Third Law of Thermodynamics

Equivalent forms

\lim_{T\to 0}\left(\frac{\partial S}{\partial T}\right) \to 0

S_0 = 0 \text{ for a non-degenerate ground state}

Fixes the otherwise-arbitrary zero of entropy and makes absolute zero an unreachable asymptote rather than a destination.

Symbol	Name	SI unit	Dimension	Range
$T$	Temperature Absolute temperature approaching zero	K	Θ	0 – 600
$S₀$	Residual entropyoutput Entropy remaining at absolute zero	J/K	M L² T⁻² Θ⁻¹	0 – 1e-22
$g₀$	Ground-state degeneracy Number of equivalent lowest-energy microstates	dimensionless	1	1 – 8

Unit systems

SI:: S in J/K
natural:: S in units of k_B
CGS:: S in erg/K

Where it holds

Applies to systems that reach internal equilibrium as

T \to 0

; glasses freeze with frozen-in configurational entropy and only approximate it.

Discovery

Walther Nernst · 1906

Nernst formulated the heat theorem in 1906 from low-temperature electrochemistry; Planck sharpened it to S → 0, and Nernst won the 1920 Nobel Prize for it.

Try this

Why can you never reach absolute zero?

Each cooling step removes a fraction of the remaining heat, but the entropy you must extract shrinks to zero — so it would take infinitely many steps to actually hit 0 K.

Research status: stable

Common misconceptions

The Third Law does not say absolute zero is forbidden by energy — it says it is unattainable in finite steps; and residual entropy (ice, CO) is real where the ground state is degenerate.

Derivation

From

S = k_B \ln W

, as

T \to 0

the system occupies only its ground state with multiplicity

g_{0}

, so

W \to g_{0} and S \to k_B \ln g_{0}

.

The unattainability statement follows: removing the last

\Delta S

requires infinitely many finite cooling steps because

\Delta S

available per step

\to 0

.

Limiting cases

Non-degenerate ground state

(g_{0} = 1)

⟶

S_{0} = 0

: a perfect crystal has zero entropy at absolute zero

Degenerate ground state (e.g. ice,

g_{0}

> 1)⟶ Residual entropy

S_{0} = k_B \ln g_{0}

> 0 (Pauling's ice entropy)

T \to 0

⟶ Heat capacities C_p,

C_v \to 0

and thermal expansion

\to 0

Boltzmann Entropy→Entropy Change→Heat Capacity at Constant Volume→Sackur-Tetrode Equation→