Ecuación nacida

Equation for Gibbs free energy of solvation

La ecuación de Born se puede utilizar para estimar el componente electrostático de la energía libre de solvatación de Gibbs de un ion. Es un modelo electrostático que trata el solvente como un medio dieléctrico continuo (por lo tanto, es un miembro de una clase de métodos conocidos como métodos de solvatación continua ).

Fue derivado por Max Born . ^{[1] [2]}

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

dónde:

• N _A = constante de Avogadro
• z = carga de iones
• e = carga elemental , 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:

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

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:

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

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

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

References

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

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