Math 2.0

Classical thermodynamics, heat transfer Definition

Thermodynamic square

Builds on Internal energyEnthalpyHelmholtz free energyGibbs free energyEntropyFundamental thermodynamic relationMaxwell relations

Definition

Consider a closed simple system with entropy SS, volume VV, temperature TT and pressure pp. The thermodynamic square is the square diagram with these labels:

  1. Corners: −S-S at the top left, VV at the top right, TT at the bottom right, −p-p at the bottom left.
  2. Sides: internal energy UU on top, Helmholtz free energy FF right, Gibbs free energy GG bottom, enthalpy HH left.

It is read with this rule. For a potential on a side, its natural variables are the two adjacent corners, taken without sign. For such a variable vv, let cvc_v be the label, with its sign, at the corner diagonally opposite to vv. The differential of the potential is the sum of cv dvc_v \, dv over its two natural variables vv.

Remarks

Example: GG lies between TT and pp. The corner opposite TT is −S-S, and the corner opposite pp is VV, so dG=−S dT+V dpdG = -S \, dT + V \, dp, which agrees with the fundamental relation. In the same way the square gives dU=T dS−p dVdU = T \, dS - p \, dV, dF=−S dT−p dVdF = -S \, dT - p \, dV and dH=T dS+V dpdH = T \, dS + V \, dp, where SS is the entropy. Non-example: a square with FF and HH swapped is not the thermodynamic square, since its rule would give dF=T dS+V dpdF = T \, dS + V \, dp, which is false. The Maxwell relations can also be read off from the corners of the square.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  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
  41. Thermodynamic square definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Thermodynamic_square
About this page
TagGG80GYTSP5
URIm2://GG80GYTSP5
Kinddefinition
WritingW0 stub
MSC80A10
DefinesThermodynamic square
UsesInternal energy, Enthalpy, Helmholtz free energy, Gibbs free energy, Entropy, Fundamental thermodynamic relation, Maxwell relations
Changed3 November 2026

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