21-Mat-A3 Structure and Characterization of Materials · Dec-12-Mtl-A3 2018
Question 7 of 8
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, December 2018 — 12-Mtl-A3, Structure and Characterization of Materials. Three hours, open book, any non-communicating calculator permitted. Eight questions constitute a complete exam paper; all eight are solved here.
Reference texts. The answers below are keyed to the standard undergraduate materials-science references recommended for this syllabus code:
W. D. Callister Jr. and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. — crystal structure and defects (Ch. 4), mechanical properties and fracture (Ch. 6, 8), phase diagrams (Ch. 9), diffraction (Ch. 3), polymer structure and mechanical behaviour (Ch. 14–15).
B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction, 3rd ed. — structure-factor selection rules for cubic lattices.
Part (a) — copolymer architectures. A copolymer contains two (or more) distinct repeat-unit (mer) types, denoted here A and B, arranged along the chain in one of several characteristic patterns:
Random copolymer: the two mer types are distributed with no regular sequence along the chain, e.g. $\cdots$A–A–B–A–B–B–A$\cdots$, governed by the relative reactivity and feed ratio of the two monomers.
Alternating copolymer: the two mer types strictly alternate, A–B–A–B–A–B$\cdots$, a special (and comparatively rare) limiting case of ordered copolymerization.
Block copolymer: long, chemically homogeneous runs (blocks) of one mer type are joined end-to-end to runs of the other, e.g. A–A–A–A–A–B–B–B–B–B$\cdots$; each block can retain properties characteristic of its own homopolymer (e.g. microphase separation in SBS thermoplastic elastomers).
Graft copolymer: a main chain (backbone) of one mer type has side chains (branches) of the other mer type covalently attached at intervals along its length, giving a comb-like architecture.
Part (b) Given.
Quantity
Value
Number-average degree of polymerization, $\bar X_n$
2500
Weight-average degree of polymerization, $\bar X_w$
2000
Number-average molecular weight, $\bar M_n$
81,500 g/mol
Weight-average molecular weight, $\bar M_w$
67,200 g/mol
Find. Whether this data set can describe a real random poly(ethylene-propylene) copolymer.
Approach. Check whether the stated averages satisfy the one relationship that must always hold for any real molecular-weight distribution: $\bar M_w\geq\bar M_n$ (equivalently $\bar X_w\geq\bar X_n$), before even considering the ethylene/propylene composition.
Test the fundamental inequality. For any distribution of chain lengths, the weight average is a second-moment (variance-weighted) average and the number average is a first-moment (simple) average; it is a basic statistical identity that $\bar M_w/\bar M_n = \bar X_w/\bar X_n = \text{PDI}\geq1$, with equality only for a perfectly monodisperse (every chain identical) sample. Checking the given numbers: $$\text{PDI (from }X\text{)} = \frac{\bar X_w}{\bar X_n} = \frac{2000}{2500} = 0.80$$ $$\text{PDI (from }M\text{)} = \frac{\bar M_w}{\bar M_n} = \frac{67{,}200}{81{,}500} = 0.825$$
Conclusion. Both computed ratios are less than 1, which is impossible for any real polymer sample — a weight-average value can never fall below the corresponding number-average value, regardless of composition or copolymer architecture. $$\boxed{\text{Not possible: both stated PDIs} < 1}$$
Cross-check for completeness (not the deciding factor, but consistent). The implied average mer masses, $\bar M_n/\bar X_n=81{,}500/2500=32.6$ g/mol and $\bar M_w/\bar X_w=67{,}200/2000=33.6$ g/mol, both fall sensibly between pure polyethylene's mer mass (28.05 g/mol) and pure polypropylene's (42.08 g/mol) — e.g. 32.6 g/mol implies about 32% propylene by mole fraction — so the composition numbers alone look plausible. The data set is rejected purely on the $\bar M_w\geq\bar M_n$ statistical requirement, independent of whether the implied composition is chemically reasonable.
