Reversible step-growth polymerization of PET — conversion, chain length, PDI: 23-Chem-B8 Polymer Engineering, December 2015, Q1

Question 1: Reversible step-growth polymerization of PET — conversion, chain length, PDI (25 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. Self-condensation of BHET modelled as a reversible A–B step polymerization that releases a small molecule (ethylene glycol) which is not removed (closed system). Data:

Quantity	Value
Propagation rate constant	$k_p = 4\times10^{4}\exp(-1.5\times10^{4}/1.98\,T)$ L·mol$^{-1}$·min$^{-1}$
Equilibrium constant	$K_p=k_p/k_p'=0.5$ (T-independent)
Initial monomer concentration	$c_0 = 4.58$ mol/L
Reaction time	$t = 10$ min
Temperatures	280 °C (553.15 K) and 200 °C (473.15 K)

Find. Fractional conversion $p$, number-average chain length $\bar X_n$, and polydispersity index (PDI) at each temperature after 10 min.

Integrated conversion–time curves. At 280 °C the reaction is fast enough to approach the equilibrium ceiling $p_{eq}=0.414$ within 10 min; at 200 °C it is still climbing ($p=0.166$). Because the condensate is not removed, conversion cannot exceed $p_{eq}$.

Approach. Treat the reactive end-groups as a single reversible bimolecular reaction, write the net rate in terms of conversion $p$, evaluate $k_p$ at each temperature by Arrhenius, integrate to $t=10$ min, then convert $p$ to $\bar X_n=1/(1-p)$ and $\text{PDI}=1+p$.

Rate law in terms of conversion. With functional-group concentration $c=c_0(1-p)$, ester bonds $=c_0p$ and released glycol $=c_0p$, the net forward rate is $$-\frac{dc}{dt}=k_p c^2 - k_p'(c_0p)^2 \;\Longrightarrow\; \frac{dp}{dt}=k_p c_0\!\left[(1-p)^2-\frac{p^2}{K_p}\right]$$ since $k_p'=k_p/K_p$. Setting $dp/dt=0$ gives the equilibrium ceiling $K_p=p_{eq}^2/(1-p_{eq})^2$, i.e. $p_{eq}=\sqrt{K_p}/(1+\sqrt{K_p})=0.414$.
Rate constants (Arrhenius). The pre-exponent $1.98$ is $R$ in cal·mol$^{-1}$·K$^{-1}$, so the activation energy is $1.5\times10^4$ cal/mol. Evaluating: $$k_p(553.15\,\text{K})=4\times10^4 e^{-15000/(1.98\cdot553.15)}=0.0451\ \text{L mol}^{-1}\text{min}^{-1},\quad k_p(473.15\,\text{K})=4.45\times10^{-3}$$ Hence $k_pc_0=0.206\ \text{min}^{-1}$ at 280 °C but only $0.0204\ \text{min}^{-1}$ at 200 °C — a ten-fold difference in effective rate.
Integrate to 10 min. Numerically integrating the conversion ODE (Euler, $\Delta t\to0$) gives $$p_{280}=\boxed{0.413}\quad(\text{essentially at } p_{eq}),\qquad p_{200}=\boxed{0.166}$$ The hot case has all but reached the reversible ceiling; the cold case is still rising (see figure).
Chain length and dispersity. By the Carothers relation $\bar X_n=1/(1-p)$ and, for the most-probable step-growth distribution, $\text{PDI}=\bar X_w/\bar X_n = 1+p$: $$\bar X_{n,280}=\frac{1}{1-0.413}=1.70,\ \ \text{PDI}=1.41;\qquad \bar X_{n,200}=\frac{1}{1-0.166}=1.20,\ \ \text{PDI}=1.17$$

Quantity	280 °C	200 °C
Conversion $p$ (10 min)	0.413	0.166
Average chain length $\bar X_n=1/(1-p)$	1.70	1.20
Polydispersity index $1+p$	1.41	1.17

Check

The low conversions are physically correct, not an error: with the glycol condensate retained in the melt, the reaction is throttled by its own equilibrium ($p_{eq}=0.414$). Real PET manufacture drives $p\to0.99\!+$ precisely by pulling the glycol off under vacuum, which removes the reverse term entirely. The numbers above describe the sealed-ampoule kinetics the question specifies.

Topic: Reversible (equilibrium-limited) step-growth polymerization kinetics.

Key relations: reversible rate $dp/dt=k_pc_0[(1-p)^2-p^2/K_p]$; equilibrium conversion $p_{eq}=\sqrt{K_p}/(1+\sqrt{K_p})$; Carothers $\bar X_n=1/(1-p)$; Flory most-probable $\text{PDI}=1+p$; Arrhenius $k_p=A e^{-E_a/RT}$.

Why this works: Step-growth builds molecular weight only at very high conversion, so any process that caps $p$ caps $\bar X_n$. Retaining the condensate installs a reverse (depolymerization) reaction whose rate rises with $p$; the two rates balance at $p_{eq}$, which for $K_p<1$ is well below unity. Temperature enters only through $k_p$ (the ratio $K_p$ is stated constant), so it sets how fast $p_{eq}$ is approached, not its value — exactly what the two curves show.

Common pitfall: Using the irreversible result $\bar X_n=1/(1-p)=1+kc_0t$ and getting a large chain length; that ignores the reverse reaction and violates the stated $K_p$. A second trap is reading $1.98$ as anything but $R$ in cal-units, which corrupts $E_a$.

Source: Odian, Principles of Polymerization 4th ed., §2.4–2.6 (step-growth kinetics & equilibrium); Rudin & Choi, Elements of Polymer Science 3rd ed., Ch. 5.

Quantity	Value
Propagation rate constant	\(k_p = 4\times10^{4}\exp(-1.5\times10^{4}/1.98\,T)\) L·mol\(^{-1}\)·min\(^{-1}\)
Equilibrium constant	\(K_p=k_p/k_p'=0.5\) (T-independent)
Initial monomer concentration	\(c_0 = 4.58\) mol/L
Reaction time	\(t = 10\) min
Temperatures	280 °C (553.15 K) and 200 °C (473.15 K)

Quantity	280 °C	200 °C
Conversion \(p\) (10 min)	0.413	0.166
Average chain length \(\bar X_n=1/(1-p)\)	1.70	1.20
Polydispersity index \(1+p\)	1.41	1.17

Question 1 of 6: Reversible step-growth polymerization of PET — conversion, chain length, PDI

Question 1: Reversible step-growth polymerization of PET — conversion, chain length, PDI (25 marks)