Math 2.0

Classical thermodynamics, heat transfer Thermodynamic potentials and relations Proposition

Gibbs–Duhem equation

Builds on Euler relation for internal energyFundamental thermodynamic relationEntropyChemical potentialIntensive and extensive properties

Statement

Let a simple system have internal energy U(S,V,N1,…,Nr)U(S, V, N_1, \dots, N_r), a differentiable and extensive function of its entropy SS, volume VV and amounts of substance N1,…,NrN_1, \dots, N_r. Let TT, pp and μ1,…,μr\mu_1, \dots, \mu_r be its temperature, pressure and chemical potentials. Then for any change between neighbouring equilibrium states, S dT−V dp+∑i=1rNi dμi=0.S \, dT - V \, dp + \sum_{i=1}^{r} N_i \, d\mu_i = 0.

Remarks

Differentiate the Euler relation U=TS−pV+∑iμiNiU = TS - pV + \sum_i \mu_i N_i. This gives dU=T dS+S dT−p dV−V dp+∑i(μi dNi+Ni dμi)dU = T \, dS + S \, dT - p \, dV - V \, dp + \sum_i (\mu_i \, dN_i + N_i \, d\mu_i). Subtract the fundamental relation dU=T dS−p dV+∑iμi dNidU = T \, dS - p \, dV + \sum_i \mu_i \, dN_i to get the equation. Example: for one substance, dividing by NN gives dμ=−s dT+v dpd\mu = -s \, dT + v \, dp, where s=S/Ns = S/N and v=V/Nv = V/N. So at constant temperature, (∂μ/∂p)T=v(\partial \mu / \partial p)_T = v. For a two-component mixture at constant TT and pp, it gives N1 dμ1+N2 dμ2=0N_1 \, d\mu_1 + N_2 \, d\mu_2 = 0, so the two chemical potentials cannot change independently.

Going deeper

For a student

For a system with rr substances there are r+2r + 2 intensive variables, but only r+1r + 1 of them are independent. This count underlies the Gibbs phase rule. In chemistry the equation relates the activities of the components of a mixture, so one can be computed from measurements of the others.

Reading path

To understand this page from the ground up, read these 30 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…statementGibbs–Duhem equation, statement, no evidence levelGibbs–Duhem equationstatement
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
  30. Gibbs–Duhem equation statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gibbs%E2%80%93Duhem_equation
About this page
Tag57YCS65Z5T
URIm2://57YCS65Z5T
Kindstatement, proposition
WritingW0 stub
MSC80A10, 80A50
StatusThis copy of the site has no build data. Run m2 status 57YCS65Z5T.
AboutChemical potential
UsesEuler relation for internal energy, Fundamental thermodynamic relation, Entropy, Chemical potential, Intensive and extensive properties
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Changed3 November 2026

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