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At 25°C the graphite–diamond transition occurs at 1600 MPa (megapascals, 1 MPa =10 b). Using the standard state (298 K, 0.1 MPa) data below, predict the pressure at which the transformation occurs when temperature is 1000°C.

	Graphite	Diamond
α (K–1)	1.05 × 10−05	7.50 × 10−06
β (MPa−1)	3.08 × 10−05	2.27 × 10−06
S° (J/K-mol)	5.74	2.38
V (cm3/mol)	5.2982	3.417

Answer: We can use the Clapeyron equation to determine the slope of the phase boundary. Then, assuming that ΔS and ΔV are independent of temperature, we can extrapolate this slope to 1000°C to find the pressure of the phase transition at that temperature.

First, we calculate the volumes of graphite and diamond at 1600 MPa as (eqn. 2.140):

(3.5)

where ΔP is the difference between the pressure of interest (1600 MPa in this case) and the reference pressure (0.1 MPa). Doing so, we find the molar volumes to be 5.037 for graphite and 3.405 for diamond, so ΔV_r is −1.6325 cc/mol. The next step will be to calculate ΔS at 1600 MPa. The pressure dependence of entropy is given by equation 2.106: (∂S/∂P)_T = −αV. Thus, to determine the effect of pressure we integrate:

(3.6)

(We use S_p to indicate the entropy at the pressure of interest and S° the entropy at the reference pressure.) We need to express V as a function of pressure, so we substitute eqn. 3.5 into 3.6:

(3.7)

The reference pressure, P_ref, is negligible compared with P₁ (0.1 MPa vs 1600 MPa), so that this simplifies to:

For graphite, S_p is 5.66 J/K-mol, for diamond it is 2.34 J/K-mol, so ΔS_r at 1600 MPa is −3.32 J-K⁻¹-mol⁻¹.

The Clapeyron slope is therefore:

One distinct advantage of the SI units is that cm³ = J/MPa, so the above units are equivalent to K/MPa. From this, the pressure of the phase change at 1000°C can be calculated as:

The Clapeyron slope we calculated (solid line) is compared with the experimentally determined phase boundary in Figure 3.3. Our calculated phase boundary is linear whereas the experimental one is not. The curved nature of the observed phase boundary indicates ΔV and ΔS are pressure- and temperature-dependent. This is indeed the case, particularly for graphite. A more accurate estimate of the volume change requires that β be expressed as a function of pressure.

Figure 3.3 Comparison of the graphite–diamond phase boundary calculated from thermodynamic data and the Clapeyron slope (solid line) with the experimentally observed phase boundary (dashed line).