Helmholtz energy function

Hermann Ludwig Ferdinand von Helmholtz Definition of A (for arbeit):

$A=U-TS$

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

$dA=dU-TdS-SdT$

From the Second law of thermodynamics one obtains

$dA=TdS-pdV-TdS-SdT$

thus one arrives at

$dA=-pdV-SdT$

leading finally to

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

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

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

Good for use in the Canonical ensemble.