Helmholtz energy function

Hermann Ludwig Ferdinand von Helmholtz Definition of 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

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

leading finally to

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

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}

Good for use in the Canonical ensemble.