14:440:407 ch9
Question 9.1 Consider the sugar–water phase diagram of Figure 9.1.
(a) How much sugar will dissolve in 1500 g water at 90°C (194°F)?
(b) If the saturated liquid solution in part (a) is cooled to 20°C (68°F), some of the sugar will precipitate out as
a solid. What will be the composition of the saturated liquid solution (in wt% sugar) at 20°C?
(c) How much of the solid sugar will come out of solution upon cooling to 20°C?
Solution
(a) We are ask ed to d etermine h ow m uch sug ar will d issolve in 100 0 g of water at 9 0°C. From the
solubility limit curve in Figure 9.1, at 90°C the maximum concentration of sugar in the syrup is about 77 wt%. It is
now possible to calculate the mass of sugar using Equation 4.3 as
Csugar (wt%) =
77 wt% =
msugar
msugar + mwater
msugar
msugar + 1500 g
× 100
× 100
Solving for msugar yields msugar = 5022 g
(b) Again using this same plot, at 20°C the so lubility limit (or the concentration of the saturated solution)
is about 64 wt% sugar.
(c) The mass of sugar in this saturated solution at 20°C (m' sugar ) may also be ca lculated using Equation
4.3 as follows:
64 wt% =
m' sugar
m' sugar + 1500 g
× 100
which yields a val ue for m'sugar of 26 67 g. S ubtracting the latter from the former of t hese sugar concentrations
yields the amount of sugar that precipitated out of the solution upon cooling m"sugar ; that is
m" sugar = msugar − mÕsugar = 5022 g − 2667 g = 2355 g
Question 9.3 Cite three variables that determine the microstructure of an alloy.
Solution
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.
Question 9.4 What thermodynamic condition must be met for a state of equilibrium to exist?
Solution
In order for a syste m to ex ist in a state o f equ ilibrium th e free en ergy must b e a min imum fo r so me
specified combination of temperature, pressure, and composition.
Question 9.5 Consider a specimen of ice that is at 210°C and 1 atm pressure. Using Figure 9.2, the pressure–
temperature phase diagram for H2O, determine the pressure to which the specimen must be raised or
lowered to cause it (a) to melt, and (b) to sublime.
Solution
The figure below shows the pressure-temperature phase diagram for H 2O, Figure 10.2; a v ertical line has
been constructed at -10°C, and the location on this line at 1 atm pressure (point B) is also noted.
(a) Melting occurs, (by changing pressure) as, moving vertically (upward) at this temperature, we cross the
Ice-Liquid p hase bo undary. This occ urs at approxim ately 570 atm; th us, th e pressure of the s pecimen must be
raised from 1 to 570 atm.
(b) In order to determine the pressure at which sublimation occurs at this temperature, we move vertically
downward from 1 at m u ntil we cross th e Ice-Vapor ph ase boundary. Th is in tersection occurs at ap proximately
0.0023 atm.
Question 9.8 Cite the phases that are present and the phase compositions for the following alloys:
(a) 90 wt% Zn-10 wt% Cu at 400°C (750°F)
(b) 75 wt% Sn-25 wt% Pb at 175°C (345°F)
(c) 55 wt% Ag-45 wt% Cu at 900°C (1650°F)
(d) 30 wt% Pb-70 wt% Mg at 425°C (795°F)
(e) 2.12 kg Zn and 1.88 kg Cu at 500°C (930°F)
(f) 37 lbm Pb and 6.5 lbm Mg at 400°C (750°F)
(g) 8.2 mol Ni and 4.3 mol Cu at 1250°C (2280°F)
(h) 4.5 mol Sn and 0.45 mol Pb at 200°C (390°F)
Solution
This problem asks that we cite the phase or phases present for several alloys at specified temperatures.
(a) That portion of the Cu-Zn phase diagram (Figure 9.19) that pertains to this problem is sh own below;
the point labeled “A” represents the 90 wt% Zn-10 wt% Cu composition at 400°C.
As m ay be no ted, point A l ies wi thin t he ε and η ph ase field. A tie lin e has b een co nstructed at 400°C; its
intersection with the ε−ε + η phase boundary is at 87 wt% Zn, which corresponds to the composition of the ε phase.
Similarly, the tie-line intersection with the ε + η−η phase boundary occurs at 97 wt% Zn, which is the composition
of the η phase. Thus, the phase compositions are as follows:
Cε = 87 wt% Zn-13 wt% Cu
Cη = 97 wt% Zn-3 wt% Cu
(b) That portion of the Pb-Sn phase diagram (Figure 9.8) that pertains to this problem is shown below; the
point labeled “B” represents the 75 wt% Sn-25 wt% Pb composition at 175°C.
As may be noted, point B lies within the α + β phase field. A tie line has been constructed at 175°C; its intersection
with the α−α + β phase boundary is at 16 wt% Sn, which corresponds to the composition of the α phase. Similarly,
the tie-lin e i ntersection with the α + β−β phase boundary o ccurs at 97 wt% Sn , wh ich is t he co mposition of t he
β phase. Thus, the phase compositions are as follows:
Cα = 16 wt% Sn-84 wt% Pb
Cβ = 97 wt% Sn-3 wt% Pb
(c) The Ag-Cu phase diagram (Figure 9.7) is shown below; t he point labeled “C” represents the 55 wt%
Ag-45 wt% Cu composition at 900°C.
As m ay be no ted, point C l ies wi thin t he Liquid p hase field. Th erefore, on ly th e liq uid phase is present; its
composition is 55 wt% Ag-45 wt% Cu.
(d) The Mg-Pb phase diagram (Figure 9.20) is shown below; the point labeled “D” represents the 30 wt%
Pb-70 wt% Mg composition at 425°C.
As may be noted, point D lies within the α phase field. Therefore, only the α phase is present; its composition is 30
wt% Pb-70 wt% Mg.
(e) For an alloy composed of 2.12 kg Zn and 1.88 kg Cu and at 500°C, we must first determine the Zn and
Cu concentrations, as
C Zn =
2.12 kg
× 100 = 53 wt%
2.12 kg + 1.88 kg
CCu =
1.88 kg
× 100 = 47 wt%
2.12 kg + 1.88 kg
That p ortion o f t he C u-Zn p hase di agram (Figure 9.19) t hat pert ains t o t his pro blem is sho wn below; t he poi nt
labeled “E” represents the 53 wt% Zn-47 wt% Cu composition at 500°C.
As may be noted, point E lies within the β + γ phase field. A tie line has been constructed at 500°C; its intersection
with the β−β + γ phase boundary is at 4 9 wt% Zn, which corresponds to the composition of the β phase. Similarly,
the tie-lin e in tersection with th e β + γ−γ phase bounda ry occurs at 58 wt% Zn , which is th e co mposition of the
γ phase. Thus, the phase compositions are as follows:
Cβ = 49 wt% Zn-51 wt% Cu
Cγ = 58 wt% Zn-42 wt% Cu
(f) For an alloy composed of 37 lbm Pb and 6.5 lbm Mg and at 400 °C, we must first determine the Pb and
Mg concentrations, as
CPb =
37 lb m
× 100 = 85 wt%
37 lb m + 6.5 lb m
CMg =
6.5 lb m
× 100 = 15 wt%
37 lb m + 6.5 lb m
That p ortion of t he M g-Pb phase diagram (Fi gure 9.20) t hat pe rtains t o t his p roblem is sho wn below; t he point
labeled “F” represents the 85 wt% Pb-15 wt% Mg composition at 400°C.
As m ay b e n oted, po int F lies with in th e L + Mg 2Pb phase fi eld. A t ie l ine has been co nstructed at 400 °C; it
intersects the vertical line at 81 wt% Pb, which corresponds to the composition of Mg2Pb. Furthermore, the tie line
intersection with the L + Mg 2Pb-L phase boundary is at 93 wt% Pb, which is the composition of t he liquid phase.
Thus, the phase compositions are as follows:
CMg Pb = 81 wt% Pb-19 wt% Mg
2
CL = 93 wt% Pb-7 wt% Mg
(g) For an alloy composed of 8.2 mol Ni and 4.3 mol Cu and at 1250°C, it is first n ecessary to determine
the Ni and Cu concentrations, which we will do in wt% as follows:
n'Ni = n m Ni ANi = (8.2 mol)(58.69 g/mol) = 481.3 g
' =n
nCu
m Cu ACu = (4.3 mol)(63.55 g/mol) = 273.3 g
CNi =
481.3 g
× 100 = 63.8 wt%
481.3 g + 273.3 g
CCu =
273.3 g
× 100 = 36.2 wt%
481.3 g + 273.3 g
The C u-Ni phase d iagram ( Figure 9.3a) is sho wn below; th e point labeled “G” rep resents th e 63 .8 wt% Ni -36.2
wt% Cu composition at 1250°C.
As may be noted, point G lies with in the α phase field. Therefore, only the α phase is present; its com position is
63.8 wt% Ni-36.2 wt% Cu.
(h) For an alloy composed of 4.5 mol Sn and 0.45 mol Pb a nd at 200°C, it is fi rst necessary to determine
the Sn and Pb concentrations, which we will do in weight percent as follows:
n’Sn = n mSn ASn = (4.5 mol)(118.71 g/mol) = 534.2 g
' =n
nPb
m Pb APb = (0.45 mol)(207.2 g/mol) = 93.2 g
CSn =
534.2 g
× 100 = 85.1 wt%
534.2 g + 93.2 g
CPb =
93.2 g
× 100 = 14.9 wt%
534.2 g + 93.2 g
That portion of the Pb-Sn phase diagram (Figure 9.8) that pertains to this problem is shown below; the point labeled
“H” represents the 85.1 wt% Sn-14.9 wt% Pb composition at 200°C.
As may be noted, point H lies within the β + L phase field. A tie line has been constructed at 200°C; its intersection
with the L−β + L phase boundary is at 74 wt% Sn, which corresponds to the composition of the L phase. Similarly,
the tie-line intersection with the β + L−β phase boundary occurs at 97.5 wt% Sn, which is t he composition of the
β phase. Thus, the phase compositions are as follows:
Cβ = 97.5 wt% Sn-2.5 wt% Pb
CL = 74 wt% Sn-26 wt% Pb
Question 9.9 Is it possible to have a copper–nickel alloy that, at equilibrium, consists of a liquid phase of
composition 20 wt% Ni–80 wt% Cu and also an α phase of composition 37 wt% Ni–63 wt% Cu? If so, what
will be the approximate temperature of the alloy? If this is not possible, explain why.
Solution
It
is not possible to have a Cu-Ni alloy, which at equilibrium, consists of a liq uid phase of composition 20
wt% Ni-80 wt% Cu and an α phase of composition 37 wt% Ni-63 wt% Cu. From Figure 9.3a, a single tie line does
not exist within the α + L region that intersects the phase boundaries at the given compositions. At 20 wt% Ni, the
L-(α + L) ph ase bo undary is at abo ut 1200°C, whe reas at 37 wt% Ni t he ( L + α)-α phase boundary is at about
1230°C.
Question 9.11
A copper-nickel alloy of composition 70 wt% Ni-30 wt% Cu is slowly heated from a
temperature of 1300°C (2370°F).
(a) At what temperature does the first liquid phase form?
(b) What is the composition of this liquid phase?
(c) At what temperature does complete melting of the alloy occur?
(d) What is the composition of the last solid remaining prior to complete melting?
Solution
Shown below is the Cu-Ni phase diagram (Figure 9.3a) and a vertical line con structed at a co mposition of
70 wt% Ni-30 wt% Cu.
(a) U pon heating from 1300°C, the first liqui d phase form s at the te mperature at which this ve rtical line
intersects the α-(α + L) phase boundary--i.e., about 1345°C.
(b) The composition of this liquid phase corresponds to the intersection with the (α + L)-L phase boundary,
of a tie line constructed across the α + L phase region at 1345°C--i.e., 59 wt% Ni;
(c) Complete melting of the alloy occurs at the intersection of this same vertical line at 70 wt% Ni with the
(α + L)-L phase boundary--i.e., about 1380°C;
(d) Th e composition of th e last so lid remaining prior to complete melting corresponds to the intersection
with α-(α + L) phase b oundary, of t he tie line constructed across t he α + L phase re gion at 1380°C--i.e., about 79
wt% Ni.
Question 9.12 A 50 wt% Pb-50 wt% Mg alloy is slowly cooled from 700°C (1290°F) to 400°C (750°F).
(a) At what temperature does the first solid phase form?
(b) What is the composition of this solid phase?
(c) At what temperature does the liquid solidify?
(d) What is the composition of this last remaining liquid phase?
Solution
Shown below is the Mg-Pb phase diagram (Figure 9.20) and a vertical line constructed at a composition of
50 wt% Pb-50 wt% Mg.
(a) Upon cooling from 700°C, 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., about 560°C;
(b) The composition of this solid phase corresponds to the intersection with the α-(α + L) phase boundary,
of a tie line constructed across the α + L phase region at 560°C--i.e., 21 wt% Pb-79 wt% Mg;
(c) Complete solidification of the alloy occurs at th e intersection of th is same vertical line at 50 wt% Pb
with the eutectic isotherm--i.e., about 465°C;
(d) The com position of the last liquid phase remaining prior to complete solidification corresponds to the
eutectic composition--i.e., about 67 wt% Pb-33 wt% Mg.
Question 9.15 A 1.5-kg specimen of a 90 wt% Pb–10 wt% Sn alloy is heated to 250°C (480°F); at this
temperature it is entirely an α-phase solid solution (Figure 9.8). The alloy is to be melted to the extent that
50% of the specimen is liquid, the remainder being the α phase. This may be accomplished either by heating
the alloy or changing its composition while holding the temperature constant.
(a) To what temperature must the specimen be heated?
(b) How much tin must be added to the 1.5-kg specimen at 250°C to achieve this state?
Solution
(a) Pro bably the easiest way to solve this part of th e problem is b y trial and error--that is, o n the Pb-Sn
phase diagram (Figure 9.8), moving vertically at the given composition, through the α + L region until th e tie-line
lengths on both sides of the given composition are the same. This occurs at approximately 295°C (560°F).
(b) We can also produce a 50% liquid solution at 250°C, by adding Sn to the alloy. At 250°C and within
the α + L phase region
Cα = 14 wt% Sn-86 wt% Pb
CL = 34 wt% Sn-66 wt% Pb
Let C0 be the new alloy composition to give Wα = WL = 0.5. Then,
Wα = 0.5 =
34 − C 0
CL − C0
=
C L − Cα
34 − 14
And so lving fo r C0 gi ves 2 4 wt % S n. N ow, l et mSn b e th e mass o f Sn add ed to th e allo y to ach ieve th is new
composition. The amount of Sn in the original alloy is
(0.10)(1.5 kg) = 0.15 kg
Then, using a modified form of Equation 4.3
⎡ 0.15 kg + m ⎤
Sn ×100 = 24
⎢
⎥
⎣ 1.5 kg + mSn ⎦
And, solving for mSn (the mass of tin to be added), yields mSn = 0.276 kg.
Question 9.17 A 90 wt% Ag-10 wt% Cu alloy is heated to a temperature within the β + liquid phase region.
If the composition of the liquid phase is 85 wt% Ag, determine:
(a) The temperature of the alloy
(b) The composition of the β phase
(c) The mass fractions of both phases
Solution
(a) In o rder to determine the temperature of a 90 wt% Ag-10 wt% Cu alloy for whic h β and liquid phases
are present with the liquid phase of composition 85 wt% Ag, we need to construct a tie line across th e β + L phase
region of Figure 9.7 that intersects the liquidus line at 85 wt% Ag; this is possible at about 850°C.
(b) The composition of the β phase at th is temperature is determin ed from the intersection of this same tie
line with solidus line, which corresponds to about 95 wt% Ag.
(c) The mass fractions of the two phases are determined using the lever rule, Equations 9.1 and 9.2 with C0
= 90 wt% Ag, CL = 85 wt% Ag, and Cβ = 95 wt% Ag, as
Wβ =
WL =
C0 − CL
90 − 85
=
= 0.50
95 − 85
Cβ − C L
Cβ − C 0
Cβ − C L
=
95 − 90
= 0.50
95 − 85
Question 9.22 For 11.20 kg of a magnesium-lead alloy of composition 30 wt% Pb-70 wt% Mg, is it possible,
at equilibrium, to have α and Mg2Pb phases having respective masses of 7.39 kg and 3.81 kg? If so, what will
be the approximate temperature of the alloy? If such an alloy is not possible, explain why.
Solution
Yes,
it is possible to have a 30 wt% Pb-70 wt% Mg alloy which has masses of 7.39 kg and 3.81 kg for the
α and Mg2Pb phases, respectively. In order to demonstrate this, it is first n ecessary to determine the mass fraction
of each phase as follows:
Wα =
mα
7.39 kg
=
= 0.66
mα + mMg 2Pb
7.39 kg + 3.81 kg
WMg 2Pb = 1.00 − 0.66 = 0.34
Now, if we apply the lever rule expression for Wα
Wα =
CMg 2Pb − C 0
CMg 2Pb − Cα
Since the Mg2Pb phase exists only at 81 wt% Pb, and C0 = 30 wt% Pb
Wα = 0.66 =
81 − 30
81 − Cα
Solving fo r Cα fr om t his exp ression y ields Cα = 3 .7 wt% Pb. Th e p osition along th e α−(α + Mg 2Pb) phas e
boundary of Figure 9.20 corresponding to this composition is approximately 190°C.
Question 9.26 It is desirable to produce a copper-nickel alloy that has a minimum noncold-worked tensile
strength of 350 MPa (50,750 psi) and a ductility of at least 48%EL. Is such an alloy possible? If so, what
must be its composition? If this is not possible, then explain why.
Solution
From
Figure 9.6a, a t ensile st rength g reater t han 3 50 M Pa (5 0,750 psi) i s po ssible for c ompositions
between about 22.5 a nd 98 wt% Ni. On the other hand, according to Figure 9.6 b, ductilities greater than 48%EL
exist for compositions less than about 8 wt% and greater than about 98 wt% Ni. Therefore, the stipulated criteria are
met only at a composition of 98 wt% Ni.
Question 9.27 A 45 wt% Pb–55 wt% Mg alloy is rapidly quenched to room temperature from an elevated
temperature in such a way that the high-temperature microstructure is preserved. This microstructure is
found to consist of the α phase and Mg2Pb, having respective mass fractions of 0.65 and 0.35. Determine the
approximate temperature from which the alloy was quenched.
Solution
We are as ked to determine the approxim ate temperature from which a 45 wt% Pb-55 wt% Mg alloy was
quenched, gi ven t he m ass fract ions o f α and M g2Pb pha ses. We can write a lever -rule expressi on for the m ass
fraction of the α phase as
Wα = 0.65 =
CMg 2Pb − C0
CMg 2Pb − Cα
The value of C0 is stated as 45 wt% Pb-55 wt% Mg, and CMg Pb is 81 wt% Pb-19 wt% Mg, which is independent
2
of temperature (Figure 9.20); thus,
0.65 =
81 − 45
81 − Cα
which yields
Cα = 25.6 wt% Pb
The temperature at which the α–(α + Mg 2Pb) phase boundary (Figure 9.20) has a value of 2 5.6 wt% Pb is about
360°C (680°F).
Question 9.32 For a copper-silver alloy of composition 25 wt% Ag-75 wt% Cu and at 775°C (1425°F) do the
following:
(a) Determine the mass fractions of α and β phases.
(b) Determine the mass fractions of primary α and eutectic microconstituents.
(c) Determine the mass fraction of eutectic α.
Solution
(a) This portion of the problem asks that we determine the mass fractions of α and β phases for an 25 wt%
Ag-75 wt% C u allo y (at 775°C). In order to do t his it is n ecessary to employ th e lev er rule u sing a tie lin e th at
extends entirely across the α + β phase field. From Figure 9.7 and at 775°C, Cα = 8.0 wt% Ag, Cβ = 91.2 wt% Ag,
and Ceutectic = 71.9 wt% Sn. Therefore, the two lever-rule expressions are as follows:
Wα =
Wβ =
Cβ − C0
Cβ − Cα
=
91.2 − 25
= 0.796
91.2 − 8.0
25 − 8.0
C 0 − Cα
=
= 0.204
91.2 − 8.0
Cβ − Cα
(b) Now it is necessary to determine the mass fractions of primary α and eutectic microconstituents for this
same alloy. This requires us to utilize the lever rule and a tie line that extends from the maximum solubility of Ag in
the α phase at 775°C (i.e., 8.0 wt% Ag) to the eutectic composition (71.9 wt% Ag). Thus
Wα' =
71.9 − 25
C eutectic Ź− C 0
=
= 0.734
71.9 − 8.0
Ceutectic Ź− Cα
We =
25 − 8.0
C0 − Cα
=
= 0.266
71.9 − 8.0
C eutectic − Cα
(c) And, finally, we are asked to compute the mass fraction of eutectic α, Weα. This quantity is simply the
difference between the mass fractions of total α and primary α as
Weα = Wα – Wα' = 0.796 – 0.734 = 0.062
Question 9.34 Consider the hypothetical eutectic phase diagram for metals A and B, which is similar to that
for the lead-tin system, Figure 9.8. Assume that (1) α and β phases exist at the A and B extremities of the
phase diagram, respectively; (2) the eutectic composition is 47 wt% B-53 wt% A; and (3) the composition of
the β phase at the eutectic temperature is 92.6 wt% B-7.4 wt% A. Determine the composition of an alloy that
will yield primary α and total α mass fractions of 0.356 and 0.693, respectively.
Solution
We are given a hypothetical eu tectic phase diagram for w hich Ceutectic = 47 wt% B, Cβ = 92. 6 wt% B at
the eutectic te mperature, a nd als o that Wα' = 0.356 a nd Wα = 0.693; fro m th is we are ask ed to d etermine th e
composition of the alloy. Let us write lever rule expressions for Wα' and Wα
Wα =
Wα' =
Cβ Ź− C 0
Cβ − Cα
=
92.6 − C 0
= 0.693
92.6 − C α
47 − C 0
C eutectic Ź− C 0
=
= 0.356
Ceutectic Ź− Cα
47 − C α
Thus, we have two simultaneous equations with C0 and Cα as unknowns. Solving them for C0 gives C0 = 32.6 wt%
B.
Question 9.35 For an 85 wt% Pb-15 wt% Mg alloy, make schematic sketches of the microstructure that
would be observed for conditions of very slow cooling at the following temperatures: 600°C (1110°F), 500°C
(930°F), 270°C (520°F), and 200°C (390°F). Label all phases and indicate their approximate compositions.
Solution
The illustration below is th e Mg-Pb phase diagram (Figure 9.20). A v ertical line at a composition of 8 5
wt% Pb-15 wt% Mg has been drawn, and, in addition, horizontal arrows at th e four temperatures called for in the
problem statement (i.e., 600°C, 500°C, 270°C, and 200°C).
On the basis of the locations of the four temperature-composition points, schematic sketches of the four respective
microstructures along with phase compositions are represented as follows:
Question 9.37 For a 30 wt% Zn-70 wt% Cu alloy, make schematic sketches of the microstructure that would
be observed for conditions of very slow cooling at the following temperatures: 1100°C (2010°F), 950°C
(1740°F), 900°C (1650°F), and 700°C (1290°F). Label all phases and indicate their approximate compositions.
Solution
The illu stration b elow is th e Cu -Zn ph ase diagram (Fig ure 9 .19). A v ertical lin e at a composition of 30
wt% Zn-70 wt% Cu has been drawn, and, in addition, horizontal arrows at th e four temperatures called for in th e
problem statement (i.e., 1100°C, 950°C, 900°C, and 700°C).
On the basis of the locations of the four temperature-composition points, schematic sketches of the four respective
microstructures along with phase compositions are represented as follows:
Question 9.39 The room-temperature tensile strengths of pure lead and pure tin are 16.8 MPa and 14.5 MPa,
respectively.
(a) Make a schematic graph of the room-temperature tensile strength versus composition for all
compositions between pure lead and pure tin. (Hint: you may want to consult Sections 9.10 and 9.11, as well
as Equation 9.24 in Problem 9.64.)
(b) On this same graph schematically plot tensile strength versus composition at 150°C.
(c) Explain the shapes of these two curves, as well as any differences between them.
Solution
The (a) and (b) portions of the problem ask that we make schematic plots on the same graph for the tensile
strength versus composition for lead-tin alloys at both room temperature and 150°C; such a graph is shown below.
(c) Upon consultation of the Pb-Sn phase diagram (Figure 9.8) we note that, at room temperature (20°C),
about 1.5 wt% o f Sn is solu ble in Pb (with in th e α-phase re gion at t he l eft ext remity of t he pha se di agram).
Similarly, only about 1 wt% of Pb is soluble in Sn (within the β-phase region at the left extremity). Thus, there will
a solid-solution strengthening effect on both ends of the phase diagram—strength increases slightly with additions of
Sn to Pb [in the α phase region (left-hand side)] and with additions of Pb to Sn [in the β phase region (right-hand
side)]; these effects are noted in the a bove figure. T his figure also shows that the tensile stren gth of pure lead is
greater than pure tin, which is in agreement with tensile strength values provided in the problem statement.
In addition, at room temperature, for compositions between about 1.5 wt% Sn and 99 wt% Sn, both α and β
phase will co exist, (Figu re 9.8), Fu rthermore, for co mpositions within th is ran ge, ten sile stren gth will d epend
(approximately) on the tensile strengths of each of th e α and β phases as well as their pha se fractions in a manner
described by Equation 9.24 for the elastic modulus (Problem 9.64). That is, for this problem
(TS) alloy ≅ (TS)α Vα + (TS)β Vβ
in wh ich TS a nd V d enote ten sile stren gth an d vo lume fraction, respectively, an d the s ubscripts rep resent th e
alloy/phases. Also, mass fractions of the α and β phases change linearly with ch anging composition (according to
the lever rule). Furthermore, although there is some disparity between the densities of Pb and Sn (11.35 versus 7.27
g/cm3), weight and volume fractions of the α and β phases will also be similar (see Equation 9.6).
At
150°C, th e cu rve will b e sh ifted t o sign ificantly lo wer ten sile streng ths in asmuch as ten sile stren gth
diminishes with increasing temperature (Section 6.6, Figure 6 .14). In addition, according to Figure 9.8, solubility
limits for both α and β phases increase—for the α phase from 1.5 to 10 wt% Sn, and for the β phase from 1 to about
2 wt% Pb. Thus, the compositional ranges over which solid-solution strengthening occurs increase somewhat from
the ro om-temperature ranges; these effects are also note d on the 150°C curve above . Furtherm ore, at 150 °C, it
would be expected that the tensile strength of lead will be greater than that of tin; and for compositions over which
both α and β phases coexist, strength will decrease approximately linearly with increasing Sn content.
Question 9.46 Compute the mass fractions of α ferrite and cementite in pearlite.
Solution
This problem asks that we compute the mass fractions of α ferrite and cementite in pearlite. The lever-rule
expression for ferrite is
Wα =
C Fe C − C 0
3
CFe C − Cα
3
and, since CFe C = 6.70 wt% C, C0 = 0.76 wt% C, and Cα = 0.022 wt% C
3
Wα =
6.70 − 0.76
= 0.89
6.70 − 0.022
Similarly, for cementite
C 0 − Cα
0.76 − 0.022
=
= 0.11
WFe C =
3
6.70 − 0.022
C Fe C − Cα
3
Question 9.48 What is the carbon concentration of an iron–carbon alloy for which the fraction of total ferrite
is 0.94?
Solution
This problem asks that we compute the carbon concentration of an iron-carbon alloy for which the fraction
of total ferrite is 0.94. Application of the lever rule (of the form of Equation 9.12) yields
Wα = 0.94 =
CFe 3C − C0'
CFe 3C − Cα
=
6.70 − C 0'
6.70 − 0.022
and solving for C 0'
C 0' = 0.42 wt% C
Question 9.49 What is the proeutectoid phase for an iron–carbon alloy in which the mass fractions of total
ferrite and total cementite are 0.92 and 0.08, respectively? Why?
Solution
In this problem we are given values of Wα and WFe C (0.92 and 0.08, respectively) for an iron-carbon alloy
3
and then are asked to specify the proeutectoid phase. Employment of the lever rule for total α leads to
Wα = 0.92 =
CFe 3C − C 0
CFe 3C − Cα
=
6.70 − C 0
6.70 − 0.022
Now, solving for C0, the alloy composition, leads to C0 = 0.56 wt% C. Therefore, the proeutectoid phase is α-ferrite
since C0 is less than 0.76 wt% C.
Question 9.50 Consider 1.0 kg of austenite containing 1.15 wt% C, cooled to below 727°C (1341°F).
(a) What is the proeutectoid phase?
(b) How many kilograms each of total ferrite and cementite form?
(c) How many kilograms each of pearlite and the proeutectoid phase form?
(d) Schematically sketch and label the resulting microstructure.
Solution
(a) The proeutectoid phase will be Fe 3C since 1.15 wt% C is greater t han the eutectoid composition (0.76
wt% C).
(b) For this portion of the problem, we are ask ed to determine how much total ferrite and cementite form.
Application of the appropriate lever rule expression yields
Wα =
CFe 3C − C 0
CFe 3C − Cα
=
6.70 − 1.15
= 0.83
6.70 − 0.022
which, when multiplied by the total mass of the alloy (1.0 kg), gives 0.83 kg of total ferrite.
Similarly, for total cementite,
WFe 3C =
C 0 − Cα
1.15 − 0.022
=
= 0.17
CFe 3C − Cα
6.70 − 0.022
And the mass of total cementite that forms is (0.17)(1.0 kg) = 0.17 kg.
(c) No w we are ask ed to calcu late h ow much p earlite an d th e pro eutectoid ph ase (cem entite) fo rm.
Applying Equation 9.22, in which C1' = 1.15 wt% C
Wp =
6.70 − C 1'
6.70 − 1.15
=
= 0.93
6.70 − 0.76 6.70 − 0.76
which corresponds to a mass of 0.93 kg. Likewise, from Equation 9.23
WFe 3C' =
C1' − 0.76
5.94
=
1.15 − 0.76
= 0.07
5.94
which is equivalent to 0.07 kg of the total 1.0 kg mass.
(d) Schematically, the microstructure would appear as:
Question 9.51 Consider 2.5 kg of austenite containing 0.65 wt% C, cooled to below 727°C (1341°F).
(a) What is the proeutectoid phase?
(b) How many kilograms each of total ferrite and cementite form?
(c) How many kilograms each of pearlite and the proeutectoid phase form?
(d) Schematically sketch and label the resulting microstructure.
Solution
(a) Ferrite is the proeutectoid phase since 0.65 wt% C is less than 0.76 wt% C.
(b) For this portion of the problem, we are ask ed to determine how much total ferrite and cementite form.
For ferrite, application of the appropriate lever rule expression yields
Wα =
CFe 3C − C 0
6.70 − 0.65
=
= 0.91
6.70 − 0.022
CFe 3C − Cα
which corresponds to (0.91)(2.5 kg) = 2.27 kg of total ferrite.
Similarly, for total cementite,
WFe 3C =
0.65 − 0.022
C 0 − Cα
=
= 0.09
6.70 − 0.022
CFe 3C − Cα
Or (0.09)(2.5 kg) = 0.23 kg of total cementite form.
(c) Now consider the amounts of pearlite and proeutectoid ferrite. Using Equation 9.20
Wp =
C 0' − 0.022
0.74
=
0.65 − 0.022
= 0.85
0.74
This corresponds to (0.85)(2.5 kg) = 2.12 kg of pearlite.
Also, from Equation 9.21,
Wα' =
0.76 − 0.65
= 0.15
0.74
Or, there are (0.15)(2.5 kg) = 0.38 kg of proeutectoid ferrite.
(d) Schematically, the microstructure would appear as:
Question 9.53 The microstructure of an iron–carbon alloy consists of proeutectoid ferrite and pearlite; the
mass fractions of these two microconstituents are 0.286 and 0.714, respectively. Determine the concentration
of carbon in this alloy.
Solution
This prob lem asks th at we dete rmine the carbon concentr ation in an iron -carbon allo y, g iven th e mass
fractions of proeutectoid ferrite and pearlite. From Equation 9.20
W p = 0.714 =
C 0' − 0.022
0.74
which yields C 0' = 0.55 wt% C.
Question 9.55 The microstructure of an iron-carbon alloy consists of proeutectoid ferrite and pearlite; the
mass fractions of these microconstituents are 0.20 and 0.80, respectively. Determine the concentration of
carbon in this alloy.
Solution
We are asked in this problem to determine the concentration of carbon in an alloy for which Wα' = 0.20 and
Wp = 0 .80. If we let
C 0' equal the carbon concent ration i n the allo y, em ployment of the app ropriate lever rule
expression, Equation 9.20, leads to
Wp =
Solving for C 0' yields C 0' = 0.61 wt% C.
C 0' − 0.022
= 0.80
0.74
Question 9.61 The mass fraction of eutectoid cementite in an iron-carbon alloy is 0.104. On the basis of this
information, is it possible to determine the composition of the alloy? If so, what is its composition? If this is
not possible, explain why.
Solution
This problem asks whether or not it is possible to determine the com position of an iron-carbon alloy for
which the mass fraction of eutectoid cem entite is 0.104; and if so, to cal culate the composition. Yes, it is possible
to determ ine the alloy com position; an d, in fact, t here are two possible answ ers. For the first, t he e utectoid
cementite exists in addition
to proeutectoi d cem entite. For t his case the m ass fraction of eutectoid cem entite
(WFe C'') is just the difference between total cementite and proeutectoid cementite mass fractions; that is
3
WFe C'' = WFe C – WFe C'
3
3
3
Now, it is possible to wr ite ex pressions for WFe C (of the form of E quation 9.12) and WFe C' (Equation 9.23) in
3
3
terms of C0, the alloy composition. Thus,
C 0 − Cα
C − 0.76
WFe C" =
− 0
3
5.94
C Fe C − Cα
3
=
C 0 − 0.022
C − 0.76
− 0
= 0.104
6.70 − 0.022
5.94
And, solving for C0 yields C0 = 1.11 wt% C.
For t he second possi bility, we ha ve a hypoeutectoid alloy wherein
cementite. Thus, it is necessary to set up a lever
all of the cem entite is eutectoi d
rule expression wherein the m ass fraction of t otal ce mentite i s
0.104. Therefore,
WFe 3C =
C 0 − Cα
C − 0.022
= 0
= 0.104
CFe 3C − Cα
6.70 − 0.022
And, solving for C0 yields C0 = 0.72 wt% C.
Question 9.66 A steel alloy is known to contain 93.8 wt% Fe, 6.0 wt% Ni, and 0.2 wt% C.
(a) What is the approximate eutectoid temperature of this alloy?
(b) What is the proeutectoid phase when this alloy is cooled to a temperature just below the eutectoid?
(c) Compute the relative amounts of the proeutectoid phase and pearlite.
Assume that there are no alterations in the positions of other phase boundaries with the addition of Ni.
Solution
(a) From Figure 9.34, the eutectoid temperature for 6.0 wt% Ni is approximately 650°C (1200°F).
(b)
From Figu re 9.35, t he eutectoi d c omposition is approximately 0. 62 wt% C . Since the c arbon
concentration in the alloy (0.2 wt%) is less than the eutectoid (0.62 wt% C), the proeutectoid phase is ferrite.
(c) Ass ume that the α–(α + Fe3C) phas e bo undary is at a negligible carb on co ncentration. M odifying
Equation 9.21 leads to
Wα' =
0.62 − C '0
0.62 − 0.20
=
= 0.68
0.62 − 0
0.62
Likewise, using a modified Equation 9.20
Wp =
C 0' − 0
0.20
=
= 0.32
0.62 − 0 0.62
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