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| '''Helmholtz energy function''' ([[Hermann Ludwig Ferdinand von Helmholtz]])
| | [[Hermann Ludwig Ferdinand von Helmholtz]] |
| Definition of <math>A</math> (for ''arbeit''): | | Definition: |
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| :<math>A:=U-TS</math> | | :<math>\left.A\right.=U-TS</math> |
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| where ''U'' is the [[internal energy]], ''T'' is the [[temperature]] and ''S'' is the [[Entropy|entropy]].
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| ''(TS)'' is a ''conjugate pair''. The differential of this function is | | ''(TS)'' is a ''conjugate pair''. The differential of this function is |
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| :<math>\left.dA\right.=dU-TdS-SdT</math> | | :<math>\left.dA\right.=dU-TdS-SdT</math> |
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| From the [[Second law of thermodynamics | second law of thermodynamics]] one obtains | | From the [[Second law of thermodynamics]] one obtains |
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| :<math>\left.dA\right.=TdS -pdV -TdS-SdT</math> | | :<math>\left.dA\right.=TdS -pdV -TdS-SdT</math> |
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| thus one arrives at | | thus one arrives at |
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| :<math>\left.dA\right.=-pdV-SdT</math>. | | :<math>\left.dA\right.=-pdV-SdT</math> |
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| For ''A(T,V)'' one has the following ''total differential''
| | leading finally to |
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| :<math>dA=\left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV</math>
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| The following equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and
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| [[Statistical mechanics | statistical mechanics]]:
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| :<math>\left.A\right.=-k_B T \ln Q_{NVT}</math> | | :<math>\left.A\right.=-k_B T \ln Q_{NVT}</math> |
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| 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]].
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| ==Ideal gas==
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| :''Main article: [[Ideal gas Helmholtz energy function]]''
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| ==Quantum correction==
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| 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>):
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| :<math>\frac{A-A_{ {\mathrm{classical}} }}{N} = \frac{\hbar^2}{24m(k_BT)^2} \langle F^2 \rangle </math>
| | For ''A(T,V)'' one has the following ''total differential'' |
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| where <math>\langle F^2 \rangle</math> is the mean squared force on any one atom due to all the other atoms.
| | :<math>dA=\left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV</math> |
| ==See also==
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| *[[Canonical ensemble]]
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| *[[Grand canonical ensemble]]
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| *[[Computing the Helmholtz energy function of solids]]
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| ==References==
| | Good for use in the [[Canonical ensemble]]. |
| <references/>
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| [[Category: Classical thermodynamics]] | |