Math 2.0

Classical thermodynamics, heat transfer Thermodynamic potentials and relations Theorem

Maxwell relations

Builds on Fundamental thermodynamic relationInternal energyEnthalpyHelmholtz free energyGibbs free energySymmetry of second derivativesPartial derivative

Statement

Let a closed simple system have temperature TT, entropy SS, pressure pp and volume VV. Suppose that its internal energy U(S,V)U(S, V), enthalpy H(S,p)H(S, p), Helmholtz free energy F(T,V)F(T, V) and Gibbs free energy G(T,p)G(T, p) are twice continuously differentiable in their natural variables. Then the partial derivatives satisfy

  1. (∂T∂V)S=−(∂p∂S)V\left(\dfrac{\partial T}{\partial V}\right)_S = -\left(\dfrac{\partial p}{\partial S}\right)_V,
  2. (∂T∂p)S=(∂V∂S)p\left(\dfrac{\partial T}{\partial p}\right)_S = \left(\dfrac{\partial V}{\partial S}\right)_p,
  3. (∂S∂V)T=(∂p∂T)V\left(\dfrac{\partial S}{\partial V}\right)_T = \left(\dfrac{\partial p}{\partial T}\right)_V,
  4. (∂S∂p)T=−(∂V∂T)p\left(\dfrac{\partial S}{\partial p}\right)_T = -\left(\dfrac{\partial V}{\partial T}\right)_p.

A subscript names the variable held fixed.

Remarks

The fundamental relation gives dU=T dS−p dVdU = T \, dS - p \, dV, dH=T dS+V dpdH = T \, dS + V \, dp, dF=−S dT−p dVdF = -S \, dT - p \, dV and dG=−S dT+V dpdG = -S \, dT + V \, dp. So, for instance, T=(∂U/∂S)VT = (\partial U / \partial S)_V and −p=(∂U/∂V)S-p = (\partial U / \partial V)_S. Each relation is the equality of mixed second derivatives of one potential: relation 1 for UU, 2 for HH, 3 for FF and 4 for GG. Example: for an ideal gas, p=nRT/Vp = nRT / V, so relation 3 gives (∂S/∂V)T=nR/V(\partial S / \partial V)_T = nR / V. Integrating, an isothermal expansion from V1V_1 to V2V_2 raises the entropy by nRln⁡(V2/V1)nR \ln(V_2 / V_1).

Going deeper

For a student

The relations turn derivatives of the entropy, which cannot be measured directly, into derivatives of pp, VV and TT, which can. With them one derives, for example, the difference Cp−CV=TVα2/κTC_p - C_V = T V \alpha^2 / \kappa_T of the heat capacities, where α\alpha is the thermal expansion coefficient and κT\kappa_T the isothermal compressibility. For systems with more variables, such as magnetic systems or open systems, each potential gives further Maxwell relations in the same way.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

Show the path as a graph
Enthalpy, concept, no evidence levelEnthalpyconceptEntropy, concept, no evidence levelEntropyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGibbs free energy, concept, no evidence levelGibbs free energyconceptHeat, concept, no evidence levelHeatconceptHeat capacity ratio, concept, no evidence levelHeat capacity ratioconceptHeat engine, concept, no evidence levelHeat engineconceptHelmholtz free energy, concept, no evidence levelHelmholtz free energyconceptIdeal 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 workconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementFirst law of thermodynamics, statement, no evidence levelFirst law of thermody…statementPartial derivative, concept, no evidence levelPartial derivativeconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptKelvin–Planck statement of the second law, statement, no evidence levelKelvin–Planck stateme…statementCarnot's theorem, statement, no evidence levelCarnot's theoremstatementPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionCarnot cycle, example, no evidence levelCarnot cycleexampleHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionClausius theorem, statement, no evidence levelClausius theoremstatementIterated derivative, definition, no evidence levelIterated derivativedefinitionSecond law of thermodynamics, statement, no evidence levelSecond law of thermod…statementSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementFundamental thermodynamic relation, statement, no evidence levelFundamental thermodyn…statementMaxwell relations, statement, no evidence levelMaxwell relationsstatement
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. Enthalpy concept
  2. Entropy concept
  3. Euclidean space object
  4. Function concept
  5. Gibbs free energy concept
  6. Heat concept
  7. Heat capacity ratio concept
  8. Heat engine concept
  9. Helmholtz free energy concept
  10. Ideal gas concept
  11. Internal energy concept
  12. Isentropic process concept
  13. Isothermal process concept
  14. Reversible process concept
  15. State function concept
  16. Temperature concept
  17. Thermal efficiency concept
  18. Thermodynamic cycle concept
  19. Thermodynamic system concept
  20. Thermodynamic temperature concept
  21. Thermodynamic work concept
  22. Fréchet derivative definition
  23. Function of several variables concept
  24. Clausius statement of the second law statement
  25. First law of thermodynamics statement
  26. Partial derivative concept
  27. Limit of a function of several variables concept
  28. Kelvin–Planck statement of the second law statement
  29. Carnot's theorem statement
  30. Partial derivative definition
  31. Differentiable function of several variables definition
  32. Carnot cycle example
  33. Higher partial derivatives definition
  34. Differentiable on an open set definition
  35. Clausius theorem statement
  36. Iterated derivative definition
  37. Second law of thermodynamics statement
  38. Symmetry of second derivatives statement
  39. Fundamental thermodynamic relation statement
  40. Maxwell relations statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Maxwell_relations
About this page
TagMAHDE6JZYT
URIm2://MAHDE6JZYT
Kindstatement, theorem
WritingW0 stub
MSC80A10, 80A05
StatusThis copy of the site has no build data. Run m2 status MAHDE6JZYT.
AboutThermodynamic potential
UsesFundamental thermodynamic relation, Internal energy, Enthalpy, Helmholtz free energy, Gibbs free energy, Symmetry of second derivatives, Partial derivative
Same as
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by