Absolute temperature scale

Definition and physical interpretation

An absolute temperature scale assigns to each equilibrium state a positive number $T>0$ such that for any two thermal reservoirs at temperatures $T_H$ and $T_C$, a reversible engine operating between them satisfies the Carnot relation

where $Q_H$ is the heat absorbed from the hot reservoir and $Q_C$ is the heat delivered to the cold reservoir (both taken as positive magnitudes). This characterization is a consequence of the second law and the existence of reversible cycles.

Physically, the scale is “absolute” because it does not depend on the properties of any particular substance (unlike a gas thermometer calibration); instead, it is tied to universal constraints on cyclic energy conversion. The notion of “same temperature” is operationally captured by thermal equilibrium and formalized by the zeroth law , while the numerical scale is fixed by the second law.

Entropy-based characterization

For a simple compressible system described by a fundamental relation $S=S(U,V,N)$, the absolute temperature can be defined intrinsically by

Equivalently, if one uses the energy representation $U=U(S,V,N)$ (see energy fundamental relation ), then

This makes explicit that $T$ is the variable thermodynamically conjugate to entropy .

Key properties and conventions

• Scale uniqueness: The Carnot characterization fixes temperature only up to a multiplicative constant. Choosing units (Kelvin) fixes that constant; in statistical mechanics this choice is often packaged into the value of the Boltzmann constant $k_B$.

• Inverse temperature: It is common to use the inverse temperature $\beta=1/(k_B T)$, especially under the canonical ensemble convention .

• Absolute zero: The scale has a natural lower bound at $T=0$ (“absolute zero”), tied to unattainability statements of the third law . The operational meaning is subtle: $T=0$ is a limiting concept rather than an ordinary equilibrium point.

• Natural units: In some conventions (see natural units ), one sets $k_B=1$, so temperature has the same units as energy and $\beta=1/T`.