Helmholtz energy function

Hermann Ludwig Ferdinand von Helmholtz Definition:

$\left.A\right.=U-TS$

(TS) is a conjugate pair. The differential of this function is

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

but from the Second law equation 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

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

Good for $NVT$