Question

Starting with the definition of Helmholtz Energy

A=U-TS

prove that the change in Helmholtz Energy for a process at constant temperature is the total work (expansion and non-expansion work).

Answer 1

Ans:

The Helmholtz free energy, A,

     A = U – TS

We know that,

dU = TdS – pdV + dw_other ---------- (1)

Now, by using the definition of Helmholtz free energy,

dA = - SdT – pdV + dw_other ---------------- (2)

Any process at constant temperature have,

dA_T = - pdV + dw_other = dw_{all, reversible} -------------- (3)

Hence, from the above equation we can say that the change in Helmholtz Energy for a process at constant temperature is the total work that is expansion and non-expansion work.