# Third law of thermodynamics

The third law of thermodynamics (or Nernst's theorem after the experimental work of Walther Nernst in 1906 [1]) states that the entropy of a system approaches a minimum (that of its ground state) as one approaches the temperature of absolute zero. One can write

${\displaystyle \lim _{T\rightarrow 0}{\frac {S(T)}{N}}=0}$

where ${\displaystyle N}$ is the number of particles. Note that there are systems whose ground state entropy is not zero, for example metastable states or glasses, or systems with weakly or non-coupled spins that are not subject to an ordering field.

## Implications

The heat capacity (for either pressure or volume) tends to zero as one approaches absolute zero. From

${\displaystyle C_{p,V}(T)=T\left.{\frac {\partial S}{\partial T}}\right\vert _{p,V}}$

one has

${\displaystyle S(T)-S(0)=\int _{0}^{x}{\frac {C_{p,V}(T)}{T}}~\mathrm {d} T}$

thus ${\displaystyle C\rightarrow 0}$ as ${\displaystyle T\rightarrow 0}$, otherwise the integrand would become infinite.

Similarly for the thermal expansion coefficient

${\displaystyle \alpha :={\frac {1}{V}}\left.{\frac {\partial V}{\partial T}}\right\vert _{p}=-{\frac {1}{V}}\left.{\frac {\partial S}{\partial p}}\right\vert _{T}\rightarrow 0}$