## Vinet equation of state

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

1.  Assume the Helmholtz free energy (F) as a function of volume (V).
2. Obtain the pressure (P) as a function of V with KT0 and KT0' by differentiating F by V.
3. Obtain the isothermal bulk modulus at P (KT)  by multiplying the V derivative of P by V.
4. Obtain the isothermal bulk modulus at zero pressure (KT0) by substituting V = V0 into the formula in Step 3.
5. Obtain the P derivative of KT (KT') by differentiating KT by P
6. Obtain KT' at zero P (KT0') by substituting V = V0 into the formula in Step 5.
7. Replace unknown parameters in the formula of P in Step 2 by KT0 and KT0' using the formulas obtained in Steps 4 and 6.

Rose et al.  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 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:

(3)

where F0 is a constant.
By expressing Eq. (3) by the volumes at zero and high pressures, V0 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), KT is:

(9)

The KT0 is obtained by substituting V = V0 to Eq. (9):

(10)

KT' is obtained by dividing the V derivative of KT by the V derivative of P.

(11)

The V derivative of KT is:

(12)

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

(13)

KT' at zero P (KT0') is obtained by substituting V = V0 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.  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.