Third law of thermodynamics

One common formulation (Nernst heat theorem) is:

• For any reversible isothermal change between equilibrium states, the entropy change tends to zero as the temperature tends to absolute zero (on the absolute temperature scale ).

A closely related and widely used consequence is that, for a system in thermodynamic equilibrium with fixed composition and fixed internal constraints,

where $S$ is the thermodynamic entropy and $S_0$ is a constant independent of the continuous thermodynamic coordinates $X$ (such as volume or pressure). For a “perfect crystal” with a unique ground state, one often takes $S_0=0$, which is an entropy normalization convention compatible with the third law.

A complementary formulation is the unattainability principle: no finite sequence of thermodynamic processes can reach $T=0$.

Physical interpretation

At very low temperatures, accessible microscopic configurations collapse toward the ground-state set, so there is (typically) no extensive configurational disorder left to contribute to entropy. When the ground state has a degeneracy $g$, the third-law limit allows a “residual entropy”

linking macroscopic entropy to microscopic state counting via the Boltzmann constant .

Key relations

• Computing absolute entropies from heat capacities. If the third law fixes $S_0$, then along a path at fixed volume,

  $$S(T) = S_0 + \int_{0}^{T} \frac{C_V(T')}{T'}\,dT' \;+\; \sum_{\text{transitions}} \Delta S,$$

  where $C_V$ is the heat capacity at constant volume and the sum accounts for entropy jumps at phase transitions. An analogous formula uses the heat capacity at constant pressure along a constant-pressure path.

• Low-temperature constraint. For equilibrium systems that obey the third law, $C_V$ and $C_P$ cannot remain finite and nonzero all the way to $T=0^+$ without making the integral of $C/T$ diverge; in many familiar systems they vanish as $T\to 0^+$, strongly constraining admissible equations of state near absolute zero.

• Statistical-mechanical parameter. In ensemble language, $T\to 0^+$ corresponds to the inverse temperature $\beta\to +\infty$, emphasizing that the third law concerns the extreme low-energy limit.