Interiors of the Earth and planets are under high-pressure (*P*) and high-temperature (*T*). At HP, matters
are compressed, and have higher density (*ρ*), namely have smaller volumes (*V*) than ambient pressure. Except for some special matters such as water between 273 and 277 K,
*V* increases with increasing *T* due to thermal expansion. The *ρ*, which is a mass (*m*) divided by *V*, is one of an essential parameter to discuss mantle dynamics, because mantle convection [Wiki] is driven by heterogeneity of *ρ*. With a given *m*, one of these quantities (*P*, *V*,
*T*) are a function of the other two quantities. An equation of state (EOS) [Wiki] is a relation of these
parameters.

In thermodynamics, *P* is defined by a *V* derivative of Helmholtz free energy, [Wiki] *F*:

(1)

Since *P* is primarily a function of *V* as shown in Eq. (1), we first discuss equations of state at constant *T*, or isothermal equations of state, where we consider the increasing rate of *F* with decreasing *V*.