Appendix C: Chemical Potential and Fugacity

The chemical potential of O2 (\(\mu_{\text{O2}}\)) is related to oxygen fugacity (\(f_{\text{O2}}\)) by

(17)\[\log_{10} \frac{f_{\text{O2}}}{P_\text{ref}} = \frac{1}{\ln 10} \frac{\mu_{\text{O2}} - g_{\text{O2}}^{\circ}\!\left(P_\text{ref}, T\right)}{\text{R}\,T}\]

where \(g_{\text{O2}}^{\circ}\!\left(P_\text{ref},T\right)\) is the standard state Gibbs energy of O2 gas at reference pressure \(P_\text{ref}\) (typically 1 bar), \(\text{R}\) is the gas constant (8.314 J/mol·K), and \(T\) is the temperature (K) of interest. Because \(f_{\text{O2}}\) is dependent on thermodynamic convention, common practice is to report \(f_{\text{O2}}\) relative to a buffer such that

(18)\[\Delta_{\text{buffer}} = \log_{10} \left(\frac{f_{\text{O2}}}{f_{\text{buffer}}} \right) = \frac{1}{\ln 10} \frac{\mu_{\text{O2}} - g_{\text{O2,buffer}}\left(P, T\right)}{\text{R}\,T}\]

where \(g_{\text{O2,buffer}}\left(P, T\right)\) is the partial molar Gibbs energy of oxygen in the chosen buffer. Writing the buffering reaction in the general form

(19)\[\text{oxidized assemblage} = \text{reduced assemblage} + \text{O}_2\]

the buffered partial molar Gibbs energy of oxygen is

(20)\[g_{\text{O2,buffer}} \! \left(P, T\right) = g_{\text{oxidized assemblage}} \! \left(P, T\right) - g_{\text{reduced assemblage}} \! \left(P, T\right)\]

For example, for the quartz-fayalite-magnetite (QFM) buffer, the reaction is

(21)\[2 \, \text{mt} + 3 \, \text{q} = 3 \, \text{fa} + \text{O}_2\]

and the buffered partial molar Gibbs energy of oxygen is

(22)\[g_{\text{O2,QFM}} \! \left(P, T\right) = 3\,g_{\text{fa}}^{\circ}\!\left(P, T\right) - 2\,g_{\text{mt}}^{\circ}\!\left(P, T\right) - 3\,g_{\text{q}}^{\circ}\!\left(P, T\right)\]

Warning

It is bad practice to calculate \(\Delta_{\text{buffer}}\) values using tabulated \(f_{\text{O2}}\) values for buffers because the tabulated values may not be consistent with the thermodynamic data being used in Perple_X.

Example, calculation of oxygen fugacity and Delta QFM

From the NIL16 example, the best central model is:

  • \(P\) = 9690. bar

  • \(T\) = 1175.5 K

  • \(\mu_{\text{O2}}\) = -422705. J/mol

To evaluate f(O2) (Eq 17):

  1. using the Perple_X program FRENDLY with the hp633ver.dat thermodynamic data file:

    • \(g_{\text{O2}}^{\circ}\!\left(1\,\text{bar}, 1175.5\,\text{K}\right)\) = -203013. J/mol.

  2. Substituting this value into Eq 17 yields

    • \(\log_{10} \frac {f_{\text{O2}}} {P_\text{ref}}\) = -9.763

Warning

FRENDLY outputs \(g_i^\circ\) in units of kJ/mol, MC_fit outputs \(\mu_{i}\) in units of J/mol.

To evaluate Delta(QFM) (Eq 18):

  1. Although it is possible to evaluate Eq 22 in FRENDLY as the Gibbs energy of an unbalanced reaction, a useful short cut is to uncomment the ready-made make definition

    | QFM = 2 mt + 3 q  -3 fa         | qfm O2 buffer
| DQF(J/mol) = 0

    in the header of hp633ver.dat (uncommenting consists of deleting the “|” characters at the beginning of the lines and, to be safe, making the input left-justified). Once this has been done, FRENDLY yields

    • \(g_{\text{O2,QFM}} \! \left(P, T \right)\) = -488834.9 J/mol.

  2. Substituting this value into Eq 18 yields

    • \(\Delta_{\text{QFM}}\) = 2.939

Warning

Comment out the QFM buffer definition in the thermodynamic data file after use. Its prescence as a thermodynamic entity may affect free energy minimization calculations in MC_fit, MEEMUM, and VERTEX.