Math 2.0

Classical thermodynamics, heat transfer Thermodynamic potentials and relations Proposition

Fundamental thermodynamic relation

Builds on Internal energyEntropyThermodynamic temperatureReversible processFirst law of thermodynamicsSecond law of thermodynamics

Statement

Let a closed simple system, such as a fluid of fixed composition, pass between two neighbouring equilibrium states. Let UU be its internal energy, SS its entropy, TT its thermodynamic temperature, pp its pressure and VV its volume. Then dU=T dS−p dV.dU = T \, dS - p \, dV. If the amounts N1,…,NrN_1, \dots, N_r of the substances may also change, then dU=T dS−p dV+∑i=1rμi dNidU = T \, dS - p \, dV + \sum_{i=1}^{r} \mu_i \, dN_i, where μi\mu_i is the chemical potential of substance ii.

Remarks

For a reversible change, the second law gives δQ=T dS\delta Q = T \, dS and the work done by the fluid is δW=p dV\delta W = p \, dV. The first law dU=δQ−δWdU = \delta Q - \delta W then gives the relation. Since every term involves only state functions, the relation holds for any change between the two equilibrium states, reversible or not. For an irreversible change, though, T dST \, dS is no longer the heat and p dVp \, dV is no longer the work. Equivalently, T=(∂U/∂S)VT = (\partial U / \partial S)_V and p=−(∂U/∂V)Sp = -(\partial U / \partial V)_S. Example: for an ideal gas at constant temperature, dU=0dU = 0, so dS=p dV/T=nR dV/VdS = p \, dV / T = nR \, dV / V and ΔS=nRln⁡(V2/V1)\Delta S = nR \ln(V_2 / V_1).

Going deeper

For a student

The fundamental relation is the master equation of equilibrium thermodynamics. It shows that UU is a thermodynamic potential with natural variables SS and VV. Legendre transforms of UU give the enthalpy, the Helmholtz free energy and the Gibbs free energy, with analogous differentials. The equality of mixed second derivatives of these potentials gives the Maxwell relations.

Reading path

To understand this page from the ground up, read these 24 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 gasconceptInternal energy, concept, no evidence levelInternal energyconceptIsentropic process, concept, no evidence levelIsentropic processconceptIsothermal process, concept, no evidence levelIsothermal processconceptReversible process, concept, no evidence levelReversible processconceptState function, concept, no evidence levelState functionconceptTemperature, 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 workconceptFirst law of thermodynamics, statement, no evidence levelFirst law of thermody…statementClausius 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…statementFundamental thermodynamic relation, statement, no evidence levelFundamental thermodyn…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. Internal energy concept
  7. Isentropic process concept
  8. Isothermal process concept
  9. Reversible process concept
  10. State function concept
  11. Temperature concept
  12. Thermal efficiency concept
  13. Thermodynamic cycle concept
  14. Thermodynamic system concept
  15. Thermodynamic temperature concept
  16. Thermodynamic work concept
  17. First law of thermodynamics statement
  18. Clausius statement of the second law statement
  19. Kelvin–Planck statement of the second law statement
  20. Carnot's theorem statement
  21. Carnot cycle example
  22. Clausius theorem statement
  23. Second law of thermodynamics statement
  24. Fundamental thermodynamic relation statement

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation
About this page
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Kindstatement, proposition
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AboutInternal energy, Entropy
UsesInternal energy, Entropy, Thermodynamic temperature, Reversible process, First law of thermodynamics, Second law of thermodynamics
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Changed3 November 2026

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