21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2017 — 10-Met-A4, Structure of Materials. Three hours, closed book, one approved calculator (Casio or Sharp). Seven questions of 20 marks each (Roman numerals I–VII); the rubric asks for any five, with only the first five in the answer book marked. All seven are solved here, because this set is a study resource rather than an exam script. All necessary equations, constants and an error-function table are provided in the exam's own appendix (reproduced where used below).
Note on the exam title. The printed exam header reads 10-Met-A4, Structure of Materials. Only parts of Question VI (grain-size strengthening) touch mechanical properties directly; the paper as a whole is a broad introductory materials-science survey — electron structure, bonding, crystal structure, crystallographic planes, solid solubility, XRD, diffusion and phase diagrams — and is answered as such below.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
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.
A perfectly defect-free crystal would have the lowest possible internal energy, but it would also have zero configurational entropy. The Gibbs free energy of the crystal, $G=H-TS$, is minimized (at any $T>0$) not at zero defects but at a small, finite equilibrium defect concentration: introducing the first few vacancies costs relatively little enthalpy $H$ (energy $Q_D$ per defect) while producing a very large increase in configurational entropy $S$ (many ways to arrange a few vacancies among many sites), so $-TS$ initially falls faster than $H$ rises and $G$ decreases. Beyond the equilibrium concentration, further vacancies cost more enthalpy than the now-saturating entropy term can offset, and $G$ turns back up. The result, given directly in the appendix, is the Arrhenius-type equilibrium defect density $$N_D=N\exp\!\left(-\frac{Q_D}{kT}\right)$$ where $N$ is the total number of atomic sites and $Q_D$ the defect formation energy. Increasing temperature increases the equilibrium defect density, because more thermal energy $kT$ is available to pay the formation-energy cost $Q_D$.
A substitutional impurity atom occupies (replaces) a regular lattice site of the host, displacing a host atom; it forms when the impurity atom's size and chemistry are reasonably close to the host's (e.g. Ni dissolved in Cu — both FCC, similar atomic radii, forming a continuous substitutional solid solution; also Zn or Pb in Cu, Question IV.3). An interstitial impurity atom instead squeezes into one of the small open spaces (interstices) between the host atoms on their regular sites, without displacing any host atom; it forms only when the impurity atom is much smaller than the host (e.g. carbon dissolved interstitially in $\gamma$-iron, austenite, the basis of carburizing and steel hardening, Question VI.1). Schematically, a substitutional atom simply appears as an oversized or undersized sphere sitting on the regular lattice grid in place of a host atom, while an interstitial atom appears as a small sphere squeezed between the regular lattice sites, locally straining the surrounding lattice outward.
The extent to which one element dissolves substitutionally in another (Hume-Rothery rules) is governed by four factors: (1) atomic size — appreciable solid solubility requires the atomic radii to differ by less than about 15%; larger differences produce excessive lattice strain that the solvent cannot accommodate. (2) Crystal structure — extensive (complete) solubility requires the solute and solvent to share the same crystal structure. (3) Electronegativity — the more electropositive one element and the more electronegative the other, the greater the tendency to form a compound instead of a solid solution. (4) Valence — other factors equal, a metal of lower valence more readily dissolves a metal of higher valence than the reverse.
| Element | Atom radius (nm) | Size mismatch vs. Cu | Crystal structure | Electronegativity | Valence |
|---|---|---|---|---|---|
| Zinc | 0.133 | +3.9% | HCP (differs from Cu's FCC) | 1.7 (vs. Cu 1.8) | +2 |
| Lead | 0.175 | +36.7% | FCC (matches Cu) | 1.6 (vs. Cu 1.8) | +2, +4 |
Zinc satisfies the dominant Hume-Rothery criterion — its size mismatch with Cu is only $(0.133-0.128)/0.128=3.9\%$, comfortably inside the 15% rule — and its electronegativity (1.7) is very close to Cu's (1.8), discouraging compound formation. Its crystal structure (HCP) differs from Cu's (FCC), which caps the solubility short of being complete, but the favourable size and electronegativity factors dominate. Lead, by contrast, has a size mismatch of $(0.175-0.128)/0.128=36.7\%$ — more than double the 15% threshold — despite matching Cu's FCC structure; this large size mismatch alone is normally sufficient to suppress solid solubility almost entirely. Predicted result: zinc has substantially greater solid solubility in copper than lead does, which matches the real Cu–Zn system (brass, extensive solid solution, up to ∼35 wt% Zn at room temperature) against the real Cu–Pb system (essentially negligible mutual solid solubility, forming instead a monotectic with discrete Pb-rich particles).
