where α(T) is the thermal expansivity ambient P as a function of T. By assuming a constant thermal expansivity α0, we have:
Or by assuming that α is a linear function of T, we have:
where α0 is α at ambient P and T, and α1 is the first T derivative of α at ambient P. Then, the system is compressed from V1 to V3 to increase P from P0 to P3. By using BM2-EOS, we have:
By using BM3-EOS, we have:
By combining Eq (2) or (3) and Eqs (5) and (6), we have high-temperature 3rd–order Brich-Murnaghan EOS. Note that K'T,0 is assumed to be independent from T because of experimental difficulty.
It is noted that experimental data along this path (Path HC) are difficult to obtain than along Path CH, because a matter tends to melt at high T corresponding to the mantle at ambient P. In practice, α(T) and KT,0(T) are estimated by extrapolating experimental data obtained at high T but lower than the mantle T at ambient P.