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Gibbs energy

## Description

Gibbs energy G [J] is exergy which cannot be created internally (subscript i), but in contrast to internal-energy (diU/dt = 0) is not conserved but is dissipated (diG/dt < 0) in irreversible energy transformations at constant temperature and (barometric) pressure, T,p. Exergy is available as work in reversible energy transformations (100 % efficiency), and can be partially conserved when the exergonic transformation is coupled to an endergonic transformation.

Abbreviation: G [J]

Reference: Energy Figure 8.5. Gibbs energy as a function of advancement of a transformation (0 = -1 A + 1 B) in a closed isothermal system at constant pressure, for μA° = μB° = 0 kJ·mol-1 (modified from Gnaiger 2020 BEC MitoPathways - see Footnote 1).

## Gibbs energy as a function of advancement

```Communicated by Gnaiger E 2022-10-19
```
In a transformation tr 0 = -1 A +1 B proceeding in a system with volume V at constant barometric pressure p, the Gibbs energy of reactants A and B is
```Eq. 1:  G = µA·nA + µB·nB [J]
```
A small change dtrG at constant chemical potentials µi is due to a small advancement of a transformation tr, in closed or open isothermal systems, exchanging heat in equilibrium with an external thermostat at constant temperature,
```Eq. 2:  dtrG = µA·dtrnA + µB·dtrnB [J]
```
where the advancement dtrξi (i = A or B) is
```Eq. 3:  dtrξi = dtrnA·νA-1 = dtrnB·νB-1 [mol]
```
The total change of Gibbs energy dG is the sum of all partial transformations, dG = ΣdtriG, where tri = 1 to N transformation types — not to be confused with the internal Gibbs energy change diG due to internal transformations (i) only.
The isomorphic force of transformation ΔtrFX is the derivative of exergy per advancement (Gibbs force, compare affinity of reaction),
```Eq. 4:  ΔtrFX = ∂G/∂trξX [J·mol-1]
```
Note that ∂G ≝ dtrG. Then inserting Eq. 2 and Eq. 3 into Eq. 4, the force of transformation is expressed as
```Eq. 5:  ΔtrFX = (µA·dtrnA + µB·dtrnB)/dtrξi [J·mol-1]
```
Using Eq. 3, Eq. 5 can be rewritten as
```Eq. 6:  ΔtrFX = µA·dtrnA/(dtrnA·νA-1) + µB·dtrnB/(dtrnB·νB-1) [J·mol-1]
```
This yields the force as the sum of stoichiometric potentials, summarized in Figure 8.5 (Chapter 8; Gnaiger 2020 BEC MitoPathways),
```Eq. 7:  ΔtrFX = µA·νA + µB·νB [J·mol-1]
```
In general,
```Eq. 8:  ΔtrFX = Σµi·νi = ΣFtri[J·mol-1]
```
It may arouse curiosity, why the sign of difference Δ is used in the symbol, whereas the equation suggest a sum Σ in contrast to a difference. This is best explained by the fact that in various conventional contexts — such as the classical formulation of the pmF — the stoichiometric numbers (-1 and +1) are omitted, which yields a difference Δ as an equivalence,
```Eq. 9:  ΔtrFX ≡ µB - µA [J·mol-1]
```
The conceptual importance of the stoichiometric numbers is emphasized by defining the term stoichiometric potential (Gnaiger 2020), analogous to combining dtrnA·νA-1 in the expression of advancement (Eq. 3; see Eqs. 7 and 8),
```Eq. 10:  Ftri = µi·νi [J·mol-1]
```
To get acquainted with the meaning of subscripts such as 'tr' used above, consult »Abbreviation of iconic symbols.

References

Footnote 1

The original Figure 8.5 shows ∂trξX = ∂trnXνX-1 instead of dtrξX = dtrnXνX-1. The formal inconsistency was pointed out by Marin Kuntic during the BEC tutorial-Living Communications: pmF to pmP (Schroecken 2022 Sep 30-Oct 03).

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Units
• Joule [J]; 1 J = 1 N·m = 1 V·C; 1 cal = 4.184 J
Fundamental relationships
» Energy
» Exergy
» Extensive quantity
Contrast
» Force
» Pressure
» Intensive quantity
Forms of energy
» Internal-energy dU
» Enthalpy dH
» Heat deQ
» Bound energy dB
Forms of exergy
» Helmholtz energy dA
» Gibbs energy dG
» Work deW
» Dissipated energy diD

MitoPedia concepts: Ergodynamics