The definition of the Helmoltz Free Energy is AUTS using the

The definition of the Helmoltz Free Energy is

A=U-TS

using the defintions of U show that

dA=-PdV-SdT

Equations I have: dU=TdS-PdV, deltaU=q-w, deltaU=0 at equilibrium, deltaU=q-PdeltaV, U=H-PV

Solution

we know that   dU=TdS-PdV

and   A=U-TS

so, differentiating the second equation, we get:

dA = dU - TdS - SdT = (dU - TdS) - SdT = -PdV - SdT