Nernst theorem

The Nernst Theorem , often Nernst heat theorem called (after the German physicist Walther Nernst ) is another name for the third law of thermodynamics . It says that the absolute zero point of the temperature cannot be reached.

The theorem can be proven with the help of quantum mechanics (see below).

As expected, it was not refuted in experiments , since it was only possible to get closer and closer to absolute zero, but never to reach it.

formulation

The theorem was established by Nernst in 1905 and deals with the change in the entropy of a chemical reaction at a temperature of zero Kelvin : it approaches zero. ${\displaystyle S}$

The formulation was made more precise by Max Planck in 1911 . After that, the entropy becomes independent of thermodynamic parameters and thus constant when the temperature approaches zero:

${\displaystyle \lim _{T\to 0}S(T,p,V,\dots )=S(T=0)=S_{0}=k_{\mathrm {B} }\cdot \ln g}$,

where is the Boltzmann constant and is the degeneracy of the ground state . ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle g}$

If the basic state of the system is not degenerate, then and therefore applies . Thus, the entropy of a system disappears when the temperature approaches zero. ${\displaystyle g=1}$ ${\displaystyle S_{0}=0}$

Proof of canonical distribution

${\displaystyle S=-k_{\mathrm {B} }\,\operatorname {Sp} \,\rho \ln \rho }$

First, the statistical operator is replaced by its representation in the canonical distribution. is here the empirical temperature. ${\displaystyle \rho }$ ${\displaystyle T={\frac {1}{k_{\mathrm {B} }\beta }}}$

${\displaystyle S=-k_{\mathrm {B} }\,\operatorname {Sp} {\frac {e^{-\beta H}}{\operatorname {Sp} e^{-\beta H}}}\left(-\beta H-\ln \operatorname {Sp} e^{-\beta H}\right)}$

If you evaluate the track using the operators, you get:

${\displaystyle S=-k_{\mathrm {B} }\,\sum _{n}{\frac {e^{-\beta E_{n}}}{\sum _{m}e^{-\beta E_{m}}}}\left(-\beta E_{n}-\ln \sum _{m}e^{-\beta E_{m}}\right)}$

Now the energy of the ground state is subtracted from each level.

${\displaystyle S=-k_{\mathrm {B} }\,\sum _{n}{\frac {e^{-\beta \left(E_{n}-E_{g}\right)}}{\sum _{m}e^{-\beta \left(E_{m}-E_{g}\right)}}}\left(-\beta \left(E_{n}-E_{g}\right)-\ln \sum _{m}e^{-\beta \left(E_{m}-E_{g}\right)}\right)}$

It now applies to (corresponds to ): ${\displaystyle \beta \rightarrow \infty }$ ${\displaystyle T\rightarrow 0}$

${\displaystyle \lim _{T\rightarrow 0}e^{-\beta \left(E_{n}-E_{g}\right)}={\begin{cases}1,&{\text{wenn }}E_{n}=E_{g}\\0,&{\text{wenn }}E_{n}>E_{g}\end{cases}}}$

If you insert this knowledge into the above double-sum representation, you get the formulation of the Nernst theorem according to Planck:

${\displaystyle \lim _{T\rightarrow 0}S=k_{\mathrm {B} }\,\ln g}$,

where indicates the degeneracy of the ground state, i.e. the number of those who are equal . ${\displaystyle g}$ ${\displaystyle E_{n}}$ ${\displaystyle E_{g}}$

See also

• First law of thermodynamics
• Second law of thermodynamics