AP Chemistry 2.3-2.4: Structure of Ionic Solids and Metals

Crystalline solid

A solid whose atoms, ions, or molecules occupy a regular, repeating three-dimensional arrangement.

Ionic solid

A crystalline solid composed of cations and anions held together by electrostatic attractions in an ionic lattice.

What physical properties are characteristic of ionic solids?

Ionic solids are generally hard, brittle, and have high melting points. They do not conduct electricity as solids because their ions cannot move, but they conduct when molten or dissolved because the ions are mobile.

Why are ionic solids often high-melting and brittle?

Strong attractions between oppositely charged ions require substantial energy to overcome. If layers shift, like charges can be brought next to one another, causing strong repulsion and cleavage rather than bending.

Metallic solid

A solid consisting of metal atoms whose nuclei are arranged in a lattice and whose valence electrons are delocalized throughout the structure.

What properties result from metallic bonding?

Delocalized electrons give metals high electrical and thermal conductivity and metallic luster. The ability of metal atoms or layers to shift while remaining bonded produces malleability and ductility.

Why do metals generally deform instead of shattering when subjected to pressure?

Metallic bonding is nondirectional: the delocalized electrons continue to attract the metal nuclei even as layers shift. Thus, metals can be malleable and ductile rather than brittle.

Why do particles in most crystalline solids adopt regular arrangements?

Efficient packing maximizes attractive interactions and minimizes the total potential energy of the particles. The repeating arrangement is therefore often more stable than a disordered arrangement.

Unit cell

The smallest repeating portion of a crystal lattice that reproduces the entire three-dimensional structure when translated in all directions.

Lattice point

A point in a crystal lattice representing the position of an atom, ion, or equivalent environment in the repeating structure.

Why do identical lattice points in a unit cell have equivalent environments?

A lattice is generated by repeating the unit cell through translation. Therefore, corresponding lattice points experience the same neighboring arrangement and represent equivalent positions in the crystal.

How is the number of atoms contributed by a unit-cell position determined?

A corner atom contributes $1/8$, an atom centered on a face contributes $1/2$, an atom centered on an edge contributes $1/4$, and an atom entirely inside the cell contributes $1$.

Coordination number

The number of nearest neighboring particles in contact with, or directly surrounding, a particle in a crystal structure.

Simple cubic unit cell

A cubic unit cell with atoms only at its eight corners. Since each corner contributes $1/8$, it contains $8(1/8)=1$ atom per unit cell, has coordination number 6, and has an approximate packing efficiency of 52%.

What relationship connects atomic radius $r$ and edge length $a$ in a simple cubic unit cell?

Adjacent corner atoms touch along an edge, so $a=2r$ and $r=a/2$.

Body-centered cubic (BCC) unit cell

A cubic unit cell with atoms at all eight corners and one atom in the center. It contains $8(1/8)+1=2$ atoms per cell, has coordination number 8, and is about 68% efficiently packed.

What relationship connects atomic radius $r$ and edge length $a$ in a BCC unit cell?

Atoms touch along the body diagonal. Because the body diagonal has length $\sqrt{3}a$ and spans four radii, $\sqrt{3}a=4r$, so $r=\frac{\sqrt{3}a}{4}$.

Face-centered cubic (FCC) unit cell

A cubic unit cell with atoms at the corners and at the center of each face. It contains $8(1/8)+6(1/2)=4$ atoms per cell, has coordination number 12, and is about 74% efficiently packed.

What relationship connects atomic radius $r$ and edge length $a$ in an FCC unit cell?

Atoms touch along a face diagonal. Since the face diagonal has length $\sqrt{2}a$ and spans four radii, $\sqrt{2}a=4r$, so $r=\frac{\sqrt{2}a}{4}$.

Cubic closest packing (CCP)

Another name for the FCC arrangement. Its close-packed layers follow an $ABCABC\ldots$ stacking sequence, and each atom has coordination number 12.

How do hexagonal closest packing and cubic closest packing differ?

Both have close-packed layers and coordination number 12. HCP repeats layers as $ABAB\ldots$, whereas CCP/FCC repeats them as $ABCABC\ldots$.

Why do close-packed structures have a coordination number of 12?

Each particle contacts six particles in its own hexagonal layer, three in the layer above, and three in the layer below, for a total of $6+3+3=12$ nearest neighbors.

How can the density of a crystalline metal be calculated from its unit cell?

Use $\rho=\frac{m_{\text{unit cell}}}{V_{\text{unit cell}}}$. The unit-cell mass is $nM/N_A$, where $n$ is the number of atoms per cell, $M$ is molar mass, and $N_A$ is Avogadro's number; for a cubic cell, $V=a^3$.

A metal has a measured density and unit-cell edge length. How can these data help identify its unit-cell type?

Calculate a predicted density for candidate structures using $\rho=\frac{nM}{N_Aa^3}$, where $n=1$ for simple cubic, $n=2$ for BCC, and $n=4$ for FCC. The structure whose predicted density agrees with the measured value is supported.

Why can a unit-cell edge length be used to determine an atomic radius?

The crystal geometry specifies which atoms touch along an edge, face diagonal, or body diagonal. Equating that geometric distance to the appropriate number of radii allows $r$ to be calculated from the measured edge length.

What geometric quantities define a general unit cell?

A unit cell is specified by three edge lengths, $a$, $b$, and $c$, and three interaxial angles, often written $\alpha$, $\beta$, and $\gamma$.

How many crystal lattice systems and distinct unit-cell types are recognized?

There are seven lattice systems and fourteen distinct unit-cell types.

Why is the structure of an ionic crystal more complicated than that of a pure metal?

An ionic crystal contains at least two types of ions, usually with different charges and radii. Its stable arrangement must accommodate both ion sizes and satisfy the compound's stoichiometric ratio.

What factors primarily determine the structure of an ionic crystal?

The most important factors are the relative sizes of the cations and anions and the stoichiometric ratio of their numbers in the compound.

What two principles generally favor stability in an ionic crystal?

A stable structure tends to place each ion in contact with as many oppositely charged ions as possible while maintaining contact between cations and anions. The achievable arrangement is strongly influenced by relative ionic sizes and the cation-to-anion ratio.