Determine the number of atoms per unit cell for each metal.
(b) Tungsten

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Identify the crystal structure of tungsten. Tungsten crystallizes in a body-centered cubic (BCC) structure.

Understand the BCC structure: In a BCC unit cell, there is one atom at each corner of the cube and one atom in the center of the cube.

Calculate the contribution of corner atoms: Each corner atom is shared by eight adjacent unit cells, so each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell. Since there are 8 corners, the total contribution from corner atoms is \( 8 \times \frac{1}{8} = 1 \) atom.

Calculate the contribution of the center atom: The atom at the center of the BCC unit cell is not shared with any other unit cell, so it contributes 1 whole atom.

Add the contributions from the corner and center atoms to find the total number of atoms per unit cell: \( 1 + 1 = 2 \) atoms per unit cell.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. It defines the arrangement of atoms in a solid and is characterized by its dimensions and angles. Understanding the unit cell is crucial for determining the number of atoms it contains, which can vary based on the type of crystal structure (e.g., face-centered cubic, body-centered cubic).

Atomic Packing Factor (APF)

The Atomic Packing Factor (APF) is a measure of how efficiently atoms are packed in a unit cell. It is calculated as the volume occupied by the atoms in the unit cell divided by the total volume of the unit cell. Different metals have different APFs depending on their crystal structures, which influences their physical properties and density.

Coordination Number

The coordination number is the number of nearest neighbor atoms surrounding a central atom in a crystal structure. It provides insight into the arrangement of atoms and the bonding characteristics within the metal. For example, tungsten has a body-centered cubic structure with a coordination number of 8, indicating that each tungsten atom is surrounded by eight other tungsten atoms.

Determine the number of atoms per unit cell for each metal.(b) Tungsten