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physics
thermodynamics

Materials Science and Engineering An Introduction 8th edition William D. Callister Jr., David G. Rethwisch - Solutions

Determine the Miller indices for the planes shown in the following unit cell:
Cite the indices of the direction that results from the intersection of each of the following pair of planes within a cubic crystal: (a) (100) and (010) planes, (b) (111) and (111 ) planes, ,and (c) (101 ) and (001) planes.
Sketch the atomic packing of (a) the (100) plane for the BCC crystal structure, and (b) the (201) plane for the FCC crystal structure.
Consider the reduced-sphere unit cell shown in Problem 3.20, having an origin of the coordinate system positioned at the atom labeled with an O. For the following sets of planes, determine which are equivalent:(a) (001 ), (010), and, (1 00)(b) (11 0), (101 ), (01 1), and (1 1 0)(c) (1 1 1 ), (1 11
Here are shown the atomic packing schemes for several different crystallographic directions for some hypothetical metal. For each direction the circles represent only those atoms contained within a unit cell, which circles are reduced from their actual size.(a) To what crystal system does the unit
Below are shown three different crystallographic planes for a unit cell of some hypothetical metal. The circles represent atoms:(a) To what crystal system does the unit cell belong?(b) What would this crystal structure be called?(c) If the density of this metal is 8.95 g/cm3, determine its atomic
Convert the (010) and (101) planes into the four-index Miller-Bravais scheme for hexagonal unit cells.
Show that the atomic packing factor for BCC is 0.68.
Determine the indices for the planes shown in the hexagonal unit cells below:
Sketch the (11 01) and (112 0) planes in a hexagonal unit cell.
(a) Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius R.(b) Compute and compare linear density values for these same two directions for silver.
(a) Derive linear density expressions for BCC [110] and [111] directions in terms of the atomic radius R. (b) Compute and compare linear density values for these same two directions for tungsten.
(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius R. (b) Compute and compare planar density values for these same two planes for nickel.
a) Derive planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R. (b) Compute and compare planar density values for these same two planes for vanadium.
(a) Derive the planar density expression for the HCP (0001) plane in terms of the atomic radius R. (b) Compute the planar density value for this same plane for magnesium.
Explain why the properties of polycrystalline materials are most often isotropic
Using the data for molybdenum in Table 3.1, compute the interplanar spacing for the (111) set of planes.
Determine the expected diffraction angle for the first-order reflection from the (113) set of planes for FCC platinum when monochromatic radiation of wavelength 0.1542 nm is used.
Show that the atomic packing factor for HCP is 0.74.
Using the data for aluminum in Table 3.1, compute the interplanar spacings for the (110) and (221) sets of planes.
The metal iridium has an FCC crystal structure. If the angle of diffraction for the (220) set of planes occurs at 69.22° (first-order reflection) when monochromatic x radiation having a wavelength of 0.1542 nm is used, compute (a) the interplanar spacing for this set of planes, and (b) the atomic
The metal rubidium has a BCC crystal structure. If the angle of diffraction for the (321) set of planes occurs at 27.00° (first-order reflection) when monochromatic x radiation having a wavelength of 0.0711 nm is used, compute (a) the interplanar spacing for this set of planes, and (b) the atomic
For which set of crystallographic planes will a first-order diffraction peak occur at a diffraction angle of 46.21° for BCC iron when monochromatic radiation having a wavelength of 0.0711 nm is used?
Figure 3.22 shows an x-ray diffraction pattern for Î±-iron taken using a diffractometer and monochromatic x-radiation having a wavelength of 0.1542 nm; each diffraction peak on the pattern has been indexed. Compute the interplanar spacing for each set of planes indexed; also determine $Figure 3.22 shows an x-ray diffraction pattern for Î±-iron taken$
Figure 3.25 shows the first four peaks of the x-ray diffraction pattern for copper, which has an FCC crystal structure; monochromatic x-radiation having a wavelength of 0.1542 nm was used.(a) Index (i.e., give h, k, and l indices) for each of these peaks.(b) Determine the interplanar spacing for
Would you expect a material in which the atomic bonding is predominantly ionic in nature to be more or less likely to form a non crystalline solid upon solidification than a covalent material? Why? (See Section 2.6.)
Iron has a BCC crystal structure, an atomic radius of 0.124 nm, and an atomic weight of 55.85 g/mol. Compute and compare its theoretical density with the experimental value found inside the front cover.
Calculate the radius of an iridium atom, given that Ir has an FCC crystal structure, a density of 22.4 g/cm3, and an atomic weight of 192.2 g/mol.
Calculate the radius of a vanadium atom, given that V has a BCC crystal structure, a density of 5.96 g/cm3, and an atomic weight of 50.9 g/mol.
Calculate the fraction of atom sites that are vacant for lead at its melting temperature of 327°C (600 K). Assume an energy for vacancy formation of 0.55 eV/atom.
What is the composition, in atom percent, of an alloy that contains 98 g tin and 65 g of lead?
What is the composition, in atom percent, of an alloy that contains 99.7 lbm copper, 102 lbm zinc, and 2.1 lbm lead?
What is the composition, in atom percent, of an alloy that consists of 97 wt% Fe and 3 wt% Si?
Calculate the number of atoms per cubic meter in aluminum.
The concentration of carbon in an iron-carbon alloy is 0.15 wt%. What is the concentration in kilograms of carbon per cubic meter of alloy?
Determine the approximate density of a high-leaded brass that has a composition of 64.5 wt% Cu, 33.5 wt% Zn, and 2.0 wt% Pb.
Calculate the unit cell edge length for an 85 wt% Fe-15 wt% V alloy. All of the vanadium is in solid solution, and, at room temperature the crystal structure for this alloy is BCC.
Some hypothetical alloy is composed of 12.5 wt% of metal A and 87.5 wt% of metal B. If the densities of metals A and B are 4.27 and 6.35 g/cm3, respectively, whereas their respective atomic weights are 61.4 and 125.7 g/mol, determine whether the crystal structure for this alloy is simple cubic,
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, N1, given the concentration of that element specified in weight percent, C1. This computation is possible
Calculate the number of vacancies per cubic meter in iron at 850°C. The energy for vacancy formation is 1.08 eV/atom. Furthermore, the density and atomic weight for Fe are 7.65 g/cm3 and 55.85 g/mol, respectively.
Gold forms a substitutional solid solution with silver. Compute the number of gold atoms per cubic centimeter for a silver-gold alloy that contains 10 wt% Au and 90 wt% Ag. The densities of pure gold and silver are 19.32 and 10.49 g/cm3, respectively.
Germanium forms a substitutional solid solution with silicon. Compute the number of germanium atoms per cubic centimeter for a germanium-silicon alloy that contains 15 wt% Ge and 85 wt% Si. The densities of pure germanium and silicon are 5.32 and 2.33 g/cm3, respectively.
Sometimes it is desirable to be able to determine the weight percent of one element, C1, that will produce a specified concentration in terms of the number of atoms per cubic centimeter, N1, for an alloy composed of two types of atoms. This computation is possible using the following
Molybdenum forms a substitutional solid solution with tungsten. Compute the weight percent of molybdenum that must be added to tungsten to yield an alloy that contains 1.0 × 1022 Mo atoms per cubic centimeter. The densities of pure Mo and W are 10.22 and 19.30 g/cm3, respectively.
Niobium forms a substitutional solid solution with vanadium. Compute the weight percent of niobium that must be added to vanadium to yield an alloy that contains 1.55 × 1022 Nb atoms per cubic centimeter. The densities of pure Nb and V are 8.57 and 6.10 g/cm3, respectively.
Silver and palladium both have the FCC crystal structure, and Pd forms a substitutional solid solution for all concentrations at room temperature. Compute the unit cell edge length for a 75 wt% Ag-25 wt% Pd alloy. The room-temperature density of Pd is 12.02 g/cm3, and its atomic weight and atomic
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 3.54 at the end of Chapter 3.)
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 3.55 at the end of Chapter 3.)
(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 small-angle grain boundary is less than for a high-angle one. Why is this so?
Calculate the activation energy for vacancy formation in aluminum, given that the equilibrium number of vacancies at 500°C (773 K) is 7.57 × 1023 m-3. The atomic weight and density (at 500°C) for aluminum are, respectively, 26.98 g/mol and 2.62 g/cm3.
(a) Briefly describe a twin and a twin boundary. (b) Cite the difference between mechanical and annealing twins.
For each of the following stacking sequences found in FCC metals, cite the type of planar defect that exists: (a) . . . A B C A B C B A C B A . . . (b) . . . A B C A B C B C A B C . . . Now, copy the stacking sequences and indicate the position(s) of planar defect(s) with a vertical dashed line.
(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
(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.
For an ASTM grain size of 8, approximately how many grains would there be per square inch at (a) A magnification of 100, and (b) Without any magnification?
Determine the ASTM grain size number if 25 grains per square inch are measured at a magnification of 600.
Determine the ASTM grain size number if 20 grains per square inch are measured at a magnification of 50.
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.Which of these elements would you expect to form the following with copper:
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
Derive the following equations:(a) Equation 4.7a(b) Equation 4.9a(c) Equation 4.10a(d) Equation 4.11b
What is the composition, in atom percent, of an alloy that consists of 30 wt% Zn and 70 wt% Cu?
What is the composition, in weight percent, of an alloy that consists of 6 at% Pb and 94 at% Sn?
Calculate the composition, in weight percent, of an alloy that contains 218.0 kg titanium, 14.6 kg of aluminum, and 9.7 kg of vanadium.
Aluminum-lithium alloys have been developed by the aircraft industry to reduce the weight and improve the performance of its aircraft. A commercial aircraft skin material having a density of 2.55 g/cm3 is desired. Compute the concentration of Li (in wt%) that is required.
Iron and vanadium both have the BCC crystal structure and V forms a substitutional solid solution in Fe for concentrations up to approximately 20 wt% V at room temperature. Determine the concentration in weight percent of V that must be added to iron to yield a unit cell edge length of 0.289 nm.
Briefly explain the difference between self-diffusion and interdiffusion.
Show that
Determine the carburizing time necessary to achieve a carbon concentration of 0.45 wt% at a position 2 mm into an iron-carbon alloy that initially contains 0.20 wt% C. The surface concentration is to be maintained at 1.30 wt% C, and the treatment is to be conducted at 1000°C. Use the diffusion
An FCC iron-carbon alloy initially containing 0.35 wt% C is exposed to an oxygen-rich and virtually carbon-free atmosphere at 1400 K (1127°C). Under these circumstances the carbon diffuses from the alloy and reacts at the surface with the oxygen in the atmosphere; that is, the carbon concentration
Nitrogen from a gaseous phase is to be diffused into pure iron at 700°C. If the surface concentration is maintained at 0.1 wt% N, what will be the concentration 1 mm from the surface after 10 h? The diffusion coefficient for nitrogen in iron at 700°C is 2.5 × 10-11 m2/s.
Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemental impurity; furthermore, assume each impurity level is constant throughout its side of the diffusion couple. For this
For a steel alloy it has been determined that a carburizing heat treatment of 10-h duration will raise the carbon concentration to 0.45 wt% at a point 2.5 mm from the surface. Estimate the time necessary to achieve the same concentration at a 5.0-mm position for an identical steel and at the same
Cite the values of the diffusion coefficients for the interdiffusion of carbon in both α-iron (BCC) and γ-iron (FCC) at 900°C. Which is larger? Explain why this is the case.
Using the data in Table 5.2, compute the value of D for the diffusion of zinc in copper at 650ºC.
At what temperature will the diffusion coefficient for the diffusion of copper in nickel have a value of 6.5 × 10-17m2/s. Use the diffusion data in Table 5.2.
The preexponential and activation energy for the diffusion of iron in cobalt are 1.1 × 10-5 m2/s and 253,300 J/mol, respectively. At what temperature will the diffusion coefficient have a value of 2.1 × 10-14 m2/s?
Self-diffusion involves the motion of atoms that are all of the same type; therefore it is not subject to observation by compositional changes, as with interdiffusion. Suggest one way in which self-diffusion may be monitored.
The diffusion coefficients for iron in nickel are given at two temperatures:(a) Determine the values of D0 and the activation energy Qd.(b) What is the magnitude of D at 1100ºC (1373 K)?
The diffusion coefficients for silver in copper are given at two temperatures(a) Determine the values of D0 and Qd.(b) What is the magnitude of D at 875°C?
Below is shown a plot of the logarithm (to the base 10) of the diffusion coefficient versus reciprocal of the absolute temperature, for the diffusion of iron in chromium. Determine values for the activation energy and preexponential.
Carbon is allowed to diffuse through a steel plate 15 mm thick. The concentrations of carbon at the two faces are 0.65 and 0.30 kg C/m3 Fe, which are maintained constant. If the preexponential and activation energy are 6.2 × 10-7 m2/s and 80,000 J/mol, respectively, compute the temperature at
The steady-state diffusion flux through a metal plate is 5.4 × 10-10 kg/m2-s at a temperature of 727°C (1000 K) and when the concentration gradient is -350 kg/m4. Calculate the diffusion flux at 1027°C (1300 K) for the same concentration gradient and assuming an activation energy for diffusion
At approximately what temperature would a specimen of γ-iron have to be carburized for 2 h to produce the same diffusion result as at 900°C for 15 h?
(a) Calculate the diffusion coefficient for copper in aluminum at 500ºC.(b) What time will be required at 600ºC to produce the same diffusion result (in terms of concentration at a specific point) as for 10 h at 500ºC?
A copper-nickel diffusion couple similar to that shown in Figure 5.1a is fashioned. After a 700-h heat treatment at 1100°C (1373 K) the concentration of Cu is 2.5 wt% at the 3.0-mm position within the nickel. At what temperature must the diffusion couple need to be heated to produce this same
A diffusion couple similar to that shown in Figure 5.1a is prepared using two hypothetical metals A and B. After a 30-h heat treatment at 1000 K (and subsequently cooling to room temperature) the concentration of A in B is 3.2 wt% at the 15.5-mm position within metal B. If another heat treatment is
(a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion.
The outer surface of a steel gear is to be hardened by increasing its carbon content. The carbon is to be supplied from an external carbon-rich atmosphere, which is maintained at an elevated temperature. A diffusion heat treatment at 850°C (1123 K) for 10 min increases the carbon concentration to
An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the
Phosphorus atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background concentration of P in this silicon material is known to be 5 × 1019 atoms/m3. The predeposition treatment is to be conducted at 950°C for 45 minutes; the surface
Aluminum atoms are to be diffused into a silicon wafer using both predeposition and drive in heat treatments; the background concentration of Al in this silicon material is known to be 3 × 1019 atoms/m3. The drive-in diffusion treatment is to be carried out at 1050°C for a period of 4.0 h, which
Briefly explain the concept of steady state as it applies to diffusion.
(a) Briefly explain the concept of a driving force. (b) What is the driving force for steady-state diffusion?
The purification of hydrogen gas by diffusion through a palladium sheet was discussed in Section 5.3. Compute the number of kilograms of hydrogen that pass per hour through a 5 mm-thick sheet of palladium having an area of 0.20 m2 at 500°C. Assume a diffusion coefficient of 1.0 × 10-8 m2/s, that
A sheet of steel 1.5 mm thick has nitrogen atmospheres on both sides at 1200°C and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is 6 × 10-11 m2/s, and the diffusion flux is found to be 1.2 × 10-7 kg/m2-s. Also, it
A sheet of BCC iron 1 mm thick was exposed to a carburizing gas atmosphere on one side and a decarburizing atmosphere on the other side at 725°C. After having reached steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were
When Î±-iron is subjected to an atmosphere of hydrogen gas, the concentration of hydrogen in the iron, CH (in weight percent), is a function of hydrogen pressure, pH2 (in MPa), and absolute temperature (T) according toFurthermore, the values of D0 and Qd for this diffusion system are 1.4 Ã—
It is desired to enrich the partial pressure of hydrogen in a hydrogen-nitrogen gas mixture for which the partial pressures of both gases are 0.1013 MPa (1 atm). It has been proposed to accomplish this by passing both gases through a thin sheet of some metal at an elevated temperature; inasmuch as
A gas mixture is found to contain two diatomic A and B species for which the partial pressures of both are 0.05065 MPa (0.5 atm). This mixture is to be enriched in the partial pressure of the A species by passing both gases through a thin sheet of some metal at an elevated temperature. The
The wear resistance of a steel shaft is to be improved by hardening its surface. This is to be accomplished by increasing the nitrogen content within an outer surface layer as a result of nitrogen diffusion into the steel. The nitrogen is to be supplied from an external nitrogen-rich gas at an

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