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Chapter 9: Problem 72

The compounds \(\mathrm{Na}_{2} \mathrm{O},\) CdS, and \(\mathrm{ZrI}_{4}\) all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case?

Short Answer

Expert verified
For the given compounds, the fractions of tetrahedral holes occupied by cations are: \(Na_2O: 1\) (all tetrahedral holes are occupied), \(CdS: \frac{1}{2}\) (half of the tetrahedral holes are occupied), and \(ZrI_4: \frac{1}{8}\) (one-eighth of the tetrahedral holes are occupied).

Step by step solution

01

Recall Formula for Tetrahedral Holes

The formula for the tetrahedral hole's fraction is given by \[\frac{\text{Number of cations}}{\text{Number of tetrahedral holes}}.\] Since the cubic closest packed structure has two tetrahedral holes for each anion, we can rewrite the formula as: \[\frac{\text{Number of cations}}{2 \times \text{Number of anions}}.\]
02

Find Tetrahedral Hole Fraction for Na2O

In Na₂O, there are two sodium (Na) cations for each oxygen (O) anion. Applying the formula from Step 1, we get:\[\frac{2 \text{ Na}}{2 \times \text{1 O}} = \frac{2}{2} = 1.\] Hence, all tetrahedral holes are occupied by sodium cations in Na2O.
03

Find Tetrahedral Hole Fraction for CdS

In CdS, there is one cadmium (Cd) cation for each sulfur (S) anion. Applying the formula from Step 1, we get:\[\frac{1 \text{ Cd}}{2 \times \text{1 S}} = \frac{1}{2}.\] Hence, half of the tetrahedral holes are occupied by cadmium cations in CdS.
04

Find Tetrahedral Hole Fraction for ZrI4

In ZrI₄, there is one zirconium (Zr) cation for every four iodine (I) anions. Applying the formula from Step 1, we get: \[\frac{1 \text{ Zr}}{2 \times \text{4 I}} = \frac{1}{8}.\] Hence, one-eighth of the tetrahedral holes are occupied by zirconium cations in ZrI4. To summarize, the fractions of tetrahedral holes occupied by cations in each case are: - Na2O: 1 (all tetrahedral holes are occupied) - CdS: 1/2 (half of the tetrahedral holes are occupied) - ZrI4: 1/8 (one-eighth of the tetrahedral holes are occupied)

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Most popular questions from this chapter

You and a friend each synthesize a compound with the formula \(\mathrm{XeCl}_{2} \mathrm{F}_{2}\). Your compound is a liquid and your friend's compound is a gas (at the same conditions of temperature and pressure). Explain how the two compounds with the same formulas can exist in different phases at the same conditions of pressure and temperature.

The molar heat of fusion of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is \(9.92 \mathrm{kJ} / \mathrm{mol}\). Its molar heat of vaporization is \(30.7 \mathrm{kJ} / \mathrm{mol}\). Calculate the heat required to melt 8.25 g benzene at its normal melting point. Calculate the heat required to vaporize 8.25 g benzene at its normal boiling point. Why is the heat of vaporization more than three times the heat of fusion?

The CsCl structure is a simple cubic array of chloride ions with a cesium ion at the center of each cubic array (see Exercise 69 ). Given that the density of cesium chloride is \(3.97 \mathrm{g} / \mathrm{cm}^{3},\) and assuming that the chloride and cesium ions touch along the body diagonal of the cubic unit cell, calculate the distance between the centers of adjacent \(\mathrm{Cs}^{+}\) and \(\mathrm{Cl}^{-}\) ions in the solid. Compare this value with the expected distance based on the sizes of the ions. The ionic radius of \(\mathrm{Cs}^{+}\) is \(169 \mathrm{pm},\) and the ionic radius of \(\mathrm{Cl}^{-}\) is \(181 \mathrm{pm}\).

In each of the following groups of substances, pick the one that has the given property. Justify each answer. a. highest boiling point: \(\mathrm{CCl}_{4}, \mathrm{CF}_{4}, \mathrm{CBr}_{4}\) b. lowest freezing point: \(\mathrm{LiF}, \mathrm{F}_{2}, \mathrm{HCl}\) c. smallest vapor pressure at \(25^{\circ} \mathrm{C}: \mathrm{CH}_{3} \mathrm{OCH}_{3}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{3}\) d. greatest viscosity: \(\mathrm{H}_{2} \mathrm{S}, \mathrm{HF}, \mathrm{H}_{2} \mathrm{O}_{2}\) e. greatest heat of vaporization: \(\mathrm{H}_{2} \mathrm{CO}, \mathrm{CH}_{3} \mathrm{CH}_{3}, \mathrm{CH}_{4}\) f. smallest enthalpy of fusion: \(\mathrm{I}_{2}, \mathrm{CsBr}, \mathrm{CaO}\)

Compare and contrast the structures of the following solids. a. \(\mathrm{CO}_{2}(s)\) versus \(\mathrm{H}_{2} \mathrm{O}(s)\) b. \(\mathrm{NaCl}(s)\) versus \(\mathrm{CsCl}(s) ;\) see Exercise 69 for the structures.
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