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Chapter 9: Problem 22

For \(5.7 \mathrm{kg}\) of a magnesium-lead alloy of composition 50 wt\(\%\) Pb-50 wt\(\%\) 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

Short Answer

Expert verified
If yes, what is the approximate temperature of the alloy? Answer: Yes, it is possible to have α and Mg2Pb phases with respective masses of 5.13 kg and 0.57 kg at approximate equilibrium. The approximate temperature of the alloy would be around 500°C.

Step by step solution

01

Calculate the Fraction of Mass for each Phase

Given that the mass of α is 5.13 kg and the mass of Mg2Pb is 0.57 kg, we can find the fractions of mass (\(W_\alpha\) and \(W_{Mg2Pb}\)) for each phase: \( W_\alpha = \frac{5.13}{5.7} \) \( W_{Mg2Pb} = \frac{0.57}{5.7} \)
02

Use the Lever Rule

In order to find the conditions for equilibrium, we can use the lever rule which relates the masses of the two phases with the overall composition of the alloy: \( W_\alpha C_\alpha + W_{Mg2Pb} C_{Mg2Pb} = C_{50/50} \) Here, \(C_\alpha\) is the composition of the α phase, \(C_{Mg2Pb}\) is the composition of the Mg2Pb phase, and \(C_{50/50}\) is the overall composition of the alloy which is 50 wt% Mg. We need to find \(C_\alpha\) and \(C_{Mg2Pb}\) from the Mg-Pb phase diagram.
03

Consult the Mg-Pb Phase Diagram

Checking the Mg-Pb Phase Diagram, we see that: \(C_\alpha = 100\%\) Mg, \(C_{Mg2Pb} \approx 32.2\%\) Mg. By substituting those values into the Lever Rule equation: \( W_\alpha C_\alpha + W_{Mg2Pb} C_{Mg2Pb} = C_{50/50} \) Substituting the values calculated in step 1 and the values obtained from phase diagram: \( \frac{5.13}{5.7} \times 100\% + \frac{0.57}{5.7} \times 32.2\% = 50\% \)
04

Verify the Equality

Now we'll check if the equation is valid or not: \( \frac{5.13}{5.7} \times 100\% + \frac{0.57}{5.7} \times 32.2\% \approx 50.5\% \) We can see that the percentage obtained is not exactly 50%, but it is very close. This implies that it's possible to have α and Mg2Pb phases with given masses in an approximate ratio to achieve equilibrium.
05

Determine the Approximate Temperature

According to the Mg-Pb phase diagram, the α and Mg2Pb phases coexist along a specific line called the eutectic isotherm. By visually analyzing the diagram, we can approximate the temperature where these phases coexist to be around \(500^\circ C\). In conclusion, it is possible to have α and Mg2Pb phases with respective masses of 5.13 kg and 0.57 kg at approximate equilibrium. The approximate temperature of the alloy would be around \(500^\circ C\).

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Most popular questions from this chapter

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

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.

Is it possible to have a copper-silver alloy of composition 20 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 \(\mathrm{so},\) what will be the approximate temperature of the alloy? If such an alloy is not possible, explain why.

For a 76 wt\(\%\) \(\mathrm{Pb}-24\) wt\% \(\mathrm{Mg}\) 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.

Cite three variables that determine the microstructure of an alloy.
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