## Thermal EOS

### Mie-Grüneisen EOS

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).

(1)

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

(2)

ΔPth 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, ΔPth is related to the internal energy [Wiki] (E) increase by T increase at constant V (thermal energy [Wiki], ΔEth) by the following equation:

(5)

where γth 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], CV:

(6)

Therefore, the thermal energy is expressed as

(7)

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

(8)

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:

(9)