None of the 40+ equations that have been proposed to describe material properties at the pressures of the Earth’s core and mantle have escaped serious criticism. In this paper, some basic algebraic and thermodynamic constraints are reviewed, with the conclusion that the next step should be a re-examination of the relationship between the dependence of the bulk modulus, K, on pressure, P, that is $K^{\prime }\equiv \mathrm{d}K/dP$ , and the normalized (dimensionless) pressure, $P/K$ . A linear relationship between $1/K^{\prime }$ and $P/K$ terminating at the infinite pressure asymptote, at which these quantities become equal, has been used for analysing properties at extreme pressure, but may be inadequate for calculations requiring precise derivatives of an equation of state. A new analysis indicates that $\mathrm{d}\left(1/K^{\prime }\right)/\mathrm{d}\left(P/K\right)$ increases with compression (or $P/K$ ), but there are, at present, no reliable equations representing this. Relationships between higher derivatives of K and the thermodynamic Grüneisen parameter offer the prospect of a resolution of the problem and hence a new generation of fundamentally-based equations of state. Although an earlier conclusion that a completely general ‘universal’ equation is not possible, in principle, is confirmed in this study, the fundamental relationships present strong constraints for the forms of other proposed equations.

A convenient summary of thermodynamic relationships is presented and applied to the properties of iron at high temperatures and pressure, with appeal to seismological data on the Earth’s core and the theory of the electronic structure of iron. The Grüneisen parameter is found to have some temperature dependence with a high temperature asymptote that is approached reasonably closely in the core. Numerical estimates are given of thermodynamic properties of iron or of core alloy at mid‐core conditions (243 GPa, 4500 K).