Palladium crystallizes with a face-centered cubic structure. It has a density of 12.0 g/cm³, a radius of 138 pm, and a molar mass of 106.42 g/mol. Use these data to calculate Avogadro's number.

Verified step by step guidance

1. Identify the type of crystal structure: Palladium has a face-centered cubic (FCC) structure.
2. Calculate the volume of the unit cell: In an FCC structure, the edge length (a) of the unit cell is related to the atomic radius (r) by the formula: $a = \frac{4r}{\sqrt{2}}$. Convert the radius from picometers to centimeters before using it in the formula.
3. Determine the mass of the unit cell: Use the density formula $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$ to find the mass of the unit cell. Rearrange the formula to solve for mass: $\text{Mass} = \text{Density} \times \text{Volume}$. Use the given density and the calculated volume of the unit cell.
4. Calculate the number of atoms per unit cell: In an FCC structure, there are 4 atoms per unit cell.
5. Calculate Avogadro's number: Use the formula $\text{Avogadro's number} = \frac{\text{Molar mass} \times \text{Number of atoms per unit cell}}{\text{Mass of the unit cell}}$. Use the molar mass of palladium and the mass of the unit cell calculated in the previous steps.

Key Concepts

Face-Centered Cubic Structure

A face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the cube faces. In this arrangement, each unit cell contains four atoms, as each corner atom is shared among eight unit cells and each face atom is shared between two. Understanding this structure is crucial for calculating the volume occupied by the atoms in the unit cell, which is necessary for determining Avogadro's number.

Density and Molar Mass Relationship

Density is defined as mass per unit volume, and it plays a key role in relating the mass of a substance to its volume. The formula for density (density = mass/volume) can be rearranged to find the volume of a given mass. In the context of calculating Avogadro's number, knowing the density and molar mass allows us to determine the number of atoms in a given volume of the substance, which is essential for this calculation.

Avogadro's Number

Avogadro's number (approximately 6.022 x 10²³) is the number of atoms, ions, or molecules in one mole of a substance. It provides a bridge between the macroscopic scale of substances and the microscopic scale of individual particles. To calculate Avogadro's number using the given data, one must relate the number of atoms in a unit cell to the molar mass and density, allowing for the determination of how many atoms are present in one mole of palladium.