Born equation

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

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