Chapter 12

Explain the difference between cubic closest-packed and hexagonal closest- packed arrangements of identical spheres.

Which unit cell has the greater packing efficiency, simple cubic or body- centered cubic?

The unit cell in iron metal is either fec or bec, depending on temperature. Are the fcc form of iron and the bec form allotropes? Explain your answer.

At low temperatures, the unit cell of calcium metal is found to be fcc, a closest-packed crystal lattice. At higher temperatures, the unit cell of calcium metal is found to be bcc, a crystal lattice that is not a closest- packed structure. What might be a reason for this difference?

Europium, one of the lanthanide elements used in television screens, crystallizes in a crystal lattice built on bcc unit cells, with a unit cell edge of \(240.6 \mathrm{pm}\). Calculate the radius of a europium atom.

Nickel has an fec unit cell with an edge length of \(350.7 \mathrm{pm}\). Calculate the radius of a nickel atom.

What is the length of an edge of the unit cell when barium (atomic radius \(222 \mathrm{pm}\) ) crystallizes in a crystal lattice of bcc unit cells?

What is the length of an edge of the unit cell when aluminum (atomic radius \(143 \mathrm{pm}\) ) crystallizes in a crystal lattice of fec unit cells?

Sodium metal crystallizes with a body-centered cubic structure at normal atmospheric pressure. The atomic radius of a sodium atom is \(186 \mathrm{pm}\) and the density of \(\mathrm{Na}\) is \(0.971 \mathrm{g} / \mathrm{cm}^{3} .\) At 613,000 atm ( 9 million psi) the structure of Na metal becomes fcc. Assuming the radius of Na is unchanged, what is the density of the fce form of Na?

Describe the structural differences between substitutional and interstitial alloys and give an example of each alloy.

Describe how homogeneous alloys and intermetallic compounds are similar and how they are different.

White gold was originally developed to give the appearance of platinum. One formulation of white gold contains \(25 \%\) nickel and \(75 \%\) gold. Which is more malleable, white gold or pure gold? Explain your answer.

Magnesium and hafnium have nearly the same atomic radius (within \(1 \mathrm{pm}\) ). Does substitution of \(25 \%\) of the magnesium by hafnium in an alloy increase or decrease the density of the alloy in comparison with pure magnesium? Explain your answer.

Why are the alloys that second-row nonmetals - such as \(\mathrm{B}\), \(\mathrm{C}\), and \(\mathrm{N}\)- -form with transition metals more likely to be interstitial than substitutional?

Hydrogen Storage Hydrogen is an attractive alternative fuel to hydrocarbons because it has a high fuel value and does not produce carbon dioxide when burned. One challenge to developing a hydrogen economy is storing hydrogen. Many transition metals absorb hydrogen by breaking the \(\mathrm{H}-\mathrm{H}\) bond and storing \(\mathrm{H}\) atoms in the interstices between the metal atoms. a. What is the minimum radius for the host metal for a H atom (radius \(37 \mathrm{pm}\) ) to fit in a tetrahedral hole of an fcc unit cell? b. If the \(H\) is present as a hydride ion, \(H^{-}\) (ionic radius \(=\) \(146 \mathrm{pm}),\) is the hydrogen more likely to be found as an interstitial alloy in a tetrahedral or octahedral hole or as a substitutional alloy? Hint: Consult Appendix 3 for atomic radii of metals.

Metal borides MB \(_{x}\) exist for many transition metals. a. What is the minimum radius for the host metal for a B atom (radius \(88 \mathrm{pm}\) ) to fit in an octahedral hole of an foc unit cell? b. What is the value of \(x\) if an average of one octahedral hole is occupied per unit cell?

Dental Fillings Dental fillings are mixtures of several alloys, including one with the formula \(\mathrm{Ag}_{3} \mathrm{Sn}\). Silver (radius \(144 \mathrm{pm}\) ) and tin \((140 \mathrm{pm})\) both crystallize in an fcc unit cell. Is this alloy likely to be a substitutional alloy or an interstitial alloy?

What is the formula of vanadium carbide if the vanadium atoms have a cubic closest-packed structure and two of the octahedral holes are occupied by C atoms?

If the unit cell of a substitutional alloy of copper and tin has the same unit cell edge as the unit cell of copper, will the alloy have a greater density than copper? Explain your answer.

If the unit cell of an interstitial alloy of vanadium and carbon has the same unit cell edge as the unit cell of vanadium, will the alloy have a greater density than vanadium? Explain your answer.

Crystals of both \(\mathrm{LiCl}\) and \(\mathrm{KCI}\) have the rock salt structure. In the unit cell of \(\mathrm{LiCl}\), adjacent \(\mathrm{Cl}\)" ions touch each other. In \(\mathrm{KCI}\) , they don't. Why?

Does the absolute size of an octahedral hole in an \(\mathrm{fee}\) lattice of halide ions change as we move down group 17 from fluoride to iodide?

If some of the sulfide ions in zinc sulfide are replaced by selenide ions, will the selenide ions occupy the same sites as the sulfide ions?

Why can't sodium fluoride have exactly the same structure as calcium fluoride?

As the cation-anion radius ratio increases for an ionic compound with the rock salt crystal structure, is the calculated density more likely to be greater than, or less than, the measured value?

As the cation-anion radius ratio increases for an ionic compound with the rock salt crystal structure, is the length of the unit cell edge calculated from ionic radii likely to be greater than, or less than, the observed unit cell edge length?

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?

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

The sphalerite structure of ZnS changes to a rock salt structure above \(15 \mathrm{GPa}\) a. Describe the differences between these two structures. b. Explain how both structures are consistent with the observed stoichiometry of ZnS.

The unit cell of rhenium trioxide (ReO \(_{3}\) ) 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 O atoms in \(\operatorname{Re} \mathrm{O}_{3}\) are 137 and \(73 \mathrm{pm}\), respectively. Calculate the density of \(\mathrm{ReO}_{3}\)

With reference to Figure \(\mathrm{P} 12.55,\) calculate the density of simple cubic CsCl.

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

In the fullerene known as buckminsterfullerene, \(C_{60}\) 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 \(C_{60}\) molecule as a sphere of 60 carbon atoms, what is the radius of the \(C_{60}\) molecule?

Ice under Pressure Kurt Vonnegut's novel Cat's Cradle describes an imaginary, high-pressure form of ice called "ice-nine." With the assumption that ice-nine has a cubic closest-packed arrangement of oxygen atoms with hydrogen atoms in the appropriate holes, what type of hole will accommodate the \(\mathrm{H}\) atoms?

Can a polymer be composed of more than one type of monomer? Explain why or why not.

Compare the large hydrocarbon \(\mathrm{C}_{24} \mathrm{H}_{50}\) with polycthylenc. What structural feature(s) do they share in common? How do the structures of these compounds differ?

Polyethylene is prepared from the monomer ethylene, \(\mathrm{C}_{2} \mathrm{H}_{4} .\) About how many monomers are needed to make a polymer with a molar mass of \(100,000 \mathrm{g} / \mathrm{mol} ?\)

Synthetic rubber is prepared from butadicnc, \(\mathrm{C}_{4} \mathrm{H}_{6}\). About how many monomers are needed to make a polymer with a molar mass of \(100,000 \mathrm{g} / \mathrm{mol} ?\)

Non-Biodegradable Polymers Biodegradable polymers are important in applications such as dissolving sutures, but biocompatible polymers that resist degradation can serve as drug delivery systems in implanted devices. Poly(caprolactone) (PCL, Figure P12.88) degrades slowly in the human body but is very permeable to the contraceptive levonorgestrel. This combination is used in implantable contraceptive devices that are effective for many years. a. Why is levonorgestrel soluble in PCL? b. PCL does eventually degrade by reacting with water. What are the products of biodegradation? c. Is PCL an addition or a condensation polymer?

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?

The unit cell of an oxide of uranium consists of cubic closest-packed uranium ions with oxide ions in all the tetrahedral holes. What is the formula of the oxide?

The phase diagram for titanium is shown in Figure P12.91. a. Which structure does Ti metal have at \(1500 \mathrm{K}\) and \(6 \mathrm{GPa}\) of pressure? b. How many phase changes does Ti metal undergo as pressure is increased at \(725^{\circ} \mathrm{C} ?\)

Silver nanoparticles are embedded in clothing fabric to kill odor-causing bacteria. Silver crystallizes in an fcc unit cell. Aow many unit cells are present in a cubic silver particle with an edge length of \(25 \mathrm{nm} ?\) b. How many silver atoms are there in this particle? c. If there are \(1360 \mu \mathrm{g}\) of silver per sock, how many of these particles does this mass correspond to?

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 bec iron given 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 only about \(90 \%\) of 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 of MgSr is \(390 \mathrm{pm}\). What is the density of MgSr?

Gold and silver can be separately alloyed with zinc to form AuZn (unit cell edge \(319 \mathrm{pm}\) ) and \(\mathrm{Ag}\) Zn (unit cell edge \(316 \mathrm{pm}) .\) The two alloys have the same unit cell. Which alloy is more dense?

Manganese steels are a mixture of iron, manganese, and carbon. Is the manganese likely to occupy holes in the austenite foc unit cell, or are manganese steels substitutional alloys?

Aluminum forms alloys with lithium \((LiAl),\) gold \(\left(\mathrm{Au} \mathrm{Al}_{2}\right)\) and titanium \(\left(\mathrm{Al}_{3} \mathrm{Ti}\right) .\) On the basis of their crystal lattices, each of these alloys is considered a substitutional alloy. a. Do these alloys fit the general size requirements for substitutional alloys? The atomic radii for Li, Al, Au, and Ti are \(152,143,144,\) and \(147 \mathrm{pm},\) respectively. b. If the unit cell of LiAl is bec, what is the density of LiAl?

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