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Ionic Bonding

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Crystallography and Crystal Chemistry

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Abstract

Here we introduce the concept of ionic bonding, including electronegativity, the Madelung constant, lattice energy, the Born Haber cycle, and Pauling’s Rules. The chapter ends with the empirical application of Fajans’s Rules to estimate the ionic fraction of bonds and a comparison between ionic and covalent bond strengths. Brief biographies of Erwin Madelung, Max Born, and Linus Pauling are also included.

In my own field, x-ray crystallography, we used to work out the structure of minerals by various dodges which we never bothered to write down, we just used them. Then Linus Pauling came along to the laboratory, saw what we were doing and wrote out what we now call Pauling’s Rules. We had all been using Pauling’s Rules for about three or four years before Pauling told us what the rules were.

―J.D. Bernal

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Notes

  1. 1.

    He is also credited with the discovery of cerium (Ce) and selenium (Se) and being the first to isolate silicon (Si) and thorium (Th).

  2. 2.

    which means that χ, as derived by Pauling, has implicit units of eV½

  3. 3.

    His derived value of Eion for the C-I bond also didn’t match the calculated value, but as that was inexplicably negative, Pauling wrote the discrepancy off as experimental error.

  4. 4.

    the energy change that results from adding an electron to an isolated atom.

  5. 5.

    In 1930, Mayer married Maria Goeppert, who was a student of Max Born’s. She won the Nobel Prize in Physics in 1963, becoming only the second woman to do so (the first was Marie Curie 60 years earlier).

  6. 6.

    Some older texts give this equation in cgs (Gaussian) units and so omit the Coulomb constant (4πε0).

  7. 7.

    Lattice energy, ΔE, and lattice enthalpy, ΔH, are related but distinct concepts. Both are measures of the strength of the forces between the ions in an ionic solid. Specifically, lattice energy is the energy required to separate an ionic solid into gaseous ions. It is the heat given off or absorbed when a reaction is run at constant volume. Lattice enthalpy further takes into account that work must be performed against an outer pressure, P, such that ΔE = ΔH − PΔV, where ΔV is the change in volume. Enthalpy is the total energy of a system that can be converted into heat, and lattice enthalpy is the heat given off or absorbed when an ionic solid is separated into gaseous ions at constant pressure. Accordingly, the difference between ΔE and ΔH is small for reactions involving only liquids and solids because there is little if any change in volume during the reaction; however, the difference can be significant for reactions that involve gases and/or high pressures, especially if the reaction involves a change in the amount of gas. The sign of ΔH determines whether a reaction is exothermic (ΔH < 0) or endothermic (ΔH > 0).

  8. 8.

    This means that bonds aren’t completely broken even upon melting.

  9. 9.

    These are the radii which multivalent ions would have if they were to retain their actual electron distributions but interact as if they were univalent. According to Pauling, runivalent = rmultivalent Z2/(n-1) where Z is the charge and n is the Born exponent.

  10. 10.

    There are actually two other polymorphs of TiO2, namely anatase and brookite, but both are metastable in ambient conditions. There are another five high-pressure phases as well.

  11. 11.

    Experimental dipole moments are only available for molecular (gas phase) diatomic molecules.

  12. 12.

    If you have never seen two tipsy chemists settling this argument in a barroom brawl, then you’re not going to the right pubs.

  13. 13.

    Arnold Somerfield never won a Nobel Prize himself, although he was nominated no fewer than 81 times! He was also the doctoral advisor for at least six future Nobel laureates (including Pauling, Heisenberg, Pauli, Debye, Bethe, and Rabi) – more than anyone else in history.

  14. 14.

    Of course, this means that there were no oxides in the dataset, although these equations are often applied (erroneously) to oxides.

  15. 15.

    As we have already seen, most materials scientists would consider ionic/covalent bonds broken upon melting (rather than requiring boiling), and we could in fact just as easily use melting temperatures here.

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Authors and Affiliations

  1. Micron School of Materials Science and Engineering, Boise State University, Boise, ID, USA

    Rick Ubic

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Review Questions

Review Questions

  1. 1.

    Compare/contrast the terms crystal radius, effective ionic radius, metallic radius, covalent radius, and Van der Waals radius.

  2. 2.

    Define/describe the term coordination polyhedron.

  3. 3.

    Define/describe the Madelung constant.

  4. 4.

    Define/describe the Buckingham potential.

  5. 5.

    Explain ionic bonding and describe how the structure and properties of ionically bonded materials result from the characteristics of ionic bonding.

  6. 6.

    Calculate the minimum stable radius ratio for octahedral (sixfold) coordination (e.g., the NaCl structure). Use appropriate diagrams as necessary.

  7. 7.

    Use Evjen’s summation method to calculate the Madelung constant of NaCl. Use sufficient terms for a model including eight unit cells (2 × 2 × 2) and draw these cells out, labeling each set of “nearest” neighbours.

  8. 8.

    A Sr2+ cation is bonded to 12 equidistant oxygen ions in SrTiO3 (Z = 1). What is its electrostatic bond valence strength?

  9. 9.

    In cubic perovskite SrTiO3 (a = 3.905 Å) the atom positions are:

    Sr: 0,0,0

    Ti: ½,½,½

    O: ½,½,0, ½,0,½, 0,½,½

    1. (a)

      Use the bond lengths, r, with the parameters:

      ro (Sr2+–O2−) = 2.118 Å b = 0.37

      ro (Ti4+–O2−) = 1.815 Å b = 0.37

      to calculate the bond valence sums (effective charges) of all three ionic species. Hint: Make sure you include all the “nearest-neighbour” bonds (e.g., there are six of them for oxygen). It may help to draw out more than one unit cell to see them all.

    2. (b)

      Which species are overbonded? Which are underbonded?

  10. 10.

    For each of the compounds below:

    1. (a)

      Determine the cation and anion charges (ZC and ZA), effective ionic radii (rC and rA), and coordinations (NC and NA). Then, use Pauling’s rules to calculate s, the electrostatic bond valence strengths, in the given compounds. Finally, use values from Fig. 15.1 in Eq. 15.31 to calculate the percent ionicity (P) of the bonds in each compound.

      Hint::

      Use Shannon’s ionic radii data for sixfold coordination.

      Note::

      Pauling originally used the univalent radii (rather than crystal radii or effective ionic radii) to calculate radius ratios; however, the success rate of the predictions does not depend strongly on which set of radii are used, as long as both the cation and anion radii come from a set that was derived using consistent assumptions.

 

ZC

ZA

rC

rA

rC/rA

NC

s

NA

P

NaCl

         

CsBr

         

TiO2

         

KI

         

ZnO

         
  1. (b)

    Comment on the s results for the different compounds (e.g., what does a higher s mean?).

  2. (c)

    Carefully chosen examples, like those above, can make the radius-ratio concept look like an accurate predictive tool; however, it is often in error, particularly for complex structures. In fact, ZnO is a good example of its limitations in that the coordinations which would be predicted for it in this way are actually incorrect. Discuss three reasons that might explain this apparent anomaly.

  1. 11.

    Explain why lattice energies calculated from the Born-Landé, Born-Mayer, or Kapustinskii equations are typically underestimations.

  2. 12.

    Forsterite, Mg2SiO4, is an ortho-silicate with the olivine crystal structure. Given univalent radii data, r(Mg+) = 0.82 Å, r(Si+) = 0.65 Å, and r(O−) = 1.76 Å, use Pauling’s rules and your knowledge of bond hybridization to describe the forsterite crystal structure in terms of:

    1. (a)

      The preferred oxygen coordination around Mg

    2. (b)

      The preferred oxygen coordination around Si

    3. (c)

      The bond valence, s, of Mg-O bonds and Si-O bonds. Which are stronger?

  3. 13.

    Lead orthosilicate, Pb2SiO4, forms a glass of density ρ = 7.36 g/cm3.

    1. (a)

      What is the oxygen density (ρO) of this glass (in g/cm3)?

    2. (b)

      How does this compare with the oxygen density, ρO, in fused silica (ρ ~ 2.2 g/cm3)?

    3. (c)

      Assuming univalent ionic radii, r(Pb+) = 1.06 Å, r(Si+) = 0.65 Å, and r(O−) = 1.76 Å, predict the coordination environments of Pb and Si.

    4. (d)

      Given your answers above, where in the structure would you predict that the lead ions are going? Sketch the resultant structure.

    5. (e)

      Determine the density (g/cm3) of crystalline Pb2SiO4 having four formula units per unit cell (Z = 4) and being approximately orthorhombic with lattice constants a = 4.7 Å, b = 12 Å, c = 6.1 Å.

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Ubic, R. (2024). Ionic Bonding. In: Crystallography and Crystal Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-49752-0_15

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