Gibbs Function as a Condition for Spontaneous Reaction

Decrease in the Gibbs Function as a Condition for Spontaneous Reaction  

The expression U + PV naturally arises in thermodynamics and is called the enthalpy, H thus, H = U + PV. The entire expression U - TS + PV, which was shown to naturally develop by this argument, is called the Gibbs free energy, G thus, G = U + PV - TS = H - TS. Thus  

This is true for a reversible process (essentially at equilibrium) carried out at constant T and P. Therefore, under these conditions, the maximum work over and above that associated with the volume change is given by the decrease in the Gibbs free energy (GFE). 

for a given increment of a process, dU is the same whether it is brought about reversibly or irreversibly  

The conclusion is that more heat is rejected by the system and hence absorbed by the surroundings in the irreversible case. Specifically, the magnitude of the extra heat is that of the work w r, which could have been realized in a reversible process. Hence, since -dG = w r, (at constant T and P), -dG is the energy available from the process and represents either useful work if the process is permitted to occur reversibly or extra heat rejected to the environment (which, importantly, can never be used isothermally to do the work otherwise possible). Since this energy, 

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