Hello Mr.G, great question! Thermodynamics can be vexing, and I had to return to some textbooks to answer this!

As you said, Gibbs free energy represents the maximum amount of reversible work that can be performed on or by a system. Gibbs free energy is relevant for systems at constant temperature and pressure, and can be written as
G = U - TS + PV
where G is Gibbs free energy, U is the internal energy, T is the temperature of the system, S is the entropy of the system, P is the pressure of the system, and V is the volume of the system. What we really care about when we are calculating maximum work is delta G (I will write dG), and the equation looks like this:
dG = dU - TdS + PdV.

Note here that the temperature and pressure are fixed for the system, and U, S, and V are allowed to vary.

As for enthalpy, H, one helpful way I have seen it derived is by considering the "effective work" performed on the system. The effective work is defined as the total work on the system, minus pressure-volume work, which is either done on the system by the environment or done by the system on the environment. The positive sign in the equation below comes from the cancellation of subtracting pressure-volume work, which by convention has a negative sign (work done by system).
dW' = dW - (-PdV) = dW + PdV
Using the first law (dU = dW + dQ), we can write
dW' = dU - dQ + PdV
and we can define dH = dU + PdV to get dW' = dH - dQ, or dH = dW' + dQ.

Here, enthalpy is defined (at constant pressure) as a mix of the heat transfer to/from the system and the effective work done on the system.

Now let's combine it with the dG equation!
dG = dU + PdV - TdS
dG = dH - TdS

Gibbs free energy is only valid for systems with constant T and P, so the dH term is really just dH = dW'.

So we see here that at constant temperature and pressure, the maximum amount of available work to the system is the effective work minus temperature times the change in entropy
dG = dH - TdS = dW' - TdS.

The sign of dG can be positive or negative depending on both the sign and value of dH and dS. For spontaneous processes, we need dG to be negative, which leaves us with these options:

dS > 0 and dH < 0: dG is always negative, and is less than dH
dS > 0 and dH > 0: dG is negative at high temperatures relative to dH (dG is negative when dH < TdS). dG will be less than dH unless dH = 0.
dS < 0 and dH < 0: dG is negative at low temperatures relative to dH (dG is negative when dH > TdS). dG can be more positive than dH at higher temperatures, and will eventually become positive when the temperature gets high enough.

It is in this third case, when dS < 0 and dH < 0, that dG can be greater than dH and even positive at high temperatures. In this circumstance, the process can be spontaneous or not depending on the system temperature.

I hope this helps! Good luck, Mr. G!

Dear Mr. G,

The universe will reach equilibrium when the entropy of the universe is maximized.

Entropy can be roughly thought of as a quantity that describes disorder, and is a function of the energy, the available volume, and the amount of matter in the universe.

Theoretically, if you knew the exact location and energy of every particle in the universe, you could use the entropy function to calculate the total entropy of the universe.

All real processes increase the entropy of the universe, so if you wanted to know whether a process would occur, you could calculate the entropy of the universe before and after the process. If you calculated that the entropy increased, you would know that the process would occur. So, for the universe, the relevant thermodynamic function is the entropy function, because it lets you know what processes are expected to happen.

However, when describing a small part of the universe--a system like an engine or a specific chemical reaction--it is more convenient to consider different thermodynamic quantities. To do this, thermodynamics splits the universe into two parts: the system, and the surroundings (meaning the rest of the universe).

If we consider a system at constant temperature and pressure, then the relevant thermodynamic function describing whether processes within the system will occur spontaneously is the Gibbs free energy, G = H-TS, where G, H, T, and S are all properties of the system. If the system is at equilibrium, G is minimized. Any process within the system that results in a decrease in G, and therefore a negative delta G, will spontaneously occur. When this happens, the Gibbs free energy decreases, and there is the opportunity for the system to do work on the rest of the universe.

If the process occurs infinitely slowly and reversibly (and no energy is lost as heat), then the amount of work that can be done is equal to the change in G of the system. For a real process, though, the amount of work is less than the change in Gibbs free energy, because a real process loses some of the energy.

A negative delta G for the system occurs as long as delta H - T delta S is negative, where H, T, and S are the enthalpy, temperature, and entropy of the system (not the universe). So, for a process in the system, any time the T delta S term in the equation is larger than the H term, which can occur if the process results in a large increase in entropy, or if the temperature is really high, then the delta G is negative and the process is spontaneous. Alternatively, if T delta S is negative but the delta H term is even more negative, then the process also occurs spontaneously. So, for a spontaneous process with a negative delta G, the delta H can be any value--even more negative, less negative, or positive--as long as the entropy change for the process corresponds to a value of delta S that makes the quantity delta H - T delta S a negative value.

In summary, the value of delta G for a process in a system depends on both the enthalpy AND the entropy of the process, and so both of these terms determine the amount of work that a process can do on the surroundings.