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TS · Thermodynamics and statistical mechanics

GRE · GRE Subject Test · GRE 物理 · 知识点 4

训练
4.1

热力学与统计力学

Two engines can obey energy conservation, yet only one obeys the entropy 熵 limit on efficiency.

Prerequisites: 19, 20, 34, 35, 36.

  • Apply the first and second laws with a sign convention
  • Relate ensembles, Boltzmann factors and entropy
  • Analyse ideal gases, heat engines and quantum statistics
词汇 训练
English 中文 拼音
entropy/ˈentrəpi/ 熵 shāng
4.2

Choose the system and model

State the first-law sign convention: ΔU=Q−W when W is work done by the system.

4.3

Use the governing relation

For a reversible engine between reservoirs, efficiency is bounded by 1−Tc/Th using absolute temperatures.

4.4

Apply the conditions

Boltzmann weights are exp(−E/kT). Probabilities require division by the partition function 配分函数.

词汇 训练
English 中文 拼音
partition function/pɑːˈtɪʃn ˈfʌŋkʃn/ 配分函数 pèi fēn hán shù
4.5

Check the conclusion

Distinguish Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac assumptions. Fermions cannot share an identical single-particle state.

4.6

Worked method

Use work W done by the gas as positive. The first law 热力学第一定律 is

$$\Delta U=Q-W.$$
For reversible isothermal expansion of an ideal gas, U depends only on T, so
$$W=\int_{V_i}^{V_f}p\,dV=nRT\ln(V_f/V_i),\qquad Q=W.$$
For n = 0.50 mol, T = 300 K and doubled volume,
$$W=nRT\ln 2=(0.50\,\mathrm{mol})(8.314\,\mathrm{J/(mol\,K)})(300\,\mathrm K)\ln2=864\,\mathrm J.$$
$$\Delta S=Q_{rev}/T=(864\,\mathrm J)/(300\,\mathrm K)=2.88\,\mathrm{J/K}.$$
The gas expands and receives heat. The reservoir loses the same entropy in this reversible limit.

Thermodynamics and statistical mechanics: GRE original diagram
Thermodynamics and statistical mechanics: original GRE teaching diagram.
词汇 训练
English 中文 拼音
first law/fɜːst lɔː/ 热力学第一定律 rè lì xué dì yí dìng lǜ
4.7

Check conditions and vocabulary

Celsius temperature ratios cannot be used in the Carnot efficiency formula.

entropy: A state quantity connected to multiplicity and reversible heat transfer.

partition function: The sum of statistical weights used to normalise probabilities.

该知识点的互动课程

逐步完成，配合即时检查练习。

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