21.23.^56 Fe crystallizes in a body-centered unit cell having
sides 2.8664 Å on a side. Its atomic mass is 55.9349 g/mol
and its density is 7.8748 g/cm^3. From this information, calcu-
late Avogadro’s number, NA. (This is one of the more accurate
ways of determining NA.)
21.6 Miller Indices
21.24.For a simple cubic lattice, what Miller indices describe
the plane(s) that contain two of the three crystal axes?
21.25.For a simple cubic lattice, what is the ratio of the d
spacings for the (100), (110), and (111) planes?
21.26.For a face-centered cubic lattice, what are the Miller
indices of the plane made by the atoms centered in the faces
of the unit cells as exemplified in Figure 21.36?
21.27.Because crystals exist in three dimensions, 3-D dia-
grams are often necessary to illustrate concepts. Using the sin-
gle unit cell in Figure 21.21, draw a 2 2 2 set of eight
cubic unit cells and draw the same plane in all unit cells.
21.28.Consider Figure 21.21. If the lower rightmost corner
of the unit cell were selected arbitrarily as the origin, what
would be the Miller indices of the indicated plane? Compare
your answer to the solution of Example 21.7.
21.29.Any one plane can be described by more than one set
of Miller indices, if negative indices are used. For a cubic unit
cell the (111) plane is equivalent to what other plane whose
indices are expressed in terms of all positive numbers? You
may have to draw a few unit cells to determine an answer.
21.30.The aluminum-nickel alloy AlNi has a simple cubic lat-
tice with a unit cell parameter of 2.88 Å. If X rays having a
wavelength of 1.544 Å were used, at what angles would the
X rays be diffracted by (a)the (100) plane of atoms; (b)the
(110) plane of atoms; (c)the (210) plane of atoms?
21.31.A powdered sample diffracts X rays ( 1.5418 Å) at
angles of 15.7°, 18.2°, 26.1°, 31.1°, and 32.6°. What type of
cubic crystal is it, and what is the unit cell parameter?
21.32.Predict the angles of diffraction of X rays having
1.54056 Å by KBr, which has the sodium chloride structure
and a unit cell parameter of 6.59 Å. Consult Table 21.3.
21.33.Use geometric arguments to illustrate why the (111)
plane of a body-centered cubic lattice does not cause de-
tectable diffraction of X rays.
21.34.Explain why the X-ray diffraction pattern of CuZn, a
1 1 stoichiometric form of brass that has a body-centered
cubic unit cell, is sometimes mistakenly interpreted as simple
cubic. (Consider the scattering factors of the atoms.)
21.35.A given X-ray diffraction pattern is composed of dif-
fractions that are roughly the same intensity. Explain whether
or not this sole fact supports the possible identification of the
sample as (a)KBr (b)CsF (c)NaCl (d)MgO.21.7 Predicting Unit Cells
21.36.Predict the unit cells for the following materials: (a)
potassium bromide, KBr; (b)cesium fluoride, CsF; (c)barium
oxide, BaO.
21.37.Predict the unit cells for the following materials: (a)
titanium sulfide, TiS 2 ; (b)barium fluoride, BaF 2 ; (c)potassium
sulfate, K 2 SO 4.
21.38.Sulfur, S, has some interesting solid-solid phase
changes relatively close to room temperature. At room tem-
perature it has an orthorhombic unit cell, but it is monoclinic
at temperatures not much higher than boiling water. Why isn’t
elemental sulfur hcp or fcc?
21.39.Explain why the element carbon does not have a face-
centered cubic or hexagonal close-packed unit cell even
though we typically designate the element carbon with the
monatomic formula C.
21.40.What is the coordination number in the cesium chlo-
ride cubic structure?
21.41.Determine the coordination number(s) of the ions in
the fluorite and rutile unit cells. Why are there two unequal
coordination numbers, whereas for cesium chloride, sodium
chloride, and zincblende unit cells there is only one coordina-
tion number?
21.42.Which solid phase (that is, which allotrope) of carbon
is more stable, graphite or diamond? (You should consult some
of the tables in the thermodynamics section of this text.) Both
solid phases exist under normal conditions of pressure and
temperature. Explain why this is so, given that one solid phase
is more thermodynamically stable than the other. Do their unit
cells provide any suggestion for their relative stabilities?21.8 & 21.9 Lattice Energies, Defects,
and Semiconductors
21.43.Write the specific chemical reactions whose enthalpy
change (or negative thereof) represent the lattice energy of
(a)potassium fluoride, KF; (b)magnesium selenide, MgSe;
(c)sodium oxide, Na 2 O; (d)sodium peroxide, Na 2 O 2.
21.44.Explain why lattice energy is considered a form of po-
tentialenergy.
21.45.Write Born-Haber cycles showing the relationship be-
tween the formation reaction and the lattice energy definitions
of each of the ionic compounds in exercise 21.43. You may
need to review the definition of “formation reaction” from ear-
lier in the text.Exercises for Chapter 21 763Figure 21.36 What are the Miller indices of the indicated plane? See
exercise 21.26.