# Recombination of Poly(Acrylic Acid) Radicals in Acidic Aqueous Solutions: A Pulse Radiolysis Study
## 3 Results and Discussion
### Radical Decay Kinetics at pH 2
#### 3.4.2 Radical Recombination Rate Constants and Their Dependence on Experimental Parameters  
Second-order kinetics stipulates that the reciprocal half-life of radicals should be directly proportional to the initial radical concentration. In Figure 6 results of kinetic analysis for decay of PAA radicals have been presented in the second-order coordinates, for various polymer concentrations (2.5-50 mmol dm\({}^{-3}\) in the terms of monomer units). While there is considerable scatter in the data, especially for the lowest PAA concentrations (due to the low molar absorption coefficient and low yield of polymer radicals resulting from strong competition of mutual \({}^{\bullet}\)OH and H\({}^{\bullet}\) recombination, resulting in weak signals), the data can be approximated with linear functions, and the slopes correspond to the second-order rate constant \(2k\). While for the highest polymer concentrations used in this study (10-50 mmol dm\({}^{-3}\)) the slopes were similar, below this concentration range the slope tended to increase with decreasing PAA concentration, indicating an increase in the recombination rate constant of radicals. Such an effect was not observed for OAA and PA (data not shown). This observation can be explained by the fact that if a constant concentration of polymer radicals is created in the system by electron pulse, then the lower polymer concentration, the higher average initial number of radicals on each macromolecule. This is illustrated in Figure 7, where the observed second-order rate constant calculated for each experiment (i.e., for each combination of dose and polymer concentration) is plotted as a function of average initial number of radicals on a chain, \(Z_{R0}\). Figure 5: Pulse radiolysis of Ar-saturated aqueous solution containing 50 mmol dm\({}^{-3}\) PAA, pH 2, dose per pulse 62 Gy, average initial number of radicals on a chain 1.64. Experimental data on the decay of the PAA radicals and the fits based on three kinetic models: classical first- and second-order, and dispersive model (Plonka model, Equation (7)). It is clear that, in contrast to Figure 6, here all data points for various polymer concentrations can be described by a single dependence, notwithstanding the data scatter at 2.5 and 5 mmol dm\({}^{-3}\). It seems that indeed the average number of radicals per chain is the decisive factor governing the radical decay kinetics. When this number is low, radicals decay mainly by intermolecular recombination. However, for \(Z_{R0}\) higher than around 5, there is a distinct upward tendency in the rate constant, which can be attributed to the fact that if many radicals are located simultaneously on each chain, their main mode of recombination is the reaction with their neighbors at the same macromolecule. The lower the interradical distance along the chain, the faster the reaction, since the radicals in proximity require rearrangements of shorter chain segments. These findings are in line with earlier experimental data on radicals of poly(ethylene oxide) [89] and, when  
Figure 6: Kinetic analysis for decay of PAA radicals observed during pulse radiolysis of Ar-saturated aqueous solutions containing 2.5–50 mmol dm\({}^{-3}\) PAA, pH 2, dose per pulse 11–67 Gy. Reciprocal half-life of radicals as a function of the initial radical concentration \(c_{0}\). Polymer concentrations, in mmol dm\({}^{-3}\) of monomer units, are given in the graph. Figure 7: Kinetic analysis for decay of PAA radicals observed during pulse radiolysis of Ar-saturated aqueous solutions containing 2.5–50 mmol dm\({}^{-3}\) PAA, pH 2, dose per pulse 11–67 Gy. Second-order rate constant calculated for each experiment as a function of average initial number of radicals on a chain. Polymer concentrations, in mmol dm\({}^{-3}\) of monomer units, are given in the graph. intramolecular recombination is concerned, also with Monte Carlo simulations by the Cooperative Motion Algorithm [92].