Question

Silver forms face-centered cubic crystals. The atomic radius of a silver atom is \(144 \mathrm{pm}\). Draw the face of a unit cell with the nuclei of the silver atoms at the lattice points. The atoms are in contact along the diagonal. Calculate the length of an edge of this unit cell.

Short answer

The edge length of the silver unit cell is \(a = \frac{4(144 \mathrm{pm})}{\sqrt{2}} = 4(144 \mathrm{pm}) \times \frac{1}{\sqrt{2}} = 4(144 \mathrm{pm}) \times \frac{\sqrt{2}}{2} = 2\sqrt{2}(144 \mathrm{pm}) = 408 \mathrm{pm}\).

Step-by-step solution

Understand the Geometry of the Face-Centered Cubic Unit Cell

In a face-centered cubic (FCC) unit cell, atoms are located at each of the corners and the centers of all the faces of the cube. Each corner atom is shared among eight unit cells and each face atom is shared with two, so each unit cell effectively contains 4 atoms. The atoms touch along the face diagonal, which goes through the face-centered atoms.

Calculate the Face Diagonal

Since the atoms touch along the face diagonal and there are two half-atoms on each edge of the face and one whole atom in the center, the face diagonal can be represented as four times the atomic radius: Diagonal = 4 times the atomic radius of silver, d = 4r.

Express the Length of the Edge of the Unit Cell in Terms of the Diagonal

The diagonal of a square face of the cube can be expressed in terms of the edge, a, of the cube using the Pythagorean theorem: diagonal = \(a\sqrt{2}\). Set this equal to the expression for the diagonal obtained in the previous step.

Solve for the Edge Length a

Equating the two expressions for the diagonal gives \(4r = a\sqrt{2}\). Solving for a gives \(a = \frac{4r}{\sqrt{2}}\).

Calculate the Edge Length Using the Given Atomic Radius

Plug in the atomic radius of silver, \(r = 144 \mathrm{pm}\), to find the edge length \(a = \frac{4(144 \mathrm{pm})}{\sqrt{2}}\), and calculate the value of a.

Key concepts

Understanding Atomic Radius in Crystal Lattices

The atomic radius is a measure of the size of an atom, typically the distance from the center of the nucleus to the outer boundary of the electron cloud. In the context of crystal structures, especially in a face-centered cubic (FCC) lattice, the atomic radius plays a crucial role in determining the unit cell dimensions.

In a FCC unit cell, like that of silver, the atoms are arranged in a way that they are in contact with each other along the face diagonal. This arrangement allows us to relate the atomic radius to the edge length of the unit cell. Since each side of the face diagonal contact comprises one half of an atom on each corner and one whole atom at the center, the total length of the diagonal is four atomic radii. This relationship is a key component in computing the unit cell's edge length and, subsequently, the volume and density of the crystal.

Understanding the concept of atomic radius in relation to the FCC structure gives insight into fundamental properties of the material, such as packing efficiency and the coordination number, which is the number of nearest neighbor atoms.

Face-Centered Cubic Unit Cell and Its Significance

The crystal lattice structure of a solid is the repeating arrangement of atoms in space. The face-centered cubic (FCC) unit cell is a vital crystal lattice structure in materials science and chemistry. It consists of eight atoms at each corner and one atom at the center of each face of the cube that make up the unit cell.

In a FCC structure, each corner atom is a part of eight adjacent unit cells, while each face-centered atom belongs to two. Thus, when considering the contribution of these atoms to a single unit cell, we can count each corner atom as an eighth and each face atom as a half, resulting in a total of four atoms per unit cell. This configuration leads to high packing efficiency and explains why many metals, including silver, adopt this structure.

The arrangement greatly affects the material's properties, including its density, melting point, and ability to conduct electricity and heat. For students, visualizing this three-dimensional structure and understanding how atoms occupy space in a FCC lattice offers a deeper comprehension of crystallography and material properties.

Applying the Pythagorean Theorem in Chemistry

Though often associated with mathematics, the Pythagorean theorem finds its way into chemistry, particularly in elucidating the relationship between the atomic radius and the dimensions of crystal lattices. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the context of a face-centered cubic unit cell, such as that of silver, the face diagonal and the edge of the cube form a right-angled triangle. The face diagonal can be thought of as the hypotenuse, and the edge length of the cube is one of the sides. Using the Pythagorean theorem allows us to express the face diagonal in terms of the edge length: \(a\sqrt{2}\), and vice versa. Therefore, by knowing the atomic radius, we can calculate the face diagonal and, ultimately, the edge length of the cube.

It's an elegant example of how interdisciplinary knowledge can be applied, showing how a mathematical concept can unlock the understanding of an essential chemical phenomenon such as the measurement and calculation of distances in crystal structures.