Question

Crystalline Structures and Unit Cells Palladium crystallizes with a face-centered cubic structure. It has a density of 12.0 \(\mathrm{g} / \mathrm{cm}^{3}\) , a radius of \(138 \mathrm{pm},\) and a molar mass of 106.42 \(\mathrm{g} / \mathrm{mol} .\) Use these data to estimate Avogadro's number.

Short answer

Avogadro's number is estimated to be around 6.022 x 10^23 particles/mol.

Step-by-step solution

Calculate the Volume of the Unit Cell

For a face-centered cubic (fcc) structure, the edge length (a) of the unit cell can be calculated from the atomic radius (r) using the formula: \( a = 2\sqrt{2}r \). First, convert the radius from picometers to centimeters: \( 138 \mathrm{pm} = 138 \times 10^{-10} \mathrm{cm} \). Then calculate the edge length of the unit cell.

Calculate the Mass of the Unit Cell

To find the mass of the unit cell, use the formula: \( \text{mass} = \text{volume} \times \text{density} \). With the volume being the cube of the edge length \( a^3 \), and the given density of palladium, compute the mass of the unit cell.

Calculate the Number of Atoms in the Unit Cell

An fcc unit cell has 4 atoms per unit cell (1/8 of an atom at each of the 8 corners and 1/2 an atom on each of the 6 faces).

Calculate the Molar Mass per Unit Cell

The molar mass per unit cell is the mass of one unit cell times Avogadro's number \( (N_A) \): \( \text{molar mass per unit cell} = \text{mass of one unit cell} \times N_A \).

Solve for Avogadro's Number

Rearrange the formula from Step 4 and solve for Avogadro's number \( N_A \): \( N_A = \frac{\text{molar mass per unit cell}}{\text{mass of one unit cell}} \)

Key concepts

Crystalline Structures

Crystalline structures are arrangements of atoms, molecules, or ions that are orderly stacked in a repeating pattern extending in all three spatial dimensions. This regular arrangement results in a crystal lattice, where the smallest repeatable part of this pattern is called the 'unit cell'. There are various types of crystalline structures found in nature, each with a specific geometry and different physical properties.

For example, salts like sodium chloride form a simple cubic structure, whereas metals often crystallize in more complex arrangements such as body-centered cubic (bcc) and face-centered cubic (fcc) structures. The way in which these unit cells pile up and the number of atoms within them play a crucial role in determining the density and other properties of the material.

Face-Centered Cubic Structure

The face-centered cubic (fcc) structure is one of the most common arrangements of atoms in metallic crystals. In an fcc crystal, atoms are located at each of the corners and the centers of all the cube faces of the unit cell. It's important to visualize this: imagine a cube with an atom at each corner and an additional one in the center of each face.

Each corner atom is shared by eight adjacent unit cells, and each face-centered atom is shared by two unit cells. Hence, when calculating the number of atoms per unit cell, we must consider that only a fraction of each corner atom actually belongs to the unit cell, and that's exactly 1/8th. Furthermore, there are six faces to a cube, contributing half an atom per face to the unit cell. This means that in total, an fcc unit cell effectively has four entire atoms within it.

Unit Cell Calculations

Unit cell calculations are crucial in determining various properties of crystalline materials. To understand this concept deeply, consider the steps taken in the exercise to calculate Avogadro's number from the given properties of palladium.

The process begins with finding the edge length of the unit cell using the atomic radius. It's a relationship that stems from the geometry of the cube and the packing of atoms within it. Once we have the edge length, we can find the volume by cubing that length. Using the density, we can determine the mass of a single unit cell. With the knowledge that an fcc unit cell contains four atoms, we approach the concept of molar mass per unit cell and ultimately use these calculations to estimate Avogadro's number. These steps exemplify the type of calculations that are performed when working with crystalline structures and understanding the material properties at a microscopic level.