The Born equation can be used for estimating the electrostatic component of Gibbs free energy of solvation of an ion. It is an electrostatic model that treats the solvent as a continuous dielectric medium (it is thus one member of a class of methods known as continuum solvation methods).

It was derived by Max Born.^[1][2]

$${\displaystyle \Delta G=-{\frac {N_{A}z^{2}e^{2)){8\pi \varepsilon _{0}r_{0))}\left(1-{\frac {1}{\varepsilon _{r))}\right)}$$

• N_A = Avogadro constant
• z = charge of ion
• e = elementary charge, 1.6022×10⁻¹⁹ C
• ε₀ = permittivity of free space
• r₀ = effective radius of ion
• ε_r = dielectric constant of the solvent

Derivation

The energy U stored in an electrostatic field distribution is:

$${\displaystyle U={\frac {1}{2))\varepsilon _{0}\varepsilon _{r}\int |{\bf {E))|^{2}dV}$$

${\displaystyle |{\bf {E))|={\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{r}r^{2))))$

${\displaystyle dV}$

${\displaystyle dV=4\pi r^{2}dr}$

${\displaystyle U}$

$${\displaystyle U={\frac {1}{2))\varepsilon _{0}\varepsilon _{r}\int _{r_{0))^{\infty }\left({\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{r}r^{2))}\right)^{2}4\pi r^{2}dr={\frac {z^{2}e^{2)){8\pi \varepsilon _{0}\varepsilon _{r}r_{0))))$$

$${\displaystyle {\frac {\Delta G}{N_{A))}=U(\varepsilon _{r})-U(\varepsilon _{r}=1)=-{\frac {z^{2}e^{2)){8\pi \varepsilon _{0}r_{0))}\left(1-{\frac {1}{\varepsilon _{r))}\right)}$$