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The Born-Lande' equation is used to theoretically calculate the lattice energy, $\Delta U$, of ionic compounds. It is often cited as such in literature,

$$\Delta U = -\frac{k_Az_1z_2Me^2}{4 \pi \epsilon_\circ r_\circ}\left(1-\frac{1}{n} \right)$$

In which, $n$ is the Born exponent, $k_A$ is Avogradro's constant, $M$ is the Madelung constant, $e$ is the elementary charge, $\epsilon_\circ$ is the permittivity of free space, $z_1$ is the magnitude of the relative charge of the anion and $z_2$ is the magnitude of the relative charge of the cation with $r_\circ$ being their closest distance apart.

However, the most apparent limitations of the Born-Lande' equation I'm able to think of are as follows.

  • The corresponding Born exponent, $n$, and Madelung constant, $M$, must be determined empirically or theoretically for each ionic compound.
  • The bonds between the cation and anion are assumed not to possess any covalent character. However, the difference in electronegativity between the cation and the anion results in covalent characteristics which I believe can be further explained through Fajans' rule. Moreover, I suspect it is the primary reason for the discrepancy between its calculated lattice energies and that of which are determined through the Born-Haber cycle.

Therefore, I would like to ask the following questions.

  • What other limitations does the Born-Lande' equation have?
  • Why are the lattice energies of ionic compounds determined from the Born-Haber cycle considerably higher than that predicted by the Born-Lande' equation?
  • How exactly is Fajans' rule related to the deviation from perfectly ionic behaviour?
  • Is it possible to quantify the discrepancy in the values of the lattice energies in terms of the covalent character of the ionic compound?
  • If electrons have particle-wave duality, how are we able to define ionic radii to determine $r_\circ$ if the location of an electron is described by its probabilistic wave-function?
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    $\begingroup$ Nice question. I agree with premise that partial covalent bonding could be a better explanation. // The Madelung constant, M, is determined by the crystal structure. So you could create a table of the appropriate values by normalizing to the nearest neighbor distance $r_0$. // The value of $r_0$ would have to be determined experimentally by say x-ray diffraction. // Wikipedia says that $n$ depends on "the compressibility of the solid" which makes no sense to me since $r_0$ would already have been determined experimentally. // basically the fudge factor is $n' = 1 - 1/n = (n-1)/n$ $\endgroup$
    – MaxW
    May 17, 2020 at 20:33
  • $\begingroup$ Wikipedia says that n is "typically a number between 5 and 12" so $0.9167> 1-1/n >0.8$ – Max $\endgroup$
    – MaxW
    May 17, 2020 at 21:23
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    $\begingroup$ (a) The limitation arises from the assumption that the repulsive term is $\sim 1/r^n$. A different/ more exact expression could possibly be obtained depending on the type of atoms involved. This may need an accurate MO calculation. (b) The values for NaCl from both methods seem to be close. Do you have examples where the comparison is poor and there are no reasons for this? (e) Atomic/ion radii can be determined experimentally or from theory. Electrons in atoms are constrained by the positive nuclear charge so are 'local' in that sense. $\endgroup$
    – porphyrin
    May 18, 2020 at 12:56

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