Premed · Premed · General Chemistry 1

Lecture 18: Solids and Crystal Structures

General Chemistry I

Learning Objectives

By the end of this lecture, students will be able to:

1. Classify solids as crystalline or amorphous
2. Describe the four types of crystalline solids (ionic, molecular, covalent network, metallic) and their properties
3. Describe the basic cubic unit cells (simple cubic, body-centered cubic, face-centered cubic)
4. Calculate the number of atoms per unit cell and relate unit cell dimensions to atomic radius
5. Calculate the density of a crystalline solid from unit cell parameters
6. Describe metallic bonding using the electron sea model and band theory

Lecture Content

I. Crystalline vs. Amorphous Solids

Crystalline solids have their atoms, ions, or molecules arranged in a regular, repeating three-dimensional pattern known as long-range order. This ordered arrangement gives crystals sharp melting points (since all the bonds break at similar energies) and well-defined geometric faces and angles. Examples include NaCl, diamond, ice, and iron. Amorphous solids lack long-range order; their constituent particles are arranged randomly, resembling a frozen liquid. These materials soften gradually over a temperature range rather than melting sharply. Glass, rubber, most plastics, and amorphous silicon are common examples. Amorphous solids are sometimes referred to as "supercooled liquids," though this terminology is debated.

II. Types of Crystalline Solids

A. Ionic Solids

In ionic solids, the lattice positions are occupied by cations and anions held together by electrostatic attraction. These solids are hard but brittle and have high melting points. They are poor electrical conductors in the solid state (ions are immobilized) but conduct well when molten or dissolved in water (ions become mobile). NaCl (melting point 801 C), CaF2, and MgO (melting point 2852 C) are representative examples.

B. Molecular Solids

Molecular solids have discrete molecules at their lattice positions, held together by intermolecular forces -- London dispersion, dipole-dipole, or hydrogen bonding. Because these forces are relatively weak, molecular solids are typically soft with low to moderate melting points. They are poor electrical conductors in all phases. Familiar examples include ice (H2O, mp 0 C), dry ice (CO2, sublimes at -78.5 C), sucrose, and I2.

C. Covalent Network Solids

In covalent network solids, atoms are connected by a continuous network of covalent bonds extending throughout the entire crystal. These solids are extremely hard, have very high melting points, and are generally poor conductors of electricity (with graphite as the notable exception). Diamond, in which each carbon is sp3 hybridized and bonded to four other carbons in a tetrahedral arrangement, is the hardest natural substance with a melting point of approximately 3550 C. Graphite features sp2-hybridized carbons arranged in planar hexagonal sheets; weak London forces between layers allow them to slide, making graphite useful as a lubricant and pencil lead. Delocalized pi electrons within each layer enable electrical conduction parallel to the sheets. Silicon dioxide (quartz, SiO2), in which each silicon bonds to four oxygens and each oxygen bridges two silicons, melts at 1710 C.

D. Metallic Solids

Metallic solids consist of metal cations arranged in a lattice and surrounded by a delocalized "sea" of valence electrons. This electron sea accounts for the excellent electrical and thermal conductivity of metals, as the mobile electrons can carry charge and heat efficiently. Metals are malleable and ductile because layers of cations can slide past one another without disrupting the electron sea. Their melting points and hardness vary widely (sodium melts at 98 C; tungsten at 3422 C). Metals are lustrous because free electrons can absorb and re-emit photons of light.

III. Crystal Lattices and Unit Cells

A crystal lattice is an infinite, regular array of points representing the positions of atoms, ions, or molecules. The unit cell is the smallest repeating unit that, when stacked in three dimensions, generates the entire crystal. The geometry of the unit cell defines the crystal system. Seven crystal systems exist, but this course focuses on the cubic system, in which all edges are equal and all angles are 90 degrees.

IV. Cubic Unit Cells

A. Simple Cubic (SC)

The simple cubic unit cell has atoms at each of its 8 corners. Because each corner atom is shared among 8 adjacent unit cells, only 1/8 of each atom belongs to any given cell, yielding 8 x (1/8) = 1 atom per unit cell. The coordination number (number of nearest neighbors) is 6. Atoms touch along the cell edge, so the edge length equals 2r (where r is the atomic radius). With a packing efficiency of just 52%, simple cubic is the least efficiently packed arrangement. Only polonium crystallizes in this structure.

B. Body-Centered Cubic (BCC)

The BCC unit cell has 8 corner atoms plus 1 atom at the center of the cube, giving 8(1/8) + 1 = 2 atoms per unit cell. The coordination number is 8. Atoms touch along the body diagonal, yielding the relationship 4r = a*sqrt(3), or a = 4r/sqrt(3). The packing efficiency is 68%. Iron (at room temperature), chromium, tungsten, sodium, potassium, and barium all adopt BCC structures.

C. Face-Centered Cubic (FCC) / Cubic Close-Packed (CCP)

The FCC unit cell has 8 corner atoms plus 6 face-centered atoms, each shared between 2 cells, giving 8(1/8) + 6(1/2) = 4 atoms per unit cell. The coordination number is 12. Atoms touch along the face diagonal: 4r = asqrt(2), or a = 2rsqrt(2). With a packing efficiency of 74%, FCC achieves the maximum packing density possible for identical spheres (tied with HCP). Copper, aluminum, silver, gold, lead, and nickel all crystallize in this structure.

<image>A three-panel diagram of the three cubic unit cells. Panel A (Simple Cubic): a cube with atoms at each corner. One atom highlighted to show it is shared among 8 cubes. An expanded view shows atoms touching along the edge, with the relationship a = 2r labeled. Text: "1 atom/unit cell, CN = 6, 52% packing." Panel B (Body-Centered Cubic): a cube with atoms at corners plus one full atom at the center. A body diagonal is drawn showing atoms touching along it, with 4r = asqrt(3) labeled. Text: "2 atoms/unit cell, CN = 8, 68% packing." Panel C (Face-Centered Cubic): a cube with atoms at corners and at the center of each face. A face diagonal is drawn showing atoms touching along it, with 4r = asqrt(2) labeled. Text: "4 atoms/unit cell, CN = 12, 74% packing." Each panel shows both the space-filling model and the ball-and-stick model with the unit cell edges outlined.</image>

V. Close-Packed Structures

Maximum packing efficiency (74%) can be achieved in two ways. Hexagonal close-packed (HCP) uses ABAB... layer stacking, with alternating layers; magnesium, zinc, and titanium adopt this structure. Cubic close-packed (CCP) uses ABCABC... layer stacking and is equivalent to the FCC unit cell; copper, aluminum, and gold are examples. Both arrangements have a coordination number of 12. Within close-packed structures, the gaps between atoms create two types of interstitial holes: octahedral holes (surrounded by 6 atoms, coordination number 6) and tetrahedral holes (surrounded by 4 atoms, coordination number 4). In ionic crystals, the smaller ions frequently occupy these holes within the close-packed arrangement of the larger ions.

VI. Calculating Density from Unit Cell Data

The density of a crystalline solid can be calculated directly from unit cell information using: density = mass of unit cell / volume of unit cell. The mass of the unit cell equals (number of atoms per unit cell) x (molar mass / Avogadro's number), and the volume of a cubic cell is a^3. Combining these gives d = (Z x M) / (N_A x a^3), where Z is the number of atoms per unit cell, M is the molar mass in g/mol, N_A is 6.022 x 10^23, and a is the edge length in cm. This calculation can work in either direction -- determining the crystal structure from a known density and atomic radius, or predicting density from crystallographic data.

VII. Ionic Crystal Structures

Ionic crystals arrange oppositely charged ions in a lattice, with the specific structure depending on the radius ratio (r_cation / r_anion). Several common structures illustrate the variety. The rock salt structure (NaCl) features an FCC arrangement of Cl- with Na+ filling all octahedral holes, giving a coordination number of 6:6. The cesium chloride structure (CsCl) has a simple cubic arrangement of Cl- with Cs+ at the center (this is not BCC, since the two ions are different), with coordination number 8:8. The zinc blende structure (ZnS) places S^2- in an FCC arrangement with Zn^2+ occupying half the tetrahedral holes, yielding coordination number 4:4. The fluorite structure (CaF2) has Ca^2+ in an FCC arrangement with F- occupying all tetrahedral holes, giving coordination number 8:4.

VIII. Metallic Bonding Models

A. Electron Sea Model

The electron sea model envisions metal cations arranged in a regular lattice and surrounded by a delocalized sea of valence electrons that are free to move throughout the entire metal. This simple picture explains electrical conductivity (mobile electrons carry charge), thermal conductivity (electrons transfer kinetic energy), malleability and ductility (cation layers can slide without breaking bonds), and metallic luster (free electrons absorb and re-emit photons).

B. Band Theory (Brief Introduction)

When a large number of metal atoms combine, their atomic orbitals merge into continuous bands of closely spaced energy levels. The valence band, at lower energy, is filled or partially filled with electrons. The conduction band, at higher energy, is empty or partially filled. In conductors, the valence and conduction bands overlap or the valence band is only partially filled, so electrons move freely into higher energy states. In insulators, a large band gap separates the fully occupied valence band from the empty conduction band, preventing electron flow at normal temperatures. Semiconductors have a small band gap; their conductivity increases with temperature as thermal energy promotes electrons across the gap. Semiconductors can be intrinsic (pure elements like Si and Ge) or doped -- n-type doping adds electron-rich atoms (like P in Si) to provide extra electrons, while p-type doping adds electron-poor atoms (like B in Si) to create electron vacancies called holes.

<image>A band theory energy diagram comparing conductors, semiconductors, and insulators. Three side-by-side energy diagrams with a vertical "Energy" axis. Left (Conductor, e.g., Cu): the valence band and conduction band overlap, with electrons filling up to partway through the overlapping region. A note: "Electrons easily move to higher energy states." Center (Semiconductor, e.g., Si): the valence band is fully filled (shaded), a small band gap (labeled ~1 eV) separates it from the empty conduction band. Arrows show electrons jumping the gap with thermal energy. Right (Insulator, e.g., diamond): the valence band is fully filled, a large band gap (labeled ~5.5 eV) separates it from the empty conduction band. A note: "Gap too large for electrons to cross at normal temperatures."</image>

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