Estimation of $f\mathrm{O}{\phantom{X}}_2$$f\ce{O2}$ based on Fe-Mg exchange in olivine-orthopyroxene in magnetite buffered assemblages

This is based on the following equilibrium (see classic paper of Frost 1982, JPet)

$6\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4+\mathrm{O}{\phantom{X}}_2=3\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6+2\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4$$6\ce{Fe_2SiO4} + \ce{O2} = 3\ce{Fe2Si2O6} + 2\ce{Fe3O4}$ (1)

Oxygen fugacity can be expressed as follows

$\log f\mathrm{O}{\phantom{X}}_2=3\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\log a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}+\mathrm{\Delta }G{°}_1/(\ln \left(10\right)·RT)$$\log f\ce{O_2} = 3 \log a_{\ce{Fe2Si2O6}}^{opx} + 2 \log a_{\ce{Fe3O4}}^{mt} - 6\log a_{\ce{Fe2SiO4}}^{ol} + \Delta G°_1/ (\ln(10)·RT)$ [Eq. 1]

In order to derive this expression, let's start as

$\mathrm{\Delta }{G}^o+RT\ln K=0$$\Delta G^o + RT\ln K =0$

where

$K=\frac{(a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}{)}^3+(a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}{)}^2}{(a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}{)}^6+a_{\mathrm{O}{\phantom{X}}_2}^{fl}}$$K = \frac{(a_{\ce{Fe2Si2O6}}^{opx})^3 + (a_{\ce{Fe3O4}}^{mt})^2}{ (a_{\ce{Fe2SiO4}}^{ol})^6+a_{\ce{O_2}}^{fl}}$

and

$RT\ln K=3RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2RT\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}-RT\ln a_{\mathrm{O}{\phantom{X}}_2}$$RT\ln K = 3RT \ln a_{\ce{Fe2Si2O6}}^{opx} + 2RT \ln a_{\ce{Fe3O4}}^{mt} - 6RT\ln a_{\ce{Fe2SiO4}}^{ol} - RT\ln a_{\ce{O_2}}$

the activity of $\mathrm{O}{\phantom{X}}_2$$\ce{O_2}$ is

$\ln a_{\mathrm{O}{\phantom{X}}_2}=3\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}+\mathrm{\Delta }G{°}_1/RT$$\ln a_{\ce{O_2}} = 3 \ln a_{\ce{Fe2Si2O6}}^{opx} + 2 \ln a_{\ce{Fe3O4}}^{mt} - 6\ln a_{\ce{Fe2SiO4}}^{ol} + \Delta G°_1/RT$

but we want ${\log }_{10}$$\log_{10}$ units instead so,

$\log f\mathrm{O}{\phantom{X}}_2=3\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\log a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}+\mathrm{\Delta }G{°}_1/(\ln \left(10\right)·RT)$$\log f\ce{O_2} = 3 \log a_{\ce{Fe2Si2O6}}^{opx} + 2 \log a_{\ce{Fe3O4}}^{mt} - 6\log a_{\ce{Fe2SiO4}}^{ol} + \Delta G°_1/ (\ln(10)·RT)$ [Eq. 1]

Formal derivation

The above treatment is obscure in the definition of $\mathrm{\Delta }G{°}_1$$\Delta G°_1$ because it is calculated at the P,T conditions for all endmembers except for $\mathrm{O}{\phantom{X}}_2$$\ce{O2}$ that is computed at 1 bar. The derivation is as follows

$6\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4+\mathrm{O}{\phantom{X}}_2=3\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6+2\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4$$6\ce{Fe_2SiO4} + \ce{O2} = 3\ce{Fe2Si2O6} + 2\ce{Fe3O4}$ (1)

for this reaction in solid solution we have

$3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^o+RT\ln K=0$$3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o - \mu _{\ce{O2}}^o + RT\ln K = 0$

Which is strictly

$\mathrm{\Delta }{G}^o+RT\ln K=0$$\Delta G^o + RT\ln K =0$

then

$3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^o+3RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2RT\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}-RT\ln a_{\mathrm{O}{\phantom{X}}_2}$$3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o - \mu _{\ce{O2}}^o + 3RT \ln a_{\ce{Fe2Si2O6}}^{opx} + 2RT \ln a_{\ce{Fe3O4}}^{mt} - 6RT\ln a_{\ce{Fe2SiO4}}^{ol} - RT\ln a_{\ce{O_2}}$

but noting that

${\mu }_{\mathrm{O}{\phantom{X}}_2}={\mu }_{\mathrm{O}{\phantom{X}}_2}^o+RT\ln a_{\mathrm{O}{\phantom{X}}_2}$$\mu _{\ce{O2}} = \mu _{\ce{O2}}^o + RT\ln a_{\ce{O_2}}$

that simplifies to

$3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}+3RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2RT\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o - \mu _{\ce{O2}} + 3RT \ln a_{\ce{Fe2Si2O6}}^{opx} + 2RT \ln a_{\ce{Fe3O4}}^{mt} - 6RT\ln a_{\ce{Fe2SiO4}}^{ol}$

then

${\mu }_{\mathrm{O}{\phantom{X}}_2}=3g_{fs}^o+2g_{mt}^o-6g_{fa}^o+3RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2RT\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$\mu _{\ce{O2}} = 3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o + 3RT \ln a_{\ce{Fe2Si2O6}}^{opx} + 2RT \ln a_{\ce{Fe3O4}}^{mt} - 6RT\ln a_{\ce{Fe2SiO4}}^{ol}$

we normally report this value as $f_{\mathrm{O}{\phantom{X}}_2}$$f_{\ce{O2}}$, that by definition is

$f_{\mathrm{O}{\phantom{X}}_2}=\exp \left(\frac{{\mu }_{\mathrm{O}{\phantom{X}}_2}-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}}{RT}\right)$$f_{\ce{O2}} = \exp(\frac{\mu _{\ce{O2}}-\mu _{\ce{O2}}^{1bar}}{RT})$

$f_{\mathrm{O}{\phantom{X}}_2}=\exp \left(\frac{3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}+3RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2RT\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6RT\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}}{RT}\right)$$f_{\ce{O2}} = \exp(\frac{{3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o -\mu _{\ce{O2}}^{1bar} + 3RT \ln a_{\ce{Fe2Si2O6}}^{opx} + 2RT \ln a_{\ce{Fe3O4}}^{mt} - 6RT\ln a_{\ce{Fe2SiO4}}^{ol}}}{RT})$

$f_{\mathrm{O}{\phantom{X}}_2}=\exp \left(\frac{3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}}{RT}\right)+3a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$f_{\ce{O2}} = \exp(\frac{3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o -\mu _{\ce{O2}}^{1bar}}{RT}) + 3 a_{\ce{Fe2Si2O6}}^{opx} + 2a_{\ce{Fe3O4}}^{mt} - 6 a_{\ce{Fe2SiO4}}^{ol}$

or in $\ln$$\ln$ units

$\ln f_{\mathrm{O}{\phantom{X}}_2}=\frac{3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}}{RT}+3\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$\ln f_{\ce{O2}} = \frac{3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o -\mu _{\ce{O2}}^{1bar}}{RT} + 3\ln a_{\ce{Fe2Si2O6}}^{opx} + 2\ln a_{\ce{Fe3O4}}^{mt} - 6\ln a_{\ce{Fe2SiO4}}^{ol}$

and more frequently ($\ln x=lo{g}_{10}x\ln \left(10\right))$$\ln x = log_{10}x\ln(10))$

${\log }_{10}{f}_{\mathrm{O}{\phantom{X}}_2}\ln \left(10\right)=\frac{3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}}{RT}+3\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\ln a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\ln a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$\log_{10} f_{\ce{O2}} \ln(10) = \frac{3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o -\mu _{\ce{O2}}^{1bar}}{RT} + 3\ln a_{\ce{Fe2Si2O6}}^{opx} + 2\ln a_{\ce{Fe3O4}}^{mt} - 6\ln a_{\ce{Fe2SiO4}}^{ol}$

${\log }_{10}{f}_{\mathrm{O}{\phantom{X}}_2}=\frac{3g_{fs}^o+2g_{mt}^o-6g_{fa}^o-{\mu }_{\mathrm{O}{\phantom{X}}_2}^{1bar}}{\ln \left(10\right)RT}+3\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{Si}{\phantom{X}}_2\mathrm{O}{\phantom{X}}_6}^{opx}+2\log a_{\mathrm{Fe}{\phantom{X}}_3\mathrm{O}{\phantom{X}}_4}^{mt}-6\log a_{\mathrm{Fe}{\phantom{X}}_2\mathrm{SiO}{\phantom{X}}_4}^{ol}$$\log_{10} f_{\ce{O2}} = \frac{3 g_{fs}^o + 2g_{mt}^o - 6g_{fa}^o -\mu _{\ce{O2}}^{1bar}}{\ln(10)RT} + 3\log a_{\ce{Fe2Si2O6}}^{opx} + 2\log a_{\ce{Fe3O4}}^{mt} - 6\log a_{\ce{Fe2SiO4}}^{ol}$

which is Eq.1