Problem 110 - Chemistry: The Central Science in SI Units, Global Edition

Question

What type of lattice-primitive cubic, body-centered cubic, or face-centered cubic-does each of the following (a) NaF, (b) Ag, (c) Fe, (d) \(\mathrm{CsCl}\) structure types possess: (e) \(\mathrm{ScF}_{3} ?\)

Step-by-Step Solution

Answer

(a) NaF has a face-centered cubic (fcc) lattice structure. (b) Ag has a face-centered cubic (fcc) lattice structure. (c) Fe has a body-centered cubic (bcc) lattice structure. (d) CsCl has a primitive cubic (simple cubic) lattice structure. (e) ScF3 has a complex arrangement related to face-centered cubic (fcc) lattice structure.

Step 1: (a) NaF Lattice Type

NaF has an alternating arrangement of Na and F ions in a cubic manner with each ion surrounded by six ions of the other type. This arrangement is characteristic of a face-centered cubic (fcc) lattice structure.

Step 2: (b) Ag Lattice Type

Silver (Ag) has a cubic arrangement where atoms are located at the corners and in the centers of the faces of the cell. This is a characteristic of the face-centered cubic (fcc) lattice structure.

Step 3: (c) Fe Lattice Type

Iron (Fe) has a cubic arrangement with an atom at each corner and an additional atom in the center of the cell. This arrangement is typical of a body-centered cubic (bcc) lattice structure.

Step 4: (d) CsCl Lattice Type

The CsCl structure consists of one Cs ion located at the corners of the cell and a Cl ion in the center of the cubic arrangement. This type of arrangement is characteristic of a primitive cubic (simple cubic) lattice structure.

Step 5: (e) ScF3 Lattice Type

ScF3 has a complex arrangement where the Sc ions are located at the corners and centers of the faces of the cell, while the F ions are located at the quarter and three-quarter positions along the cell edges. This type of arrangement does not fit any of the three simple lattice types described (primitive cubic, bcc, or fcc). However, its closest relation would be a face-centered cubic (fcc) lattice structure, as its basis contains FCC-type surroundings for Sc and F ions.

Key Concepts

Face-Centered Cubic

A face-centered cubic (fcc) lattice is a crystal structure where atoms are located at each of the corners and the centers of all the faces of the cube. This structure is incredibly efficient in terms of space utilization, so it's quite common in metals like silver (Ag) and many salts like sodium fluoride (NaF).

In an fcc arrangement, each atom is surrounded by 12 others. This high coordination number means atoms can "pack" together closely, which maximizes the density of the material. The arrangement is:

• 4 atoms per unit cell.
• One-eighth of an atom at each of the 8 corners (totals to 1 atom).
• One-half of an atom on each of the 6 faces (totals to 3 atoms).

The efficiency of packing in the fcc structure leads to a high packing efficiency of about 74%. This means a significant fraction of the space within the crystal lattice is occupied by atoms. Such dense packing affects the properties of the substance, often yielding highly malleable and ductile materials.

Because of its tightly packed structure, materials with an fcc lattice are usually soft and have low melting points, facilitating easy shaping and molding, which is beneficial in various industrial applications.

Body-Centered Cubic

The body-centered cubic (bcc) structure is another type of crystal lattice found in various metals, including iron (Fe). This structure is characterized by an atom at each corner of the cube and one atom situated at the very center of the cube.

With a bcc structure:

• 2 atoms per unit cell.
• One-eighth of an atom contributed by each of the 8 corner atoms (totals to 1 atom).
• One whole atom located in the center of the cube.

In contrast to the fcc lattice, the packing efficiency in a bcc structure is about 68%. Consequently, this structure is less densely packed than fcc. This affects material properties, as bcc structures tend to make materials harder and stronger, qualities seen in steel and other alloys that incorporate iron.

The coordination number for bcc is 8, meaning each atom is directly adjacent to 8 others. This arrangement provides a balance between density and strength, making bcc-structured materials suitable for use in construction and manufacturing where durability is essential. They are typically brittle compared to fcc due to the lower coordination number and packing efficiency.

Primitive Cubic

The primitive cubic lattice, also known as simple cubic, is the simplest form of crystal lattice but is rare due to its inefficiency in packing atoms. In this structure, atoms are positioned only at each corner of the cube, contributing minimally to the space-filling attributes of the lattice.

For a primitive cubic structure:

• 1 atom per unit cell.
• Each of the 8 corners contributes one-eighth of an atom (totals to 1 atom).

The packing efficiency for this structure is quite low at approximately 52%. This reflects the large amount of empty space within the structure, making materials with a primitive cubic lattice structure relatively porous.

Due to its low coordination number of 6 and inefficiency in packing, the primitive cubic structure results in materials that are generally soft, brittle, and have a low density. Such properties are not common in naturally occurring materials but can be found in some synthetic compounds, like cesium chloride (CsCl), where specific chemical interactions dictate this arrangement despite its inefficiency.