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A new synchrotron X-ray diffraction study of chromium oxide Cr2O3 (eskolaite) with the corundum-type structure has been carried out in a Kawai-type multi-anvil apparatus to pressure of 15 GPa and temperatures of 1873 K. Fitting the Birch–Murnaghan equation of state (EoS) with the present data up to 15 GPa yielded: bulk modulus (K 0,T0), 206 ± 4 GPa; its pressure derivative K′0,T , 4.4 ± 0.8; (∂K 0,T /∂T) = ‒0.037 ± 0.006 GPa K‒1; a = 2.98 ± 0.14 × 10−5 K−1 and b = 0.47 ± 0.28 × 10‒8 K‒2, where α 0,T = a + bT is the volumetric thermal expansion coefficient. The thermal expansion of Cr2O3 was additionally measured at the high-temperature powder diffraction experiment at ambient pressure and α 0,T0 was determined to be 2.95 × 10−5 K−1. The results indicate that coefficient of the thermal expansion calculated from the EoS appeared to be high-precision because it is consistent with the data obtained at 1 atm. However, our results contradict α 0 value suggested by Rigby et al. (Brit Ceram Trans J 45:137–148, 1946) widely used in many physical and geological databases. Fitting the Mie–Gruneisen–Debye EoS with the present ambient and high-pressure data yielded the following parameters: K 0,T0 = 205 ± 3 GPa, K′0,T = 4.0, Gruneisen parameter (γ 0) = 1.42 ± 0.80, q = 1.82 ± 0.56. The thermoelastic parameters indicate that Cr2O3 undergoes near isotropic compression at room and high temperatures up to 15 GPa. Cr2O3 is shown to be stable in this pressure range and adopts the corundum-type structure. Using obtained thermoelastic parameters, we calculated the reaction boundary of knorringite formation from enstatite and eskolaite. The Clapeyron slope (with $${\text{d}}P/{\text{d}}T = - 0.014$$ GPa/K) was found to be consistent with experimental data.

A simplest equation within the framework of the Mie-Gruneisen–Einstein approach is considered. Pressure estimation values are presented that are derived by conventional arithmetic and algebraic calculations as a function of temperature and volume. The equation under consideration complies with the Mie-Gruneisen–Debye model at high temperature. Different versions of an equation of state (EoS) of MgO proposed by Speziale et al. (J Geophys Res 106B:515–528, 2001) as a pressure standard at high temperatures are subject to analyses. In the literature, at least four versions of Speziale et al. EoS of MgO are discussed; the discrepancy between them reaching a few GPa at T > 2,000 K and P > 100 GPa. Our analyses of these equations suggest that the volume dependence of the Debye temperature is accepted arbitrarily and does not agree with the definition of the Gruneisen parameter, γ = −(∂lnΘ/∂lnV) T . Pressure as a function of temperature and volume in the Mie-Gruneisen–Einstein approach or the Gao pressure calculator can be used to estimate true pressure at compression x = V/V 0