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

$\left.dA\right.=dU-TdS-SdT$

From the second law of thermodynamics one obtains

$\left.dA\right.=TdS-pdV-TdS-SdT$

thus one arrives at

$\left.dA\right.=-pdV-SdT$.

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

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

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

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

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 ):

${\frac {A-A_{\mathrm {classical} }}{N}}={\frac {\hbar ^{2}}{24m(k_{B}T)^{2}}}\langle F^{2}\rangle$

where $\langle F^{2}\rangle$ 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