ABSTRACT
Equations of State (EOS) for many reference materials are needed not only for the realization of a reliable Practical Pressure Scale, but also for many applications in Geo-and Planetary-Sciences. More generally, the EOS of solids are related to all the other thermo-physical properties of solids, which implies, that a reliable formulation of EOS should be based on a complete thermodynamic modelling, which includes not only p(V, T ), but also all the other thermo-physical properties, as for instants the Free Energy, F (V, T ), the Internal Energy, U(V, T ), the Entropy, S(V, T ) and their partial derivatives, as for instance the (isothermal) Bulk Modulus, KT (p, T ) , Thermal Volume Expansion Coefficient or, in other words, the (thermal) Volume Expansivity, α(p, T ) , and the Heat Capacities, Cp(p, T ) and CV (p, T ). One may notice, that I have represented here these thermo-physical quantities either as functions of volume and temperature or as functions of pressure and temperature. The more convenient thermodynamic variable in applications is usually pressure, and volume is more convenient in theoretical modelling. In any case, the EOS p(V, T ) or its inverted form V (p, T ) , allow to interchange these variables. One must notice however, that F (V, T ) is only a thermodynamic Potential, which gives a complete definition of the thermodynamic system, if it is represented as a function of volume and temperature. In fact, the Free Energy has here a more fundamental meaning than any of the other thermo-physical quantities, because it is related directly to the quantum statistical modelling, which starts from a calculation of the energies for all the possible quantum
states of the solid, En(V ), and uses this information in the calculation of the Partition Function:
Z(V, T ) =< e−H/(kT ) >= Σe−En(V )/(kT )
The Free Energy is than given by the simple relation:
F (V, T ) = −kT ln(Z(V, T )) Before I discuss the usual approximations made in the calculation of the partition function (in section 3), I present in section 2 a review of the most commonly used Parametric EOS Forms, which represent with temperature dependent ”parameters” V0(T ), K0(T ), K
′ 0(T ), K
′′ 0 (T ) ... the pressure as a
function of volume or in reversed form the volume as a function of pressure. These ”parameters” denote the volume, the isothermal bulk modulus and its first and higher order isothermal pressure derivatives all for zero pressure. Mostly the difference of these values for zero and ambient pressure is much smaller the experimental uncertainty and therefore not explicitly noted.
