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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Implications

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{p,V}(T)= T \left. \frac{\partial S}{\partial T} \right\vert_{p,V} }

one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(T) - S(0) = \int_0^x \frac{C_{p,V}(T)}{T} ~\mathrm{d}T}

thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C \rightarrow 0} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \rightarrow 0} , otherwise the integrand would become infinite.

Similarly for the thermal expansion coefficient

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}

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)