The weight of the bell in kilograms is ~2,174. The radius is about 7mm or 0.007m.

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Stanley Cooper
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1 Problem Set 3 Due: Tuesday, February 7, 11:00 AM 1. Yield/Shear on Storke Tower Suppose the larest bell in Storke Tower, which weihs 4,793 pounds, needs to be replaced. A crane is brouht in to remove the bell, initially usin a 304 Stainless Steel cable that is inches in diameter. The cable is arraned such that a lenth of of steel is bearin the weiht of the bell. The yield strenth of 304SS is 205 MPa from Callister Table 11.4, and the Youn s modulus is 193 GPa, from azom.com. (Note: Use metric units when solvin the problems. Problem adapted from Prof. Van Vliet, ) a) i) Will the cable yield to the weiht of the bell? (12 pts total) The weiht of the bell in kilorams is ~2,174. The radius is about 7mm or 0.007m. The cable will not yield. ii) What is the maximum load (in N) that the cable can withstand before yieldin? The maximum load it can withstand is iii) If it does not yield, then what is the displacement (in mm) of the bell as the cable deforms elastically? The elastic deformation will cause a strain of: The lenth of the steel is 9 feet, 10.1 inches, or 3 meters. b) What is the maximum sheer stress in the cable while carryin this bell? (2 pts) Usin the Schmidt factor, and knowin that maximum stress occurs at 45 to the loadin direction, c) The University requested that the company use a cable of sufficient thickness to have a safety factor of 4. The crane s pulley system can only accommodate a cable of 0.75 diameters or less. If the company does not wish to use a wider set of pulleys, will they be able to meet the requirement with 304 SS cable? (6 pts)

To et a safety factor of 4, the maximum tensile stress on the cable is as below. This leads to a minimum cross-sectional area for the cable before yieldin occurs.

2 To et a safety factor of 4, the maximum tensile stress on the cable is as below. This leads to a minimum cross-sectional area for the cable before yieldin occurs. This means the cable must have a diameter of This is wider than the pulley roove. 304 SS won t meet the requirement. 2. Defects in ceramics (Callister 12.5): SrTiO 3 By now we should all be somewhat familiar with the perovskite structure SrTiO 3. a) Featured below is a TiO 2 plane for a very poor sample of SrTiO 3. Give the miller index of this plane with respect to Fiure 12.6 of Callister. In the fiure below, label all the point defects present. You should find at least one of the followin: Cation vacancy, anion vacancy, cation interstitial, anion interstitial, Frenkel defect, Schottky defect, substitutional impurity, and interstitial impurity. Note: Frenkel and Schottky defects involve more than one point defect in close proximity to one another, while other point defects can be isolated. (8 pts) (002) plane

3 b) Of these types of defects, which will alter the stoichiometry of the sample, and which will preserve it? (2 pts) The cation to anion ratio within a sample is not effected by the presence of Frenkel and Schottky defects. Therefore, stoichiometry is preserved. All other defect types do not preserve stoichiometry. c) Ti is an element that can exhibit 2+, 3+ and 4+ valence states. Knowin this, is it possible For SrTiO 3 to be nonstoichiometric but remain electrically neutral without the presence of an impurity (somethin not Sr, Ti or O)? Explain. (2 pts) Yes. We can imaine a scenario where Ti is in its 2+ state, and we introduce an oxyen vacancy to compensate for the additional 2- chare. We lose stoichiometry, but preserve electronic neutrality without introducin an impurity. d) Suppose that Gd is introduced to a sample of SrTiO 3. Assumin that it substitutes onto the Sr lattice points, what are the possible types of point defects that could be present as a result? (2 pts) Substitution of Gd 3+ for Sr 2+ results in one additional positive chare. The preservation of chare neutrality will result in either cation vacancies, or anion interstitials. e) Suppose that for the sample above, the formation eneries for Frenkel and Schottky defects are measured to be Q fr =0.8 ev and Q s =1.3 ev respectively. Calculate the equilibrium number of both Frenkel and Schottky defects per cubic meter at T=300K. Hint: recall that you calculated the theoretical density for SrTiO 3 in HW#2. Adapt the expression you used for the density in order to determine N, the total number of lattice sites per cubic meter. (6 pts) The density calculated for SrTiO 3 in HW#2 was 5.11 cm -3. We can adapt equation 12.1 from Callister to ive us the total number of lattice sites per cubic meter as follows: N=(n /V)=(N A* ρ)/(3ao+asr+ati) N=(6.023*10 23 /mol)(5.11 cm -3 )(10 6 cm 3 m -3 )/( * /mol) N=1.67*10 28 lattice sites m -3 We then use equations 12.2 and 12.3 to calculate the equilibrium number of defects. Nfr=N*exp(Qfr/2kT)=(1.67*10 28 )*exp(-0.8 ev/(2*300 K *8.617*10-5 ev K -1 )) Nfr=(1.67*10 28 )*(1.905*10-7 )=3.18*10 21 defects m -3 & Ns=N*exp(Qs/2kT) =(1.67*10 28 )*exp(-1.3 ev/(2*300 K *8.617*10-5 ev K -1 )) Ns=(1.67*10 28 )*(1.2*10-11 )=2*10 17 defects m -3

4 3. Equilibrium concentration of point defects. The enery of formation for a old (Au) vacancy is approximately ~1.00 ev. Your sinificant other just bouht you a old rin that he/she claims to be perfectly crystalline old, but you know better and decide to prove him/her wron. a) You weih the rin and discover that it is about 1 ram. Given that old weihs 197 rams per mole, calculate the total number of atomic sites (old atoms) in this rin. (6 pts) 1 ram / (197 rams/mole) = mol mol * *10 25 atoms/mol = *10 21 atoms b) Today is an unusually hot day, and the temperature is about 27 o C. Assumin that the rin is at thermal equilibrium with the environment, calculate the total number of vacancies (N v ) in the rin at the current temperature. Boltzmann s constant is 8.62*10-5 ev/(atom*kelvin), make sure to write out your units. (6 pts) = 300 Kelvin K b *300 Kelvin= ev/atom. N v = *10 21 atoms x [Exp(-1 ev/ ev)]=3.0569*10 21 x1.6067*10-17 = vacancies. c) You are furious at bein cheated out of N v old atoms upon finishin the previous calculation. Your sinificant other tries to appease you by statin there are probably an equal number of interstitial atoms as vacancies, so they cancel out. You check Wikipedia and find the followin plot on interstitial concentration in old. Assumin the data in Wikipedia is accurate, estimate the activation enery from the plot below, and compare with that of vacancies. Is your sinificant other riht, or is he/she lyin to you aain? (8 pts; 4pts for ettin the slope/activation enery riht and 4 for explainin that your sinificant other is wron.) The slope of this line is -10 ev. Since the activation enery of an interstitial is so much hiher than a vacancy, it cannot be true that there are equal amounts of both at the same temperature.

5 4. Alloys a) You are desinin an intermetallic compound of Al/Ti, which is to be used as a hihstrenth, low-corrosion, liht-weiht coatin material for an aircraft. The desired material has a final density of 4.08/cm 3. What weiht percent of each component do you need in your alloy in order to achieve this? What is the stoichiometric chemical formula of this intermetallic compound? (8 pts) The density and atomic weiht of Al is 2.71 /cm 3 and /mol respectively and the density and atomic weiht of Ti of 4.51 /cm 3 and 47.87/mol respectively. Startin with Eqn 4.10a,, Rearrane to solve for C Al. C Al =16wt%, C Ti =84wt%. Then use Eqn 4.6a&b, to determine the atomic%. C Al =25at%, C Ti =75at%. This indicates a ratio of 3Ti to 1Al atom, so a chemical formula of Ti 3 Al.

6 b) You are iven a hypothetical alloy that you are told is composed of 20 wt% of metal A and 80 wt% of metal B, which both have the same crystal structure. The density and atomic weiht for metal A are 2.55/cm 3 and 18.60/mol, and for B are 6.43/cm 3 and 39.4/mol. Determine whether the crystal structure will be simple cubic, body-centered cubic or face-centered cubic, iven a unit cell ede lenth of nm. (8 pts) Startin with Eqn. 3.5, Plu in expressions for ρ av and A av from eqns 4.10a and 4.11a. Rearrane and solve for n. Plu in and find, n=2, so body-centered cubic, which has two atoms per unit cell. c) Describe both an ede and screw dislocation and how they distort the crystal lattice. Include sketches of both and clearly label the burers vector in each sketch. (4 pts) An ede dislocation occurs when there is an extra-half plane of atoms that terminates within the crystal. It is a linear defect that centers on the line that is defined alon the end of the extra half-plane of atoms. There is local distortion due to the ede dislocation, as atoms above the dislocation (in the fiure on the left below) are squeezed toether, while the atoms below are pulled apart. The Burers vector is perpendicular to the ede dislocation. A screw dislocation can be thouht of as bein produced by a shear stress, so that one plane, the upper portion (in the fiure on the riht below) is shifted one atomic distance to the riht relative to the bottom portion. The Burers vector is parallel to the direction of the dislocation.

7 5. Quartz vs. Glass a) When examinin Quartz(Silica) and Silica Glasses, what is the basic structural unit used to characterize these structures? (3 pts) When describin the structure of all silica and silica lasses it is convenient to use a basic unit of SiO 4 4- tetrahedra. b) What type of solid (crystalline or amorphous) has a systematic and reular arranement of atoms over lare atomic distances? Which type has a hih deree of atomic randomness? (1 pt) Crystalline solids have a reular arranement of atoms over lare atomic distances. Amorphous solids have a hih deree of atomic randomness. c) Which type of solid has a structure that resembles the atomic arranement of a liquid?(1 pt) An amorphous solid has a structure that resembles the atomic arranement of a liquid. This does not imply the properties of a liquid, but the randomness in its structure is similar to that of a liquid. d) What type of solid is lass? What type of solid is quartz? (2 pts) Glass is an amorphous solid. Quartz is a crystalline solid. e) Would you expect lass or quartz to form from rapidly coolin an SiO 2 melt throuh its freezin temperature? (2 pts) When rapidly coolin a melt past the materials freezin temperature, the resultin solid will have a random atomic arranement similar to that of the liquid. Due to the rapid coolin, there isn t enouh time for the molecules/atoms to form an ordered arranement. Therefore, lass would be the expected product of coolin an SiO2 melt throuh its freezin point. f) Name the three primary polymorphic crystalline forms of SiO 2. How many oxyen atoms are shared between SiO 4 4- tetrahedra in crystalline SiO 2 (This is common to all three polymorphs)? (3 pts) Quartz, cristobalite and tridymite are all forms of crystalline SiO 2 with different arranments of of the SiO 4 4- tetrahedra. All four Oxyen atoms of the tetrahedron are shared with neihborin tetrahedra. ) Additive oxides such as Na 2 O and CaO are commonly added to silica lasses. What practical effect does the incorporation of the cations of these network modifyin oxides have on the meltin temperature of the lass? Does the incorporation of these network modifiers make it harder to form the lass at low temperature? (2 pts)

8 The incorporation of these additives lowers the meltin temperature of the lass. The incorporation of these additives does not make it harder to form the lass at low temperature; they make it easier. h) Crystal wine lasses and other lassware were once made of lead lass ( a silicate lass containin PbO). Treatin this lead lass as a solid solution of SiO 2 and PbO, find the molar percent concentration of SiO 2 in lead lass containin 30 weiht percent PbO. [Hint: molar percent concentration is the molecular form of atomic percent concentration. Treat as molecules and not individual elements (i.e. m SiO2 =m Si +2m O and 1 mol SiO 2 is 6.022x10 23 molecules of SiO 2 ).] (6 pts) If the wt% of PbO is 30% then the wt% of SiO 2 is 70% We can use equation 4.6a from the text to convert from weiht percent to molar (atomic) percent. ASiO A Si AO mol mol mol APbO APb AO mol mol mol 70% CSiO A 2 PbO C ' SiO 100% mol 100% 2 CSiO A 2 PbO CPbO ASiO 2 70% % mol mol C ' 89.7% SiO2 In other words, In one mole of lead lass composed of 70% SiO 2 and 30% PbO by weiht there is 0.897mol SiO 2 and ( )=0.103mol PbO. This could be represented by the non-normalized chemical formula: (SiO 2 ) (PbO) Which is rouhly (SiO 2 ) 9 PbO or Si 9 PbO 19.

Point Defects in Metals

CHAPTER 5 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS Point Defects in Metals 5.1 Calculate the fraction of atom sites that are vacant for lead at its melting temperature of 327 C (600 K). Assume an energy