Question

Lead adopts the face-centered cubic unit cell in its crystal structure. If the edge length of the unit cell is \(495 \mathrm{pm},\) find the atomic radius of lead.

Short answer

The atomic radius of lead is approximately 175.1 pm.

Step-by-step solution

Understand FCC 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. The edge length of an FCC unit cell and the atomic radius are related.

Write Relation Between Edge Length and Atomic Radius

For an FCC unit cell, the relationship between the edge length (a) and the atomic radius (r) is given by the formula: Given: a = 495 pm

Solve for Atomic Radius

Rearrange the formula to solve for r: 2.262 For lead: r = \(\frac{495}{\sqrt{8}}\)

Simplify the Expression

Compute the value of the expression: 2.262 Simplified: \(r \approx 175.1\mathrm{pm}\)

Key concepts

Atomic Radius

The atomic radius is a measurement of the size of an atom. It is typically defined as the distance from the center of the nucleus to the outer boundary of the electron cloud. In a face-centered cubic (FCC) unit cell, the atomic radius can be related to the edge length of the unit cell. This relation allows us to calculate the atomic radius if we know the edge length.

An important aspect to remember is that the atomic radius can vary slightly based on the atom's environment. For our exercise, we use a specific equation that relates the atomic radius to the edge length in an FCC structure, helping us find the radius of a lead atom.

Crystal Structure

A crystal structure describes the ordered arrangement of atoms in a crystalline solid. Key points:

• In an FCC unit cell, atoms are positioned at each corner and at the center of each face of the cube.
• This arrangement creates a highly efficient packing structure, where each face atom is shared between two unit cells, and each corner atom is shared among eight unit cells.
• The presence of atoms at each of these positions defines the characteristic geometry and properties of the FCC crystal system.

Understanding these positions and their implications is crucial for solving problems related to the crystal structure, such as finding the atomic radius.

Edge Length

The edge length of a unit cell represents the distance between two parallel edges of the cube. In an FCC structure, this edge length is crucial to determining other properties of the crystal. The relationship between the edge length (a) and the atomic radius (r) in an FCC unit cell is derived from the geometry of the cube. For an FCC unit cell:
\[a = 2r\sqrt{2}\]
In our exercise:

• We know the edge length to be 495 pm.
• We use the relation: \[a = 2r\sqrt{2}\]
• By rearranging, we find: \[r = \frac{495}{\sqrt{8}} \approx 175.1 \text{ pm}\]

This formula helps us calculate the atomic radius from the known edge length of the lead unit cell.