## Equation of state

### Why "equation of state"?

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.