## Derivation of 2nd-order Birch-Murnaghan equation of state

In equations of state (EOS) at constant temperature, the pressure, P, is expressed by a function of the volume, V.

Since P is the V derivative of the Helmholtz free energy, F, at constant temperature:

(1)

F should be considered in order to derive the EOS.

In the case of the Burch-Murnaghan EOS, compression of matter is expressed by the finite strain, f:

(2)

Since the strain energy is proportional to squared displacement in the Fick's law, an increase in F by compression is expressed by the squared finite strain for the first approximation:

(3)

By substituting Eq. (3) into Eq. (1), we have,

(4)

As shown in another page, the parameter "a" is give by:

(5)

The V derivative of f is:

(6)

By substituting Eqs (5) and (6) into (4), we have

(7)

We obtain the 2nd-order Birch-Murnaghan equation of state from Eq. (7):

(8)

This equation has a formula of difference between the 7/3 and 5/3 powers of V0/V. This formula is originated from the formula of the finite strain. Therefore, appropriateness of definition of the finite strain is responsible for this EOS. Validity of this EOS should be examined by experimental data.

The explanation in this page was constructed by referring to P. 69-70 in Poirier . The explanation in Poirier  is not very easy to understand probably because of the limited space in the book. This paper and the page to derive the parameter "a" explain BM2-EOS in more detail.

Reference:

Poirier, J. P. Introduction to the physics of the Earth's Interior 2nd edition, Cambridge University Press, Cambridge, pp. 312, 2000.