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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]

\Delta G=-{\frac {N_{A}z^{2}e^{2}}{8\pi \varepsilon _{0}r_{0}}}\left(1-{\frac {1}{\varepsilon _{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:

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

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

and the volume element

dV

can be expressed as

dV=4\pi r^{2}dr

, the energy

U

can be written as:

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:

{\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)

References

^ Born, M. (1920-02-01). "Volumen und Hydratationswärme der Ionen". Zeitschrift für Physik (in German). 1 (1): 45–48. Bibcode:1920ZPhy....1...45B. doi:10.1007/BF01881023. ISSN 0044-3328. S2CID 92547891.
^ Atkins; De Paula (2006). Physical Chemistry (8th ed.). Oxford university press. p. 102. ISBN 0-7167-8759-8.

External links

aspects about this equation

source: http://en.wikipedia.org/wiki/Born_equation

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