Metallic magnesium has a hexagonal close-packed structure...

Metallic magnesium has a hexagonal close-packed structure and a density of $1.74 \mathrm{~g} / \mathrm{cm}^{3}$. Assume magnesium atoms to be spheres of radius $r$. Because magnesium has a close-packed structure, $74.1 \%$ of the space is occupied by atoms. Calculate the volume of each atom; then find the atomic radius, $r$. The volume of a sphere is equal to $4 \pi r^{3} / 3$.

Key Concepts

Sphere Geometry

Understanding sphere geometry is important in this context because atoms are modeled as spheres when calculating their individual volumes. The formula for the volume of a sphere (4?r³/3) is a key mathematical tool used to relate the atomic volume to the atomic radius.

Packing Efficiency

Packing efficiency refers to the fraction of the total volume of a crystal that is actually occupied by the constituent atoms. In close-packed metals, a high packing efficiency (approximately 74.1% for HCP) is used to relate the measured density of the material to the volume occupied by individual atoms.

Density and Atomic Volume Calculations

This concept involves linking the macroscopic density of a substance to the microscopic volume occupied by each atom. By combining the known density with the atomic packing efficiency and crystal structure, one can calculate the volume per atom, which is critical for determining atomic dimensions.

Crystalline Structures

This concept covers the ordered, periodic arrangement of atoms in a solid. Understanding crystalline structures is essential when studying material properties, as the arrangement directly affects characteristics such as density, strength, and thermal conductivity.

Close-Packed Structures

Close-packed structures, such as hexagonal close-packed (HCP) and face-centered cubic (FCC), are arrangements where atoms are packed together with minimal empty space. This concept is crucial for understanding how atoms efficiently occupy volume in a crystalline solid and influences the calculation of properties like density and atomic packing fraction.

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