Eq. (5) in the derivation of the 2^nd-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, K_T, 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:

K_T by 2^nd 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:

Final procedure

By substituting Eqs. (6) and (3) into Eq. (8), we have

From Eq. (9), we have Eq. (1).