Born equation

{{Short description|Equation for Gibbs free energy of solvation}}

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.{{cite journal |last=Born |first=M. |date=1920-02-01 |title=Volumen und Hydratationswärme der Ionen |url=https://doi.org/10.1007/BF01881023 |journal=Zeitschrift für Physik |language=de |volume=1 |issue=1 |pages=45–48 |doi=10.1007/BF01881023 |bibcode=1920ZPhy....1...45B |s2cid=92547891 |issn=0044-3328 }}{{cite book |title=Physical Chemistry |last1=Atkins |last2=De Paula |year=2006 |publisher=Oxford university press |isbn=0-7167-8759-8 |page=[https://archive.org/details/atkinsphysicalch00pwat/page/102 102] |edition=8th |url-access=registration |url=https://archive.org/details/atkinsphysicalch00pwat/page/102 }}

$\Delta G =- \frac{N_\text{A} z^2 e^2}{8 \pi \varepsilon_0 r_0}\left(1-\frac{1}{\varepsilon_\text{r}}\right)$

where:

N_A = Avogadro constant
z = charge of ion
e = elementary charge, 1.6022{{e|−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: $U=\frac{1}{2} \varepsilon_0 \varepsilon_\text{r} \int |{\bf{E}}|^2 dV$ Knowing the magnitude of the electric field of an ion in a medium of dielectric constant ε_r is $|{\bf{E}}|=\frac{z e}{4 \pi \varepsilon_0 \varepsilon_{r} r^2}$ and the volume element $dV$ can be expressed as $dV=4\pi r^2 dr$ , the energy $U$ can be written as: $U=\frac{1}{2} \varepsilon_0 \varepsilon_\text{r} \int_{r_0}^\infty \left(\frac{z e}{4 \pi \varepsilon_0 \varepsilon_\text{r} r^2}\right)^2 4\pi r^2 dr=\frac{z^2 e^2}{8\pi \varepsilon_0 \varepsilon_\text{r} r_0}$ Thus, the energy of solvation of the ion from gas phase ({{nowrap|1=ε_r = 1}}) to a medium of dielectric constant ε_r is: $\frac{\Delta G}{N_\text{A}} = U(\varepsilon_\text{r} )- U(\varepsilon_\text{r}=1)=- \frac{z^2 e^2}{8 \pi \varepsilon_0 r_0}\left(1-\frac{1}{\varepsilon_\text{r}}\right)$

References

External links

[https://www.hindawi.com/journals/isrn/2012/204104/ aspects about this equation]

Category:Enthalpy

Category:Max Born