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 $${Maximum work } = \, - Δ {G}_{{{rxn}}} = \, - ( {Δ {H}_{{{rxn}}} {-}{ T}Δ {S}_{{{rxn}}} } ) { constant T},{P}$$ Maximum work = - Δ G rxn = - Δ H rxn - T Δ S rxn constant T , 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 d( Δ G_rxn )/dT = - Δ S_rxn ( constant P ) d Δ G rxn / dT = - Δ S rxn constant P 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.