## Thermal equation of state

An equation of state that includes not only the relation between P and V but also T is called the thermal equation of state.Thermal equations of state of mantle minerals are essential because the Earth's interior is under not only high-pressure but also high temperature conditions. The temperatures at the top of the asthenosphere, the 660-km discontinuity and the core-mantle boundary are considered to be 1700, 2000 [Katsura et al., 2010] and 4000~5000 K [Hernlund et al. 2005; Nomura et al, 2014].

There are two paths to reach from ambient conditions (T = T0, V = V0 and P = P0 = 0) to target conditions (T = T3, V = V3 and P = P3). One path is that T is first increased from T0 to T3 at P = P0 = 0 (Condition 1 in Fig. 1, 2 and 3) to increase V from V0 to V1 due to thermal expansion, and then V is decreased from V1 to V3 at constant T = T3 to increase P from P0 = 0 to P3. This path is called Path HC (first heating and then compression) here. The other path is that V is first decreased from V0 to V3 at a constant T = T0 (Condition 2 in Fig. 1, 2 and 3) to increase P from P0 to P2, and then T is increased from T0 to T3 at constant V = V3 to further increase P from P2 to P3. This path is called Path CH (first compression and then heating) here. We will discuss Path HC and Path CH in separate pages.

Fig. 1. Path HC and Path CH in a V-T space.

In Path HC, the system first goes to the condition 1 (V1, T3, P0) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3). In Path CH, the system first goes to the condition 2 (V3, T0 P2) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3).

Pig. 2. Path HC and Path CH in a P-V space.

In Path HC, the system first goes to the condition 1 (V1, T3, P0) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3). In Path CH, the system first goes to the condition 2 (V3, T0 P2) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3).

Fig. 3. Path HC and Path CH in a P-T space.

In Path HC, the system first goes to the condition 1 (V1, T3, P0) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3). In Path CH, the system first goes to the condition 2 (V3, T0 P2) from the initial condition (V0, T0, P0), and then to the final condition (V3, T3, P3).