Math 2.0

Classical thermodynamics, heat transfer Definition

Intensive and extensive properties

Builds on Thermodynamic systemState functionThermodynamic equilibrium

Definition

Let XX be a state function defined for homogeneous systems in equilibrium. For a real number λ>0\lambda > 0, let λΣ\lambda \Sigma be the system of the same material in the same condition as Σ\Sigma. In λΣ\lambda \Sigma all amounts of matter are multiplied by λ\lambda.

  1. XX is extensive if X(λΣ)=λX(Σ)X(\lambda \Sigma) = \lambda X(\Sigma) for all Σ\Sigma and all λ>0\lambda > 0.
  2. XX is intensive if X(λΣ)=X(Σ)X(\lambda \Sigma) = X(\Sigma) for all Σ\Sigma and all λ>0\lambda > 0.

Remarks

Volume is extensive: doubling the amount of a gas at fixed temperature and pressure doubles its volume. Temperature is intensive: the doubled system has the same temperature. The square of the volume is neither: under scaling by λ\lambda it is multiplied by λ2\lambda^2, so it fails both clauses. The ratio of two extensive quantities is intensive, for example the density m/Vm/V or the molar volume V/NV/N. For a simple fluid, U(λS,λV,λN)=λU(S,V,N)U(\lambda S, \lambda V, \lambda N) = \lambda U(S, V, N), so the internal energy is a homogeneous function of degree 11 of its extensive variables.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Intensive_and_extensive_properties
About this page
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MSC80A05, 80A10
DefinesIntensive and extensive properties
UsesThermodynamic system, State function, Thermodynamic equilibrium
Changed3 November 2026

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