Third law of thermodynamics

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

\lim_{T \rightarrow 0} \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.


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

C_{p,V}(T)= T \left. \frac{\partial S}{\partial T} \right\vert_{p,V}

one has

S(T) - S(0) = \int_0^x \frac{C_{p,V}(T)}{T} ~\mathrm{d}T

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

Similarly for the thermal expansion coefficient

\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


  1. P. T. Landsberg "A comment on Nernst's theorem", Journal of Physics A: Mathematical and General 22 pp. 139-141 (1989)