An element crystallizes in a face-centered cubic lattice....

An element crystallizes in a face-centered cubic lattice. The edge of the unit cell is $0.408 \mathrm{nm}$, and the density of the crystal is $10.49 \mathrm{~g} / \mathrm{cm}^{3}$. Calculate the atomic weight of the element and identify the element.

Key Concepts

Face-Centered Cubic (FCC) Lattice

This lattice structure is characterized by atoms positioned at each cube corner and the centers of all the faces, leading to a total of 4 atoms per unit cell. The FCC arrangement is important in determining how densely atoms pack in a crystal and influences many material properties, including density and atomic coordination.

Unit Cell Geometry

The unit cell is the basic repeating unit that defines the crystal lattice. Its edge length determines the volume of the cell and, when paired with the lattice type (such as FCC), allows for calculation of the number of atoms per unit cell. Understanding the geometry of the unit cell is crucial for linking microscopic structure with macroscopic properties like density.

Crystal Density

Density is the mass per unit volume of a material and for crystals, it is calculated using the unit cell's mass and volume. The density provides a direct relation between the atomic mass, the number of atoms per unit cell, and the cell volume. Thus, by combining these quantities, one can deduce information such as the atomic weight.

Atomic Weight Calculation

This concept involves using the density of the crystal along with the unit cell geometry and the known number of atoms in the cell to calculate the mass of one mole of atoms (atomic weight). The method typically involves equating the mass obtained from microscopic parameters (atoms per cell multiplied by the atomic mass divided by Avogadro's number) to the macroscopic mass determined from density and cell volume.

Avogadro's Number

Avogadro’s number is a fundamental constant that denotes the number of atoms or molecules in one mole of a substance. It is used to convert between the atomic scale (individual atoms and their masses) and macroscopic measurements (grams per mole), thereby linking the microscopic structure of the crystal with its measured density and atomic weight.

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