Helmholtz energy function

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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 [1]):

\frac{A-A_{ {\mathrm{classical}} }}{N} = \frac{\hbar^2}{24m(k_BT)^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

References