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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 molar volume and Gibbs free energy of forsterite at 100 MPa and 298 K?

Answer: Let's deal with volume first. We want to know how the molar volume (43.79 cc/mol) changes as the pressure increases from the reference value (0.1 MPa) to 1 GPa. The compressibility is defined as:

(2.12)

So the change in volume for an incremental increase in pressure is given by:

(2.137)

To find the change in volume over a finite pressure interval, we rearrange and integrate:

Performing the integral, we have:

(2.138)

This may be rewritten as:

(2.139)

However, the value of P−P^o is of the order of 10^–2, and in this case, the approximation holds, so that eqn. 2.139 may be written as:

(2.140)

equation 2.140 is a general expression that expresses volume as a function of pressure when β is known, small, and is independent of temperature and pressure. Furthermore, in situations where P > P^o, this can be simplified to:

(2.141)

Using equation 2.141, we calculate a molar volume of 43.54 cc/mol (identical to the value obtained using eqn. 2.139). The volume change, ΔV, is 0.04 cc/mol.

The change in free energy with volume is given by:

so that the free energy change as a consequence of a finite change is pressure can be obtained by integrating:

Into this we may substitute eqn. 2.141:

(2.142)

Using eqn. 2.142 we calculate a value of ΔG of 4.37 kJ/mol.