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Chapter 12: Problem 5

On the basis of ionic charge and ionic radii given in Table \(12.3,\) predict crystal structures for the following materials: (a) \(\mathrm{CaO},\) (b) \(\mathrm{MnS}\) (c) \(\mathrm{KBr},\) and (d) \(\mathrm{CsBr}\). Justify your selections.

Short Answer

Expert verified
Answer: The likely crystal structures and coordination numbers for each material are: (a) CaO: Rock salt structure (NaCl-type) - Octahedral coordination with a coordination number of 6. (b) MnS: Zinc blende structure (ZnS-type) - Tetrahedral coordination with a coordination number of 4. (c) KBr: Rock salt structure (NaCl-type) - Octahedral coordination with a coordination number of 6. (d) CsBr: Cesium chloride structure (CsCl-type) - Cubic coordination with a coordination number of 8.

Step by step solution

01

Find the formula of each material

To predict the crystal structure of each material, we must first write the chemical formula for CaO, MnS, KBr, and CsBr. (a) CaO: Calcium has a +2 charge, and Oxygen has a -2 charge. (b) MnS: Manganese has a +2 charge, and Sulfur has a -2 charge. (c) KBr: Potassium has a +1 charge, and Bromine has a -1 charge. (d) CsBr: Cesium has a +1 charge, and Bromine has a -1 charge.
02

Calculate ionic radii ratios

We can use the provided ionic radii in Table 12.3 to find the ratio of cation radius to anion radius (rc/ra) for each material. (a) CaO: rc (Ca虏鈦�) = 100 pm, and ra (O虏鈦�) = 140 pm, so the radius ratio (rc/ra) = 100/140 = 0.714 (b) MnS: rc (Mn虏鈦�) = 83 pm, and ra (S虏鈦�) = 184 pm, so the radius ratio (rc/ra) = 83/184 = 0.451 (c) KBr: rc (K鈦�) = 138 pm, and ra (Br鈦�) = 194 pm, so the radius ratio (rc/ra) = 138/194 = 0.711 (d) CsBr: rc (Cs鈦�) = 170 pm, and ra (Br鈦�) = 194 pm, so the radius ratio (rc/ra) = 170/194 = 0.876
03

Determine the coordination numbers and crystal structures based on the critical radius ratio

Using the critical radius ratio, we can predict the coordination numbers and the crystal structures for each material. (a) CaO: The radius ratio (0.714) suggests a coordination number of 6. Therefore, the crystal structure is likely to be an octahedral structure, like the rock salt structure (NaCl-type). (b) MnS: The radius ratio (0.451) suggests a coordination number of 4. Therefore, the crystal structure is likely to be a tetrahedral structure, like the zinc blende structure (ZnS-type). (c) KBr: The radius ratio (0.711) suggests a coordination number of 6. Therefore, the crystal structure is likely to be an octahedral structure, like the rock salt structure (NaCl-type). (d) CsBr: The radius ratio (0.876) suggests a coordination number of 8. Therefore, the crystal structure is likely to be a cubic structure, like the cesium chloride structure (CsCl-type).
04

Conclusion

To summarize, the predicted crystal structures for the given materials are: (a) CaO: Rock salt structure (NaCl-type) - Octahedral coordination (b) MnS: Zinc blende structure (ZnS-type) - Tetrahedral coordination (c) KBr: Rock salt structure (NaCl-type) - Octahedral coordination (d) CsBr: Cesium chloride structure (CsCl-type) - Cubic coordination

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Most popular questions from this chapter

(a) A three-point transverse bending test is conducted on a cylindrical specimen of aluminum oxide having a reported flexural strength of \(390 \mathrm{MPa}(56,600 \mathrm{psi})\). If the specimen radius is \(2.5 \mathrm{~mm}\) (0.10 in.) and the support point separation distance is 30 \(\mathrm{mm}\) (1.2 in.), predict whether you would expect the specimen to fracture when a load of \(620 \mathrm{~N}\left(140 \mathrm{lb}_{\mathrm{f}}\right)\) is applied. Justify your prediction. (b) Would you be \(100 \%\) certain of the prediction in part (a)? Why or why not?

The unit cell for \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) has hexagonal symmetry with lattice parameters \(a=0.4961 \mathrm{~nm}\) and \(c=1.360 \mathrm{~nm}\). If the density of this material is \(5.22 \mathrm{~g} / \mathrm{cm}^{3}\), calculate its atomic packing factor. For this computation assume ionic radii of \(0.062 \mathrm{~nm}\) and \(0.140 \mathrm{~nm}\), respectively, for \(\mathrm{Cr}^{3+}\) and \(\mathrm{O}^{2-}\).

Iron sulfide (FeS) may form a crystal structure that consists of an HCP arrangement of \(\mathrm{S}^{2-}\) ions. (a) Which type of interstitial site will the \(\mathrm{Fe}^{2+}\) ions occupy? (b) What fraction of these available interstitial sites will be occupied by \(\mathrm{Fe}^{2+}\) ions?

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).

Calculate the number of Frenkel defects per cubic meter in zinc oxide at \(1000^{\circ} \mathrm{C}\). The energy for defect formation is \(2.51 \mathrm{eV}\), whereas the density for \(\mathrm{ZnO}\) is \(5.55 \mathrm{~g} / \mathrm{cm}^{3}\) at \(\left(1000^{\circ} \mathrm{C}\right)\)
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