Here, we prove Eq. (3) in the 3rd-order Birch-Murnaghan equation of state (BM3-EOS). For simplicity, the parameter "3b/2a" is expressed by "m" in this page. Our goal is therefore:
1. The Helmholtz free energy is expanded to the 3rd order of the finite strain f.
2. The KT and F are differentiated by V to a higher order.
We refer equations in the page for the parameter "a" as (PA Eq. ??).
2nd V derivative of f
At P = 0, V = V0. Hence we have
KT' and 2nd V derivative of P
The relations between KT and V derivative of P are given as PA Eq. (4) - (7). Here we obtain relations between the P derivative of KT, KT', and 2nd V derivative of P.
From the definition, we have:
At P = 0, we have
2nd V derivative of F with f
We differentiate BM3 Eq. (2) by V with replacing 3b/2a by m.
Eq. (8) is once again differentiated by V.
At P = 0, f = 0. We also substitute Eq. (3) and PA-Eq. (3) into Eq. (10).
By equating Eqs (6) and (11), and substituting BM2-Eq. (5) into it, we have:
By simplifying Eq. (12), we have Eq. (1).
The proof of the parameter "3b/2a" was constructed by referring to section 6.2 (p. 162-167) in Anderson . However, this book is very difficult to understand because its topic is very specific. P. 72 of Poirier  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.
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.