Unattainability formulation of the third law

Statement

(Unattainability principle.) For a thermodynamic system governed by the second law of thermodynamics , it is impossible to reach the state $T=0$ (absolute zero temperature ) from any initial state with $T>0$ by a finite sequence of thermodynamic operations.

Common equivalent operational readings include:

• No finite number of (idealized) reversible or irreversible steps can take a system from $T>0$ to $T=0$.
• Any cooling protocol that respects the second law requires resources (time, steps, or auxiliary reservoirs) that diverge as $T\to 0$.

Key hypotheses

• Thermodynamic description in equilibrium (or quasi-static limits) with well-defined temperature.
• Validity of the second law (e.g., via Kelvin–Planck/Clausius statements; see Kelvin–Planck/Clausius equivalence ).
• “Finite process” interpreted as finitely many steps with finite reservoirs/controls; idealizations are allowed, but not infinite concatenations.

Conclusions

• $T=0$ is a limit point that cannot be attained by finite operations starting from $T>0$.
• The approach to $T=0$ is constrained even if entropy decreases are allowed via heat extraction; in particular, the ability to use Carnot-like refrigeration cycles is limited as the cold temperature decreases (compare Carnot theorem and Carnot efficiency formula ).

Proof idea / significance

A standard route links unattainability to the second law through refrigeration/Carnot-cycle reasoning: extracting heat from an ever-colder reservoir while maintaining consistency with entropy production becomes increasingly “costly,” forcing an infinite limiting procedure as $T\to 0$.
Conceptually, unattainability is one of the main operational formulations of the third law; it complements entropy-based formulations (e.g., limits of entropy as $T\to 0$) by focusing on what can be achieved by physical processes rather than equilibrium state functions alone.