Properties and Crystalline Structures of Solids

Learning Objectives and Course Overview

• Course Identification: CHEM 1155 — Week 2: Properties of Solids (Chapter 10: Liquids & Solids).

• Learning Target 1.2 (LS): Define the major types of solids:

  • Atomic solids

  • Molecular solids

  • Metallic solids

  • Network solids

  • Ionic solids

  • Amorphous solids

• Learning Target 1.3 (LS): Describe characteristic differences between major types of solids in terms of physical properties, including:

  • Melting point

  • Malleability

  • Electrical and thermal conductivity

• Learning Target 1.4 (LS): Define the three fundamental types of cubic unit cells.

• Learning Target 7.1 (LS): Use the geometry of cubic unit cells to calculate densities, molar masses, and atomic radii of metallic solids.

Classifications and Characteristics of Solid Types

• Atomic Solids: Consist of individual atoms held together by weak dispersion forces (e.g., solid noble gases). Characterized by extremely low melting points and non-conductive behavior.

• Molecular Solids: Composed of neutral molecules held together by intermolecular forces (London dispersion forces, dipole-dipole interactions, or hydrogen bonds). Characterized by relatively low-to-moderate melting points, brittleness, and poor electrical conductivity.

• Metallic Solids: Formed by positive metal cations surrounded by a delocalized "sea of electrons". Characterized by variable-to-high melting points, high malleability, high ductility, and excellent thermal and electrical conductivity.

• Network Covalent Solids: Consist of atoms connected throughout a multi-dimensional network entirely by strong covalent bonds (e.g., diamond, quartz, silicon carbide). Characterized by extremely high melting points, exceptional hardness, and low electrical conductivity (with exceptions such as graphite).

• Ionic Solids: Composed of alternating cations and anions held in a crystal lattice by strong electrostatic attractions. Characterized by high melting points, hard and brittle mechanical properties, and electrical conductivity when molten or dissolved in aqueous solution (but non-conductive in the solid state).

• Amorphous Solids: Lack a long-range, ordered repeating lattice structure (e.g., glass, synthetic polymers). Characterized by gradual softening over a temperature range rather than a sharp, distinct melting point.

Cubic Unit Cell Geometry

• A unit cell represents the smallest repeating structural unit that displays the full symmetry and stoichiometry of a crystalline lattice.

• Crystalline metallic solids typically adopt one of three primary cubic unit cell geometries:

• Primitive Cubic (Simple Cubic - SC):

  • Lattice points/atoms are located exclusively at the $8$ corners of the cube.

  • Atoms contact each other along the edges of the cell.

• Body-Centered Cubic (BCC):

  • Atoms are located at all $8$ corners of the cube, plus $1$ full atom situated at the geometric center of the cube.

  • Atoms contact each other along the body diagonal of the cell.

• Face-Centered Cubic (FCC):

  • Atoms are located at all $8$ corners of the cube, plus $1$ atom centered on each of the $6$ cube faces.

  • Atoms contact each other along the face diagonals of the cell.

Spatial Sharing and Counting Atoms in Unit Cells

• Atoms positioned on corners, edges, or faces of a unit cell are shared with adjacent neighboring unit cells in three-dimensional space:

• Corner Positions:

  • A corner position is shared equally among $8$ adjacent unit cells that meet at that point.

  • Fractional contribution of each corner position to a single unit cell: $\frac{1}{8}$ of an atom.

• Face Positions:

  • A face-centered position is shared equally between $2$ adjacent unit cells that share that boundary face.

  • Fractional contribution of each face position to a single unit cell: $\frac{1}{2}$ of an atom.

• Body Center Position:

  • A body-centered position lies completely inside a single unit cell and is not shared with any neighboring cell.

  • Fractional contribution of a central body position: $1$ full atom.

• Net Atom Counts per Unit Cell Type:

  • Primitive Cubic (SC):

  • Calculation: $(8\text{ corners}) \times \left(\frac{1}{8}\text{ atom per corner}\right) = 1\text{ net atom per unit cell}$

• Body-Centered Cubic (BCC):

  • Calculation: $\left(\frac{1}{8} \times 8\right) + 1 = 1 + 1 = 2\text{ net atoms per unit cell}$

• Face-Centered Cubic (FCC):

  • Calculation: $\left(\frac{1}{8} \times 8\right) + \left(\frac{1}{2} \times 6\right) = 1 + 3 = 4\text{ net atoms per unit cell}$

Unit Cell Packing Arrangements in Metallic Elements

• The crystalline packing arrangement adopted by metallic elements across the periodic table depends on atomic structure, electron configuration, and packing efficiency:

• Primitive (Simple Cubic):

  • Extremely rare due to low packing efficiency ($52\%$ space occupancy).

  • Example element: Polonium ($\text{Po}$).

• Body-Centered Cubic (BCC):

  • Packing efficiency of $68\%$.

  • Adopted by Group 1 alkali metals ($\text{Li}$, $\text{Na}$, $\text{K}$, $\text{Rb}$, $\text{Cs}$, $\text{Fr}$), as well as Barium ($\text{Ba}$), Vanadium ($\text{V}$), Chromium ($\text{Cr}$), Iron ($\text{Fe}$), Niobium ($\text{Nb}$), Molybdenum ($\text{Mo}$), Tantalum ($\text{Ta}$), and Tungsten ($\text{W}$).

• Cubic Close Packing (CCP) / Face-Centered Cubic (FCC):

  • Maximum close-packing efficiency of $74\%$.

  • Adopted by Aluminum ($\text{Al}$), Calcium ($\text{Ca}$), Strontium ($\text{Sr}$), Nickel ($\text{Ni}$), Copper ($\text{Cu}$), Palladium ($\text{Pd}$), Silver ($\text{Ag}$), Platinum ($\text{Pt}$), Gold ($\text{Au}$), Lead ($\text{Pb}$), and Actinium ($\text{Ac}$).

• Hexagonal Close Packing (HCP):

  • Maximum close-packing efficiency of $74\%$ in a non-cubic hexagonal lattice arrangement.

  • Adopted by Beryllium ($\text{Be}$), Magnesium ($\text{Mg}$), Scandium ($\text{Sc}$), Titanium ($\text{Ti}$), Cobalt ($\text{Co}$), Zinc ($\text{Zn}$), Yttrium ($\text{Y}$), Zirconium ($\text{Zr}$), Technetium ($\text{Tc}$), Ruthenium ($\text{Ru}$), Rhodium ($\text{Rh}$), Cadmium ($\text{Cd}$), Indium ($\text{In}$), Hafnium ($\text{Hf}$), Rhenium ($\text{Re}$), Osmium ($\text{Os}$), Iridium ($\text{Ir}$), and Thallium ($\text{Tl}$).

Density and Structural Calculations for Unit Cells

• Calculating the physical density ($d$) of a metallic crystal requires determining both the mass and volume of a single unit cell:   $\text{Density } (d) = \frac{\text{Mass of unit cell}}{\text{Volume of unit cell}}$

• Step 1: Calculate Unit Cell Volume ($V$):

  • Volume of a cubic cell with edge length $a$ is given by $V = a^3$.

  • Unit Conversion Requirement: Always convert the unit cell edge length from picometers ($\text{pm}$) to centimeters ($\text{cm}$) before cubing to avoid large scale-factor errors.

  • Conversion equivalence:     $100\,\text{pm} = 1.00 \times 10^{-8}\,\text{cm}$     $1\,\text{pm} = 1.00 \times 10^{-10}\,\text{cm}$

  • Expected Magnitude: Typical cubic unit cell volumes range from:     $\approx 1 \times 10^{-23}\,\text{cm}^3 \text{ to } 1 \times 10^{-22}\,\text{cm}^3$

• Step 2: Calculate Mass per Atom:

  • Convert molar mass ($\text{g\,mol}^{-1}$) to mass per individual atom ($\text{g\,atom}^{-1}$) using Avogadro’s number ($N_A = 6.022 \times 10^{23}\,\text{atoms\,mol}^{-1}$):     $\text{Mass per atom} = \frac{\text{Molar mass } (\text{g\,mol}^{-1})}{6.022 \times 10^{23}\,\text{atoms\,mol}^{-1}}$

  • Expected Magnitude: The mass of an individual metal atom typically ranges from:     $\approx 1 \times 10^{-23}\,\text{g\,atom}^{-1} \text{ to } 1 \times 10^{-22}\,\text{g\,atom}^{-1}$

• Step 3: Calculate Total Unit Cell Mass:

  • Multiply the mass per single atom by the net number of atoms contained within the specific unit cell type ($Z$):     $\text{Mass of unit cell} = Z \times \text{Mass per atom}$

  • Number of atoms ($Z$) by cell type:

  • Simple Cubic (SC): $Z = 1$

  • Body-Centered Cubic (BCC): $Z = 2$

  • Face-Centered Cubic (FCC): $Z = 4$

• Step 4: Compute Crystal Density:

  • Divide total unit cell mass by unit cell volume:     $\text{Density } (d) = \frac{\text{Mass of unit cell } (\text{g})}{\text{Volume of unit cell } (\text{cm}^3)}$

Academic Schedule and Upcoming Deliverables

• Wednesday 09/09:

  • Chapter 10 Liquids & Solids — Open Work Time

• Thursday 09/10:

  • Chapter 10 Liquids & Solids — Structure of Crystalline Solids

  • Key Topics: 3D representations of unit cells, calculations related to cubic cell density, identifying ionic unit cells

• Monday 09/14:

  • Learning Curve 01

  • Homework 02

  • Submission Deadline: 11:59 PM

• Tuesday 09/15:

  • Spontaneity & Standard Molar Entropy

  • Key Topics: Defining entropy & statistical thermodynamics, defining spontaneity in chemical reactions, predicting standard molar entropy from molecular shape