# Dissecting steric and polar substituent effects in linear free energy Relationships: Re-Assessment of the Taft equation with Temperature-Dependent kinetic modeling

Sindi Baco, Mélanie Mignot, Christoph Held, Julien Legros, Sébastien Leveneur

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Sindi Baco, Mélanie Mignot, Christoph Held, Julien Legros, Sébastien Leveneur. Dissecting steric and polar substituent effects in linear free energy Relationships: Re-Assessment of the Taft equation with Temperature-Dependent kinetic modeling. Chemical Engineering Science, 2025, 320, part C, pp.122700. 10.1016/j.ces.2025.122700. hal-05286167

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# Dissecting steric and polar substituent effects in linear free energy Relationships: Re-Assessment of the Taft equation with Temperature-Dependent kinetic modeling

Sindi Baco<sup>a,b,c</sup>, Mélanie Mignot<sup>c,d</sup>, Christoph Held<sup>b</sup>, Julien Legros<sup>c,d</sup>, Sébastien Leveneur<sup>a,d,*</sup>

## ARTICLE INFO

### Keywords:

Linear Free Energy Relationships

Taft equation

Kinetic modeling

Esterification

Saponification

Biomass valorization

## ABSTRACT

Kinetic models are vital for developing process flow diagrams that support cost, risk, and environmental assessments, but constructing them is time-consuming, particularly when screening multiple substituents or solvents. Linear Free Energy Relationships (LFERs) can reduce experimental effort, yet most existing formulations neglect temperature effects. In this work, we quantify the temperature dependence (20–50 °C) of Taft’s polar (σ*) and steric (Es) substituent parameters for methyl, ethyl, n-propyl, and 2-chloroethyl groups using kinetic data from esterification and saponification of levulinate derivatives in aqueous media. Both σ* and Es varied linearly with temperature, with the strongest sensitivity observed for 2-chloroethyl (–0.007 °C⁻¹ for σ*, 0.008 °C⁻¹ for Es). Incorporating these temperature coefficients into LFER-based kinetic models enhances their predictive accuracy for reaction rates, enabling more efficient process design in biomass valorization and related chemical systems.

## 1. Introduction

The development of biorefineries is increasing, as they can contribute to a more sustainable energy mix, reduce greenhouse gas emissions and other pollutants, and decrease dependence on fossil resources. To properly evaluate the economic, environmental, and safety aspects of such processes, it is necessary to establish a process flow diagram (PFD) (Di Menno Di Bucchianico et al., 2025). The first step in developing a PFD is to elaborate a kinetic model, which can be time-consuming, especially in the field of biomass valorization.

In this context, the concept of Linear Free Energy Relationships (LFERs) could be useful, as they provide equations that correlate the kinetics (rate constant, k) or equilibrium (K<sub>eq</sub>) of chemical reactions with free energy quantities using empirically derived parameters (Zahrt et al., 2020). These equations help elucidate reaction mechanisms and allow the prediction of thermodynamic and kinetic behavior based on structural features of reactants and solvents, as well as the influence of solvents on reaction kinetics and equilibria.

The origins of the LFER concept date back to the 1930s, when Eyring, Evans, and Polanyi developed the transition state theory. By studying a series of structurally related atom-transfer reactions, Evans and Polanyi observed that the activation energy varies linearly with the reaction enthalpy (Eq. (1)) (De Oliveira et al., 2016; Evans and Polanyi, 1938, 1936, 1935).

$$E_a = E_0 + \alpha \cdot \Delta H_R \quad (1)$$

where, E<sub>a</sub> is the activation energy, ΔH<sub>R</sub> is the enthalpy change of the reaction, α is a proportionality constant (typically between 0 and 1) that reflects the resemblance of the transition state to the reactants or products, and E<sub>0</sub> is a constant offset.

These series typically consist of reactions where either the molecular structure is slightly altered or the reaction medium is varied, while the reactive functional group or center remains constant.

Transition state theory posits that reactants pass through a high-energy intermediate—an activated complex—before forming products.

<footnote><sup>a</sup> INSA Rouen Normandie, UNIROUEN, Normandie Univ, LSPC, UR4704, F-76000 Rouen, France</footnote>

<footnote><sup>b</sup> Laboratory of Thermodynamics, Department of Biochemical and Chemical Engineering, TU Dortmund University, Emil-Figge. Str.70, 44227 Dortmund, Germany</footnote>

<footnote><sup>c</sup> INSA Rouen Normandie, Univ Rouen Normandie, CNRS, Normandie Univ, COBRA UMR 6014, INC3M FR 3038, F-76000 Rouen, France</footnote>

<footnote><sup>d</sup> IRCELYON, Institut de Recherches sur la Catalyse et l'Environnement de Lyon, UMR5256 CNRS-UCBL, Villeurbanne 69626, France</footnote>

<footnote>* Corresponding author at: IRCELYON, Institut de Recherches sur la Catalyse et l'Environnement de Lyon, UMR5256 CNRS-UCBL, Villeurbanne 69626, France. E-mail address: sebastien.leveneur@ircelyon.univ-lyon1.fr (S. Leveneur).</footnote>
The concentration of this activated complex is assumed to be in quasi-equilibrium ($K_{eq}^\ddagger$) with the reactants. This complex exists at the point of maximum Gibbs free energy along the reaction coordinate, characterized by an excess free energy $\Delta G^\ddagger$.

Under this framework:

– The equilibrium constant $K_{eq}$ of the overall reaction relates to the Gibbs free energy change $\Delta G_R$ as:

$$ \ln(K_{eq}) = - \frac{\Delta G_R}{2.3 \cdot R_g \cdot T} \qquad (2) $$

– The rate constant k, according to absolute rate theory, relates to the activation free energy, $\Delta G^\ddagger$:

$$ \ln \left( \frac{k}{R_{N_A \cdot h}^\frac{T}{\Delta G^\ddagger}} \right) = - \frac{\Delta G^\ddagger}{2.3 \cdot R \cdot T} \qquad (3) $$

–Or equivalently:

$$ k = \frac{R \cdot T}{N_A \cdot h} \cdot e^{-\frac{\Delta G^\ddagger}{2.3 \cdot R \cdot T}} \qquad (4) $$

Here, R is the ideal gas constant, T is the temperature, $N_A$ is Avogadro's number, and h is Planck's constant. Note that $\Delta G_R$ and $\Delta G^\ddagger$ both represent Gibbs free energy changes but refer to the overall reaction and the transition state, respectively.

LFERs are often used to assess how substituents or solvents affect reaction rates or equilibria (Chapman and Shorter, 1972; Zahrt et al., 2020). When comparing two series of reactions, $\alpha$ and $\beta$, their respective rate or equilibrium constants may follow a linear relation:

$$ \log(k_\beta) = m\log(k_\alpha) + c \qquad (5) $$

By normalizing to a standard reaction (with subscript ref), this can be reformulated as:

$$ \log \left( \frac{k_\beta}{k_\beta^{\text{ref}}} \right) = m \log \left( \frac{k_\alpha}{k_\alpha^{\text{ref}}} \right) \qquad (6) $$

This form is particularly useful whether the variation is in substituents or solvents, as it enables both mechanistic insight and predictive capability (Finiels et al., 1996; Minakata et al., 2014; Shukla and DeShong, 2008; Strathmann, 2011; Wu et al., 2016).

In the case of substituent effects, LFERs examine how variations in substituent groups (–R) affect reactivity. This approach applies to congeneric series—compounds with the same functional group or reaction center (e.g., –COOH, –NH₂, –CO) but differing in the attached –R group (Fig. 1).

Several well-known equations arise from this approach:

* Single-parameter: Hammett equation (Hammett, 1938)

* Multi-parameter: Taft (Taft, 1953a, 1953b, 1953c, 1952a, 1952b), Charton (Charton, 1976, 1975), Grunwald–Winstein (Grunwald and Winstein, 1948), and Swain–Lupton equations (Gardner Swain and Lupton, 1968)

<figure><img src="image_3.png" /><figcaption>Fig. 1. Reactivity when –R is attached to the reaction center Y and when –R is attached to the functional group A.</figcaption></figure>

These models assess the impact of polar, resonance, and steric effects of –R on reactivity.

Hammett and Taft equations are the most popular LFER equations for considering the substituent effects.

The pioneering research on establishing relationships between substituents and kinetics was done by Hammett in 1937. He considered the effects of the substituent in meta- and para-positions in benzene structure on ester hydrolysis, realizing that the variation in the free-energy of activation is proportional to the change in Gibbs free energy. Fig. 2 shows the para and meta positions of the substituent R in the benzene molecule, where A is a longer or shorter chain and Z is the reaction center (Hammett, 1938, 1937, 1935).

In his studies, Hammett proposed evaluating the effect of substituents upon the rate or the equilibrium constant by considering the following Eq. (7):

$$ -R \cdot T \cdot \ln(k) + R \cdot T \cdot \ln(k_{\text{ref}}) = \Delta F = \frac{A}{d^2} \cdot \left( \frac{B_1}{\varepsilon} + B_2 \right) \qquad (7) $$

where,

* k is the rate or equilibrium constant of the reaction with a substituted reactant;

* $k_{\text{ref}}$ is the rate or equilibrium constant ref the reaction with a non-substituted reactant, which is considered to be the reference;

* $\Delta F$ is the free energy variation (or its kinetic equivalent);

* d is the distance between the reaction center and the reacting group;

* $\varepsilon$ is the dielectric constant of the medium;

* A is a constant depending only on the type of the substituent and its position in the ring;

* $B_1$ and $B_2$ are constants depending only on the type of the reaction.

Eq. (7) considers the same reaction where only the substrate changes by introducing a substituent. The novelty proposed by Hammett consists of evaluating the effect of this variation by two separate constants: one depending only on the substituent and the other only on the reaction. The quantification of the A constant for a series of substituents in a chosen reaction becomes easier if Eq. (8) assumes the following form:

$$ \log \left( \frac{k}{k_{\text{ref}}} \right) = \sigma(-R) \cdot \rho \qquad (8) $$

where $\sigma(-R)$ and $\rho$ are, respectively, the substituent and reaction constants, expressed as:

$$ \sigma(-R) = - \frac{A}{2.3 \cdot R_g} \qquad (9) $$

$$ \rho = - \frac{1}{d^2 \cdot T} \left( \frac{B_1}{\varepsilon} + B_2 \right) \qquad (10) $$

Experimental observations allow knowing the product $\sigma(-R) \cdot \rho$, but estimating the individual values of $\sigma(-R)$ and $\rho$ separately can be

<figure><img src="image_4.png" /><figcaption>Fig. 2. Para and meta substitutions in benzene.</figcaption></figure>
challenging. For reference, where no substituents are introduced and R = H, $\sigma(-R)$ assumes the value of zero. The $\sigma(-R)$ values are validated when they satisfy the linearity of Eq. (8). Some values have been established and summarized in Table 1.

The $\sigma$ values quantify the polar electronic effects of substituent groups. In general, meta $\sigma$ constants reflect primarily the inductive (field) effects of the substituent, as resonance interactions are minimal in that position. In contrast, para $\sigma$ constants capture the combined influence of both inductive and resonance effects, due to possible delocalization through the aromatic π-system.

In Table 1, the highest $\sigma$ value in the para position is + 1.270 for the nitro group (–NO₂), while the lowest is the amino group (–NH₂) with a value of –0.660. Within the considered substituents, –NO₂ is the most electron-attracting, thus the most acid-strengthening, while –NH₂ is the strongest electron-donor. By studying only substituents in –meta and –para positions, Hammett gave a first quantification of the polar effect, leaving out the steric contribution that would have been present if substituents in ortho position were also considered.

Once $\sigma(-R)$ values were identified for a set of substituents, different values of $\rho$ were also evaluated by changing the type of reaction. In the linear relationship, $\rho$ represents the slope, and it can assume a positive or negative value. For reactions involving the benzene derivative as a substituent, it was found that $\rho$ assumes a positive value for nucleophilic reactions and a negative one for electrophilic reactions. In terms of absolute value, $\rho$ turns to depend on the distance from the substituent R to the reaction center Z, thus on the length and distance of the A group (Fig. 2). In particular, $\rho$ decreases when the side chain gets longer (i.e., by introducing an A group) and its dependence on this can be quantified as follows:

$$ \pi_A = \frac{\rho_{AZ}}{\rho_Z} \qquad (11) $$

The $\pi_A$ is known as the transmission factor and indicates the chain A's to transmit polar effects. As mentioned the reaction constant also depends on other factors such as temperature, pressure and solvent.

To conclude, the reaction constant $\rho$ measures the susceptibility of the reaction to the influence of the substituents and for this reason, it can

be considered as a sensitivity factor for the polar effect $\sigma$. In particular, if:

* $\rho > 0$, the reaction is more sensitive to substituents than benzoic acid, indicating negative charge build-up to the reaction center Z.

* $\rho = 0$, the reaction shows no sensitivity to substituents, and no charge is built or lost.

* $\rho < 0$, the reaction involves a build-up of positive charge at the reaction center Z.

Table 1 shows that $\rho$ values for acid-catalyzed reactions are typically small in absolute value, and their sign can change, while basic-catalyzed reactions have a positive and high value. Based on the definition of $\rho$, this suggests that basic catalyzed reactions are more sensitive to the polar effects than the same acid-catalyzed systems.

Table 1 shows that $\sigma$ values vary based on whether the substituent is an electron-donating group (EDG) or an electron-withdrawing group (EWG). EDG groups stabilize negative charge and destabilize positive charge in the transition state or intermediate by withdrawing electron density through inductive or resonance effects.

EWG groups of substituents stabilize positive charge and destabilize negative charge in the transition state or intermediate by donating electron density through inductive or resonance effects. Typically, these groups have negative $\sigma$ values.

A notable step forward in understanding the effects of substituents located near the reaction center—such as ortho-substituents in aromatic or aliphatic systems—occurred in 1952 when R. W. Taft developed a linear free energy relationship capable of quantifying polar, steric, and resonance effects independently (Taft, 1952b, 1952a).

The development of the Taft equation was based on the Hammett equation and the earlier mechanistic work of C. K. Ingold (Ingold, 1930a). Ingold proposed that the relationship between the rate constants for ester hydrolysis under acidic and basic conditions could be described primarily by polar effects, even though each rate constant is also influenced by steric hindrance (Day and Ingold, 1941; Groocock et al., 1930; Ingold, 1957, 1930a, 1930b; Ingold and Ingold, 1932).

Taft studied the esterification and hydrolysis reaction of esters with a

<table><caption>**Table 1**
Values of $\rho$, $\sigma_{\text{meta}}$ and $\sigma_{\text{(para)}}$ for a set of reaction substituent group –R (Hammett, 1940).</caption><thead><tr><th>ρ</th><th>σ<sub>meta</sub>(-R)</th><th>σ<sub>para</sub>(-R)</th><th>Substituent(-R)</th><th>Reaction</th></tr></thead><tbody><tr><td>+ 1.000</td><td>–</td><td>–0.268</td><td>–CH<sub>3</sub>O</td><td rowspan="2">Ionization of benzoic acids, water, 25 °C</td></tr><tr><td>+ 1.000</td><td>–0.069</td><td>–0.268</td><td>–CH<sub>3</sub></td></tr><tr><td>+ 1.000</td><td>–0.069</td><td>–0.151</td><td>–C<sub>2</sub>H<sub>5</sub></td><td rowspan="2">–CH<sub>3</sub>O</td></tr><tr><td>+ 1.000</td><td>–0.069</td><td>–0.151</td><td>–i-C<sub>2</sub>H<sub>5</sub></td></tr><tr><td>+ 1.000</td><td>0.000</td><td>0.000</td><td>–H (no substituent)</td><td rowspan="2">–CH<sub>3</sub>O</td></tr><tr><td>+ 1.000</td><td>+0.115</td><td>0.000</td><td>–Cl</td></tr><tr><td>+ 1.000</td><td>+0.227</td><td>0.000</td><td>–Br</td><td rowspan="2">–Cl</td></tr><tr><td>+ 1.000</td><td>+0.232</td><td>0.000</td><td>–F</td></tr><tr><td>+ 1.000</td><td>+0.337</td><td>+0.062</td><td>–I</td><td rowspan="2">–F</td></tr><tr><td>+ 1.000</td><td>+0.352</td><td>+0.062</td><td>–NO<sub>2</sub></td></tr><tr><td>+ 2.498</td><td>–0.161</td><td>–0.660</td><td>–NH<sub>2</sub></td><td rowspan="2">Alkaline hydrolysis of ethyl benzoates, 87.83 % ethanol, 30 °C</td></tr><tr><td>+ 2.498</td><td>–0.161</td><td>–0.660</td><td>–C<sub>2</sub>H<sub>3</sub>O</td></tr><tr><td>+ 2.498</td><td>–0.161</td><td>–0.170</td><td>–CH<sub>3</sub></td><td rowspan="2">–CH<sub>3</sub>S</td></tr><tr><td>+ 2.498</td><td>–0.170</td><td>–0.170</td><td>–CH<sub>3</sub>S</td></tr><tr><td>+ 2.498</td><td>+0.276</td><td>+0.009</td><td>–C<sub>6</sub>H<sub>5</sub></td><td rowspan="2">–C<sub>6</sub>H<sub>5</sub></td></tr><tr><td>+ 2.498</td><td>+0.276</td><td>+0.009</td><td>–I</td></tr><tr><td>– 0.946</td><td>–0.678</td><td>+0.640</td><td>–C<sub>6</sub>H<sub>5</sub>-N=N</td><td rowspan="2">Phenolate ions with ethylene oxide, 98 % ethanol, 70.4 °C</td></tr><tr><td>– 0.946</td><td>+0.678</td><td>+0.640</td><td>–CN</td></tr><tr><td>– 0.771</td><td>–0.211</td><td>+0.640</td><td>–C<sub>6</sub>H<sub>5</sub>-N=N</td><td rowspan="2">Phenolate ions with propylene oxide, 98 % ethanol, 70.4 °C</td></tr><tr><td>– 0.771</td><td>–0.211</td><td>+0.640</td><td>–CN</td></tr><tr><td>– 0.467</td><td>–0.027</td><td>–0.066</td><td>–(CH<sub>3</sub>)<sub>2</sub>N</td><td rowspan="2">Acid hydrolysis of aryl-sulfuric acids, water, 78.6 °C</td></tr><tr><td>– 0.467</td><td>+0.141</td><td>+0.619</td><td>–CH<sub>2</sub>CH<sub>2</sub>CO<sub>2</sub>H</td></tr><tr><td>– 0.467</td><td>+0.218</td><td>+0.619</td><td>–CH<sub>2</sub>CH<sub>2</sub>CO<sub>2</sub>H</td><td rowspan="2">–CH<sub>2</sub>CH<sub>2</sub>CO<sub>2</sub>H</td></tr><tr><td>– 0.467</td><td>+0.306</td><td>+0.874</td><td>–C<sub>6</sub>H<sub>5</sub></td></tr><tr><td>– 0.467</td><td>+0.355</td><td>+0.728</td><td>–CH<sub>3</sub>CO</td><td rowspan="2">–CH<sub>3</sub>CO</td></tr><tr><td>– 0.467</td><td>+0.381</td><td>+1.126</td><td>–CO<sub>2</sub>H</td></tr><tr><td>– 0.467</td><td>+0.381</td><td>+1.126</td><td>–CHO</td><td></td></tr></tbody></table>
wide range of aliphatic or o-benzoate substituents. He proposed the relationship in Eq. (12), known as the Taft equation:

$$ \log\left(\frac{k_{-R}(T)}{k_{ref}(T)}\right) = \rho^* \cdot \sigma^*(-R) + \delta \cdot E_S(-R) + \Psi \quad (12) $$

where,

* $k_{-R}(T)$ is the rate constant at a temperature T of a reaction involving the substituent -R and $k_{ref}(T)$ is the rate constant at the same temperature T of the same reaction but involving the reference substituent.

* $\sigma^*(-R)$ represents the net polar effect of the substituent -R on the functional group or reaction center. As in the Hammett equation, it measures the inductive electron-withdrawing or -donating towards the functional group or reaction center, and it only depends on the nature of -R.

* $\rho^*$ is the reaction constant that measures the importance of the polar effect on a given reaction series. As in the Hammett equation, this value can depend on several operating conditions, such as the nature of the reaction center or functional group and temperature.

* $E_S(-R)$ represents the total steric effect due to the substituent -R on the functional group or reaction center. From the literature, it only depends on the nature of -R.

* $\delta$ is a constant value evaluating the importance of the steric effect for a given reaction series. This value also depends on the nature of the reaction center/functional group and temperature.

* $\Psi$ is a parameter taking into account the resonance effect between the substituent and the reaction center.

The reaction constants $\rho^*$ and $\delta$ describe how sensitive a reaction is to polar and steric effects of the substituent, respectively. A positive $\rho^*$ means the reaction is accelerated by electron-withdrawing groups ($\sigma^*>0$) and slowed by electron-donating groups ($\sigma^*<0$), consistent with negative charge build-up in the transition state. In contrast, a negative $\rho^*$ indicates the opposite trend, with electron-donating groups accelerating the rate due to the build-up of positive charge. For the steric sensitivity constant $\delta$, it is important to note that $E_S$ values become more negative as bulk increases. Thus, a positive $\delta$ means that increasing steric bulk decreases the reaction rate, implying greater steric congestion in the transition state, whereas a negative $\delta$ means that bulk increases the reaction rate, suggesting that steric hindrance is relieved or plays a smaller role in the transition state.

The polar effect $\sigma^*(-R)$ and the steric effect $E_S(-R)$ are quantitatively estimated as shown in equations Eqs (13) and (14) (Ingold, 1930a; MacPhee et al., 1978; Taft, 1956):

$$ \sigma^*(-R) = \frac{1}{2.48} \left[ \log\left(\frac{k_{-R}(T)}{k_{ref}(T)}\right)_B - \log\left(\frac{k_{-R}(T)}{k_{ref}(T)}\right)_A \right] \quad (13) $$

$$ E_S(-R) = \log\left(\frac{k_{-R}(T)}{k_{ref}(T)}\right)_A \quad (14) $$

In Eq. (13), the value of 2.48 acts as a weight factor, aiming to bring the $\sigma^*(-R)$ to be on the same scale as Hammett's $\sigma(-R)$. Instead, the subscripts A and B indicate the respective reaction media, which are acidic and basic. In most reactions, Taft used strong mineral acids (i.e., HCl and $H_2SO_4$) and strong bases (i.e., NaOH and KOH). The temperature was kept constant at 25 °C, and the solvent medium was not always the same. Thus, most classical LFER and Taft analyses are strictly isothermal at only one temperature, highlighting the need to evaluate the sensitivity of substituent parameters, e.g., $E_S(-R)$ and $\sigma^*(-R)$, to temperature.

The Taft equation was developed primarily through studies on the synthesis and hydrolysis of esters, as steric effects strongly influence these reactions due to the proximity of substituents to the reaction center (Meyer, 1894; Newman, 1950). The $k_{-R}$ is indicative of a given

ester RCOOR' hydrolysis (or its correspondent synthesis) while $k_{ref}$ is indicative of the same reaction, carried out with the reference ester MeCOOR'. Therefore, in the Taft equation, the reference group is the methyl group (-CH$_3$), while in the Hammett equation, it was -H. A crucial point to emphasize is that rate constants of reactions in acidic and basic media must be obtained for a series of esters where the structure of only one component is changed (i.e., by introducing the -R group).

On one hand, Eqs (13) and (14) illustrate the influence of Ingold's idea. In this framework, the basic medium ratio $\left(\frac{k_{-R}(T)}{k_0(T)}\right)_B$ reflects the combined effects of polarity, steric hindrance, and resonance, whereas the acidic medium ratio $\left(\frac{k_{-R}(T)}{k_0(T)}\right)_A$ is assumed to capture only steric and resonance effects. Consequently, the difference in Eq. (13), i.e., $\sigma^*(-R)$, is interpreted as the net polar effect. However, this interpretation relies on the critical assumption that resonance and steric contributions remain constant across both acidic and basic conditions—an assumption that may not hold rigorously, especially in acid media where the nature of the transition state may alter resonance interactions (Fig. 3).

As shown in Fig. 3, the two intermediates differ by only two protons, which are known to be very limited in dimension. This notion leads to considering the difference in steric effects in the transition states of reactions in acidic and basic media, where the same -R group is introduced, and this effect is constant and independent of the latter. In other words, this means that the difference in the steric contributions to the free energy of formation of the two intermediates is not the same, but it is independent of -R. This difference is then assumed to be zero because the starting structures are equal in acidic and basic reactions. For the same reason and because both transition states are saturated at the carboxylate carbon, the resonance effect of -R to -CH$_3$ is also considered zero.

On the other hand, Taft based part of his work on Hammett's results. As shown in Table 1, $\rho$ assumes values around +2.00 in basic conditions, while $\rho$ drops to values around 0.00 for acidic hydrolysis, meaning that the polar effect due to the -R substituent is smaller for acidic than basic reactions. Table 2 shows a class of reactions for which the values of $E_S(-R)$ and $\sigma^*(-R)$ were obtained as an average of the different literature sources available at that time. Taft substituent parameters, evaluated at 25 °C are summarized in Tables 3 and 4.

Values in Table 3 show that when the -R group is methyl, ethyl, i-propyl and t-butyl, the $\sigma^*(-R)$ is negative and tends to decrease progressively, meaning that branching at the $\alpha$-carbon improves the electron-releasing effect of these groups in the mentioned sequence. Branching at the $\beta$-carbon, leads to a serial decrease of $\sigma^*(-R)$ but less significantly. The polar effect for the ethyl group is -0.1, for n-propyl is -0.115 and for n-butyl is -0.13. Both Tables 3 and 4 show the opposite trend characterizing the halogens, which are known to be highly electronegative. For this reason, they exert a high electron-attracting influence, which is typical of positive and high $\sigma^*(-R)$.

Introducing bigger substituents on the $\alpha$-carbon or $\beta$-carbon, reduces the system's reactivity due to the increasing steric hindrance of the -R

<figure><img src="image_5.png" /><figcaption>Fig. 3. Tetrahedral intermediates positively (1.) and negatively (2.) charged.</figcaption></figure>
<table><caption>**Table 2**
Some of the reactions and operating conditions studied by Taft for aliphatic systems.</caption><thead><tr><th>Acid</th><th>Base</th></tr></thead><tbody><tr><td>Hydrolysis of ethyl esters in 70 vol-% aqueous acetone, 25 °C (Davies and Evans, 1940; Haskell and Hammett, 1949; Smith and Myers, 1942)</td><td>Hydrolysis of ethyl esters in 70 vol-% aqueous acetone, 25 °C (Hartman and Borders, 1937)</td></tr><tr><td>Esterification of carboxylic acids with methanol, 25 °C (Hartman and Borders, 1937; Hartman and Gassmann, 1940; Smith and Burn, 1944; Smith and Hurley, 1950)</td><td>Hydrolysis of ethyl esters in 85 vol-% aqueous ethanol, 25 °C (Evans et al., 1938; Ingold and Nathan, 1936; Kindler, 1928; Smith and Levenson, 1940)</td></tr><tr><td>Esterification of carboxylic acids with ethanol, 25 °C (Bhide and Sudborough, 1925)</td><td>Hydrolysis of benzyl esters in 60 vol-% aqueous acetone, 25 °C (Tommila, 1941)</td></tr><tr><td>Hydrolysis of ethyl esters in 60 vol-% aqueous acetone, 25 °C (Newling and Hinshelwood, 1936)</td><td>Hydrolysis of methyl esters in water, 25 °C</td></tr><tr><td>Hydrolysis of methyl esters in water, 25 °C</td><td>Hydrolysis of ethyl esters in 60 vol-% water, aqueous acetone, 25 °C (Charton, 1975)</td></tr><tr><td>Esterification of benzoic acids in cyclohexanol, 25 °C (Newling and Hinshelwood, 1936)</td><td>Hydrolysis of n-propyl esters in water, 25 °C</td></tr><tr><td></td><td>Hydrolysis of i-propyl esters in water, 25 °C</td></tr></tbody></table>

<table><caption>**Table 3**
Taft parameters for a class of aliphatic substituents at 25 °C (Taft, 1956).</caption><thead><tr><th>Substituent group (–R)</th><th>E<sub>s</sub>(–R)</th><th>σ<sup>*</sup>(–R)</th></tr></thead><tbody><tr><td>–Cl<sub>3</sub>C</td><td>–2.06</td><td>+2.65</td></tr><tr><td>–Cl<sub>2</sub>CH</td><td>–1.54</td><td>+1.94</td></tr><tr><td>–CH<sub>3</sub>CO</td><td>–</td><td>+1.65</td></tr><tr><td>–ClCH<sub>2</sub></td><td>–0.24</td><td>+1.05</td></tr><tr><td>–H</td><td>+1.24</td><td>+0.49</td></tr><tr><td>–C<sub>6</sub>H<sub>5</sub>CH<sub>2</sub></td><td>–0.38</td><td>+0.215</td></tr><tr><td>–(CH<sub>3</sub>)(C<sub>6</sub>H<sub>5</sub>)CH</td><td>–1.19</td><td>+0.11</td></tr><tr><td>–C<sub>6</sub>H<sub>5</sub>(CH<sub>2</sub>)<sub>2</sub></td><td>–0.38</td><td>+0.08</td></tr><tr><td>–CH<sub>3</sub></td><td>0.00</td><td>0.00</td></tr><tr><td>–C<sub>2</sub>H<sub>5</sub></td><td>–0.07</td><td>–0.10</td></tr><tr><td>–n-C<sub>3</sub>H<sub>7</sub></td><td>–0.36</td><td>–0.115</td></tr><tr><td>–i-C<sub>4</sub>H<sub>9</sub></td><td>–0.93</td><td>–0.125</td></tr><tr><td>–n-C<sub>4</sub>H<sub>9</sub></td><td>–0.39</td><td>–0.13</td></tr><tr><td>–i-C<sub>3</sub>H<sub>7</sub></td><td>–0.47</td><td>–0.19</td></tr><tr><td>–(C<sub>2</sub>H<sub>5</sub>)<sub>2</sub>CH</td><td>–1.98</td><td>–0.225</td></tr><tr><td>–t-C<sub>4</sub>H<sub>9</sub></td><td>–1.54</td><td>–0.30</td></tr><tr><td>–cyclo-C<sub>5</sub>H<sub>9</sub></td><td>–0.51</td><td>–</td></tr><tr><td>n-C<sub>5</sub>H<sub>11</sub></td><td>–0.40</td><td>–</td></tr></tbody></table>

<table><caption>**Table 4**
Taft parameters for a class of o-substituted benzoates at 25 °C (Taft, 1956).</caption><thead><tr><th>Substituent group (–R)</th><th>E<sub>s</sub></th><th>σ<sup>*</sup></th></tr></thead><tbody><tr><td>–OCH<sub>3</sub></td><td>+0.99</td><td>–0.22</td></tr><tr><td>–OC<sub>2</sub>H<sub>5</sub></td><td>+0.90</td><td>–0.18</td></tr><tr><td>–F</td><td>+0.49</td><td>+0.41</td></tr><tr><td>–Cl</td><td>+0.18</td><td>+0.37</td></tr><tr><td>–Br</td><td>0.00</td><td>+0.38</td></tr><tr><td>–CH<sub>3</sub></td><td>0.00</td><td>0.00</td></tr><tr><td>–I</td><td>–0.20</td><td>+0.38</td></tr><tr><td>–NO<sub>2</sub></td><td>–0.75</td><td>+0.97</td></tr><tr><td>–C<sub>6</sub>H<sub>5</sub></td><td>–0.90</td><td>+0.60</td></tr></tbody></table>

group. In Table 4, $E_s(–R)$ assumes progressively more negative values as the –R size increases. For $C_2H_5$, $nC_3H_7$, $nC_4H_9$, and $nC_5H_{11}$, the steric Taft parameters are –0.07, –0.36, –0.39, and –0.40, respectively. This is also true when comparing the reactivity of the system with an aromatic and an aliphatic substituent. In the case of cycloC<sub>5</sub>H<sub>9</sub> and nC<sub>5</sub>H<sub>11</sub>, the respective $E_s(–R)$ values are –0.51 and –0.40, which validates this statement. Finally, the steric effect increases (i.e. $E_s(–R)$ is more

negative) when multiple substituents are introduced.

Despite their deep historical roots in physical organic chemistry, LFER-based models such as the Taft equation have increasing relevance in chemical reaction engineering. The ability to quantify steric and polar effects systematically offers valuable input for building transferable kinetic models, screening reactant candidates, and enabling predictive scale-up strategies. This becomes particularly useful in biomass valorization, where structurally diverse feedstocks and operational variability complicate traditional model development.

With his equation, Taft gave an important contribution to the scientific community toward understanding the link between structure and reactivity. Recently, this equation has been successfully tested for different reaction systems: esterification over homogeneous catalyst (Sahu and Pandit, 2019; Vojtko and Tomčík, 2014), esterification over heterogeneous catalyst (Lilja et al., 2002), alkoxysilanes hydrolysis (Issa et al., 2019), carboxylic acid perhydrolysis (Leveneur et al., 2009), hydrogenation of alkyl levulicates over Ru/C (Delgado et al., 2022) or epoxidation of vegetable oils (Leveneur, 2017).

However, some investigations were missing, such as the dependence of polar and steric effects on temperature or their dependence on the solvent media. In Taft’s work on esterification and hydrolysis of esters, polar and steric effects have been estimated at a fixed temperature of 25 °C. However, an attempt to re-evaluate only the $E_s(–R)$ value at 40 °C was made by MacPhee et al. (MacPhee et al., 1978).

In a recent study by our group, we found that Taft steric and polar substituent parameters for the ethyl substituent were relatively sensitive to reaction temperature (Baco et al., 2024a). We found that the polar ($\sigma^*(-Et)$) and steric ($E_s(–Et)$) Taft parameters for the ethyl group vary linearly with temperature in the range of 20 to 50 °C. The $\sigma^*(-Et)$ value decreases from –0.205 at 20 °C to –0.169 at 50 °C, indicating a slight reduction in the polar effect. In contrast, the $E_s$ value changes from –0.052 to –0.122 over the same temperature range, showing a more pronounced increase in the steric effect. Thus, the temperature sensitivity is greater for the steric parameter ($E_s(–Et)$) than for the polar parameter ($\sigma^*(-Et)$). Overall, increasing the temperature within this range enhances steric contributions while diminishing polar effects. The other motivation of this study (Baco et al., 2024a) came from the fact that kinetic models used for the estimation of Taft substituent parameters lacked of unicity.

Another important unsolved problem is whether the trend observed for the ethyl substituent is consistent with other substituents, and how significant the temperature’s impact is on the Taft substituent parameters.

In this study, the Taft substituent parameters $\sigma^*(-R)$ and $E_s(–R)$ of methyl, ethyl, n-propyl, and 2-chloroethyl groups were re-estimated in a temperature range of 20–50 °C. The re-estimation of these parameters requires knowledge of the rate constants in acid and basic media (Eqs. (13)-(14)). Hence, the first step was to develop detailed kinetic models of levulinic acid esterification via $H_2SO_4$ by the corresponding alcohol and levulinate saponification. In the second step, we calculated $\sigma^*(-R)$ and $E_s(–R)$ for each substituent group across the temperature range.

The choice of the mentioned substituent groups is due, on the one hand, to the desire to study the effect of introducing an increasingly longer carbon chain into the reacting system and, on the other hand, to investigate the role of an already substituted group (i.e. 2-chloroethyl). Esters such as n-propyl levulinate are interesting high-value molecules employed in many different market sectors; therefore, their synthesis is of high interest (Alamgir Ahmad et al., 2022; Vinod and Dutta, 2023; Yamanaka et al., 2025). Moreover, kinetic models for these reactions are not always reliable or consistent; therefore, an evaluation of Taft parameters based on data already available in the literature could lead to uncertainties since different solvents, catalysts, and operating conditions are typically used (Alamgir Ahmad et al., 2022; Bhavsar and Yadav, 2018; Iborra et al., 2019). Finally, although any carboxylic acid could serve as the reference for Taft analysis, levulinic acid (LA) and its
esters were selected due to their growing importance in biomass valorization.

## 2. Experimental section

### 2.1. Chemicals

The following chemicals were used without further purification: levulinic acid (LA) (>= 97 wt%), and 2-chloroethanol (>= 99 wt%) were purchased from Sigma Aldrich. Methanol absolute (>= 99.7 wt%), ethanol absolute (>= 99.7 wt%), 1-propanol (>= 99.5 wt%), and acetone (=99.8 wt%) were supplied by VWR Chemicals. Methyl levulinate (>= 98 wt%), ethyl levulinate (>= 98 wt%) and propyl levulinate (>= 98 wt%) were purchased from Acros Organics. Sulfuric acid (>= 98 wt%) was obtained from Honeywell Fukla Chemicals. 2-chloroethyl levulinate (>= 87 wt%) has been synthesized and purified in the laboratory following a specific protocol, described below. Magnesium sulfate (>=98.0 wt%), dichloromethane (>=99.9 wt%) and sodium hydrogen carbonate (>=99.9 wt%) were obtained by Sigma Aldrich. Finally, sodium hydroxide pellets were purchased from Panreac. Distilled water (W) was used.

### 2.2. Esterification and saponification procedures

Esterification reactions were conducted in a 300 mL jacketed glass reactor, operating in batch mode at atmospheric pressure. The reactor was equipped with a circulating water jacket connected to a thermostatic bath for temperature control, a condenser to minimize solvent loss due to evaporation, and a mechanical stirrer operating at 450 rpm to ensure the homogeneity of the reaction mixture.

For each experiment, a mixture of the alcohol and solvent (water) was first charged into the reactor. Once the system reached the set temperature, a preheated acidic solution of levulinic acid and sulfuric acid was introduced into the vessel. The reaction temperature was maintained throughout the experiment. Samples were withdrawn every 10 min during the first 30 min and then hourly thereafter, and were stored in a refrigerator to quench the reaction before analysis by GC-FID.

The maximum reaction temperature was limited by the boiling point of the alcohol under atmospheric pressure. For example, during levulinic acid esterification with ethanol, the reaction was capped at 70 °C, as ethanol boils at approximately 78 °C at 1 bar.

The experimental matrices for esterification using methanol, ethanol, n-propanol, and 2-chloroethanol are provided in Tables S1.1–S1.4.

Saponification reactions were performed in an 800 mL jacketed glass reactor under the same batch and atmospheric conditions. The setup included a reflux condenser, a mechanical stirrer (450 rpm), and a conductivity probe for online monitoring. Water was used as the heat transfer medium.

A known amount of solid NaOH was dissolved in water to prepare the alkaline solution, which was then transferred to the reactor. After reaching a stable temperature and baseline conductivity, a preheated ester was added via syringe. A rapid drop in conductivity indicated the start of the reaction. Conductivity and temperature were logged periodically for 10–15 min, until conductivity stabilized, indicating reaction completion.

Experimental matrices for the saponification of methyl, ethyl, n-propyl, and 2-chloroethyl levulinate are presented in Tables S2.1–S2.4.

The methodology for conductivity measurements is detailed in Section S3.

### 2.3. Analytical methods

Several analytical techniques were initially tested to quantify esterification and transesterification products, including aqueous NaOH titration, pH measurements, and gas chromatography with flame ionization detection (GC-FID). Among these, GC-FID was selected as the most accurate and reliable method for ester quantification.

GC analyses were performed using a Bruker Scion GC436 system equipped with a flame ionization detector (FID), an autosampler, and a capillary column (Rxi-5 ms, 30 m × 0.25 mm i.d., 0.25 µm film thickness). Helium (99.99 %) was used as the carrier gas at a constant flow rate of 1.2 mL·min⁻¹. An injection volume of 1 µL was used in split mode (20:1).

The column temperature was initially held at 40 °C for 1 min, then increased at a rate of 20 °C·min⁻¹ up to 200 °C. The injector and detector temperatures were both maintained at 250 °C. These parameters, optimized based on the volatility and polarity of the compounds, are summarized in Table 5.

To establish retention times for each compound, standard solutions of pure reference chemicals were prepared. Approximately 20 mg of each analyte was dissolved in acetone and diluted to 10 mL. Aliquots (50 µL) were further diluted with 950 µL of acetone, vortexed to ensure homogeneity, and injected into the GC-FID system.

The retention times (RTs) of the relevant components are listed in Table 6.

### 2.4. Synthesis and purification of 2-chloroethyl levulinate

2-Chloroethyl levulinate was synthesized via acid-catalyzed esterification of levulinic acid with 2-chloroethanol. A fixed amount of levulinic acid was placed in a round-bottom flask fitted with a magnetic stirrer and a condenser, and submerged in a water bath maintained at ~ 80 °C. A few drops of concentrated sulfuric acid were added to catalyze the reaction.

In parallel, the reactants (levulinic acid and 2-chloroethanol) were preheated separately until they reached the desired temperature. The preheated 2-chloroethanol was then added dropwise to the acid mixture. The reaction was allowed to proceed under reflux for 8 h.

After completion, the reaction mixture was transferred to a separatory funnel, and a few milliliters of dichloromethane were added to facilitate extraction. The organic phase was washed three times with saturated sodium hydrogen carbonate solution to neutralize residual acid. The mixture was allowed to stand for phase separation, after which the organic layer was collected.

To remove traces of water, anhydrous magnesium sulfate was added to the organic phase and stirred. The drying agent was subsequently removed by vacuum filtration, and the filtrate was concentrated using a rotary evaporator (rotavapor) operated at 60 °C and 67 mbar for 3–4 h.

The resulting viscous organic phase, rich in 2-chloroethyl levulinate, was subjected to NMR analysis for structural confirmation and purity assessment (Fig. 3).

The ¹H NMR spectrum in Fig. 4. confirms the successful synthesis of 2-chloroethyl levulinate, showing characteristic signals corresponding to the methylene protons adjacent to the ester (–CH₂O–), the chloroethyl group (–CH₂Cl), and the methylene groups adjacent to the carbonyl (–CH₂CO–).

### 2.5. Kinetic modeling

Athena Visual Studio was used to solve the ordinary differential equations (ODEs) and algebraic equations associated with the reaction kinetics and to estimate kinetic and equilibrium parameters. The

<table><caption>**Table 5**
GC-FID method parameters for esterification and transesterification analysis.</caption><tr><td>Detector temperature</td><td>250 °C</td></tr><tr><td>Oven temperature</td><td>40 °C (1 min) – 20 °C min<sup>-1</sup> – 200 °C</td></tr><tr><td>Injection temperature</td><td>250 °C</td></tr><tr><td>Injection volume</td><td>1 µL</td></tr><tr><td>Split ratio</td><td>20:1</td></tr></table>
<table><caption>**Table 6**
Retention times of key compounds.</caption><thead><tr><th>Chemical species</th><th>Retention time (min)</th></tr></thead><tbody><tr><td>Acetone</td><td>2.70</td></tr><tr><td>Levulinic acid (LA)</td><td>6.03</td></tr><tr><td>Methyl levulinate (ML)</td><td>7.14</td></tr><tr><td>Ethyl levulinate (EL)</td><td>7.84</td></tr><tr><td>Propyl levulinate (PL)</td><td>8.23</td></tr><tr><td>2-chloroethyl levulinate (2-ClEL)</td><td>9.03</td></tr></tbody></table>

software employs Bayesian inference for parameter estimation. ODEs and algebraic equations were solved simultaneously using the DDAPLUS package via a damped Newton method, while GREGPLUS was used to obtain optimal parameter values along with 95 % confidence intervals, expressed via highest posterior density (HPD). GREGPLUS also provided the correlation matrix (Stewart and Caracotsios, 2007).

where, $k_{est.RL}$ is the rate constant and $K_{est.RL}$ is the equilibrium constant of esterification. The superscript w indicates that these constants were estimated in water solvent.

The temperature dependence of the equilibrium constant was modeled via the van't Hoff equation:

$$ \ln \frac{K_{est.RL}(T)}{K_{est.RL}(T_{ref})} = \frac{-\Delta H_{R}}{R} \left( \frac{1}{T} - \frac{1}{T_{ref}} \right) \quad (16) $$

where, $K_{est.RL}(T)$ and $K_{est.RL}(T_{ref})$ are the equilibrium constants at a temperature T and a reference temperature, respectively. The term $\Delta H_{R}$ is the reaction enthalpy, and R is the universal gas constant.

The hydronium ion concentration, $[H_3O^+]$, was calculated using the following expression, which accounts for sulfuric acid dissociation and the carboxylic acid equilibrium:

$$ [H_3O^+] = \frac{1}{2} [H_2SO_4]_0 + \sqrt{\frac{[H_2SO_4]_0^2}{4} + 2 \cdot K_{II,w} \cdot [H_2SO_4]_0 \cdot [H_2O] + K_{III,w} \cdot [H_2O] \cdot [LA]} \quad (17) $$

All experiments were conducted under isothermal conditions. The material balance equations and derivations are provided in the Supplementary Information (Section S4).

As shown in our previous study (Baco et al., 2022), the rate expression for esterification in water was defined as:

$$ r_{esterification} = k_{est.RL} * \frac{[H_3O^+]}{[H_2O]} * \left( [LA] * [ROH] - \frac{1}{K_{est.RL}} * [RL] * [H_2O] \right) \quad (15) $$

where, $[H_2SO_4]_0$ is the initial concentration of sulfuric acid; $K_{II,w}$ is the second dissociation of sulfuric acid and $K_{III,w}$ is the carboxylic acid dissociation (Table S4.1 in SI) (Baco et al., 2022). Eq. (17) was used to calculate the concentration of hydroxonium ion during the modeling.

A modified Arrhenius equation was used to reduce correlation between rate constants and activation energies (Buzzi-Ferraris, 1999):

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** NMR spectra for 2-chloroethyl levulinate.</figcaption></figure>
$$ \ln(\mathbf{k}_{\text{reaction,RL}}(\mathbf{T})) = \left[ \ln(\mathbf{k}_{\text{reaction,RL}}(\mathbf{T}_{\text{ref}})) + \frac{\mathbf{E}_{\text{a,reaction,RL}}}{\mathbf{R} \cdot \mathbf{T}_{\text{ref}}} \cdot \left(1 - \frac{\mathbf{T}_{\text{ref}}}{\mathbf{T}}\right) \right] \quad (18) $$

Using Eq. (18) decreases the correlation between the rate constant and activation energy.

From this model, the following parameters, a.k.a kinetic and thermodynamic constants, were estimated:

$\ln(\mathbf{k}_{\text{reaction,RL}}(\mathbf{T}_{\text{ref}}))$: rate constant at reference temperature,

$\mathbf{E}_{\text{a,reaction,RL}}/\mathbf{R} \cdot \mathbf{T}_{\text{ref}}$: activation term,

$\ln(\mathbf{k}_{\text{est,RL}}(\mathbf{T}_{\text{ref}}))$: equilibrium constant at reference temperature,

$\frac{\Delta H_R}{\mathbf{R} \cdot \mathbf{T}_{\text{ref}}}$: enthalpy term.

The alkyl levulinate concentration was used as the observable for model fitting. The contribution of hydronium ions was separated from the intrinsic rate constant by explicitly calculating $[\text{H}_3\text{O}^+]$ via acid dissociation equilibria.

For saponification, the rate was expressed as:

$$ \mathbf{r}_{\text{saponification}} = \mathbf{k}_{\text{sap,RL}} * \left( \frac{[\text{RL}] * [\text{OH}^-]}{\mathbf{K}_{\text{sap,RL}}} * [\text{ROH}] * [\text{LA}^-] \right) \quad (19) $$

The concentration of hydroxide ions ($[\text{OH}^-]$) was used as the observable. The same modified Arrhenius formulation (Eq. (18)) was used to model temperature dependence, and Eq. (16) was used for the expression of the equilibrium constant.

The following constants were estimated $\ln(\mathbf{k}_{\text{sap,RL}}(\mathbf{T}_{\text{ref}}))$, $\frac{\mathbf{E}_{\text{a,sap,RL}}}{\mathbf{R} \cdot \mathbf{T}_{\text{ref}}}$, $\ln(\mathbf{K}_{\text{sap,RL}}(\mathbf{T}_{\text{ref}}))$ and $\frac{\Delta H_R}{\mathbf{R} \cdot \mathbf{T}_{\text{ref}}}$.

For both reaction systems, the objective function minimized by GREGPLUS was:

$$ \text{OF} = (C_{\text{exp}} - C_{\text{estimated}})^2 \quad (20) $$

# 3. Results and discussion

## 3.1. Experimental results

To investigate the kinetics and thermodynamics of esterification reactions, the evolution of alkyl levulinate concentrations (PL, BL, and 2-CIEL) was monitored. For comparative purposes, the ester concentration was normalized to the initial levulinic acid concentration, i.e., $\frac{[\text{RL}]}{[\text{LA}]_0}$. In saponification experiments, the normalized hydroxide concentration $\frac{[\text{OH}^-]}{[\text{OH}^-]_0}$ was considered.

Repeatability tests confirmed that the results were reproducible across runs (S4).

<figure><img src="image_7.png" /><figcaption>Fig. 5. Normalized alkyl levulinate concentration with time for the esterification of methanol, ethanol, n-propanol and 2-chloroethanol at 50 °C and 0.1 mol/L of H₂SO₄.</figcaption></figure>

Fig. 5 shows the normalized concentrations of alkyl levulinates (RL) over time for esterification with five alcohols: methanol, ethanol, n-propanol, and 2-chloroethanol, all conducted at 50 °C with 0.1 mol·L⁻¹ sulfuric acid.

As shown in Fig. 5 and supported by previous studies (Marchetti and Errazu, 2008; Pappu et al., 2013; Sahu and Pandit, 2019), reactivity generally decreases with increasing alkyl chain length or branching.

This is consistent with established steric effects, where bulkier alcohols reduce the nucleophilic attack on the carbonyl carbon.

While steric hindrance is the dominant factor in esterification kinetics, polarity effects also play a role, especially under basic conditions (Alonso et al., 2009; Bowden et al., 1968; Liu et al., 2006a, 2006b; Taft, 1952a). Lotero et al. showed that longer-chain acids reduce carbonyl electrophilicity via inductive effects, making nucleophilic attack more energetically demanding (Liu et al., 2006b).

Fig. 6 shows the normalized hydroxide concentration over time for the saponification of methyl, ethyl, propyl and 2-chloroethyl levulinates at 40 °C, under equimolar conditions and in the presence of excess aqueous NaOH.

The observed saponification kinetics align with trends typically seen in esterification reactions: the reaction rate generally decreases with increasing alkyl chain length. Among the tested esters, methyl levulinate (ML) exhibits the fastest saponification rate, followed by ethyl (EL), 2-chloroethyl (2-CIEL) and propyl (PL).

2-chloroethyl levulinate undergoes saponification faster than propyl levulinate, while the corresponding esterification reaction with 2-chloroethanol proceeds more slowly than with n-propanol.

Literature data support these trends. Fukuto and Metcalf (1959) (Fukuto and Metcalf, 1959) observed a marked decrease in saponification rates from methyl to ethyl esters of alkyl phosphonates, with less pronounced differences for longer chains. Similarly, Hudson and Keay (1956) (Hudson and Keay, 1956) reported a clear reduction in hydrolysis rates for dimethyl, diethyl, and dipropyl methylphosphonates. Additionally, polar effects are likely significant in these basic conditions. According to Taft's polar substituent constant, polar effects are more pronounced under basic rather than acid-catalyzed conditions and should not be neglected in interpreting kinetic behavior (Bowden et al., 1968; Taft, 1952a).

## 3.2. Kinetic modeling

Athena Visual Studio was used to develop the concentration-based kinetic models. GREGPLUS package was used to minimize the objective function: $\text{OF} = (c_{\text{exp}} - c_{\text{estimated}})^2$ where $c_{\text{exp}}$ is the concentration of the species followed experimentally while $c_{\text{estimated}}$ is the concentration of the species simulated with the software.

For the saponification kinetic modeling, we observed that the reaction was quasi-irreversible and that the equilibrium constant was effectively independent of temperature. Consequently, the modeling gave better results when the term $\frac{\Delta H_R}{RT_{\text{ref}}}$ was omitted.

### 3.2.1. Levulinic acid esterification with methanol

Table 7 reports the estimated parameters for methyl levulinate (ML) synthesis at $T_{\text{ref}} = 44$ °C, along with their 95 % highest posterior density (HPD) credible intervals.

The low HPD values indicate that the estimated parameters are reliable. This is confirmed by the parity plot in Fig. 7A, where the predicted values closely match the experimental methyl levulinate concentrations. A representative model fit for experiment Run 4 M, including 95 % prediction intervals, is shown in Fig. 7B.

It can be observed that the rate constant, equilibrium constant, activation energy, and reaction enthalpy are only slightly correlated (Table 8), and the correlations are far from ± 1, indicating good overall parameter identifiability.
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Normalized OH⁻ concentration with time for the saponification of methyl, ethyl, propyl and 2-chloroethyl levulinate at 40 °C and a ratio between reagents of 1:1. All reactions were performed in excess of NaOH aqueous solution.</figcaption></figure>

<table><caption>**Table 7**
Estimated values at $T_{ref}$ of 44.17 °C and Bayesian statistical data for LA esterification with methanol (Baco et al., 2024b).</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>est.ML</sub> (T<sub>ref</sub>))</td><td>L<sup>2</sup>*mol<sup>-2</sup>*min<sup>-1</sup></td><td>-3.57</td><td>0.03</td></tr><tr><td>E<sub>a.est.ML</sub>/RT<sub>ref</sub></td><td>—</td><td>24.78</td><td>1.72</td></tr><tr><td>ln(K<sub>est.ML</sub> (T<sub>ref</sub>))</td><td>—</td><td>1.87</td><td>0.23</td></tr><tr><td>ΔH<sub>R.ML</sub>/RT<sub>ref</sub></td><td>—</td><td>27.14</td><td>12.11</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>A. Parity plot for LA esterification with MeOH in water.</figcaption></figure>

<caption>**Fig. 7A.** Parity plot for LA esterification with MeOH in water.</caption>

### 3.2.2. Levulinic acid esterification with ethanol

Table 9 presents the estimated kinetic and thermodynamic parameters for ethyl levulinate (EL) synthesis at $T_{ref}$ = 64 °C, together with their 95 % highest posterior density (HPD) credible intervals. The low HPD values for most parameters indicate a reliable estimation, although the relatively larger interval for the reaction enthalpy suggests a higher uncertainty for.

<figure><img src="image_10.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the synthesis of ML in water solvent (Run 4M).</figcaption></figure>

<caption>**Fig. 7B.** Fit of Model to the experimental concentrations with prediction intervals for the synthesis of ML in water solvent (Run 4 M).</caption>

<table><caption>**Table 8**
Correlation matrix for ML synthesis in water solvent (Baco et al., 2024b).</caption><thead><tr><th></th><th>ln(k<sub>est.ML</sub> (T<sub>ref</sub>))</th><th>E<sub>a.est.ML</sub>/RT<sub>ref</sub></th><th>ln(K<sub>est.ML</sub> (T<sub>ref</sub>))</th><th>ΔH<sub>R.ML</sub>/RT<sub>ref</sub></th></tr></thead><tbody><tr><td>ln(k<sub>est.ML</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>E<sub>a.est.ML</sub>/RT<sub>ref</sub></td><td>-0.506</td><td>1</td><td>0</td><td>0</td></tr><tr><td>ln(K<sub>est.ML</sub> (T<sub>ref</sub>))</td><td>-0.707</td><td>0.367</td><td>1</td><td>0</td></tr><tr><td>ΔH<sub>R.ML</sub>/RT<sub>ref</sub></td><td>0.401</td><td>-0.776</td><td>-0.278</td><td>1</td></tr></tbody></table>

ΔH<sub>R.EL</sub>/RT<sub>ref</sub>. The parity plot in Fig. 8A shows good agreement between simulated and experimental EL concentrations, supporting the quality of the fit. A representative model fit for experiment Run 6E, including 95 % prediction intervals, is given in Fig. 8B.
<table><caption>**Table 9**
Estimated parameter values at $T_{ref}=64$ °C for EL synthesis (Baco et al., 2024b).</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>est.EL</sub> (T<sub>ref</sub>))</td><td>L<sup>2</sup>·mol<sup>-2</sup>·min<sup>-1</sup></td><td>-2.46</td><td>0.08</td></tr><tr><td>E<sub>a.est.ML</sub> / RT<sub>ref</sub></td><td></td><td>21.82</td><td>2.42</td></tr><tr><td>ln(K<sub>est.ML</sub> (T<sub>ref</sub>))</td><td></td><td>1.14</td><td>0.18</td></tr><tr><td>ΔH<sub>R.EL</sub> / RT<sub>ref</sub></td><td></td><td>18.18</td><td>5.80</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>A. Parity plot for LA esterification with EtOH.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the synthesis of EL in water solvent (Run 6E)</figcaption></figure>

<caption>**Fig. 8B.** Fit of Model to the experimental concentrations with prediction intervals for the synthesis of EL in water solvent (Run 6E).</caption>

<table><caption>**Table 10**
Correlation matrix for EL synthesis (Baco et al., 2024b).</caption><thead><tr><th></th><th>ln(k<sub>est.EL</sub> (T<sub>ref</sub>))</th><th>E<sub>a.est.ML</sub> / RT<sub>ref</sub></th><th>ln(K<sub>est.EL</sub> (T<sub>ref</sub>))</th><th>ΔH<sub>R.EL</sub> / RT<sub>ref</sub></th></tr></thead><tbody><tr><td>ln(k<sub>est.EL</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>E<sub>a.est.ML</sub> / RT<sub>ref</sub></td><td>0.074</td><td>1</td><td>0</td><td>0</td></tr><tr><td>ln(K<sub>est.EL</sub> (T<sub>ref</sub>))</td><td>-0.229</td><td>-0.065</td><td>1</td><td>0</td></tr><tr><td>ΔH<sub>R.EL</sub> / RT<sub>ref</sub></td><td>-0.073</td><td>-0.209</td><td>0.866</td><td>1</td></tr></tbody></table>

<table><caption>**Table 11**
Estimated parameter values at $T_{ref} = 62.5$ °C and Bayesian statistical data for PL synthesis in water.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>est.PL</sub> (T<sub>ref</sub>))</td><td>L·mol<sup>-1</sup>·min<sup>-1</sup></td><td>-2.62</td><td>0.07</td></tr><tr><td>E<sub>a.est.PL</sub> / RT<sub>ref</sub></td><td></td><td>27.43</td><td>1.86</td></tr><tr><td>ln(K<sub>est.PL</sub> (T<sub>ref</sub>))</td><td></td><td>2.35</td><td>0.28</td></tr><tr><td>ΔH<sub>R.PL</sub> / RT<sub>ref</sub></td><td></td><td>21.02</td><td>7.99</td></tr></tbody></table>

The correlation matrix (Table 10) indicates minimal correlation among most parameters, with the only notable association being between ln(K<sub>est.ML</sub> (T<sub>ref</sub>)) and ΔH<sub>R.EL</sub>/RT<sub>ref</sub> (0.866), as expected from the van't Hoff relationship.

### 3.2.3. Levulinic acid esterification with n-propanol

Table 11 presents the estimated kinetic and thermodynamic constants for propyl levulinate (PL) synthesis at a reference temperature of 62.5 °C. These include the rate constant, activation energy, equilibrium constant, and reaction enthalpy. The associated 95 % highest posterior density (HPD) intervals are relatively narrow, indicating good parameter identifiability.

The correlation matrix in Table 12 shows moderate to low correlation among the parameters. Although some interdependencies exist, especially between activation energy and other parameters, none of the coefficients exceed 0.95 (Toch et al., 2015).

Fig. 9A compares experimental and simulated PL concentrations, yielding an R² of 0.98. Fig. 9B illustrates the model fit for experimental run 2P. The model performs well overall, although it underestimates PL concentration beyond 180 min.

### 3.2.4. Levulinic acid esterification with 2-chloroethanol

The same modeling procedure was applied to LA esterification with 2-chloroethanol at 65 °C. Table 13 summarizes the estimated parameters, all with relatively narrow HPD intervals, indicating reliable estimation.

Figs. 10A and 10B confirm good model performance with an R² of 0.98. The correlation matrix (Table 14) reveals stronger interdependence between activation energy and both the rate and equilibrium constants, particularly a high correlation (0.961) between activation energy and rate constant. Despite this, parameter estimates remain within acceptable independence thresholds for practical modeling purposes.

The observed temperature-dependent increase in equilibrium constant suggests the reaction is endothermic. However, this behavior is not universal across esterifications.

### 3.2.5. Methyl levulinate saponification

Table 15 presents the estimated kinetic parameters for methyl levulinate (ML) saponification at $T_{ref}=42$ °C, together with their 95 % highest posterior density (HPD) credible intervals.

Both the rate constant and the activation energy are estimated with high precision, as indicated by the small HPD intervals (< 2 % of the

<table><caption>**Table 12**
Correlation matrix of estimated constants for PL synthesis.</caption><thead><tr><th></th><th>ln(k<sub>est.PL</sub> (T<sub>ref</sub>))</th><th>E<sub>a.est.PL</sub> / RT<sub>ref</sub></th><th>ln(K<sub>est.PL</sub> (T<sub>ref</sub>))</th><th>ΔH<sub>R.PL</sub> / RT<sub>ref</sub></th></tr></thead><tbody><tr><td>ln(k<sub>est.PL</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>E<sub>a.est.PL</sub> / RT<sub>ref</sub></td><td>0.610</td><td>1</td><td>0</td><td>0</td></tr><tr><td>ln(K<sub>est.PL</sub> (T<sub>ref</sub>))</td><td>-0.647</td><td>-0.527</td><td>1</td><td>0</td></tr><tr><td>ΔH<sub>R.PL</sub> / RT<sub>ref</sub></td><td>0.031</td><td>-0.769</td><td>0.114</td><td>1</td></tr></tbody></table>
<figure><img src="image_13.png" /><figcaption>A. Parity plot for LA esterification with n-propanol in water solvent.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the LA esterification with n-propanol (Run 2P).</figcaption></figure>

<caption>Fig. 9B. Fit of Model to the experimental concentrations with prediction intervals for the LA esterification with n-propanol (Run 2P).</caption>

<caption>Table 13</caption>

<table><caption>Estimated parameter values at T<sub>ref</sub> = 65 °C and Bayesian statistical data for 2-CIEL synthesis.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</td><td>L<sup>2</sup>.mol<sup>-2</sup>.min<sup>-1</sup></td><td>-3.31</td><td>0.16</td></tr><tr><td>E<sub>a.est.2-CIEL</sub>/RT<sub>ref</sub></td><td></td><td>28.52</td><td>4.16</td></tr><tr><td>ln(K<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</td><td></td><td>0.48</td><td>0.32</td></tr><tr><td>ΔH<sub>R.2-CIEL</sub>/RT<sub>ref</sub></td><td></td><td>6.29</td><td>8.22</td></tr></tbody></table>

estimated values). The parity plot in Fig. 11A confirms that the model reproduces the experimental [OH⁻] data very accurately, with R²≈0.996. A representative model fit for Run 1SM, including 95 % prediction intervals, is shown in Fig. 11B.

<figure><img src="image_15.png" /><figcaption>A. Parity plot for LA esterification with 2-chloroethanol in water solvent.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the LA esterification with 2ClEtOH (Run10-2ClEL).</figcaption></figure>

<caption>Fig. 10B. Fit of Model to the experimental concentrations with prediction intervals for the LA esterification with 2ClEtOH (Run10-2ClEL).</caption>

The correlation matrix indicates a moderate correlation between the activation energy and rate constant (0.790), as often observed in Arrhenius-type fits, and a negative correlation between.

ln(K<sub>sap.ML</sub>(T<sub>ref</sub>)) and the kinetic constants. Overall, the small HPD values and high R² confirm that the kinetic model accurately describes the ML saponification data (Table 16.).

### 3.2.6. Ethyl levulinate saponification

Table 17 summarizes the estimated kinetic parameters for ethyl levulinate (EL) saponification at T<sub>ref</sub> = 41.7 °C, along with their 95 % highest posterior density (HPD) credible intervals.

The small HPD intervals indicate high precision in the estimated rate constant and activation energy, with slightly higher uncertainty for the equilibrium constant. Fig. 12A shows that the parity plot yields an R² of 0.985, confirming a strong agreement between simulated and
<table><caption>**Table 14**
Correlation matrix of estimated constant for 2-CIEL synthesis.</caption><thead><tr><th></th><th>ln(k<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</th><th>E<sup>c</sup><sub>a.2-CIEL</sub>/RT<sub>ref</sub></th><th>ln(K<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</th><th>ΔH<sub>R.2-CIEL</sub>/RT<sub>ref</sub></th></tr></thead><tbody><tr><td>ln(k<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>E<sup>c</sup><sub>a.2-CIEL</sub>/RT<sub>ref</sub></td><td>0.961</td><td>1</td><td>0</td><td>0</td></tr><tr><td>ln(K<sub>est.2-CIEL</sub> (T<sub>ref</sub>))</td><td>-0.806</td><td>-0.798</td><td>1</td><td>0</td></tr><tr><td>ΔH<sub>R.2-CIEL</sub>/RT<sub>ref</sub></td><td>0.707</td><td>-0.726</td><td>0.866</td><td>1</td></tr></tbody></table>

<table><caption>**Table 15**
Estimated parameter values at T<sub>ref</sub> = 42 °C and Bayesian statistical data for ML saponification.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>sap.ML</sub> (T<sub>ref</sub>))</td><td>L·mol<sup>-1</sup>·min<sup>-1</sup></td><td>4.30</td><td>0.07</td></tr><tr><td>E<sub>a.sap.ML</sub>/RT<sub>ref</sub></td><td>—</td><td>13.78</td><td>1.28</td></tr><tr><td>ln(K<sub>sap.ML</sub> (T<sub>ref</sub>))</td><td>—</td><td>7.74</td><td>3.48</td></tr></tbody></table>

<figure><img src="image_17.png" /><figcaption>A. Parity plot for ML saponification in water solvent.</figcaption></figure>

<caption>**Fig. 11A.** Parity plot for ML saponification in water solvent.</caption>

experimental [OH⁻] values. Fig. 12B presents the model fit for Run 5SE, including 95 % prediction intervals, which closely match the experimental data.

All correlation values are below 0.90, indicating no severe parameter correlation and supporting the reliability of the parameter estimation, as previously observed for ML saponification (Table 18).

### 3.2.7. Propyl levulinate saponification

Saponification of PL was modeled at T<sub>ref</sub> = 27.6 °C. Table 19 summarizes the estimated parameters. Figs. 13A and 13B show good agreement between simulated and experimental concentrations (R² = 0.96).

Table 20 shows low to moderate correlations between estimated parameters. No strong linear dependencies are evident.

#### 2-chloroethyl levulinate saponification

Saponification of 2-CIEL was studied at T<sub>ref</sub> = 42 °C. Estimated parameters are listed in Table 24, with model fits shown in Figs. 14A and 14B. The model performance is strong across the observed time range.

Table 21 shows no significant correlation between estimated parameters, suggesting a stable and well-defined model. Table 22 shows that the estimated parameters are not correlated.

<figure><img src="image_18.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the saponification of ML (Run 1SM).</figcaption></figure>

<caption>**Fig. 11B.** Fit of Model to the experimental concentrations with prediction intervals for the saponification of ML (Run 1SM).</caption>

<table><caption>**Table 16**
Correlation matrix for ML saponification.</caption><thead><tr><th></th><th>ln(k<sub>sap.ML</sub> (T<sub>ref</sub>))</th><th>E<sub>a.sap.ML</sub>/RT<sub>ref</sub></th><th>ln(K<sub>sap.ML</sub> (T<sub>ref</sub>))</th></tr></thead><tbody><tr><td>ln(k<sub>sap.ML</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td></tr><tr><td>E<sub>a.sap.ML</sub>/RT<sub>ref</sub></td><td>0.790</td><td>1</td><td>0</td></tr><tr><td>ln(K<sub>sap.ML</sub> (T<sub>ref</sub>))</td><td>-0.812</td><td>-0.592</td><td>1</td></tr></tbody></table>

<table><caption>**Table 17**
Estimated parameter values at T<sub>ref</sub> = 41.7 °C and Bayesian statistical data for EL saponification.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>sap.EL</sub> (T<sub>ref</sub>))</td><td>L·mol<sup>-1</sup>·min<sup>-1</sup></td><td>3.03</td><td>0.07</td></tr><tr><td>E<sub>a.sap.EL</sub>/RT<sub>ref</sub></td><td>—</td><td>14.24</td><td>0.74</td></tr><tr><td>ln(K<sub>sap.EL</sub> (T<sub>ref</sub>))</td><td>—</td><td>7.91</td><td>1.19</td></tr></tbody></table>

Both Figs. 15A and 15B are obtained by calculating the rate constant evolution with temperature using the Arrhenius equation Eq. (18) and the estimated parameters obtained at a reference temperature via Athena Visual Studio. These plots show kinetic trends in agreement with what was expected. It is shown that in both reactions a longer carbon chain typically slows down the reaction of esterification or saponification, meaning in this case that k<sub>reaction.ML</sub> > k<sub>reaction.EL</sub> > k<sub>reaction.PL</sub>, even if saponification reactions have very high values of rate constants
<figure><img src="image_19.png" /><figcaption>A. Parity plot for EL saponification in water solvent.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the saponification of EL (Run 5SE)</figcaption></figure>

<caption>**Fig. 12B.** Fit of Model to the experimental concentrations with prediction intervals for the saponification of EL (Run 5SE).</caption>

<table><caption>**Table 18**
Normalized covariance matrix for EL saponification at 41.7 °C.</caption><thead><tr><th></th><th>ln(k<sub>sap,EL</sub> (T<sub>ref</sub>))</th><th>E<sub>a,sap,EL</sub> / RT<sub>ref</sub></th><th>ln(K<sub>sap,EL</sub> (T<sub>ref</sub>))</th></tr></thead><tbody><tr><td>ln(k<sub>sap,EL</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td></tr><tr><td>E<sub>a,sap,EL</sub> / RT<sub>ref</sub></td><td>0.630</td><td>1</td><td>0</td></tr><tr><td>ln(K<sub>sap,EL</sub> (T<sub>ref</sub>))</td><td>-0.275</td><td>0.771</td><td>1</td></tr></tbody></table>

compared to the esterification, which confirms that the nature of these two reactions is different. In fact, at 20 °C, kinetic constants for saponification reactions are in a range of [5; 30] L·mol⁻¹·min⁻¹ while esterification reactions, at the same temperature, show a kinetic constant in a range of [8·10⁻⁴; 3.5·10⁻³] L·mol⁻¹·min⁻¹. Surprisingly, 2-chloroethanol shows the lowest reactivity in esterification reactions; however, in saponification reactions, 2-chloroethyl levulinate is characterized by a rate constant higher than that of ethanol.

<table><caption>**Table 19**
Estimated parameter values at T<sub>ref</sub> = 27.6 °C and Bayesian statistical data for PL saponification.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln(k<sub>sap,PL</sub> (T<sub>ref</sub>))</td><td>L·mol<sup>-1</sup>·min<sup>-1</sup></td><td>1.61</td><td>0.08</td></tr><tr><td>E<sub>a,sap,PL</sub> / RT<sub>ref</sub></td><td>—</td><td>12.97</td><td>1.77</td></tr><tr><td>ln(K<sub>sap,PL</sub> (T<sub>ref</sub>))</td><td>—</td><td>8.03</td><td>0.98</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>A. Parity plot for PL saponification reaction in water solvent.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the PL saponification (Run 7SP).</figcaption></figure>

<caption>**Fig. 13B.** Fit of Model to the experimental concentrations with prediction intervals for the PL saponification (Run 7SP).</caption>

<table><caption>**Table 20**
Correlation matrix of estimated constants for PL saponification.</caption><thead><tr><th></th><th>ln(k<sub>sap,PL</sub> (T<sub>ref</sub>))</th><th>E<sub>a,sap,PL</sub> / RT<sub>ref</sub></th><th>ln(K<sub>sap,PL</sub> (T<sub>ref</sub>))</th></tr></thead><tbody><tr><td>ln(k<sub>sap,PL</sub> (T<sub>ref</sub>))</td><td>1</td><td>0</td><td>0</td></tr><tr><td>E<sub>a,sap,PL</sub> / RT<sub>ref</sub></td><td>-0.322</td><td>1</td><td>0</td></tr><tr><td>ln(K<sub>sap,PL</sub> (T<sub>ref</sub>))</td><td>-0.551</td><td>0.4961</td><td>1</td></tr></tbody></table>
<figure><img src="image_23.png" /><figcaption>A. Parity plot for 2-CIEL saponification reaction in water solvent.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>B. Fit of Model to the experimental concentrations with prediction intervals for the 2-CIEL saponification (Run 4SCE).</figcaption></figure>

<caption>Fig. 14B. Fit of Model to the experimental concentrations with prediction intervals for the 2-CIEL saponification (Run 4SCE).</caption>

<caption>Table 21</caption>

<table><caption>Estimated parameter values at $T_{ref}$ = 42.0 °C and Bayesian statistical data for 2-CIEL saponification.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Estimated value</th><th>95 % HPD</th></tr></thead><tbody><tr><td>ln($k_{sap,2-CIEL}$ ($T_{ref}$))</td><td>L·mol<sup>-1</sup>·min<sup>-1</sup></td><td>3.20</td><td>0.09</td></tr><tr><td><sup>E<sub>a,sap,2-CIEL</sub></sup>&frasl;<sub>RT<sub>ref</sub></sub></td><td></td><td>6.91</td><td>2.20</td></tr><tr><td>ln($K_{sap,2-CIEL}$ ($T_{ref}$))</td><td>—</td><td>7.83</td><td>4.13</td></tr></tbody></table>

At a constant temperature, the steps between the kinetic constants of the different substituents are approximately equidistant for esterification. E.g., at 50 °C, $k_{est,ML}$ = 0.044 L·mol⁻¹·min⁻¹ while $k_{est,EL}$ = 0.033 L·mol⁻¹·min⁻¹ and $k_{est,PL}$ = 0.026 L·mol⁻¹·min⁻¹. At the same temperature, the $k_{est,2-CIEL}$ is only 0.01 L·mol⁻¹·min⁻¹. Saponification reactions exhibit a non-equidistant change in kinetics. The $k_{sap,ML}$ is by far the highest with a value at 50 °C equal to 103.4 L·mol⁻¹·min⁻¹, followed by

<table><caption>Table 22
Correlation matrix of estimated constants for 2-CIEL saponification.</caption><thead><tr><th></th><th>ln($k_{sap,2-CIEL}$ ($T_{ref}$))</th><th><sup>E<sub>a,sap,2-CIEL</sub></sup>&frasl;<sub>RT<sub>ref</sub></sub></th><th>ln($K_{sap,2-CIEL}$ ($T_{ref}$))</th></tr></thead><tbody><tr><td>ln($k_{sap,2-CIEL}$ ($T_{ref}$))</td><td>1</td><td>0</td><td>0</td></tr><tr><td><sup>E<sub>a,sap,2-CIEL</sub></sup>&frasl;<sub>RT<sub>ref</sub></sub></td><td>-0.322</td><td>1</td><td>0</td></tr><tr><td>ln($K_{sap,2-CIEL}$ ($T_{ref}$))</td><td>-0.551</td><td>0.4961</td><td>1</td></tr></tbody></table>

<figure><img src="image_25.png" /><figcaption>A. Estimated rate constant evolution with temperature for LA esterification reaction with alkyl alcohols by considering concentration-based estimated parameters and Eq. (18).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>B. Estimated rate constant evolution with temperature for AL saponification by considering concentration-based estimated parameters and Eq. (18).</figcaption></figure>

<caption>Fig. 15B. Estimated rate constant evolution with temperature for AL saponification by considering concentration-based estimated parameters and Eq. (18).</caption>
$k_{sap.2-CIEL} = 35.4 \text{ L·mol}^{-1}\text{·min}^{-1}$, $k_{sap.EL} = 29.7 \text{ L·mol}^{-1}\text{·min}^{-1}$, and $k_{sap.PL} = 12.7 \text{ L·mol}^{-1}\text{·min}^{-1}$. The evidence of a large gap in terms of reactivity when going from methyl to ethyl levulinates, is in agreement with what is claimed by Fukuto et al. (Fukuto and Metcalf, 1959).

### 3.3. Temperature effect on Taft substituent parameters for the ethyl, propyl, and 2-chloroethyl levulinate synthesis

Taft substituent parameters were estimated for ethyl, propyl, and 2-chloroethyl levulinate in the temperature range [20; 50]°C following Eqs (13) and (14). Figs. 16 and 17 show the evolution of Taft substituent parameters $\sigma^*(-R)$ and $E_s(-R)$ with temperature, respectively. For each -R group, $\sigma^*$ and $E_s$ vary linearly with temperature.

Fig. 16 shows that $\sigma^*$ values decrease approximately linearly with increasing temperature for all substituents, with 2-chloroethyl displaying the strongest temperature dependence (slope: -0.0068 °C⁻¹). This is consistent with its strong inductive effect, which diminishes slightly at higher temperatures. For comparison, $\sigma^*$ for n-propyl changes from -0.189 at 20 °C to -0.279 at 50 °C, aligning with Taft's original estimate of -0.16 at 25 °C. 2-chloroethyl $\sigma^*$ drops from 0.261 to 0.043 over the same range; although our estimate at 25 °C (0.25) is lower than Taft's 0.47, the trend supports increasing polarity with decreasing temperature. Notably, ethyl shows a small positive slope (0.001 °C⁻¹), indicating a mild increase in $\sigma^*$ with temperature — opposite to the trend observed for the other groups.

The ethyl group's small positive slope likely reflects its near-neutral polar character at ambient temperature, making it more sensitive to subtle solvent polarity and transition-state stabilization changes with increasing temperature. Unlike strongly polar substituents, the temperature-dependent dielectric decrease in water/alcohol mixtures can slightly enhance ethyl's apparent inductive effect, leading to a mild increase in $\sigma^*$.

Despite the observed trends, the temperature dependence of $\sigma^*$ is relatively minor, with all slopes in the range of $10^{-3} \text{ °C}^{-1}$, suggesting the polar effect remains largely stable across moderate temperatures.

Fig. 17 presents the evolution of Taft's steric parameter, Es. Here, 2-chloroethyl again exhibits the strongest temperature dependence (slope: 0.008 °C⁻¹), followed by n-propyl (0.006 °C⁻¹), while n-ethyl shows a decreasing trend (-0.002 °C⁻¹).

For each substituent R-, we modeled the temperature-dependence of Taft parameters over 20–50 °C as

$$ \sigma^*(-R) = a_\sigma \cdot T + b_\sigma \qquad (21) $$

$$ E_s(-R) = a_{Es} \cdot T + b_{Es} \qquad (22) $$

To propagate kinetic uncertainty to ($a_x, b_x$), we approximated the posterior for (lnk, Ea/(RTref)) of each fit as bivariate normal, using the reported 95 % HPD widths to set the standard deviations (SD=HPD/3.92) and the within-pair correlation from the fit. For each posterior draw, we computed ($k_{-R}(T)$)$_A$ (esterification) and ($k_{-R}(T)$)$_B$ (saponification) for $T \in \{20,25,30,35,40,45,50\}$ °C, then using Eqs (13) and (14).

We then fit $y = aT + b$ by least squares to obtain one draw of ($a_x, b_x$). Posterior summaries (95 % credible interval from the 2.5th–97.5th percentiles) were reported for $a_x$ and $b_x$. In Table 23, we reported the values of $a_\sigma, b_\sigma, a_{Es}, b_{Es}$ and the associated half-width credible interval at 95 %.

In literature, Taft substituent parameters are available only at 25 °C. For the Es parameter, literature values at 40 °C are available from MacPhee et al. (MacPhee et al., 1978).

Table 24 shows a comparison of data obtained from this work and those of MacPhee et al. and Taft.

At 25 °C, $E_s(-Et)$ obtained by Taft is very similar to the one obtained in this study, however $\sigma^*(-Et)$ obtained by Taft is two times lower than ours. In the same way, at 25 °C, $E_s(-Pr)$ estimated in this study and the one of Taft are differing on a factor of 0.007, thus can be considered very similar. A relatively limited difference is shown for $\sigma^*(-nPr)$ when comparing it at 25 °C. On the contrary, both $\sigma^*(-2-CIEt)$ and $E_s(-2-CIEt)$ reveal different values both at 25 °C and 40 °C when comparing the estimated parameters obtained from this study and those in the literature. An explanation for this is the different approach used in this study when estimating the kinetic, which considered the separation in the contribution of the hydroxonium concentration and the reversibility of the rate of esterification. At 40 °C, the values obtained by MacPhee et al. and ours are similar for ethyl and propyl levulinates. For these two R groups, as well as for 2-chloroethyl levulinate, the dependence of the steric Taft substituent parameter on temperature obtained in this study is the same as that obtained from MacPhee et al.

From a mechanistic perspective, the observed differences between $\sigma$ and Es trends can be linked to distinct thermodynamic contributions. Polar effects ($\sigma^*$) mainly reflect enthalpic stabilization or destabilization of the transition state through electronic interactions, which explains their relatively weak temperature dependence. By contrast, steric effects

<figure><img src="image_27.png" /><figcaption>Fig. 16. Taft polar substituent values for Et-, n-Pr- and 2-CIEt- with temperature.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Fig. 17. Taft steric substituent values for Et-, n-Pr- and 2-ClEt- with temperature.</figcaption></figure>

<table><caption>**Table 23**

Linear temperature dependence of Taft parameters, with 95% credible interval half-widths.</caption><thead><tr><th rowspan="2">Substituent -R</th><th colspan="3">σ<sup>*</sup>(-R) = a<sub>σ</sub>.T + b<sub>σ</sub></th><th colspan="3">Es(-R) = a<sub>Es</sub>.T + b<sub>Es</sub></th></tr><tr><th>a<sub>σ</sub> [°C<sup>-1</sup>]</th><th>half-width (95 % CI)</th><th>b<sub>σ</sub></th><th>half-width (95 % CI)</th><th>a<sub>Es</sub> [°C<sup>-1</sup>]</th><th>half-width (95 % CI)</th><th>b<sub>Es</sub></th><th>half-width (95 % CI)</th></tr></thead><tbody><tr><td>Ethyl</td><td>0.001</td><td>0.001</td><td>-0.227</td><td>0.056</td><td>-0.002</td><td>0.002</td><td>-0.016</td><td>0.130</td></tr><tr><td>Propyl</td><td>-0.003</td><td>0.001</td><td>-0.111</td><td>0.048</td><td>0.006</td><td>0.002</td><td>-0.550</td><td>0.105</td></tr><tr><td>2-Chloroethyl</td><td>-0.007</td><td>0.002</td><td>0.405</td><td>0.091</td><td>0.008</td><td>0.003</td><td>-1.064</td><td>0.211</td></tr></tbody></table>

<table><caption>**Table 24**

Polar and steric effects at 25 °C and 40 °C.</caption><thead><tr><th></th><th>T (°C)</th><th>Es, calc</th><th>Es, lit</th><th>σ<sup>*</sup>, calc</th><th>σ<sup>*</sup>, lit</th></tr></thead><tbody><tr><td>Ethyl</td><td>25</td><td>-0.069</td><td>-0.070 (168)</td><td>-0.197</td><td>-0.10 (168)</td></tr><tr><td>n-Propyl</td><td>25</td><td>-0.367</td><td>-0.360</td><td>-0.205</td><td>-0.16</td></tr><tr><td>2-Chloroethyl</td><td>25</td><td>-0.856</td><td>-0.24</td><td>0.222</td><td>0.47</td></tr><tr><td>Ethyl</td><td>40</td><td>-0.105</td><td>-0.08</td><td>-0.178</td><td>Not found</td></tr><tr><td>n-Propyl</td><td>40</td><td>-0.285</td><td>-0.310</td><td>-0.251</td><td>Not found</td></tr><tr><td>2-Chloroethyl</td><td>40</td><td>-0.732</td><td>-0.180</td><td>0.111</td><td>Not found</td></tr></tbody></table>

(Es) are closely associated with entropic penalties due to restricted conformational freedom and steric crowding, consistent with the stronger temperature dependence observed for n-propyl and 2-chloroethyl substituents. For the smaller ethyl group, steric restrictions are minimal, and solvent-solute interactions may dominate, leading to the slight decrease in Es with temperature.

## 4. Conclusions

The esterification of levulinic acid with methanol, ethanol, n-propanol, and 2-chloroethanol, along with the corresponding saponification reactions, was investigated in acidic and basic aqueous media. The Taft parameters for the polar (σ*) and steric (Es) contributions for ethyl, propyl, and 2-chloroethyl groups were evaluated over a range of temperatures from 20 °C to 50 °C; therefore, the impact of temperature on these parameters was established.

Kinetic models were developed using a Bayesian approach and fitted to experimental data obtained under isothermal batch conditions. The models showed good agreement with measured concentrations, and the estimated kinetic parameters were well-identified and minimally correlated. However, in some esterification cases involving longer-chain

alcohols, slight underestimation of alkyl levulinate concentrations was observed, likely due to challenges in reaching equilibrium in aqueous media.

From the estimated rate constants, Taft's polar (σ*) and steric (Es) substituent parameters were calculated for ethyl, propyl, and 2-chloroethyl groups across the 20–50 °C temperature range. All substituents exhibited a linear dependence of σ* and Es on temperature. Notably, 2-chloroethyl showed the highest sensitivity, with slopes of -0.007 °C⁻¹ (σ*) and 0.008 °C⁻¹ (Es). Ethyl exhibited an opposite trend in σ*, increasing with temperature, whereas σ* values for the other groups decreased.

In general, the steric effect Es showed greater temperature dependence than the polar effect σ*. Despite these trends, the absolute temperature influence on Taft parameters was small, with all slopes on the order of 10⁻³ °C⁻¹.

A continuation of this work involves evaluating Taft substituent parameters at higher reaction temperatures, i.e., above 50 °C, to verify linearity, as well as examining the effect of solvent on these parameters. Another expansion is to consider bulkier (i-Pr, t-Bu) or strongly electron-donating (–OMe, –NH₂) substituents, which would increase generality. In addition, the methodology presented here could be extended beyond homogeneous esterification and saponification. For example, applying temperature-dependent LFER analysis to heterogeneous catalytic systems would provide insights into substituent effects under more industrially relevant conditions. Similarly, testing different solvent environments could reveal how solvation modulates both steric and polar contributions. Integrating this framework into reactor-scale kinetic models would help assess its value for process design and scale-up in biomass valorization.
## CRediT authorship contribution statement

**Sindi Baco:** Writing – review & editing, Methodology, Investigation.
**Mélanie Mignot:** Resources. **Christoph Held:** Supervision, Software, Methodology. **Julien Legros:** Supervision, Resources. **Sébastien Leveneur:** Writing – review & editing, Supervision, Software, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors thank INSA Rouen Normandy, University of Rouen Normandy, the Centre National de la Recherche Scientifique (CNRS), European Regional Development Fund (ERDF) No. HN0001343, Labex SynOrg (ANR-11-LABX-0029), Carnot Institute I2C, the graduate school for research XL Chem (ANR-18-EURE-0020 XL CHEM) and Region Normandie for their support. GC/FID was financed by ERDF RIN Green Chem 2019NU01FOBC08 No. 17P04374. This research was funded, in whole or in part, by the ANR (French National Research Agency) and the DFG (German Research Foundation) through the project MUST (Microfluidics for Structure-reactivity relationships aided by Thermodynamics & kinetics) [ANR-20-CE92-0002– Project number 446436621].

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ces.2025.122700.

## Data availability

The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.

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