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Chapter 9: Problem 62

The mass fraction of eutectoid ferrite in an iron-carbon alloy is \(0.82\). 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.

Short Answer

Expert verified
If so, what is the composition? A: Yes, we can determine the composition of the alloy. The composition of the alloy is approximately 1.062 wt.% Carbon.

Step by step solution

01

Understand the eutectoid ferrite in iron-carbon alloy

In iron-carbon alloys, eutectoid ferrite is one of the phases formed during the eutectoid reaction. The eutectoid reaction occurs at the eutectoid point on the iron-carbon phase diagram, where Austenite (0.76 wt.% C) transforms into Ferrite (0.022 wt.% C) and Cementite (6.7 wt.% C). So, eutectoid ferrite refers to the ferrite phase not exceeding 0.022 wt.% C content.
02

Apply the Lever Rule

The Lever Rule is a mass balance technique used to find the relative mass fractions of two phases in a two-phase region. The Lever rule can be expressed as: \(\textrm{Mass fraction of phase 1} = \frac{\textrm{Composition of phase 2} - \textrm{Average composition}}{\textrm{Composition of phase 2} - \textrm{Composition of phase 1}}\) Since we are given the mass fraction of eutectoid ferrite, we must determine the composition of the other phase which is eutectoid cementite. Given mass fraction of eutectoid ferrite (\(w_\textrm{ferrite}\)): \(0.82\). We will use the lever rule to determine the mass fraction of eutectoid cementite (\(w_\textrm{cementite}\)) and then find the composition of the alloy.
03

Calculate the mass fraction of eutectoid cementite

To find the mass fraction of eutectoid cementite, we can simply subtract 1 minus the mass fraction of eutectoid ferrite. \(w_\textrm{cementite} = 1 - w_\textrm{ferrite} = 1 - 0.82 = 0.18\) Now we have the mass fraction of both phases, eutectoid ferrite (0.82) and eutectoid cementite (0.18).
04

Determine the composition of the alloy

We can now apply the lever rule to find the average composition (\(C_\textrm{avg}\)) of the alloy. Using the lever rule with mass fractions: \(w_\textrm{ferrite} = \frac{C_\textrm{cementite} - C_\textrm{avg}}{C_\textrm{cementite} - C_\textrm{ferrite}}\) Substituting the known values, we get: \(0.82 = \frac{0.067 - C_\textrm{avg}}{0.067 - 0.00022}\) Solving for \(C_\textrm{avg}\): \(C_\textrm{avg} = 0.067 - 0.82 (0.067 - 0.00022) \approx 0.01062\) The composition of the alloy is about 1.062 wt.% Carbon.

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Most popular questions from this chapter

For a copper-silver alloy of composition 25 \(\mathrm{wt} \% \mathrm{Ag}-75 \mathrm{wt} \% \mathrm{Cu}\) and at \(775^{\circ} \mathrm{C}\left(1425^{\circ} \mathrm{F}\right)\), do the following: (a) Determine the mass fractions of \(\alpha\) and \(\beta\) phases. (b) Determine the mass fractions of primary \(\alpha\) and eutectic microconstituents. (c) Determine the mass fraction of eutectic \(\alpha\).

For alloys of two hypothetical metals \(\mathrm{A}\) and B, there exist an \(\alpha\), A-rich phase and a \(\beta\), Brich 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. $$ \begin{array}{lcc} \hline \text { Alloy Composition } & \text { Fraction } & \text { Fraction } \\\ & \alpha \text { Phase } & \beta \text { Phase } \\ \hline 60 \mathrm{wt} \% \mathrm{~A}-40 \mathrm{wt} \% \text { B } & 0.57 & 0.43 \\ 30 \mathrm{wt} \% \text { A-70 wt \% B } & 0.14 & 0.86 \\ \hline \end{array} $$

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

A \(30 \mathrm{wt} \% \mathrm{Sn}-70 \mathrm{wt} \% \mathrm{~Pb}\) alloy is heated to a temperature within the \(\alpha\) + liquid phase region. If the mass fraction of each phase is \(0.5\), estimaten (a) The temperature of the alloy (b) The compositions of the two phases

The mass fraction of \(e\) utectoid 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.
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