Third law of thermodynamics

The third law of thermodynamics (or Nernst's theorem after the experimental work of Walther Nernst in 1906 ) 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

$\underset{T\to 0}{lim}\frac{S(T)}{N}=0$

where $N$ 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

$C_{p,V}(T)=T{\frac{\partial S}{\partial T}|}_{p,V}$

one has

$S(T)-S(0)=\underset{0}{\overset{x}{\int }}\frac{C_{p,V}(T)}{T}\mathrm{d}T$

thus $C\to 0$ as $T\to 0$, otherwise the integrand would become infinite.

Similarly for the thermal expansion coefficient

$\alpha :=\frac{1}{V}{\frac{\partial V}{\partial T}|}_p=-\frac{1}{V}{\frac{\partial S}{\partial p}|}_T\to 0$

References

Related reading

• P. T. Landsberg "A comment on Nernst's theorem", Journal of Physics A: Mathematical and General 22 pp. 139-141 (1989)
• Lluís Masanes and Jonathan Oppenheim "A general derivation and quantification of the third law of thermodynamics", Nature Communications 8 14538 (2017)