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. Δ G = − N A z 2 e 2 8 π ε 0 r 0 ( 1 − 1 ε r ) {\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:
- NA = Avogadro constant
- z = charge of ion
- e = elementary charge, 1.6022×10−19 C
- ε0 = permittivity of free space
- r0 = effective radius of ion
- εr = dielectric constant of the solvent
Derivation
The energy U stored in an electrostatic field distribution is: U = 1 2 ε 0 ε r ∫ | E | 2 d V {\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 εr is | E | = z e 4 π ε 0 ε r r 2 {\displaystyle |{\bf {E}}|={\frac {ze}{4\pi \varepsilon _{0}\varepsilon _{r}r^{2}}}} and the volume element d V {\displaystyle dV} can be expressed as d V = 4 π r 2 d r {\displaystyle dV=4\pi r^{2}dr} , the energy U {\displaystyle U} can be written as: U = 1 2 ε 0 ε r ∫ r 0 ∞ ( z e 4 π ε 0 ε r r 2 ) 2 4 π r 2 d r = z 2 e 2 8 π ε 0 ε r r 0 {\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 (εr = 1) to a medium of dielectric constant εr is: Δ G N A = U ( ε r ) − U ( ε r = 1 ) = − z 2 e 2 8 π ε 0 r 0 ( 1 − 1 ε r ) {\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)}
External links
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. https://doi.org/10.1007/BF01881023 ↩
Atkins; De Paula (2006). Physical Chemistry (8th ed.). Oxford university press. p. 102. ISBN 0-7167-8759-8. 0-7167-8759-8 ↩