Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 17-Phys-B7 Structure of Materials, National Examination
December 2018 — a closed-book examination (Casio or Sharp approved calculators only; all
necessary equations, constants and diagrams supplied in the paper's own appendix). Candidates
attempt any five of the seven questions, each worth 20 marks; every question is nonetheless
answered in full below so the paper remains a complete study resource.
Reference texts. W. D. Callister Jr. & D. G. Rethwisch, Materials
Science and Engineering: An Introduction, 10th ed. (atomic bonding, crystal structure and
packing, point defects, diffusion, dislocations and slip, mechanical properties, phase diagrams
and the lever rule, precipitation hardening, X-ray diffraction).
Check — two points on the printed paper. (1) Question I.2
prints the ion as "Cl- (Z = 16)"; Z = 16 is sulfur, not chlorine
(Cl is Z = 17) — a printed typo. The
electronic structure below uses the correct Z = 17 for chlorine. (2) Question III.1(d)
prints the hexagonal plane as $(2\bar{2}10)$, i.e. Miller–Bravais indices $h=2,\,k=-2,\,i=1$;
the third index of a valid Miller–Bravais symbol is never independent — it is fixed by
$i=-(h+k)$, here $i=-(2-2)=0$, not 1. Reducing the self-consistent index $(2\bar200)$ by its common
factor of 2 gives $(1\bar100)$, a standard prism-type plane. The drawing below uses the
symmetry-equivalent, non-degenerate face $(10\bar10)$ of the same $\{1\bar100\}$ family (chosen
because it renders visibly in the cell projection used here) and states this substitution
explicitly.
Check — figure-label correction. Point y cannot lie in the "α+γ" field for a 1.5 wt% C alloy (1.5 wt% C is above the 0.76 wt% eutectoid composition, so this is a HYPEREUTECTOID steel and its cooling path never re-enters the α+γ field at all). The printed diagram shows point y sits between the 727°C eutectoid line and the γ/(γ+Fe₃C) solvus ("Acm" line) that runs from (2.14 wt%C, 1147°C) down to (0.76 wt%C, 727°C) — i.e. y is in the two-phase γ+Fe₃C field, consistent with question 4 asking for "pearlite and PROEUTECTOID CEMENTITE" (not proeutectoid ferrite, which would only apply to a hypoeutectoid alloy below 0.76 wt% C). This field is used throughout below.
Part 1 — invariant points and reactions. The diagram has THREE invariant (three-phase, zero-degree-of-freedom) horizontal lines:
Invariant points on the Fe–Fe₃C diagram
Reaction
Temperature
Composition
Peritectic (δ+L→γ)
1493°C
δ and L meet near 0.1–0.5 wt% C (not separately labelled on this print)
Eutectic
1147°C
4.30 wt% C
Eutectoid
727°C
0.76 wt% C
The question specifically asks for the eutectic and eutectoid REACTIONS:
(the eutectoid product, alternating lamellae of α and Fe₃C, is given the special name "pearlite").
Part 2 — phase(s) at point x. At 1.5 wt% C and 1100°C, point x lies inside the single-phase γ (austenite) field (bounded below by the 912°C/727°C region and above/right by the γ/(γ+L) and γ/(γ+Fe₃C) boundaries, which at 1.5 wt% C sit well above 1100°C and well below 1100°C respectively). Because only ONE phase is present, its composition must equal the overall alloy composition:
Point x
Phase present
Composition
γ (austenite) — single phase
1.5 wt% C (= overall alloy composition)
Part 3 — microstructure evolution on cooling. Given. 1.5 wt% C alloy (hypereutectoid, since 1.5 > 0.76 wt% C), cooled along the dashed line from 1100°C (point x) through point y to point z below 727°C. Find. A schematic microstructure at each point. Approach. Track which phase field each point falls in and, for point z, apply the standard hypereutectoid-steel cooling sequence (proeutectoid Fe₃C nucleates at the prior-austenite grain boundaries first, then the remaining austenite — whose composition has drifted down the Acm line to the eutectoid 0.76 wt% C — transforms entirely to pearlite at 727°C).
At point x (1100°C). Single-phase γ: uniform, equiaxed austenite grains, no second phase anywhere.
At point y (between the Acm line and 727°C). Two-phase γ+Fe₃C: proeutectoid cementite has begun precipitating as a thin, continuous film decorating the PRIOR γ grain boundaries, with untransformed γ still filling the grain interiors.
At point z (below 727°C). Two-phase α+Fe₃C, but with a distinctive hypereutectoid microstructure: the grain-boundary proeutectoid-cementite network laid down above 727°C is retained, and the grain INTERIORS — which were still γ at 0.76 wt% C the instant the alloy crossed 727°C — have all transformed to pearlite (fine alternating lamellae of α and Fe₃C).
At point x: single-phase γ (austenite), uniform grains.
At point y: γ grains outlined by a thin proeutectoid-Fe₃C grain-boundary film.
At point z: pearlite (lamellar α+Fe₃C, magnified inset) filling the former γ grains, still outlined by the retained proeutectoid-Fe₃C network.
Find. Mass fraction of pearlite and of proeutectoid cementite at point z.
Approach. Apply the lever rule across the tie line just ABOVE 727°C (the proeutectoid-cementite amount is fixed at that instant and does not change further as the remaining γ converts to pearlite on crossing 727°C); the tie line spans from the eutectoid composition (the composition of the γ that is about to become pearlite) to the cementite composition.
Proeutectoid cementite fraction. By the lever rule, using the segment of the tie line on the γ-rich (pearlite) side:
$$W_{\text{Fe}_3\text{C}^\prime} = \frac{C_0-C_{\text{eutectoid}}}{C_{\text{Fe}_3\text{C}}-C_{\text{eutectoid}}} = \frac{1.5-0.76}{6.70-0.76} = \frac{0.74}{5.94} = \boxed{0.1246\ (12.5\%)}.$$
Pearlite fraction. The remainder of the microstructure is pearlite:
$$W_{\text{pearlite}} = 1-W_{\text{Fe}_3\text{C}^\prime} = 1-0.1246 = \boxed{0.8754\ (87.5\%)}.$$
