Calcium crystallizes in a face-centered cubic unit cell at room temperature that has an edge length of 5.588 Å. (b) Calculate the density of Ca metal at this temperature.

Verified step by step guidance

1. Identify the type of unit cell: Calcium crystallizes in a face-centered cubic (FCC) unit cell.
2. Calculate the volume of the unit cell: Use the edge length (a = 5.588 Å) to find the volume, V = a^3.
3. Determine the number of atoms per unit cell: In an FCC unit cell, there are 4 atoms per unit cell.
4. Calculate the mass of the atoms in the unit cell: Use the molar mass of calcium (40.08 g/mol) and Avogadro's number (6.022 x 10^23 atoms/mol) to find the mass of 4 atoms.
5. Calculate the density: Use the formula density = mass/volume, ensuring units are consistent (convert Å^3 to cm^3 if necessary).

Key Concepts

Face-Centered Cubic (FCC) Structure

In a face-centered cubic (FCC) structure, atoms are located at each corner and the centers of all the cube faces. This arrangement results in a high packing efficiency, with each unit cell containing four atoms. Understanding the FCC structure is crucial for calculating properties like density, as it determines how many atoms are present in a given volume.

Density Calculation

Density is defined as mass per unit volume (density = mass/volume). To calculate the density of a substance, one must know the mass of the atoms in the unit cell and the volume of the unit cell. For FCC structures, the volume can be derived from the edge length, while the mass can be calculated using the molar mass and Avogadro's number.

Molar Mass and Avogadro's Number

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole. Avogadro's number (approximately 6.022 x 10²³) is the number of atoms or molecules in one mole of a substance. These concepts are essential for converting between the mass of atoms in the unit cell and the number of moles, which is necessary for density calculations.