In this path, *V* is first decreased from *V*0 to *V*3 at a constant *T* = *T*0 to increase *P* from *P*0 to *P*2 (Condition 2 in the above figures).

To express this pressure increase, we simply adopt BM3-EOS in this page.

(1)

Then, *T* is increased from *T*0 to *T*3 at constant *V* = *V*3 to further increase *P* from *P*2 to *P*3. The pressure increase
Δ*P*_{th} = *P*3 – *P*2 is expressed by

(2)

Δ*P*_{th} or the *T* derivative of *P* at constant *V*, (∂*P* /∂*P*)* _{V}* is called the thermal
pressure. The thermal pressure is equal to the product of thermal expansivity [Wiki] and isothermal bulk modulus [Wiki]:

(3)

By assuming than the thermal pressure is independent from *P* and *T*, we have,

(4)

As shown in a separate page, Δ*P*_{th} is related to the internal energy [Wiki] (*E*) increase by *T* increase at constant *V* (**thermal
energy** [Wiki], Δ*E*_{th}) by the
following equation:

(5)

where *γ*_{th} is the thermodynamic Grüneisen paramter [Wiki]. Eq. (5) is the **Mie-Grüneisen EOS** [Wiki]. The *T* derivative of *E* at constant *V* is the isobaric heat capacity [Wiki],
*C _{V}*:

(6)

Therefore, the thermal energy is expressed as

(7)

Based on the **Debye approximation** [Wiki], *E* is expressed by the following formula.

(8)

where *N*_{A} is the Avogadro number [Wiki], *k*_{B} is the Boltzmann constant [Wiki], and *θ*_{D} is the Debye temperature [Wiki].

Anyway, the final *P* = *P*3 is express the sum of *P* by compression (*P*2) and the thermal pressure (Δ*E*_{th}) as:

(9)