Math 2.0

Classical thermodynamics, heat transfer Thermodynamic potentials and relations Proposition

Bridgman's thermodynamic equations

Builds on Jacobian determinantPartial derivativeEntropyEnthalpyGibbs free energyHeat capacityMaxwell relations

Statement

Let a closed simple system have its equilibrium states described by temperature TT and pressure pp. Let every state quantity be a continuously differentiable function of (T,p)(T, p). This applies to the volume VV, the entropy SS, the enthalpy HH and the Gibbs free energy GG. For state quantities x,zx, z set (∂x)z=∂(x,z)∂(T,p),(\partial x)_z = \frac{\partial(x, z)}{\partial(T, p)}, the Jacobian determinant of (x,z)(x, z) with respect to (T,p)(T, p). Then:

  1. For state quantities x,y,zx, y, z with (∂y)z≠0(\partial y)_z \ne 0, the partial derivative of xx with respect to yy at fixed zz is (∂x∂y)z=(∂x)z(∂y)z\left(\dfrac{\partial x}{\partial y}\right)_z = \dfrac{(\partial x)_z}{(\partial y)_z}.
  2. (∂x)z=−(∂z)x(\partial x)_z = -(\partial z)_x for all x,zx, z.
  3. With the heat capacity Cp=T(∂S/∂T)pC_p = T (\partial S / \partial T)_p at constant pressure, the following hold: (∂T)p=1(\partial T)_p = 1, (∂V)p=(∂V/∂T)p(\partial V)_p = (\partial V / \partial T)_p, (∂S)p=Cp/T(\partial S)_p = C_p / T, (∂H)p=Cp(\partial H)_p = C_p, (∂G)p=−S(\partial G)_p = -S, (∂V)T=−(∂V/∂p)T(\partial V)_T = -(\partial V / \partial p)_T, (∂S)T=(∂V/∂T)p(\partial S)_T = (\partial V / \partial T)_p, (∂G)T=−V(\partial G)_T = -V.

Remarks

Item 1 is the chain rule for Jacobians: ∂(x,z)/∂(y,z)=(∂x/∂y)z\partial(x, z)/\partial(y, z) = (\partial x / \partial y)_z, and Jacobians multiply under change of variables. Item 2 holds because swapping two rows changes the sign of a determinant. In item 3, (∂H)p=Cp(\partial H)_p = C_p uses dH=T dS+V dpdH = T \, dS + V \, dp, and (∂S)T=−(∂S/∂p)T=(∂V/∂T)p(\partial S)_T = -(\partial S / \partial p)_T = (\partial V / \partial T)_p uses a Maxwell relation. The entries for GG use dG=−S dT+V dpdG = -S \, dT + V \, dp. Example: the cooling of a substance under adiabatic expansion. One has (∂T/∂p)S=(∂T)S/(∂p)S=(∂S)T/(∂S)p=T(∂V/∂T)p/Cp(\partial T / \partial p)_S = (\partial T)_S / (\partial p)_S = (\partial S)_T / (\partial S)_p = T (\partial V / \partial T)_p / C_p. Since (∂V/∂T)p>0(\partial V / \partial T)_p > 0 for most substances, reducing the pressure at constant entropy lowers the temperature.

Going deeper

For a student

Percy Bridgman's table lists (∂x)z(\partial x)_z for all pairs among T,p,V,S,U,H,F,GT, p, V, S, U, H, F, G in terms of CpC_p, (∂V/∂T)p(\partial V / \partial T)_p, (∂V/∂p)T(\partial V / \partial p)_T and the state variables. Every first derivative of one state quantity with respect to another at fixed third then follows from item 1. The method is a systematic use of the Jacobian calculus together with the Maxwell relations.

Reading path

To understand this page from the ground up, read these 45 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 concept
  8. Heat capacity ratio concept
  9. Heat engine concept
  10. Helmholtz free energy concept
  11. Ideal gas concept
  12. Internal energy concept
  13. Isentropic process concept
  14. Isothermal process concept
  15. Reversible process concept
  16. State function concept
  17. Temperature concept
  18. Thermal efficiency concept
  19. Thermodynamic cycle concept
  20. Thermodynamic system concept
  21. Thermodynamic temperature concept
  22. Thermodynamic work concept
  23. Fréchet derivative definition
  24. Total derivative concept
  25. Function of several variables concept
  26. Clausius statement of the second law statement
  27. First law of thermodynamics statement
  28. Limit of a function of several variables concept
  29. Partial derivative concept
  30. Kelvin–Planck statement of the second law statement
  31. Carnot's theorem statement
  32. Differentiable function of several variables definition
  33. Jacobian matrix concept
  34. Partial derivative definition
  35. Carnot cycle example
  36. Differentiable on an open set definition
  37. Jacobian determinant concept
  38. Higher partial derivatives definition
  39. Clausius theorem statement
  40. Iterated derivative definition
  41. Second law of thermodynamics statement
  42. Symmetry of second derivatives statement
  43. Fundamental thermodynamic relation statement
  44. Maxwell relations statement
  45. Bridgman's thermodynamic equations statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Bridgman%27s_thermodynamic_equations
About this page
Tag3PMZ8MDCYN
URIm2://3PMZ8MDCYN
Kindstatement, proposition
WritingW0 stub
MSC80A10, 26B12
StatusThis copy of the site has no build data. Run m2 status 3PMZ8MDCYN.
AboutThermodynamic potential
UsesJacobian determinant, Partial derivative, Entropy, Enthalpy, Gibbs free energy, Heat capacity, Maxwell relations
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Changed3 November 2026

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