In this path, *V* is increased from *V*0 to *V*1 by heating from *T*0 to *T*3 (Condition 1 in the above figures.) due to
thermal expansion [Wiki].

(1)

where *α*(*T*) is the thermal expansivity ambient *P* as a function of *T.* By assuming a constant
thermal expansivity *α*_{0}, we have:

(2)

Or by assuming that *α* is a linear function of *T*, we have:

(3)

where *α*_{0} is *α* at ambient *P* and *T*, and *α*_{1} is the first *T* derivative of *α* at ambient *P*. Then, the
system is compressed from *V*1 to *V*3 to increase *P* from *P*0 to *P*3. By using BM2-EOS, we have:

(4)

By using BM3-EOS, we have:

(5)

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:

(6)

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*.