The flow to obtain Vinet equation of state is as follows:

- Assume the Helmholtz free energy (
*F*) as a function of volume (*V*). - Obtain the pressure (
*P*) as a function of*V*with*K*_{T0}and*K*_{T0}' by differentiating*F*by*V*. - Obtain the isothermal bulk modulus at
*P*(*K*_{T}) by multiplying the*V*derivative of*P*by*V*. - Obtain the isothermal bulk modulus at zero pressure (
*K*_{T0}) by substituting*V*=*V*_{0}into the formula in Step 3. - Obtain the
*P*derivative of*K*_{T}(*K*_{T}') by differentiating*K*_{T}by*P*. - Obtain
*K*_{T}' at zero*P*(*K*_{T0}') by substituting*V*=*V*_{0}into the formula in Step 5. - Replace unknown parameters in the formula of
*P*in Step 2 by*K*_{T0}and*K*_{T0}' using the formulas obtained in Steps 4 and 6.

Rose *et al*. [1983]
proposed that the binding energy of metals can be well approximated by the following function:

(1)

where *a* is reduction of the atomic spacing:

(2)

where *r*_{0} 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:

(3)

where *F*_{0} is a constant.

By expressing Eq. (3) by the volumes at zero and high pressures, *V*_{0} and *V* as:

(4)

(5)

By substituting Eqs. (4) and (5) into Eq. (3), we have:

(6)

Therefore, *P* is:

(7)

By differentiating Eq. (7) by *V* at constant *T*, we have:

(8)

From Eq. (8), *K*_{T} is:

(9)

The *K*_{T0} is obtained by substituting *V* = *V*_{0} to Eq. (9):

(10)

*K*_{T}' is obtained by dividing the *V* derivative of *K*_{T} by the *V* derivative of *P*.

(11)

The *V* derivative of *K*_{T} is:

(12)

By substituting Eqs. (12) and (8) into Eq. (11), we have:

(13)

*K*_{T}' at zero *P* (*K*_{T0}') is obtained by substituting *V* = *V*_{0} into Eq. (13).

(14)

From Eq. (7) with Eqs. (10) and (14), we have:

(15)

Finally, we have the Vinet equation of state:

(16)

Thus, the mathematical derivation of the Vinet equation of state is clear from the assumption of the formula of *F*. Although Rose *et al*. [1983] proposed the potential (Eq. 1)
based on metal data, Vinet et al. [1987] 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.