Helmholtz energy function: Difference between revisions

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[[Hermann Ludwig Ferdinand von Helmholtz]]  
'''Helmholtz energy function''' ([[Hermann Ludwig Ferdinand von Helmholtz]])
Definition of '''A''' (for ''arbeit''):
Definition of <math>A</math> (for ''arbeit''):


:<math>\left.A\right.=U-TS</math>
:<math>A:=U-TS</math>


where ''U'' is the [[internal energy]], ''T'' is the [[temperature]] and ''S'' is the [[Entropy|entropy]].
where ''U'' is the [[internal energy]], ''T'' is the [[temperature]] and ''S'' is the [[Entropy|entropy]].
Line 27: Line 27:


where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], and <math>Q_{NVT}</math> is the [[Canonical ensemble | canonical ensemble partition function]].
where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], and <math>Q_{NVT}</math> is the [[Canonical ensemble | canonical ensemble partition function]].
==Ideal gas==
:''Main article: [[Ideal gas Helmholtz energy function]]''
==Quantum correction==
A quantum correction can be calculated by making use of the [[Wigner-Kirkwood expansion]] of the partition function, resulting in (Eq. 3.5 in <ref>[http://dx.doi.org/10.1080/00268977900102921 J.G. Powles and G. Rickayzen "Quantum corrections and the computer simulation of molecular fluids", Molecular Physics '''38''' pp. 1875-1892 (1979)]</ref>):
:<math>\frac{A-A_{ {\mathrm{classical}} }}{N} = \frac{\hbar^2}{24m(k_BT)^2} \langle F^2 \rangle </math>
where <math>\langle F^2 \rangle</math> is the mean squared force on any one atom due to all the other atoms.
==See also==
==See also==
*[[Canonical ensemble]]
*[[Canonical ensemble]]
*[[Grand canonical ensemble]]
*[[Grand canonical ensemble]]
*[[Computing the Helmholtz energy function of solids]]


==References==
<references/>
[[Category: Classical thermodynamics]]
[[Category: Classical thermodynamics]]

Latest revision as of 18:53, 20 February 2015

Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz) Definition of (for arbeit):

where U is the internal energy, T is the temperature and S is the entropy. (TS) is a conjugate pair. The differential of this function is

From the second law of thermodynamics one obtains

thus one arrives at

.

For A(T,V) one has the following total differential

The following equation provides a link between classical thermodynamics and statistical mechanics:

where is the Boltzmann constant, T is the temperature, and is the canonical ensemble partition function.

Ideal gas[edit]

Main article: Ideal gas Helmholtz energy function

Quantum correction[edit]

A quantum correction can be calculated by making use of the Wigner-Kirkwood expansion of the partition function, resulting in (Eq. 3.5 in [1]):

where is the mean squared force on any one atom due to all the other atoms.

See also[edit]

References[edit]