# 14:635:407:02 Homework III Solutions

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1 14:635:407:0 Homework III Solutions 4.1 Calculate the fraction of atom sites that are vacant for lead at its melting temperature of 37 C (600 K). Assume an energy for vacancy formation of 0.55 ev/atom. In order to compute the fraction of atom sites that are vacant in lead at 600 K, we must employ Equation 4.1. As stated in the problem, Q v = 0.55 ev/atom. Thus, N v N = exp Q v = exp kt 0.55 ev / atom ( ev /atom- K) (600 K) = Calculate the number of vacancies per cubic meter in iron at 850C. 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 g/mol, respectively. Determination of the number of vacancies per cubic meter in iron at 850C (113 K) requires the utilization of Equations 4.1 and 4. as follows: N v = N exp Q v = N A Fe kt A Fe exp Q v kt And incorporation of values of the parameters provided in the problem statement into the above equation leads to N v = ( atoms /mol)(7.65 g /cm 3 ) g /mol exp 1.08 ev/ atom ( ev /atom K) (850C + 73 K) = cm -3 = m -3

2 4.5 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 termedd an octahedral interstitial site. On the other hand, with BCC the larger site type is found at 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 BCCC 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. In the drawing below is shown the atoms on the (100) facee of an FCC unit cell; the interstitial site is at the center of the edge. The diameter of an atom that will just fit into this site (r) is just thee difference between that unit cell edge length (a) and the radii of the two host atoms that are located on either side of the site (R); that is r = a R However, for FCC a is related to R according to Equation 3.1 as a R ; therefore, solving for r from the above equation gives r = a R = R R = 0.41R

3 A (100) face of a BCC unit cell is shown below. The interstitial atom thatt just fits into this t interstitial site is shown by the small circle. It is situated in the plane of this (100) face, midway between the two vertical unitt cell edges, and one quarter of the distance between the bottom and top cell edges. From the right triangle that is defined by the three arrows we may write a + a 4 = (R r) However, from Equation 3.3, a = 4R R, and, therefor re, making thiss substitution, the above equation takes the form 3 4R 3 + 4R4 4 3 = R + Rr + r After rearrangement the following f quadratic equation results: r + Rr 0.667R = 0 And upon solving for r:

4 r (R) (R) (4)(1)(0.667R ) R.58R And, finally R.58R r() 0.91R R.58R r().91r Of course, only the r(+) root is possible, and, therefore, r = 0.91R. Thus, for a host atom of radius R, the size of an interstitial site for FCC is approximately 1.4 times that for BCC.

5 4.15 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? In order to compute the concentration in kg/m 3 of C in a 0.15 wt% C wt% Fe alloy we must employ Equation 4.9 as C C " = C C C C C C Fe Fe 10 3 From inside the front cover, densities for carbon and iron are.5 and 7.87 g/cm 3, respectively; and, therefore C C " = g/cm g/cm = 11.8 kg/m Determine the approximate density of a high-leaded brass that has a composition of 64.5 wt% Cu, 33.5 wt% Zn, and.0 wt% Pb. In order to solve this problem, Equation 4.10a is modified to take the following form: ave = C Cu Cu 100 C Zn Zn C Pb Pb And, using the density values for Cu, Zn, and Pb i.e., 8.94 g/cm 3, 7.13 g/cm 3, and g/cm 3 (as taken from inside the front cover of the text), the density is computed as follows: ave = wt% 8.94 g /cm wt% 7.13 g /cm 3.0 wt% g /cm 3 = 8.7 g/cm 3

6 4.18 Some hypothetical alloy is composed of 1.5 wt% of metal A and 87.5 wt% of metal B. If the densities of metals A and B are 4.7 and 6.35 g/cm 3, respectively, whereas their respective atomic weights are 61.4 and 15.7 g/mol, determine whether the crystal structure for this alloy is simple cubic, face-centered cubic, or body-centered 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 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 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 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 = ave a3 N A A ave Expressions for A ave and ave are found in Equations 4.11a and 4.10a, respectively, which, when incorporated into the above expression yields n = 100 C A C a B 3 N A A B 100 C A C B A A A B

7 Substitution of the concentration values (i.e., C A = 1.5 wt% and C B = 87.5 wt%) as well as values for the other parameters given in the problem statement, into the above equation gives n = 100 ( wt% 4.7 g/cm wt% -8 nm) 3 ( atoms/mol) 6.35 g/cm wt% 61.4 g/mol 87.5 wt% 15.7 g/mol =.00 atoms/unit cell Therefore, on the basis of this value, the crystal structure is body-centered cubic. 4.0 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 19.3 and g/cm 3, respectively. To solve this problem, employment of Equation 4.18 is necessary, using the following values: C 1 = C Au = 10 wt% 1 = Au = 19.3 g/cm 3 = Ag = g/cm 3 A 1 = A Au = g/mol Thus N Au = C Au A Au Au N A C Au A Au Ag (100 C Au ) = ( atoms /mol) (10 wt%) (10 wt%)( g /mol) g /mol 19.3 g /cm 3 ( wt%) g /cm3 = atoms/cm 3

8 4.6 Cite the relative Burgers vector dislocation line orientations for edge, screw, and mixed dislocations. The Burgers vector and dislocation line are perpendicular for edge dislocations, parallel for screw dislocations, and neither perpendicular nor parallel for mixed dislocations. 4.7 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.) 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 nearest-neighbor 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 3.54, planar densities for FCC (100) and (111) planes are 4R and 1 R 3, respectively that is and R R (where R is the atomic radius). Thus, since the planar density for (111) is greater, it will have the lower surface energy. 4.8 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.) 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 nearest-neighbor atoms, and the more atomic bonds in that plane that are satisfied, and, consequently, the lower the surface energy. From the 3 solution to Problem 3.55, the planar densities for BCC (100) and (110) are 16R and 3 8R, respectively that is and. Thus, since the planar density for (110) is greater, it will have the lower surface energy. R R

9 4.9 (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 grainn boundary iss less than for a high-angle 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 small-angle grain boundary energy is lower thann for a high-angle one because more atoms bond across the boundary for the small-angle, and, thus, there are fewer unsatisfied bonds (a) Briefly describe a twin and a twin boundary. (b) Cite the difference between mechanical and annealing twins. (a) A twin boundary is an interface such that atoms on one side are located at mirror image positions of those atoms situated on the other boundary side. The region on one side of this boundary is called a twin. (b) Mechanical twins are produced as a result of mechanical deformation and generally occur in BCC and HCP metals. Annealing twins form during annealing heat treatments, most often in FCC metals 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) off planar defect(s) with a vertical dashed line. (a) The interfacial defect that exists for this stacking sequence iss a twin boundary, which occurs at the indicated position. The stacking sequence on one side of this position is mirrored on thee other side. (b) The interfacial defect that exists within this FCC stacking sequencee is a stacking fault, which occurs between the two lines. Within this region, the stacking sequence is HCP.

10 14:440:407 Section 0 Fall 010 SOLUTION OF HOMEWORK 04 Question 5.3: (a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion. Solution of 5.3: (a) With vacancy diffusion, atomic motion is from one lattice site to an adjacent vacancy. Self-diffusion and the diffusion of substitutional impurities proceed via this mechanism. On the other hand, atomic motion is from interstitial site to adjacent interstitial site for the interstitial diffusion mechanism. (b) Interstitial diffusion is normally more rapid than vacancy diffusion because: (1) interstitial atoms, being smaller, are more mobile; and () the probability of an empty adjacent interstitial site is greater than for a vacancy adjacent to a host (or substitutional impurity) atom. Question 5.4: Briefly explain the concept of steady state as it applies to diffusion. Solution of 5.4: Steady-state diffusion is the situation wherein the rate of diffusion into a given system is just equal to the rate of diffusion out, such that there is no net accumulation or depletion of diffusing species-i.e., the diffusion flux is independent of time. Question 5.6: 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 0.0 m at 500C. Assume a diffusion coefficient of m /s, that the concentrations at the highand low-pressure sides of the plate are.4 and 0.6 kg of hydrogen per cubic meter of palladium, and that steadystate conditions have been attained. Solution of 5.6: This problem calls for the mass of hydrogen, per hour, that diffuses through a Pd sheet. It first becomes necessary to employ both Equations 5.1a and 5.3. Combining these expressions and solving for the mass yields M = JAt = DAt C x = ( m /s)(0.0 m kg /m3 ) (3600 s/h) m = kg/h

11 Question 5.8: 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 75C. After having reached steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were determined to be 0.01 and wt%. Compute the diffusion coefficient if the diffusion flux is kg/m -s. Hint: Use Equation 4.9 to convert the concentrations from weight percent to kilograms of carbon per cubic meter of iron. Solution of 5.8: Let us first convert the carbon concentrations from weight percent to kilograms carbon per meter cubed using Equation 4.9a. For 0.01 wt% C C " C C = C C C C 10 3 Fe C Fe = g/cm g/cm Similarly, for wt% C kg C/m 3 C C " = g/cm g/cm Now, using a rearranged form of Equation 5.3 = kg C/m 3 x D = J A x B C A C B = ( kg/m 10 -s) 3 m kg/m kg/m 3 = m /s

12 Question 5.11: Determine the carburizing time necessary to achieve a carbon concentration of 0.45 wt% at a position mm into an iron carbon alloy that initially contains 0.0 wt% C. The surface concentration is to be maintained at 1.30 wt% C, and the treatment is to be conducted at 1000C. Use the diffusion data for -Fe in Table 5.. Solution of 5.11: In order to solve this problem it is first necessary to use Equation 5.5: C x C 0 x = 1 erf C s C 0 Dt wherein, C x = 0.45, C 0 = 0.0, C s = 1.30, and x = mm = 10-3 m. Thus, C x C 0 C s C 0 = = 0.73 = 1 erf x Dt or x erf = = Dt By linear interpolation using data from Table 5.1 z erf(z) z z = From which z = = x Dt Now, from Table 5., at 1000C (173 K) D = ( m 148, 000 J/mol /s) exp (8.31 J/mol- K)(173 K) Thus, Solving for t yields = m /s = 3 m () ( m /s) (t) t = s = 19.7 h

13 Question 5.1: 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 (117C). 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 at the surface position is maintained essentially at 0 wt% C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be 0.15 wt% after a 10-h treatment? The value of D at 1400 K is m /s. Solution of 5.1: This problem asks that we determine the position at which the carbon concentration is 0.15 wt% after a 10-h heat treatment at 135 K when C 0 = 0.35 wt% C. From Equation 5.5 C x C 0 C s C 0 = Thus, x erf = Dt Using data in Table 5.1 and linear interpolation x = = 1 erf Dt z erf (z) z z = And, Which means that And, finally x Dt z = = x = (0.400) Dt = (0.8004) ( m /s)( s) = m = 1.6 mm Note: this problem may also be solved using the Diffusion module in the VMSE software. Open the Diffusion module, click on the Diffusion Design submodule, and then do the following:

14 1. Enter the given data in left-hand window that appears. In the window below the label D Value enter the value of the diffusion coefficient viz. 6.9e-11.. In the window just below the label Initial, C0 enter the initial concentration viz In the window the lies below Surface, Cs enter the surface concentration viz Then in the Diffusion Time t window enter the time in seconds; in 10 h there are (60 s/min)(60 min/h)(10 h) = 36,000 s so enter the value 3.6e4. 5. Next, at the bottom of this window click on the button labeled Add curve. 6. On the right portion of the screen will appear a concentration profile for this particular diffusion situation. A diamond-shaped cursor will appear at the upper left-hand corner of the resulting curve. Click and drag this cursor down the curve to the point at which the number below Concentration: reads 0.15 wt%. Then read the value under the Distance:. For this problem, this value (the solution to the problem) is ranges between 1.4 and 1.30 mm.

15 Question 5.15: 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.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 carburizing temperature. Solution of 5.15: This problem calls for an estimate of the time necessary to achieve a carbon concentration of 0.45 wt% at a point 5.0 mm from the surface. From Equation 5.6b, x Dt = constant But since the temperature is constant, so also is D constant, and x = constant t or x 1 = x t 1 t Thus, (.5 mm) 10 h = (5.0 mm) t from which t = 40 h Question 5.16: 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. Solution 5.16: We are asked to compute the diffusion coefficients of C in both and iron at 900C. Using the data in Table 5., D = ( m 80, 000 J/mol /s) exp (8.31 J/mol- K)(1173 K) = m /s D = ( m 148, 000 J/mol /s) exp (8.31 J/mol- K)(1173 K) = m /s The D for diffusion of C in BCC iron is larger, the reason being that the atomic packing factor is smaller than for FCC iron (0.68 versus 0.74 Section 3.4); this means that there is slightly more interstitial void space in the BCC Fe, and, therefore, the motion of the interstitial carbon atoms occurs more easily.

16 Question 5.18: At what temperature will the diffusion coefficient for the diffusion of copper in nickel have a value of m /s. Use the diffusion data in Table 5.. Solution of 5.18: Solving for T from Equation 5.9a Q T = d R(ln D ln D 0 ) and using the data from Table 5. for the diffusion of Cu in Ni (i.e., D 0 = m /s and Q d = 56,000 J/mol), we get T = 56,000 J/mol (8.31 J/mol- K) ln ( m /s) ln ( m /s) = 115 K = 879C Note: this problem may also be solved using the Diffusion module in the VMSE software. Open the Diffusion module, click on the D vs 1/T Plot submodule, and then do the following: 1. In the left-hand window that appears, there is a preset set of data for several diffusion systems. Click on the box for which Cu is the diffusing species and Ni is the host metal. Next, at the bottom of this window, click the Add Curve button.. A log D versus 1/T plot then appears, with a line for the temperature dependence of the diffusion coefficient for Cu in Ni. At the top of this curve is a diamond-shaped cursor. Click-and-drag this cursor down the line to the point at which the entry under the Diff Coeff (D): label reads m /s. The temperature at which the diffusion coefficient has this value is given under the label Temperature (T):. For this problem, the value is 1153 K.

17 Question 5.1: The diffusion coefficients for iron in nickel are given at two temperatures: T (K) D (m /s) (a) Determine the values of D 0 and the activation energy Q d. (b) What is the magnitude of D at 1100ºC (1373 K)? Solution of 5.1: (a) Using Equation 5.9a, we set up two simultaneous equations with Q d and D 0 as unknowns as follows: ln D 1 lnd 0 Q d 1 R T 1 ln D lnd 0 Q d 1 R T Now, solving for Q d in terms of temperatures T 1 and T (173 K and 1473 K) and D 1 and D ( and m /s), we get = (8.31 J/mol- K) Q d = R ln D 1 ln D 1 T 1 1 T ln ( ) ln ( ) K K = 5,400 J/mol Now, solving for D 0 from Equation 5.8 (and using the 173 K value of D) D 0 = D 1 exp Q d RT 1 = ( m 5, 400 J/mol /s)exp (8.31 J/mol- K)(173 K) = m /s (b) Using these values of D 0 and Q d, D at 1373 K is just D = ( m 5, 400 J/mol /s)exp (8.31 J/mol- K)(1373 K) = m /s

18 Note: this problem may also be solved using the Diffusion module in the VMSE software. Open the Diffusion module, click on the D0 and Qd from Experimental Data submodule, and then do the following: 1. In the left-hand window that appears, enter the two temperatures from the table in the book (viz. 173 and 1473, in the first two boxes under the column labeled T (K). Next, enter the corresponding diffusion coefficient values (viz. 9.4e-16 and.4e-14 ). 3. Next, at the bottom of this window, click the Plot data button. 4. A log D versus 1/T plot then appears, with a line for the temperature dependence for this diffusion system. At the top of this window are give values for D 0 and Q d ; for this specific problem these values are m /s and 5 kj/mol, respectively 5. To solve the (b) part of the problem we utilize the diamond-shaped cursor that is located at the top of the line on this plot. Click-and-drag this cursor down the line to the point at which the entry under the Temperature (T): label reads The value of the diffusion coefficient at this temperature is given under the label Diff Coeff (D):. For our problem, this value is m /s.

19 Question 5.4: 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/m 3 Fe, which are maintained constant. If the preexponential and activation energy are m /s and 80,000 J/mol, respectively, compute the temperature at which the diffusion flux is kg/m -s. Solution 5.4: Combining Equations 5.3 and 5.8 yields J = D C x C = D 0 x exp Q d RT Solving for T from this expression leads to T = Q d 1 R ln D 0 C J x And incorporation of values provided in the problem statement yields = 80, 000 J/mol J/mol- K ln ( m /s)(0.65 kg/m kg/m 3 ) ( kg/m -s)( m) = 1044 K = 771C

20 Question 5.30: 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 850C (113 K) for 10 min increases the carbon concentration to 0.90 wt% at a position 1.0 mm below the surface. Estimate the diffusion time required at 650C (93 K) to achieve this same concentration also at a 1.0-mm position. Assume that the surface carbon content is the same for both heat treatments, which is maintained constant. Use the diffusion data in Table 5. for C diffusion in -Fe. Solution of 5.30: In order to compute the diffusion time at 650C to produce a carbon concentration of 0.90 wt% at a position 1.0 mm below the surface we must employ Equation 5.6b with position (x) constant; that is Or Dt = constant D 850 t 850 = D 650 t 650 In addition, it is necessary to compute values for both D 850 and D 650 using Equation 5.8. From Table 5., for the diffusion of C in -Fe, Q d = 80,000 J/mol and D 0 = m /s. Therefore, D 850 = ( m 80, 000 J/mol /s)exp (8.31 J/mol- K)( K) = m /s D 650 = ( m 80, 000 J/mol /s)exp (8.31 J/mol- K)( K) = m /s Now, solving the original equation for t 650 gives t 650 = D 850 t 850 D 650 = ( m /s)(10 min) m /s = 63.9 min

### In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus,

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