In this path, V is first decreased from V0 to V3 at a constant T = T0 to increase P from P0 to P2 (Condition 2 in the above figures).
To express this pressure increase, we simply adopt BM3-EOS in this page.
Then, T is increased from T0 to T3 at constant V = V3 to further increase P from P2 to P3. The pressure increase ΔPth = P3 – P2 is expressed by
By assuming than the thermal pressure is independent from P and T, we have,
Therefore, the thermal energy is expressed as
Based on the Debye approximation [Wiki], E is expressed by the following formula.
where NA is the Avogadro number [Wiki], kB is the Boltzmann constant [Wiki], and θD is the Debye temperature [Wiki].
Anyway, the final P = P3 is express the sum of P by compression (P2) and the thermal pressure (ΔEth) as: