Editing Helmholtz energy function

[[Hermann Ludwig Ferdinand von Helmholtz]] 
Definition of A (for ''arbeit''):

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

where ''U'' is the [[internal energy]], ''T'' is the [[temperature]] and ''S'' is the [[Entropy|entropy]].
''(TS)'' is a ''conjugate pair''. The differential of this function is

:<math>\left.dA\right.=dU-TdS-SdT</math>

From the [[Second law of thermodynamics]]  one obtains

:<math>\left.dA\right.=TdS -pdV -TdS-SdT</math>

thus one arrives at

:<math>\left.dA\right.=-pdV-SdT</math>

leading finally to 

:<math>\left.A\right.=-k_B T \ln Q_{NVT}</math>


For ''A(T,V)'' one has the following ''total differential''

:<math>dA=\left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV</math>

[[Category: Classical thermodynamics]]
Good for use in the [[Canonical ensemble]].
@@ Line 1: / Line 1: @@
-'''Helmholtz energy function''' ([[Hermann Ludwig Ferdinand von Helmholtz]])
+[[Hermann Ludwig Ferdinand von Helmholtz]]
-Definition of <math>A</math> (for ''arbeit''):
+Definition of A (for ''arbeit''):
-:<math>A:=U-TS</math>
+:<math>\left.A\right.=U-TS</math>
 where ''U'' is the [[internal energy]], ''T'' is the [[temperature]] and ''S'' is the [[Entropy|entropy]].
@@ Line 9: / Line 9: @@
 :<math>\left.dA\right.=dU-TdS-SdT</math>
-From the [[Second law of thermodynamics | second law of thermodynamics]]  one obtains
+From the [[Second law of thermodynamics]]  one obtains
 :<math>\left.dA\right.=TdS -pdV -TdS-SdT</math>
@@ Line 15: / Line 15: @@
 thus one arrives at
-:<math>\left.dA\right.=-pdV-SdT</math>.
+:<math>\left.dA\right.=-pdV-SdT</math>
-For ''A(T,V)'' one has the following ''total differential''
+leading finally to
-:<math>dA=\left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV</math>
-The following equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and
-[[Statistical mechanics | statistical mechanics]]:
 :<math>\left.A\right.=-k_B T \ln Q_{NVT}</math>
-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>
+For ''A(T,V)'' one has the following ''total differential''
-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==
-*[[Canonical ensemble]]
-*[[Grand canonical ensemble]]
-*[[Computing the Helmholtz energy function of solids]]
-==References==
-<references/>
 [[Category: Classical thermodynamics]]
+Good for use in the [[Canonical ensemble]].