Born equation

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).

The equation was derived by Max Born.^[1][2] $${\displaystyle \Delta G=-{\frac {N_{\text{A}}z^{2}e^{2}}{8\pi \varepsilon _{0}r_{0}}}\left(1-{\frac {1}{\varepsilon _{\text{r}}}}\right)}$$ where:

• 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:$${\displaystyle U={\frac {1}{2}}\varepsilon _{0}\varepsilon _{\text{r}}\int |{\bf {E}}|^{2}dV}$$Knowing the magnitude of the electric field of an ion in a medium of dielectric constant ε_r is ${\displaystyle |{\bf {E}}|={\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{r}r^{2}}}}$ and the volume element ${\displaystyle dV}$ can be expressed as ${\displaystyle dV=4\pi r^{2}dr}$, the energy ${\displaystyle U}$ can be written as: $${\displaystyle U={\frac {1}{2}}\varepsilon _{0}\varepsilon _{\text{r}}\int _{r_{0}}^{\infty }\left({\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{\text{r}}r^{2}}}\right)^{2}4\pi r^{2}dr={\frac {z^{2}e^{2}}{8\pi \varepsilon _{0}\varepsilon _{\text{r}}r_{0}}}}$$ Thus, the energy of solvation of the ion from gas phase (ε_r = 1) to a medium of dielectric constant ε_r is:$${\displaystyle {\frac {\Delta G}{N_{\text{A}}}}=U(\varepsilon _{\text{r}})-U(\varepsilon _{\text{r}}=1)=-{\frac {z^{2}e^{2}}{8\pi \varepsilon _{0}r_{0}}}\left(1-{\frac {1}{\varepsilon _{\text{r}}}}\right)}$$