INTRODUCTION TO
`INORGANIC
`CHEMISTRY
`
`Wikibook
`Penn State University
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`Book: Introduction to Inorganic Chemistry
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`This text was compiled on 07/17/2022
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`TABLE OF CONTENTS
`
`Inorganic chemistry is the study of the synthesis, reactions, structures and properties of compounds of the elements. Inorganic
`chemistry encompasses the compounds - both molecular and extended solids - of everything else in the periodic table, and overlaps
`with organic chemistry in the area of organometallic chemistry, in which metals are bonded to carbon-containing ligands and
`molecules. Inorganic chemistry is fundamental to many practical technologies including catalysis and materials (structural,
`electronic, magnetic etc.), energy conversion and storage, and electronics. Inorganic compounds are also found in biological
`systems where they are essential to life processes.
`
`1: Review of Chemical Bonding
`1.1: Prelude to Chemical Bonding
`1.2: Valence Bond Theory- Lewis Dot Structures, the Octet Rule, Formal Charge, Resonance, and the Isoelectronic Principle
`1.3: The Shapes of Molecules (VSEPR Theory) and Orbital Hybridization
`1.4: Bond Polarity and Bond Strength
`1.5: Discussion Questions
`1.6: Problems
`1.7: References
`
`2: Molecular Orbital Theory
`2.1: Prelude to Molecular Orbital Theory
`2.2: Constructing Molecular Orbitals from Atomic Orbitals
`2.3: Orbital Symmetry
`2.4: σ, π, and δ orbitals
`2.5: Diatomic Molecules
`2.6: Orbital Filling
`2.7: Periodic Trends in π Bonding
`2.8: Three-center Bonding
`2.9: Building up the MOs of More Complex Molecules- NH₃, P₄
`2.10: Homology of σ and π orbitals in MO diagrams
`2.11: Chains and Rings of π-Conjugated Systems
`2.12: Discussion Questions
`2.13: Problems
`2.14: References
`
`3: Acid-Base Chemistry
`3.1: Prelude to Acid-Base Chemistry
`3.2: Brønsted and Lewis Acids and Bases
`3.3: Hard and Soft Acids and Bases
`3.4: The Electrostatic-Covalent (ECW) Model for Acid-Base Reactions
`3.5: Frustrated Lewis Pairs
`3.6: Discussion Questions
`3.7: Problems
`3.8: References
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`4: Redox Stability and Redox Reactions
`4.1: Prelude to Redox Stability and Redox Reactions
`4.2: Balancing Redox Reactions
`4.3: Electrochemical Potentials
`4.4: Latimer and Frost Diagrams
`4.5: Redox Reactions with Coupled Equilibria
`4.6: Pourbaix Diagrams
`4.7: Discussion Questions
`4.8: Problems
`4.9: References
`
`5: Coordination Chemistry and Crystal Field Theory
`5.1: Prelude to Coordination Chemistry and Crystal Field Theory
`5.2: Counting Electrons in Transition Metal Complexes
`5.3: Crystal Field Theory
`5.4: Spectrochemical Series
`5.5: π-Bonding between Metals and Ligands
`5.6: Crystal Field Stabilization Energy, Pairing, and Hund's Rule
`5.7: Non-octahedral Complexes
`5.8: Jahn-Teller Effect
`5.9: Tetrahedral Complexes
`5.10: Stability of Transition Metal Complexes
`5.11: Chelate and Macrocyclic Effects
`5.12: Ligand Substitution Reactions
`5.13: Discussion Questions
`5.14: Problems
`5.15: References
`
`6: Metals and Alloys- Structure, Bonding, Electronic and Magnetic Properties
`6.1: Prelude to Metals and Alloys
`6.2: Unit Cells and Crystal Structures
`6.3: Bravais Lattices
`6.4: Crystal Structures of Metals
`6.5: Bonding in Metals
`6.6: Conduction in Metals
`6.7: Atomic Orbitals and Magnetism
`6.8: Ferro-, Ferri- and Antiferromagnetism
`6.9: Hard and Soft Magnets
`6.10: Discussion Questions
`6.11: Problems
`6.12: References
`
`7: Metals and Alloys - Mechanical Properties
`7.1: Defects in Metallic Crystals
`7.2: Work Hardening, Alloying, and Annealing
`7.3: Malleability of Metals and Alloys
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`7.4: Iron and Steel
`7.5: Amorphous Alloys
`7.6: Discussion Questions
`7.7: Problems
`7.8: References
`
`8: Ionic and Covalent Solids - Structures
`8.1: Prelude to Ionic and Covalent Solids - Structures
`8.2: Close-packing and Interstitial Sites
`8.3: Structures Related to NaCl and NiAs
`8.4: Tetrahedral Structures
`8.5: Layered Structures and Intercalation Reactions
`8.6: Bonding in TiS₂, MoS₂, and Pyrite Structures
`8.7: Spinel, Perovskite, and Rutile Structures
`8.8: Discussion Questions
`8.9: Problems
`8.10: References
`
`9: Ionic and Covalent Solids - Energetics
`9.1: Ionic Radii and Radius Ratios
`9.2: Structure Maps
`9.3: Energetics of Crystalline Solids- The Ionic Model
`9.4: Born-Haber Cycles for NaCl and Silver Halides
`9.5: Kapustinskii Equation
`9.6: Discovery of Noble Gas Compounds
`9.7: Stabilization of High and Low Oxidation States
`9.8: Alkalides and Electrides
`9.9: Resonance Energy of Metals
`9.10: Prelude to Ionic and Covalent Solids - Energetics
`9.11: The Strange Case of the Alkali Oxides
`9.12: Lattice Energies and Solubility
`9.13: Discussion Questions
`9.14: Problems
`9.15: References
`
`10: Electronic Properties of Materials - Superconductors and Semiconductors
`10.1: Prelude to Electronic Properties of Materials - Superconductors and Semiconductors
`10.2: Metal-Insulator Transitions
`10.3: Superconductors
`10.4: Periodic Trends- Metals, Semiconductors, and Insulators
`10.5: Semiconductors- Band Gaps, Colors, Conductivity and Doping
`10.6: Semiconductor p-n Junctions
`10.7: Diodes, LEDs and Solar Cells
`10.8: Amorphous Semiconductors
`10.9: Discussion Questions
`10.10: Problems
`10.11: References
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`11: Basic Science of Nanomaterials
`11.1: Prelude to Basic Science of Nanomaterials
`11.2: Physics and Length Scales- Cavity Laser, Coulomb Blockade, Nanoscale Magnets
`11.3: Semiconductor Quantum Dots
`11.4: Synthesis of Semiconductor Nanocrystals
`11.5: Surface Energy
`11.6: Nanoscale Metal Particles
`11.7: Applications of Nanomaterials
`11.8: Discussion Questions
`11.9: Problems
`11.10: References
`
`12: Resources for Students and Teachers
`12.1: VIPEr- Virtual Inorganic Pedagogical Electronic Resource- A Community for Teachers and Students of Inorganic
`Chemistry
`12.2: Beloit College/University of Wisconsin Video Lab Manual
`12.3: Atomic and Molecular Orbitals (University of Liverpool)
`12.4: Interactive 3D Crystal Structures (University of Liverpool)
`12.5: Appendix 1- Periodic Tables
`12.6: Appendix 2- Selected Thermodynamic Values
`12.7: Appendix 3- Bond Enthalpies
`
`13: Metals and Alloys - Mechanical Properties
`13.1: Prelude to Metals and Alloys - Mechanical Properties
`
`Index
`
`Glossary
`
`Book: Introduction to Inorganic Chemistry (Wikibook) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by
`Chemistry 310 (Wikibook) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit
`history is available upon request.
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`CHAPTER OVERVIEW
`
`6: Metals and Alloys- Structure, Bonding, Electronic and Magnetic Properties
`Learning Objectives
`Identify and assign unit cells, coordination numbers, asymmetric units, numbers of atoms contained within a unit cell, and
`the fraction of space filled in a given structure.
`Relate molecular orbital theory to the delocalization of valence electrons in metals.
`Understand the concepts of electron wavelength and density of states.
`Understand the consequences of the nearly free electron model for the band structure of metals and their conductivity.
`Explain why some metals are magnetic and others are diamagnetic, and how these phenomena relate to bonding and orbital
`overlap.
`Use the Curie-Weiss law to explain the temperature dependence of magnetic ordering.
`Acquire a physical picture of different kinds of magnetic ordering and the magnetic hysteresis loops of ferro- and
`ferrimagnets.
`
`It should come as no surprise that the properties of extended solids are also connected to their structures, and so to understand what
`they do we should begin with their crystal structures. Most of the metals in the periodic table have relatively simple structures and
`so this is a good place to begin.
`6.1: Prelude to Metals and Alloys
`6.2: Unit Cells and Crystal Structures
`6.3: Bravais Lattices
`6.4: Crystal Structures of Metals
`6.5: Bonding in Metals
`6.6: Conduction in Metals
`6.7: Atomic Orbitals and Magnetism
`6.8: Ferro-, Ferri- and Antiferromagnetism
`6.9: Hard and Soft Magnets
`6.10: Discussion Questions
`6.11: Problems
`6.12: References
`
`6: Metals and Alloys- Structure, Bonding, Electronic and Magnetic Properties is shared under a CC BY-SA 4.0 license and was authored,
`remixed, and/or curated by Chemistry 310 (Wikibook) via source content that was edited to conform to the style and standards of the LibreTexts
`platform; a detailed edit history is available upon request.
`
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`6.1: Prelude to Metals and Alloys
`In the chemistry of molecular compounds, we are accustomed to the idea that properties depend strongly on structure. For example
`we can rationalize the polarity of the water molecule based on its shape. We also know that two molecules with the same
`composition (e.g., ethanol and dimethyl ether) have very different properties based on the bonding arrangements of atoms. It should
`come as no surprise that the properties of extended solids are also connected to their structures, and so to understand what they do
`we should begin with their crystal structures. Most of the metals in the periodic table have relatively simple structures and so this is
`a good place to begin. We will see in Chapter 8 that the structures of more complex compounds are also in many cases related to
`the simple structures of metals and alloys.
`
`Over 2/3 of the elements in the periodic table exist in their pure form as metals. All elemental metals (except the three - Cs, Ga, Hg - that are
`liquid) are crystalline solids at room temperature, and most have one of three simple crystal structures.
`
`6.1: Prelude to Metals and Alloys is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Chemistry 310
`(Wikibook) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available
`upon request.
`
`6.1.1
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`https://chem.libretexts.org/@go/page/189559
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`6.2: Unit Cells and Crystal Structures
`Crystals can be thought of as repeating patterns, much like wallpaper or bathroom tiles, but in three dimensions. The fundamental
`repeating unit of the crystal is called the unit cell. It is a three dimensional shape that can be repeated over and over by unit
`translations to fill space (and leave minimal gaps) in the structure. Some possible unit cells are shown in the tiling pattern at the
`right, along with arrows that indicate unit translation vectors. In three dimensions, the hexagonal or rhombic unit cells of this
`pattern would be replaced by three dimensional boxes that would stack together to fill all space. As shown in the figure, the origin
`of the unit cell is arbitrary. The same set of boxes will fill all space no matter where we define the origin of the lattice. We will see
`that pure metals typically have very simple crystal structures with cubic or hexagonal unit cells. However the crystal structures of
`alloys can be quite complicated.
`When considering the crystal structures of metals and alloys, it is not sufficient to think of each atom and its neighboring ligands as
`an isolated system. Instead, think of the entire metallic crystal as a network of atoms connected by a sea of shared valence
`electrons. The electrons are delocalized because there are not enough of them to fill each "bond" between atoms with an electron
`pair. For example, in the crystal structures of s-block and p-block metals, each atom has either 8 or 12 nearest neighbors, but the
`maximum number of s + p electrons is 8. Thus, there are not enough to put two electrons between each pair of atoms. Transition
`metals can also use their d-orbitals in bonding, but again there are never enough electrons to completely fill all the "bonds."
`
`Possible unit cells in a periodic tile pattern. The arrows connect translationally equivalent points (lattice points) in the pattern.
`
`The atoms in a metal lattice arrange themselves in a certain pattern which can be represented as a 3D box structure known as the
`unit cell which repeats across the entire metal.
`
`Simple Cubic
`
`Simple cubic.svg
`
`1 atom/cell
`
`Body Centered Cubic
`
`Face Centered Cubic
`
`Hexagonal Close Packed
`
`Lattice body centered cubic.svg
`
`Lattice face centered cubic.svg
`
`Hexagonal close packed.png
`
`2 atoms/cell
`
`4 atoms/cell
`
`2 atoms/cell
`
`Metal atoms can be approximated as spheres, and therefore are not 100 % efficient in packing, the same way a stack of cannonballs
`has some empty spaces between the balls. Different unit cells have different packing efficiencies. The number of atoms that is
`included in the unit cell only includes the fractions of atoms inside of the box. Atoms on the corners of the unit cell count as ⅛ of
`an atom, atoms on a face count as ½, an atom in the center counts as a full atom. Using this, let's calculate the number of atoms in a
`simple cubic unit cell, a face centered cubic (fcc) unit cell, and a body centered cubic (bcc) unit cell.
`Simple Cubic:
`8 corner atoms × ⅛ = 1 atom/cell. The packing in this structure is not efficient (52%) and so this structure type is very rare for
`metals.
`Body Centered Cubic, bcc:
`
`6.2.1
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`(8 corner atoms × ⅛) + (1 center atom × 1)= 2 atoms/cell. The packing is more efficient (68%) and the structure is a common one
`for alkali metals and early transition metals. Alloys such as brass (CuZn) also adopt these structures.
`Face Centered Cubic, fcc (also called Cubic Close Packed, ccp):
`(8 corner atoms × ⅛)+ (6 face atoms × ½)= 4 atoms/cell. This structure, along with its hexagonal relative (hcp), has the most
`efficient packing (74%). Many metals adopt either the fcc or hcp structure.
`Hexagonal Close Packed, hcp:
`Like the fcc structure, the packing density of hcp is 74%.
`
`The unit cell of a bcc metal contains two atoms.
`
`Calculating the packing fraction. The packing fractions of the crystal structures shown above can be calculated by dividing the
`volume of the atoms in the cell by the volume of the cell itself. The volume of the atoms in the cell is equal to the number of
`3
`atoms/cell times the volume of a sphere, (4/3)πr . The volume of the cubic cells is found by cubing the side length. As an example,
`let's calculate the packing efficiency of a simple cubic unit cell. As we saw earlier in the section, a simple cubic unit cell contains
`one atom. The side length of the simple cubic unit cell is 2r, since the centers of each atom occupy the corners of the unit cell.
`
`The same method can be applied to bcc and fcc structures.
`
`Face-centered cubic stack of cannonballs.
`
`6.2: Unit Cells and Crystal Structures is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Chemistry 310
`(Wikibook) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available
`upon request.
`
`6.2.2
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`https://chem.libretexts.org/@go/page/183325
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`Packing efficiency =
`= 0.523
`(1 atom) × ( )π
`4
`3
`r
`3
`(2r)
`3
`(6.2.1)
`

`

`6.3: Bravais Lattices
`Crystal lattices can be classified by their translational and rotational symmetry. In three-dimensional crytals, these symmetry
`operations yield 14 distinct lattice types which are called Bravais lattices. In these lattice diagrams (shown below) the dots
`represent lattice points, which are places where the whole structure repeats by translation. For example, in the body-centered cubic
`(bcc) structure of sodium metal, which is discussed below, we put one atom at the corner lattice points and another in the center of
`the unit cell. In the NaCl structure, which is discussed in Chapter 8, we place one NaCl formula unit on each lattice point in the
`face-centered cubic (fcc) lattice. That is, one atom (Na or Cl) would be placed on the lattice point and the other one would be
`placed halfway between. Similarly, in the cubic diamond structure, we place one C unit around each lattice point in the fcc lattice.
`2
`The fourteen Bravais lattices fall into seven crystal systems that are defined by their rotational symmetry. In the lowest symmetry
`system (triclinic), there is no rotational symmetry. This results in a unit cell in which none of the edges are constrained to have
`equal lengths, and none of the angles are 90º. In the monoclinic system, there is one two-fold rotation axis (by convention, the b-
`axis), which constrains two of the angles to be 90º. In the orthorhombic system, there are three mutually perpendicular two-fold
`axes along the three unit cell directions. Orthorhombic unit cells have three unequal unit cell edges that are mutually
`perpendicular. Tetragonal unit cells have a four-fold rotation axis which constrains all the angles to be 90º and makes the a and b
`axes equivalent. The rhombohedral system has a three-fold axis, which constrains all the unit cell edges and angles to be equal,
`and the hexagonal system has a six-fold axis, which constrains the a and b lattice dimensions to be equal and the angle between
`them to be 120º. The cubic system has a three-fold axis along the body diagonal of the cube, as well as two-fold axes along the
`three perpendicular unit cell directions. In the cubic system, all unit cell edges are equal and the angles between them are 90º.
`The translational symmetry of the Bravais lattices (the lattice centerings) are classified as follows:
`Primitive (P): lattice points on the cell corners only (sometimes called simple)
`Body-Centered (I): lattice points on the cell corners with one additional point at the center of the cell
`Face-Centered (F): lattice points on the cell corners with one additional point at the center of each of the faces of the cell
`Base-Centered (A, B, or C): lattice points on the cell corners with one additional point at the center of each face of one pair of
`parallel faces of the cell (sometimes called end-centered)
`Not all combinations of the crystal systems and lattice centerings are unique. There are in total 7 × 6 = 42 combinations, but it can
`be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a
`monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-
`centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.
`When the fourteen Bravais lattices are combined with the 32 crystallographic point groups, we obtain the 230 space groups.
`These space groups describe all the combinations of symmetry operations that can exist in unit cells in three dimensions. For two-
`dimensional lattices there are only 17 possible plane groups, which are also known as wallpaper groups.
`
`Crystal family
`
`Lattice system
`
`Schönflies
`
`14 Bravais Lattices
`
`Primitive
`
`Base-centered
`
`Body-centered
`
`Face-centered
`
`C
`
`C
`
`i 2
`
`h
`
`Triclinic
`
`Monoclinic, simple
`
`Monoclinic, centered
`
`rhombohedral
`
`hexagonal
`
`D
`2h
`
`D
`4h
`
`D
`3d
`
`D
`6h
`
`O
`h
`
`Orthorhombic, simple
`
`Tetragonal, simple
`
`Rhombohedral
`
`Hexagonal
`
`Cubic, simple
`
`Orthorhombic, base-
`centered
`
`Orthorhombic, body-
`centered
`
`Orthorhombic, face-centered
`
`Tetragonal, body-centered
`
`Cubic, body-centered
`
`Cubic, face-centered
`
`Triclinic
`
`Monoclinic
`
`Orthorhombic
`
`Tetragonal
`
`Hexagonal
`
`Cubic
`
`6.3: Bravais Lattices is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Chemistry 310 (Wikibook) via source
`content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
`
`6.3.1
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`https://chem.libretexts.org/@go/page/183326
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`6.4: Crystal Structures of Metals
`
`The Crystalline Nature of Metals
`All metallic elements (except Cs, Ga, and Hg) are crystalline solids at room temperature. Like ionic solids, metals and alloys have a
`very strong tendency to crystallize, whether they are made by thermal processing or by other techniques such as solution reduction
`or electroplating. Metals crystallize readily and it is difficult to form a glassy metal even with very rapid cooling. Molten metals
`have low viscosity, and the identical (essentially spherical) atoms can pack into a crystal very easily. Glassy metals can be made,
`however, by rapidly cooling alloys, particularly if the constituent atoms have different sizes. The different atoms cannot pack in a
`simple unit cell, sometimes making crystallization slow enough to form a glass.
`
`Body-centered cubic
`
`6.4.1
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`hcp (left) and fcc (right) close-packing of spheres
`
`Crystal structures
`Most metals and alloys crystallize in one of three very common structures: body-centered cubic (bcc), hexagonal close packed
`(hcp), or cubic close packed (ccp, also called face centered cubic, fcc). In all three structures the coordination number of the metal
`atoms (i.e., the number of equidistant nearest neighbors) is rather high: 8 for bcc, and 12 for hcp and ccp. We can contrast this with
`the low coordination numbers (i.e., low valences - like 2 for O, 3 for N, or 4 for C) found in nonmetals. In the bcc structure, the
`nearest neighbors are at the corners of a cube surrounding the metal atom in the center. In the hcp and ccp structures, the atoms
`pack like stacked cannonballs or billiard balls, in layers with a six-coordinate arrangement. Each atom also has six more nearest
`neighbors from layers above and below. The stacking sequence is ABCABC... in the ccp lattice and ABAB... in hcp. In both cases,
`it can be shown that the spheres fill 74% of the volume of the lattice. This is the highest volume fraction that can be filled with a
`lattice of equal spheres.
`
`Atoms in metallic crystals have a tendency to pack in dense arrangments that fill space efficiently. The simple square packing
`(above) upon which the simple cubic structure is based is inefficient and thus rare among metallic crystal structures. Body- or face-
`centered structures fill space more efficiently and more common.
`
`6.4.2
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`Periodic trends in structure and metallic behavior
`Remember where we find the metallic elements in the periodic table - everywhere except the upper right corner. This means that as
`we go down a group in the p-block (let's say, group IVA, the carbon group, or group VA, the nitrogen group), the properties of the
`elements gradually change from nonmetals to metalloids to metals. The carbon group nicely illustrates the transition. Starting at the
`top, the element carbon has two stable allotropes - graphite and diamond. In each one, the valence of carbon atoms is exactly
`satisfied by making four electron pair bonds to neighboring atoms. In graphite, each carbon has three nearest neighbors, and so
`there are two single bonds and one double bond. In diamond, there are four nearest neighbors situated at the vertices of a
`tetrahedron, and so there is a single bond to each one.
`The two elements right under carbon (silicon and germanium) in the periodic table also have the diamond structure (recall that
`these elements cannot make double bonds to themselves easily, so there is no graphite allotrope for Si or Ge). While diamond is a
`good insulator, both silicon and germanium are semiconductors (i.e., metalloids). Mechanically, they are hard like diamond. Like
`carbon, each atom of Si and Ge satisfies its valence of four by making single bonds to four nearest neighbors.
`The next element under germanium is tin (Sn). Tin has two allotropes, one with the diamond structure, and one with a slightly
`9
`distorted bcc structure. The latter has metallic properties (metallic luster, malleability), and conductivity about 10 times higher
`than Si. Finally, lead (Pb), the element under Sn, has the ccp structure, and also is metallic. Note the trends in coordination number
`and conducting properties:
`
`Element
`
`C
`
`Si
`
`Ge
`
`Sn
`
`Pb
`
`Structure
`
`graphite, diamond
`
`diamond
`
`diamond
`
`diamond, distorted bcc
`
`ccp
`
`Coord. no.
`
`3, 4
`
`4
`
`4
`
`4, 8
`
`12
`
`Conductivity
`
`semimetal, insulator
`
`semiconductor
`
`semiconductor
`
`semiconductor, metal
`
`metal
`
`The elements C, Si, and Ge obey the octet rule, and we can easily identify the electron pair bonds in their structures. Sn and Pb, on
`the other hand, adopt structures with high coordination numbers. They do not have enough valence electrons to make electron pair
`bonds to each neighbor (this is a common feature of metals). What happens in this case is that the valence electrons become
`"smeared out" or delocalized over all the atoms in the crystal. It is best to think of the bonding in metals as a crystalline
`arrangement of positively charged cores with a "sea" of shared valence electrons gluing the structure together. Because the
`electrons are not localized in any particular bond between atoms, they can move in an electric field, which is why metals conduct
`electricity well. Another way to describe the bonding in metals is nondirectional. That is, an atom's nearest neighbors surround it in
`every direction, rather than in a few particular directions (like at the corners of a tetrahedron, as we found for diamond). Nonmetals
`(insulators and semiconductors), on the other hand, have directional bonding. Because the bonding is non-directional and
`coordination numbers are high, it is relatively easy to deform the coordination sphere (i.e., break or stretch bonds) than it is in the
`case of a nonmetal. This is why elements like Pb are much more malleable than C, Si, or Ge.
`
`6.4: Crystal Structures of Metals is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Chemistry 310
`(Wikibook) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available
`upon request.
`
`6.4.3
`
`https://chem.libretexts.org/@go/page/183327
`
`Scramoge Technology Ltd.
`Ex. 2019 - Page 15
`
`

`

`6.5: Bonding in Metals
`The electron pair description of chemical bonds, which was the basis of the octet rule for p-block compounds, breaks down for
`metals. This is illustrated well by Na metal, the structure of which is shown at the left. Na has too few valence electrons to make
`electron pair bonds between each pair of atoms. We could think of the Na unit cell as having eight no-bond resonance structures in
`which only one Na-Na bond per cell contains a pair of electrons.
`
`Sodium metal crystallizes in the body-centered cubic structure, in which each atom has eight nearest neighbors. Since the electronic configuration
`1
`of Na is [Ar]3s , there are only two valence electrons per unit cell that are shared among eight Na-Na bonds. This means that the Na-Na bond
`order is 1/8 in Na metal.
`
`A more realistic way to describe the bonding in metals is through band theory. The evolution of energy bands in solids from
`simple MO theory (Chapter 2) is illustrated at the right for a chain of six Na atoms, each of which has one 3s valence orbital and
`contributes one valence electron. In general, n atomic orbitals (in this case the six Na 3s orbitals) will generate n molecular orbitals
`with n-1 possible nodes. In Chapter 2, we showed that the energy versus internuclear distance graph for a two hydrogen atom
`system has a low energy level and a high energy level corresponding to the bonding and antibonding molecular orbitals,
`respectively. These two energy levels were well separated from each other, and the two electrons in H energetically prefer the
`2
`lower energy level. If more atoms are introduced to the system, there will be a number of additional levels between the lowest and
`highest energy levels.
`
`MO picture for a linear chain of six Na atoms. Three of the six MOs can accommodate all six valence electrons. Adding more atoms to the chain
`makes more molecular orbitals of intermediate energy, which eventually merge into a continuous band of orbitals. For Na, the 3s band is always
`half-filled because each MO can accommodate two electrons.
`
`22
`In band theory, the atom chain is extrapolated to a very large number - on the order of 10 atoms in a crystal - so that the different
`combinations of bonding and anti-bonding orbitals create "bands" of possible energy states for the metal. In the language of
`physics, this approach of building the bands from discrete atomic orbitals is called the "tight-binding" approximation. The number
`of atoms is so large that the energies can be thought of as a continuum rather than a series of distinct levels. A metal will only
`partially fill this band, as there are fewer valence electrons than there are energy states to fill. In the case of Na metal, this results in
`a half-filled 3s band.
`
`6.5.1
`
`https://chem.libretexts.org/@go/page/183328
`
`Scramoge Technology Ltd.
`Ex. 2019 - Page 16
`
`

`

`Nearly free electron model
`In metals, the valence electrons are delocalized over many atoms. The total energy of each electron is given by the sum of its
`kinetic and potential energy:
`
`E = KE + PE
`2
`E ≈ p /2m + V
`
`where p is the momentum of the electron (a vector quantity), m is its mass, and V is an average potential that the electron feels
`from the positive cores of the atoms. This potential holds the valence electrons in the crystal but, in the free electron model, is
`essentially uniform across the crystal.
`
`Electron wavelength and wavenumber
`What are the consequences of this model for band theory? For a hypothetical infinite chain (i.e., a 1D crystal) of Na atoms, the
`molecular orbitals at the bottom of the 3s band are fully bonding and the wavelength of electrons (2x the distance between nodes)
`in these orbitals is very long. At the top of the band, the highest orbital is fully antibonding and the wavelength is 2 times the
`distance between atoms (2a), since there