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Covalent, Metallic, and Secondary Bonding

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Crystallography and Crystal Chemistry

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Abstract

This chapter introduces the concepts of bonding, coordination, and packing fraction. We start with covalent bonding, encompassing both valence bond theory and molecular orbital theory, then move to metallic bonding, the Hume-Rothery Rules, and band theory; and we finish with Van der Waals forces and hydrogen bonding. A biographical sketch of Johannes Diderik van der Waals is also included.

One is almost tempted to say… at last I can almost see a bond. But that will never be, for a bond does not really exist at all: it is a most convenient fiction which, as we have seen, is convenient both to experimental and theoretical chemists.

― Charles Alfred Coulson

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Notes

  1. 1.

    The valence level is the outermost electron shell of an atom, containing valence electrons which are accessible for the formation of chemical bonds.

  2. 2.

    Electrons are delocalized if they are associated with more than one bond. Valence bond theory does not allow for bond delocalization, which is one of its weaknesses.

  3. 3.

    Paramagnetic materials are weakly attracted by external magnetic fields whereas diamagnetic materials are repelled by magnetic fields.

  4. 4.

    As for single atoms, the first superscript in the molecular notation is the multiplicity (the number of unpaired electrons plus one). Instead of the symbols S, P, D, or F to indicate a total orbital angular momentum quantum number equal to 0, 1, 2, or 3, we use the Greek equivalents, Σ, Π, Δ, or Φ. The subscript “g” indicates the state’s parity. If inverting the electron’s position with respect to the molecular mass centre leaves the wavefunction unchanged, it is labeled “g” for gerade (even); otherwise, it is “u” for ungerade (odd). Finally, for the Σ states, a superscript is used to indicate whether the state is symmetric, “+”, or antisymmetric, “−”, upon reflection through any plane containing the internuclear axis.

  5. 5.

    The ultimate tensile strength, or simply tensile strength, is the maximum stress that a material can withstand before breaking while being stretched or pulled.

  6. 6.

    Remember, metallic bonding isn’t fully broken until the metal boils.

  7. 7.

    More strictly, we mean here only recoverable, or elastic, deformation, as this kind of deformation involves only the stretching/compression of atomic spacings.

  8. 8.

    Hume-Rothery earned a first-class honours degree in chemistry at Oxford and a PhD from the Royal School of Mines (now part of Imperial College London) despite being totally deaf from the age of 18. He went on to found the Department of Metallurgy (which is now the Department of Materials) at the University of Oxford in the mid-1950s. One of his students there was William “Bill” Pearson.

  9. 9.

    In 1819, Pierre Louis Dulong and Alexis Thérèse Petit found that the heat capacity of a mole of many solid elements is about 3R or ~ 25 J/K (the gas constant, R, had not yet been defined).

  10. 10.

    A direct semiconductor is one for which the peak in the valence-band energy aligns with the minimum in the conduction-band energy. An indirect semiconductor is one for which the peak in the valence-band energy does not align with the minimum in the conduction-band energy.

  11. 11.

    Not to be confused with metalloids, which, although there is no standard definition, are typically described as elements with properties between (or a mixture of) those of metals and nonmetals. The six most commonly recognised metalloids include B, Si, Ge, As, Sb, and Te. Other elements sometimes included are C, Al, Se, Po, and At. Although there is some overlap in the sets of semimetals and metalloids, the terms are not synonymous. Unlike metalloids, semimetals can also be compounds, like HgTe; and unlike metals, semimetals have both electron and hole charge carriers but typically in smaller numbers than in a metal. The electrical properties of semimetals are between those of metals and semiconductors.

  12. 12.

    Strongly correlated materials are defined by the fact that the behaviour of their electrons cannot be described effectively without considering interaction. Theoretical models of the electronic structure of these strongly correlated materials must include electronic correlation (interaction) to be accurate. Such materials can show unusual electronic or magnetic properties, including multiferroic behaviour, metal-insulator transitions, high-Tc superconductivity, heavy-fermion behaviour, half-metallicity, and spin-charge separation. Many transition-metal oxides fall into this category.

  13. 13.

    Charles formulated the original law in unpublished work during the 1780s. Joseph Louis Gay-Lussac mentions it in his “Recherches sur la dilatation des gaz et des vapeurs” of 1802.

  14. 14.

    Clapeyron never explains why he choose the letter R for his constant, and some have seen it as an homage to Henri Victor Regnault, who provided much of the empirical data for its derivation (although it probably just meant rapport “ratio.”)

  15. 15.

    A mole of gas contains NA molecules/atoms, where NA = 6.02214076 × 1023.

  16. 16.

    The term “ideal gas” has been introduced by Rudolf Clausius in 1864. [35]

  17. 17.

    There is some debate as to whether Van der Waals forces can be said to include forces which act intramolecularly (between two different parts of the same molecule) as well as intermolecularly; however, it is probably logical to argue that a Van der Waals force should be one which can be described by the Van der Waals equation (Eq. 14.22), and specifically the constant a, which is only valid for intermolecular interactions. In this sense, all Van der Waals solids and even graphite can be considered molecular solids.

  18. 18.

    In Dutch names, the first “tussenvoegsel” (in this case “van”) should be lowercase only if it is preceded by the first name, so we would refer to Johannes Diderik van der Waals but Van der Waals forces. German literature of the time referred to “van der Waalsschen Kräfte” or “van der Waalsschen Kohäsionskräfte,” so “van der Waals forces” seems to have been the accepted form then, but that might have simply been in keeping with the usage of the German “von,” which is only ever capitalized when it occurs at the beginning of a sentence.

  19. 19.

    Note that this is the only definition for Van der Waals forces given in some texts.

  20. 20.

    Helium has the lowest melting point of any element. It can only be solidified below 1 K and even then only by simultaneously applying a pressure of more than 25 atm. Given these extreme conditions, there is significant error in measurements and so no general agreement on Tm. The melting of He has not been shown to require heat, i.e., it is not endothermic; however, there is no evidence to suggest that it is exothermic either. Its ΔHf seems to be about 0.

  21. 21.

    And only about half of all hydrogen bonds are broken upon vaporization

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Author information

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.

    Explain covalent bonding and describe how the structure and properties of covalently-bonded materials result from the characteristics of covalent bonding.

  2. 2.

    Compare/contrast σ bond and π bonds.

  3. 3.

    The choice of unit cell is not unique and does not always reveal all of symmetry characteristics of a structure. Consider a hard-sphere model of a body-centred tetragonal structure (tI with one atom per unit cell) with an arbitrary c/a ratio (c/a = δ). Assuming that spheres touch, three cases can be distinguished:

    • Case I: Spheres touch along <001>

    • Case II: Spheres touch along <111>

    • Case III: Spheres touch along <100>

      1. (a)

        Calculate the packing fraction as a function of δ for each case.

      2. (b)

        Plot the packing fraction as a function of δ. Use linear axes with 0 ≤ δ ≤ 3 and 0 ≤ Fp ≤ 1 for your graph.

      3. (c)

        Calculate the values of δ at which realistic transitions from one case to another occur and indicate them on your graph? What is the Bravais lattice in each case? What is the Bravais lattice when δ = 1? Illustrate your answers.

      4. (d)

        Calculate the packing fractions at these transitions and indicate them on your graph.

      5. (e)

        Based on your calculations/graph above, what is the maximum packing fraction which occurs?

  4. 4.

    Consider melting temperature (Tm), tensile strength (σUTS), elastic modulus (E), and thermal expansion coefficient (α) and describe how bonding affects each. Reference an energy vs displacement diagram where appropriate.

  5. 5.

    Briefly explain metallic bonding and describe how the structure and properties of metallically-bonded materials result from the characteristics of metallic bonding.

  6. 6.

    Given that Na (r = 1.9 Å) forms in a BCC structure while Mg (r = 1.6 Å) forms with an HCP structure, compare the relative strength of metallic bonding in Na and Mg metals. Based on your conclusion, how would you expect the melting temperature of Mg to compare to that of Na? Why? Consider the atomic number, electron configuration, atomic size, and coordination.

  7. 7.

    (a) What is the ground-state electron configuration of Mg? Does it have a filled outer orbital? Using words and appropriate diagrams, explain why Mg can form bonds.

    (b) Sketch the likely electron configuration (orbitals) around MgH2. What types of bonds form (be specific and specify geometry!)?

    (c) Based on the relative strength of metallic bonding, how would you expect the melting temperature of Mg to compare to that of Na? Why (consider the electron configuration, atomic number, atomic size, and coordination)?

  8. 8.

    What information can be inferred from a curve of energy vs separation?

  9. 9.

    A model for the energy of an inert gas solid is given by the Lennard-Jones potential:

$$ E=4\upvarepsilon \left[{\left(\frac{\upsigma}{r}\right)}^{12}-{\left(\frac{\upsigma}{r}\right)}^6\right]\kern0.24em \mathrm{or}\kern0.24em E=\upvarepsilon \left[{\left(\frac{r_0}{r}\right)}^{12}-2{\left(\frac{r_0}{r}\right)}^6\right] $$

where -ε is the equilibrium bond energy, σ is the spacing at which E = 0, r is an arbitrary interatomic spacing, and r0 is the equilibrium separation.

(a) With data in the table below, plot the repulsive energy (first term), attractive energy (second term), and total energy as a function of separation for Ar. The curves should be on the same plot. Include only the range 3 Å ≤ r ≤ 5 Å and − 0.1 eV ≤ E ≤ 0.15 eV and be sure to label all three curves.

(b) Again with data from the table below, plot the total energy as a function of interatomic separation for Ne, Ar, Kr, and Xe. The curves should be on the same plot. Include only the range 2.5 Å ≤ r ≤ 5 Å and − 0.04 eV ≤ E ≤ 0.08 eV and be sure to label all four curves.

(c) Determine the equilibrium atomic spacing, r0, for each solid in the table, mark this value on your plots from part b, and comment on the trend you notice between r0 and bond energy.

(d) Calculate the Van der Waal’s radius of Ne.

HINT: Noble-gas solids are Van der Waals solids.

Solid

ε (eV)

σ (Å)

Ne

0.0031

2.85

Ar

0.0104

3.31

Kr

0.0140

3.55

Xe

0.0180

3.87

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Ubic, R. (2024). Covalent, Metallic, and Secondary Bonding. In: Crystallography and Crystal Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-49752-0_14

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