Domain: Polymer kinetics

A dilatometric experiment was carried out to study the bulk free-radical polymerization
of styrene. A dilatometer of total volume \(49.0~\mathrm{cm^{3}}\), fitted with a capillary
tube of radius \(0.10~\mathrm{cm}\), was charged with freshly distilled styrene containing
\(0.1\%\) (by weight) of \(2,2'\text{-}azo\text{-}bis\text{-}isobutyronitrile\) as the
initiator. The filled dilatometer was immersed in an oil bath maintained at
\(70 \pm 1^{\circ}\mathrm{C}\) such that the capillary extended above the liquid surface.

After thermal equilibrium was established, polymerization commenced, leading to a
progressive decrease in the total volume of the reaction mixture. This volume change was
monitored by recording the fall in the liquid column height \(h\) in the capillary tube as
a function of reaction time.

At \(70^{\circ}\mathrm{C}\), the measured decrease in liquid height was
\(0.83~\mathrm{cm}\) after \(300~\mathrm{s}\), increasing to \(1.68~\mathrm{cm}\) after
\(600~\mathrm{s}\) and \(5.03~\mathrm{cm}\) after \(1800~\mathrm{s}\). With continued
polymerization, the fall in height reached \(10.05~\mathrm{cm}\) at \(3600~\mathrm{s}\),
\(15.10~\mathrm{cm}\) at \(5400~\mathrm{s}\), and \(20.13~\mathrm{cm}\) after
\(7200~\mathrm{s}\).

Using these dilatometric data, determine the kinetic parameter
\(\left(\dfrac{k_p}{k_t}\right)^{1/2}\) for the polymerization of styrene at
\(70^{\circ}\mathrm{C}\). The initiator efficiency is \(f = 0.60\), and the initiator
decomposition rate constant at this temperature is
\(k_d = 4.0 \times 10^{-5}~\mathrm{s^{-1}}\).
The densities at \(70^{\circ}\mathrm{C}\) are
\(0.860~\mathrm{g\,cm^{-3}}\) for styrene and
\(1.046~\mathrm{g\,cm^{-3}}\) for polystyrene.

Grading rubric:
criterion: Calculate polymer volume (Vp) as exactly 40.287 cm³ based on initial mass and polymer density.
weight: 20
criterion: Explicityly determine volume contraction (ΔV) as 8.713 cm³ via the difference between V0 and Vp.
weight: 20
criterion: Explicitly derive total theoretical fall in liquid height (ht) as 277.22 cm using ΔV and capillary area. 
weight: 20
criterion: Identify initial monomer concentration (M0) as exactly 8.27 mol L⁻¹ using molar mass and density.
weight: 10
criterion: Applies only the differential steady-state rate expression Rp = kp M0 (2 f kd [I]0 / kt)^(1/2) and does not introduce any integrated or logarithmic kinetic formulations.
weight: 10
criterion: States to treats the initiator concentration as constant and equal to [I]0  with no calculation or use of time-dependent initiator decay
weight: 10
criterion: Explicitly state final kinetic value as 2.9 * 10^-2 with mandatory units of L^1/2 mol^-1/2 s^-1/2.
weight: 10

Answer:
2.9 × 10^-2 L^1/2 mol^-1/2 s^-1/2