Gold crystallizes in a face-centered cubic lattice. The edge...

Gold crystallizes in a face-centered cubic lattice. The edge of the unit cell has a length of $407.86 \mathrm{pm}$. The density of gold is $19.31 \mathrm{~g} / \mathrm{cm}^{3}$. Use these data and the atomic mass of gold to calculate the value of Avogadro's number.

Key Concepts

Face-Centered Cubic Lattice

A face-centered cubic lattice is a type of crystal structure where atoms are located at each of the corners and the centers of all the faces of the cube. This arrangement leads to a specific count of atoms per unit cell (typically four), which is essential when relating the microscopic structure of a material to its macroscopic properties such as density.

Unit Cell

A unit cell is the smallest repeating geometrical structure that fully describes the lattice of a crystalline material. It defines the basic dimensions and arrangement of atoms in the crystal, and its volume, along with the number of atoms it contains, is used to link the material's microscopic mass distribution to its overall density.

Density and Its Relation to Atomic Structure

Density is the mass per unit volume of a material and serves as a bridge between macroscopic measurements and microscopic atomic properties. By knowing the density along with the unit cell volume and the number of atoms per unit cell, one can calculate the mass of a single atom, which is critical for determining fundamental constants.

Atomic Mass

Atomic mass is the mass of an individual atom expressed in atomic mass units (amu) or grams, and it allows the conversion between the mass of a sample and the number of atoms present. It is crucial for connecting the measured mass of a macroscopic sample to the mass of one mole of atoms, as encapsulated by Avogadro’s number.

Avogadro's Number

Avogadro's number is a fundamental constant that represents the number of atoms or molecules in one mole of a substance. It provides the key link between the microscopic world of atoms and the macroscopic quantities used in chemistry, and it is often determined experimentally by relating measurable quantities such as density, unit cell dimensions, and atomic mass.

Unit Conversion

Unit conversion is the process of converting measurements from one system of units to another, ensuring consistency across calculations. In problems linking crystallography and density, it is important to convert units for lengths, volumes, and masses appropriately to maintain dimensional accuracy and facilitate the correct derivation of physical constants.

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