Compute the atomic packing factor for the rock salt crystal structure in which \(r_{\mathrm{C}} / r_{\mathrm{A}}=0.414\).

Short Answer

Expert verified
Answer: The atomic packing factor (APF) for the rock salt crystal structure is approximately 0.68.

Step by step solution

01

Calculate the number of cations and anions in the unit cell

Firstly, we need to determine the number of cations and anions in the unit cell. In a rock salt crystal structure, both types of ions are present at the corners and face centers of an FCC lattice. Each corner ion is shared by eight adjacent unit cells, and each face-centered ion is shared by two adjacent unit cells. Hence, the number of cations and anions within one unit cell are: Number of Cations = \(\frac{8\,\text{corner ions}}{8} + \frac{6\,\text{face-centered ions}}{2} = 4\) Number of Anions = \(\frac{8\,\text{corner ions}}{8} + \frac{6\,\text{face-centered ions}}{2} = 4\)
02

Calculate the volume occupied by cations and anions in the unit cell

Now, we will calculate the volume occupied by the cations (V\(_\mathrm{C}\)) and anions (V\(_\mathrm{A}\)) in the unit cell. The volume of a sphere can be calculated using the formula: \(V_{\text{sphere}} = \frac{4}{3}\pi r^3\) We will use this formula to find the volume occupied by cations and anions and then sum them: \(V_{\text{C}} = 4\,\text{cations} \times \frac{4}{3}\pi r_{\mathrm{C}}^3\) \(V_{\text{A}} = 4\,\text{anions} \times \frac{4}{3}\pi r_{\mathrm{A}}^3\) \(V_{\text{total}} = V_{\text{C}} + V_{\text{A}} = 4\,\frac{4}{3}\pi (r_{\mathrm{C}}^3 + r_{\mathrm{A}}^3)\)
03

Calculate the volume of the unit cell

To find the volume of the unit cell (V\(_{\text{cell}}\)), we need first to determine the unit cell edge length (a) and then cube this length. As we have an FCC lattice, the relationship between the edge length (a) and the anion radius is: \(a = 2r_{\mathrm{A}} + 2r_{\mathrm{C}}\) Now, we will cube edge length to find the volume of the unit cell: \(V_{\text{cell}} = a^3 = (2r_{\mathrm{A}} + 2r_{\mathrm{C}})^3\)
04

Calculate the atomic packing factor

Finally, we will calculate the atomic packing factor (APF) by dividing the total volume occupied by atoms with the volume of the unit cell: \(\text{APF} = \frac{V_{\text{total}}}{V_{\text{cell}}} = \frac{4\,\frac{4}{3}\pi (r_{\mathrm{C}}^3 + r_{\mathrm{A}}^3)}{(2r_{\mathrm{A}} + 2r_{\mathrm{C}})^3}\) Given that \(\frac{r_{\mathrm{C}}}{r_{\mathrm{A}}}=0.414\), we can substitute \(r_{\mathrm{C}}\) with \(0.414r_{\mathrm{A}}\): \(\text{APF} = \frac{4\,\frac{4}{3}\pi ((0.414r_{\mathrm{A}})^3 + r_{\mathrm{A}}^3)}{(2r_{\mathrm{A}} + 2(0.414)r_{\mathrm{A}})^3}\) Now, simplify the expression and calculate the APF: \(\text{APF} = \frac{4\,\frac{4}{3}\pi (1.414^3) r_{\mathrm{A}}^3}{(2(1.414) r_{\mathrm{A}})^3} \approx 0.68\) Hence, the atomic packing factor for the rock salt crystal structure is approximately 0.68.

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Cite one reason why ceramic materials are, in general, harder yet more brittle than metals.

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