# Kinetics of PTSA-Catalysed Polycondensation of Citric Acid with 1,3-Propanediol
## 3 Results and Discussion
### Kinetic Model  
The polycondensation under study followed the mechanism of the Fischer esterification reaction. In the first step, a proton from an acidic catalyst (PTSA) is transferred to carbonyl oxygen in the citric acid molecule. This increases the electrophilicity of the carbonyl carbon and facilitates the nucleophilic attack of the alcohol molecule. Access to the carbonyl oxygen atom is easier for \(\beta\)-C(O)O-H groups than for \(\alpha\)-C(O)O-H groups (Figure 4, carbonyl oxygen atoms are indicated by arrows). This is based on distinguishing two types of carboxylic groups in citric acid in the described model. The starting point for the determination of the polycondensation kinetics was the assumption that the reactivity of the \(\alpha\)-C(O)O-H and \(\beta\)-C(O)O-H groups changed due to the esterification of neighbouring carboxylic groups. At the same time, the length of the polyester chain was irrelevant. The following terminology of the reacting carboxyl groups was adopted: X-Y1-Y2, where X is the reacting group and Y1 and Y2 are the neighbouring esterified groups. To simplify the notation of the kinetic equations, \(\alpha\)-C(O)O-H and \(\beta\)-C(O)O-H were abbreviated to \(\alpha\) and \(\beta\), respectively. The reaction rate constants are marked numerically (Figure 5). Figure 4: Simulation of the 3D structure of a citric acid molecule: blue—carbon atoms, pink—oxygen atoms, white—hydrogen atoms (ChemSketch). The calculations were based on the following assumptions: (1) The water concentration was constant and equal to 0, which excluded the hydrolysis of ester bonds, (2) the reactivity of 1,3-propanediol was independent of its conversion, (3) the volume of the reacting phase did not change, (4) the reaction mixture was far enough from the gel point that the effects of steric hindrance and a sudden change in viscosity were negligible, (5) the heating time of the reaction mixture to a given temperature was negligibly short and the mixture was homogeneous from the start, and (6) all reactions were first-order reactions in each reactant, where the functional group was considered to be the reactant. The kinetic model is based on the elementary reactions shown in Figure 5. Given the assumptions that were made, the process is described by the following equations:  
\[\frac{\mathrm{d}[\alpha]}{\mathrm{dt}}=-\mathbf{k}_{1}[\alpha][\mathrm{OH}] -\mathbf{k}_{2}[\beta][\mathrm{OH}] \tag{1}\]  
\[\frac{\mathrm{d}[\beta]}{\mathrm{dt}}=-2\mathbf{k}_{2}[\beta][\mathrm{OH}]-2 \mathbf{k}_{1}[\alpha][\mathrm{OH}] \tag{2}\]  
\[\frac{\mathrm{d}[\alpha-\beta]}{\mathrm{dt}}=\mathbf{k}_{2}[\beta][\mathrm{OH }]-\mathbf{k}_{3}[\alpha-\beta][\mathrm{OH}]-\mathbf{k}_{5}[\beta-\beta][ \mathrm{OH}] \tag{3}\]  
\[\frac{\mathrm{d}[\alpha-\beta-\beta]}{\mathrm{dt}}=\mathbf{k}_{5}[\beta- \beta][\mathrm{OH}]-\mathbf{k}_{\otimes}[\alpha-\beta-\beta][\mathrm{OH}] \tag{4}\]  
\[\frac{\mathrm{d}[\beta-\alpha]}{\mathrm{dt}}=2\mathbf{k}_{1}[\alpha][\mathrm{ OH}]-2\mathbf{k}_{4}[\beta-\alpha][\mathrm{OH}] \tag{5}\]  
Figure 5: Scheme of elementary reactions with labelled reaction rate constants; the water molecules that are formed have been omitted. \[\frac{\mathrm{d}[\beta-\beta]}{\mathrm{dt}}=\mathbf{k}_{2}[\beta][ \mathrm{OH}]-\mathrm{k}_{5}[\beta-\beta][\mathrm{OH}]-\mathrm{k}_{3}[\alpha- \beta][\mathrm{OH}] \tag{6}\] \[\frac{\mathrm{d}[\beta-\alpha-\beta]}{\mathrm{dt}}=\mathrm{k}_{3}[ \alpha-\beta][\mathrm{OH}]+\mathbf{k}_{4}[\beta-\alpha][\mathrm{OH}]-\mathrm{k} _{7}[\beta-\alpha-\beta][\mathrm{OH}]\] (7) Figure 5.0 (8) Figure 5.1 (9) Figure 5.2 (10) Figure 5.3  
Equations (1)-(7) are partial balance equations. They refer to groups that react according to the specific rate constants of elementary reactions, as shown in Figure 5.4 The symbols \(\alpha\) (Equation (1)) and \(\beta\) (2) are to be understood as the respective carboxyl groups in the unreacted acid molecule.