In this path, V is increased from V0 to V1 by heating from T0 to T3 (Condition 1 in the above figures.) due to thermal expansion [Wiki].

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:

where K_T,0(T) is the isothermal bulk modulus [Wiki] at zero P as a function of T. For simplicity, K_T,0(T) is approximated as a linear function of T. Namely:

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 K_T,0(T) are estimated by extrapolating experimental data obtained at high T but lower than the mantle T at ambient P.