Thermochemical and pressure-volume-temperature systematics of data on solids, examples: Tungsten and MgO

Abstract

Data systematization using the constraints from the equation

$$Cp = Cv + \alpha _P {}^2V_T K_T T$$

where C _p, C _v, α _p, K _T and V are respectively heat capacity at constant pressure, heat capacity at constant volume, isobaric thermal expansion, isothermal bulk modulus and molar volume, has been performed for tungsten and MgO. The data are

$$K_T (W) = 1E - 5/(3.1575E - 12 + 1.6E - 16T + 3.1E - 20T^2 )$$

$$\alpha _P (W) = 9.386E - 6 + 5.51E - 9T$$

$$C_P (W) = 24.1 + 3.872E - 3T - 12.42E - 7T^2 + 63.96E - 11T^3 - 89000T^{ - 2} $$

$$K_T (MgO) = 1/(0.59506E - 6 + 0.82334E - 10T + 0.32639E - 13T^2 + 0.10179E - 17T^3 $$

$$\alpha _P (MgO) = 0.3754E - 4 + 0.7907E - 8T - 0.7836/T^2 + 0.9148/T^3 $$

$$C_P (MgO) = 43.65 + 0.54303E - 2T - 0.16692E7T^{ - 2} + 0.32903E4T^{ - 1} - 5.34791E - 8T^2 $$

For the calculation of pressure-volume-temperature relation, a high temperature form of the Birch-Murnaghan equation is proposed

$$P = 3K_T (1 + 2f)^{5/2} (1 + 2\xi f)$$

Where

$$K_T = 1/(b_0 + b_1 T + b_2 T^2 + b_3 T^3 )$$

$$f = (1/2)\{ [V(1,T)/V(P,T)]^{2/3} - 1\} $$

$$\xi = ({3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4})[K'_0 + K'_1 \ln ({T \mathord{\left/ {\vphantom {T {300}}} \right. \kern-\nulldelimiterspace} {300}}) - 4]$$

where in turn

$$V(1,T) = V_0 [\exp (\int\limits_{300}^T {\alpha dT)]} $$

.

The temperature dependence of the pressure derivative of the bulk modulus (K′₁) is estimated by using the shock-wave data. For tungsten the data are K′₀ = 3.5434, K′₁ = 0.032; for MgO K′₀ = 4.17 and K′₁ = 0.1667.

For calculating the Gibbs free energy of a solid at high pressure and at temperatures beyond that of melting at 1 atmosphere, it is necessary to define a high-temperature reference state for the fictive solid.