The GCE model from (Grimm et al. 2026) is adapted from the original model in Schlichting & Young (2022) with the addition of carbon following Werlen et al. (2025b,a).

In total, the chemical network is defined with 19 linearly independent reactions involving 26 phase components.

Chemical reactions are allowed within the gas and silicate phases as well as between the metal-silicate-gas phases.

Reactions within the silicate phase are:
Na2SiO3,silicate ⇌Na2Osilicate + SiO2,silicate
(R1)
MgSiO3,silicate ⇌MgOsilicate + SiO2,silicate
(R2)
FeOsilicate + 1
2 Simetal ⇌Femetal + 1
(R3)
2 SiO2,silicate
FeSiO3,silicate ⇌FeOsilicate + SiO2,silicate
(R4) Reactions between the metal and the silicate phase include:
Ometal + 1
2 Simetal ⇌1
(R5)
2 SiO2,silicate
2 Hmetal ⇌H2,silicate
(R6)
Simetal + 2 H2Osilicate ⇌SiO2,silicate + 2 H2,silicate
(R7)
Cmetal + Ometal ⇌COsilicate (R8) Reactions among gas-phase species are:
COgas + 1
2 O2,gas ⇌CO2,gas
(R9)
CH4,gas + 1
2 O2,gas ⇌2 H2,gas + COgas
(R10)
H2gas + 1
2 O2,gas ⇌H2Ogas
(R11) Reactions representing magma ocean–atmosphere exchange are:
FeOsilicate ⇌Fegas + 1
(R12)
2 O2,gas
MgOsilicate ⇌Mggas + 1
(R13)
2 O2,gas
SiO2,silicate ⇌SiOgas + 1
(R14)
2 O2,gas
Na2Osilicate ⇌2 Nagas + 1
(R15)
2 O2,gas
H2,silicate ⇌H2,gas
(R16)
H2Osilicate ⇌H2Ogas (R17)
COsilicate ⇌COgas (R18)
CO2,silicate ⇌CO2,gas
(R19) Note that any additional reaction that can be formed as a linear combination of these basis reactions is also permitted.

The reaction space is therefore not limited to the expressions explicitly listed above.

Chemical equilibrium among the 19 reactions is determined by solving the following condition for the molar fractions xi of all participating species:
� ∆ˆG◦
�
�
rxn
νg ln(P/P ◦)
νi ln xi +
= 0, (A.1)
RT + �
g
i
where xi denotes the mole fraction of species i within its respective phase, and νi are the associated stoichiometric coefficients.

The term ∆ˆG◦ rxn represents the standard Gibbs free energy change of the reaction, R is the ideal gas constant, T is the temperature, P is the pressure at the atmosphere–magma ocean interface (AMOI), and P ◦is the reference pressure, set here to 1 bar.

The summation over g accounts for gas-phase species only, which introduces the explicit pressure dependence.

In addition, the system includes seven elemental mass balance constraints for H, C, O, Na, Si, Mg, and Fe, as well as three phase normalization conditions for the gas, silicate, and metal phases.

The complete system therefore comprises 29 coupled nonlinear equations.

These equations are solved simultaneously for the mole fractions xi,k of the 26 species and the total mole numbers Nk of the three phases.

The thermodynamic parameters underlying the chemical reaction network are adopted from the compilation presented in Schlichting & Young (2022).

In addition to their framework, we explicitly account for carbon dissolution into the metallic phase by implementing metal–silicate partition coefficients from Blanchard et al. (2022).

Hydrogen solubility in silicate melt is described using the revised formulation introduced by Werlen et al. (2026b), which combines the experimental constraints of Hirschmann (2012) with DFT–MD results from Gilmore & Stixrude (2026).

Deviations from ideal solution behavior are considered only within the metallic phase.