Materials Science and Engineering. An Introduction

William D. Callister

Chapter 4

Imperfections in Solids - all with Video Answers

Educators

Chapter Questions

Problem 1

Calculate the fraction of atom sites that are vacant for copper at its melting temperature of $1084^{\circ} \mathrm{C}(1357 \mathrm{~K})$. Assume an energy for vacancy formation of $0.90 \mathrm{eV} /$ atom.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 2

Calculate the number of vacancies per cubic meter in gold at $900^{\circ} \mathrm{C}$. The energy for vacancy formation is $0.98 \mathrm{eV} /$ atom. Furthermore, the density and atomic weight for Au are $18.63 \mathrm{~g} / \mathrm{cm}^3$ (at $900^{\circ} \mathrm{C}$ ) and $196.9 \mathrm{~g} / \mathrm{mol}$, respectively.

Manik Pulyani

Numerade Educator

Problem 3

Calculate the energy for vacancy formation in silver, given that the equilibrium number of vacancies at $800^{\circ} \mathrm{C}(1073 \mathrm{~K})$ is $3.6 \times 10^{23} \mathrm{~m}^{-3}$. The atomic weight and density (at $800^{\circ} \mathrm{C}$ ) for silver are, respectively, $107.9 \mathrm{~g} / \mathrm{mol}$ and $9.5 \mathrm{~g} / \mathrm{cm}^3$.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 4

Below, atomic radius, crystal structure, electronegativity, and the most common valence are tabulated, for several elements; for those that are nonmetals, only atomic radii are indicated.
$$
\begin{array}{ccccc}
\hline & \begin{array}{c}
\text { Atomic } \\
\text { Radius } \\
\text { (nmm })
\end{array} & \begin{array}{c}
\text { Crystal } \\
\text { Structure }
\end{array} & \begin{array}{c}
\text { Electro- } \\
\text { negativity }
\end{array} & \text { Valence } \\
\hline \mathrm{Ni} & 0.1246 & \mathrm{FCC} & 1.8 & +2 \\
\mathrm{C} & 0.071 & & & \\
\mathrm{H} & 0.046 & & & \\
\mathrm{O} & 0.060 & & & \\
\mathrm{Ag} & 0.1445 & \mathrm{FCC} & 1.9 & +1 \\
\mathrm{Al} & 0.1431 & \mathrm{FCC} & 1.5 & +3 \\
\mathrm{Co} & 0.1253 & \mathrm{HCP} & 1.8 & +2 \\
\mathrm{Cr} & 0.1249 & \mathrm{BCC} & 1.6 & +3 \\
\mathrm{Fe} & 0.1241 & \mathrm{BCC} & 1.8 & +2 \\
\mathrm{Pt} & 0.1387 & \mathrm{FCC} & 2.2 & +2 \\
\mathrm{Zn} & 0.1332 & \mathrm{HCP} & 1.6 & +2 \\
\hline
\end{array}
$$
Which of these elements would you expect to form the following with nickel:
(a) A substitutional solid solution having complete solubility
(b) A substitutional solid solution of incomplete solubility
(c) An interstitial solid solution

Cathy Geisel

Numerade Educator

Problem 5

For both FCC and BCC crystal structures, there are two different types of interstitial sites. In each case, one site is larger than the other, and is normally occupied by impurity atoms. For FCC, this larger one is located at the center of each edge of the unit cell; it is termed an octahedral interstitial site. On the other hand, with BCC the larger site type is found at $0 \frac{1}{2} \frac{1}{4}$ positions-that is, lying on (100) faces, and situated midway between two unit cell edges on this face and one-quarter of the distance between the other two unit cell edges; it is termed a tetrahedral interstitial site. For both FCC and BCC crystal structures, compute the radius $r$ of an impurity atom that will just fit into one of these sites in terms of the atomic radius $R$ of the host atom.

Manik Pulyani

Numerade Educator

Problem 6

Derive the following equations:
(a) Equation 4.7 a
(b) Equation 4.9 a
(c) Equation 4.10 a
(d) Equation 4.11 b

Manik Pulyani

Numerade Educator

Problem 7

What is the composition, in atom percent, of an alloy that consists of $92.5 \mathrm{wt} \% \mathrm{Ag}$ and $7.5 \mathrm{wt} \% \mathrm{Cu}$ ?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 8

What is the composition, in weight percent, of an alloy that consists of $5 \mathrm{at} \% \mathrm{Cu}$ and $95 \mathrm{at} \% \mathrm{Pt}$ ?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 9

Calculate the composition, in weight percent, of an alloy that contains 105 kg of iron, 0.2 kg of carbon, and 1.0 kg of chromium.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 10

What is the composition, in atom percent, of an alloy that contains 33 g copper and 47 g zinc?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 11

What is the composition, in atom percent, of an alloy that contains $44.5 \mathrm{lb}_{\mathrm{m}}$ of silver, $83.7 \mathrm{lb}_{\mathrm{m}}$ of gold, and $5.3 \mathrm{lb}_{\mathrm{m}}$ of Cu ?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 12

What is the composition, in atom percent, of an alloy that consists of $5.5 \mathrm{wt} \% \mathrm{~Pb}$ and $94.5 \mathrm{wt} \% \mathrm{Sn}$ ?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 13

Convert the atom percent composition in Problem 4.11 to weight percent.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 14

Calculate the number of atoms per cubic meter in lead.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 15

The concentration of silicon in an iron-silicon alloy is $0.25 \mathrm{wt} \%$. What is the concentration in kilograms of silicon per cubic meter of alloy?

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 16

Determine the approximate density of a Ti-6Al-4V titanium alloy that has a composition of $90 \mathrm{wt} \% \mathrm{Ti}, 6 \mathrm{wt} \% \mathrm{Al}$, and $4 \mathrm{wt} \% \mathrm{~V}$.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 17

Calculate the unit cell edge length for an $80 \mathrm{wt} \% \mathrm{Ag}-20 \mathrm{wt} \% \mathrm{Pd}$ alloy. All of the palladium is in solid solution, the crystal structure for this alloy is FCC, and the roomtemperature density of Pd is $12.02 \mathrm{~g} / \mathrm{cm}^3$.

Manik Pulyani

Numerade Educator

Problem 18

Some hypothetical alloy is composed of $25 \mathrm{wt} \%$ of metal A and $75 \mathrm{wt} \%$ of metal B. If the densities of metals A and B are 6.17 and $8.00 \mathrm{~g} / \mathrm{cm}^3$, respectively, whereas their respective atomic weights are 171.3 and $162.0 \mathrm{~g} / \mathrm{mol}$, determine whether the crystal structure for this alloy is simple cubic, face-centered cubic, or body-centered cubic. Assume a unit cell edge length of 0.332 nm .

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 19

For a solid solution consisting of two elements (designated as 1 and 2 ), sometimes it is desirable to determine the number of atoms per cubic centimeter of one element in a solid solution, $N_1$, given the concentration of that element specified in weight percent, $C_1$. This computation is possible using the following expression:

$$
N_1=\frac{N_{\mathrm{A}} C_1}{\frac{C_1 A_1}{\rho_1}+\frac{A_1}{\rho_2}\left(100-C_1\right)}
$$

where
$N_{\mathrm{A}}=$ Avogadro's number
$\rho_1$ and $\rho_2=$ densities of the two elements
$A_1=$ the atomic weight of element 1
Derive Equation 4.18 using Equation 4.2 and expressions contained in Section 4.4.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

Molybdenum forms a substitutional solid solution with tungsten. Compute the number of molybdenum atoms per cubic centimeter for a molybdenum-tungsten alloy that contains $16.4 \mathrm{wt} \%$ Mo and $83.6 \mathrm{wt} \% \mathrm{~W}$. The densities of pure molybdenum and tungsten are 10.22 and $19.30 \mathrm{~g} / \mathrm{cm}^3$, respectively.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 21

Niobium forms a substitutional solid solution with vanadium. Compute the number of niobium atoms per cubic centimeter for a niobium-vanadium alloy that contains 24 $\mathrm{wt} \% \mathrm{Nb}$ and $76 \mathrm{wt} \% \mathrm{~V}$. The densities of pure niobium and vanadium are 8.57 and 6.10 $\mathrm{g} / \mathrm{cm}^3$, respectively.

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 22

Sometimes it is desirable to be able to determine the weight percent of one element, $C_1$, that will produce a specified concentration in terms of the number of atoms per cubic centimeter, $N_1$, for an alloy composed of two types of atoms. This computation is possible using the following expression:

$$
C_1=\frac{100}{1+\frac{N_{\mathrm{A}} \rho_2}{N_1 A_1}-\frac{\rho_2}{\rho_1}}
$$

where

$$
\begin{aligned}
N_{\mathrm{A}} & =\text { Avogadro's number } \\
\rho_1 \text { and } \rho_2 & =\text { densities of the two elements } \\
A_1 \text { and } A_2 & =\text { the atomic weights of the two } \\
& \text { elements }
\end{aligned}
$$

Derive Equation 4.19 using Equation 4.2 and expressions contained in Section 4.4.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

Gold forms a substitutional solid solution with silver. Compute the weight percent of gold that must be added to silver to yield an alloy that contains $5.5 \times 10^{21} \mathrm{Au}$ atoms per cubic centimeter. The densities of pure Au and Ag are 19.32 and $10.49 \mathrm{~g} / \mathrm{cm}^3$, respectively.

Narayan Hari

Numerade Educator

Problem 24

Germanium forms a substitutional solid solution with silicon. Compute the weight percent of germanium that must be added to silicon to yield an alloy that contains $2.43 \times 10^{21} \mathrm{Ge}$ atoms per cubic centimeter. The densities of pure Ge and Si are 5.32 and $2.33 \mathrm{~g} / \mathrm{cm}^3$, respectively.

Hunza Gilgit

Numerade Educator

Problem 25

Iron and vanadium both have the BCC crystal structure, and V forms a substitutional solid solution for concentrations up to approximately $20 \mathrm{wt} \% \mathrm{~V}$ at room temperature. Compute the unit cell edge length for a $90 \mathrm{wt} \% \mathrm{Fe}-10 \mathrm{wt} \% \mathrm{~V}$ alloy.

Manik Pulyani

Numerade Educator

Problem 26

Cite the relative Burgers vector-dislocation line orientations for edge, screw, and mixed dislocations.

Hunza Gilgit

Numerade Educator

Problem 27

For an FCC single crystal, would you expect the surface energy for a (100) plane to be greater or less than that for a (111) plane? Why? (Note: You may want to consult the solution to Problem 3.53 at the end of Chapter 3.)

Manik Pulyani

Numerade Educator

Problem 28

For a BCC single crystal, would you expect the surface energy for a (100) plane to be greater or less than that for a (110) plane? Why? (Note: You may want to consult the solution to Problem 3.54 at the end of Chapter 3.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

(a) For a given material, would you expect the surface energy to be greater than, the same as, or less than the grain boundary energy? Why?
(b) The grain boundary energy of a smallangle grain boundary is less than for a highangle one. Why is this so?

Hunza Gilgit

Numerade Educator

Problem 30

(a) Briefly describe a twin and a twin boundary.
(b) Cite the difference between mechanical and annealing twins.

Hunza Gilgit

Numerade Educator

Problem 31

For each of the following stacking sequences found in FCC metals, cite the type of planar defect that exists:
(a) $\ldots A B C A B C B A C B A \ldots$
(b) $\ldots A B C A B C B C A B C \ldots$

Now, copy the stacking sequences and indicate the position(s) of planar defect(s) with a vertical dashed line.

Hunza Gilgit

Numerade Educator

Problem 32

(a) Using the intercept method, determine the average grain size, in millimeters, of the specimen whose microstructure is shown in Figure 4.14(b); use at least seven straight-line segments.
(b) Estimate the ASTM grain size number for this material.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

(a) Employing the intercept technique, determine the average grain size for the steel specimen whose microstructure is shown in Figure 9.25(a); use at least seven straight-line segments.
(b) Estimate the ASTM grain size number for this material.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

For an ASTM grain size of 6, approximately how many grains would there be per square inch at
(a) a magnification of 100 , and
(b) without any magnification?

Hunza Gilgit

Numerade Educator

Problem 35

Determine the ASTM grain size number if 30 grains per square inch are measured at a magnification of 250 .

Hunza Gilgit

Numerade Educator

Problem 36

Determine the ASTM grain size number if 25 grains per square inch are measured at a magnification of 75 .

Hunza Gilgit

Numerade Educator