Chapter 13: Solids and Modern Materials – Study Notes

Chapter 13: Solids and Modern Materials

X-ray Diffraction

X-ray diffraction is a key technique for investigating the structure of crystalline solids. It allows chemists to determine the arrangement of atoms and the distances between them by analyzing how X-rays are diffracted by the crystal lattice.

• X-ray diffraction reveals the atomic arrangement within crystals by producing interference patterns.

• Atomic planes within crystals are typically separated by about 100 pm.

• The path length and angle of reflection of X-rays provide information about the distance between planes of atoms.

• Bragg's Law relates the wavelength of incident X-rays and the angle of reflection to the distance between atomic planes:

$n\lambda = 2d\sin(\theta)$

• Example: Given $\lambda = 179$ pm, $\theta = 35.7^\circ$, $n = 1$, calculate $d$ (the separation between layers of atoms).

Crystal Lattice

The crystal lattice is the regular, repeating arrangement of atoms, ions, or molecules in a solid. The smallest repeating unit is called the unit cell.

• Unit cell: The smallest portion of a crystal lattice that shows the full symmetry of the arrangement.

• Unit cells are repeated in three dimensions to build up the entire crystal.

• Unit cells are classified by their symmetry.

Types of Unit Cells and Lattice Parameters

There are seven types of three-dimensional primitive lattices, each described by six lattice parameters: the lengths of the cell edges ($a$, $b$, $c$) and the angles between them ($\alpha$, $\beta$, $\gamma$).

• Lattice parameters are typically measured in Ångströms (Å) or picometers (pm).

Lattice Type	Edge Lengths	Angles
Cubic	$a = b = c$	$\alpha = \beta = \gamma = 90^\circ$
Tetragonal	$a = b \neq c$	$\alpha = \beta = \gamma = 90^\circ$
Orthorhombic	$a \neq b \neq c$	$\alpha = \beta = \gamma = 90^\circ$
Rhombohedral	$a = b = c$	$\alpha = \beta = \gamma \neq 90^\circ$
Hexagonal	$a = b \neq c$	$\alpha = \beta = 90^\circ, \gamma = 120^\circ$
Monoclinic	$a \neq b \neq c$	$\alpha = \gamma = 90^\circ, \beta \neq 90^\circ$
Triclinic	$a \neq b \neq c$	$\alpha \neq \beta \neq \gamma \neq 90^\circ$

Unit Cells: Coordination Number and Packing Efficiency

The coordination number is the number of nearest neighbors to a particle in a crystal. Packing efficiency is the fraction of volume in a unit cell occupied by the constituent particles.

Cubic Cell Name	Atoms per Unit Cell	Coordination Number	Edge Length (in terms of r)	Packing Efficiency
Simple Cubic	1	6	$2r$	52%
Body-Centered Cubic	2	8	$4r/\sqrt{3}$	68%
Face-Centered Cubic	4	12	$2\sqrt{2}r$	74%

Cubic Unit Cells

Cubic unit cells are the most common and are characterized by all edges being equal and all angles being 90°.

• Simple Cubic (Primitive): 8 particles at corners, 1/8 of each inside the cell, total 1 atom per cell.

• Body-Centered Cubic (BCC): 8 corners + 1 center, total 2 atoms per cell.

• Face-Centered Cubic (FCC): 8 corners + 6 faces, total 4 atoms per cell.

Fraction of each particle within the cube:

• Corner: 1/8

• Edge: 1/4

• Face: 1/2

• Fully inside: 1

Closest-Packed Structures

Efficient packing of spheres leads to two main types of closest-packed structures:

• Hexagonal Closest Packing (HCP): ABAB... stacking pattern.

• Cubic Closest Packing (CCP): ABCABC... stacking pattern, equivalent to FCC.

Classifications of Solids

Solids are classified based on the nature of their constituent particles and the forces holding them together.

• Molecular solids: Molecules held by intermolecular forces (e.g., CO2, H2O).

• Ionic solids: Ions held by strong electrostatic forces (e.g., NaCl).

• Atomic solids: Atoms held by various forces (nonbonding, metallic, or covalent).

Atomic Solids Subtypes

• Nonbonding: Weak dispersion forces, very low melting points (e.g., noble gases).

• Metallic: Metallic bonding, variable melting points, good conductors.

• Network Covalent: Covalent bonds, very high melting points, hard (e.g., diamond, quartz).

Molecular Solids

Molecular solids are composed of molecules held together by intermolecular forces such as dispersion forces, dipole-dipole attractions, and hydrogen bonds.

• Generally have low melting points due to weak forces.

• More symmetric molecules pack more efficiently, leading to higher melting points.

• Some molecular solids can exist in different forms (polymorphs), which is important in pharmaceuticals.

Ionic Solids

Ionic solids consist of cations and anions arranged in a lattice and held together by strong electrostatic (Coulombic) forces.

• High melting points.

• Stability increases with higher coordination number (more close cation-anion interactions).

• Coordination number depends on the relative sizes of the ions.

Metallic Bonding

Metallic solids are composed of metal atoms that release their valence electrons, forming a 'sea' of mobile electrons around fixed metal cations.

• This electron sea model explains properties such as electrical conductivity, malleability, and ductility.

Examples of Ionic Structures

Structure	Coordination Number	Key Features
Cesium Chloride (CsCl)	8	Cs+ in center, Cl- at corners, simple cubic arrangement
Rock Salt (NaCl)	6	Na+ in octahedral holes, Cl- in FCC arrangement
Zinc Blende (ZnS)	4	Zn2+ in tetrahedral holes, S2- in FCC arrangement
Fluorite (CaF2)	8 for Ca2+, 4 for F-	Ca2+ in FCC, F- in tetrahedral holes

Network Covalent Atomic Solids – Carbon

• Graphite: Carbon atoms in sheets, each bonded to three others (sp2), sheets held by dispersion forces, high melting point, electrical conductor parallel to sheets.

• Diamond: Each carbon atom bonded to four others (sp3), forming a rigid 3D network, very high melting point, electrical insulator, extremely hard.

Practice Problems and Examples

• Calculate the separation between atomic layers using Bragg's Law.

• Determine the radius of an atom in a unit cell given the cell volume.

• Estimate the density of ionic solids from their structure and atomic radii.

• Classify solids as molecular, ionic, metallic, or network covalent based on their properties.