Question

# A hypothetical AX type of ceramic material is known to have a density of 2.65 g/cm3 and a unit cell of cubic symmetry with a cell edge length of 0.43 nm. The atomic weights of the A and X elements are 86.6 and 40.3 g/mol, respectively. On the basis of this information, which crystal structures is possible for this material? a. cesium chloride b. zirconium oxide c. zinc blende d. rock salt

Answer

The only possible crystal structure for this material is rock salt.

- Calculate the total number of atoms in the unit cell using Avogadro's number. A: The unit cell volume can be calculated as V = a^3, where a is the length of the cell edge in nm. V = (0.43 nm)^3 = 0.079507 nm^3. The number of unit cells per cm^3 can be calculated as N = (1 cm/10 nm)^3. N = 1x10^24 cm^-3. The number of atoms per unit cell can be calculated as n = Z(A) + Z(X), where Z is the number of atoms per formula unit and A and X are the atomic symbols for the elements in the compound. n = 1 + 1 = 2. The total number of atoms in the unit cell can be calculated as N x n = (1x10^24 cm^-3) x 2 = 2x10^24 atoms.
- Calculate the theoretical density of the material based on its crystal structure. A: The theoretical density can be calculated as p = (Z(A) x w(A) + Z(X) x w(X)) / (N x V), where p is the density in g/cm^3, w is the atomic weight in g/mol, and all other variables are as previously defined. For the cesium chloride structure, Z(A) = 1, Z(X) = 1, and the theoretical density is 4.05 g/cm^3. For the zirconium oxide structure, Z(A) = 1, Z(X) = 2, and the theoretical density is 5.68 g/cm^3. For the zinc blende structure, Z(A) = 1, Z(X) = 1, and the theoretical density is 3.63 g/cm^3. For the rock salt structure, Z(A) = 1, Z(X) = 1, and the theoretical density is 2.78 g/cm^3.
- Compare the theoretical density to the given density to determine the possible crystal structures. A: The given density of the material is 2.65 g/cm^3. Based on the theoretical calculations, the only possible crystal structure is rock salt, which has a theoretical density of 2.78 g/cm^3.