Gibbs energy

high-resolution terminology - matching measurements at high-resolution

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. Exergy is available as work in reversible energy transformations, 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 transformation in a closed isothermal system at constant pressure (modified from Gnaiger 2020 BEC MitoPathways).

Gibbs energy as a function of advancement

Communicated by Gnaiger E 2022-10-18

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 dG at constant chemical potentials µ_i is due to a small advancement of the transformation, 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 isomorphic force of transformation Δ_trF_X is the derivative of exergy per advancement,

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_trn_B·ν_B^-1) [J·mol^-1]

which 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 = ΣFtr_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 the difference,

Eq. 9:  Δ_trF_X ≡ µ_B - µ_A [J·mol^-1]

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

Eq. 10:  F_tri = µ_i·ν_i [J·mol^-1]

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