Gold has cubic crystals whose unit cell has an edge length of 407.9 pm. The density of the metal

step 1 Here, unit cell of gold crystal has age length, 407 .9 p .m. density of the metal is 19 .3 gram

Gold has cubic crystals whose unit cell has an edge length...

Gold has cubic crystals whose unit cell has an edge length of $407.9 \mathrm{pm}$. The density of the metal is $19.3 \mathrm{~g} / \mathrm{cm}^{3}$. From these data and the atomic mass, calculate the number of gold atoms in a unit cell, assuming all atoms are at lattice points. What type of cubic lattice does gold have?

Key Concepts

Crystalline Unit Cell

The unit cell is the smallest repeating geometric pattern that defines the structure of a crystal. In crystallography, the properties of the whole crystal can be derived from the shape, size, and atomic arrangement within the unit cell, which is crucial for understanding material properties such as density and symmetry.

Density and Mass Calculations in Crystals

In solid state physics, the mass density of a crystal can be related to the number of atoms per unit cell and the volume of that cell. By using the atomic mass and the cell dimensions, one can derive the mass within a unit cell and compare it to the measured density to obtain structural information, such as the number of atoms per unit cell.

Lattice Points and Atomic Contributions

Atoms in a crystal lattice are positioned at specific points known as lattice points. In a cubic unit cell, atoms located at the corners, edges, and faces contribute fractions of their total mass to the cell due to sharing with adjacent unit cells. Understanding these contributions is essential to correctly count the effective number of atoms within a unit cell.

Cubic Lattice Structures

Cubic lattice structures are characterized by their high symmetry and include variations such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). Each type has a different number of atoms per unit cell based on the positions of the lattice points, which directly affects the crystal's density and packing efficiency.

Face-Centered Cubic (FCC) Structure

A face-centered cubic structure is one of the common cubic arrangements where atoms are located at each of the corners and the centers of all the faces of the cube. This arrangement results in an effective count of four atoms per unit cell and is known for its high packing efficiency and stability in many metals.

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