Question 5 of 8: Phase Diagram: Iron–Carbon (20 marks)

Question V — Phase Diagram: Iron–Carbon (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The printed Fe–Fe₃C diagram's own labelled values: pure-Fe melting 1538°C; peritectic 1493°C; $\delta\to\gamma$ transus (0%C) 1394°C; eutectic 1147°C at 2.14/4.30 wt%C; $\gamma\to\alpha$ transus (0%C) 912°C; eutectoid 727°C at 0.76/0.022 wt%C; cementite fixed at 6.70 wt%C.

[Figure not reproduced: Fe–Fe₃C metastable phase diagram redrawn from the paper's own printed labels (all values read directly from the printed diagram). See the official exam paper.]

V.1 — Invariant reactions

Approach. An invariant reaction in a binary system is a three-phase, zero-degree-of-freedom equilibrium (Gibbs phase rule: $F=C-P+1=2-3+1=0$ at fixed pressure), so it occurs at one fixed temperature and fixed compositions for all three phases. The Fe–Fe₃C system has three: a peritectic (low-C, high-T), a eutectic, and a eutectoid.

V.2 — Maximum carbon solubility in α-ferrite and austenite

Approach. Both maxima occur where the relevant phase field is widest along the composition axis — for $\alpha$ this is exactly at the eutectoid temperature (727°C), and for $\gamma$ this is at the eutectic temperature (1147°C), both read directly off the diagram's own printed labels.

1. α-ferrite. The $\alpha/(\alpha+\gamma)$ solvus reaches its rightmost extent exactly at the eutectoid point: $$\boxed{C_{\alpha,max}=0.022\ \text{wt\%C at }727^{\circ}\text{C}}$$
2. γ-austenite. The $\gamma/(\gamma+L)$ and $\gamma/(\gamma+\text{Fe}_3\text{C})$ boundaries meet at the eutectic point, the widest extent of the $\gamma$ field: $$\boxed{C_{\gamma,max}=2.14\ \text{wt\%C at }1147^{\circ}\text{C}}$$

The huge difference (0.022 vs. 2.14 wt%C, a factor of ≈100) is the physical basis of steel heat treatment: austenite can dissolve nearly 100× more interstitial carbon than ferrite because the FCC $\gamma$ structure's octahedral interstices are larger than BCC $\alpha$'s — quenching austenite traps that carbon supersaturated in solution (forming martensite) because the low-temperature equilibrium phase (ferrite) cannot hold nearly as much.

V.3 — Lever rule: 99.65 wt%Fe–0.35 wt%C, just below 727°C

Given. $C_0=0.35$ wt%C; just below 727°C the two-phase field is $\alpha+\text{Fe}_3\text{C}$ with $C_\alpha=0.022$, $C_{Fe_3C}=6.70$ wt%C.

Find. $W_\alpha$, $W_{Fe_3C}$.

Approach. Apply the lever rule directly across the $\alpha+\text{Fe}_3\text{C}$ tie-line spanning the full 0.022–6.70 wt%C range.

1. Lever-rule fractions. $$W_\alpha=\frac{C_{Fe_3C}-C_0}{C_{Fe_3C}-C_\alpha}=\frac{6.70-0.35}{6.70-0.022}=\frac{6.35}{6.678}=\boxed{0.951\ (95.1\%)}$$ $$W_{Fe_3C}=\frac{C_0-C_\alpha}{C_{Fe_3C}-C_\alpha}=\frac{0.35-0.022}{6.678}=\frac{0.328}{6.678}=\boxed{0.0491\ (4.91\%)}$$
2. Check. $W_\alpha+W_{Fe_3C}=0.951+0.0491=1.000$. ✓

This is a hypoeutectoid steel far on the ferrite-rich side of the eutectoid composition (0.35 vs. 0.76 wt%C), so the result — overwhelmingly ferrite (95.1%) with only a small cementite fraction (4.9%) — is exactly the expected qualitative picture: most of this alloy's microstructure just below 727°C is proeutectoid ferrite plus a comparatively small volume of pearlite (whose own overall composition also averages to 0.35 wt%C once its internal $\alpha$/Fe₃C split is folded back in).

V.4 — Equilibrium microstructures of 1 wt%C steel

Approach. Locate $C_0=1$ wt%C (hypereutectoid, since 1 > 0.76 eutectoid) on the phase diagram at each of the three temperatures and read off which phase field it falls in.

1. (a) 1000°C. Since $C_0=1$ wt%C $< C_{\gamma,max}=2.14$ wt%C at this temperature (below the eutectic but the $\gamma$ field is still wide), the alloy is entirely single-phase $\boxed{\gamma\text{ (austenite)}}$ — no proeutectoid phase yet.
2. (b) 750°C. This lies between the A$_{cm}$ (the $\gamma/(\gamma+\text{Fe}_3\text{C})$ boundary, which at 1 wt%C sits above 727°C) and the 727°C eutectoid line, so the alloy is hypereutectoid two-phase: $\boxed{\gamma+\text{proeutectoid Fe}_3\text{C (grain-boundary network)}}$.
3. (c) 500°C. Below 727°C, all remaining austenite (which had been enriched back down to the eutectoid composition 0.76 wt%C by continued Fe₃C rejection between 750°C and 727°C) transforms via the eutectoid reaction to pearlite: $\boxed{\text{proeutectoid Fe}_3\text{C network}+\text{pearlite }(\alpha+\text{Fe}_3\text{C lamellae})}$.