On the use of thermal equations of state and the extrapolation at high temperature and pressure for geophysical and petrological applications

Abstract

The equation that relates pressure, temperature and volume and is described by parameters that are function of temperature at 1 bar (hereafter called thermal equation of state, TEOS), has practical computational advantages for petrological and geophysical applications over the equation that considers explicitly a thermal pressure. Some considerations that justify the use of the TEOS are discussed here. (1) The assumption that the parameters are function of temperature is perhaps better understood by looking at the Helmholtz energy function that is implicitly assumed in the case of an equation of state (EOS) derived from interatomic potentials. A test case shows that the Helmholtz energy related to the Vinet EOS and the Helmholtz energy from the Debye model are very similar. (2) The TEOS should be able to reproduce thermal expansion (α), isothermal bulk modulus (KT) and heat capacity (Cp and Cv) at high P, T computed from a lattice vibration model. The generalized Rydberg EOS applied to MgO is able to fit reasonably well the properties computed using Jacobs' lattice dynamics formulation (T range = 300-3000 K, P range = 1 bar-1500 kbar). (3) It is shown that in the case of MgO, the TEOS can be used quite successfully for extrapolation that goes beyond the P, T range of the measured/given data. Some physical constraints need to be applied to the derivation of the volume, bulk modulus and derivative of the bulk modulus with pressure at 1 bar. (4) The pressure dependence of the reference parameters in the TEOS that was inferred several decades ago is only apparent. A numerical computation demonstrates that the combined pressure effect in the terms defining the partial derivative of the reference V and K (and K') over temperature cancels out, making the reference parameters independent of pressure at any condition.