Third law of thermodynamics: Difference between revisions
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Carl McBride (talk | contribs) (New page: 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 gr...) |
Carl McBride (talk | contribs) (→References: Added a recent publication) |
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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 | The '''third law of thermodynamics''' (or '''Nernst's theorem''' after the experimental work of Walther Nernst in 1906 <ref>[http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1906 W. Nernst "Ueber die Berechnung chemischer Gleichgewichte aus thermischen Messungen" Königliche Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse. Nachrichten, pp. 1-40 (1906)]</ref>) 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 | ||
:<math>\lim_{T \rightarrow 0} \frac{S(T)}{N} = 0</math> | :<math>\lim_{T \rightarrow 0} \frac{S(T)}{N} = 0</math> | ||
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where <math>N</math> 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. | where <math>N</math> 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== | ==Implications== | ||
The [[heat capacity]] (for either [[pressure]] or volume) tends to zero as one approaches absolute zero. | The [[heat capacity]] (for either [[pressure]] or volume) tends to zero as one approaches absolute zero. From | ||
:<math>C_{p,V}(T)= T \left. \frac{\partial S}{\partial T} \right\vert_{p,V} </math> | :<math>C_{p,V}(T)= T \left. \frac{\partial S}{\partial T} \right\vert_{p,V} </math> | ||
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thus <math>C \rightarrow 0</math> as <math>T \rightarrow 0</math>, otherwise the integrand would become infinite. | thus <math>C \rightarrow 0</math> as <math>T \rightarrow 0</math>, otherwise the integrand would become infinite. | ||
Similarly for [[thermal expansion coefficient]] | Similarly for the [[thermal expansion coefficient]] | ||
:<math>\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</math> | :<math>\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</math> | ||
==References== | ==References== | ||
<references/> | |||
;Related reading | |||
*[http://dx.doi.org/10.1088/0305-4470/22/1/021 P. T. Landsberg "A comment on Nernst's theorem", Journal of Physics A: Mathematical and General '''22''' pp. 139-141 (1989)] | |||
*[http://dx.doi.org/10.1038/ncomms14538 Lluís Masanes and Jonathan Oppenheim "A general derivation and quantification of the third law of thermodynamics", Nature Communications '''8''' 14538 (2017)] | |||
[[category: classical thermodynamics]] | [[category: classical thermodynamics]] | ||
[[category: quantum mechanics]] | [[category: quantum mechanics]] | ||
Latest revision as of 17:28, 14 March 2017
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
- 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 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[edit]
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[edit]
- Related reading