The flow to obtain Vinet equation of state is as follows:
Rose et al.  proposed that the binding energy of metals can be well approximated by the following function:
where a is reduction of the atomic spacing:
where r0 and r are the interatomic distances at zero and high pressures, and l is the scaling length. Based on this relation, F of a matter of interest can be expressed as:
where F0 is a constant.
By expressing Eq. (3) by the volumes at zero and high pressures, V0 and V as:
By substituting Eqs. (4) and (5) into Eq. (3), we have:
Therefore, P is:
By differentiating Eq. (7) by V at constant T, we have:
From Eq. (8), KT is:
The KT0 is obtained by substituting V = V0 to Eq. (9):
KT' is obtained by dividing the V derivative of KT by the V derivative of P.
The V derivative of KT is:
By substituting Eqs. (12) and (8) into Eq. (11), we have:
KT' at zero P (KT0') is obtained by substituting V = V0 into Eq. (13).
From Eq. (7) with Eqs. (10) and (14), we have:
Finally, we have the Vinet equation of state:
Thus, the mathematical derivation of the Vinet equation of state is clear from the assumption of the formula of F. Although Rose et al.  proposed the potential (Eq. 1) based on metal data, Vinet et al.  confirmed validity of the equation of state (Eq. 16) by various material such as H2, D2, Xe, Rb, Mo, NaCl, MgO, magnetite and so on.
Rose, J. H., J. R. Smith, and J. Ferrante, Universal features of bonding in metals, Phys. Rev. B Condens. Matter, 28, 1835, 1983
Vinet, P., Ferrante J., Rose, J. H. & Smith, J. R., Compressibility of Solids, J. Geophys. Res. B, 02, 9319-9325, 1987.