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Chapter 9: Problem 20

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 wt\(\% $$\mathrm{B}-87 \mathrm{wt} \% \mathrm{A},\) what is the composition of the \(\beta\) phase?

Short Answer

Expert verified
Question: Calculate the composition of the β phase for an A-B binary alloy system with an overall composition of 40 wt% B-60 wt% A, mass fractions of α and β phases as 0.66 and 0.34, respectively, and α phase composition of 13 wt% B and 87 wt% A. Answer: The composition of the β phase is 92.41 wt% B and 7.59 wt% A.

Step by step solution

01

List given information

We have been given the following information: 1. Overall alloy composition: \(40 \mathrm{wt} \% \mathrm{B}-60 \mathrm{wt} \% \mathrm{A}\) 2. Mass fractions of α and β phases: 0.66 and 0.34, respectively. 3. Composition of α phase: 13 wt\(\%\) B and 87 wt\(\%\) A.
02

Applying the Lever Rule

The Lever Rule states that for a given two-phase alloy system, the ratio of the mass fractions of the two phases is equal to the ratio of the difference in composition between the overall alloy composition and each of the phase compositions. Mathematically, this can be represented as follows: \(\frac{W_\alpha}{W_\beta} = \frac{C_\beta - C_{overall}}{C_{overall} - C_\alpha}\) Where \(W_\alpha\) and \(W_\beta\) are mass fractions of α and β phase, respectively, \(C_\alpha\) and \(C_\beta\) are compositions of α and β phase, respectively (weight percent B), \(C_{overall}\) is the overall alloy composition (weight percent B).
03

Substitute given values

We have: \(W_\alpha\) = 0.66 \(W_\beta\) = 0.34 \(C_\alpha\) = 13 wt\(\%\) B \(C_{overall}\) = 40 wt\(\%\) B Let's substitute these values in the equation and solve for \(C_\beta\): \(\frac{0.66}{0.34} = \frac{C_\beta - 40}{40 - 13}\)
04

Solve for \(C_\beta\)

Rearranging the equation and solving for \(C_\beta\): \(C_\beta - 40 = \frac{0.66}{0.34} (40 - 13)\) \(C_\beta - 40 = 1.941176 (27)\) \(C_\beta = 1.941176 (27) + 40\) \(C_\beta = 52.41176 + 40\) \(C_\beta = 92.41176\) Thus, the composition of the β phase is 92.41 wt\(\%\) B and \(7.59 \mathrm{wt} \% \mathrm{A}\).

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Most popular questions from this chapter

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

Cite three variables that determine the microstructure of an alloy.

Two intermetallic compounds, \(A_{3} B\) and \(\mathrm{AB}_{3},\) exist for elements \(\mathrm{A}\) and \(\mathrm{B}\). If the compositions for \(A_{3} B\) and \(A B_{3}\) are 91.0 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 A is zirconium, identify element B.

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.

It is desirable to produce a copper-nickel alloy that has a minimum noncold- 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.A 60 wt\(\% $$\mathrm{Pb}-40\) wt\(\%\) \(\mathrm{Mg}\) alloy is rapidly quenched to room temperature from an elevated temperature in such a way that the hightemperature 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.
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