To calculate the density of copper (Cu) given its atomic radius and atomic weight, we need to consider the crystal structure and the relationship between these parameters.
Copper has a face-centered cubic (FCC) crystal structure. In an FCC lattice, each corner of the cube is occupied by an atom, and there is an additional atom at the center of each face. This arrangement results in a packing efficiency of 74%.
Given that the atomic radius of copper is 0.1278 nm, we can calculate the edge length of the cubic unit cell using the relationship:
Edge length (a) = 4 * Atomic radius
a = 4 * 0.1278 nm = 0.5112 nm
The volume of the unit cell (V) can be calculated as:
V = a^3
V = (0.5112 nm)^3 = 0.1332 nm^3
Next, we need to determine the number of copper atoms in the unit cell. In an FCC structure, there are four atoms per unit cell.
Now, we can calculate the volume occupied by a single copper atom using the atomic radius:
Volume of one atom = (4/3) * π * (Atomic radius)^3
Volume of one atom = (4/3) * π * (0.1278 nm)^3
The density (ρ) of copper can be calculated using the following formula:
Density = (Atomic weight * Number of atoms) / Volume of unit cell
Plugging in the values:
Density = (63.5 g/mol * 4) / (0.1332 nm^3 * Volume of one atom)
Density = (63.5 g/mol * 4) / (0.1332 nm^3 * [(4/3) * π * (0.1278 nm)^3])
Now, we need to convert nm^3 to cm^3 to match the density unit:
1 nm = 10^(-7) cm
(0.1332 nm^3) * (10^(-7) cm/nm)^3 = 0.1332 * 10^(-21) cm^3
Substituting the values and calculating:
Density ≈ (63.5 g/mol * 4) / (0.1332 * 10^(-21) cm^3 * [(4/3) * π * (0.1278 nm)^3])
Density ≈ 8.92 g/cm^3
Therefore, the approximate density of copper is 8.92 g/cm^3.