Math 2.0

Classical thermodynamics, heat transfer Definition

Conjugate variables

Builds on Intensive and extensive propertiesInternal energyEntropyThermodynamic temperaturePartial derivative

Definition

Let a system in equilibrium have internal energy U=U(X1,…,Xk)U = U(X_1, \dots, X_k), a differentiable function of extensive variables X1,…,XkX_1, \dots, X_k. For each ii, the variable conjugate to XiX_i is the intensive variable Yi=∂U∂Xi,Y_i = \frac{\partial U}{\partial X_i}, where the partial derivative is taken with the other XjX_j fixed. The pairs (Yi,Xi)(Y_i, X_i) are the conjugate pairs, and dU=∑i=1kYi dXidU = \sum_{i=1}^{k} Y_i \, dX_i.

Remarks

For a simple fluid with entropy SS, volume VV and particle number NN, one has dU=T dS−p dV+μ dN.dU = T \, dS - p \, dV + \mu \, dN. So thermodynamic temperature TT is conjugate to SS, the pressure pp to VV up to sign, and the chemical potential μ\mu to NN. Each product TST S, pVp V and μN\mu N has the dimension of energy. Temperature and pressure are not a conjugate pair: both are intensive, and neither is the derivative of UU with respect to the other. Other pairs are magnetic field and magnetic moment, and surface tension and area.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Entropy, concept, no evidence levelEntropyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptIntensive and extensive properties, concept, no evidence levelIntensive and extensi…conceptInternal energy, concept, no evidence levelInternal energyconceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptConjugate variables, definition, no evidence levelConjugate variablesdefinition
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.
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  1. Entropy concept
  2. Euclidean space object
  3. Function concept
  4. Intensive and extensive properties concept
  5. Internal energy concept
  6. Thermodynamic temperature concept
  7. Function of several variables concept
  8. Partial derivative concept
  9. Conjugate variables definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugate_variables_(thermodynamics)
About this page
TagW3PMXQGA4Y
URIm2://W3PMXQGA4Y
Kinddefinition
WritingW0 stub
MSC80A05, 80A10
DefinesConjugate variables
UsesIntensive and extensive properties, Internal energy, Entropy, Thermodynamic temperature, Partial derivative
Changed3 November 2026

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