 — The sum of statistical weights used to normalise probabilities.
- Apply the first and second laws with a sign convention.
- Relate ensembles, Boltzmann factors and entropy.
- Analyse ideal gases, heat engines and quantum statistics.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.