CHAPTER 5 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS

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1 CHAPTER 5 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS Vacancies and SelfInterstitials 5.1 Calculate the fraction of atom sites that are vacant for copper at its melting temperature of 1084 C (1357 K). Assume an energy for vacancy formation of 0.90 ev/atom. In order to compute the fraction of atom sites that are vacant in copper at 1357 K, we must employ Equation 5.1. As stated in the problem, Q v 0.90 ev/atom. Thus, N v N exp Q v 0.90 ev /atom exp kt ( ev/atom K) (1357 K)
2 5.2 Calculate the energy for vacancy formation in silver, given that the equilibrium number of vacancies at 800 C (1073 K) is m 3. The atomic weight and density (at 800 C) for silver are, respectively, g/mol and 9.5 g/cm 3. This problem calls for the computation of the activation energy for vacancy formation in silver. Upon examination of Equation 5.1, all parameters besides Q v are given except N, the total number of atomic sites. However, N is related to the density, (ρ), Avogadro's number (N A ), and the atomic weight (A) according to Equation 5.2 as N N A ρ Pb A Pb ( atoms /mol)(9.5 g /cm 3 ) g /mol atoms/cm atoms/m 3 Now, taking natural logarithms of both sides of Equation 5.1, ln N v lnn Q v kt and, after some algebraic manipulation N Q v kt ln v N Now, inserting values for the parameters given in the problem statement leads to Q v ( m ev/atom K)(800 C K) ln m ev/atom
3 Point Defects in Ceramics 5.3 Calculate the fraction of lattice sites that are Schottky defects for cesium chloride at its melting temperature (645 C). Assume an energy for defect formation of 1.86 ev. In order to solve this problem it is necessary to use Equation 5.4 and solve for the N s /N ratio. Rearrangement of this expression and substituting values for the several parameters leads to N s N exp Q s 2kT exp 1.86 ev (2)( ev/k)( K)
4 5.4 Using the data given below that relate to the formation of Schottky defects in some oxide ceramic (having the chemical formula MO), determine the following: (a) the energy for defect formation (in ev), (b) the equilibrium number of Schottky defects per cubic meter at 0 C, and (c) the identity of the oxide (i.e., what is the metal M?) T ( C) ρ (g/cm 3 ) N s (m 3 ) ? The (a) portion of the problem asks that we compute the energy for defect formation. To begin, let us combine a modified form of Equation 5.2 and Equation 5.4 as N s N exp Q s 2kT N A ρ A M + A exp Q s O 2kT Inasmuch as this is a hypothetical oxide material, we don't know the atomic weight of metal M, nor the value of Q s in the above equation. Therefore, let us write equations of the above form for two temperatures, T 1 and T 2. These are as follows: N s1 N A ρ 1 A M + A exp Q s O 2kT 1 (5.S1a) N s2 N A ρ 2 A M + A exp Q s O 2kT 2 (5.S1b) Dividing the first of these equations by the second leads to
5 N s1 N s2 N A ρ 1 A M + A exp Q s O 2kT 1 N A ρ 2 A M + A exp Q s O 2kT 2 which, after some algebraic manipulation, reduces to the form N s1 ρ 1 exp Q s N s2 ρ 2 2k 1 T 1 1 T 2 (5.S2) Now, taking natural logarithms of both sides of this equation gives ln N s1 N ln ρ 1 s2 ρ Q s 1 2 2k T 1 1 T 2 and solving for Q s leads to the expression Q s 2k ln N s1 N ln ρ 1 s2 ρ 2 1 T 1 1 T 2 Let us take T C and T C, and we may compute the value of Q s as Q s (2)( ev/k) ln 9 m g/cm m 3 ln 3.40 g/cm K K 7.70 ev (b) It is now possible to solve for N s at 0 C using Equation 5.S2 above. This time let's take T 1 0 C and T C. Thus, solving for N s1, substituting values provided in the problem statement and Q s determined above yields
6 N s1 N s2 ρ 1 ρ 2 exp Q s 1 2k T 1 1 T 2 ( m 3 )( 3.45 g/cm 3 ) 3.50 g/cm 3 exp 7.70 ev 1 (2)( ev/k) K K m 3 (c) And, finally, we want to determine the identity of metal M. This is possible by computing the atomic weight of M (A M ) from Equation 5.S1a. Rearrangement of this expression leads to N A ρ 1 A M + A N s1 exp Q s O 2kT 1 And, after further algebraic manipulation N A ρ 1 Q N s1 exp s 2kT 1 A M + A O And, solving this expression for A M gives A M N A ρ 1 Q N s1 exp s 2kT 1 A O Now, assuming that T C, the value of A M is A M ( ions/mol)( 3.50 g/cm 3 )(10 6 cm 3 /m 3 ) ( ions/m ev ) exp (2)( ev/k)( K) g/mol g/mol
7 Upon consultation of the periodic table in Figure 2.6, the divalent metal (i.e., that forms M 2+ ions) that has an atomic weight closest to g/mol is magnesium. Thus, this metal oxide is MgO.
8 Impurities in Solids 5.5 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. Atomic Crystal Electro Element Radius (nm) Structure negativity Valence Ni FCC C H O Ag FCC Al FCC Co HCP Cr BCC Fe BCC Pt FCC Zn HCP 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 In this problem we are asked to cite which of the elements listed form with Ni the three possible solid solution types. For complete substitutional solubility the following criteria must be met: 1) the difference in atomic radii between Ni and the other element (ΔR%) must be less than ±15%, 2) the crystal structures must be the same, 3) the electronegativities must be similar, and 4) the valences should be the same, or nearly the same. Below are tabulated, for the various elements, these criteria.
9 Crystal ΔElectro Element ΔR% Structure negativity Valence Ni FCC 2+ C 43 H 63 O 52 Ag +16 FCC Al +15 FCC Co +0.6 HCP 0 2+ Cr +0.2 BCC Fe 0.4 BCC 0 2+ Pt +11 FCC Zn +7 HCP (a) Pt is the only element that meets all of the criteria and thus forms a substitutional solid solution having complete solubility. At elevated temperatures Co and Fe experience allotropic transformations to the FCC crystal structure, and thus display complete solid solubility at these temperatures. (b) Ag, Al, Co, Cr, Fe, and Zn form substitutional solid solutions of incomplete solubility. All these metals have either BCC or HCP crystal structures, and/or the difference between their atomic radii and that for Ni are greater than ±15%, and/or have a valence different than 2+. (c) C, H, and O form interstitial solid solutions. These elements have atomic radii that are significantly smaller than the atomic radius of Ni.
10 5.6 (a) Suppose that CaO is added as an impurity to Li 2 O. If the Ca 2+ substitutes for Li +, what kind of vacancies would you expect to form? How many of these vacancies are created for every Ca 2+ added? (b) Suppose that CaO is added as an impurity to CaCl 2. If the O 2 substitutes for Cl, what kind of vacancies would you expect to form? How many of these vacancies are created for every O 2 added? (a) For Ca 2+ substituting for Li + in Li 2 O, lithium vacancies would be created. For each Ca 2+ substituting for Li +, one positive charge is added; in order to maintain charge neutrality, a single positive charge may be removed. Positive charges are eliminated by creating lithium vacancies, and for every Ca 2+ ion added, a single lithium vacancy is formed. (b) For O 2 substituting for Cl  in CaCl 2, chlorine vacancies would be created. For each O 2 substituting for a Cl , one negative charge is added; negative charges are eliminated by creating chlorine vacancies. In order to maintain charge neutrality, one O 2 ion will lead to the formation of one chlorine vacancy.
11 Specification of Composition 5.7 What is the composition, in atom percent, of an alloy that consists of 92.5 wt% Ag and 7.5 wt% Cu? In order to compute composition, in atom percent, of a 92.5 wt% Ag7.5 wt% Cu alloy, we employ Equation 5.9 as C ' Ag C Ag A Cu C Ag A Cu + C Cu A Ag (92.5)(63.55 g /mol) (92.5)(63.55 g /mol) + (7.5)(107.87g /mol) 87.9 at% C ' Cu C Cu A Ag C Ag A Cu + C Cu A Ag (7.5)( g /mol) (92.5)(63.55 g /mol) + (7.5)(107.87g /mol) 12.1 at%
12 5.8 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. The concentration, in weight percent, of an element in an alloy may be computed using a modified form of Equation 5.6. For this alloy, the concentration of iron (C Fe ) is just C Fe m Fe m Fe + m C + m Cr 105 kg 105 kg kg kg wt% Similarly, for carbon C C 0.2 kg 105 kg kg kg 0.19 wt% And for chromium C Cr 1.0 kg 105 kg kg kg 0.94 wt%
13 5.9 What is the composition, in atom percent, of an alloy that contains 44.5 lb m of silver, 83.7 lb m of gold, and 5.3 lb m of Cu? To determine the concentrations, in atom percent, of the AgAuCu alloy, it is first necessary to convert the amounts of Ag, Au, and Cu into grams. m ' Ag (44.5 lb m )(453.6 g/lb m ) 20,185 g m ' Au (83.7 lb m )(453.6 g/lb m ) 37,966 g m ' Cu (5.3 lb m )(453.6 g/lb m ) 2, 404 g These masses must next be converted into moles (Equation 5.7), as n mag m ' Ag A Ag 20,185 g g /mol mol n mau 37,966 g g /mol mol n mcu 2, 404 g g /mol 37.8 mol Now, employment of a modified form of Equation 5.8, gives C ' Ag n mag n mag + n mau + n mcu mol mol mol mol 44.8 at% C ' Au mol mol mol mol 46.2 at%
14 C ' Cu 37.8 mol mol mol mol 9.0 at%
15 5.10 Convert the atom percent composition in Problem 5.9 to weight percent. The composition in atom percent for Problem 5.9 is 44.8 at% Ag, 46.2 at% Au, and 9.0 at% Cu. Modification of Equation 5.10 to take into account a threecomponent alloy leads to the following C Ag C ' Ag C ' Ag A Ag A Ag + C ' Au A Au + C ' Cu A Cu (44.8) ( g /mol) (44.8)( g /mol) + (46.2) ( g /mol) + (9.0) (63.55 g /mol) 33.3 wt% C Au C ' Ag C ' Au A Au A Ag + C ' Au A Au + C ' Cu A Cu (46.2) ( g /mol) (44.8)( g /mol) + (46.2) ( g /mol) + (9.0) (63.55 g /mol) 62.7 wt% C Cu C ' Ag C ' Cu A Cu A Ag + C ' Au A Au + C ' Cu A Cu (9.0) (63.55 g /mol) (44.8)( g /mol) + (46.2) ( g /mol) + (9.0) (63.55 g /mol) 4.0 wt%
16 5.11 Determine the approximate density of a Ti6Al4V titanium alloy that has a composition of 90 wt% Ti, 6 wt% Al, and 4 wt% V. In order to solve this problem, Equation 5.13a is modified to take the following form: ρ ave C Ti ρ Ti + C Al ρ Al + C V ρ V And, using the density values for Ti, Al, and V i.e., 4.51 g/cm 3, 2.71 g/cm 3, and 6.10 g/cm 3 (as taken from inside the front cover of the text), the density is computed as follows: ρ ave 90 wt% 4.51 g /cm wt% 2.71 g /cm wt% 6.10 g /cm g/cm 3
17 5.12 Some hypothetical alloy is composed of 25 wt% of metal A and 75 wt% of metal B. If the densities of metals A and B are 6.17 and 8.00 g/cm 3, respectively, whereas their respective atomic weights are and g/mol, determine whether the crystal structure for this alloy is simple cubic, facecentered cubic, or bodycentered cubic. Assume a unit cell edge length of nm. In order to solve this problem it is necessary to employ Equation 3.5; in this expression density and atomic weight will be averages for the alloy that is ρ ave na ave V C N A Inasmuch as for each of the possible crystal structures, the unit cell is cubic, then V C a 3, or ρ ave na ave a 3 N A And, in order to determine the crystal structure it is necessary to solve for n, the number of atoms per unit cell. For n 1, the crystal structure is simple cubic, whereas for n values of 2 and 4, the crystal structure will be either BCC or FCC, respectively. When we solve the above expression for n the result is as follows: n ρ ave a3 N A A ave Expressions for A ave and ρ ave are found in Equations 5.14a and 5.13a, respectively, which, when incorporated into the above expression yields n C A + C a B 3 N A ρ A ρ B C A + C B A A A B
18 Substitution of the concentration values (i.e., C A 25 wt% and C B 75 wt%) as well as values for the other parameters given in the problem statement, into the above equation gives n ( wt% 6.17 g/cm wt% 8 nm) 3 ( atoms/mol) 8.00 g/cm 3 25 wt% g/mol + 75 wt% g/mol 1.00 atom/unit cell Therefore, on the basis of this value, the crystal structure is simple cubic.
19 5.13 Molybdenum forms a substitutional solid solution with tungsten. Compute the number of molybdenum atoms per cubic centimeter for a molybdenumtungsten alloy that contains 16.4 wt% Mo and 83.6 wt% W. The densities of pure molybdenum and tungsten are and g/cm 3, respectively. To solve this problem, employment of Equation 5.21 (which derivation is called for in problem W5.16 on the book s Web site) is necessary, using the following values: C 1 C Mo 16.4 wt% ρ 1 ρ Mo g/cm 3 ρ 2 ρ W 19.3 g/cm 3 A 1 A Mo g/mol Thus N Mo C Mo A Mo ρ Mo N A C Mo + A Mo ρ W ( C Mo ) ( atoms /mol) (16.4 wt%) (16.4 wt%)(95.94 g /mol) g /mol g /cm 3 + ( 16.4 wt%) 19.3 g /cm atoms/cm 3
20 5.14 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 N A ρ 2 N 1 A 1 ρ 2 ρ 1 (5.22) where N A Avogadro s number ρ 1 and ρ 2 densities of the two elements A 1 and A 2 the atomic weights of the two elements Derive Equation 5.22 using Equation 5.2 and expressions contained in Section 5.6. The number of atoms of component 1 per cubic centimeter is just equal to the atom fraction of component 1 (c ' 1 ) times the total number of atoms per cubic centimeter in the alloy (N). Thus, using the equivalent of Equation 5.2, we may write N 1 c 1 ' N c 1 ' N A ρ ave A ave Realizing that c ' 1 C 1 ' and C 2 ' C 1 ' and substitution of the expressions for ρ ave and A ave, Equations 5.13b and 5.14b, respectively, leads to N 1 c 1 ' N A ρ ave A ave
21 N A C 1 ' ρ 1 ρ 2 C 1 ' ρ 2 A 1 + ( C 1 ' )ρ 1 A 2 And, solving for C 1 ' C 1 ' N 1 ρ 1 A 2 N A ρ 1 ρ 2 N 1 ρ 2 A 1 + N 1 ρ 1 A 2 Substitution of this expression for C ' 1 into Equation 5.10a, which may be written in the following form C 1 C 1 ' A 1 C 1 ' A 1 + C 2 ' A 2 C 1 ' A 1 C 1 ' A 1 + ( C 1 ' )A 2 yields C N A ρ 2 N 1 A 1 ρ 2 ρ 1 the desired expression.
22 5.15 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 Ge atoms per cubic centimeter. The densities of pure Ge and Si are 5.32 and 2.33 g/cm 3, respectively. To solve this problem, employment of Equation 5.22 is necessary, using the following values: N 1 N Ge 2.43 x atoms/cm 3 ρ 1 ρ Ge 5.32 g/cm 3 ρ 2 ρ Si 2.33 g/cm 3 A 1 A Ge g/mol A 2 A Si g/mol Thus C Ge 1 + N A ρ Si ρ Si N Ge A Ge ρ Ge 1 + ( atoms /mol)(2.33 g /cm 3 ) ( atoms /cm 3 ) (72.64 g /mol) 2.33 g /cm g /cm wt%
23 5.16 Iron and vanadium both have the BCC crystal structure, and V forms a substitutional solid solution for concentrations up to approximately 20 wt% V at room temperature. Compute the unit cell edge length for a 90 wt% Fe 10 wt% V alloy. First of all, the atomic radii for Fe and V (using the table inside the front cover) are and nm, respectively. Also, using Equation 3.5 it is possible to compute the unit cell volume, and inasmuch as the unit cell is cubic, the unit cell edge length is just the cube root of the volume. However, it is first necessary to calculate the density and average atomic weight of this alloy using Equations 5.13a and 5.14a. Inasmuch as the densities of iron and vanadium are 7.87g/cm 3 and 6.10 g/cm 3, respectively, (as taken from inside the front cover), the average density is just ρ ave C V + C Fe ρ V ρ Fe 10 wt% 6.10 g /cm wt% 7.87 g /cm g/cm 3 And for the average atomic weight A ave C V + C Fe A V A Fe 10 wt% g /mole + 90 wt% g /mol g/mol Now, V C is determined from Equation 3.5 as V C na ave ρ ave N A
24 (2 atoms / unit cell)(55.32 g / mol) (7.65 g /cm 3 )( atoms /mol) cm 3 /unit cell And, finally a (V C ) 1/3 ( cm 3 /unit cell) 1/ x 108 cm nm
25 Interfacial Defects 5.17 For an FCC single crystal, would you expect the surface energy for a () plane to be greater or less than that for a (111) plane? Why? (Note: You may want to consult the solution to Problem W3.46 at the end of Chapter 3.) The surface energy for a crystallographic plane will depend on its packing density [i.e., the planar density (Section 3.11)] that is, the higher the packing density, the greater the number of nearestneighbor atoms, and the more atomic bonds in that plane that are satisfied, and, consequently, the lower the surface energy. From the 1 solution to Problem W3.46, planar densities for FCC () and (111) planes are 4R 2 and 1 2R 2 3, respectively that is and R 2 R 2 (where R is the atomic radius). Thus, since the planar density for (111) is greater, it will have the lower surface energy.
26 5.18 (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? (a) The surface energy will be greater than the grain boundary energy. For grain boundaries, some atoms on one side of a boundary will bond to atoms on the other side; such is not the case for surface atoms. Therefore, there will be fewer unsatisfied bonds along a grain boundary. (b) The smallangle grain boundary energy is lower than for a highangle one because more atoms bond across the boundary for the smallangle, and, thus, there are fewer unsatisfied bonds.
27 5.19 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) The interfacial defect that exists for this stacking sequence is a twin boundary, which occurs at the indicated position. The stacking sequence on one side of this position is mirrored on the other side. (b) The interfacial defect that exists within this FCC stacking sequence is a stacking fault, which occurs between the two lines. Within this region, the stacking sequence is HCP.
28 Grain Size Determination 5.20 (a) Employing the intercept technique, determine the average grain size for the steel specimen whose microstructure is shown in Figure 10.29(a); use at least seven straightline segments. (b) Estimate the ASTM grain size number for this material. (a) This portion of the problem calls for a determination of the average grain size of the specimen which microstructure is shown in Figure 10.29(a). Seven line segments were drawn across the micrograph, each of which was 60 mm long. The average number of grain boundary intersections for these lines was 6.3. Therefore, the average line length intersected is just 60 mm mm Hence, the average grain diameter, d, is d ave. line length intersected magnification 9.5 mm mm (b) This portion of the problem calls for us to estimate the ASTM grain size number for this same material. The average grain size number, n, is related to the number of grains per square inch, N, at a magnification of according to Equation However, the magnification of this micrograph is not x, but rather 90. Consequently, it is necessary to use Equation 5.20 M 2 N M 2 n 1 where N M the number of grains per square inch at magnification M, and n is the ASTM grain size number. Taking logarithms of both sides of this equation leads to the following: log N M M + 2 log (n 1) log 2 Solving this expression for n gives n M log N M + 2 log + 1 log 2
29 From Figure 10.29(a), N M is measured to be approximately 4, which leads to n log log log 2 2.7
30 5.21 For an ASTM grain size of 6, approximately how many grains would there be per square inch at (a) a magnification of, and (b) without any magnification? (a) This part of problem asks that we compute the number of grains per square inch for an ASTM grain size of 6 at a magnification of. All we need do is solve for the parameter N in Equation 5.19, inasmuch as n 6. Thus N 2 n grains/in. 2 (b) Now it is necessary to compute the value of N for no magnification. In order to solve this problem it is necessary to use Equation 5.20: M 2 N M 2 n 1 where N M the number of grains per square inch at magnification M, and n is the ASTM grain size number. Without any magnification, M in the above equation is 1, and therefore, 1 2 N And, solving for N 1, N 1 320,000 grains/in. 2.
31 of Determine the ASTM grain size number if 25 grains per square inch are measured at a magnification In order to solve this problem we make use of Equation 5.20 viz. M 2 N M 2 n 1 where N M the number of grains per square inch at magnification M, and n is the ASTM grain size number. Solving the above equation for n, and realizing that N M 25, while M 75, we have n M log N M + 2 log + 1 log 2 75 log log log
32 DESIGN PROBLEMS Specification of Composition 5.D1 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 2.47 g/cm 3 is desired. Compute the concentration of Li (in wt%) that is required. This problem calls for us to compute the concentration of lithium (in wt%) that, when added to aluminum, will yield a density of 2.47 g/cm 3. of this problem requires the use of Equation 5.13a, which takes the form ρ ave C Li + C Li ρ Li ρ Al inasmuch as C Li + C Al. According to the table inside the front cover, the respective densities of Li and Al are and 2.71 g/cm 3. Upon solving for C Li from the above equation, we get C Li ρ Li (ρ Al ρ ave ) ρ ave (ρ Al ρ Li ) ()(0.534 g /cm3 )(2.71 g /cm g /cm 3 ) (2.47 g /cm 3 )(2.71 g /cm g /cm 3 ) 2.38 wt%
33 5.D2 Gallium arsenide (GaAs) and indium arsenide (InAs) both have the zinc blende crystal structure and are soluble in each other at all concentrations. Determine the concentration in weight percent of InAs that must be added to GaAs to yield a unit cell edge length of nm. The densities of GaAs and InAs are and g/cm 3, respectively. This problem asks that we determine the concentration (in weight percent) of InAs that must be added to GaAs to yield a unit cell edge length of nm. The densities of GaAs and InAs were given in the problem statement as and g/cm 3, respectively. To begin, it is necessary to employ Equation 3.6, and solve for the unit cell volume, V C, for the InAsGaAs alloy as V C n' A ave ρ ave N A where A ave and ρ ave are the atomic weight and density, respectively, of the InAsGaAs alloy. Inasmuch as both of these materials have the zinc blende crystal structure, which has cubic symmetry, V C is just the cube of the unit cell length, a. That is V C a 3 ( nm) 3 ( cm) cm 3 It is now necessary to construct expressions for A ave and ρ ave in terms of the concentration of indium arsenide, C InAs using Equations 5.14a and 5.13a. For A ave we have A ave C InAs + ( C InAs ) A InAs A GaAs C InAs g /mol + ( C InAs ) g /mol whereas for ρ ave
34 ρ ave C InAs + ( C InAs ) ρ InAs ρ GaAs C InAs g /cm 3 + ( C InAs ) g /cm 3 Within the zinc blende unit cell there are four formula units, and thus, the value of n' in Equation 3.6 is 4; hence, this expression may be written in terms of the concentration of InAs in weight percent as follows: V C x cm 3 n' A ave ρ ave N A (4 fu /unit cell) C InAs + ( C ) InAs g /mol g /mol C InAs g /cm 3 + ( C ) ( fu /mol) InAs g /cm 3 And solving this expression for C InAs leads to C InAs 46.1 wt%.
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