Chapter 18

What is the formula of the oxide that crystallizes with \(\mathrm{Fe}^{3+}\) ions in one-fourth of the octahedral holes, \(\mathrm{Fe}^{3+}\) ions in one- eighth of the tetrahedral holes, and \(\mathrm{Mg}^{2+}\) in one-fourth of the octahedral holes of a cubic closest-packed arrangement of oxide ions \(\left(\mathrm{O}^{2-}\right) ?\)

What is the chemical formula of the compound that crystallizes in a simple cubic arrangement of fluoride ions with \(\mathrm{Ba}^{2+}\) ions occupying half of the cubic holes?

A compound of uranium and oxygen consists of a cubic close-packed arrangement of uranium ions with oxide ions in all the tetrahedral holes. What is the formula of this compound?

A mixture of gallium and arsenic is a widely used semiconductor. The arsenide ions are in a cubic close-packed arrangement and half the tetrahedral holes are occupied by the gallium ions. What is the formula of this compound?

The crystal structure of olivine- \(\mathrm{M}_{2} \mathrm{SiO}_{4}(\mathrm{M}=\mathrm{Mg}\) Fe) - can be viewed as a ccp arrangement of oxide ions with silicon(IV) ions in tetrahedral holes and metal ions in octahedral holes. a. What fraction of each type of hole is occupied? b. The unit cells of \(\mathrm{Mg}_{2} \mathrm{SiO}_{4}\) and \(\mathrm{Fe}_{2} \mathrm{SiO}_{4}\) have volumes of \(2.91 \times 10^{-26} \mathrm{cm}^{3}\) and \(3.08 \times 10^{-26} \mathrm{cm}^{3} .\) Why is the volume of \(\mathrm{Fe}_{2} \mathrm{SiO}_{4}\) larger?

The cadmium(II) sulfide (CdS) mineral hawleyite, has a sphalerite structure, and its density at \(25^{\circ} \mathrm{C}\) is \(4.83 \mathrm{g} / \mathrm{cm}^{3} .\) A hypothetical form of CdS with the rock salt structure would have a density of \(5.72 \mathrm{g} / \mathrm{cm}^{3} .\) Why should the rock salt structure of CdS be denser? The ionic radii of \(\mathrm{Cd}^{2+}\) and \(S^{2-}\) are 95 pm and 184 pm, respectively.

There are two crystalline forms of manganese(II) sulfide \((\mathrm{MnS}):\) the \(\alpha\) form has a rock salt structure, whereas the \(\beta\) form has a sphalerite structure. a. Describe the differences between the two structures of \(\mathrm{MnS}\) b. The ionic radii of \(\mathrm{Mn}^{2+}\) and \(\mathrm{S}^{2-}\) are 67 and \(184 \mathrm{pm}\), respectively. Which type of hole in a ccp lattice of sulfide ions could theoretically accommodate a \(\mathrm{Mn}^{2+}\) \(\sin 3\)

The unit cell of rhenium trioxide \(\left(\mathrm{ReO}_{3}\right)\) consists of a cube with rhenium atoms at the corners and an oxygen atom on each of the 12 edges. The atoms touch along the edge of the unit cell. The radii of Re and \(\mathrm{O}\) atoms in \(\mathrm{ReO}_{3}\) are 137 and \(73 \mathrm{pm},\) respectively. a. Sketch the unit cell of \(\operatorname{Re} \mathrm{O}_{3}\) b. Calculate the density of \(\operatorname{Re} \mathrm{O}_{3}\) c. Calculate the percent empty space in a unit cell of \(\mathrm{ReO}_{3}\)

Crystalline potassium bromide (KBr) has a rock salt structure and a density of \(2.75 \mathrm{g} / \mathrm{cm}^{3} .\) Calculate the edge length of its unit cell.

Which of the following properties are associated with ceramics and which are associated with metals? (a) ductile; (b) thermal insulator; (c) electrically conductive; (d) malleable

Replacing \(\mathrm{Al}^{3+}\) ions in kaolinite \(\left[\mathrm{Al}_{2}\left(\mathrm{Si}_{2} \mathrm{O}_{5}\right)(\mathrm{OH})_{4}\right]\) with \(\mathrm{Mg}^{2+}\) ions yields the mineral antigorite. What is its formula?

What is the formula of the silicate mineral talc, obtained by replacing \(\mathrm{Al}^{3+}\) ions in pyrophyllite \(\left[\mathrm{Al}_{4} \mathrm{Si}_{8} \mathrm{O}_{20}(\mathrm{OH})_{4}\right]\) with \(\mathrm{Mg}^{2+}\) ions?

Kaolinite \(\left[\mathrm{Al}_{2}\left(\mathrm{Si}_{2} \mathrm{O}_{5}\right)(\mathrm{OH})_{4}\right]\) is formed by weathering of the mineral \(\mathrm{KAlSi}_{3} \mathrm{O}_{8}\) in the presence of carbon dioxide and water, as described by the following unbalanced reaction: \(\mathrm{KAISi}_{3} \mathrm{O}_{8}(s)+\mathrm{H}_{2} \mathrm{O}(\ell)+\mathrm{CO}_{2}(g) \rightarrow\) $$ \mathrm{Al}_{2}\left(\mathrm{Si}_{2} \mathrm{O}_{5}\right)(\mathrm{OH})_{4}(s)+\mathrm{SiO}_{2}(s)+\mathrm{K}_{2} \mathrm{CO}_{3}(a q) $$ Balance the reaction and determine whether or not it is a redox reaction.

Albite, a feldspar mineral with an ideal composition of $$\mathrm{NaAlSi}_{3} \mathrm{O}_{8}$$ can be converted to jadeite $$\left(\mathrm{NaAlSi}_{2} \mathrm{O}_{6}\right)$$ and quartz. Write a balanced chemical equation describing this transformation.

The calcium silicate mineral grossular is also formed under pressure in a reaction between anorthite \(\left(\mathrm{CaAl}_{2} \mathrm{Si}_{2} \mathrm{O}_{8}\right)\) gehlenite \(\left(\mathrm{Ca}_{2} \mathrm{Al}_{2} \mathrm{SiO}_{7}\right),\) and wollastonite \(\left(\mathrm{CaSiO}_{3}\right)\) (EQUATION CANNOT COPY) a. Balance this chemical equation. b. Express the composition of gehlenite the way mineralogists often do: as the percentage of the metal and metalloid oxides in it- that is, $$96 \mathrm{CaO}, \% \mathrm{Al}_{2} \mathrm{O}_{3}$$ $$\text { and } \% \mathrm{SiO}_{2}$$

The ceramic material barium titanate \(\left(\mathrm{BaTiO}_{3}\right)\) is used in devices that measure pressure. The radii of \(\mathrm{Ba}^{2+}, \mathrm{T} \mathrm{i}^{4+},\) and \(\mathrm{O}^{2-}\) are \(135,60.5,\) and \(140 \mathrm{pm},\) respectively. If the \(\mathrm{O}^{2-}\) ions are in a closest-packed structure, which hole(s) can accommodate the metal cations?

The mixed metal oxide \(\operatorname{LiMnTiO}_{4}\) has a structure with cubic closest-packed oxide ions and metal ions in both octahedral and tetrahedral holes. Which metal ion is most likely to be found in the tetrahedral holes? The ionic radii of \(\mathrm{Li}^{+}, \mathrm{Mn}^{3+}, \mathrm{Ti}^{4+},\) and \(\mathrm{O}^{2-}\) are \(76,67,60.5,\) and \(140 \mathrm{pm}\) respectively.

Why does an amorphous solid not produce an XRD scan with sharp peaks?

Why can we not use X-ray diffraction to determine the structures of compounds in solution?

Why are X-rays rather than microwaves chosen for diffraction studies of crystalline solids?

The spacing between the layers of ions in sylvite (the mineral form of \(\mathrm{KCl}\) ) is larger than in halite (NaCl). Which crystal will diffract X-rays of a given wavelength through larger \(2 \theta\) values?

Silver halides are used in black-and-white photography. In which compound would you expect to see a larger distance between ion layers, AgCl or AgBr? Which compound would you expect to diffract X-rays through larger values of \(2 \theta\) if the same wavelength of X-ray were used?

Cobalt(II) oxide is used as a pigment in ceramics. It has the same type of crystal structure as NaCl. When cobalt(II) oxide is exposed to X-rays with \(\lambda=154 \mathrm{pm},\) the XRD pattern contains strong peaks at \(2 \theta=42.38^{\circ}, 65.68^{\circ},\) and \(92.60^{\circ} .\) Determine the values of \(n\) to which these peaks correspond, and calculate the spacing between the crystal layers.

Pyrophyllite \(\left[\mathrm{Al}_{2} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\right]\) is a silicate mineral with a layered structure. The distance between the layers is \(1855 \mathrm{pm} .\) What is the smallest angle of diffraction of X-rays with \(\lambda=154 \mathrm{pm}\) from this solid?

Minnesotaite \(\left[\mathrm{Fe}_{3} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\right]\) is a silicate mineral with a layered structure similar to that of kaolinite. The distance between the layers in minnesotaite is \(1940 \pm 10 \mathrm{pm}\) What is the smallest angle of diffraction of X-rays with \(\lambda=154\) pm from this solid?

A unit cell consists of a cube that has an ion of element X at each corner, an ion of element \(Y\) at the center of the cube, and an ion of element \(Z\) at the center of each face. What is the formula of the compound?

An element crystallizes in the cubic closest-packed structure. The length of an edge of the unit cell is \(408 \mathrm{pm}\). The density of the element is \(19.27 \mathrm{g} / \mathrm{cm}^{3} .\) Identify the element.

What is the packing efficiency of the Si atoms in pure Si if the radius of one Si atom is \(117 \mathrm{pm} ?\) The density of pure silicon is \(2.33 \mathrm{g} / \mathrm{cm}^{3}\)

The colored lights on many electronic devices are light-emitting diodes (LEDs). One of the compounds used to make them is aluminum phosphide (AlP), which crystallizes in a sphalerite crystal structure. a. If AlP were an ionic compound, would the ionic radii of \(\mathrm{Al}^{3+}\) and \(\mathrm{P}^{3-}\) be consistent with the size requirements of the ions in a sphalerite crystal structure? b. If AlP were a covalent compound, would the atomic radii of Al and P be consistent with the size requirements of atoms in a sphalerite crystal structure?

Under the appropriate reaction conditions, small cubes of molybdenum, \(4.8 \mathrm{nm}\) on a side, can be deposited on carbon surfaces. These "nanocubes" are made of bcc arrays of Mo atoms. a. If the edge of each nanocube corresponds to 15 unit cell lengths, what is the effective radius of a molybdenum atom in these structures? b. What is the density of each molybdenum nanocube? c. How many Mo atoms are in each nanocube?

In the fullerene known as buckminsterfullerene, molecules of \(\mathrm{C}_{60}\) form a cubic closest-packed array of spheres with a unit cell edge length of \(1410 \mathrm{pm}\). a. What is the density of crystalline \(C_{60} ?\) b. If we treat each \(\mathrm{C}_{60}\) molecule as a sphere of 60 carbon atoms, what is the radius of the \(C_{60}\) molecule? c. \(C_{60}\) reacts with alkali metals to form \(\mathrm{M}_{3} \mathrm{C}_{60}\) (where \(\mathrm{M}=\mathrm{Na} \text { or } \mathrm{K}) .\) The crystal structure of \(\mathrm{M}_{3} \mathrm{C}_{60}\) contains cubic closest-packed spheres of \(C_{60}\) with metal ions in holes. If the radius of a \(\mathrm{K}^{+}\) ion is \(138 \mathrm{pm},\) which type of hole is a \(\mathrm{K}^{+}\) ion likely to occupy? What fraction of the holes will be occupied? d. Under certain conditions, a different substance, \(\mathrm{K}_{6} \mathrm{C}_{60}\) can be formed in which the \(\mathrm{C}_{60}\) molecules have a bcc unit cell. Calculate the density of a crystal of \(\mathrm{K}_{6} \mathrm{C}_{60}\)

The center of Earth is composed of a solid iron core within a molten iron outer core. When molten iron cools, it crystallizes in different ways depending on pressure - in a bcc unit cell at low pressure and in a hexagonal unit cell at high pressures like those at Earth's center. a. Calculate the density of bcc iron, assuming that the radius of an iron atom is \(126 \mathrm{pm}\). b. Calculate the density of hexagonal iron, given a unit cell volume of \(5.414 \times 10^{-23} \mathrm{cm}^{3}\) c. Seismic studies suggest that the density of Earth's solid core is less than that of hexagonal Fe. Laboratory studies have shown that up to \(4 \%\) by mass of Si can be substituted for Fe without changing the hcp crystal structure built on hexagonal unit cells. Calculate the density of such a crystal.

The unit cell of an alloy with a 1: 1 ratio of magnesium and strontium is identical to the unit cell of CsCl. The unit cell edge length of MgSr is \(390 \mathrm{pm}\). a. What is the density of MgSr? b. Find the atomic radii of Mg and Sr in Appendix 3. Do atoms touch along the body diagonal of the unit cell? c. Why doesn't the formula of the alloy allow us to distinguish between a Mg atom in the cubic hole of simple cubic unit cell of Sr atoms and a Sr atom in the cubic hole of a simple cubic arrangement of Mg atoms? d. \(\mathrm{MgSr}\) is a good electrical conductor. Do you expect a MgSr alloy to have a partially filled valence band or overlapping conduction and valence bands?

Substitutional alloys may form when the difference in atomic radii between the alloying elements is less than \(15 \%\) a. Predict which of the following alloys has the greatest mismatch in atomic radii: AuZn, AgZn, or CuZn. b. Find the atomic radii of \(\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au},\) and \(\mathrm{Zn}\) in Appendix 3, and calculate the percent difference in their atomic radii. Are all three alloys expected to form substitutional alloys? c. If gold is alloyed with silver in a 1: 1 ratio, do the atoms still touch along the face diagonal of a face-centered cubic unit cell?

Removing two electrons from \(\mathrm{S}_{8}\) yields the dication \(\mathrm{S}_{8}^{2+}\) Will all of the sulfur atoms be in one plane in the \(\mathrm{S}_{8}^{2+}\) cation?

The alloy Cu \(_{3}\) Al crystallizes in a bcc unit cell. Propose a way that the Cu and Al atoms could be allocated between bcc unit cells that is consistent with the formula of the alloy.