A hypothetical AX type of ceramic material is known to have a density of \(2.65 \mathrm{~g} / \mathrm{cm}^{3}\) and a unit cell of cubic symmetry with a cell edge length of \(0.43 \mathrm{~nm}\). The atomic weights of the A and \(X\) elements are \(86.6\) and \(40.3 \mathrm{~g} / \mathrm{mol}\), respectively. On the basis of this information, which of the following crystal structures is (are) possible for this material: rock salt, cesium chloride, or zinc blende? Justify your choice(s).

For each crystal structure, we will find the number of formula units (Z) per unit cell. - Rock Salt: 4 formula units per unit cell - Cesium Chloride: 1 formula unit per unit cell - Zinc Blende: 4 formula units per unit cell

To calculate the molecular weight of the AX compound, we will sum the atomic weights of the A and X elements: Molecular weight of AX = Atomic weight of A + Atomic weight of X = \(86.6 + 40.3 = 126.9 \frac{\mathrm{g}}{\mathrm{mol}}\)

With the molecular weight of AX and Avogadro's number (6.0221×10^23 1/mol), we can calculate the theoretical density for each crystal structure using the formula: Theoretical density = \(\frac{Z * Molecular Weight * 10^{24}}{ N_a * a^{3}}\) Where: - Z = Number of formula units per unit cell - Molecular Weight = Molecular weight of the AX compound - \(N_a\) = Avogadro's number (\(6.0221 \times 10^{23} \frac{1}{\mathrm{mol}}\)) - a = Unit cell edge length (\(0.43 nm = 4.3 \times 10^{-8} \mathrm{cm}\)) Calculating the theoretical density for each crystal structure: Rock Salt: \(\frac{4 * 126.9 * 10^{24}}{ (6.0221 \times 10^{23}) * (4.3 \times 10^{-8})^3} = 2.585 \frac{\mathrm{g}}{\mathrm{cm^3}}\) Cesium Chloride: \(\frac{1 * 126.9 * 10^{24}}{ (6.0221 \times 10^{23}) * (4.3 \times 10^{-8})^3} = 0.646 \frac{\mathrm{g}}{\mathrm{cm^3}}\) Zinc Blende: \(\frac{4 * 126.9 * 10^{24}}{ (6.0221 \times 10^{23}) * (4.3 \times 10^{-8})^3} = 2.585 \frac{\mathrm{g}}{\mathrm{cm^3}}\) We know the actual density of the material is 2.65 g/cm³. Comparing the calculated theoretical density values with the actual density: - Rock Salt: Differs by approximately 0.065 g/cm³ - Cesium Chloride: Differs by approximately 2.004 g/cm³ - Zinc Blende: Differs by approximately 0.065 g/cm³

Based on the comparison of theoretical density values to the actual density of the AX compound, both Rock Salt, and Zinc Blende are possible crystal structures for this material, as their theoretical densities are very close to the actual density. On the other hand, the Cesium Chloride structure has a much lower theoretical density and is not a possible crystal structure for this material.