# Kinetics of PTSA-Catalysed Polycondensation of Citric Acid with 1,3-Propanediol
## 3 Results and Discussion
### Kinetic Model
The coefficient of 2 appearing in Equations (2) and (5) results from the fact that in a given elementary reaction, the reactivity of two carboxylic groups simultaneously changes into an esterified or adjacent-to-esterified one. Equations (8) and (9) are total balances of \(\alpha\)-C(O)O-H and \(\beta\)-C(O)O-H groups, respectively. Equation (10) describes the total balance of hydroxyl groups. Table 3 shows the determined reaction rate constants for the polycondensation of citric acid and 1,3-propanediol at temperatures of 120, 135, and 150 \({}^{\circ}\)C. The calculations were carried out using the Scilab program. The changes in the total concentrations of carboxyl and hydroxyl groups over time are shown in Figure 5.5  
The natural logarithm of each of the seven rate constants as a function of the inverse of the absolute temperature can be described by a linear relationship (Figure 7) according to the Arrhenius equation:  
Figure 5.6  
where: \(A\)--pre-exponential factor, i.e., rate constant at an infinitely high temperature, \(E_{a}\)--activation energy (J/mol), \(R\)--universal gas constant (8.314 J/(mol\(\cdot\)K)), and \(T\)--temperature (K). The values of the activation energy and pre-exponential factor were determined based on the results obtained for the developed kinetic model (Table 4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\mathbf{k_{1}}\) & \(\mathbf{k_{2}}\) & \(\mathbf{k_{3}}\) & \(\mathbf{k_{4}}\) & \(\mathbf{k_{5}}\) & \(\mathbf{k_{6}}\) & \(\mathbf{k_{7}}\) \\ \hline \(E_{a}\left[\frac{\mathrm{d}}{\mathrm{mol}}\right]\) & 63.89 & 71.96 & 67.18 & 69.94 & 70.60 & 74.29 & 95.48 \\ \hline \(A\cdot 105\left[\frac{\mathrm{d}m^{3}}{\mathrm{mol}\cdot\mathrm{s}}\right]\) & 0.15 & 7.54 & 0.30 & 1.12 & 1.60 & 1.53 & 384.34 \\ \hline \end{tabular}
\end{table} 
Table 4: Values of the activation energy (\(E_{a}\)) and pre-exponential factors (\(A\)) for the developed model. Figure 6: Changes in the total concentrations of carboxyl and hydroxyl groups over time—comparisons of experimental data and the model. The values of the seven reaction rate constants correspond to the expected course of polycondensation. The fastest reacting carboxyl group is \(\beta\)-C(O)O-H in the unreacted acid molecule, followed by the (\(\beta\)-\(\beta\)) and (\(\beta\)-\(\alpha\)) groups, according to the established nomenclature. This results from the \(\beta\)-C(O)O-H group being more accessible than the \(\alpha\)-C(O)O-H group, even when one adjacent group is esterified. The steric hindrance created by the oligomer/polymer chain is more negligible when the substitution of the \(\beta\)-C(O)O-H group is involved. As a result, the rate constant \(\mathrm{k_{5}}\) is greater than \(\mathrm{k_{4}}\). In the case of the esterification of the \(\alpha\)-C(O)O-H groups, the unreacted acid molecule is most favoured, followed by the molecule with one esterified group and the molecule with two esterified groups. Access of the hydroxyl group of 1,3-propanediol to the partially reacted acid is hindered by the presence of oligomer/polymer chains. Esterification of the (\(\alpha\)-\(\beta\)-\(\beta\)) group contributes significantly to the gelation process. The similar activation energy values for all of the reactions indicate similar mechanisms. The selectivity of the process does not depend on temperature, which means that the preferred direction of attack of the hydroxyl group on the carboxyl group, which is understood as the choice of a given type of group, does not change. The \(\beta\)-C(O)O-H group is the most rapidly esterified in the unreacted acid molecule. The available literature lacks information on the polycondensation kinetics of citric acid and 1,3-propanediol. Kinetic studies have referred to reactions without a catalyst, which makes it impossible to compare the results. The most similar is the polyester obtained from sebacic acid and glycerol. The monomer's functionality is the opposite, i.e., a bifunctional (dicarboxylic) acid and a trifunctional (trihydroxylic) alcohol.