Born-Landé equation

The Born-Landé equation (named after Max Born and Alfred Landé ), also Madelung equation (after Erwin Madelung ), is an extension of the electrostatic Coulomb model to ion crystals (i.e. salts ) and allows the calculation of lattice energies .

The Coulomb model assumes opposing point charges in a regular arrangement, since ions are positively or negatively charged towards the outside. However, if the approach is sufficiently close, the electron shells of the ions meet and repel each other due to their identical “charge”. The size of the repulsion depends on the electron density of the ions and is measured using the Born exponent . This is determined experimentally from the compressibility of the ion crystals, but average values ​​can also be used for the calculation. ${\displaystyle n}$

${\displaystyle U_{0}=-{\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {N_{\mathrm {A} }\cdot A\cdot a\cdot b\cdot e^{2}}{d_{0}}}\cdot {\biggl (}1-{\frac {1}{n}}{\biggr )}}$

The Madelung constant should be marked with instead of with to avoid confusion with the molar mass . ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle M}$

symbol	Surname	unit
${\displaystyle U_{0}}$	Lattice energy	J / mol
${\displaystyle N_{\mathrm {A} }}$	Avogadro's constant = 6.022141 · 10 23 mol −1	1 / mol
${\displaystyle A}$	Madelung constant of the lattice type	dimensionless
${\displaystyle a}$	Number of elementary charges of the cation	dimensionless
${\displaystyle b}$	Number of elementary charges of the anion	dimensionless
${\displaystyle e}$	Elementary charge = 1.602176 · 10 −19 C	C.
${\displaystyle \varepsilon _{0}}$	Permittivity of the vacuum = 8.854185 · 10 −12 C 2 · (J · m) −1	C / (V · m) = C² / (J · m)
${\displaystyle d_{0}}$	Ion distance in equilibrium (from X-ray diffraction experiments or approximated as the sum of the ion radii )	m
${\displaystyle n}$	Born exponent	dimensionless

Further refinements are based on the inclusion of the temperature:

• Zero point energy
• Conversion of the result to temperatures above absolute zero

The larger the covalent bond component, the worse the results will correspond to reality. This is because the model is based on the consideration of pure ions.

example

The lattice energy of barium oxide is to be calculated. Barium is twice positively charged in the crystal lattice and the oxide ions twice negative. The amounts of the ion charges are entered:

${\displaystyle U_{0}=-{\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {N_{\mathrm {A} }\cdot A\cdot |+2|\cdot |-2|\cdot e^{2}}{d_{0}}}\cdot {\biggl (}1-{\frac {1}{n}}{\biggr )}}$

Barium oxide crystallizes in the NaCl type. Its Madelung constant is 1.7475. We also insert the numerical value of the Avogadro constant, the permittivity and the elementary charge:

${\displaystyle U_{0}=-{\frac {1}{4\pi (8{,}854\cdot 10^{-12}\;\mathrm {\frac {As}{Vm}} )}}\cdot {\frac {6{,}022\cdot 10^{23}\;{\frac {1}{\mathrm {mol} }}\cdot 1{,}7475\cdot 4\cdot (1{,}602\cdot 10^{-19}\;\mathrm {As} )^{2}}{d_{0}}}\cdot {\biggl (}1-{\frac {1}{n}}{\biggr )}}$

We determine the equilibrium distance between the ions from the sum of the ion radii. This amounts to:

${\displaystyle d_{0}=150\;\mathrm {pm} +140\;\mathrm {pm} =290\cdot 10^{-12}\;\mathrm {m} }$

${\displaystyle U_{0}=-{\frac {1}{4\pi (8{,}854\cdot 10^{-12}\;\mathrm {\frac {As}{Vm}} )}}\cdot {\frac {6{,}022\cdot 10^{23}\;{\frac {1}{\mathrm {mol} }}\cdot 1{,}7475\cdot 4\cdot (1{,}602\cdot 10^{-19}\;\mathrm {As} )^{2}}{290\cdot 10^{-12}\;\mathrm {m} }}\cdot {\biggl (}1-{\frac {1}{n}}{\biggr )}}$

${\displaystyle n}$ is the mean of the Born exponent of the cation and the anion:

${\displaystyle n={\frac {n_{\mathrm {b} }(\mathrm {Ba} ^{2+})+n_{\mathrm {b} }(\mathrm {O} ^{2-})}{2}}={\frac {12+7}{2}}=9{,}5}$

${\displaystyle U_{0}=-{\frac {1}{4\pi (8{,}854\cdot 10^{-12}\;\mathrm {\frac {As}{Vm}} )}}\cdot {\frac {6{,}022\cdot 10^{23}\;{\frac {1}{\mathrm {mol} }}\cdot 1{,}7475\cdot 4\cdot (1{,}602\cdot 10^{-19}\;\mathrm {As} )^{2}}{290\cdot 10^{-12}\;\mathrm {m} }}\cdot {\biggl (}1-{\frac {1}{9{,}5}}{\biggr )}}$

The units are reduced to voltampereseconds (energy, joules) per mol:

${\displaystyle U_{0}=-2995648{,}76\;\mathrm {\frac {J}{mol}} \approx -2995{,}65\;\mathrm {\frac {kJ}{mol}} }$

See also

• Kapustinskii equation
• Born-Haber cycle

literature

• M. Born, A. Landé: Ber. Prussia. Akad. Wiss. Berlin No. 45, 1918, p. 1048 ff.