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 reaction

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

Maximum work

=

- Δ

G rxn

=

-

Δ

H rxn

-

T

Δ

S rxn

constant T

, P

∆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 equation

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

d

Δ

G rxn

/ dT = - Δ

S rxn

constant P

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.