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CHEM2522 · Sustainable Chemical Manufacture

Q: Why are there two different average molar masses?

Because a synthetic polymer is a distribution of chain lengths, and how you average depends on what you weight by. The number-average Mn weights every chain equally (it is what colligative and end-group methods 'see'), so the many short chains pull it down. The weight-average Mw weights by mass — the Mi^2 term on top — so long chains count more, and it is what light-scattering methods 'see'. Their ratio, the dispersity Đ = Mw/Mn, tells you how broad the distribution is.

Q: What is the Kuhn length actually telling me?

It repackages a real chain — with fixed bond angles and hindered rotation — as an equivalent freely-jointed chain of N rigid segments each of length b, so the simple random-flight result Ree = N^(1/2) b applies. A stiffer or more sterically crowded backbone (bulkier side groups) has a longer Kuhn length b and therefore fewer, longer segments. In calculations you usually get L = Nb from the contour length and Ree from experiment, then solve for N and b.

Q: What is the difference between SEC and MALDI-ToF?

SEC (size-exclusion chromatography / GPC) separates chains by hydrodynamic volume — the largest elute first — but it is a relative method that needs calibration against standards, so it gives Mn, Mw and Đ relative to those standards. MALDI-ToF is a soft-ionisation mass spectrometry method that is absolute: it measures true masses, with peaks spaced by the monomer mass, and lets you read off DP directly. Use SEC for the distribution and dispersity, MALDI for absolute masses.

Q: Can Sia help me with the molar-mass and Kuhn-length calculations?

Yes. Sia can set up an Mn/Mw/Đ problem, keep the number- versus mass-weighting straight, and check your arithmetic, and it can walk the Kuhn-length random-flight relations (L = Nb, Ree = N^(1/2) b, Rg = Ree/sqrt(6)) step by step on a fresh polymer. It explains the method and checks your working; it does not do graded assessment for you, and University of Sydney academic-integrity rules apply.

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Chapter 7 of 13 · CHEM2522

Introduction to Polymers

Week 7 opens the polymer half of CHEM2522. You learn how a chain's random-flight conformation is described (end-to-end distance, Kuhn length, radius of gyration, contour length, and how these scale with solvent quality), how to quantify a molar-mass distribution with the number- and weight-average masses Mn and Mw and the dispersity Đ, and how these are measured by size-exclusion chromatography (SEC) and MALDI-ToF. The molar-mass and Kuhn-length calculations are prime datasheet-formula exam questions.

In this chapter

What this chapter covers

01Monomer to macromolecule; degree of polymerisation n; how chain length changes physical state
02Chain conformation: the random-flight / freely-jointed model; end-to-end distance and contour length L = Nb
03Kuhn length b and N rigid segments: Ree = N^(1/2) b; radius of gyration Rg = Ree/sqrt(6)
04Solvent scaling Rg ~ m^v: v = 1/2 theta solvent, 3/5 good solvent (expanded coil), 1/3 poor solvent (globule)
05Number-average Mn = sum(ni Mi)/sum(ni); weight-average Mw = sum(ni Mi^2)/sum(ni Mi); dispersity D = Mw/Mn
06Tacticity, branching and crystallinity; glass transition Tg vs melting Tm
07SEC/GPC: separates by hydrodynamic volume (largest elute first), a relative method needing calibration
08MALDI-ToF: soft ionisation, an absolute method; peaks spaced by the monomer mass

Worked example · free

Mn, Mw and dispersity of a two-population polystyrene

Q [4 marks]. A polystyrene sample is an equal-number mixture of two populations: chains of DP = 500 and chains of DP = 1500. The styrene repeat mass is 104.15 g/mol. Using the datasheet formulas, find the molar mass of each population, then Mn, Mw and the dispersity Đ. (4 marks)

+1Molar mass of each population: M(A) = 500 × 104.15 = 52 075 g/mol; M(B) = 1500 × 104.15 = 156 225 g/mol (end groups neglected).
+1Number-average (equal numbers, so a simple mean): Mn = (52 075 + 156 225) ÷ 2 = 208 300 ÷ 2 = 104 150 g/mol.
+1Weight-average with equal numbers n each: Mw = sum(ni Mi^2)/sum(ni Mi) = (52 075^2 + 156 225^2) ÷ (52 075 + 156 225) = 27 118 056 250 ÷ 208 300 = 130 187 g/mol.
+1Dispersity: Đ = Mw ÷ Mn = 130 187 ÷ 104 150 = 1.25. (Đ > 1, as it must be for any real distribution.)

M(A) = 52 075 g/mol, M(B) = 156 225 g/mol; Mn = 104 150 g/mol; Mw = 130 187 g/mol; Đ = Mw/Mn = 1.25.

Sia tip — Watch which average is weighted how: Mn weights by number of chains, Mw weights by mass (it uses Mi^2 on top), so Mw ≥ Mn and Đ ≥ 1 always — if you ever get Đ < 1 you have swapped a formula. Ask Sia to rerun this with an equal-mass (instead of equal-number) mixture so you feel how the weighting changes Mn, Mw and Đ.

Glossary

Key terms

Degree of polymerisation (DP, n): The number of monomer repeat units in a chain; the chain molar mass is roughly DP × (monomer mass) plus end groups.
Kuhn length (b): The length of a rigid statistical segment in the random-flight model; a real chain of contour length L = Nb behaves like N freely-jointed segments, giving end-to-end distance Ree = N^(1/2) b. Bulkier side groups give a larger b.
Radius of gyration (Rg): A size measure of the coil, the root-mean-square distance of segments from the chain's centre of mass; for a random-flight chain Rg = Ree/sqrt(6).
Number-average molar mass (Mn): Mn = sum(ni Mi)/sum(ni), the mean molar mass weighted by the number of chains; dominated by the many small chains.
Weight-average molar mass (Mw): Mw = sum(ni Mi^2)/sum(ni Mi) = sum(wi Mi)/sum(wi), the mean weighted by mass; always ≥ Mn and more sensitive to the large chains.
Dispersity (Đ): Đ = Mw/Mn, the breadth of the molar-mass distribution (≥ 1; = 1 monodisperse). Step-growth polymers approach 2; controlled radical methods reach ≈ 1.1-1.4.

FAQ

Introduction to Polymers FAQ

Why are there two different average molar masses?

Because a synthetic polymer is a distribution of chain lengths, and how you average depends on what you weight by. The number-average Mn weights every chain equally (it is what colligative and end-group methods 'see'), so the many short chains pull it down. The weight-average Mw weights by mass — the Mi^2 term on top — so long chains count more, and it is what light-scattering methods 'see'. Their ratio, the dispersity Đ = Mw/Mn, tells you how broad the distribution is.

What is the Kuhn length actually telling me?

It repackages a real chain — with fixed bond angles and hindered rotation — as an equivalent freely-jointed chain of N rigid segments each of length b, so the simple random-flight result Ree = N^(1/2) b applies. A stiffer or more sterically crowded backbone (bulkier side groups) has a longer Kuhn length b and therefore fewer, longer segments. In calculations you usually get L = Nb from the contour length and Ree from experiment, then solve for N and b.

What is the difference between SEC and MALDI-ToF?

SEC (size-exclusion chromatography / GPC) separates chains by hydrodynamic volume — the largest elute first — but it is a relative method that needs calibration against standards, so it gives Mn, Mw and Đ relative to those standards. MALDI-ToF is a soft-ionisation mass spectrometry method that is absolute: it measures true masses, with peaks spaced by the monomer mass, and lets you read off DP directly. Use SEC for the distribution and dispersity, MALDI for absolute masses.

Can Sia help me with the molar-mass and Kuhn-length calculations?

Yes. Sia can set up an Mn/Mw/Đ problem, keep the number- versus mass-weighting straight, and check your arithmetic, and it can walk the Kuhn-length random-flight relations (L = Nb, Ree = N^(1/2) b, Rg = Ree/sqrt(6)) step by step on a fresh polymer. It explains the method and checks your working; it does not do graded assessment for you, and University of Sydney academic-integrity rules apply.

Study strategy

Exam move

Week 7 is where the datasheet-formula marks concentrate, so drill the two calculation families until they are automatic. For molar mass, always write M(each population) first, then Mn (number-weighted), then Mw (mass-weighted, remember the Mi^2 on top), then Đ = Mw/Mn, and sanity-check Đ ≥ 1. Practise both the equal-number and equal-mass variants because they give different answers from the same masses. For conformation, keep the chain of relations in order — contour length L = Nb, Ree = N^(1/2) b, Rg = Ree/sqrt(6) — and know the solvent scaling exponents (1/2 theta, 3/5 good, 1/3 poor). Learn SEC (relative, needs calibration, largest elute first) versus MALDI (absolute, monomer-spaced peaks) as a clean compare-and-contrast, a common short-answer item.

Working through Introduction to Polymers in CHEM2522? Sia is AskSia’s AI Chemistry tutor — ask any CHEM2522 Introduction to Polymers question and get a clear, step-by-step explanation grounded in how CHEM2522 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.

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