Gibbs energy

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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 (d_iU/dt = 0) is not conserved but is dissipated (d_iG/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 μ_A° = μ_B° = 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 = µ_A·n_A + µ_B·n_B [J]

A small change d_trG at constant chemical potentials µ_i 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:  d_trG = µ_A·d_trn_A + µ_B·d_trn_B [J]

where the advancement d_trξ_i (i = A or B) is

Eq. 3:  d_trξ_i = d_trn_A·ν_A^-1 = d_trn_B·ν_B^-1 [mol]

The total change of Gibbs energy dG is the sum of all partial transformations, dG = Σd_{tr_i}G, where tr_i = 1 to N transformation types — not to be confused with the internal Gibbs energy change d_iG due to internal transformations (i) only.

The isomorphic force of transformation Δ_trF_X is the derivative of exergy per advancement (Gibbs force, compare affinity of reaction),

Eq. 4:  Δ_trF_X = ∂G/∂_trξ_X [J·mol^-1]

Note that ∂G ≝ d_trG. Then inserting Eq. 2 and Eq. 3 into Eq. 4, the force of transformation is expressed as

Eq. 5:  Δ_trF_X = (µ_A·d_trn_A + µ_B·d_trn_B)/d_trξ_i [J·mol^-1]

Using Eq. 3, Eq. 5 can be rewritten as

Eq. 6:  Δ_trF_X = µ_A·d_trn_A/(d_trn_A·ν_A^-1) + µ_B·d_trn_B/(d_trn_B·ν_B^-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:  Δ_trF_X = µ_A·ν_A + µ_B·ν_B [J·mol^-1]

In general,

Eq. 8:  Δ_trF_X = Σµ_i·ν_i = ΣF_{tr_i} [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:  Δ_trF_X ≡ µ_B - µ_A [J·mol^-1]

The conceptual importance of the stoichiometric numbers is emphasized by defining the term stoichiometric potential (Gnaiger 2020), analogous to combining d_trn_A·ν_A^-1 in the expression of advancement (Eq. 3; see Eqs. 7 and 8),

Eq. 10:  F_tri = µ_i·ν_i [J·mol^-1]

To get acquainted with the meaning of subscripts such as 'tr' used above, consult »Abbreviation of iconic symbols.

References

Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5th ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002

Footnote 1

The original Figure 8.5 shows ∂_trξ_X = ∂_trn_X∙ν_X^-1 instead of d_trξ_X = d_trn_X∙ν_X^-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).