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Description
Gibbs energy G [J] is exergy which cannot be created internally (subscript i), but in contrast to internal-energy (d_{i}U/dt = 0) is not conserved but is dissipated (d_{i}G/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
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_{tr}G 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_{tr}G = µ_{A}·d_{tr}n_{A} + µ_{B}·d_{tr}n_{B} [J]
- where the advancement d_{tr}ξ_{i} (i = A or B) is
Eq. 3: d_{tr}ξ_{i} = d_{tr}n_{A}·ν_{A}^{-1} = d_{tr}n_{B}·ν_{B}^{-1} [mol]
- The total change of Gibbs energy dG is the sum of all partial transformations, dG = Σd_{tri}G, where tr_{i} = 1 to N transformation types — not to be confused with the internal Gibbs energy change d_{i}G due to internal transformations (i) only.
- The isomorphic force of transformation Δ_{tr}F_{X} is the derivative of exergy per advancement (Gibbs force, compare affinity of reaction),
Eq. 4: Δ_{tr}F_{X} = ∂G/∂_{tr}ξ_{X} [J·mol^{-1}]
- Note that ∂G ≝ d_{tr}G. Then inserting Eq. 2 and Eq. 3 into Eq. 4, the force of transformation is expressed as
Eq. 5: Δ_{tr}F_{X} = (µ_{A}·d_{tr}n_{A} + µ_{B}·d_{tr}n_{B})/d_{tr}ξ_{i} [J·mol^{-1}]
- Using Eq. 3, Eq. 5 can be rewritten as
Eq. 6: Δ_{tr}F_{X} = µ_{A}·d_{tr}n_{A}/(d_{tr}n_{A}·ν_{A}^{-1}) + µ_{B}·d_{tr}n_{B}/(d_{tr}n_{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_{tr}F_{X} = µ_{A}·ν_{A} + µ_{B}·ν_{B} [J·mol^{-1}]
- In general,
Eq. 8: Δ_{tr}F_{X} = Σµ_{i}·ν_{i} = ΣF_{tri} [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_{tr}F_{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_{tr}n_{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 ∂_{tr}ξ_{X} = ∂_{tr}n_{X}∙ν_{X}^{-1} instead of d_{tr}ξ_{X} = d_{tr}n_{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).
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- Units
- Joule [J]; 1 J = 1 N·m = 1 V·C; 1 cal = 4.184 J
- Units
- Fundamental relationships
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- » Heat d_{e}Q
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- Forms of energy
- Forms of exergy
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- » Gibbs energy dG
- » Work d_{e}W
- » Dissipated energy d_{i}D
- Forms of exergy
MitoPedia concepts:
Ergodynamics