Math 2.0

Classical thermodynamics, heat transfer Thermodynamic potentials and relations Proposition

Euler relation for internal energy

Builds on Internal energyEntropyIntensive and extensive propertiesFundamental thermodynamic relationChemical potentialGibbs free energyGrand potential

Statement

Let a simple system have internal energy U(S,V,N1,…,Nr)U(S, V, N_1, \dots, N_r), a differentiable function of its entropy SS, volume VV and amounts of substance N1,…,NrN_1, \dots, N_r. Let T=∂U/∂ST = \partial U / \partial S, p=−∂U/∂Vp = -\partial U / \partial V and μi=∂U/∂Ni\mu_i = \partial U / \partial N_i be its temperature, pressure and chemical potentials. Suppose that UU is extensive, that is, U(λS,λV,λN1,…,λNr)=λ U(S,V,N1,…,Nr)U(\lambda S, \lambda V, \lambda N_1, \dots, \lambda N_r) = \lambda \, U(S, V, N_1, \dots, N_r) for all λ>0\lambda > 0. Then U=TS−pV+∑i=1rμiNi.U = TS - pV + \sum_{i=1}^{r} \mu_i N_i.

Remarks

Differentiate the homogeneity identity with respect to λ\lambda and set λ=1\lambda = 1; this is Euler's theorem on homogeneous functions of degree 11. The partial derivatives of UU are those of the fundamental relation. Two consequences: the Gibbs free energy is G=U−TS+pV=∑iμiNiG = U - TS + pV = \sum_i \mu_i N_i, and the grand potential is U−TS−∑iμiNi=−pVU - TS - \sum_i \mu_i N_i = -pV. Example: for a one-component system, G=μNG = \mu N, so the chemical potential is the Gibbs free energy per particle. Extensivity is needed: it fails for small systems where surface energy matters and for systems with long-range forces, such as self-gravitating gases.

Going deeper

For a student

The Euler relation turns the differential form of the fundamental relation into an integrated form. Differentiating it and subtracting the fundamental relation gives the Gibbs–Duhem equation. It also explains why Ω=−pV\Omega = -pV for a homogeneous fluid, a fact used constantly in statistical mechanics.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Chemical potential, concept, no evidence levelChemical potentialconceptEntropy, concept, no evidence levelEntropyconceptGibbs free energy, concept, no evidence levelGibbs free energyconceptGrand potential, concept, no evidence levelGrand potentialconceptHeat, concept, no evidence levelHeatconceptHeat capacity ratio, concept, no evidence levelHeat capacity ratioconceptHeat engine, concept, no evidence levelHeat engineconceptIdeal gas, concept, no evidence levelIdeal gasconceptIntensive and extensive properties, concept, no evidence levelIntensive and extensi…conceptInternal 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 workconceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementFirst law of thermodynamics, statement, no evidence levelFirst law of thermody…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…statementEuler relation for internal energy, statement, no evidence levelEuler relation for in…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. Chemical potential concept
  2. Entropy concept
  3. Gibbs free energy concept
  4. Grand potential concept
  5. Heat concept
  6. Heat capacity ratio concept
  7. Heat engine concept
  8. Ideal gas concept
  9. Intensive and extensive properties concept
  10. Internal energy concept
  11. Isentropic process concept
  12. Isothermal process concept
  13. Reversible process concept
  14. State function concept
  15. Temperature concept
  16. Thermal efficiency concept
  17. Thermodynamic cycle concept
  18. Thermodynamic system concept
  19. Thermodynamic temperature concept
  20. Thermodynamic work concept
  21. Clausius statement of the second law statement
  22. First law of thermodynamics statement
  23. Kelvin–Planck statement of the second law statement
  24. Carnot's theorem statement
  25. Carnot cycle example
  26. Clausius theorem statement
  27. Second law of thermodynamics statement
  28. Fundamental thermodynamic relation statement
  29. Euler relation for internal energy statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation
About this page
TagB6DK7G8K3G
URIm2://B6DK7G8K3G
Kindstatement, proposition
WritingW0 stub
MSC80A10, 80A05
StatusThis copy of the site has no build data. Run m2 status B6DK7G8K3G.
AboutInternal energy
UsesInternal energy, Entropy, Intensive and extensive properties, Fundamental thermodynamic relation, Chemical potential, Gibbs free energy, Grand potential
Changed3 November 2026

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