Domain: Photopolymerization

The photoinitiated free-radical polymerization of methyl methacrylate \((\text{MMA})\) is conducted in a \(10\%\) \((\text{w/v})\) solution under monochromatic irradiation \((\lambda = 3130 \  \text{angstrom})\) emanating from a mercury arc source in the presence of a photosensitizing agent. Actinometric quantification reveals that the radiant energy absorbed by the reactive system proceeds at a rate of \(1.2 \times 10^5  \ \text{erg L}^{-1} \text{s}^{-1}\), establishing the photonic excitation flux governing radical generation.

Assuming the overall quantum efficiency \((\phi)\) of primary radical formation to be \(0.60\), and adopting the kinetic chain parameter composite \((k^2_p / k_t)^{1/2} = 0.102 \ \text{L}^{1/2}\text{mol}^{-1/2} \ \text{s}^{-1/2} )\) at \(60^\circ \text{C}\), quantitatively determine the rate of polymerization under these conditions. Take planck's constant \((h = 6.63 \times 10^{-27} \text{erg.s})\).

Grading rubric:
criterion: Expresses the photonic excitation flux (I_a) using the specific photochemical unit 'Einsteins per liter-second' (or explains that 1 Einstein = 1 mole of photons).
weight: 10
criterion: Explains that the quantum efficiency (φ = 0.60) is less than unity primarily due to the 'Solvent Cage Effect' (or primary recombination), where geminate radical pairs recombine before escaping the solvation shell.
weight: 20
criterion: Justifies the factor of 2 in the initiation rate equation (R_i = 2φI_a) by explicitly citing the homolytic cleavage of the photoinitiator into two reactive radical fragments.
weight: 10
criterion: Deduces that the kinetic parameter given, combined with the rate equation used, implies the termination rate law is defined as -d[R•]/dt = 2k_t[R•]^2 (bimolecular termination), explicitly noting the convention of the factor '2' representing the destruction of two radicals.
weight: 20
criterion: Explicitly derives the molar concentration of the monomer ([M] ≈ 1.0 M) by calculating the molar mass of Methyl Methacrylate (C_5H_8O_2) as approximately 100.12 g/mol.
weight: 10
criterion: Derives the final rate expression R_p = (k_p/k_t^(1/2))[M](R_i/2)^(1/2) starting from the Steady-State Approximation (R_i = R_t), demonstrating the algebraic isolation of the radical concentration [R•].
weight: 20
criterion: Reports the final rate of polymerization as 1.40 x 10^-5 mol L^-1 s^-1 .
weight: 10

Answer:
1.40 × 10^-5 mol L^-1 s^-1