Deciphering the physical meaning of Gibbs’s maximum work equation

Abstract

J. Willard Gibbs derived the following equation to quantify the maximum work possible for a chemical reactionMaximum work=-ΔGrxn=-ΔHrxn-TΔSrxnconstant T,P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{Maximum work }} = \\, - \\Delta {\ ext{G}}_{{{\ ext{rxn}}}} = \\, - \\left( {\\Delta {\ ext{H}}_{{{\ ext{rxn}}}} {-}{\ ext{ T}}\\Delta {\ ext{S}}_{{{\ ext{rxn}}}} } \\right) {\ ext{ constant T}},{\ ext{P}}$$\\end{document}∆Hrxn is the enthalpy change of reaction as measured in a reaction calorimeter and ∆Grxn 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 ∆Grxn. But what is T∆Srxn, 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∆Srxn 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 equationdΔGrxn/dT=-ΔSrxnconstant P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{d}}\\left( {\\Delta {\ ext{G}}_{{{\ ext{rxn}}}} } \\right)/{\ ext{dT}} = - \\Delta {\ ext{S}}_{{{\ ext{rxn}}}} \\left( {\ ext{constant P}} \\right)$$\\end{document}While this equation illustrates a relationship between ∆Grxn and ∆Srxn, I don’t understand how this is so, especially since orbital electron energies that I hypothesize are responsible for ∆Grxn are not directly involved in the entropy determination of atoms and molecules that are responsible for ∆Srxn. I write this paper to both share my progress and also to seek help from any who can clarify this for me.