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Essentials of Materials Science & Engineering - SI Version

Donald R. Askeland, Pradeep P. Fulay

Chapter 4

Imperfections in the Atomic and Ionic Arrangements - all with Video Answers

Educators

Chapter Questions

02:37

Problem 1

Calculate the number of vacancies per $\mathrm{cm}^3$ expected in copper at $1080^{\circ} \mathrm{C}$ (just below the melting temperature). The energy for vacancy formation is $20,000 \mathrm{cal} / \mathrm{mol}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:37

Problem 2

The fraction of lattice points occupied by vacancies in solid aluminum at $660^{\circ} \mathrm{C}$ is $10^{-3}$. What is the energy required to create vacancies in aluminum?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:57

Problem 3

The density of a sample of FCC palladium is $11.98 \mathrm{~g} / \mathrm{cm}^3$ and its lattice parameter is $3.8902 \mathrm{~A}$ Calculate
(a) the fraction of the lattice points that contain vacancies; and
(b) the total number of vacancies in a cubic centimeter of $\mathrm{Pd}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:30

Problem 4

The density of a sample of HCP beryllium is $1.844 \mathrm{~g} / \mathrm{cm}^3$ and the lattice parameters are $a_0=$ $0.22858 \mathrm{~nm}$ and $c_0=0.35842 \mathrm{~nm}$. Calculate
(a) the fraction of the lattice points that contain vacancies; and
(b) the total number of vacancies in a cubic centimeter.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:53

Problem 5

BCC lithium has a lattice parameter of $3.5089 \times$ $10^{-8} \mathrm{~cm}$ and contains one vacancy per 200 unit cells. Calculate
(a) the number of vacancies per cubic centimeter; and
(b) the density of Li.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
07:44

Problem 6

FCC lead $(\mathrm{Pb})$ has a lattice parameter of 0.4949 $\mathrm{nm}$ and contains one vacancy per $500 \mathrm{~Pb}$ atoms. Calculate
(a) the density; and
(b) the number of vacancies per gram of $\mathrm{Pb}$.

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
03:47

Problem 7

A niobium alloy is produced by introducing tungsten substitutional atoms in the BCC structure; eventually an alloy is produced that has a lattice parameter of $0.32554 \mathrm{~nm}$ and a density of $11.95 \mathrm{~g} / \mathrm{cm}^3$. Calculate the fraction of the atoms in the alloy that are tungsten.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:30

Problem 8

Tin atoms are introduced into a FCC copper crystal, producing an alloy with a lattice parameter of $3.7589 \times 10^{-8} \mathrm{~cm}$ and a density of $8.772 \mathrm{~g} / \mathrm{cm}^3$. Calculate the atomic percentage of tin present in the alloy.

Narayan Hari
Narayan Hari
Numerade Educator
03:53

Problem 9

We replace 7.5 atomic percent of the chromium atoms in its BCC crystal with tantalum. X-ray diffraction shows that the lattice parameter is $0.29158 \mathrm{~nm}$. Calculate the density of the alloy.

Madi Sousa
Madi Sousa
Numerade Educator
01:18

Problem 10

Suppose we introduce one carbon atom for every 100 iron atoms in an interstitial position in BCC iron, giving a lattice parameter of $0.2867 \mathrm{~nm}$. For the $\mathrm{Fe}-\mathrm{C}$ alloy, find the density and the packing factor.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:46

Problem 11

The density of BCC iron is $7.882 \mathrm{~g} / \mathrm{cm}^3$ and the lattice parameter is $0.2866 \mathrm{~nm}$ when hydrogen atoms are introduced at interstitial positions. Calculate
(a) the atomic fraction of hydrogen atoms; and
(b) number of unit cells on average that contain hydrogen atoms.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:33

Problem 12

Suppose one Schottky defect is present in every tenth unit cell of $\mathrm{MgO} . \mathrm{MgO}$ has the sodium chloride crystal structure and a lattice parameter of $0.396 \mathrm{~nm}$. Calculate
(a) the number of anion vacancies per $\mathrm{cm}^3$; and (b) the density of the ceramic.

Madi Sousa
Madi Sousa
Numerade Educator
04:58

Problem 13

$\mathrm{ZnS}$ has the zinc blende structure. If the density is $3.02 \mathrm{~g} / \mathrm{cm}^3$ and the lattice parameter is $0.59583 \mathrm{~nm}$, determine the number of Schottky defects
(a) per unit cell; and
(b) per cubic centimeter.

Freddie Montague
Freddie Montague
Numerade Educator
01:19

Problem 14

What are the Miller indices of the slip directions:
(a) on the (111) plane in an FCC unit cell?
(b) on the (011) plane in a BCC unit cell?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:10

Problem 15

What are the Miller indices of the slip planes in FCC unit cells that include the $[101]$ slip direction?

Ameer Said
Ameer Said
Numerade Educator
01:06

Problem 16

What are the Miller indices of the $\{110\}$ slip planes in BCC unit cells that include the [111] slip direction?

Ameer Said
Ameer Said
Numerade Educator
01:13

Problem 17

Calculate the length of the Burgers vector in the following materials:
(a) $\mathrm{BCC}$ niobium;
(b) FCC silver; and
(c) diamond cubic silicon.

Manik Pulyani
Manik Pulyani
Numerade Educator
09:37

Problem 18

Determine the interplanar spacing and the length of the Burgers vector for slip on the expected slip systems in FCC aluminum. Repeat, assuming that the slip system is a (110) plane and a [1T1] direction. What is the ratio between the shear stresses required for slip for the two systems? Assume that $\mathrm{k}=2$ in Equation 4-2.

Ameer Said
Ameer Said
Numerade Educator
03:40

Problem 19

Determine the interplanar spacing and the length of the Burgers vector for slip on the (110)/[1 11$]$ slip system in BCC tantalum. Repeat, assuming that the slip system is a $(111) /[1 \overline{1} 0]$ system. What is the ratio between the shear stresses required for slip for the two systems? Assume that $\mathrm{k}=2$ in Equation 4-2.

Ameer Said
Ameer Said
Numerade Educator
01:27

Problem 20

How many grams of aluminum, with a dislocation density of $10^{10} \mathrm{~cm} / \mathrm{cm}^3$, are required to give a total dislocation length that would stretch from New York City to Los Angeles $(4860 \mathrm{~km})$ ?

Narayan Hari
Narayan Hari
Numerade Educator
03:40

Problem 21

Compare the $c / a$ ratios for the following HCP metals, determine the likely slip processes in each, and estimate the approximate critical resolved shear stress. Explain. (See data in Appendix A.)
(a) zine
(b) magnesium
(c) titanium
(d) zirconium
(e) rhenium
(f) beryllium

Ameer Said
Ameer Said
Numerade Educator
00:30

Problem 22

A single crystal of an FCC metal is oriented so that the $[001]$ direction is parallel to an applied stress of $35 \mathrm{MPa}$. Calculate the resolved shear $[0 \overline{1} 1]$, and $[10 \overline{1}]$ slip directions. Which slip system (s) will become active first?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:19

Problem 23

A single crystal of a BCC metal is oriented so that the $[001]$ direction is parallel to the applied stress. If the critical resolved shear stress required for slip is $83 \mathrm{MPa}$, calculate the magnitude of the applied direction on the (110), (011), and (101) slip planes.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:35

Problem 24

Why is it that single crystal and polycrystalline copper are both ductile, however, single crystal, but not polycrystalline, zinc can exhibit considerable ductility?

Supratim Pal
Supratim Pal
Numerade Educator
00:52

Problem 25

Why is it that cross slip in BCC and FCC metals is easier than that in HCP metals? How does this influence the ductility of BCC, FCC, and HCP metals?

Ameer Said
Ameer Said
Numerade Educator
03:28

Problem 26

The strength of titanium is found to be $448 \mathrm{MPa}$ when the grain size is $17 \times 10^{-6} \mathrm{~m}$ and $565 \mathrm{MPa}$ when the grain size is $0.8 \times 10^{-6} \mathrm{~m}$. Determine
(a) the constants in the Hall-Petch equation; and
(b) the strength of the titanium when the grain size is reduced to $0.2 \times 10^{-6} \mathrm{~m}$.

Surendra Kumar
Surendra Kumar
Numerade Educator
02:06

Problem 27

A copper-zinc alloy has the following properties
$$
\begin{array}{lc}
\text { Grain Diameter (mm) } & \text { Strength (MPa) } \\
\hline 0.015 & 170 \mathrm{MPa} \\
0.025 & 158 \mathrm{MPa} \\
0.035 & 151 \mathrm{MPa} \\
0.050 & 145 \mathrm{MPa}
\end{array}
$$
Determine
(a) the constants in the Hall-Petch equation; and
(b) the grain size required to obtain a strength of $200 \mathrm{MPa}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:09

Problem 28

For an ASTM grain size number of 8 , calculate the number of grains per square inch
(a) at a magnification of 100 and
(b) with no magnification.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:12

Problem 29

Determine the ASTM grain size number if 20 grains/square inch are observed at a magnification of 400.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:32

Problem 30

Determine the ASTM grain size number if 25 grains/square inch are observed at a magnification of 50 .

Hunza Gilgit
Hunza Gilgit
Numerade Educator
04:56

Problem 31

Determine the ASTM grain size number for the materials in: Figure 4-15 and Figure 4-18.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:01

Problem 32

The yield stress of several samples of a steel containing $0.12 \%$ carbon with different grain sizes was measured. The data are shown here.
$$
\begin{array}{lcc}
\begin{array}{l}
\text { Sample } \\
\text { ID }
\end{array} & \begin{array}{c}
\text { Grain-Size Inverse } \\
\text { Square Root } \\
\left(\boldsymbol{d}^{-1 / 2}\right)
\end{array} & \begin{array}{c}
\text { Yield Stress } \\
(\mathrm{MPa})
\end{array} \\
\hline \text { A } & 24 & 500 \\
\text { B } & 19 & 420 \\
\text { C } & 12 & 320 \\
\text { D } & 10 & 250 \\
\text { E } & 6 & 190
\end{array}
$$
(a) Calculate the grain size of each steel sample in micrometers.
(b) Which sample has the grain size of $27.7 \mu \mathrm{m}$ ?
(c) Fit these data to a straight line and calculate the constants $\sigma_0$ and $\mathrm{K}$ for the Hall-Petch equation.
(d) What is the grain size of the sample that has the highest yield strength?
(e) A sample of this steel with $15 \mu \mathrm{m}$ grain size is produced. What will be the yield stress of this sample?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:43

Problem 33

A researcher working in the nano-science area develops a sample of $0.12 \%$ carbon steel such that the value of $d^{-1 / 2}$ is 110 . What will be the grain size of this steel? Can she use the Hall-Petch relationship developed for this steel in the previous problem to predict the yield stress of this sample?

Prabhu Ramji
Prabhu Ramji
Numerade Educator

Problem 34

What is meant by the term strain hardening?

Check back soon!

Problem 35

Which mechanism of strengthening is the HallPetch equation related to?

Check back soon!
02:42

Problem 36

Pure copper is strengthened by addition of small concentration of $\mathrm{Be}$. What mechanism of strengthening is this related to?
C. Design Problems

Ramesh Singh
Ramesh Singh
Numerade Educator
00:55

Problem 37

The density of pure aluminum calculated from crystallographic data is expected to be $2.69955 \mathrm{~g} / \mathrm{cm}^3$.
(a) Design an aluminum alloy that has a density of $2.6450 \mathrm{~g} / \mathrm{cm}^3$.
(b) Design an aluminum alloy that has a density of $2.7450 \mathrm{~g} / \mathrm{cm}^3$.

Himanshu Garg
Himanshu Garg
Numerade Educator
00:33

Problem 38

You would like to use a metal plate with good weldability. During the welding process, the metal next to the weld is heated almost to the melting temperature and, depending on the welding parameters, may remain hot for some period of time. Design an alloy that will minimize the loss of strength in this "heat-affected zone" during the welding process.

Kristela Garcia
Kristela Garcia
Numerade Educator