Hermann Ludwig Ferdinand von Helmholtz
Definition:
![{\displaystyle \left.A\right.=U-TS}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f15813977ba06c5a8bd39d23b2d20f5956462cd7)
(TS) is a conjugate pair. The differential of this function is
![{\displaystyle \left.dA\right.=dU-TdS-SdT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a80973203f702857e6c95eefa853790cda050f)
From the Second law of thermodynamics one obtains
![{\displaystyle \left.dA\right.=TdS-pdV-TdS-SdT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd956a7763d4b2bdacc331721729169d74852feb)
thus one arrives at
![{\displaystyle \left.dA\right.=-pdV-SdT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c7ad3e3c04cac366e2a026805aeebfe9f65b8)
leading finally to
![{\displaystyle \left.A\right.=-k_{B}T\ln Q_{NVT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02b79d6d95deaa5891e56df3c445e303e1c1ea54)
For A(T,V) one has the following total differential
![{\displaystyle dA=\left({\frac {\partial A}{\partial T}}\right)_{V}dT+\left({\frac {\partial A}{\partial V}}\right)_{T}dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/720b12677dd53511517d647bdbbb60f55ea28ed8)
Good for use in the Canonical ensemble.