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Different derivation of G and F

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.

The physical meaning of free energy is highlighted in this video.

The free energies are “free energy”. Thermodynamic free energy is the amount of work that a system can perform. They are energies freely available for us to use! The Helmholtz free energy is the maximum work that can be obtained, and the Gibbs free energy is the maximum non-expansion work.

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Thermodynamics in Energy Engineering

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