Question 2 of 6: Living Anionic Polymerization — Chain Lengths and Poisson Kinetics

Question 2: Living Anionic Polymerization — Chain Lengths and Poisson Kinetics (20 points: 12.5 + 12.5)

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.

Find. (a) X̄_n, X̄_w, M̄_n, M̄_w, conversion and PDI at complete reaction; (b) the concentration of living j-mers [P_j](t) for j = 3, 7, 10.

Approach. With k_i = ∞ every initiator becomes a living chain at t = 0, so the number of chains is fixed and X̄_n is just the monomer-to-initiator ratio; the absence of termination makes the length distribution Poisson, which fixes X̄_w/X̄_n; the time evolution follows from a constant active-centre concentration driving first-order monomer decay.

Part (a) — final averages.

1. Number-average chain length. Instantaneous initiation means all [I]₀ chains start together and none are created or destroyed thereafter, so at complete conversion every monomer is shared equally among the chains:$$\bar X_n=\frac{[M]_0}{[I]_0}=\frac{2}{0.01}=\boxed{200}$$
2. Poisson breadth. A living chain adds monomers one at a time with no termination, so the number of added units is Poisson-distributed with mean λ = X̄_n−1 = 199. The weight average and PDI are$$\bar X_w=\frac{\lambda^2+3\lambda+1}{\lambda+1}=201.0,\qquad \text{PDI}=\frac{\bar X_w}{\bar X_n}=1+\frac{\lambda}{(\lambda+1)^2}=\boxed{1.005}$$the hallmark near-monodispersity of living polymerization.
3. Molecular weights. With styrene as the reference monomer,$$\bar M_n=\bar X_n M_0=200(104.15)=2.08\times10^{4}\ \text{g/mol},\quad \bar M_w=\bar X_w M_0=2.09\times10^{4}\ \text{g/mol}$$
4. Extent of polymerization. Living chains consume monomer until it is exhausted, so the final fractional conversion is $p=1-[M]_\infty/[M]_0\to\boxed{1.0\ (100\%)}$ — the X̄_n above is the complete-conversion limit.

Part (b) — time evolution of the living j-mers.

5. Monomer decay. The total active-centre concentration is constant, [P*] = [I]₀, so monomer disappears by first-order kinetics:$$-\frac{d[M]}{dt}=k_p[I]_0[M]\ \Rightarrow\ [M]=[M]_0e^{-k_p[I]_0t},\qquad k_p[I]_0=100(0.01)=1\ \text{s}^{-1}$$so [M] = 2e^−t mol/dm³.
6. Poisson kinetics of the distribution. The population balance for the living j-mer, $d[P_j]/dt=k_p[M]([P_{j-1}]-[P_j])$, is solved by a Poisson form in the “monomer clock” $\nu(t)=\int_0^t k_p[M]\,dt'=\dfrac{[M]_0}{[I]_0}\left(1-e^{-k_p[I]_0t}\right)$:$$[P_j](t)=[I]_0\,e^{-\nu}\,\frac{\nu^{\,j-1}}{(j-1)!}$$Because there is no termination, this active-centre concentration is the concentration of j-mer polymer at that instant.
7. Peaks for j = 3, 7, 10. Each curve is a travelling wave that peaks when ν = j−1:$$[P_j]_{max}=[I]_0\,e^{-(j-1)}\frac{(j-1)^{\,j-1}}{(j-1)!}$$giving 2.71×10^-3, 1.61×10^-3 and 1.32×10^-3 mol/dm³ at t = 0.0101, 0.0305 and 0.0460 s respectively. Longer chains appear later and their peak is broader and lower, because the population disperses as ν grows.