Difference between revisions of "Third law of thermodynamics"

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(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...)
 
m (Corrected typos.)
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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. Form
+
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>

Revision as of 12:22, 22 January 2010

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.

Implications

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

References

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