---
title: "Polymers: Synthesis, Structure and Properties"
book: "University Chemistry — Year 2"
subject: chemistry
language: en
chapter: 29
exercises: 12
source: https://one-course.com/books/chemistry/3/en/chapter/29-polymers-synthesis-structure-and-properties
license: CC-BY-NC-SA-4.0
credit: "One Chemistry Book, One Course (one-course.com)"
---

# Chapter 29 — Polymers: Synthesis, Structure and Properties

Nylon stockings went on sale in 1939. They came out of a laboratory that had set out, a few years earlier, to understand how small molecules join into long ones, and that had learnt along the way a lesson that surprises every chemist the first time: for a [polymer](#def-b2-polymer-synthesis-polymer) made by linking molecules end to end, a 99 % yield is not enough. This chapter explains why, describes the two ways in which chains grow, and connects the structure of the chains to the properties of the materials.

**You already know.**

The school volume: [polymers](#def-b2-polymer-synthesis-polymer), [monomers](#def-b2-polymer-synthesis-polymer) and [repeat units](#def-b2-polymer-synthesis-polymer), addition and condensation [polymers](#def-b2-polymer-synthesis-polymer), thermoplastics and thermosets. The Year 1 volume: radicals and homolysis, carbocations and carbanions, the steady-state approximation. [Chapter 24](https://one-course.com/books/chemistry/3/en/chapter/24-carboxylic-acid-derivatives#ch-b2-acyl-substitution): esters and amides. [Chapter 20](https://one-course.com/books/chemistry/3/en/chapter/20-organometallic-catalysis-elementary-steps-and-cycles#ch-b2-catalytic-cycles): insertion into a metal–carbon bond.

## 29.1 Chains and their molar masses

**Definition 29.1 (Polymer).**

A *polymer* is a substance made of macromolecules, each built by the repeated linking of small molecules, the *monomers*. The *repeat unit* is the smallest group of atoms whose repetition makes up the chain; the *degree of polymerisation* $X$ of a chain is the number of monomer-derived units it contains.

A sample of a synthetic [polymer](#def-b2-polymer-synthesis-polymer) is never made of identical chains: it is a mixture of chains of different lengths, and its molar mass is an average, which depends on how the chains are counted.

**Definition 29.2 (Molar-mass averages).**

For a sample containing $N_i$ chains of molar mass $M_i$, the *number-average molar mass* and the *mass-average molar mass* are

$$
\bar M_n = \frac{\sum_i N_i M_i}{\sum_i N_i}, \qquad
\bar M_w = \frac{\sum_i N_i M_i^2}{\sum_i N_i M_i} = \sum_i w_i M_i,
$$

where $w_i$ is the mass fraction of the chains of mass $M_i$. The *dispersity* $\text{\DH} = \bar M_w/\bar M_n$ is at least 1, and equal to 1 only if all chains have the same mass. The averages $\bar X_n$ and $\bar X_w$ of the [degree of polymerisation](#def-b2-polymer-synthesis-polymer) are defined in the same way.

The number average weights each chain once; the mass average weights each chain by its mass, so the long chains count more. That $\text{\DH} \ge 1$ is the inequality $\left(\sum N_i M_i\right)^2 \le
\sum N_i \sum N_i M_i^2$ (Cauchy–Schwarz), with equality only for equal $M_i$.

**Method 29.3 (Computing the averages).**

1. Turn the data into amounts $N_i$ (or numbers of chains) and masses $M_i$ ; from mass fractions, $N_i \propto w_i/M_i$ .
2. $\bar M_n = \sum N_i M_i / \sum N_i$ ; equivalently, $1/\bar M_n = \sum w_i/M_i$ .
3. $\bar M_w = \sum w_i M_i$ .
4. $\text{\DH} = \bar M_w/\bar M_n$ ; check that $\bar M_w \ge \bar M_n$ .

## 29.2 Step growth and chain growth

**Definition 29.4 (Step and chain growth).**

In a *step-growth polymerisation*, any two molecules carrying complementary functional groups ([monomers](#def-b2-polymer-synthesis-polymer), oligomers or long chains) can react, and the chains lengthen by joining one another: polyesters, polyamides. In a *chain-growth polymerisation*, a reactive centre (radical, ion or metal–carbon bond) adds [monomer](#def-b2-polymer-synthesis-polymer) molecules one at a time to the end of a growing chain: the [polymers](#def-b2-polymer-synthesis-polymer) of alkenes.

**Method 29.5 (Step or chain?).**

1. A [monomer](#def-b2-polymer-synthesis-polymer) with two functional groups that react with each other’s partner (acid and alcohol, acid and amine), often releasing a small molecule: step growth.
2. A [monomer](#def-b2-polymer-synthesis-polymer) with a $\ce{C=C}$ double bond (or a strained ring) and an initiator: chain growth.
3. Check with the course of the reaction: in step growth, [monomer](#def-b2-polymer-synthesis-polymer) disappears early and long chains appear only at the very end; in chain growth, long chains appear from the start while [monomer](#def-b2-polymer-synthesis-polymer) is consumed gradually.

### Step growth

Take a balanced mixture of $\ce{A-A}$ and $\ce{B-B}$ [monomers](#def-b2-polymer-synthesis-polymer) (a diamine and a diacid), or a single $\ce{A-B}$ [monomer](#def-b2-polymer-synthesis-polymer). Call $p$ the *extent of reaction*: the fraction of the functional groups of one kind that have reacted.

**Theorem 29.6 (Carothers equation).**

In a [step-growth polymerisation](#def-b2-polymer-synthesis-step-chain) of exactly balanced bifunctional [monomers](#def-b2-polymer-synthesis-polymer), the number-average [degree of polymerisation](#def-b2-polymer-synthesis-polymer), counted in [monomer](#def-b2-polymer-synthesis-polymer) units, is

$$
\bar X_n = \frac{1}{1 - p}.
$$

**Proof.** Start with $N_0$ [monomer](#def-b2-polymer-synthesis-polymer) molecules, carrying $N_0$ groups $\ce{A}$ and $N_0$ groups $\ce{B}$. Each reaction between an $\ce{A}$ and a $\ce{B}$ joins two molecules into one, so it lowers the number of molecules by one. After $pN_0$ reactions there remain $N = N_0 - pN_0 = N_0(1-p)$ molecules sharing the $N_0$ [monomer](#def-b2-polymer-synthesis-polymer) units: $\bar X_n = N_0/N = 1/(1-p)$. ∎

**Corollary 29.7 (Imbalance).**

If the groups are not balanced, with $r = N_{\ce{A}}/N_{\ce{B}} \le 1$ and $p$ the extent of reaction of the minority groups $\ce{A}$,

$$
\bar X_n = \frac{1 + r}{1 + r - 2rp}, \qquad \text{at most } \frac{1+r}{1-r} \text{ when } p = 1.
$$

**Proof.** There are $(N_{\ce{A}} + N_{\ce{B}})/2$ bifunctional [monomer](#def-b2-polymer-synthesis-polymer) molecules at the start. Each linear molecule has two chain ends, each an unreacted group; after reaction there remain $N_{\ce{A}}(1-p)$ groups $\ce{A}$ and $N_{\ce{B}} - pN_{\ce{A}}$ groups $\ce{B}$, hence $N = [N_{\ce{A}}(1-p) + N_{\ce{B}} -
pN_{\ce{A}}]/2$ molecules. Dividing, with $N_{\ce{B}} = N_{\ce{A}}/r$: $\bar X_n = (1 + 1/r)/(1 - 2p + 1/r) = (1+r)/(1 + r - 2rp)$. With $r = 1$ this is the Carothers equation. ∎

The equation explains the hook. At $p = 0.99$, a “99 % yield” of the linking reaction, $\bar X_n =
100$; a useful fibre needs about that or more, so the linking must go beyond 99 %, with pure [monomers](#def-b2-polymer-synthesis-polymer) in exactly equal amounts and the small molecule released (water, methanol) removed to keep pushing the equilibrium. An excess of 1 % of one [monomer](#def-b2-polymer-synthesis-polymer) caps $\bar X_n$ at 201 even at complete [conversion](https://one-course.com/books/chemistry/3/en/chapter/5-continuous-reactors-and-industrial-processes#def-b2-continuous-reactors-conversion).

**Theorem 29.8 (Flory distribution).**

In a [step-growth polymerisation](#def-b2-polymer-synthesis-step-chain) at extent $p$, if every group has the same reactivity whatever the length of its chain, the number fraction and the mass fraction of chains of $x$ units are

$$
n_x = (1-p)\,p^{\,x-1}, \qquad w_x = x(1-p)^2 p^{\,x-1},
$$

and $\bar X_n = 1/(1-p)$, $\bar X_w = (1+p)/(1-p)$, so that $\text{\DH} = 1 + p$, close to 2 at high [conversion](https://one-course.com/books/chemistry/3/en/chapter/5-continuous-reactors-and-industrial-processes#def-b2-continuous-reactors-conversion).

**Proof.** Follow a chain from one end (an $\ce{A-B}$ [monomer](#def-b2-polymer-synthesis-polymer), for simplicity). Each link along it is formed with probability $p$, independently. The chain has exactly $x$ units if the first $x-1$ links are formed and the next is not: probability $p^{\,x-1}(1-p)$, which is $n_x$. With $\sum_{x\ge1} p^{\,x-1} =
1/(1-p)$ and its derivatives $\sum x p^{\,x-1} = 1/(1-p)^2$ and $\sum x^2 p^{\,x-1} = (1+p)/(1-p)^3$: $\sum n_x = 1$, $\bar X_n = \sum x n_x = 1/(1-p)$, $w_x = x n_x/\bar X_n = x(1-p)^2p^{\,x-1}$, and $\bar X_w = \sum x w_x = (1-p)^2(1+p)/(1-p)^3 = (1+p)/(1-p)$. ∎

![Step-growth polymerisation (models). Left: Flory mass distributions (); the peak sits near X_n and the distribution widens as it moves out. Right: the Carothers equation; X_n stays small until p is very close to 1.](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-2e251ee6aba3.svg)

![Step-growth polymerisation (models). Left: Flory mass distributions (); the peak sits near X_n and the distribution widens as it moves out. Right: the Carothers equation; X_n stays small until p is very close to 1.](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-c607d1d94c3d.svg)

*[Step-growth polymerisation](#def-b2-polymer-synthesis-step-chain) (models). Left: Flory mass distributions ([Theorem 29.8](#thm-b2-polymer-synthesis-flory)); the peak sits near $\bar X_n$ and the distribution widens as it moves out. Right: the Carothers equation; $\bar X_n$ stays small until $p$ is very close to 1.*

### Radical chain growth

**Definition 29.9 (Steps of a chain polymerisation).**

A radical chain polymerisation runs in three kinds of steps. In *initiation*, a *radical initiator* (a molecule with a weak bond, such as a peroxide or an azo compound) splits on heating into radicals, one of which adds to a [monomer](#def-b2-polymer-synthesis-polymer). In *propagation*, the radical at the chain end adds a [monomer](#def-b2-polymer-synthesis-polymer) molecule and remains a radical. In *termination*, two radicals destroy each other, by combination (they bond) or disproportionation (one takes a hydrogen atom from the other).

*Radical polymerisation of styrene. The initiator $\ce{I-I}$ (dibenzoyl peroxide, for instance) splits into two radicals; each adds to the $\ce{CH2}$ end of styrene, giving the more stable benzylic radical; [propagation](#def-b2-polymer-synthesis-chain-steps) repeats the addition thousands of times; two chain radicals combine to end both chains. The dot marks the unpaired electron.*

**Theorem 29.10 (Rate of radical polymerisation).**

With an initiator decomposing with rate constant $k_d$ and efficiency $f$ (the fraction of radicals that start a chain), [propagation](#def-b2-polymer-synthesis-chain-steps) constant $k_p$ and [termination](#def-b2-polymer-synthesis-chain-steps) constant $k_t$ ([termination](#def-b2-polymer-synthesis-chain-steps) rate $2k_t
[\mathrm{M^\bullet}]^2$), the steady-state rate of polymerisation is

$$
R_p = k_p [\mathrm M] \sqrt{\frac{f k_d [\mathrm I]}{k_t}}.
$$

**Proof.** Let $[\mathrm{M^\bullet}]$ be the total concentration of chain radicals, whatever their length. They are created at the rate $R_i = 2fk_d[\mathrm I]$ (two radicals per initiator molecule, a fraction $f$ of which start chains) and destroyed at the rate $2k_t[\mathrm{M^\bullet}]^2$; [propagation](#def-b2-polymer-synthesis-chain-steps) turns one chain radical into another and does not change their number. The steady-state approximation on $[\mathrm{M^\bullet}]$ gives $2fk_d[\mathrm I] = 2k_t[\mathrm{M^\bullet}]^2$, so $[\mathrm{M^\bullet}] =
\sqrt{fk_d[\mathrm I]/k_t}$. [Monomer](#def-b2-polymer-synthesis-polymer) is consumed essentially by [propagation](#def-b2-polymer-synthesis-chain-steps), at $R_p =
k_p[\mathrm M][\mathrm{M^\bullet}]$. ∎

**Definition 29.11 (Kinetic chain length).**

The *kinetic chain length* $\nu$ is the average number of [monomer](#def-b2-polymer-synthesis-polymer) molecules added per radical that starts a chain: $\nu = R_p/R_i$.

**Proposition 29.12 (Chain length and initiator).**

In the steady state, $\nu = k_p[\mathrm M]/\bigl(2\sqrt{fk_dk_t[\mathrm I]}\bigr)$: more initiator gives faster polymerisation but shorter chains. With [termination](#def-b2-polymer-synthesis-chain-steps) by combination, $\bar X_n = 2\nu$; by disproportionation, $\bar X_n = \nu$.

**Proof.** $\nu = R_p/R_i = k_p[\mathrm M]\sqrt{fk_d[\mathrm I]/k_t}/(2fk_d[\mathrm I]) =
k_p[\mathrm M]/\bigl(2\sqrt{fk_dk_t[\mathrm I]}\bigr)$. Each chain started adds $\nu$ [monomers](#def-b2-polymer-synthesis-polymer) on average; combination joins two such chains into one molecule, disproportionation leaves two. ∎

**Definition 29.13 (Living polymerisation).**

A *living polymerisation* is a [chain-growth polymerisation](#def-b2-polymer-synthesis-step-chain) without [termination](#def-b2-polymer-synthesis-chain-steps) or transfer: all chains start together and keep growing as long as [monomer](#def-b2-polymer-synthesis-polymer) remains, and they resume when more [monomer](#def-b2-polymer-synthesis-polymer) is added. The anionic polymerisation of styrene initiated by butyllithium in a dry, aprotic solvent is the classic example.

**Proposition 29.14 (Narrow distributions).**

In a [living polymerisation](#def-b2-polymer-synthesis-living) in which all chains start at once, the [degree of polymerisation](#def-b2-polymer-synthesis-polymer) follows a Poisson distribution, and $\text{\DH} = 1 + \nu/(1+\nu)^2 \approx 1 + 1/\bar X_n$, where $\nu$ is the mean number of [monomers](#def-b2-polymer-synthesis-polymer) added per chain.

**Proof.** *Admitted at this level.* ∎

The distribution is that of the number of additions, independent random events at a common rate, made by each chain during the same time; the [dispersity](#def-b2-polymer-synthesis-averages) follows from the mean and variance of a Poisson law, both equal to $\nu$. A living chain of 100 units thus has $\text{\DH} \approx 1.01$, against nearly 2 for the same average made by step growth.

![Two samples with the same X_n = 50 (models). Living chains crowd around 50 (1.02); step-growth chains spread from monomer to several hundred units (1.98).](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-877e24c7c296.svg)

*Two samples with the same $\bar X_n = 50$ (models). Living chains crowd around 50 ($\text{\DH}
\approx 1.02$); step-growth chains spread from [monomer](#def-b2-polymer-synthesis-polymer) to several hundred units ($\text{\DH} \approx
1.98$).*

**Definition 29.15 (Copolymer).**

A *copolymer* is a [polymer](#def-b2-polymer-synthesis-polymer) built from two or more different [monomers](#def-b2-polymer-synthesis-polymer). Its units may follow each other at random (statistical copolymer), alternately, in long runs of each (block copolymer, made by [living polymerisation](#def-b2-polymer-synthesis-living)), or as side chains of one grafted onto a backbone of the other (graft copolymer).

## 29.3 Structure of the chains

**Definition 29.16 (Tacticity).**

In a vinyl [polymer](#def-b2-polymer-synthesis-polymer) $\ce{-[CH2-CHR]_n-}$, every $\ce{CHR}$ carbon is a stereocentre. The *tacticity* describes their relative configurations along the chain drawn as a planar zig-zag: in an *isotactic* chain all the $\ce{R}$ groups lie on the same side of the plane, in a *syndiotactic* chain they alternate, and in an *atactic* chain they are placed at random.

![Polypropene drawn as a zig-zag in the plane of the page; each methyl group is a wedge (towards the reader) or a hashed wedge (away). Isotactic: all methyls on one side; syndiotactic: alternating; atactic: irregular.](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-bf77e56f1406.svg)

*Polypropene drawn as a zig-zag in the plane of the page; each methyl group is a wedge (towards the reader) or a hashed wedge (away). [Isotactic](#def-b2-polymer-synthesis-tacticity): all methyls on one side; [syndiotactic](#def-b2-polymer-synthesis-tacticity): alternating; [atactic](#def-b2-polymer-synthesis-tacticity): irregular.*

Regular chains can pack side by side into crystalline regions; irregular ones cannot. [Isotactic](#def-b2-polymer-synthesis-tacticity) polypropene, made with the metal catalysts of the Year 3 volume, is partly crystalline, hard and high melting; [atactic](#def-b2-polymer-synthesis-tacticity) polypropene is a soft, sticky amorphous material. A [polymer](#def-b2-polymer-synthesis-polymer) is rarely fully crystalline: crystallites are embedded in amorphous regions where chains are tangled.

**Definition 29.17 (Thermal transitions).**

The *degree of crystallinity* of a [polymer](#def-b2-polymer-synthesis-polymer) is the mass fraction of it that lies in crystalline regions. The *glass transition temperature* $T_g$ is the temperature below which the amorphous regions are a rigid glass, their chain segments unable to move, and above which they become rubbery; the crystalline regions melt at a higher temperature, $T_m$.

Polystyrene and poly(methyl methacrylate) are amorphous with $T_g$ above room temperature: rigid, transparent glasses. Natural rubber has $T_g$ well below room temperature. Polyethene, with $T_g$ far below room temperature but highly crystalline, is tough and flexible: its crystallites hold the rubbery amorphous parts together.

![Modulus (stiffness) of an amorphous polymer against temperature, schematic. Below T_g the material is a glass; through the transition the modulus falls by orders of magnitude; on the rubbery plateau entangled chains behave as a rubber; at higher temperature a linear polymer flows (solid line), while a cross-linked one keeps its plateau until it decomposes (dashed).](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-ca3638941c79.svg)

*Modulus (stiffness) of an amorphous [polymer](#def-b2-polymer-synthesis-polymer) against temperature, schematic. Below $T_g$ the material is a glass; through the transition the modulus falls by orders of magnitude; on the rubbery plateau entangled chains behave as a rubber; at higher temperature a linear [polymer](#def-b2-polymer-synthesis-polymer) flows (solid line), while a cross-linked one keeps its plateau until it decomposes (dashed).*

## 29.4 Properties and uses

**Definition 29.18 (Elastomer).**

An *elastomer* is a [polymer](#def-b2-polymer-synthesis-polymer) used above its [glass transition temperature](#def-b2-polymer-synthesis-transitions) whose chains are lightly cross-linked: it can be stretched to several times its length and returns to its shape when released.

The four large classes of [polymer](#def-b2-polymer-synthesis-polymer) materials follow from structure and transitions. Thermoplastics, linear or branched, glassy or semi-crystalline at room temperature, soften on heating and can be moulded again and again: polyethene, polypropene, polystyrene, PET. [Elastomers](#def-b2-polymer-synthesis-elastomer) are lightly cross-linked chains above their $T_g$: vulcanised rubber, in which sulfur bridges link the chains. Fibres are chains aligned by drawing, with strong interactions between them: the hydrogen bonds between amide groups in nylon. Thermosets are densely cross-linked networks formed during moulding: epoxy and phenol–formaldehyde resins, which cannot be remelted. Only thermoplastics are recycled by melting; the school volume gave the scale of the problem.

![Polymer pellets in a plant, the form in which thermoplastics are sold, and fibres being drawn from a spinneret onto rotating rolls behind them (illustration).](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/img-409a072ef078.jpg)

*[Polymer](#def-b2-polymer-synthesis-polymer) pellets in a plant, the form in which thermoplastics are sold, and fibres being drawn from a spinneret onto rotating rolls behind them (illustration).*

**History — A 99 % yield is not enough.**

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/img-56ea3e715b45.jpg)

Wallace Carothers, a young university instructor recruited into an industrial laboratory for fundamental research, built long molecules from well-known reactions, and his results strongly supported the then-contested idea that [polymers](#def-b2-polymer-synthesis-polymer) are ordinary molecules, only very long. His team made polyesters, then, from 1934, polyamides; the equation that bears his name explains why only extreme [conversions](https://one-course.com/books/chemistry/3/en/chapter/5-continuous-reactors-and-industrial-processes#def-b2-continuous-reactors-conversion) gave fibres. Nylon went into production in 1939. (Photograph: unknown photographer, public domain; Wikimedia Commons.)

**Safety.**

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-0f2415719429.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-4b496db0d6c3.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-afc03563fb34.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-b3bf7bb196be.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-748906458fb6.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-3/b2-polymer-synthesis/fig-76745c6dd418.svg)

Styrene is flammable and a health hazard. Dibenzoyl peroxide is an explosive oxidiser when dry and is kept damp and cool. Hexane-1,6-diamine and hexanedioic acid are corrosive to the eyes.

## 29.5 Exercises

**Exercise 29.1 ★.**

Draw the [repeat units](#def-b2-polymer-synthesis-polymer) of polyethene, polypropene, poly(vinyl chloride), polystyrene and nylon-6,6.

**Solution of Exercise 29.1.**

Polyethene $\ce{-[CH2-CH2]-}$; polypropene $\ce{-[CH2-CH(CH3)]-}$; poly(vinyl chloride) $\ce{-[CH2-CHCl]-}$; polystyrene $\ce{-[CH2-CH(C6H5)]-}$; nylon-6,6 $\ce{-[NH(CH2)6NH-CO(CH2)4CO]-}$.

**Exercise 29.2 ★.**

Step or chain growth: PET from ethane-1,2-diol and benzene-1,4-dicarboxylic acid; polystyrene; nylon-6,6; poly(methyl methacrylate); poly(lactic acid) from 2-hydroxypropanoic acid?

**Solution of Exercise 29.2.**

PET: step growth (polyester, water released). Polystyrene: chain growth. Nylon-6,6: step growth. Poly(methyl methacrylate): chain growth (a $\ce{C=C}$ [monomer](#def-b2-polymer-synthesis-polymer)). Poly(lactic acid) from the hydroxy acid: step growth, an $\ce{A-B}$ [monomer](#def-b2-polymer-synthesis-polymer).

**Exercise 29.3 ★.**

A polystyrene has $\bar X_n = 1000$. Compute $\bar M_n$, neglecting the end groups.

**Solution of Exercise 29.3.**

$\bar M_n = 1000 \times 104.15\,\mathrm{g}/\mathrm{mol} \approx 104\,\mathrm{kg}/\mathrm{mol}$ (styrene $\ce{C8H8}$, $104.152\,\mathrm{g}/\mathrm{mol}$).

**Exercise 29.4 ★.**

A polypropene chain drawn as a planar zig-zag has its methyl groups on the side of the reader, then away, then towards, then away, and so on. Name its [tacticity](#def-b2-polymer-synthesis-tacticity). Could it crystallise?

**Solution of Exercise 29.4.**

[Syndiotactic](#def-b2-polymer-synthesis-tacticity). Yes: a regular chain can pack into crystallites.

**Exercise 29.5 ★★.**

A sample is a mixture of equal masses of two fractions, of molar masses $10\,\mathrm{kg}/\mathrm{mol}$ and $100\,\mathrm{kg}/\mathrm{mol}$. Compute $\bar M_n$, $\bar M_w$ and the [dispersity](#def-b2-polymer-synthesis-averages).

**Solution of Exercise 29.5.**

Mass fractions 0.5 and 0.5. $1/\bar M_n = 0.5/10 + 0.5/100 = 0.055$, $\bar M_n \approx
18.2\,\mathrm{kg}/\mathrm{mol}$; $\bar M_w = 0.5 \times 10 + 0.5 \times 100 = 55\,\mathrm{kg}/\mathrm{mol}$; $\text{\DH} \approx
3.0$.

**Exercise 29.6 ★★.**

Compute $\bar X_n$ for a balanced [step-growth polymerisation](#def-b2-polymer-synthesis-step-chain) at $p = 0.98$ and $p = 0.995$.

**Solution of Exercise 29.6.**

$1/(1-0.98) = 50$; $1/(1-0.995) = 200$.

**Exercise 29.7 ★★.**

A diacid and a diamine are mixed with a 1 % excess (in moles) of the diamine. What is the largest $\bar X_n$ that can be reached?

**Solution of Exercise 29.7.**

$r = 1/1.01$; at $p = 1$, $\bar X_n = (1+r)/(1-r) = (1.01 + 1)/(1.01 - 1) = 201$.

**Exercise 29.8 ★★.**

In a radical polymerisation the initiator concentration is doubled. By what factor do the rate of polymerisation and the [kinetic chain length](#def-b2-polymer-synthesis-chain-length) change?

**Solution of Exercise 29.8.**

$R_p \propto [\mathrm I]^{1/2}$: multiplied by $\sqrt2 \approx 1.41$. $\nu \propto [\mathrm I]^{-1/2}$: divided by $\sqrt 2$, about 0.71 times its former value.

**Exercise 29.9 ★★.**

Poly(vinyl chloride) is rigid; garden hoses made of it are flexible because they contain a plasticiser, a small molecule mixed between the chains. Explain with the glass transition.

**Solution of Exercise 29.9.**

Pure PVC has its glass transition above room temperature: a rigid glass. The small plasticiser molecules sit between the chains, separate them and let segments move at lower temperature: $T_g$ falls below room temperature, and the material is rubbery and flexible in use.

**Exercise 29.10 ★★★.**

Show that the Flory mass distribution $w_x = x(1-p)^2p^{\,x-1}$, viewed as a function of a continuous $x$, is largest at $x = -1/\ln p$, and that this is close to $\bar X_n$ when $p$ is close to 1.

**Solution of Exercise 29.10.**

$\dfrac{\mathrm d}{\mathrm dx}\bigl(x p^{\,x-1}\bigr) = p^{\,x-1}(1 + x\ln p)$, zero for $x = -1/\ln p$ (a maximum: the sign goes from $+$ to $-$). For $p$ close to 1, $\ln p \approx -(1-p)$, so $x \approx
1/(1-p) = \bar X_n$.

**Exercise 29.11 ★★★.**

PET is made in two stages: dimethyl benzene-1,4-dicarboxylate with excess ethane-1,2-diol gives bis(2-hydroxyethyl) benzene-1,4-dicarboxylate and methanol; then this diester is heated under vacuum and polymerises, releasing ethane-1,2-diol. Write both stages and explain how the second stage solves the stoichiometry problem of the Carothers equation.

**Solution of Exercise 29.11.**

Stage 1, [transesterification](https://one-course.com/books/chemistry/3/en/chapter/24-carboxylic-acid-derivatives#def-b2-acyl-substitution-transesterification): $\ce{C6H4(COOCH3)2 + 2HOCH2CH2OH -> C6H4(COOCH2CH2OH)2 + 2CH3OH}$, methanol distilled off. Stage 2: each diester molecule carries two $\ce{CH2CH2OH}$ ends; one end attacks an ester of another molecule and releases a molecule of ethane-1,2-diol, which is pumped off. The [monomer](#def-b2-polymer-synthesis-polymer) carries both kinds of group in the right ratio, as an $\ce{A-B}$ [monomer](#def-b2-polymer-synthesis-polymer) does: the balance $r = 1$ is built in, and removing the diol drives $p$ towards 1.

**Exercise 29.12 ★★★.**

In a radical copolymerisation of [monomers](#def-b2-polymer-synthesis-polymer) 1 and 2, the instantaneous composition of the [copolymer](#def-b2-polymer-synthesis-copolymer) is given by $\dfrac{\mathrm d[\mathrm M_1]}{\mathrm d[\mathrm M_2]} = \dfrac{[\mathrm M_1](r_1[\mathrm M_1]
+ [\mathrm M_2])}{[\mathrm M_2]([\mathrm M_1] + r_2[\mathrm M_2])}$, where $r_1$ and $r_2$ are the reactivity ratios (exercise data: $r_1 = 2.0$, $r_2 = 0.5$). Compute the ratio of units in the [copolymer](#def-b2-polymer-synthesis-copolymer) formed from an equimolar feed. What happens if $r_1 = r_2 = 0$?

**Solution of Exercise 29.12.**

Equimolar feed: ratio $= (2.0 + 1)/(1 + 0.5) = 2$: twice as many units of [monomer](#def-b2-polymer-synthesis-polymer) 1. With $r_1 = r_2 =
0$, ratio $= [\mathrm M_1][\mathrm M_2]/([\mathrm M_2][\mathrm M_1]) = 1$ whatever the feed: each radical adds only the other [monomer](#def-b2-polymer-synthesis-polymer), an alternating [copolymer](#def-b2-polymer-synthesis-copolymer).

## 29.6 Problem: Nylon-6,6 by the Kilogram

**Problem 29.1.**

Weekend problem — the monomers and the nylon salt, the Carothers equation, a deliberate imbalance, the Flory distribution and the water released

Nylon-6,6 is made from hexane-1,6-diamine and hexanedioic acid. End groups are neglected throughout. Molar masses ($\mathrm{g}/\mathrm{mol}$): C 12.011, H 1.008, N 14.007, O 15.999.

**Part I — [Monomers](#def-b2-polymer-synthesis-polymer) and [repeat unit](#def-b2-polymer-synthesis-polymer).**

1. Write the formulas of the two [monomers](#def-b2-polymer-synthesis-polymer) .
2. Write the equation of the formation of one amide link.
3. The [monomers](#def-b2-polymer-synthesis-polymer) are first combined as a 1 : 1 salt, crystallised from solution. Why?
4. Give the [repeat unit](#def-b2-polymer-synthesis-polymer) of nylon-6,6, its formula and its molar mass.
5. What is the mean molar mass $M_0$ of a monomer-derived unit in the chain?
6. What do the two numbers in the name “6,6” count?

**Part II — The Carothers equation.**

7. Compute $\bar X_n$ and $\bar M_n$ at $p = 0.98$ .
8. The same at $p = 0.99$ .
9. Explain the remark “a 99 % yield is not enough”.
10. Which extent of reaction gives $\bar M_n = 10\,\mathrm{kg}/\mathrm{mol}$ ?
11. How is the extent pushed so high in practice?
12. Why must the [monomers](#def-b2-polymer-synthesis-polymer) be very pure?

**Part III — A deliberate imbalance.**

13. The diamine is used in 1 % molar excess. Compute $r$ .
14. Compute the largest $\bar X_n$ and $\bar M_n$ that can then be reached.
15. Compute $\bar X_n$ at $p = 0.99$ with this imbalance.
16. Which groups end the chains at the end of the reaction?
17. A little ethanoic acid is sometimes added. What does it do to the chains?
18. Why would a manufacturer limit the chain length on purpose?

**Part IV — Distribution and mass balance.**

19. For a balanced mixture at $p = 0.99$ , give the [dispersity](#def-b2-polymer-synthesis-averages) .
20. Compute $\bar M_w$ at $p = 0.99$ .
21. At $p = 0.99$ , what fraction of the molecules, and what fraction of the mass, is still unreacted [monomer](#def-b2-polymer-synthesis-polymer) ?
22. Near which chain length is the mass distribution largest?
23. Compute the mass of water released per kilogram of nylon-6,6.
24. Why does melt spinning need $\bar M_n$ within a window?
25. State the extent of reaction that gives $\bar M_n = 20\,\mathrm{kg}/\mathrm{mol}$ for a balanced mixture.

**Solution of Problem 29.1.**

**1.** $\ce{H2N(CH2)6NH2}$ and $\ce{HOOC(CH2)4COOH}$. **2.** One water per link:

$$
\mathrm{{-}COOH + H_2N{-} \longrightarrow {-}CO{-}NH{-} + H_2O}.
$$

**3.** The salt contains exactly one diamine per diacid; weighing two separate liquids or solids would never reach the exactness the Carothers equation demands. **4.** $\ce{-[NH(CH2)6NH-CO(CH2)4CO]-}$, $\ce{C12H22N2O2}$, $226.32\,\mathrm{g}/\mathrm{mol}$. **5.** The [repeat unit](#def-b2-polymer-synthesis-polymer) contains two monomer-derived units: $M_0 = 226.32/2 = 113.16\,\mathrm{g}/\mathrm{mol}$. **6.** The carbons of the diamine (6) and of the diacid (6). **7.** $\bar X_n = 50$, $\bar M_n = 50 \times 113.16 \approx 5.66\,\mathrm{kg}/\mathrm{mol}$. **8.** $\bar X_n = 100$, $\bar M_n \approx 11.3\,\mathrm{kg}/\mathrm{mol}$. **9.** With 99 % of the groups reacted, the average chain has only 100 units, too short for a strong fibre; each further step towards complete [conversion](https://one-course.com/books/chemistry/3/en/chapter/5-continuous-reactors-and-industrial-processes#def-b2-continuous-reactors-conversion) doubles or triples the chain length. **10.** $\bar X_n = 10\,000/113.16 = 88.4$; $p = 1 - 1/88.4 \approx 0.989$. **11.** By removing water: the melt is heated well above its melting point, finally under reduced pressure, so that the equilibrium keeps moving towards amide. **12.** A monofunctional impurity caps chain ends, and any impurity upsets the 1 : 1 balance. **13.** $r = 1/1.01 \approx 0.990$. **14.** $\bar X_n = (1+r)/(1-r) = 201$; $\bar M_n \approx 201 \times 113.16 \approx
22.7\,\mathrm{kg}/\mathrm{mol}$. **15.** $\bar X_n = 1.9901/(1.9901 - 2 \times 0.9901 \times 0.99) \approx 1.9901/0.0297 \approx
67$. **16.** Amine groups: when the acid groups are used up, every chain ends in $\ce{NH2}$. **17.** It turns an amine end into an amide that cannot react further: it caps chains, limiting $\bar M_n$ and stopping further growth when the [polymer](#def-b2-polymer-synthesis-polymer) is remelted. **18.** To keep a reproducible melt viscosity for spinning and moulding, which would otherwise drift as chains keep reacting in the melt. **19.** $\text{\DH} = 1 + p = 1.99$. **20.** $\bar M_w = 1.99 \times 11.3 \approx 22.5\,\mathrm{kg}/\mathrm{mol}$. **21.** Number fraction $n_1 = 1 - p = 0.01$, 1 % of the molecules; mass fraction $w_1 = (1-p)^2 =
10^{-4}$, 0.01 % of the mass. **22.** Near $x = -1/\ln 0.99 \approx 99.5$, that is near $\bar X_n = 100$ (exercise 10). **23.** $1000/226.32 = 4.42\,\mathrm{mol}$ of [repeat units](#def-b2-polymer-synthesis-polymer), two amide links each: $8.84\,\mathrm{mol}$ of water, $8.84 \times 18.015 \approx 159\,\mathrm{g}$. **24.** Chains too short give a weak fibre; chains too long give a melt too viscous to be pushed through the fine holes of the spinneret. **25.** $\bar X_n = 20\,000/113.16 = 176.7$ and $p = 1 - 1/176.7 = \boldsymbol{\approx 0.994}$.