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The '''Born–Landé equation''' is a means of calculating the [[lattice energy]] of a crystalline [[ionic compound]]. In 1918<ref>{{cite book|title=The chemical bond in inorganic chemistry : the bond valence model|year=2002|publisher=Oxford University Press|location=New York|isbn=0-19-850870-0|last=Brown|first=I. David|edition=Reprint.}}</ref> [[Max Born]] and [[Alfred Landé]] proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.<ref name = "Johnson">{{cite book|title=Metals and chemical change|year=2002|publisher=Royal Society of Chemistry|location=Cambridge|isbn=0-85404-665-8|last=Johnson|first=the Open University ; RSC ; edited by David|edition=1. publ.}}</ref>
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:<math>E =- \frac{N_AMz^+z^- e^2 }{4 \pi \epsilon_0 r_0}\left(1-\frac{1}{n}\right)</math>
where:
*''N''<sub>A</sub> =  [[Avogadro constant]];
*''M'' = [[Madelung constant]], relating to the geometry of the crystal;
*''z''<sup>+</sup> = charge number of cation
*''z''<sup>−</sup> = charge number of anion
*''e'' = [[elementary charge]], 1.6022{{e|&minus;19}} [[coulomb|C]]
*''&epsilon;''<sub>0</sub> = [[permittivity of free space]]
*:4π''&epsilon;''<sub>0</sub> = 1.112{{e|&minus;10}} C<sup>2</sup>/(J·m)
*''r''<sub>0</sub> = distance to closest ion
*''n'' = Born exponent, typically a number between 5 and 12, determined experimentally by measuring the [[compressibility]] of the solid, or derived theoretically.<ref>{{Cotton&Wilkinson4th}}</ref>
 
==Derivation==
The ionic lattice is modeled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.
 
===Electrostatic potential===
The electrostatic potential, <math>E_\text{pair}</math>, between a pair of ions of equal and opposite charge is:
 
:<math>E_\text{pair} = -\frac{z^2 e^2 }{4 \pi \epsilon_0 r}</math>
where
:<math>z</math> = magnitude of charge on one ion
:<math>e</math> = elementary charge, 1.6022{{e|&minus;19}} [[coulomb|C]]
:<math>\epsilon_0</math> = [[permittivity of free space]]
::<math>4\pi \epsilon_0</math> = 1.112{{e|&minus;10}} C²/(J m)
:<math>r</math> = distance separating the ion centers
 
For a simple lattice consisting ions with equal and opposite charge in a 1:1 ratio, interactions between one ion and all other lattice ions need to be summed to calculate <math>E_M</math>, sometimes called the [[Erwin Madelung|Madelung]] or lattice energy:
 
:<math>E_M = -\frac{z^2 e^2 M}{4 \pi \epsilon_0 r}</math>
where
:<math>M</math> = [[Madelung constant]], which is related to the geometry of the crystal
:<math>r</math> = closest distance between two ions of opposite charge
 
===Repulsive term===
Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to <math>1/r^n</math> so that the repulsive energy term, <math>E_R</math>, would be expressed:
:<math>\,E_R = \frac{B}{r^n}</math>
where
:<math>B</math> = constant scaling the strength of the repulsive interaction
:<math>r</math> = closest distance between two ions of opposite charge
:<math>n</math> = Born exponent, a number between 5 and 12 expressing the steepness of the repulsive barrier
 
===Total energy===
The total intensive potential energy of an ion in the lattice can therefore be expressed as the sum of the Madelung and repulsive potentials:
 
:<math>E(r) = -\frac{z^2 e^2 M}{4 \pi \epsilon_0 r} + \frac{B}{r^n}</math>
 
Minimizing this energy with respect to <math>r</math> yields the equilibrium separation <math>r_0</math> in terms of the unknown constant <math>B</math>:
 
:<math>\begin{align}
\frac{\mathrm{d}E}{\mathrm{d}r} &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 r^2} - \frac{n B}{r^{n+1}} \\
0 &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 r_0^2} - \frac{n B}{r_0^{n+1}} \\
r_0 &= \left( \frac{4 \pi \epsilon_0 n B}{z^2 e^2 M}\right) ^\frac{1}{n-1} \\
B &= \frac{z^2 e^2 M}{4 \pi \epsilon_0 n} r_0^{n-1}
\end{align}</math>
 
Evaluating the minimum intensive potential energy and substituting the expression for <math>B</math> in terms of <math>r_0</math> yields the Born–Landé equation:
 
:<math>E(r_0) = - \frac{M z^2 e^2 }{4 \pi \epsilon_0 r_0}\left(1-\frac{1}{n}\right)</math>
 
==Calculated lattice energies==
The Born–Landé equation gives a reasonable fit to the lattice energy <ref name = "Johnson"/>
{| class="wikitable" border="1"
|-
! Compound
! Calculated Lattice Energy
! Experimental Lattice Energy
|-
| NaCl
| −756 kJ/mol
| −787 kJ/mol
|-
| LiF
| −1007 kJ/mol
| −1046 kJ/mol
|-
| CaCl<sub>2</sub>
| −2170 kJ/mol
| −2255 kJ/mol
|}
 
==See also==
* [[Kapustinskii equation]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Born-Lande Equation}}
[[Category:Solid-state chemistry]]
[[Category:Ions]]

Latest revision as of 15:30, 12 April 2014

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