Rhodium has a density of 12.41 g/cm³ and crystallizes with the face-centered cubic unit cell. Calculate the radius of a rhodium atom.

Verified step by step guidance

1. Identify the type of unit cell: Rhodium crystallizes in a face-centered cubic (FCC) unit cell.
2. Recall the relationship between the edge length (a) of the FCC unit cell and the atomic radius (r): In an FCC unit cell, the face diagonal is equal to 4 times the atomic radius, and the face diagonal can also be expressed as \( \sqrt{2}a \).
3. Use the formula for density: \( \text{Density} = \frac{\text{mass of unit cell}}{\text{volume of unit cell}} \). The mass of the unit cell can be calculated using the number of atoms per unit cell (4 for FCC) and the molar mass of rhodium.
4. Calculate the edge length (a) of the unit cell using the density formula and the given density of rhodium.
5. Solve for the atomic radius (r) using the relationship \( r = \frac{\sqrt{2}a}{4} \).

Key Concepts

Density and Unit Cell

Density is defined as mass per unit volume, and in the context of crystalline solids, it can be related to the unit cell structure. A face-centered cubic (FCC) unit cell contains four atoms per unit cell, and the density can be calculated using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). Understanding this relationship is crucial for determining the radius of the atom.

Face-Centered Cubic Structure

In a face-centered cubic (FCC) structure, atoms are located at each corner of the cube and at the center of each face. The edge length of the cube can be related to the atomic radius, as the diagonal of the face of the cube contains four atomic radii. This geometric relationship is essential for calculating the radius of the rhodium atom from the unit cell dimensions.

Atomic Radius Calculation

The atomic radius can be calculated from the edge length of the FCC unit cell using the formula: radius = (√2 * edge length) / 4. To find the edge length, one can rearrange the density formula to solve for volume and subsequently for edge length. This calculation is key to determining the size of the rhodium atom based on its density and crystal structure.