Helmholtz energy function

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

$A:=U-TS$

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

$dA=dU-TdS-SdT$

From the second law of thermodynamics one obtains

$dA=TdS-pdV-TdS-SdT$

thus one arrives at

$dA=-pdV-SdT$.

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

$dA={(\frac{\partial A}{\partial T})}_{V}dT+{(\frac{\partial A}{\partial V})}_{T}dV$

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

$A=-k_{B}T\ln Q_{NVT}$

where $k_B$ is the Boltzmann constant, T is the temperature, and $Q_{NVT}$ 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 ):

$\frac{A-A_{\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}}{N}=\frac{{\hslash }^2}{24m(k_{B}T{)}^2}⟨F^2⟩$

where $⟨F^2⟩$ is the mean squared force on any one atom due to all the other atoms.

See also

• Canonical ensemble
• Grand canonical ensemble
• Computing the Helmholtz energy function of solids