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17-Phys-B7 Structure of Materials · December 2016

Question 7 of 7: Question VII: Phase Diagram

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 98-Phys-B7 Structure of Materials, National Examination December 2016 — a closed-book examination (Casio or Sharp approved calculators only; all necessary equations, constants, the error-function table and the Cu–Ag phase diagram are 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. This sitting numbers its questions with Roman numerals (Question I–VII) while sub-items inside each question use Arabic numerals (1., 2., 3.).

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, X-ray diffraction); D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed. (Bohr model, de Broglie wavelength, Heisenberg uncertainty).

Question VII: Phase Diagram (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 Cu–Ag eutectic diagram printed on the exam gives the invariant point directly: eutectic temperature $T_E=779^\circ\text{C}$, $\alpha$-phase eutectic composition $C_{\alpha E}=8.0\ \text{wt\%Ag}$, eutectic liquid $C_E=71.9\ \text{wt\%Ag}$, $\beta$-phase eutectic composition $C_{\beta E}=91.2\ \text{wt\%Ag}$.

Check: the liquidus/solidus/solvus compositions AWAY from the labelled eutectic point (needed for the 800 °C tie line and the 700 °C solvus limits) are not printed as numbers anywhere on the exam — they must be read off the curves themselves. They were read here from the printed diagram, calibrated against its own printed axis ticks and the three labelled points above (a self-consistent calibration: the recovered pure-Cu liquidus start, $\approx1094\,{}^\circ\text{C}$, lands within $10\,{}^\circ\text{C}$ of Cu's accepted melting point, $1085\,{}^\circ\text{C}$), then reading the curve position at the two required temperatures. Treat $C_\alpha(800^\circ\text{C})\approx8.2$, $C_L(800^\circ\text{C})\approx68.4$, and the two $700^\circ\text{C}$ solvus limits below as graph-reading estimates (±1 wt%), exactly as a candidate would read them off the printed figure.

[Figure not reproduced: Cu–Ag eutectic phase diagram (redrawn from the exam's own figure) with the 800°C $\alpha+L$ tie line (blue) used in Parts 1 & 4, and the 700°C solvus limits (green) used in Part 2. See the official exam paper.]

Part 1 — Given. Alloy composition $C_0=60\ \text{wt\%Ag}$ (40 wt% Cu) at $800^\circ\text{C}$; tie-line ends $C_\alpha=8.2$, $C_L=68.4\ \text{wt\%Ag}$.

Find. Which phase(s) are present, and their compositions.

  1. Locate the alloy on the diagram. At $800^\circ\text{C}$ (above the $779^\circ\text{C}$ eutectic isotherm), $C_0=60$ lies between the solidus ($C_\alpha=8.2$) and the liquidus ($C_L=68.4$), inside the two-phase $\alpha+L$ field.
  2. Phases present. $$\boxed{\alpha+L}$$ — a solid $\alpha$ (Cu-rich, FCC) phase and a liquid phase coexist.
  3. Phase compositions. Read directly off the tie-line ends: $$C_\alpha\approx\boxed{8.2\ \text{wt\%Ag}\ (91.8\ \text{wt\%Cu})},\qquad C_L\approx\boxed{68.4\ \text{wt\%Ag}\ (31.6\ \text{wt\%Cu})}.$$

Part 2 — Given. $T=700^\circ\text{C}$; solvus limits read from the diagram: $C_\alpha(700^\circ\text{C})\approx5.3\ \text{wt\%Ag}$, $C_\beta(700^\circ\text{C})\approx94.5\ \text{wt\%Ag}$.

Find. Maximum solubility of (a) Cu in Ag, (b) Ag in Cu, at $700^\circ\text{C}$.

  1. (a) Cu in Ag. The $\beta$ (Ag-rich) solvus sits at $C_\beta\approx94.5\ \text{wt\%Ag}$, i.e. $100-94.5=\boxed{5.5\ \text{wt\%Cu}}$ dissolved in solid Ag.
  2. (b) Ag in Cu. The $\alpha$ (Cu-rich) solvus sits directly at $$\boxed{5.3\ \text{wt\%Ag}}$$ dissolved in solid Cu.

The two limits are close in magnitude — a reasonable cross-check, since both boundaries must shrink from their respective eutectic-temperature maxima ($8.0$ and $91.2\to8.8$ wt% of the minority element) toward much smaller values as $T$ drops further from $779^\circ\text{C}$ toward $700^\circ\text{C}$.

Part 3 — Find. Definition of a eutectic reaction, and the specific reaction for Cu–Ag.

A eutectic reaction is an invariant, three-phase, isothermal reaction in which, upon cooling, a liquid of one fixed composition transforms simultaneously into two distinct solid phases of two other fixed compositions (and the reverse melting reaction occurs on heating): $$L\ \underset{\text{heating}}{\overset{\text{cooling}}{\rightleftharpoons}}\ \alpha+\beta.$$

For the Cu–Ag system, using the printed eutectic data: $$\boxed{L\,(71.9\ \text{wt\%Ag})\ \xrightarrow{\ 779^\circ\text{C}\ }\ \alpha\,(8.0\ \text{wt\%Ag})+\beta\,(91.2\ \text{wt\%Ag})}.$$

Part 4 — Given. Alloy composition $C_0=55\ \text{wt\%Ag}$ (45 wt% Cu) at $800^\circ\text{C}$; same tie line as Part 1, $C_\alpha=8.2$, $C_L=68.4\ \text{wt\%Ag}$.

Find. Mass fractions $W_\alpha$, $W_L$ (lever rule).

  1. Lever rule. $$W_\alpha=\frac{C_L-C_0}{C_L-C_\alpha} =\frac{68.4-55}{68.4-8.2}=\frac{13.4}{60.2}=\boxed{0.223\ (22.3\%)}.$$
  2. Liquid fraction. $$W_L=\frac{C_0-C_\alpha}{C_L-C_\alpha} =\frac{55-8.2}{60.2}=\frac{46.8}{60.2}=\boxed{0.777\ (77.7\%)}.$$ Check: $W_\alpha+W_L =0.223+0.777=1.000$.
Final results — Question VII
QuantityValue
Phases, 60 wt%Ag @ 800°C$\alpha$ (8.2 wt%Ag) + L (68.4 wt%Ag)
Max. Cu in Ag @ 700°C5.5 wt%
Max. Ag in Cu @ 700°C5.3 wt%
Eutectic reaction$L(71.9)\to\alpha(8.0)+\beta(91.2)$ at 779°C
$W_\alpha$, 55 wt%Ag @ 800°C22.3%
$W_L$, 55 wt%Ag @ 800°C77.7%
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