Different derivation of G and F

So far, we interpret the free energies only as determinants for reversibility. Let’s look at free energies in a slightly different viewpoint. Actually, it is also related to reversibility, but more to the energy view point. This different derivation and interpretation of G & F are related to the reason why they are called the “free” energies. Let’s look at the first and second laws. The first law is dU equals to delta Q + delta W. The second law is delta Q equals to TdS + delta lost work. Insert the second law into the first law. Then dU becomes TdS + delta lost work + delta W. The last two things together is the reversible work.

So dU is now TdS + delta reversible work. So, the reversible work is dU - TdS. So the reversible work is dU - TdS from the combination of the 1st and second laws. Let’s restrict the conditions. First, at constant temperature. The reversible work delta W is delta U - T delta S. Since T is constant here, it can be written as delta (U-TS). Here we define U - TS as F, the Helmholts free energy. Then the reversible work delta W is delta F. So here, the physical meaning of Helmholtz free energy comes out. It is the reversible or maximum work at constant temperature. Let’s restrict the condition a bit more.

Second at constant temperature and pressure, Consider, the “useful” work delta W star, defined as the reversible work exclusive of P - V work. So the reversible work is the useful work - PdV. Rearrange it. Then the useful work is the reversible work + PdV. From the 1st and 2nd laws, delta W reversible is dU - TdS. So the useful work is dU - TdS + PdV. We are under constant temperature and pressure. So it can be written as d (U-TS+PV). Let’s write the useful work function here. Delta W star is d (U-TS+PV). Change the order in the parenthesis as d (U+PV-TS). U + PV is the enthalpy H by definition. So the useful work can be written as d (H-TS).

Here we can define the Gibbs free energy G as H - TS. The useful work delta W star is delta H - TS, thus delta G. So here, the physical meaning of Gibbs free energy is that it is the useful reversible work exclusive of PV work at constant temperature and pressure. F and G are thus “free energy” available to us for use! F is just the maximum work, and G is the energy that can be used other than P - V work.