Eq. (5) in the derivation of the 2nd-order Birch-Murnaghan equation of state has the parameter "a" appears. In this page, we prove the parameter "a" has the following formula:
This proof is made by equating two expressions of the second volume (V) derivative of Helmholtz free energy, F.
One expression is from the isothermal bulk modulus, KT, at zero pressure (P = 0).
The other is from the assumption that change in Helmholtz free energy by compression is proportional to square of the finite strain.
V derivative of f
The V derivative of the finite strain at constant T:
At P = 0, V = V0. Therefore, Eq. (2) becomes:
KT by 2nd V derivative of F
Definition of the isothermal bulk modulus:
From Eq. (4), we have
At P= 0, we have
V derivative of P by V derivative of f
V derivative of P based on BM2-Eq. (4) is:
At P = 0, f = 0. Eq. (7) gives:
By substituting Eqs. (6) and (3) into Eq. (8), we have
From Eq. (9), we have Eq. (1).