# Gibbs free energy, Helmholtz free energy and their contribution to expansion and non-expansion work

In the book "An Introduction to Thermal Physics" by Daniel Schroder, I got the following expressions Helmholtz free energy : F = U - TS and Gibbs Free energy : G = H - TS = U + PV - TS

The author explained the intuition behind Gibbs free energy the following way

I found in different places (Chemistry StackExchange, Wikipedia etc.) that Gibbs free energy is the capacity to do non-expansion work and Helmholtz free energy is the capacity to do both expansion work (pressure-volume work) and non-expansion work. But in the definition of Gibbs free energy there is a pressure-volume term which Helmholtz free energy does not have. Therefore, my intuition is that it should be the other way around. What am I missing here? I would really appreciate if anyone could help me with this.

• It isn't clear why you think that. – Chet Miller Aug 16 '20 at 11:51
• F = U - TS and G = U + PV - TS. So, G has a pressure-volume term (the + PV term). Therefore, it should include expansion work, I think. – incredible sulk Aug 16 '20 at 12:54
• Are you asking how to derive the equation for the maximum work in terms of G? – Chet Miller Aug 16 '20 at 15:36
• No. I was trying to understand the physical interpretation of these terms. – incredible sulk Aug 17 '20 at 5:48

The Gibbs free energy definition $$G=U+PV-TS$$ doesn't add an expansion term, it removes it. The internal energy $$U$$ is $$U=TS-PV+\Sigma_i \mu N_i$$, where $$\mu$$ is the chemical potential and $$N_i$$ is the amount of species $$i$$. Thus, $$G=\Sigma_i \mu N_i$$, which is why we also call $$\mu$$ the partial molar Gibbs free energy of species $$i$$. The process of defining $$G$$ thus strips away the factors associated with heating and expansion work.