Deciphering the physical meaning of Gibbs’s maximum work equation

J. Willard Gibbs derived the following equation to quantify the maximum work possible for a chemical reaction

\({\text{Maximum work }} = \, - \Delta {\text{G}}_{{{\text{rxn}}}} = \, - \left( {\Delta {\text{H}}_{{{\text{rxn}}}} {-}{\text{ T}}\Delta {\text{S}}_{{{\text{rxn}}}} } \right) {\text{ constant T}},{\text{P}}\)

∆H_rxn is the enthalpy change of reaction as measured in a reaction calorimeter and ∆G_rxn the change in Gibbs energy as measured, if feasible, in an electrochemical cell by the voltage across the two half-cells. To Gibbs, reaction spontaneity corresponds to negative values of ∆G_rxn. But what is T∆S_rxn, absolute temperature times the change in entropy? Gibbs stated that this term quantifies the heating/cooling required to maintain constant temperature in an electrochemical cell. Seeking a deeper explanation than this, one involving the behaviors of atoms and molecules that cause these thermodynamic phenomena, I employed an “atoms first” approach to decipher the physical underpinning of T∆S_rxn and, in so doing, developed the hypothesis that this term quantifies the change in “structural energy” of the system during a chemical reaction. This hypothesis now challenges me to similarly explain the physical underpinning of the Gibbs–Helmholtz equation

\({\text{d}}\left( {\Delta {\text{G}}_{{{\text{rxn}}}} } \right)/{\text{dT}} = - \Delta {\text{S}}_{{{\text{rxn}}}} \left( {\text{constant P}} \right)\)

While this equation illustrates a relationship between ∆G_rxn and ∆S_rxn, I don’t understand how this is so, especially since orbital electron energies that I hypothesize are responsible for ∆G_rxn are not directly involved in the entropy determination of atoms and molecules that are responsible for ∆S_rxn. I write this paper to both share my progress and also to seek help from any who can clarify this for me.