Math 2.0

Classical thermodynamics, heat transfer Laws of thermodynamics Fact

Second law of thermodynamics

Builds on Thermodynamic systemEntropyHeatThermodynamic temperatureReversible processClausius theorem

Statement

Let a closed thermodynamic system pass by some process from an equilibrium state AA to an equilibrium state BB. Suppose that during the process the system takes in small amounts of heat δQ\delta Q through parts of its boundary at thermodynamic temperature TT. Then the entropy SS of the system satisfies S(B)−S(A)≥∫ABδQT,S(B) - S(A) \ge \int_A^B \frac{\delta Q}{T}, with equality when the process is reversible. In particular, if the system is isolated, then S(B)≥S(A)S(B) \ge S(A).

Remarks

The inequality follows from the Clausius theorem. Close the process into a cycle by a reversible path from BB back to AA. The Clausius theorem gives ∮δQ/T≤0\oint \delta Q / T \le 0 for the cycle, and the reversible return path contributes S(A)−S(B)S(A) - S(B). Rearranging gives the inequality above. For example, when two blocks at temperatures T1>T2T_1 > T_2 exchange a small amount of heat qq inside an isolated box, the total entropy changes by q/T2−q/T1>0q/T_2 - q/T_1 > 0. The law has several equivalent classical forms: the Clausius statement and the Kelvin–Planck statement, both about cyclic machines. It is a law of nature: it is not derived from mechanics, and it fixes the direction in which spontaneous processes run.

Going deeper

For a student

The second law limits what engines and refrigerators can do. It gives Carnot's bound 1−Tc/Th1 - T_c/T_h on the efficiency of a heat engine working between temperatures Th>TcT_h > T_c. For reversible processes it gives δQ=T dS\delta Q = T \, dS, and with the first law this yields the fundamental relation dU=T dS−p dVdU = T \, dS - p \, dV. It also leads to the extremum principles: an isolated system in equilibrium has maximal entropy among the states allowed by its constraints.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Entropy, concept, no evidence levelEntropyconceptHeat, concept, no evidence levelHeatconceptHeat capacity ratio, concept, no evidence levelHeat capacity ratioconceptHeat engine, concept, no evidence levelHeat engineconceptIdeal gas, concept, no evidence levelIdeal gasconceptIsentropic process, concept, no evidence levelIsentropic processconceptIsothermal process, concept, no evidence levelIsothermal processconceptReversible process, concept, no evidence levelReversible processconceptTemperature, concept, no evidence levelTemperatureconceptThermal efficiency, concept, no evidence levelThermal efficiencyconceptThermodynamic cycle, concept, no evidence levelThermodynamic cycleconceptThermodynamic system, concept, no evidence levelThermodynamic systemconceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptThermodynamic work, concept, no evidence levelThermodynamic workconceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementKelvin–Planck statement of the second law, statement, no evidence levelKelvin–Planck stateme…statementCarnot's theorem, statement, no evidence levelCarnot's theoremstatementCarnot cycle, example, no evidence levelCarnot cycleexampleClausius theorem, statement, no evidence levelClausius theoremstatementSecond law of thermodynamics, statement, no evidence levelSecond law of thermod…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Entropy concept
  2. Heat concept
  3. Heat capacity ratio concept
  4. Heat engine concept
  5. Ideal gas concept
  6. Isentropic process concept
  7. Isothermal process concept
  8. Reversible process concept
  9. Temperature concept
  10. Thermal efficiency concept
  11. Thermodynamic cycle concept
  12. Thermodynamic system concept
  13. Thermodynamic temperature concept
  14. Thermodynamic work concept
  15. Clausius statement of the second law statement
  16. Kelvin–Planck statement of the second law statement
  17. Carnot's theorem statement
  18. Carnot cycle example
  19. Clausius theorem statement
  20. Second law of thermodynamics statement

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Second_law_of_thermodynamics
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TagT11MGWEKJ3
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Kindstatement, fact
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MSC80A05, 80A10
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AboutEntropy
UsesThermodynamic system, Entropy, Heat, Thermodynamic temperature, Reversible process, Clausius theorem
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Changed3 November 2026

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