Internal energy Helmholtz energy

Question

Internal energy satisfies the equation U= TS - PV

While Helmholtz energy satisfies the equation F= U -TS

So can we write Helmholtz energy as F=PV ?

Why and why not?

Answer 1

You should write your equation in the differentual forms:

dU=TdS - PdV

F= U -TS so dF= dU -TdS -SdT



So: dF= -pdV - SdT

With your developement, we just arrive to dF= -pdV which is totally false, dW= - pdV