# 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

## 1 Answer

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)!

Now impose a constraint on your system regarding the pressure and the temperature. Reaching the new equilibrium will force your system to go through temperature changes and volume changes until it reaches equilibrium so your system can't heat up anything (due to the reach of equilibirum temperature with heat bath) and you cannot extract mechanical work (due to the reach of equilibrium with pressure bath). What you can extract though is chemical work through electricity (non expansion work).

You actually could already do this at constant volume though so you see that in the first case, you can still extract chemical and mechanical work (maximum possible work in your book), defined by Helmoltz Free energy, but in the second case, as in the equilibrium your system as already given it's mechanical work in order to reach equilibrium, you can only extract chemical work (non expansion work) represented by Gibbs free energy. That's why the later is often used to describe chemical reactions at constant temperature and volume.

I hope this helps.