Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2022 / Paper 1 / 7

Section D: Thermodynamics.

For pure phases at the $1$ bar reference pressure, the equilibrium condition is ${\Delta }_r{G}^{\circ }={\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }=0$. ThusThe calculation uses the stipulated constant reaction enthalpy and entropy, with the fluid at its reference chemical fugacity.

Integrate the solids’ molar volume contribution from $P^{\circ }=1$ bar and use ideal-gas water chemical fugacity $f=P$. For pure solids,The logarithm is dimensionless, and $\Delta V_s=-2.28\mathrm{J}\mathrm{ba}{\mathrm{r}}^{-1}$ can be multiplied directly by pressure in bars. Therefore the dehydration equilibrium with ideal-gas fluid givesHence $T=1295.57$ K, or $1022.42^{\circ }$C. Do not also add a constant gas-volume pressure term: the water contribution has already been integrated through $RT\ln (P/P^{\circ })$. Ideal-gas behavior at $6$ kbar is a prescribed model approximation, not a realistic equation of state. The solid volume correction is distinct from it.

An activity model is required in addition to the two compositions. Use ideal multisite mineral activity: independent ideal Fe–Mg mixing on the three relevant sites per formula unit givesThe garnet/biotite activity ratio is $0.008/0.512=1/64$. All other solids are pure, so the dimensionless reaction quotient isWith the same pressure-corrected numerator as in part b,Diluting the magnesian product endmember more strongly than the reactant favors the forward dehydration and lowers its equilibrium temperature. The powers of three count Fe–Mg mixing sites, not three formula units of either mineral. Nonideal site interactions or site partitioning would change these activities. If ideal mixing of whole formula units rather than sites were intended, activities would instead be $0.8$ and $0.2$; composition alone does not select that different model. The calculation explicitly adopts the conventional ideal-site interpretation.

With water fugacity coefficient ${\varphi }_f=0.8$, use $f={\varphi }_{f}P=4800$ bar. Retain the same ideal mineral-site activities, givingHenceThe lower fluid chemical fugacity stabilizes the water-bearing products of the forward dehydration relative to the ideal-gas calculation and lowers the required temperature by about $13.19$ K. Here the stated fugacity coefficient is treated as a given constant for evaluating the equilibrium.