Question

Calcium metal has a density of \(1.55 \mathrm{~g} / \mathrm{cm}^{3}\) and crystallizes in a cubic unit cell with an edge length of \(558.2 \mathrm{pm}\). (a) How many Ca atoms are in one unit cell? (b) In which of the three cubic unit cells does calcium crystallize?

Short answer

(a) 4 Ca atoms; (b) Face-centered cubic (FCC).

Step-by-step solution

Convert Edge Length to cm

The given edge length is in picometers (pm). First, convert it to centimeters (cm). Use the conversion factor: \(1 ext{pm} = 10^{-12} ext{m} = 10^{-10} ext{cm}\). So,\[558.2 ext{pm} \times 10^{-10} = 5.582 \times 10^{-8} ext{cm}.\]

Calculate Volume of the Unit Cell

Use the formula for the volume of a cube, which is \( V = a^3 \), where \( a \) is the edge length in cm.\[V = (5.582 \times 10^{-8} ext{cm})^3 = 1.740 \times 10^{-22} ext{cm}^3.\]

Calculate Mass of One Unit Cell

The density of calcium is given as \(1.55 \text{ g/cm}^3\). Use the formula for density, \( \rho = \frac{m}{V} \), to find the mass \( m \) of the unit cell:\[m = \rho \times V = 1.55 \text{ g/cm}^3 \times 1.740 \times 10^{-22} ext{cm}^3 = 2.697 \times 10^{-22} ext{g}.\]

Calculate Number of Moles of Calcium in the Unit Cell

Use the molar mass of calcium (approximately 40.08 g/mol) to find the number of moles in the unit cell:\[\text{Moles of Ca} = \frac{2.697 \times 10^{-22} ext{g}}{40.08 ext{ g/mol}} = 6.73 \times 10^{-24} ext{mol}.\]

Calculate Number of Calcium Atoms in the Unit Cell

Use Avogadro's number (\(6.022 \times 10^{23} \) atoms/mol) to convert moles to atoms:\[\text{Number of Ca atoms} = 6.73 \times 10^{-24} ext{mol} \times 6.022 \times 10^{23} ext{ atoms/mol} \approx 4 \text{ atoms}.\]

Determine the Type of Cubic Unit Cell

Based on the calculation, the unit cell contains approximately 4 calcium atoms. In cubic crystal structures, a body-centered cubic (BCC) arrangement contains 2 atoms, while a face-centered cubic (FCC) arrangement contains 4 atoms per unit cell. Therefore, calcium crystallizes in a face-centered cubic (FCC) structure.

Key concepts

Cubic Unit Cell

A cubic unit cell is a basic repeating structural unit that makes up a crystalline solid. It is like a tiny box in which atoms are arranged in a very specific manner.
This structure forms the foundation of a crystal lattice. Cubic unit cells can be classified into three major types:

• Simple Cubic (SC): Where atoms are located at each of the cube's eight corners.
• Body-Centered Cubic (BCC): Atoms are also at the corners, with an additional atom at the very center of the cube.
• Face-Centered Cubic (FCC): Atoms are positioned at each corner and in the center of each face of the cube.

The arrangement of atoms within these cells greatly influences the properties of the material, such as its density, strength, and melting point.

Density Calculation

Density plays a significant role in understanding material properties. It's defined as mass per unit volume. The formula for density \[x = \frac{m}{V},\]where \(m\) is mass and \(V\) is volume, helps us discern how much matter is contained in a given structure.
To compute the density of a material like calcium, it's essential to first convert volume inputs to a standard unit, such as from picometers to centimeters in this case. Once the calculations yield the volume of the unit cell, multiplying by the density will provide the mass. This link between density and the crystal structure allows us to infer the arrangement of atoms in the unit cell.

Molar Mass

Molar mass is a concept used to express the mass of one mole of a substance, usually in units of grams per mole (g/mol). It's the bridge between the microscopic world (atoms and molecules) and the macroscopic world we can measure.
For example, the molar mass of calcium is around 40.08 g/mol. This means that 40.08 grams of calcium contains the same number of atoms as there are in 12 grams of carbon-12, which is Avogadro's number of atoms.
Understanding molar mass is key in converting between the mass of a sample and how many atoms or molecules it contains. This calculation is particularly useful when figuring out how many atoms are present in a unit cell of a solid structure.

Avogadro's Number

Avogadro's number is a fundamental constant that provides the link between the atomic scale and the macroscopic scale. It is defined as the number of atoms, molecules, or ion entities in one mole of a substance, approximately equal to \(6.022 \times 10^{23}\) entities per mole.
In practical terms, Avogadro's number allows chemists to count particles by weighing them. When applied to crystal structures, it enables us to determine the number of atoms in a unit cell, given the mass and molar mass of the element.
For instance, using Avogadro's number in the context of a cubic unit cell of calcium helps us find that, about four calcium atoms occupy each unit cell, pointing to a face-centered cubic arrangement.