Math 2.0

Classical thermodynamics, heat transfer Definition

Maxwell equal-area construction

Builds on Equation of statevan der Waals equationChemical potentialHelmholtz free energyThermodynamic stability conditions

Definition

Fix a temperature TT and a fixed amount of a fluid with an equation of state. Let p(V)p(V) be the isotherm at TT, and assume that pp is continuous and not monotone decreasing on an interval of volumes. The Maxwell equal-area construction chooses a pressure p∗p^* and volumes Vl<VgV_l < V_g with p(Vl)=p(Vg)=p∗p(V_l) = p(V_g) = p^* such that ∫VlVgp(V) dV=p∗(Vg−Vl).\int_{V_l}^{V_g} p(V) \, dV = p^* (V_g - V_l). The corrected isotherm equals p∗p^* on [Vl,Vg][V_l, V_g] and equals p(V)p(V) outside this interval.

Remarks

The condition says that the region between the curve and the line p=p∗p = p^* has equal areas above and below the line. It is equivalent to equality of the Gibbs free energies per mole, that is of the chemical potentials, of the liquid at VlV_l and the vapour at VgV_g. Indeed, along an isotherm the Helmholtz free energy satisfies F(Vg)−F(Vl)=−∫VlVgp dVF(V_g) - F(V_l) = -\int_{V_l}^{V_g} p \, dV, and μ=(F+pV)/n\mu = (F + pV)/n; equal chemical potentials at equal pressure p∗p^* give the displayed equation. On [Vl,Vg][V_l, V_g] the fluid is a mixture of liquid and vapour, and the parts of the original isotherm with ∂p/∂V>0\partial p / \partial V > 0, which violate the stability conditions, are removed.

Example: for the van der Waals equation in reduced variables at Tr=0.9T_r = 0.9, the construction gives a coexistence pressure pr∗≈0.65p_r^* \approx 0.65.

Non-example: for T≥TcT \ge T_c every van der Waals isotherm is monotone decreasing. There is no pair Vl<VgV_l < V_g with equal pressure, and no construction is needed. There is no liquid-vapour coexistence above the critical temperature.

Reading path

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Chemical potential, concept, no evidence levelChemical potentialconceptCompressibility, concept, no evidence levelCompressibilityconceptEntropy, concept, no evidence levelEntropyconceptEquation of state, concept, no evidence levelEquation of stateconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat, concept, no evidence levelHeatconceptHeat capacity, concept, no evidence levelHeat capacityconceptHeat 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 processconceptTemperature, concept, no evidence levelTemperatureconceptThermal efficiency, concept, no evidence levelThermal efficiencyconceptThermodynamic cycle, concept, no evidence levelThermodynamic cycleconceptThermodynamic equilibrium, concept, no evidence levelThermodynamic equilib…conceptThermodynamic system, concept, no evidence levelThermodynamic systemconceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptThermodynamic work, concept, no evidence levelThermodynamic workconceptvan der Waals equation, concept, no evidence levelvan der Waals equationconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementLimit 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 derivativedefinitionCarnot cycle, example, no evidence levelCarnot cycleexampleHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionClausius theorem, statement, no evidence levelClausius theoremstatementHessian matrix, definition, no evidence levelHessian matrixdefinitionSecond law of thermodynamics, statement, no evidence levelSecond law of thermod…statementPrinciple of maximum entropy in thermodynamics, statement, no evidence levelPrinciple of maximum …statementPrinciple of minimum energy, statement, no evidence levelPrinciple of minimum …statementThermodynamic stability conditions, statement, no evidence levelThermodynamic stabili…statementMaxwell equal-area construction, definition, no evidence levelMaxwell equal-area co…definition
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. Compressibility concept
  3. Entropy concept
  4. Equation of state concept
  5. Euclidean space object
  6. Function concept
  7. Heat concept
  8. Heat capacity concept
  9. Heat capacity ratio concept
  10. Heat engine concept
  11. Helmholtz free energy concept
  12. Ideal gas concept
  13. Internal energy concept
  14. Isentropic process concept
  15. Isothermal process concept
  16. Reversible process concept
  17. Temperature concept
  18. Thermal efficiency concept
  19. Thermodynamic cycle concept
  20. Thermodynamic equilibrium concept
  21. Thermodynamic system concept
  22. Thermodynamic temperature concept
  23. Thermodynamic work concept
  24. van der Waals equation concept
  25. Function of several variables concept
  26. Clausius statement of the second law statement
  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. Carnot cycle example
  32. Higher partial derivatives definition
  33. Clausius theorem statement
  34. Hessian matrix definition
  35. Second law of thermodynamics statement
  36. Principle of maximum entropy in thermodynamics statement
  37. Principle of minimum energy statement
  38. Thermodynamic stability conditions statement
  39. Maxwell equal-area construction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Maxwell_construction
About this page
TagEQ4QW84SRV
URIm2://EQ4QW84SRV
Kinddefinition
WritingW0 stub
MSC80A10, 82B26
DefinesMaxwell equal-area construction
UsesEquation of state, van der Waals equation, Chemical potential, Helmholtz free energy, Thermodynamic stability conditions
Changed3 November 2026

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