The concept of the **3 ^{rd}-order Birch-Murnaghan equation of state** (BM3-EOS) is almost the same as that of the 2

Difference is that change in Helmholtz free energy, *F*, is expanded not to the 2^{nd}-order (BM2 Eq. 3) but to the
3^{rd}-order the finite strain *f* (BM2 Eq. 2):

(1)

By differentiating Eq. (1) by *V* with the relations between *P* and *F* (BM2 Eq. 1), we have *P* as:

(2)

According to the calculation shown in the separate page, the factor 3*b*/2*a* is:

(3)

By substituting the definition of the finite strain (BM2 Eq. 2), its volume derivative (BM2
Eq. 6), the parameter "*a*" (BM2 Eq. 5), and the parameter "3*b*/2*a*" (Eq. 3) into Eq. (2), we have:

(4)

By simplifying Eq. (4), we have BM3-EOS:

(5)

As well as BM2-EOS, BM3-EOS is obtained based on the definition of finite strain. Hence, appropriateness of the definition of the finite strain is responsible for this EOS.

BM3-EOS becomes identical to BM2-EOS if *K*_{T0}' = 4. Therefore, BM-EOS can be justified (or, is useful for our purpose), if the matters of our interest have
*K*_{T0}' ~ 4. As is shown in the table (not yet made), *K*_{T0}' of many mantle minerals are around 4.

Poirier [2000] provided an excellent explanation about the Birch-Murnaghan equation of state (p. 66-74, Section 4.3). I imagine, however, that many current high-pressure workers need a plainer explanation. Actually, it is not easy to derive the third-order Birch-Murnaghan equation of state only based on what is written in Poirier [2000].

I made the present page to provide a mathematically easier explanation by referring to this book and also by referring to Section 6.2 in Anderson [1995].

Reference:

Anderson, O. L., Equations of state of solids for geophysics and ceramic science, Oxford University Press, New Yourk, pp. 405, 1995

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