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The minerals brucite (Mg(OH)₂) and periclase (MgO) are related by the reaction:

Which side of this reaction represents the stable phase assemblage at 600°C and 200 MPa?

Answer: We learned how to solve this sort of problem in Chapter 2: the side with the lowest Gibbs free energy will be the stable assemblage. Hence, we need only to calculate ΔG_r at 600°C and 200 MPa. To do so, we use eqn. 2.130:

(2.130)

Our earlier examples dealt with solids, which are incompressible to a good approximation so we could simply treat ΔV_r as independent of pressure. In that case, the solution to the first integral on the left was simply ΔV_r(P′–P_ref). The reaction in this case, like most metamorphic reactions, involves H₂O, which is certainly not incompressible: its volume as steam or a supercritical fluid is very much a function of pressure. Let's isolate the difficulty by dividing ΔV_r into two parts: the volume change of reaction due to the solids, in this case the difference between molar volumes of periclase and brucite, and the volume change due to H₂O. We will denote the former as ΔV^S and assume that it is independent of pressure. The second integral in eqn. 2.132 then becomes:

(3.49)

How do we solve the pressure integral above? One approach is to assume that H₂O is an ideal gas.

For an ideal gas:

so that the pressure integral becomes:

Steam is a very nonideal gas, so this approach would not yield a very accurate answer. The concept of fugacity provides us with an alternative solution. For a nonideal substance, fugacity bears the same relationship to volume as the pressure of an ideal gas. Hence, we may substitute fugacity for pressure so that the pressure integral in eqn. 2.130 becomes:

where we take the reference fugacity to be 0.1 MPa. Equation 3.49 thus becomes:

(3.50)

We can then compute fugacity using eqn. 3.44 and the fugacity coefficients in Table 3.1.

Using the data in Table 2.2 and solving the temperature integral in 2.130 as usual (eqn. 2.132), we calculate the ΔG_T,P is 3.29 kJ. As it is positive, the left side of the reaction, i.e., brucite, is stable.

The ΔS of this reaction is positive, however, implying that at some temperature, periclase plus water will eventually replace brucite. To calculate the actual temperature of the phase boundary requires a trial and error approach: for a given pressure, we must first guess a temperature, then look up a value of φ in Table 2.1 (interpolating as necessary), and calculate ΔG_r. Depending on our answer, we make a revised guess of T and repeat the process until ΔG is 0. Using a spreadsheet, however, this goes fairly quickly. Using this method, we calculate that brucite breaks down at 660°C at 200 MPa, in excellent agreement with experimental observations.