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PROBLEM SOLUTIONS
9.1 Three variables that determine the microstructure of an alloy are 1) the alloying elements present, 2) the concentrations of these alloying elements, and 3) the heat treatment of the alloy.
9.2 In order for a system to exist in a state of equilibrium the free energy must be a minimum for some specified combination of temperature, pressure, and composition.
9.3 Diffusion occurs during the development of microstructure in the absence of a concentration gradient because the driving force is different than for steady state diffusion as described in Section 5.3; for the development of microstructure, the driving force is a decrease in free energy.
9.4 For the condition of phase equilibrium the free energy is a minimum, the system is completely stable meaning that over time the phase characteristics are constant. For metastability, the system is not at equilibrium, and there are very slight (and often imperceptible) changes of the phase characteristics with time.
9.5 This problem asks that we cite the phase or phases present for several alloys at specified temperatures. (a) For an alloy composed of 15 wt% Sn-85 wt% Pb and at 100°C, from Figure 9.7, α and β phases are present, and Cα = 5 wt% Sn-95 wt% Pb Cβ = 98 wt% Sn-2 wt% Pb
(b) For an alloy composed of 25 wt% Pb-75 wt% Mg and at 425°C, from Figure 9.18, only the α phase is present; its composition is 25 wt% Pb-75 wt% Mg.
(c) For an alloy composed of 85 wt% Ag-15 wt% Cu and at 800°C, from Figure 9.6, β and liquid phases are present, and
Cβ = 92 wt% Ag-8 wt% Cu CL = 77 wt% Ag-23 wt% Cu
(d) For an alloy composed of 55 wt% Zn-45 wt% Cu and at 600°C, from Figure 9.17, β and γ phases are present, and
Cβ = 51 wt% Zn-49 wt% Cu Cγ = 58 wt% Zn-42 wt% Cu
(e) For an alloy composed of 1.25 kg Sn and 14 kg Pb and at 200°C, we must first determine the Sn and Pb concentrations, as
CSn = (^) 1.25 kg1.25 kg + 14 kg × 100 = 8.2 wt%
CPb = (^) 1.25 kg14 kg + 14 kg × 100 = 91.8 wt%
From Figure 9.7, only the α phase is present; its composition is 8.2 wt% Sn-91.8 wt% Pb.
(f) For an alloy composed of 7.6 lbm Cu and 114.4 lbm Zn and at 600°C, we must first
determine the Cu and Zn concentrations, as
CCu = 7.6 lb (^) m 7.6 lb (^) m + 144.4 lbm^ ×^ 100 = 5.0 wt%
C (^) Zn = 144.4 lb (^) m 7.6 lb (^) m + 144.4 lb (^) m^ ×^ 100 = 95.0 wt%
From Figure 9.17, only the L phase is present; its composition is 95.0 wt% Zn-5.0 wt% Cu
(g) For an alloy composed of 21.7 mol Mg and 35.4 mol Pb and at 350°C, it is first necessary to determine the Mg and Pb concentrations, which we will do in weight percent as follows:
mPb^ '^ = nmPbA (^) Pb = (35.4 mol)(207.2 g/mol) = 7335 g
mMg^ '^ = nmMgA (^) Mg = (21.7 mol)(24.3 g/mol) = 527 g
Cice = 0 wt% NaCl-100 wt% H 2 O
Cbrine = 13 wt% NaCl-87 wt% H 2 O
Thus, Wice = 0.5 = Cbrine − Co Cbrine − Cice=
13 − Co 13 − 0
Solving for Co (the concentration of salt) yields a value of 6.5 wt% NaCl-93.5 wt% H 2 O.
9.14 The melting and boiling temperatures for ice I at a pressure of 0.01 atm may be determined by moving horizontally across the pressure-temperature diagram of Figure 9.33 at this pressure. The temperature corresponding to the intersection of the Ice I-Liquid phase boundary is the melting temperature, which is approximately 1°C. On the other hand, the boiling temperature is at the intersection of the horizontal line with the Liquid-Vapor phase boundary--approximately 28°C.
9.16 (a) Coring is the phenomenon whereby concentration gradients exist across grains in polycrystalline alloys, with higher concentrations of the component having the lower melting temperature at the grain boundaries. It occurs, during solidification, as a consequence of cooling rates that are too rapid to allow for the maintenance of the equilibrium composition of the solid phase. (b) One undesirable consequence of a cored structure is that, upon heating, the grain boundary regions will melt first and at a temperature below the equilibrium phase boundary from the phase diagram; this melting results in a loss in mechanical integrity of the alloy.
9.19 It is possible to have a Cu-Ag alloy, which at equilibrium consists of an α phase of composition 5 wt% Ag-95 wt% Cu and a β phase of composition 95 wt% Ag-5 wt% Cu. From Figure 9.6 a horizontal tie can be constructed across the α + β region at 690°C which intersects the α−(α + β) phase boundary at 5 wt% Ag, and also the (α + β)-β phase boundary at 95 wt% Ag.
9.21 Upon cooling a 50 wt% Ni-50 wt% Cu alloy from 1400°C and utilizing Figure 9.2a: (a) The first solid phase forms at the temperature at which a vertical line at this composition intersects the L -(α + L ) phase boundary--i.e., at about 1320°C;
(b) The composition of this solid phase corresponds to the intersection with the L -(α + L ) phase boundary, of a tie line constructed across the α + L phase region at 1320°C--i.e., C α = 62 wt% Ni-38 wt% Cu; (c) Complete solidification of the alloy occurs at the intersection of this same vertical line at 50 wt% Ni with the (α + L )-α phase boundary--i.e., at about 1270°C; (d) The composition of the last liquid phase remaining prior to complete solidification corresponds to the intersection with the L- (α + L ) boundary, of the tie line constructed across the α + L phase region at 1270°C--i.e., CL is about 37 wt% Ni-63 wt% Cu.
9.28 It is not possible to have a 50 wt% Pb-50 wt% Mg alloy which has masses of 5.13 kg and 0.57 kg for the α and Mg 2 Pb phases, respectively. In order to demonstrate this, it is first necessary to determine the mass fraction of each phase as follows:
Wα = mα mα + mMg 2 Pb
= (^) 5.13 kg5.13 kg + 0.57 kg = 0.
WMg 2 Pb = 1.00 − 0.90 = 0.
Now, if we apply the lever rule expression for W α
Wα =
C (^) Mg 2 Pb − C (^) o CMg 2 Pb^ − Cα
Since the Mg 2 Pb phase exists only at 81 wt% Pb, and Co = 50 wt% Pb
Wα = 0.90 =^8181 −− C^50 α
Solving for C α from this expression yields C α = 46.6 wt% Pb. From Figure 9.18, the maximum concentration of Pb in the α phase in the α + Mg 2 Pb phase field is about 42 wt% Pb. Therefore, this alloy is not possible.
9.30 We are asked to determine the approximate temperature from which a Pb-Mg alloy was quenched, given the mass fractions of α and Mg 2 Pb phases. We can write a lever-rule expression for the mass fraction of the α phase as
(b) From Figure 9.32, the eutectoid composition is approximately 0.62 wt% C. Since the carbon concentration in the alloy (0.2 wt%) is less than the eutectoid, the proeutectoid phase is ferrite. (c) Assume that the α-(α + Fe 3 C) phase boundary is at a negligible carbon concentration.
Modifying Equation (9.19) leads to
Wα' = 0.62 − C (^) o^ ' 0.62 − 0 =
Likewise, using a modified Equation (9.18)
Wp = Co^ '^ − 0 0.62 − 0 =
