2.8.) For liquids and particularly gases, the effects of pressure and temperature on ΔV are significant and cannot be ignored. The reference pressure is generally 0.1 MPa. For solids, however, we can often ignore the effects of temperature and pressure on ΔV so the first integral reduces to: This is the change in entropy due to increasing the temperature from the reference state to T. The full change in entropy of reaction is then this plus the entropy change at the reference temperature:
(2.131)
Substituting this into 2.130, the second integral becomes:
(2.132)
Example 2.8 Predicting the equilibrium pressure of a mineral assemblage
Using the thermodynamic reaction and data as in Example 2.7:
determine the pressure at which these two assemblages will be in equilibrium at 1000°C. Assume that the volume change of the reaction is independent of pressure and temperature (i.e., α and β = 0).
Answer: These two assemblages will be in equilibrium if and only if the Gibbs free energy of reaction is 0. Mathematically, our problem is to solve eqn. 2.130 for P such that
Our first step is to find ΔGr for this reaction at 1000°C (1273 K) using eqn. 2.130. Heat capacity data in Table 2.2 is in the form:
(2.133)
Performing the double integral and collecting terms, and letting
(2.134)
equation 2.134 is a general solution to eqn. 2.130 when the Maier-Kelley heat capacity is used.
We found
Since we may assume the phases are incompressible, the solution to the pressure integral is:
(2.135)
Equation 2.130 may now be written as:
Let
Solving for pressure, we have
(2.136)
With
The transformation from “plagioclase peridotite” to “spinel peridotite” actually occurs around 1.0 GPa in the mantle. The difference between our result and the real world primarily reflects differences in mineral composition: mantle forsterite, enstatite and diopside are solid solutions containing Fe and other elements. The difference does not reflect our assumption that the volume change is independent of pressure. When available data for pressure and temperature dependence of the volume change are included in the solution, the pressure obtained is only marginally different: 1.54 GPa.
Example 2.9 Volume and free energy changes for finite compressibility
The compressibility (β) of forsterite (Mg2SiO4) is 8.33 × 10–6 MPa−1. Using this and the data given in Table 2.2, what is the change in
