Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2024 / Paper 3 / 9

At the one-bar standard state, pure phases have unit thermodynamic activities. The Gibbs energy of reaction balance is ${\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }=0$, givingThis uses the stipulated temperature-independent reaction enthalpy and entropy differences, taken per mole of the written reaction.

Apply the pressure correction to a mineral reaction to the solids and use the chemical fugacity of the water product. For pure solids and water vapour treated as an ideal gas,The gas contribution is already included by the logarithm and must not be counted again as an independent volume correction. With pressures in bars,

Adopt ideal independent mixing on crystallographic sites, with Mg and Fe sharing three positions per formula unit and the other endmember sites fixed. This gives the phlogopite and pyrope activitiesThe reaction quotient contains their ratio, since pyrope is a product and phlogopite a reactant. ConsequentlyThe site-mixing assumption is part of this numerical answer: compositions alone do not determine nonideal thermodynamic activities. More detailed activity coefficients or unequal site occupancies would need an additional model. The cube counts the three magnesium-bearing positions rather than three molecules of each mineral.

For fugacity coefficient $\varphi =0.75$, use $f=\varphi P=3375\mathrm{bar}$ for the water activity. The solid pressure correction still uses the actual pressure $4500\mathrm{bar}$. ThusLowering the water chemical potential favours the product side, which lowers the equilibrium temperature relative to part (c) under the stated constant-property model.