Question

\(\mathrm{NaCl}\) has \(\mathrm{FCC}\) structure. How many \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\) ions are there in unit cell?

Short answer

The unit cell contains 4 \(\backslash \text{Na}\textsuperscript{+}\) ions and 4 \(\backslash \text{Cl}\textsuperscript{–}\) ions.

Step-by-step solution

Identify the structure type

Recognize that \(\backslash \text{NaCl}\) has a Face-Centered Cubic (FCC) structure. In an FCC structure, ions are positioned at each corner and the centers of all the faces of the cube.

Determine the position of ions in the FCC unit cell

\(\backslash \text{NaCl}\) has an interpenetrating FCC lattice, meaning \(\backslash \text{Na}\textsuperscript{+}\) ions occupy an FCC lattice, and \(\backslash \text{Cl}\textsuperscript{–}\) ions occupy the octahedral holes in the lattice, or vice versa.

Calculate the number of \(\backslash \text{Na}\textsuperscript{+}\) ions in an FCC unit cell

There are 8 corner atoms, each shared by 8 unit cells, contributing \(1 \backslash eight\) of an atom per unit cell from corners: \[8 \backslash cdot \frac{1}{8} = 1\] There are 6 face-centered atoms, each shared by 2 unit cells, contributing \(1 \backslash two\) of an atom per unit cell from faces: \[6 \backslash cdot \frac{1}{2} = 3\] Therefore, there are a total of 4 \(\backslash \text{Na}\textsuperscript{+}\) ions per unit cell: \[1 + 3 = 4\]

Calculate the number of \(\backslash \text{Cl}\textsuperscript{–}\) ions in an FCC unit cell

In the interpenetrating FCC structure, there will also be 4 \(\backslash \text{Cl}\textsuperscript{–}\) ions per unit cell, occupying equivalent positions in the lattice as \( \backslash \text{Na}\textsuperscript{+}\) ions.

Key concepts

face-centered cubic lattice

A face-centered cubic (FCC) lattice is a type of crystal structure where each unit cell is organized in a specific manner. In this structure, atoms are located at each corner of the cube and at the center of each face.
Thus, the unit cell consists of:

• 8 corner atoms, shared among 8 unit cells
• 6 face-centered atoms, shared among 2 unit cells

Each corner atom contributes \(\frac{1}{8} \) of an atom to one unit cell because it is shared among eight neighboring unit cells. Similarly, each face-centered atom contributes \(\frac{1}{2} \) of an atom since it is shared between two unit cells.
To visualize this better, imagine a cube with spheres at each vertex and the center of each face. This arrangement maximizes the packing efficiency, meaning the atoms occupy a large fraction of the unit cell's volume. Understanding this setup helps in calculating the total number of atoms within a unit cell.

ions per unit cell

In the NaCl unit cell with an FCC structure, it is important to recognize where the ions are situated to calculate how many there are in each unit cell.

For \(\text{Na}^+\) ions, they are positioned in the FCC lattice corners and face centers. Specifically:

• 8 corner atoms, each shared by 8 unit cells, contributing \(\frac{1}{8} \) of an atom per unit cell
• 6 face-centered atoms, each shared by 2 unit cells, contributing \(\frac{1}{2} \) of an atom per unit cell

Summing these contributions gives the total number of \(\text{Na}^+\) ions per unit cell:
\[ 8 \cdot \frac{1}{8} + 6 \cdot \frac{1}{2} = 1 + 3 = 4\]
Similarly, the \(\text{Cl}^-\) ions occupy equivalent positions in the interpenetrating lattice. Thus, there are also 4 \(\text{Cl}^-\) ions per unit cell. This matching number of ions of both types ensures the neutrality and stability of the crystal structure.

crystallography

Crystallography is the scientific study of crystals and their structures. It involves understanding how atoms or ions are arranged in solid materials.
NaCl (table salt) is a classic example studied in crystallography. In its solid form, NaCl crystallizes in an FCC structure.
Here are some key aspects of crystallography related to NaCl and similar structures:

• Unit Cell: The smallest repeating block of the crystal lattice
• Bravais Lattices: There are 14 unique 3D lattice types, including FCC
• Coordination Number: Number of nearest neighbors; for FCC it's usually 12

Understanding the arrangement of ions within the lattice helps predict properties such asdensity, stability, and how light interacts with the crystal. Advanced methods like X-ray crystallography are used to determine the atomic structure of complex crystals by observing the diffraction patterns of X-rays passing through the crystal lattice. This field is crucial in materials science, chemistry, and physics for designing new materials and understanding existing ones.