The atomic weight, density, and atomic radius for three...

Name: Problem 16, Chapter 3: Materials Science and Engineering
Uploaded: 2023-03-31T05:54:54-08:00
Duration: 1 min 26 s
Channel: Dominador Tan
Description: The atomic weight, density, and atomic radius for three hypothetical alloys are listed in the following table. For each, determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination.

The atomic weight, density, and atomic radius for three hypothetical alloys are listed in the following table. For each, determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination.

Key Concepts

Relationship Between Atomic Radius and Lattice Parameter

The lattice parameter, or edge length of the unit cell, is directly related to the atomic radius and the type of crystal structure. For instance, in a simple cubic structure the edge length is twice the atomic radius, while in BCC and FCC structures the relationship is more complex involving geometric factors. This relationship is essential for connecting atomic dimensions with the overall dimensions of the unit cell.

Unit Cell and Atoms per Unit Cell

The unit cell is the smallest repeating geometric entity that, when stacked in three-dimensional space, builds up the entire crystal lattice. In this context, the number of atoms per unit cell is crucial: simple cubic has 1 atom, BCC has 2 atoms, and FCC has 4 atoms per unit cell. This information is used in calculations relating mass, volume, and density of the material.

Crystal Structures

This concept involves the different ways atoms can be arranged in a solid, specifically focusing on face-centered cubic (FCC), body-centered cubic (BCC), and simple cubic arrangements. Each structure has a distinct number of atoms per unit cell, coordination number, and geometric configuration, which significantly influence the material’s properties such as density, packing efficiency, and mechanical behavior.

Density Calculations in Crystals

This concept links the macroscopic property of density with the microscopic arrangement of atoms by using the formula density = (number of atoms per unit cell × atomic weight divided by Avogadro’s number) / (volume of the unit cell). By comparing calculated density values from this relationship to the given density, one can infer the likely crystal structure of an alloy.

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