Materials Science and Engineering

William D. Callister, David G. Rethwisch

Chapter 9

Phase Diagrams - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the sugar-water phase diagram of Figure 9.1.
(a) How much sugar will dissolve in $1000 \mathrm{~g}$ of water at $80^{\circ} \mathrm{C}\left(176^{\circ} \mathrm{F}\right) ?$
(b) If the saturated liquid solution in part (a) is cooled to $20^{\circ} \mathrm{C}\left(68^{\circ} \mathrm{F}\right)$, some of the sugar precipitates as a solid. What will be the composition of the saturated liquid solution (in wt \% sugar) at $20^{\circ} \mathrm{C} ?$
(c) How much of the solid sugar will come out of solution upon cooling to $20^{\circ} \mathrm{C}$ ?

Ivan Kochetkov

Ivan Kochetkov

Numerade Educator

Problem 2

At $100^{\circ} \mathrm{C}$, what is the maximum solubility of the following:
(a) $\mathrm{Pb}$ in $\mathrm{Sn}$
(b) $\mathrm{Sn}$ in $\mathrm{Pb}$

Ivan Kochetkov

Numerade Educator

Problem 3

Cite three variables that determine the microstructure of an alloy.

Ameer Said

Numerade Educator

Problem 4

What thermodynamic condition must be met for a state of equilibrium to exist?

Ameer Said

Numerade Educator

Problem 5

Consider a specimen of ice that is at $-15^{\circ} \mathrm{C}$ and 10 atm pressure. Using Figure $9.2$, the pressure-temperature phase diagram for $\mathrm{H}_{2} \mathrm{O}$, determine the pressure to which the specimen must be raised or lowered to cause it (a) to melt and (b) to sublime.

Ivan Kochetkov

Numerade Educator

Problem 6

At a pressure of $0.1$ atm, determine (a) the melting temperature for ice and (b) the boiling temperature for water.

Ameer Said

Numerade Educator

Problem 7

Given here are the solidus and liquidus temperatures for the copper-gold system. Construct the phase diagram for this system and label each region.
$$
\begin{array}{ccc}
\hline \begin{array}{c}
\text { Composition } \\
\text { (wt\% Au) }
\end{array} & \begin{array}{c}
\text { Solidus } \\
\text { Temperature }\left({ }^{\circ} \mathrm{C}\right)
\end{array} & \begin{array}{c}
\text { Liquidus } \\
\text { Temperature }\left({ }^{\circ} \boldsymbol{C}\right) \\
\hline 0
\end{array} & 1085 & 1085 \\
\hline 20 & 1019 & 1042 \\
\hline 40 & 972 & 996 \\
\hline 60 & 934 & 946 \\
\hline 80 & 911 & 911 \\
\hline 90 & 928 & 942 \\
\hline 95 & 974 & 984 \\
\hline 100 & 1064 & 1064 \\
\hline
\end{array}
$$

Ivan Kochetkov

Numerade Educator

Problem 8

How many kilograms of nickel must be added to $1.75 \mathrm{~kg}$ of copper to yield a liquidus temperature of $1300^{\circ} \mathrm{C} ?$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 9

How many kilograms of nickel must be added to $5.43 \mathrm{~kg}$ of copper to yield a solidus temperature of $1200^{\circ} \mathrm{C} ?$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 10

Cite the phases that are present and the phase compositions for the following alloys:
(a) $15 \mathrm{wt} \% \mathrm{Sn}-85 \mathrm{wt} \% \mathrm{~Pb}$ at $100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)$
(b) $25 \mathrm{wt} \% \mathrm{~Pb}-75 \mathrm{wt} \% \mathrm{Mg}$ at $425^{\circ} \mathrm{C}\left(800^{\circ} \mathrm{F}\right)$
(c) $85 \mathrm{wt} \% \mathrm{Ag}-15 \mathrm{wt} \% \mathrm{Cu}$ at $800^{\circ} \mathrm{C}\left(1470^{\circ} \mathrm{F}\right)$
(d) $55 \mathrm{wt} \% \mathrm{Zn}-45 \mathrm{wt} \% \mathrm{Cu}$ at $600^{\circ} \mathrm{C}\left(1110^{\circ} \mathrm{F}\right)$
(e) $1.25 \mathrm{~kg} \mathrm{Sn}$ and $14 \mathrm{~kg} \mathrm{~Pb}$ at $200^{\circ} \mathrm{C}\left(390^{\circ} \mathrm{F}\right)$
(f) $7.6 \mathrm{lb}_{\mathrm{m}} \mathrm{Cu}$ and $144.4 \mathrm{lb}_{\mathrm{m}} \mathrm{Zn}$ at $600^{\circ} \mathrm{C}\left(1110^{\circ} \mathrm{F}\right)$
(g) $21.7 \mathrm{~mol} \mathrm{Mg}$ and $35.4 \mathrm{~mol} \mathrm{~Pb}$ at $350^{\circ} \mathrm{C}\left(660^{\circ} \mathrm{F}\right)$
(h) $4.2 \mathrm{~mol} \mathrm{Cu}$ and $1.1 \mathrm{~mol} \mathrm{Ag}$ at $900^{\circ} \mathrm{C}\left(1650^{\circ} \mathrm{F}\right)$

Ameer Said

Numerade Educator

Problem 11

Is it possible to have a copper-silver alloy that, at equilibrium, consists of a $\beta$ phase of composition $92 \mathrm{wt} \%$ Ag-8 $\mathrm{wt} \% \mathrm{Cu}$ and also a liquid phase of composition $76 \mathrm{wt} \%$ Ag-24 wt $\% \mathrm{Cu}$ ? If so, what will be the approximate temperature of the alloy? If this is not possible, explain why.

Ameer Said

Numerade Educator

Problem 12

Is it possible to have a copper-silver alloy that, at equilibrium, consists of an $\alpha$ phase of composition $4 \mathrm{wt} \%$ Ag $-96 \mathrm{wt} \% \mathrm{Cu}$ and also a $\beta$ phase of composition $95 \mathrm{wt} \% \mathrm{Ag}-5 \mathrm{wt} \% \mathrm{Cu}$ ? If so, what will be the approximate temperature of the alloy? If this is not possible, explain why.

Robert Smith

Numerade Educator

Problem 13

A lead-tin alloy of composition $30 \mathrm{wt} \%$ Sn-70 $\mathrm{wt} \% \mathrm{~Pb}$ is slowly heated from a temperature of $150^{\circ} \mathrm{C}\left(300^{\circ} \mathrm{F}\right)$
(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?

Ameer Said

Numerade Educator

Problem 14

A $50 \mathrm{wt} \%$ Ni-50 wt $\%$ Cu alloy is slowly cooled from $1400^{\circ} \mathrm{C}\left(2550^{\circ} \mathrm{F}\right)$ to $1200^{\circ} \mathrm{C}\left(2190^{\circ} \mathrm{F}\right)$.
(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?

Ameer Said

Numerade Educator

Problem 15

A copper-zinc alloy of composition $75 \mathrm{wt} \% \mathrm{Zn}-25$ $\mathrm{wt} \% \mathrm{Cu}$ is slowly heated from room temperature.
(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?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 16

For an alloy of composition $52 \mathrm{wt} \% \mathrm{Zn}-48$ wt $\%$ Cu, cite the phases present and their mass fractions at the following temperatures: $1000^{\circ} \mathrm{C}$, $300^{\circ} \mathrm{C}, 500^{\circ} \mathrm{C}$, and $300^{\circ} \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 17

Determine the relative amounts (in terms of mass fractions) of the phases for the alloys and temperatures given in Problem 9.10.

Ameer Said

Numerade Educator

Problem 18

A $2.0-\mathrm{kg}$ specimen of an $85 \mathrm{wt} \% \mathrm{~Pb}-15 \mathrm{wt} \% \mathrm{Sn}$ alloy is heated to $200^{\circ} \mathrm{C}\left(390^{\circ} \mathrm{F}\right)$; at this temperature it is entirely an $\alpha$-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 $\alpha$ phase. This may be accomplished 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 $2.0$-kg specimen at $200^{\circ} \mathrm{C}$ to achieve this state?

Ameer Said

Numerade Educator

Problem 19

A magnesium-lead alloy of mass $7.5 \mathrm{~kg}$ consists of a solid $\alpha$ phase that has a composition just slightly below the solubility limit at $300^{\circ} \mathrm{C}\left(570^{\circ} \mathrm{F}\right)$.
(a) What mass of lead is in the alloy?
(b) If the alloy is heated to $400^{\circ} \mathrm{C}\left(750^{\circ} \mathrm{F}\right)$, how much more lead may be dissolved in the $\alpha$ phase without exceeding the solubility limit of this phase?

Ameer Said

Numerade Educator

Problem 20

Consider $2.5 \mathrm{~kg}$ of a $80 \mathrm{wt} \% \mathrm{Cu}-20 \mathrm{wt} \% \mathrm{Ag}$ copper-silver alloy at $800^{\circ} \mathrm{C}$. How much copper must be added to this alloy to cause it to completely solidify $800^{\circ} \mathrm{C} ?$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 21

A $65 \mathrm{wt} \%$ Ni-35 wt \% Cu alloy is heated to a temperature within the $\alpha+$ liquid-phase region. If the composition of the $\alpha$ phase is $70 \mathrm{wt} \% \mathrm{Ni}$, determine
(a) the temperature of the alloy
(b) the composition of the liquid phase
(c) the mass fractions of both phases

Ameer Said

Numerade Educator

Problem 22

A $40 \mathrm{wt} \% \mathrm{~Pb}-60 \mathrm{wt} \% \mathrm{Mg}$ alloy is heated to a temperature within the $\alpha+$ liquid-phase region. If the mass fraction of each phase is $0.5$, then estimate
(a) the temperature of the alloy
(b) the compositions of the two phases in weight percent
(c) the compositions of the two phases in atom percent

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 23

A copper-silver alloy is heated to $900^{\circ} \mathrm{C}$ and is found to consist of $\alpha$ and liquid phases. If the mass fraction of the liquid phase is $0.68$, determine
(a) the composition of both phases, in both weight percent and atom percent, and
(b) the composition of the alloy, in both weight percent and atom percent

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 24

For alloys of two hypothetical metals $\mathrm{A}$ and $\mathrm{B}$, there exist an $\alpha$, A-rich phase and a $\beta$, B-rich phase. From the mass fractions of both phases for two different alloys provided in the following table (which are at the same temperature), determine the composition of the phase boundary (or solubility limit) for both $\alpha$ and $\beta$ phases at this temperature.
$$
\begin{array}{lcc}
\hline \begin{array}{c}
\text { Alloy } \\
\text { Composition }
\end{array} & \begin{array}{c}
\text { Fraction } \\
\boldsymbol{\alpha} \text { Phase }
\end{array} & \text { Fraction } \\
\hline 70 \mathrm{wt} \% \mathrm{~A}-30 \mathrm{wt} \% \mathrm{~B} & 0.78 & 0.22 \\
\hline 35 \mathrm{wt} \% \mathrm{~A}-65 \mathrm{wt} \% \mathrm{~B} & 0.36 & 0.64 \\
\hline
\end{array}
$$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 25

A hypothetical A-B alloy of composition 40 $\mathrm{wt} \% \mathrm{~B}-60 \mathrm{wt} \% \mathrm{~A}$ at some temperature is found to consist of mass fractions of $0.66$ and $0.34$ for the $\alpha$ and $\beta$ phases, respectively. If the composition of the $\alpha$ phase is $13 \mathrm{wt} \%$ B-87 wt $\% \mathrm{~A}$, what is the composition of the $\beta$ phase?

Ameer Said

Numerade Educator

Problem 26

Is it possible to have a copper-silver alloy of composition $20 \mathrm{wt} \%$ Ag-80 wt $\%$ Cu that, at equilibrium, consists of $\alpha$ and liquid phases having mass fractions $W_{\alpha}=0.80$ and $W_{L}=0.20$ ? If so, what will be the approximate temperature of the alloy? If such an alloy is not possible, explain why.

Ameer Said

Numerade Educator

Problem 27

For $5.7 \mathrm{~kg}$ of a magnesium-lead alloy of composition $50 \mathrm{wt} \% \mathrm{~Pb}-50 \mathrm{wt} \% \mathrm{Mg}$, is it possible, at equilibrium, to have $\alpha$ and $\mathrm{Mg}_{2} \mathrm{~Pb}$ phases with respective masses of $5.13$ and $0.57 \mathrm{~kg} ?$ If so, what will be the approximate temperature of the alloy? If such an alloy is not possible, then explain why.

Ameer Said

Numerade Educator

Problem 28

Derive Equations $9.6 \mathrm{a}$ and $9.7 \mathrm{a}$, which may be used to convert mass fraction to volume fraction, and vice versa.

Ameer Said

Numerade Educator

Problem 29

Determine the relative amounts (in terms of volume fractions) of the phases for the alloys and temperatures given in Problems $9.10 \mathrm{a}, \mathrm{b}$, and $\mathrm{d}$.
The following table gives the approximate densities of the various metals at the alloy temperatures:
$$
\begin{array}{ccc}
\hline \text { Metal } & \text { Temperature }\left({ }^{\circ} \boldsymbol{C}\right) & \text { Density }\left(\mathrm{g} / \mathrm{cm}^{3}\right) \\
\hline \mathrm{Cu} & 600 & 8.68 \\
\hline \mathrm{Mg} & 425 & 1.68 \\
\hline \mathrm{Pb} & 100 & 11.27 \\
\hline \mathrm{Pb} & 425 & 10.96 \\
\hline \mathrm{Sn} & 100 & 7.29 \\
\hline \mathrm{Zn} & 600 & 6.67 \\
\hline
\end{array}
$$

Ameer Said

Numerade Educator

Problem 30

(a) Briefly describe the phenomenon of coring and why it occurs.
(b) Cite one undesirable consequence of coring.

Ameer Said

Numerade Educator

Problem 31

It is desirable to produce a copper-nickel alloy that has a minimum non-cold-worked tensile strength of $380 \mathrm{MPa}(55,000 \mathrm{psi})$ and a ductility of at least $45 \%$ EL. Is such an alloy possible? If so, what must be its composition? If this is not possible, then explain why.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 32

A 60 wt \% Pb-40 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 $\alpha$ phase and $\mathrm{Mg}_{2} \mathrm{~Pb}$, having respective mass fractions of $0.42$ and $0.58$. Determine the approximate temperature from which the alloy was quenched.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 33

Briefly explain why, upon solidification, an alloy of eutectic composition forms a microstructure consisting of alternating layers of the two solid phases.

Ameer Said

Numerade Educator

Problem 34

What is the difference between a phase and a microconstituent?

Ameer Said

Numerade Educator

Problem 35

Plot the mass fraction of phases present versus temperature for a $40 \mathrm{wt} \% \mathrm{Sn}-60 \mathrm{wt} \% \mathrm{~Pb}$ alloy as it is slowly cooled from $250^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 36

Is it possible to have a magnesium-lead alloy in which the mass fractions of primary $\alpha$ and total $\alpha$ are $0.60$ and $0.85$, respectively, at $460^{\circ} \mathrm{C}\left(860^{\circ} \mathrm{F}\right)$ ? Why or why not?

Ameer Said

Numerade Educator

Problem 37

For $2.8 \mathrm{~kg}$ of a lead-tin alloy, is it possible to have the masses of primary $\beta$ and total $\beta$ of $2.21$ and $2.53$ $\mathrm{kg}$, respectively, at $180^{\circ} \mathrm{C}\left(355^{\circ} \mathrm{F}\right)$ ? Why or why not?

Ameer Said

Numerade Educator

Problem 38

For a lead-tin alloy of composition $80 \mathrm{wt} \% \mathrm{Sn}-$ $20 \mathrm{wt} \% \mathrm{~Pb}$ and at $180^{\circ} \mathrm{C}\left(355^{\circ} \mathrm{F}\right)$, do the following:
(a) Determine the mass fractions of the $\alpha$ and $\beta$ phases.
(b) Determine the mass fractions of primary $\beta$ and eutectic microconstituents.
(c) Determine the mass fraction of eutectic $\beta$.

Ameer Said

Numerade Educator

Problem 39

The microstructure of a copper-silver alloy at $775^{\circ} \mathrm{C}\left(1425^{\circ} \mathrm{F}\right)$ consists of primary $\alpha$ and eutectic structures. If the mass fractions of these two microconstituents are $0.73$ and $0.27$, respectively, determine the composition of the alloy.

Ameer Said

Numerade Educator

Problem 40

A magnesium-lead alloy is cooled from $600^{\circ} \mathrm{C}$ to $450^{\circ} \mathrm{C}$ and is found to consist of primary $\mathrm{Mg}_{2} \mathrm{~Pb}$ and eutectic microconstituents. If the mass fraction of the eutectic microconstituent is $0.28$, determine the alloy composition.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 41

Consider a hypothetical eutectic phase diagram for metals $A$ and $B$ that is similar to that for the lead-tin system (Figure 9.8). Assume that: (1) $\alpha$ and $\beta$ phases exist at the A and B extremes of the phase diagram, respectively; (2) the eutectic composition is $36 \mathrm{wt} \% \mathrm{~A}-64 \mathrm{wt} \% \mathrm{~B} ;$ and (3) the composition of the $\alpha$ phase at the eutectic temperature is $88 \mathrm{wt} \%$ A-12 wt $\%$ B. Determine the composition of an alloy that will yield primary $\beta$ and total $\beta$ mass fractions of $0.367$ and $0.768$, respectively.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 42

For a 64 wt \% Zn-36 wt\% Cu alloy, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: $900^{\circ} \mathrm{C}\left(1650^{\circ} \mathrm{F}\right), 820^{\circ} \mathrm{C}$ $\left(1510^{\circ} \mathrm{F}\right), 750^{\circ} \mathrm{C}\left(1380^{\circ} \mathrm{F}\right)$, and $600^{\circ} \mathrm{C}\left(1100^{\circ} \mathrm{F}\right)$ Label all phases and indicate their approximate compositions.

Ameer Said

Numerade Educator

Problem 43

For a $76 \mathrm{wt} \%$ Pb-24 wt \% Mg alloy, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: $575^{\circ} \mathrm{C}\left(1070^{\circ} \mathrm{F}\right)$, $500^{\circ} \mathrm{C}\left(930^{\circ} \mathrm{F}\right), 450^{\circ} \mathrm{C}\left(840^{\circ} \mathrm{F}\right)$, and $300^{\circ} \mathrm{C}\left(570^{\circ} \mathrm{F}\right)$ Label all phases and indicate their approximate compositions.

Ameer Said

Numerade Educator

Problem 44

For a $52 \mathrm{wt} \% \mathrm{Zn}-48 \mathrm{wt} \% \mathrm{Cu}$ alloy, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: $950^{\circ} \mathrm{C}\left(1740^{\circ} \mathrm{F}\right), 860^{\circ} \mathrm{C}$ $\left(1580^{\circ} \mathrm{F}\right), 800^{\circ} \mathrm{C}\left(1470^{\circ} \mathrm{F}\right)$, and $600^{\circ} \mathrm{C}\left(1100^{\circ} \mathrm{F}\right)$. Label all phases and indicate their approximate compositions.

Ameer Said

Numerade Educator

Problem 45

On the basis of the photomicrograph (i.e., the relative amounts of the microconstituents) for the lead-tin alloy shown in Figure $9.17$ and the Pb-Sn phase diagram (Figure 9.8), estimate the composition of the alloy, and then compare this estimate with the composition given in the legend of Figure 9.17. Make the following assumptions: (1) The area fraction of each phase and microconstituent in the photomicrograph is equal to its volume fraction; (2) the densities of the $\alpha$ and $\beta$ phases and the eutectic structure are 11.2, 7.3, and $8.7 \mathrm{~g} / \mathrm{cm}^{3}$, respectively; and (3) this photomicrograph represents the equilibrium microstructure at $180^{\circ} \mathrm{C}\left(355^{\circ} \mathrm{F}\right)$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 46

The room-temperature tensile strengths of pure copper and pure silver are 209 and $125 \mathrm{MPa}$, respectively.
(a) Make a schematic graph of the room-temperature tensile strength versus composition for all compositions between pure copper and pure silver. (Hint: You may want to consult Sections $9.10$ and 9.11, as well as Equation $9.24$ in Problem 9.79.)
(b) On this same graph, schematically plot tensile strength versus composition at $600^{\circ} \mathrm{C}$.
(c) Explain the shapes of these two curves as well as any differences between them.

Ameer Said

Numerade Educator

Problem 47

Two intermetallic compounds, $\mathrm{A}_{3} \mathrm{~B}$ and $\mathrm{AB}_{3}$, exist for elements $A$ and $B$. If the compositions for $\mathrm{A}_{3} \mathrm{~B}$ and $\mathrm{AB}_{3}$ are $91.0 \mathrm{wt} \% \mathrm{~A}-9.0 \mathrm{wt} \% \mathrm{~B}$ and $53.0$ $\mathrm{wt} \% \mathrm{~A}-47.0 \mathrm{wt} \% \mathrm{~B}$, respectively, and element $\mathrm{A}$ is zirconium, identify element $\mathrm{B}$.

Ameer Said

Numerade Educator

Problem 48

An intermetallic compound is found in the aluminum-zirconium system that has a composition of $22.8 \mathrm{wt} \%$ Al-77.2 wt $\% \mathrm{Zr}$. Specify the formula for this compound.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 49

An intermetallic compound is found in the goldtitanium system that has a composition of $58.0$ wt $\%$ Au-42.0 wt $\%$ Ti. Specify the formula for this compound.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 50

Specify the liquidus, solidus, and solvus temperatures for the following alloys:
(a) $30 \mathrm{wt} \% \mathrm{Ni}-70 \mathrm{wt} \% \mathrm{Cu}$
(b) $5 \mathrm{wt} \% \mathrm{Ag}-95 \mathrm{wt} \% \mathrm{Cu}$
(c) $20 \mathrm{wt} \% \mathrm{Zn}-80 \mathrm{wt} \% \mathrm{Cu}$
(d) $30 \mathrm{wt} \% \mathrm{~Pb}-70 \mathrm{wt} \% \mathrm{Mg}$
(e) $3 \mathrm{wt} \%$ C-97 wt $\% \mathrm{Fe}$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 51

What is the principal difference between congruent and incongruent phase transformations?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 52

Figure $9.36$ is the tin-gold phase diagram, for which only single-phase regions are labeled. Specify temperature-composition points at which all eutectics, eutectoids, peritectics, and congruent phase transformations occur. Also, for each, write the reaction upon cooling.

Ameer Said

Numerade Educator

Problem 53

Figure $9.37$ is a portion of the copper-aluminum phase diagram for which only single-phase regions are labeled. Specify temperature-composition points at which all eutectics, eutectoids, peritectics, and congruent phase transformations occur. Also, for each, write the reaction upon cooling.

Ameer Said

Numerade Educator

Problem 54

Construct the hypothetical phase diagram for metals $\mathrm{A}$ and $\mathrm{B}$ between room temperature $\left(20^{\circ} \mathrm{C}\right)$ and $700^{\circ} \mathrm{C}$, given the following information:
- The melting temperature of metal $\mathrm{A}$ is $480^{\circ} \mathrm{C}$.
- The maximum solubility of $B$ in $A$ is 4 wt $\%$ B, which occurs at $420^{\circ} \mathrm{C}$.
- The solubility of $\mathrm{B}$ in $\mathrm{A}$ at room temperature is 0 wt $\%$ B.
- One eutectic occurs at $420^{\circ} \mathrm{C}$ and $18 \mathrm{wt} \%$ B-82 wt $\%$ A.
- A second eutectic occurs at $475^{\circ} \mathrm{C}$ and $42 \mathrm{wt} \%$ B- $58 \mathrm{wt} \% \mathrm{~A}$
- The intermetallic compound AB exists at a composition of $30 \mathrm{wt} \% \mathrm{~B}-70 \mathrm{wt} \% \mathrm{~A}$, and melts congruently at $525^{\circ} \mathrm{C}$.
- The melting temperature of metal B is $600^{\circ} \mathrm{C} .$
- The maximum solubility of $\mathrm{A}$ in $\mathrm{B}$ is $13 \mathrm{wt} \% \mathrm{~A}$, which occurs at $475^{\circ} \mathrm{C}$.
- The solubility of $\mathrm{A}$ in $\mathrm{B}$ at room temperature is $3 \mathrm{wt} \% \mathrm{~A}$

Ameer Said

Numerade Educator

Problem 55

Figure $9.38$ shows the pressure-temperature phase diagram for $\mathrm{H}_{2} \mathrm{O}$. Apply the Gibbs phase rule at points $A, B$, and $C$, and specify the number of degrees of freedom at each of the points - that is, the number of externally controllable variables that must be specified to define the system completely.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 56

Specify the number of degrees of freedom for th following alloys:
(a) $20 \mathrm{wt} \% \mathrm{Ni}-80 \mathrm{wt} \% \mathrm{Cu}$ at $1300^{\circ} \mathrm{C}$
(b) $71.9 \mathrm{wt} \%$ Ag-28.1 $\mathrm{wt} \% \mathrm{Cu}$ at $779^{\circ} \mathrm{C}$
(c) $52.7 \mathrm{wt} \% \mathrm{Zn}-47.3 \mathrm{wt} \% \mathrm{Cu}$ at $525^{\circ} \mathrm{C}$
(d) $81 \mathrm{wt} \% \mathrm{~Pb}-19 \mathrm{wt} \% \mathrm{Mg}$ at $545^{\circ} \mathrm{C}$
(e) $1 \mathrm{wt} \% \mathrm{C}-99 \mathrm{wt} \% \mathrm{Fe}$ at $1000^{\circ} \mathrm{C}$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 57

Compute the mass fractions of $\alpha$-ferrite and cementite in pearlite.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 58

(a) What is the distinction between hypoeutectoid and hypereutectoid steels?
(b) In a hypoeutectoid steel, both eutectoid and proeutectoid ferrite exist. Explain the difference between them. What will be the carbon concentration in each?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 59

What is the carbon concentration of an iron-carbon alloy for which the fraction of total cementite is $0.10$ ?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 60

What is the proeutectoid phase for an iron-carbon alloy in which the mass fractions of total ferrite and total cementite are $0.86$ and $0.14$, respectively? Why?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 61

Consider $3.5 \mathrm{~kg}$ of austenite containing $0.95 \mathrm{wt} \%$ C and cooled to below $727^{\circ} \mathrm{C}\left(1341^{\circ} \mathrm{F}\right)$.
(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.

Ali Beker

Numerade Educator

Problem 62

Consider $6.0 \mathrm{~kg}$ of austenite containing $0.45 \mathrm{wt} \%$ $\mathrm{C}$ and cooled to less than $727^{\circ} \mathrm{C}\left(1341^{\circ} \mathrm{F}\right)$.
(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.

Ali Beker

Numerade Educator

Problem 63

On the basis of the photomicrograph (i.e., the relative amounts of the microconstituents) for the ironcarbon alloy shown in Figure $9.30$ and the $\mathrm{Fe}-\mathrm{Fe}_{3} \mathrm{C}$ phase diagram (Figure 9.24), estimate the composition of the alloy, and then compare this estimate with the composition given in the figure legend of Figure 9.30. Make the following assumptions: (1) The area fraction of each phase and microconstituent in the photomicrograph is equal to its volume fraction; (2) the densities of proeutectoid ferrite and pearlite are $7.87$ and $7.84 \mathrm{~g} / \mathrm{cm}^{3}$, respectively; and (3) this photomicrograph represents the equilibrium microstructure at $725^{\circ} \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 64

On the basis of the photomicrograph (i.e., the relative amounts of the microconstituents) for the ironcarbon alloy shown in Figure $9.33$ and the $\mathrm{Fe}-\mathrm{Fe}_{3} \mathrm{C}$ phase diagram (Figure 9.24), estimate the composition of the alloy, and then compare this estimate with the composition given in the figure legend of Figure 9.33. Make the following assumptions: (1) The area fraction of each phase and microconstituent in the photomicrograph is equal to its volume fraction; (2) the densities of proeutectoid cementite and pearlite are $7.64$ and $7.84 \mathrm{~g} / \mathrm{cm}^{3}$, respectively; and (3) this photomicrograph represents the equilibrium microstructure at $725^{\circ} \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 65

Compute the mass fractions of proeutectoid ferrite and pearlite that form in an iron-carbon alloy containing $0.35 \mathrm{wt} \% \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 66

For a series of $\mathrm{Fe}-\mathrm{Fe}_{3} \mathrm{C}$ alloys with compositions ranging between $0.022$ and $0.76 \mathrm{wt} \% \mathrm{C}$ that have been cooled slowly from $1000^{\circ} \mathrm{C}$, plot the following:
(a) mass fractions of proeutectoid ferrite and pearlite versus carbon concentration at $725^{\circ} \mathrm{C}$
(b) mass fractions of ferrite and cementite versus carbon concentration at $725^{\circ} \mathrm{C}$.

Anatole Borisov

Anatole Borisov

Numerade Educator

Problem 67

The microstructure of an iron-carbon alloy consists of proeutectoid ferrite and pearlite; the mass fractions of these two microconstituents are $0.174$ and $0.826$, respectively. Determine the concentration of carbon in this alloy.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 68

The mass fractions of total ferrite and total cementite in an iron-carbon alloy are $0.91$ and $0.09$, respectively. Is this a hypoeutectoid or hypereutectoid alloy? Why?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 69

The microstructure of an iron-carbon alloy consists of proeutectoid cementite and pearlite; the mass fractions of these microconstituents are $0.11$ and $0.89$, respectively. Determine the concentration of carbon in this alloy.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 70

Consider $1.5 \mathrm{~kg}$ of a $99.7 \mathrm{wt} \% \mathrm{Fe}-0.3 \mathrm{wt} \%$ C alloy that is cooled to a temperature just below the eutectoid.
(a) How many kilograms of proeutectoid ferrite form?
(b) How many kilograms of eutectoid ferrite form?
(c) How many kilograms of cementite form?

Ali Beker

Numerade Educator

Problem 71

Compute the maximum mass fraction of proeutectoid cementite possible for a hypereutectoid iron-carbon alloy.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 72

Is it possible to have an iron-carbon alloy for which the mass fractions of total cementite and proeutectoid ferrite are $0.057$ and $0.36$, respectively? Why or why not?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 73

Is it possible to have an iron-carbon alloy for which the mass fractions of total ferrite and pearlite are $0.860$ and $0.969$, respectively? Why or why not?

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 74

Compute the mass fraction of eutectoid cementite in an iron-carbon alloy that contains $1.00 \mathrm{wt} \% \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 75

Compute the mass fraction of eutectoid cementite in an iron-carbon alloy that contains $0.87 \mathrm{wt} \% \mathrm{C}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 76

The mass fraction of eutectoid cementite in an iron-carbon alloy is $0.109$. 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.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 77

The mass fraction of eutectoid ferrite in an ironcarbon alloy is $0.71$. 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.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 78

For an iron-carbon alloy of composition 3 wt $\%$ C-97 wt \% Fe, make schematic sketches of the microstructure that would be observed for conditions of very slow cooling at the following temperatures: $1250^{\circ} \mathrm{C}\left(2280^{\circ} \mathrm{F}\right), 1145^{\circ} \mathrm{C}\left(2095^{\circ} \mathrm{F}\right)$, and $700^{\circ} \mathrm{C}\left(1290^{\circ} \mathrm{F}\right)$. Label the phases and indicate their compositions (approximate).

Ali Beker

Numerade Educator

Problem 79

Often, the properties of multiphase alloys may be approximated by the relationship
$$
E(\text { alloy })=E_{\alpha} V_{\alpha}+E_{\beta} V_{\beta}
$$
where $E$ represents a specific property (modulus of elasticity, hardness, etc.), and $V$ is the volume fraction. The subscripts $\alpha$ and $\beta$ denote the existing phases or microconstituents. Use this relationship to determine the approximate Brinell hardness of a $99.75 \mathrm{wt} \% \mathrm{Fe}-0.25 \mathrm{wt} \%$ C alloy. Assume Brinell hardnesses of 80 and 280 for ferrite and pearlite, respectively, and that volume fractions may be approximated by mass fractions.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 80

A steel alloy contains $95.7 \mathrm{wt} \% \mathrm{Fe}, 4.0 \mathrm{wt} \% \mathrm{~W}$, and $0.3 \mathrm{wt} \% \mathrm{C}$.
(a) What is the eutectoid temperature of this alloy?
(b) What is the eutectoid composition?
(c) What is the proeutectoid phase?
Assume that there are no changes in the positions of other phase boundaries with the addition of $\mathrm{W}$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 81

A steel alloy is known to contain $93.65 \mathrm{wt} \% \mathrm{Fe}$, $6.0 \mathrm{wt} \% \mathrm{Mn}$, and $0.35 \mathrm{wt} \% \mathrm{C}$.
(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 Mn.

Mohammad Mehran

Mohammad Mehran

Numerade Educator