Math 2.0

Statistical mechanics, structure of matter Definition

Ehrenfest classification of phase transitions

Builds on Phase transitionChemical potentialEntropyHeat capacity

Definition

Let two phases α\alpha and β\beta of a pure substance meet along a curve in the (T,p)(T, p) plane. Let g(T,p)g(T, p) be the equilibrium molar Gibbs free energy, equal to the chemical potential of the stable phase. Let n≥1n \ge 1 be an integer. The phase transition across the curve is of order nn in the sense of Ehrenfest if:

  1. all partial derivatives of gg with respect to TT and pp of order less than nn are continuous across the curve, and
  2. at least one partial derivative of gg of order nn has a finite jump across the curve.

Remarks

Since gg is continuous at any transition, order 11 means that s=−∂g/∂Ts = -\partial g/\partial T, the molar entropy, or v=∂g/∂pv = \partial g/\partial p, the molar volume, jumps. Boiling of water is first order: at 101.325 kPa101.325\,\mathrm{kPa} the molar volume of steam is far larger than that of liquid water. Order 22 means that ss and vv are continuous but a second derivative jumps, such as the molar heat capacity cp=−T ∂2g/∂T2c_p = -T\, \partial^2 g/\partial T^2. A non-example: at the lambda transition of liquid helium-4, ss and vv are continuous but cpc_p diverges. So no derivative of order 22 has a finite jump, clause 2 fails for n=2n = 2, and the transition has no Ehrenfest order. For this reason the modern classification only separates first-order transitions, with latent heat, from continuous transitions.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Phase_transition
About this page
TagPSFN7V2HRR
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Kinddefinition
WritingW0 stub
MSC82B26, 80A22, 80A10
DefinesEhrenfest classification of phase transitions
UsesPhase transition, Chemical potential, Entropy, Heat capacity
Changed3 November 2026

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