Born equation

 ( Enthalpy ) 

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.

$$\backslash Delta\; G\; =-\; \backslash frac\{N\_\backslash text\{A\}\; z^2\; e^2\}\{8\; \backslash pi\; \backslash varepsilon\_0\; r\_0\}\backslash left(1-\backslash frac\{1\}\{\backslash varepsilon\_\backslash text\{r\}\}\backslash right)$$ where:

• N_A = Avogadro constant
• z = charge of ion
• e = elementary charge, 1.6022 coulomb
• ε₀ = permittivity of free space
• r₀ = effective ionic radius
• ε_r = dielectric constant of the solvent

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Derivation The energy U stored in an electrostatic field distribution is:$$U=\backslash frac\{1\}\{2\}\; \backslash varepsilon\_0\; \backslash varepsilon\_\backslash text\{r\}\; \backslash int\; |\{\backslash bf\{E\}\}|^2\; dV$$Knowing the magnitude of the electric field of an ion in a medium of dielectric constant ε_r is $|\{\backslash bf\{E\}\}|=\backslash frac\{z\; e\}\{4\; \backslash pi\; \backslash varepsilon\_0\; \backslash varepsilon\_\{r\}\; r^2\}$ and the volume element $dV$ can be expressed as $dV=4\backslash pi\; r^2\; dr$, the energy $U$ can be written as: $$U=\backslash frac\{1\}\{2\}\; \backslash varepsilon\_0\; \backslash varepsilon\_\backslash text\{r\}\; \backslash int\_\{r\_0\}^\backslash infty\; \backslash left(\backslash frac\{z\; e\}\{4\; \backslash pi\; \backslash varepsilon\_0\; \backslash varepsilon\_\backslash text\{r\}\; r^2\}\backslash right)^2\; 4\backslash pi\; r^2\; dr=\backslash frac\{z^2\; e^2\}\{8\backslash pi\; \backslash varepsilon\_0\; \backslash varepsilon\_\backslash text\{r\}\; r\_0\}$$ Thus, the energy of solvation of the ion from gas phase () to a medium of dielectric constant ε_r is:$$\backslash frac\{\backslash Delta\; G\}\{N\_\backslash text\{A\}\}\; =\; U(\backslash varepsilon\_\backslash text\{r\}\; )-\; U(\backslash varepsilon\_\backslash text\{r\}=1)=-\; \backslash frac\{z^2\; e^2\}\{8\; \backslash pi\; \backslash varepsilon\_0\; r\_0\}\backslash left(1-\backslash frac\{1\}\{\backslash varepsilon\_\backslash text\{r\}\}\backslash right)$$

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External links

• aspects about this equation

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