Skip to main content
Library homepage
 
Loading table of contents menu...
Chemistry LibreTexts

Helmholtz (Free) Energy

  • Page ID
    1954
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz) \(A\) (for arbeit):

    \[\left.A\right.=U-TS\]

    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

    \[dA=\left(\dfrac{\partial A}{\partial T}\right)_V dT + \left(\dfrac{\partial A}{\partial V}\right)_T dV\]

    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.

    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]):

    \[\dfrac{A-A_{ {\mathrm{classical}} }}{N} = \dfrac{\hbar^2}{24m(k_BT)^2} \langle F^2 \rangle \]

    where \(\langle F^2 \rangle\) is the mean squared force on any one atom due to all the other atoms.

    Contributors and Attributions


    Helmholtz (Free) Energy is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?