Born equation

The energy U stored in an electrostatic field distribution is:

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

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:

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

Thus, the energy of solvation of the ion from gas phase (ε_r =1) to a medium of dielectric constant ε_r is:

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

• aspects about this equation