Here, we prove Eq. (3) in the 3rd-order Birch-Murnaghan equation of state (BM3-EOS). For simplicity, the parameter "3*b*/2*a*"
is expressed by "*m*" in this page. Our goal is therefore:

(1)

The strategy of proof of Eq. (1) is the same as that for the paramter "*a*" in BM2-EOS (BM2 Eq. 5). There are, however, two differences as follows.

1. The Helmholtz free energy is expanded to the 3^{rd} order of the finite strain *f*.

2. The *K*_{T} and *F* are differentiated by *V* to a higher order.

We refer equations in the page for the parameter "*a*" as (PA Eq. ??).

2^{nd} *V* derivative of *f*

The 1^{st} *V* derivative of *f* is given by PA Eq. (2).
This relation is further differentiated by *V*.

(2)

At *P* = 0, *V* = *V*_{0}. Hence we have

(3)

*K*_{T}' and 2^{nd} *V* derivative of *P*

The relations between *K*_{T} and *V* derivative of *P* are given as PA Eq. (4) - (7). Here we obtain relations
between the *P* derivative of *K*_{T}, *K*_{T}', and 2^{nd} *V* derivative of *P*.

From the definition, we have:

(4)

(5)

At *P* = 0, we have

(6)

*2 ^{nd}*

We differentiate BM3 Eq. (2) by *V* with replacing 3*b*/2*a* by *m*.

(7)

(8)

Eq. (8) is once again differentiated by *V*.

(9)

(10)

At *P* = 0, *f* = 0. We also substitute Eq. (3) and PA-Eq. (3) into Eq. (10).

(11)

Final procedure

By equating Eqs (6) and (11), and substituting BM2-Eq. (5) into it, we have:

(12)

By simplifying Eq. (12), we have Eq. (1).

The proof of the parameter "3*b*/2*a*" was constructed by referring to section 6.2 (p. 162-167) in Anderson [1995]. However, this book is very difficult to understand because its
topic is very specific. P. 72 of Poirier [2000] gives an explanation about BM3-EOS. However, this book just says, "after a simple, but tedious, calculation" about the parameter "*b*".

I am afraid that many high-pressure workers have never mathematically proved BM3-EOS even though it is frequently used. In my case, I made this proof 31 years after I started high-pressure research.

I hope that many high-pressure workers visit this page to understand the mathematical derivation of their frequently using EOS.

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.