Although high *T* generally increases *V* of matters, compression by high *P* is more significant for considering the Earth's interior. Therefore, we first discuss
compression at a constant *T*, namely, the isothermal EOS.

In thermodynamics, pressure (*P*) is defined as volume (*V*) derivative of Helmholtz free energy (*F*). Namely,

(1)

In this page, we first enumerate isothermal EOS that appear in articles of solid geophysics.

The simplest EOS is obtained from the definition of the isothermal bulk modulus [Wiki], *K*_{T} as:

(2)

where *V*_{0} is the volume at *P* = 0 and *K*_{T,0} is the isothermal bulk modulus at *P* = 0. However, *K*_{T} is not constant but
increases with *P*. By assuming the constant increasing rate of *K*_{T} with *P*, we have the Murnaghan's
EOS:

(3)

The most frequently used EOS at constant *T* in Study of the Earth's Interior is the: 3^{rd}-order Birch-Murnaghan EOS
[Wiki]
(BM3-EOS):

(4)

The lower order of Birch-Murnaghan EOS, namely, 2^{nd}-order Birch-Murnaghan EOS [Wiki] (BM2-EOS) is:

(5)

BM3-EOS becomes identical to BM2-EOS when *K*_{T0}' = 4. The relation *K*_{T0}' = 4 is the case in many kinds of materials.

Recently, the Vinet EOS (V-EOS) is also used.

(6)

Comparison of these EOS's is discussed in a separate page.

The above EOS's are expressed using *K*_{T0} and *K*_{0}'. The reason for these formulas is that these two parameters can be estimated by elastic wave
velocity [Wiki] measurement. Although sometimes a higher order EOS, for example, 4^{th}-order
Birch-Murnaghan EOS is referred, it is impractical, because the 2^{nd} *P* derivative of *K*_{T} is extremely difficult to determine.

The main argument of EOS was made more than half century ago. At that time, pressure ranges of laboratory experiments were very limited (at best a couple of GPa). In those days, the Earth's
structure should be estimated based on such experimental data obtained at such pressures. The 2^{nd}-order Birch-Murnaghan EOS was
some kind of miracle, because it can estimate density of minerals if one has a value of bulk modulus at ambient pressure. These days, the DAC can reach the pressure at the Earth's center, and
density of Earth's constituents can be obtained directly. Therefore, **argument about EOS has become much less important**.