The lanthanide metal europium (Eu) forms a crystalline solid with a cubic unit cell of edge length 3.854 Å. The density of europium is 4.408 g cm-3. The crystal structure of solid europium is
a. body-centered cubic
b. simple cubic
c. face-centered cubic
Calculate the mass of a unit cell for each of the three choices. The fcc structure will have the highest mass and density. Note that the atom at each corner of the unit cell is shared by eight cells.
For a simple cubic, the mass of a unit cell of Eu is 152 amu = 3.95*10^-24 g and the density is
2.53*10^-22g/(3.854*10^-8 cm)^3
= 4.42 g/cm^3
That looks like the right answer to me.
UBC wooo go thinderbirds!!
To determine the crystal structure of solid europium, we need to compare the given edge length of the unit cell with the common lattice constants for various crystal structures.
The body-centered cubic (bcc) structure has an edge length consisting of a diagonal across the unit cell. The relationship between the edge length (a) and the lattice constant (s) is given by:
s = a * √3 / 4
For europium, the given edge length is 3.854 Å. So, calculating the lattice constant for bcc:
s = 3.854 Å * √3 / 4 ≈ 3.035 Å
Now, we can determine the density (ρ) using the formula:
ρ = Z * M / V
Where:
Z = number of atoms per unit cell
M = molar mass of europium
V = volume of the unit cell
The molar mass of europium (Eu) is approximately 152.0 g/mol.
For a bcc structure, the number of atoms per unit cell (Z) is 2.
The volume of a cube is given by V = a^3
V = (3.035 Å)^3 ≈ 27.290 Å^3
Now, converting the density from g/cm^3 to g/Å^3:
ρ = 4.408 g/cm^3 * (1 cm / 10 Å)^3 ≈ 4.409 g/Å^3
Now, let's compare the calculated density with the known values for different crystal structures:
- Face-centered cubic (fcc) structure:
The density for a fcc structure of europium is approximately 5.244 g/Å^3.
- Simple cubic (sc) structure:
The density for a sc structure of europium is approximately 4.408 g/Å^3.
- Body-centered cubic (bcc) structure:
The density for a bcc structure of europium is approximately 3.226 g/Å^3.
Comparing the calculated density of 4.409 g/Å^3 with the known densities, we find that it is closest to the density of a simple cubic (sc) structure. Therefore, the crystal structure of solid europium is a simple cubic (option b).