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:

(1)

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*:

(2)

At *P* = 0, *V* = *V*_{0}. Therefore, Eq. (2) becomes:

(3)

*K*_{T} by 2^{nd} *V* derivative of *F*

Definition of the isothermal bulk modulus:

(4)

From Eq. (4), we have

(5)

At *P*= 0, we have

(6)

*V* derivative of *P* by *V* derivative of *f*

*V* derivative of *P* based on BM2-Eq. (4) is:

(7)

At *P* = 0, *f* = 0. Eq. (7) gives:

(8)

*Final procedure*

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

(9)

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