Math 2.0

Classical thermodynamics, heat transfer Laws of thermodynamics Fact

Unattainability principle

Builds on Thermodynamic temperatureEntropyIsothermal processAdiabatic processNernst heat theorem

Statement

No process can reduce the thermodynamic temperature of a system to absolute zero in a finite number of steps.

Remarks

The usual cooling method alternates two kinds of steps. An isothermal step changes an external parameter XX, such as a magnetic field, from X1X_1 to X2X_2 at fixed temperature. An adiabatic step changes it back at fixed entropy, which lowers the temperature. Draw the entropy curves T↦S(T,X1)T \mapsto S(T, X_1) and T↦S(T,X2)T \mapsto S(T, X_2). Suppose the two curves meet at T=0T = 0, as the Nernst heat theorem says. Then each cycle lowers the temperature by a smaller amount, and T=0T = 0 would need infinitely many steps. If they did not meet at T=0T = 0, a single adiabatic step could reach absolute zero. Adiabatic demagnetisation of a paramagnetic salt is the standard example of this cooling process.

Going deeper

For a student

The unattainability principle is one of the usual forms of the third law of thermodynamics. Under suitable assumptions on the system it is equivalent to the Nernst heat theorem.

Reading path

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

Show the path as a graph
Adiabatic process, concept, no evidence levelAdiabatic processconceptEnthalpy, concept, no evidence levelEnthalpyconceptEntropy, concept, no evidence levelEntropyconceptGibbs free energy, concept, no evidence levelGibbs free energyconceptIsothermal process, concept, no evidence levelIsothermal processconceptThermodynamic equilibrium, concept, no evidence levelThermodynamic equilib…conceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptNernst heat theorem, statement, no evidence levelNernst heat theoremstatementUnattainability principle, statement, no evidence levelUnattainability princ…statement
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. Adiabatic process concept
  2. Enthalpy concept
  3. Entropy concept
  4. Gibbs free energy concept
  5. Isothermal process concept
  6. Thermodynamic equilibrium concept
  7. Thermodynamic temperature concept
  8. Nernst heat theorem statement
  9. Unattainability principle statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Third_law_of_thermodynamics
About this page
TagB06S5CKP74
URIm2://B06S5CKP74
Kindstatement, fact
WritingW0 stub
MSC80A05, 80A10
StatusThis copy of the site has no build data. Run m2 status B06S5CKP74.
AboutThermodynamic temperature
UsesThermodynamic temperature, Entropy, Isothermal process, Adiabatic process, Nernst heat theorem
Changed3 November 2026

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