# Gibbs' free energy and Helmholtz free energy

This is a question I have from Atkins' physical chemistry, tenth edition, chapter 3c: Concentrating on the system.

I quote:

Criteria for spontaneous change: dA <=0 (const T,V) dG<=0 (const T,P) -Pg 132, eq: 3c.10

Summary of justification 3c.1 and 3c.2, pg 133 and 135 respectively:

The maximum non expansion work is given by Gibbs free energy while the maximum possible work is given by Helmholtz free energy.

I didn't get how Helmholtz free energy is the maximum possible work that can be done although we place a constraint of constant volume over it, but Gibbs free energy is the maximum non expansion work that can be done although the body is free to expand.

The proofs make perfect sense to me, but it doesn't seem to make sense when I think about it physically.

I mean, to do expansion work, volume will have to change, right?

• Helmholtz free energy is convenient to use for constant volume problem. And Gibbs free energy is convenient for constant pressure, e.g. chemical reaction, phase change. – user115350 Sep 7 '16 at 20:54

Consider a change of the energy of a system system, given the natural variable $S,V$ (entropy and volume): $$dE = \delta W + \delta Q$$ Where $W$ is the mechanical work (volume change) and $Q$ is the energy changes via heat flux.
We know that the mechanical work exchanged from the system with the environment is link with volume change and pressure i.e. $W(P,V)$ and the thermal exchanges are dependant on the entropy and temperature i.e. $Q(T,S)$.
Now impose a constraint on your system imposing constant temperature $T_0$. Obviously $\delta W=0$ during transformation your system will do in order to maximize it's Helmoltz free energy as the volums stays constant. Only heat exchanges will occur and $A$ will go to its maximum at equilibrium. What's left is the mechanical work you can extract from your system through volume changes (expansion work)!