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Cite the difference between atomic mass and atomic weight.

Chromium has four naturally-occurring isotopes: 4.34% of ^{50}Cr, with an atomic weight of 49.9460 amu, 83.79% of ^{52}Cr, with an atomic weight of 51.9405 amu, 9.50% of ^{53}Cr, with an atomic weight of 52.9407 amu, and 2.37% of ^{54}Cr, with an atomic weight of 53.9389 amu. On the basis of these data, confirm that the average atomic weight of Cr is 51.9963 amu.

(a) How many grams are there in one amu of a material?

(b) Mole, in the context of this book, is taken in units of gram-mole. On this basis, how many atoms are there in a pound-mole of a substance?

(a) Cite two important quantum-mechanical concepts associated with the Bohr model of the atom.

(b) Cite two important additional refinements that resulted from the wave-mechanical atomic model.

Relative to electrons and electron states, what does each of the four quantum numbers specify?

Allowed values for the quantum numbers of electrons are as follows:

The relationships between n and the shell designations are noted in Table 2.1. Relative to the subshells,

l = 0 corresponds to an s subshell

l = 1 corresponds to a p subshell

l = 2 corresponds to a d subshell

l = 3 corresponds to an f subshell

For the K shell, the four quantum numbers for each of the two electrons in the 1s state, in the order of nlm_{l}m_{s}, are 100() and 100(). Write the four quantum numbers for all of the electrons in the L and M shells, and note which correspond to the s, p, and d subshells.

Give the electron configurations for the following ions: Fe^{2+}, Al^{3+}, Cu^{+}, Ba^{2+}, Br^{-}, and O^{2-}.

Sodium chloride (NaCl) exhibits predominantly ionic bonding. The Na^{+} and Cl^{- }ions have electron structures that are identical to which two inert gases?

With regard to electron configuration, what do all the elements in Group VIIA of the periodic table have in common?

To what group in the periodic table would an element with atomic number 114 belong?

Without consulting figure or Table 2.2, determine whether each of the electron configurations given below is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. Justify your choices.

(a) 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{7}4s^{2}

(b) 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}

(c) 1s^{2}2s^{2}2p^{5}

(d) 1s^{2}2s^{2}2p^{6}3s^{2}

(e) 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{2}4s^{2}

(f) 1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}4s^{1}

(a) What electron subshell is being filled for the rare earth series of elements on the periodic table?

(b) What electron subshell is being filled for the actinide series?

Calculate the force of attraction between a K^{+} and an O^{2-} ion the centers of which are separated by a distance of 1.5 nm.

The net potential energy between two adjacent ions, E_{N}, may be represented by the sum of Equations 2.8 and 2.9; that is,

Calculate the bonding energy E_{0} in terms of the parameters A, B, and n using the following procedure:

(a). Differentiate E_{N} with respect to r, and then set the resulting expression equal to zero, since the curve of E_{N} versus r is a minimum at E_{0}.

(b). Solve for r in terms of A, B, and n, which yields r_{0}, the equilibrium interionic spacing.

(c). Determine the expression for E_{0} by substitution of r_{0} into Equation 2.11.

For a K^{+}–Cl^{–} ion pair, attractive and repulsive energies EA and ER, respectively, depend on the distance between the ions r, according to

For these expressions, energies are expressed in electron volts per K^{+}–Cl^{–} pair, and r is the distance in nanometers. The net energy E_{N} is just the sum of the two expressions above.

(a) Superimpose on a single plot E_{N}, E_{R}, and E_{A} versus r up to 1.0 nm.

(b) On the basis of this plot, determine (i) the equilibrium spacing r_{0} between the K^{+} and Cl^{–} ions, and (ii) the magnitude of the bonding energy E_{0} between the two ions.

(c) Mathematically determine the r_{0} and E_{0} values using the solutions to Problem 2.14 and compare these with the graphical results from part (b).

Consider a hypothetical X^{+}-Y^{-} ion pair for which the equilibrium interionic spacing and bonding energy values are 0.35 nm and -6.13eV, respectively. If it is known that n in Equation 2.11 has a value of 10, using the results of Problem 2.14, determine explicit expressions for attractive and repulsive energies E_{A} and E_{R} of Equations 2.8 and 2.9.

The net potential energy EN between two adjacent ions is sometimes represented by the expression

in which r is the interionic separation and C, D, and ρ are constants whose values depend on the specific material.

(a) Derive an expression for the bonding energy E_{0} in terms of the equilibrium interionic separation r_{0} and the constants D and ρ using the following procedure:

1. Differentiate E_{N} with respect to r and set the resulting expression equal to zero.

2. Solve for C in terms of D, ρ, and r_{0}.

3. Determine the expression for E_{0} by substitution for C in Equation 2.12.

(b) Derive another expression for E_{0} in terms of r_{0}, C, and ρ using a procedure analogous to the one outlined in part (a).

(a) Briefly cite the main differences between ionic, covalent, and metallic bonding.

(b) State the Pauli Exclusion Principle.

Compute the percents ionic character of the interatomic bonds for the following compounds: TiO_{2}, ZnTe, CsCl, InSb, and MgCl_{2}.

Make a plot of bonding energy versus melting temperature for the metals listed in Table 2.3. Using this plot, approximate the bonding energy for copper which has a melting temperature of 1084°C.

Using Table 2.2, determine the number of covalent bonds that are possible for atoms of the following elements: germanium, phosphorus, selenium, and chlorine.

What type(s) of bonding would be expected for each of the following materials: brass (a copper-zinc alloy), rubber, barium sulfide (BaS), solid xenon, bronze, nylon, and aluminum phosphide (AlP)?

Explain why hydrogen fluoride (HF) has a higher boiling temperature than hydrogen chloride (HCl) (19.4 vs. –85°C), even though HF has a lower molecular weight.

What is the difference between atomic structure and crystal structure?

If the atomic radius of aluminum is 0.143 nm, calculate the volume of its unit cell in cubic meters.

Show for the body-centered cubic crystal structure that the unit cell edge length a and the atomic radius R are related through a =4R/√3.

For the HCP crystal structure, show that the ideal c/a ratio is 1.633

Show that the atomic packing factor for BCC is 0.68.

Show that the atomic packing factor for HCP is 0.74.

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/cm^{3}, 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/cm^{3}, and an atomic weight of 50.9 g/mol.

Some hypothetical metal has the simple cubic crystal structure shown in figure. If its atomic weight is 70.4 g/mol and the atomic radius is 0.126 nm, compute itsdensity.

Zirconium has an HCP crystal structure and a density of 6.51 g/cm^{3}.

(a) What is the volume of its unit cell in cubic meters?

(b) If the c/a ratio is 1.593, compute the values of c and a.

Using atomic weight, crystal structure, and atomic radius data tabulated inside the front cover, compute the theoretical densities of lead, chromium, copper, and cobalt, and then compare these values with the measured densities listed in this same table. The c/a ratio for cobalt is 1.623.

Rhodium has an atomic radius of 0.1345 nm and a density of 12.41 g/cm^{3}. Determine whether it has an FCC or BCC crystal structure.

Below are listed the atomic weight, density, and atomic radius for three hypothetical alloys. For each determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your determination. A simple cubic unit cell is shown in Figure 3.24.

The unit cell for tin has tetragonal symmetry, with a and b lattice parameters of 0.583 and 0.318 nm, respectively. If its density, atomic weight, and atomic radius are 7.30 g/cm^{3}, 118.69 g/mol, and 0.151 nm, respectively, compute the atomic packing factor.

Iodine has an orthorhombic unit cell for which the a, b, and c lattice parameters are 0.479, 0.725, and 0.978 nm, respectively.

(a) If the atomic packing factor and atomic radius are 0.547 and 0.177 nm, respectively, determine the number of atoms in each unit cell.

(b) The atomic weight of iodine is 126.91 g/mol; compute its theoretical density.

Titanium has an HCP unit cell for which the ratio of the lattice parameters c/a is 1.58. If the radius of the Ti atom is 0.1445 nm,

(a) Determine the unit cell volume, and

(b) Calculate the density of Ti and compare it with the literature value.

Zinc has an HCP crystal structure, a c/a ratio of 1.856, and a density of 7.13 g/cm^{3}. Compute the atomic radius for Zn.

Rhenium has an HCP crystal structure, an atomic radius of 0.137 nm, and a c/a ratio of 1.615. Compute the volume of the unit cell for Re.

Below is a unit cell for a hypothetical metal.

(a)** **To which crystal system does this unit cell belong?

(b)** **What would this crystal structure be called?

(c)** **Calculate the density of the material, given that its atomic weight is 141 g/mol.

Sketch a unit cell for the body-centered orthorhombic crystal structure.

List the point coordinates for all atoms that are associated with the FCC unit cell(figure).

List the point coordinates of the titanium, barium, and oxygen ions for a unit cell of the perovskite crystal structure(figure).

List the point coordinates of all atoms that are associated with the diamond cubic unit cell(figure).

Sketch a tetragonal unit cell, and within that cell indicate locations of the ½ 1 ½ and ¼ ½ ¾ point coordinates.

Using the Molecule Definition Utility found in both “Metallic Crystal Structures and Crystallography” and “Ceramic Crystal Structures” modules of VMSE, located on the book’s web site [www.wiley.com/college/Callister (Student Companion Site)], generate a three-dimensional unit cell for the intermetallic compound AuCu_{3} given the following:

(1) The unit cell is cubic with an edge length of 0.374 nm,

(2) Gold atoms are situated at all cube corners, and

(3) Copper atoms are positioned at the centers of all unit cell faces.

Draw an orthorhombic unit cell, and within that cell a [121] direction.

Sketch a monoclinic unit cell, and within that cell a [011] direction.

What are the indices for the directions indicated by the two vectors in the sketch below?

Within a cubic unit cell, sketch the following directions:

(a)[110],

(b)[121],

(c)[012],

(d)[133],

(e)[111],

(f)[122],

(g)[123],

(h)[103],

Determine the indices for the directions shown in the following cubic unitcell:

Determine the indices for the directions shown in the following cubic unitcell:

For tetragonal crystals, cite the indices of directions that are equivalent to each of the following directions:

(a) [001]

(b) [110]

(c) [010]

Convert the [100] and [111] directions into the four-index Miller–Bravais scheme for hexagonal unit cells.

Determine indices for the directions shown in the following hexagonal unit cells:

Sketch the [1123] and [1010] directions in a hexagonal unit cell.

Using Equations 3.6a, 3.6b, 3.6c, and 3.6d, derive expressions for each of the three primed indices set (u′, v′, and w′) in terms of the four unprimed indices (u, v, t, and w).

(a) Draw an orthorhombic unit cell, and within that cell a (210) plane.

(b) Draw a monoclinic unit cell, and within that cell a (002) plane.

What are the indices for the two planes drawn in the sketchbelow?

Sketch within a cubic unit cell the following planes:

(a) (011),

(b) (112),

(c) (102),

(d) (131),

(e) (111),

(f) (122),

(g) (123),

(h) (013)

Determine the Miller indices for the planes shown in the following unitcell:

Determine the Miller indices for the planes shown in the following unitcell:

Determine the Miller indices for the planes shown in the following unitcell:

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 (similar to Figures 3.10b and3.11b).

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, (100)

(b) (110), (101), (011), and (110)

(c) (111), (111), (111), and (111)

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 cell belong?

(b) What would this crystal structure becalled?

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/cm^{3}, determine its atomic weight.

Convert the (010) and (101) planes into the four-index Miller–Bravais scheme for hexagonal unit cells.

Determine the indices for the planes shown in the hexagonal unit cells below:

(a)

(b)

(c)

(d)

Sketch the (1101) and (1120) 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 ofplanes.

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.

Using the data for aluminum in Table 3.1, compute the interplanar spacing’s 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 radius for an iridium atom.

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 radius for the rubidium atom.

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 shows an x-ray diffraction pattern for a-iron taken using a diffract meter 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 the lattice parameter of Fe for each of thepeaks.

The diffraction peaks shown in figure are indexed according to the reflection rules for BCC (i.e., the sum h + k + l must be even). Cite the h, k, and l indices for the first four diffraction peaks for FCC crystals consistent with h, k, and l all being either odd oreven.

Figure 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 each of the peaks.

(c) For each peak, determine the atomic radius for Cu and compare these with the value presented in Table3.1.

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?

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.

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/cm^{3} and 55.85 g/mol, respectively.

Calculate the activation energy for vacancy formation in aluminum, given that the equilibrium number of vacancies at 500°C (773 K) is 7.57 × 10^{23} m^{-3}. The atomic weight and density (at 500°C) for aluminum are, respectively, 26.98 g/mol and 2.62 g/cm^{3}.

Below, atomic radius, crystal structure, electro negativity, 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?

(a) A substitutional solid solution having complete solubility

(b) A substitutional solid solution of incomplete solubility

(c) An interstitial solidsolution

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 ½ ¼ 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.

Derive the following equations:

(a) Equation 4.7a

(b) Equation 4.9a

(c) Equation 4.10a

(d) Equation4.11b

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.

What is the composition, in atom percent, of an alloy that contains 98 g tin and 65 g of lead?

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