Helmholtz energy function: Difference between revisions
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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]]. | 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]] | ||
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==References== | ==References== | ||
<references/> | |||
[[Category: Classical thermodynamics]] | [[Category: Classical thermodynamics]] | ||
Revision as of 16:46, 21 March 2012
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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.dA\right.=-pdV-SdT} .
For A(T,V) one has the following total differential
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA=\left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV}
The following equation provides a link between classical thermodynamics and statistical mechanics:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.A\right.=-k_B T \ln Q_{NVT}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, T is the temperature, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{NVT}} is the 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 [1]):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A-A_{ {\mathrm{classical}} }}{N} = \frac{\hbar^2}{24m(k_BT)^2} \langle F^2 \rangle }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle F^2 \rangle} is the mean squared force on any one atom due to all the other atoms.