Copper crystallizes in a face-centered cubic unit cell with an edge length of 362 pm. What is the radius of a copper atom in picometers? What is the density of copper in g>cm3?

Verified step by step guidance

1. Step 1: In a face-centered cubic unit cell, the atoms are located at each of the corners and the centers of all the faces. The diagonal of the face of the cube is equal to 4 times the radius of the atom (4r). The face diagonal can be calculated using Pythagorean theorem, which states that the square of the face diagonal (d) is equal to the sum of the squares of the edge length (a). So, d = √2 * a.
2. Step 2: Once you have the face diagonal, you can find the radius of the atom by dividing the face diagonal by 4, since the face diagonal is equal to 4 times the radius of the atom. So, r = d / 4.
3. Step 3: To find the density of copper, you first need to find the volume of the unit cell. The volume (V) of a cube is equal to the cube of the edge length (a). So, V = a^3.
4. Step 4: In a face-centered cubic unit cell, there are 4 atoms per unit cell. The mass of one atom can be found using the atomic weight of copper (63.55 g/mol) and Avogadro's number (6.022 x 10^23 atoms/mol). The total mass of the unit cell is then 4 times the mass of one atom.
5. Step 5: Finally, the density (ρ) can be found by dividing the mass of the unit cell by the volume of the unit cell. So, ρ = mass / V.

Key Concepts

Face-Centered Cubic (FCC) Structure

In a face-centered cubic (FCC) structure, atoms are located at each corner of the cube and at the center of each face. This arrangement allows for a high packing efficiency, with each unit cell containing four atoms. The relationship between the edge length and atomic radius in an FCC structure is given by the formula: radius = edge length / (2√2).

Atomic Radius

The atomic radius is a measure of the size of an atom, typically defined as the distance from the nucleus to the outermost electron shell. In the context of crystalline solids, the atomic radius can be derived from the geometry of the crystal lattice, such as the FCC structure, where it directly influences the packing and density of the material.

Density Calculation

Density is defined as mass per unit volume and is a critical property of materials. For crystalline solids, density can be calculated using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). In the case of copper, knowing the atomic mass and the number of atoms per unit cell allows for the determination of its density in g/cm³.