Problem 25 - Chemistry The Science in Context

Question

What is the length of an edge of the unit cell when barium (atomic radius \(222 \mathrm{pm}\) ) crystallizes in a crystal lattice of bcc unit cells?

Step-by-Step Solution

Answer

Answer: The length of an edge of the unit cell when barium crystallizes in a bcc unit cell is approximately \(504.36\,\mathrm{pm}\).

Step 1: Recall the geometry of a bcc unit cell

In a bcc unit cell, there is one atom in the center and one-eighth of each of the eight atoms in the corners within the unit cell. This structure allows for a simple cubic arrangement of atoms along the face diagonals.

Step 2: Visualize the diagonal connecting two opposing corners

Imagine a line segment connecting two opposite corners of the unit cell, passing through the center atom. This line is the body diagonal of the cube, and its length is equal to four times the atomic radius (two radii for the center atom, and one radius for each of the corner atoms).

Step 3: Use Pythagorean theorem in 3D to calculate the diagonal length

The body diagonal of the cube can be calculated using the 3D Pythagorean theorem which relates the diagonal length (D), edge length (L), and the number of edge lengths (N) along the diagonal: \(D^2 = N^2 L^2\) In the case of a bcc unit cell, the number of edge lengths along the diagonal is \(\sqrt{3}\), as the body diagonal connects opposite corners through the center, equal to three times the distance between corner atoms: \(\left(4 \times 222\,\mathrm{pm}\right)^2 = \left(\sqrt{3} L\right)^2\)

Step 4: Solve for the edge length

Now solve the equation for L: \(4^2 \times 222^2\,\mathrm{pm^2} = 3L^2\) \(L^2 = \frac{4^2 \times 222^2}{3}\, \mathrm{pm^2}\) \(L = \sqrt{\frac{4^2 \times 222^2}{3}}\, \mathrm{pm}\) Finally, compute the edge length L by substituting the given atomic radius and solving the equation: \(L \approx 504.36\,\mathrm{pm}\)

Step 5: Report the edge length

The length of an edge of the unit cell when barium crystallizes in a bcc unit cell is approximately \(504.36\,\mathrm{pm}\).

Key Concepts

Crystal Lattice

A crystal lattice represents the symmetrical three-dimensional arrangement of atoms within a crystal. Imagine it as a repetitive pattern that atoms follow throughout a material. In the context of barium crystallizing in a body-centered cubic (bcc) structure, the lattice is composed of repeated units of bcc unit cells, each containing atoms at specific positions that maintain the overall symmetry and structure of the lattice.

This regular distribution of atoms in a lattice is pivotal for understanding the physical properties of materials, such as their thermal and electrical conductivities, strength, and density. The arrangement of atoms within the unit cell directly affects how these properties manifest in the bulk material.

Atomic Radius

The atomic radius is a measure of the size of an atom, roughly from its nucleus to the outer boundary of its electrons. This value is critical when examining how atoms pack together in a crystal structure, especially in a bcc unit cell. Each atom's radius helps determine the distances between atoms and hence the dimensions of the unit cell itself.

In a bcc lattice, the atomic radius influences not only the size of the entire unit cell but also the positioning of atoms within it. For barium, with a given atomic radius of 222 pm (picometers), we can determine how barium atoms will fit into the bcc structure and predict the unit cell's edge length accordingly.

Pythagorean Theorem in 3D

The Pythagorean theorem is not limited to two dimensions; it extends to three dimensions and is utilized to determine relationships between the lengths of a cuboid's edges and its diagonal. The 3D version of the theorem states that the square of the body diagonal's length is the sum of the squares of the three perpendicular edges that it spans.

For a unit cell, which is essentially a cube, the diagonal that runs from one corner of the cube through its center to the opposite corner is termed the body diagonal. The length of this diagonal in relation to the edge of the cube can be calculated using this extended principle of the Pythagorean theorem. When solving crystal lattice problems, such as finding the edge length of a bcc unit cell, the 3D Pythagorean theorem becomes an invaluable tool.

Geometry of BCC Unit Cell

Diving deeper into the geometry of the bcc unit cell helps to visualize and understand its complex structure. With an atom at each corner of the cube and one atom at the very center, the body-centered cubic structure is distinct from other lattice types due to its configuration. The body diagonal—the line running through the center of the cube—contains two corner atoms and one center atom.

The body diagonal is key to calculating the edge length of a bcc unit cell when the atomic radius is known. For instance, in our example with barium atoms, knowing that the body diagonal is made up of four atomic radii (two for the central atom and one for each of the corner atoms), allows us to apply the Pythagorean theorem to find the relationship between the atomic radius and the cell's edge length. With these considerations, we can derive the edge length and comprehend the spatial arrangement within a bcc crystal lattice.