Silver metal crystallizes in a face-centered cubic unit cell...

Silver metal crystallizes in a face-centered cubic unit cell with an edge length of 408 $\mathrm{pm}$ . The molar mass of silver is $107.9 \mathrm{g} / \mathrm{mol},$ and its density is 10.50 $\mathrm{g} / \mathrm{cm}^{3} .$ Use these data to calculate a value for Avogadro's number.

Key Concepts

Crystallographic Unit Cell

The unit cell is the fundamental building block of a crystal lattice. It is a small volume that, when repeated in three dimensions, recreates the entire crystal structure. Understanding its dimensions and geometry is essential for linking atomic-scale properties to macroscopic characteristics such as density. This concept is broadly applicable to materials science and solid state physics.

Face-Centered Cubic (FCC) Lattice

The face-centered cubic lattice is a common crystal structure where atoms are located at each of the corners and the centers of all the faces of the cube. This arrangement results in a specific and calculable number of atoms per unit cell (typically four atoms), which is critical for determining physical properties like density and for converting between microscopic and macroscopic quantities.

Density of a Solid

Density is defined as the mass per unit volume and is a key physical property of materials. In the context of crystallography, the density can be related to the mass of the atoms within a unit cell and the cell's volume. This relationship provides a bridge between the atomic level—such as the arrangement and number of atoms—and the observable macroscopic property of density.

Molar Mass and Avogadro's Number

Molar mass connects the mass of an individual atom or molecule to the mass of one mole of that substance, which contains Avogadro's number of entities. This concept is essential for converting between the weight of a sample and the number of atoms it contains, forming the basis for many calculations in chemistry and materials science.

Unit Conversion

Unit conversion is a critical step in solving problems in physical sciences, ensuring that all quantities are expressed in compatible units. For example, converting picometers to centimeters or other required units is necessary to maintain consistency in calculations and to accurately relate microscopic dimensions to macroscopic measurements.

Please give Ace some feedback