# Experimental study and modeling of the influence of pH on the selective extraction of organic acid by electrodialysis

Robin Caveriviere, S. Galier, H. Roux-de Balmann

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Robin Caveriviere, S. Galier, H. Roux-de Balmann. Experimental study and modeling of the influence of pH on the selective extraction of organic acid by electrodialysis. Separation and Purification Technology, 2026, 388, pp.136693. 10.1016/j.seppur.2025.136693. hal-05447342

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# Experimental study and modeling of the influence of pH on the selective extraction of organic acid by electrodialysis

R. Caveriviere, S. Galier, H. Roux-de Balmann*

Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

## ARTICLE INFO

Editor name: Gaohong He

**Keywords:**

Electrodialysis

Separation

Organic Acid

Ion-exchange membrane

Permselectivity

Modeling

## ABSTRACT

This study investigates the influence of pH on the selective transport of organic acids in electrodialysis, focusing on a reaction-transport mechanism enhancing weak acid dissociation at the anion-exchange membrane. Electrodialysis experiments using sodium acetate at pH 4.8 show that acetic acid transfer to the concentrate is sustained even after sodium depletion, confirming the acid dissociation reaction-mechanism near the membrane. The study is extended to equimolar mixtures of acetate with formate, lactate, and butyrate at various initial pH values. Results showed that the selectivity varies with the pH and acids dissociation constants. Formate/acetate selectivity increased at lower pH due to a higher dissociation degree of formate (lower pKa) compared to acetate, while acetate/lactate selectivity decreased. In contrast, acetate/butyrate selectivity remains nearly constant due to their similar pKa values. A previously developed model is extended to account for pH-dependent ion transport and solvent fluxes. Experimental data provide parameters, including hydration numbers and anion and cation permselectivities. A corrective coefficient $\sigma$ is introduced to reflect membrane-driven dissociation mechanism. The resulting model accurately predicts solute concentration and pH variations, supports the hypothesis of a constant permselectivity over the investigated pH range, and offers insight into the individual transfer of each ion. It can thus be used to determine operating conditions, including pH, that maximize selectivity in electrodialysis. Finally, these findings show that, for mixtures of monocarboxylic acids, a reduced description with a single corrective parameter $\sigma$ is sufficient to quantitatively predict composition and pH variations while treating permselectivity as constant.

## 1. Introduction

Electrodialysis (ED) has emerged as a key electromembrane process for the selective recovered of charged species in aqueous solutions. Its applicability to weak electrolytes, such as organic acids, has attracted growing interest in recent years due to its reagent-free operation and compatibility with bio-based chemical production [1–5]. However, conventional characterization of ED systems often assumes full dissociation of acids and does not sufficiently capture the interplay between local pH, membrane properties, and acid dissociation [6–9].

Recent studies have begun to address this limitation by exploring the electrodialytic separation of weak acids under varying pH conditions. For salts of polybasic acids such as phosphoric and citric acid, models based on the Nernst–Planck equations have been developed that explicitly couple ion transport with proton-transfer equilibria in the membrane and adjacent solution. These models reproduce characteristic features such as current–voltage curves and complex

chronopotentiometric responses, and have been validated experimentally for phosphate systems and related weak electrolytes [10–12]. Moreover, electrochemical impedance spectroscopy was used to reveal the emergence of homogeneous reactions during ED of multivalent organic acids [13]. These reactions, occurring in thin boundary layers near the membrane surface, were linked to localized generation of $H^+$ and $OH^-$ ions, demonstrating the presence of pH-dependent reactive layers that significantly alter system resistance and transport behavior. In another study, it has been shown that feed solution pH strongly influences the selective transport of volatile fatty acids across various anion-exchange membranes, with increased pH favoring the separation of smaller carboxylates due to changes in membrane–ion interactions [14]. Authors have pointed out that the transfer of smaller organic acids (acetate, propionate) over larger acids (butyrate, valerate) is higher with membrane using tertiary amines instead of quaternary ammonium as functional groups. Additionally, the competitive transport between inorganic (e.g., nitrate) and organic (e.g., citrate) anions has been investigated, highlighting how membrane selectivity and pH govern ion

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: helene.roux-de-balmann@utoulouse.fr (H. Roux-de Balmann).</footnote>
Nomenclature

Local electric field $E_v$, V.m⁻¹

Elementary charge $e$, C

Concentration $C$, mol.L⁻¹

Boltzmann constant $k_B$, J.K⁻¹

Dissociation constant $Ka$

Faraday constant $F$, C.eq⁻¹

Electrical current $I$, A

Flux $J$, mol.s⁻¹

Charged acid fraction $X$

Iteration $i$

Number of iterations $n^i$

Hydration number $n^H$, mol·H₂O/mol

Cell-pair number $N$

Permselectivity (A on B) $P_{A/B}$

Purity $x$, mol/mol

Quantity $n$, mol

Voltage $U$, V

Volume $V$, L

Water molar volume $V_m^{H_2O}$, m³.mol⁻¹

Greek letters

Catalyzed dissociation coefficient $\sigma$

Faradic yield $\eta$

Vacuum permittivity $\epsilon_0$, F.m⁻¹

Relative permittivity $\epsilon_R$

Recovered fraction $\Lambda$

Sub- and superscritps

Organic acid $a$

Charged organic acid $A^-$

Ion acetate $Ac^-$

Ion hydronium (proton) $H^+$

Anion $a^-$

Cation $c^+$

Acetate/ acetic acid $Ac$

Concentrate (ED) $C$

Diluate (ED) $D$

Formate $Fo^-$

Formate/ formic acid $Fo$

Acetic acid $HAc$

Lactate $Lac^-$

Lactate/lactic acid $Lac$

Membrane $m$

Sodium $Na^+$

transport [15]. Under acidic conditions, membrane resistance increased due to the predominance of neutral acid species, while higher pH promoted dissociation and transport of the citrate but reduced the selectivity between the acid and nitrate.

It has been shown in these studies that even in conditions where weak acids are not initially fully dissociated, very high extraction yields can be achieved in electrodialysis [7,13–16]. Two main hypotheses have been proposed to explain this behavior. The first one is based on the Second Wien Effect (SWE), which explains the observed enhanced ionization of weak acids under strong electric fields [17–19]. The second involves a catalytic protonation/deprotonation mechanism at the anion exchange membrane interface, similar to what happens with bipolar membranes [13,17,19].

The SWE refers to an increase in the dissociation constant ($Ka$) of weak acids in electric fields ranging from $10^8$ to $10^9$ V/m, likely to appear close to the membrane, resulting in greater formation of ionic species without affecting recombination or the dissociation of weak bases. In parallel, membrane-catalyzed dissociation theory attribute proton and hydroxide generation to functional groups within the anion exchange membrane. Tertiary amines, in particular, have been shown to facilitate proton transfer and water-splitting more effectively than the sulfonic acid groups typically found in cation-exchange membranes [20,21].

Both theories converge on the presence of a reactive interfacial zone at the membrane surface, where acid dissociation occurs. These mechanisms are further modulated by permittivity, temperature, and electric field strength, and are considered irreversible under the operating conditions of ED [22].

On the other hand, it has been shown that, under conditions where they are fully dissociated, the permselectivity between two weak acids remains essentially independent of operating parameters such as current density, time, or acid concentration and composition [23]. These findings have formed the basis for a predictive model of the selective transport of ions in ED. However, this study did not account for conditions where weak acids are not fully dissociated.

The present work aims to characterize and implement the effect of the pH into this model. We consider binary mixtures of monocarboxylic acids relevant to acidogenic effluents, and employ a model that incorporates pH-dependent acid dissociation, ion transport and solvent

fluxes. This approach allows us to directly link the experimentally measured permselectivity directly to pH and pKa, and to evaluate whether membrane permselectivity can be treated as constant over a wide pH range when designing selective recovered conditions. In this paper, we aim to clarify how pH influences selectivity and ion fluxes. This understanding is essential to extend the previous predictive model to incorporate pH effects and enable prediction of concentration and pH variation during electrodialysis of organic acids.

While previous results and existing literature have demonstrated that electrodialysis enables high extraction yields of weak acids across a wide pH range, much less is known about how pH and the associated dissociation mechanism affect the selectivity of this process. The objective of this study is to investigate the influence of initial pH on the selective transport of organic acids in electrodialysis. Special attention is given to the role of the reaction-transport mechanism that promotes weak acid dissociation at the anion-exchange membrane.

## 2. Theoretical background

Permselectivity is a key parameter in electrodialysis (ED) that characterizes the preferential transport of one ion over another through an ion-exchange membrane. It has been shown that the permselectivity depends on differences in ionic mobility and interactions with fixed groups within the membrane [14,24]. For anions, factors such as hydration energy, size, and charge density determine the transport order (e.g., formate > acetate > butyrate) [8,23]. On another hand, a previous study showed that neither current intensity, ion proportions, overall concentration, nor recovered fraction influences the permselectivity between organic acids in ED, even at high recovered fractions [23]. Permselectivity definition between two anions $A^-$ and $B^-$, is given by Eq. (1).

$$P_{A/B^-} = \frac{J_{A^-}/J_B}{C_{A^-}/C_B} = \frac{J_A^-/J_B^-}{C_A^-X_A^-/C_B^-X_B^-} \quad (1)$$

Where $J_i$ is the flux of anion $i$, $C_i$ is the ion concentration and $C_i$ is the total concentration of $i$ in the diluate (i.e., the sum of its dissociated and undissociated form). $X_i$ is the charged fraction (or dissociation degree), representing the fraction of the acid present in its dissociated form. For strong electrolytes, full dissociation is assumed, and $X_i = 1$.
However, for weak acids, the dissociation degree must be considered, as it depends on both the solution pH and the acid's dissociation constant. The charged fraction is given by the Eq. (2).

$$X_i = \frac{C_i}{C_i} = \frac{1}{1 + 10^{(pKa_i - pH)}} \quad (2)$$

This formulation emphasizes that the ion fluxes of weak acids are directly tied to their dissociation degree, which is governed by the pH of the solution and their respective pKa values. The pKa values of the different acids used in this paper are given in Table 1.

The variation in the ratio of charged fractions for the acid pairs investigated $X_A/X_B$ is illustrated in Fig. 1 as a function of pH.

As illustrated in Fig. 1, the charged fraction of formate over acetate $X_{F0}/X_{Ac}$ increases steeply as the pH drops below 5, reaching values ranging from 9 to 10 at pH 3 and below, which can significantly shift the selectivity. Similarly, the charged fraction of lactate over acetate reaches 7 to 8 below pH 3. In contrast, the charged fraction of acetate and butyrate remains comparable over the pH range due to their similar dissociation constants, resulting in a maximum value of $X_{Ac}/X_{Bu} = 1.15$ even at very low pH.

In a previous study, a model has been developed to determine the variation of solute fluxes in ED systems for such conditions where organic acids are totally dissociated [23]. The model involves the permselectivity definition (Eq. (1)) and the migration equation in ED, defined by Eq. (3), to determine the flux of each ion and the variation of their quantity over time t or recovered fraction $\Lambda$.

$$\sum_i z_i J_i = \eta \frac{N}{F} I \quad (3)$$

In this equation, $\eta$ is the Faradic yield, $N$ is the number of cell-pairs, $F$ (C/mol) is the Faraday constant, $z_i$ (eq/mol) is the valence of ion $i$ and $I$ (A) is the electric current. The Faradic yield $\eta$ is defined by the ratio between the current used for the sum of ion migration $I_i$ over the current intensity, as given Eq. (4).

$$\eta = \frac{\sum_i I_i}{I} = \frac{F}{N} \sum_i z_i J_i \quad (4)$$

The recovered fraction $\Lambda$ is defined by the ratio between the total quantity of organic acids $a$, including the dissociated and undissociated form, transferred into the concentrate $C$ and its initial quantity in the dilute $D$, as expressed by Eq. (5).

$$\Lambda = \frac{\sum_a n_a^C}{\sum_a n_a^{0,D}} = 1 - \frac{\sum_a n_a^D}{\sum_a n_a^{0,D}} \quad (5)$$

In this paper, this model is expanded to (1) take into account the solvent transfer in order to calculate concentration variations from ion quantities, (2) introduce the effect of the pH on the organic acids flux and (3) calculate the pH evolution during ED.

To generalize the model to any pH condition, three additional steps are required. First, the charged fractions $X$ of each acid must be calculated as functions of pH (Eq. (2)). Second, the system of equations governing the fluxes of cations must be solved (Eqs. (1) and (3), [23]). Indeed, at low pH and low $Na^+$ concentration, the concentrations of $Na^+$ and $H^+$ ions may become comparable, making it necessary to consider $H^+$ as a contributor to current transport. Thus, the permselectivity between these two cations, $P_{H^+/Na^+}$, must be determined to compute their respective fluxes (Eqs. (3) and (6)). Third, the solvent flux $J_{solvent}$ needs to

<table><caption>**Table 1**
: pKa and molecular weight values of weak acids.</caption><thead><tr><th>Weak acid</th><th>Acetic acid</th><th>Butyric acid</th><th>Formic acid</th><th>Lactic acid</th></tr></thead><tbody><tr><td>pKa</td><td>4.76</td><td>4.82</td><td>3.75</td><td>3.90</td></tr><tr><td>Molecular weight (g/Mol)</td><td>60.05</td><td>88.11</td><td>46.03</td><td>90.08</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Variation in the ratio of charged fractions of organic acid as function of the pH. Acetate is referred as Ac, butyrate as Bu, formate as Fo and lactate as Lac. An inset shows an enlarged view of the acetate/butyrate curve.</figcaption></figure>

be determined. It is assumed to result entirely from electro-osmosis and is calculated using Eq. (7), which defines the relationship between the solvent flux, ions hydration number $n^H$ and solute fluxes [25]. A previously established methodology is used to determine ion hydration numbers [25].

Subsequently, solute concentrations (Eq. (10)) and volumes in both compartments are determined from the solute and solvent fluxes (Eqs. (8) and (9)). Then, the concentration of $H^+$ ions and the associated pH value are obtained from the electroneutrality equation (Eqs. (11) and (12)). The equations are grouped in Table 2.

In these equations, $n_i$ (mol) is the molar quantity of $i$ and $V_m^{H_2O}$ ($m^3/mol$) is the molar volume of water. Concentration $C_A$ or purity $x_A$ of A can be presented as a function of time or as function of the recovered fraction $\Lambda$ (Eq. (5)). The molar purity in a compartment is defined by Eq. (13).

$$x_A = \frac{C_A}{\sum_i C_i} \quad (13)$$

If A is a weak acid, the purity is calculated with the sum of the concentrations of the dissociated and undissociated forms. Similarly, the yield $Y_A$ of a weak acid A in a specific compartment, can also be defined by Eq. (13).

<caption>**Table 2**</caption>

: Equations used in the model for two anions $A^-$ and $B^-$ and two cations $Na^+$ and $H^+$.

<table><thead><tr><th colspan="2">Equations</th></tr></thead><tbody><tr><td>P<sub>A-/B-</sub> = J<sub>A-/B-</sub> / C<sub>A</sub> X<sub>A</sub> / C<sub>B</sub> X<sub>B</sub></td><td>Eq. (1)</td></tr><tr><td>X<sub>A</sub> = 1 / (1 + 10<sup>(pKa<sub>A</sub> - pH)</sup>)</td><td>Eq. (2)</td></tr><tr><td>&sum;<sub>i</sub> J<sub>i</sub> = &eta; N / F I</td><td>Eq. (3)</td></tr><tr><td>&Lambda; = 1 - (sum<sub>a</sub> n<sub>a</sub><sup>D</sup>) / (sum<sub>a</sub> n<sub>a</sub><sup>0,D</sup>)</td><td>Eq. (5)</td></tr><tr><td>P<sub>H<sup>+</sup>/Na<sup>+</sup></sub> = (J<sub>H<sup>+</sup>/Na<sup>+</sup></sub> / C<sub>H<sup>+</sup>/Na<sup>+</sup></sub>)</td><td>Eq. (6)</td></tr><tr><td>J<sub>solvent</sub> = V<sub>m</sub><sup>H_2O</sup> &sum;<sub>i</sub> J<sub>i</sub> &times; n<sub>i</sub><sup>H</sup></td><td>Eq. (7)</td></tr><tr><td>J<sub>solvent</sub> = dV/dt</td><td>Eq. (8)</td></tr><tr><td>J<sub>i</sub> = dn<sub>i</sub>/dt</td><td>Eq. (9)</td></tr><tr><td>C<sub>i</sub> = n<sub>i</sub>/V</td><td>Eq. (10)</td></tr><tr><td>&sum;<sub>a</sub> z<sub>a</sub> C<sub>a</sub> = &sum;<sub>c<sub>r</sub></sub> z<sub>c<sub>r</sub></sub> C<sub>c<sub>r</sub></sub></td><td>Eq. (11)</td></tr><tr><td>pH = -log(C<sub>H<sup>+</sup></sub>)</td><td>Eq. (12)</td></tr></tbody></table>
$$Y_A = \frac{n_A}{n_A^{0,D}} \qquad (14)$$

Previous studies have shown that the composition of mixtures does not influence the hydration numbers of ions in electrodialysis [25]. Their values are thus considered constant. The Faradic yield is determined experimentally, and the values obtained range between 90% and 100%. Hence, ideal conditions are assumed by setting the value at 100%.

The model employs an iterative calculation, where the time step $\Delta t$ is sufficiently small to consider the parameters constant over each interval. The minimum number of iterations $n^i$ required to satisfy this condition was found to be $n^i > 200$, and a value of $n^i = 1000$ was then selected. In the following, the index $i$ denotes the iteration index of the calculation. For example, $pH[i]$ refers to the pH value obtained at iteration $i$.

Hydration numbers and permselectivities determined at pH ≥ 7 are assumed to be constant over the entire pH range. Since the studied pH range spans from 2 to 8, the concentration of hydroxide ions (OH⁻) is considered negligible relative to other ions. To sum up, the model relies on the following assumptions:

- The permselectivity between two ions is constant and does not depend on pH.

- Ion hydration numbers are independent of pH.

- The faradic yield is assumed at 100%.

- The $OH^-$ concentration is negligible compared to other ions.

The data, constants and initial conditions as well as the variables needed to calculate concentrations and pH for a binary mixture (A and B) are provided in Table 3, Table 4 and Table 5. The calculation steps were depicted in Fig. 2. Detailed calculations can be found in Appendix A.

## 3. Material and methods

### 3.1. Experimental set-up

Experiments were realized with synthetic mixtures of sodium acetate (Ac), butyrate (Bu), formate (Fo) and lactate (Lac) at different compositions. The electrodialysis set-up is composed by a EUR 2B-10 stack (Eurodia, France), with a cross-section of 0.02m², 10 cell pairs N leading to 0.4 m² of total membrane surface area, i.e., 0.2 m² for each type (cation or anion exchange membranes). Dilute and concentrate compartments were initially fed with 2 L of solution each. Electrode compartment was filled with 3 L of 10 g/L of sodium sulfate. A schematic diagram of the experimental set-up was depicted in a previous paper [25]. AEM and CEM used are Neosepta ACS and CMX membranes (Astom, Japan).

The electrodialysis of acetate-formate, acetate-lactate and acetate-butyrate mixtures have been realized with different conditions presented in Table 6. Unless otherwise specified, the total initial concentration for each experiment was 30 g/L and the initial molar proportions were 50/50.

Two factors guided the selection of these mixtures. First, these

<caption>**Table 3**</caption>

Data required to calculate concentrations and pH variations in ED for a mixture of two anions A⁻ and B⁻ with sodium as counter-ion.

<table><caption>**Table 4**</caption><thead><tr><th colspan="2">Required data</th></tr></thead><tbody><tr><td>Anions permselectivity:</td><td>P<sub>A</sub>/B</td></tr><tr><td>Cations Permselectivity:</td><td>P<sub>H</sub>^+ / Na<sup>+</sup></td></tr><tr><td>A<sup>-</sup> hydration number:</td><td>n<sup>H</sup><sub>A</sub> (molH<sub>2</sub>O/mol)</td></tr><tr><td>B<sup>-</sup> hydration number:</td><td>n<sup>H</sup><sub>B</sub> (molH<sub>2</sub>O/mol)</td></tr><tr><td>Na<sup>+</sup> hydration number:</td><td>n<sup>H</sup><sub>Na</sub> (molH<sub>2</sub>O/mol)</td></tr><tr><td>H<sup>+</sup> hydration number:</td><td>n<sup>H</sup><sub>H</sub> (molH<sub>2</sub>O/mol)</td></tr></tbody></table>

Data and initial conditions used to calculate concentrations and pH variations in ED for a mixture of two anions A⁻ and B⁻ with sodium as counter-ion.

<table><caption>**Table 5**</caption><thead><tr><th>Data</th><th>Initial conditions</th></tr></thead><tbody><tr><td>Faraday constant: F (C/eq)</td><td>Concentration of A<sup>-</sup>: C<sup>0</sup><sub>A</sub> (mol/L)</td></tr><tr><td>Faradic yield: η</td><td>Concentration of B<sup>-</sup>: C<sup>0</sup><sub>B</sub> (mol/L)</td></tr><tr><td>Number of cell-pair: N</td><td>Concentration of Na<sup>+</sup>: C<sup>0</sup><sub>Na</sub> (mol/L)</td></tr><tr><td>Current intensity: I (A)</td><td>pH: pH<sup>0</sup></td></tr><tr><td>Water molar volume: V<sup>H2O</sup><sub>m</sub> (m<sup>3</sup>/mol)</td><td>Volume: V<sup>0</sup></td></tr><tr><td>Number of iterations: n<sup>i</sup></td><td></td></tr><tr><td>Dissociation constant of A: K<sub>A</sub></td><td></td></tr><tr><td>Dissociation constant of B: K<sub>B</sub></td><td></td></tr></tbody></table>

Model variables used to calculate concentrations and pH variations in ED for a mixture of two anions A⁻ and B⁻ with sodium as counter-ion.

<table><thead><tr><th>Concentrations</th><th>Quantity</th><th>Flux</th><th>Miscellaneous</th></tr></thead><tbody><tr><td>A<sup>-</sup>: C<sub>A</sub> (mol/L)</td><td>A<sup>-</sup>: n<sub>A</sub> (mol)</td><td>A<sup>-</sup>: J<sub>A</sub> (mol/s)</td><td>Charged fraction of A: X<sub>A</sub></td></tr><tr><td>B<sup>-</sup>: C<sub>B</sub> (mol/L)</td><td>B<sup>-</sup>: n<sub>B</sub> (mol)</td><td>B<sup>-</sup>: J<sub>B</sub> (mol/s)</td><td>Charged fraction of B: X<sub>B</sub></td></tr><tr><td>Na<sup>+</sup>:<br>C<sub>Na</sub><sup>-</sup> (mol/L)</td><td>Na<sup>+</sup>:<br>n<sub>Na</sub><sup>-</sup> (mol)</td><td>Na<sup>+</sup>: J<sub>Na</sub><sup>-</sup> (mol/s)</td><td>Recovered fraction: A</td></tr><tr><td>H<sup>+</sup>: C<sub>H</sub><sup>-</sup> (mol/L)</td><td colspan="2">Solvent:<br>J<sub>solvent</sub> (m<sup>3</sup>/s)</td><td>Volume in dilute:<br>V (L)<br>Time: t (s)<br>pH</td></tr></tbody></table>

organic acids commonly appear in biological media as products, substrates, and/or inhibitors. Second and most interestingly, they have different pKa values (Table 1), which influence their dissociation degree and thus their ionic concentrations. Different selectivity outcomes are then expected as function of the pH.

Experiments were conducted at constant voltage when handling low-conductivity solutions, as typically encountered in acidic mixtures. Operating at fixed current intensity under such conditions can otherwise lead to a significant voltage increase, potentially damaging the membrane.

### 3.2. Analysis

Acetate, butyrate, formate and lactate were analyzed by High-Performance Liquid Chromatography (HPLC) (Jasco LC Net II/ADC, Japan) equipped with a Aminex HPX-87H column (BioRad, USA) and a UV detector (UV-2077plus) with a wavelength set at 210 nm. The column temperature was set at 65 °C, and the mobile phase was a 5 mM sulfuric acid solution. The flow rate was set at 0.6 mL/min and injection volume varied from 2 µL to 10 µL.

Sodium has been analyzed by ion chromatography (ICS 3000, Dionex, France) using a CS-12 column (Dionex IonPac, USA). Ion determination was performed using a CD20 conductimetric detector. Samples were diluted up to 200-fold using ultrapure water (resistivity >18 MΩ, cm, Elga, France) before analysis. Ion chromatography was performed in isocratic mode, with a mobile phase of 20 mM methanesulfonic acid at 0.5 mL/min at 40 °C. Injection volume has been set at 25 µL.

The balance for water and ions was checked, knowing the variation of the volume and concentrations in the 2 compartments. The maximum deviation was less than 2% for volume and 10% for ions.

The determination of ion hydration number was realized using the methodology developed in a previous paper [25] on the list of experiments given in Table 6. To limit uncertainties in the calculation of hydration numbers, measurements for which total volume variation ΔV < 15 mL were excluded.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Model flowchart of concentrations and pH variations calculation in ED for a mixture of two weak acids A and B with sodium as counter-ion.</figcaption></figure>

<table><caption>**Table 6**

Experimental conditions for the electrodialysis of weak acid mixtures. Voltage or electric current is fixed.</caption><thead><tr><th>Mixture</th><th>Experiment</th><th>Voltage (V)</th><th>Current (A)</th><th>Initial pH</th></tr></thead><tbody><tr><td rowspan="3">Acetate</td><td>1</td><td>20</td><td>-</td><td>4.8</td></tr><tr><td>2</td><td>-</td><td>2</td><td>7</td></tr><tr><td>3</td><td>20</td><td>-</td><td>4.7</td></tr><tr><td rowspan="5">Formate-acetate</td><td>4</td><td>-</td><td>2</td><td>4.7</td></tr><tr><td>5</td><td>-</td><td>2</td><td>4.2</td></tr><tr><td>6</td><td>10</td><td>-</td><td>3.7</td></tr><tr><td>7</td><td>20</td><td>-</td><td>3.7</td></tr><tr><td>8</td><td>20</td><td>-</td><td>4</td></tr><tr><td rowspan="3">80/20 (%Mol)</td><td>9</td><td>20</td><td>-</td><td>2.5</td></tr><tr><td>10</td><td>20</td><td>-</td><td>7</td></tr><tr><td>11</td><td>20</td><td>-</td><td>4.2</td></tr><tr><td rowspan="3">Acetate-lactate</td><td>12</td><td>20</td><td>-</td><td>5.6</td></tr><tr><td>13</td><td>20</td><td>-</td><td>4.2</td></tr><tr><td>14</td><td>20</td><td>-</td><td>4.7</td></tr><tr><td>Acetate-butyrate</td><td>15</td><td>-</td><td>1</td><td>8</td></tr><tr><td>20 g/L</td><td>16</td><td>20</td><td>-</td><td>4.8</td></tr></tbody></table>

### 3.3. Experimental procedure

The initial pH of the diluate solutions was set by adjusting the ratio between the pure organic acid and its sodium salt. For each targeted pH, the molar fractions of the acid and its salt were calculated prior to the experiments using the corresponding acid dissociation constant. The pH was checked before the electrodialysis experiments.

The experiments were carried out in batch mode, with each of the diluate and concentrate compartments supplied by 2 L of solution. Temperature in each compartment was maintained at $T = 24 \pm 2$ C using a Cryostat MC 600 Microcool (LAUDA).

The diluate compartment was fed with different mixtures of Ac, Bu, Fo and Lac sodium salts. The concentrate compartment was supplied with 2 L of tap water (conductivity $\chi = 0.3$ mS/cm) to avoid a high ohmic resistance at the start of the experiment. Experiments were conducted either at fixed voltage or fixed current using a stabilized power supply. Under constant-current conditions, experiments were stopped if voltage reached $U = 20$ V, to prevent membrane degradation. No significant difference in electrical resistance was observed between extractions carried out at fixed current or fixed voltage, provided the current remained below the limiting current. The minimum pH value was limited to 2.5 to avoid degradation of the experimental setup.

Previous studies indicate that membrane conditioning, especially membrane counter-ion, has an influence on solute and solvent transfer [26,27]. Knowing the ion exchange capacity of the membranes used, which is lesser than 2 meq.g-1, and the membrane surface used, one gets a maximum of 50 meq for ACS and of 70 meq for CMX membranes. Prior to each experiment, the membranes were pre-conditioned by soaking in the working solution for at least 4 h at a flow rate of 180 L·h$^{-1}$. After this period, the solution was replaced with a fresh, identical one. This conditioning step ensures complete exchange of counter-ions, allowing the ion exchange membranes (IEMs) to reach equilibrium before use. Indeed, the quantity of ions in solution is up to 10 times higher than the membrane exchange capacity [27].

### 3.4. Chemicals

Mixtures of sodium acetate (Ac), formate (Fo), butyrate (Bu) and lactate (Lac) were prepared using sodium acetate (>98%, Fisher Chemical), acetic acid (>99.8%, Acros Organics), sodium lactate (60% wt, Acros Organics), formic acid (>99%, VWR Chemicals), sodium butyrate (>99.8%, Alfa Aesar) and sodium hydroxide (>98%, Acros
Organics). Electrode compartment solution has been prepared with sodium sulfate (>99.9%, VWR Chemicals).

# 4. Results and discussion

## 4.1. Preliminary study

The limiting current in monobasic systems like NaCl, is primarily governed by electrolyte depletion and water splitting mechanisms. In the case of weak acids, the ion transport above the first limiting current is sustained by acid dissociation mechanisms occurring at the vicinity of the membrane [12,13]. To experimentally illustrate this phenomenon, electrodialysis of a sodium acetate solution at pH = 4.8, close to the pKa of acetic acid (pKa = 4.76, XAc = 0.52) was performed.

Fig. 3 shows the evolution of pH in both compartments as a function of the recovered fraction Λ.

The charged fraction XAc of acetic acid can be obtained from the measured pH using Eq. (2). Fig. 3 shows that the pH in the dilute gradually decreases from its initial value of 4.8 down to 3 at 60% extraction. This drop in pH is due to the shift of acetic acid dissociation equilibrium as acetate ions transfer to the concentrate as given by Eq. (15).

$$pH = -\log\left(Ka \times \frac{C_{AH}}{C_{A-}}\right) \qquad (15)$$

The concentrations of acetate ions Ac⁻ and acetic acid HAc are calculated based on the charged fraction and total acetate concentration. The evolution of sodium concentration, acetate ion concentration, and acetic acid concentration are plotted in Fig. 4 as a function of the recovered fraction.

One can observe in Fig. 4 that the sodium concentration decreases progressively and in parallel with the acetate ion up to a recovered fraction Λ between 40% and 59%, in accordance with the electro-neutrality, as acetate and sodium can be considered the only charge carriers. The recovered fraction at which their concentration approaches zero is directly associated with the initial charged fraction of acetate XAc = 52%.

The transfer of acetate and sodium ions does not change the pH in the concentrate (Fig. 3, Λ < 40%). However, in the dilute, the transfer of acetate changes the ratio acetate/acetic acid which steadily lowers the pH until it reaches a pH of 3. This value is specifically associated to the pH of an acetic acid solution at the measured concentration.

As the sodium concentration in the dilute becomes low, H⁺ ions start to carry the electrical current. These protons are produced at the anion-exchange membrane by the dissociation of acetic acid, following one of the two mechanisms described previously. At this stage, acid dissociation at the membrane is accompanied by the migration of H⁺ and Ac⁻ from the dilute to the concentrate compartment. Consequently, the pH

<figure><img src="image_5.png" /><figcaption>Fig. 3. Evolution of pH in the dilute and the concentrate as a function of the recovered fraction. Experiment 1: U = 20 V, CAc = 30 g/L, pHD = 4.8. Dashed line: Acetic acid pKa.</figcaption></figure>

in the dilute stabilizes at approximately 3 and remains constant until the end of the extraction process, as illustrated in Fig. 3.

One can also observe in Fig. 4 that the dissociation mechanism leads to the gradual decrease in acetic acid concentration within the dilute and its simultaneous increase in the concentrate compartment, from approximately 40% extraction onward. Once transferred to the concentrate, acetate ions recombine with protons to regenerate acetic acid. Therefore, an increase in acetic acid concentration accompanied by a drop in pH is observed in the concentrate compartment beyond the 40% to 59% extraction threshold, which is the value associated with the initial fraction of acetate ions.

These results confirm the presence of a reaction-transport mechanism that enables the dissociation of organic acids and then the transfer of their ionic forms during electrodialysis, when the counter-ion concentration becomes negligible. While this phenomenon has been previously observed for other acids, our results reaffirm its occurrence in the case of acetic acid and establish a reference case for subsequent analysis. These findings support the broader conclusion that electrodialysis can achieve high extraction yields of organic acids regardless of initial dissociation degree, owing to this localized dissociation mechanism at the membrane interface.

## 4.2. Experimental study of the impact of pH on selectivity

### 4.2.1. Formate-acetate mixture

Fig. 5 shows the variation of acetate and formate purity in the dilute and concentrate, respectively, as a function of the recovered fraction. The variations are given for three experiments conducted at initial pH values of 7, 4.7, and 3.7 in the dilute (Experiments 2, 3 and 7 from Table 6).

It can be observed that the acetate purity in the dilute increases progressively during extraction due to the preferential transfer of formate into the concentrate. Notably, for a given recovered fraction Λ, acetate purity in the dilute increases as initial pH decreases. Similarly, the formate purity in the concentrate is higher with decreasing pH. Indeed, for a recovered fraction below 60%, the formate purity ranges between 60% and 70% at pH 7, rises to 70% to 80% at pH 4.7, and to 80% to 90% at pH 3.7. By the end of extraction, the purity falls back until theoretically reaching 50% (the initial purity within the dilute) for a complete extraction.

The variation in acetate and formate purity with pH for a given Λ can be explained by the respective pKa values of formic acid (pKaF0 = 3.75) and acetic acid (pKaAc = 4.76). It has been shown on Fig. 1 that a decrease of the pH increases the dissociation degree ratio XF0/XAc. Thus, one can notice that, as expected, the formate purity in the concentrate, and acetate purity in the dilute, are higher at pH 3.7 due to the absence of electrophoretic migration of the acetic acid, mainly in its neutral form.

Fig. 5 also shows the evolution of the pH in the dilute and the concentrate at different initial pH during ED. One can observe in the dilute that the pH decreases with the recovered fraction for initial pH 3.7 and 4.7. As it has been shown and explained in Fig. 3 with a sodium acetate solution, the pH drop is associated with the transfer of charged organic acids and the sodium counter ion. Notably, once most of the sodium has been transferred and that its concentration becomes negligible, it triggers the organic acid dissociation mechanism.

The dissociation of the formic and acetic acid in the dilute and the transfer of the generated H⁺ ions in the concentrate induce a pH decrease in the compartment, as observable for the experiment realized with an initial pH of 3.7.

This indicates that the recovered fraction for which the mechanism is triggered depends on the initial pH, as it determines the available sodium concentration. At higher initial pH, the sodium concentration remains sufficient to carry the electrical current throughout the entire extraction. In contrast, at lower pH, the sodium is depleted for a lower recovered fraction, triggering the reaction-transport mechanism.
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Evolution of Sodium, Acetate ions, and Acetic acid concentrations as a function of the recovered fraction. Initial diluate pH = 4.8 (XAc = 52%). Experiment 1: U = 20 V, CAc = 30 g/L.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Acetate purity (a) and pH (b) in the diluate and formate purity (c) and pH (d) in the concentrate as a function of the recovered fraction for different initial pH values in the diluate. CAc = 30 g/L, Experiment 2: I = 2 A, pHD = 7; Experiment 3: U = 20 V, pHD = 4.7; Experiment 7: U = 20 V, pHD = 3.7.</figcaption></figure>

### 4.2.2. Lactate/acetate mixture

Fig. 6 shows how the lactate and acetate purity evolve in the diluate and concentrate for three experiments conducted at initial pH values of 7, 4.7, and 4.2 (Experiments 10, 11 and 14 from Table 6).

The results indicate that for a given recovered fraction the purity of lactate in the diluate decreases as the pH is reduced. Similarly, acetate purity in the concentrate compartment also declines with decreasing pH. This trend aligns with the dissociation degree of lactic and acetic acids, which has been previously illustrated on Fig. 1. It is noteworthy that an inversion of selectivity arises for an initial pH of 4.2. In this case, lactate purity in the diluate (or acetate purity in the concentrate) falls below the

initial value of 50%. This behavior can, once again, be explained by the difference between the pKa values of acetate and lactate which governs their respective dissociation degree. As the pH decreases, the dissociated fraction ratio $X_{Lac}/X_{Ac}$ increases (Fig. 1), reflecting a relatively higher dissociation degree of lactate compared to acetate. This shift results in a progressive increase in lactate flux, which may eventually exceed that of acetate, reaching the point where $J_{Lac} > J_{Ac}$. One can also notice that the pH variations in the diluate and the concentrate follow the same pattern than with previous mixtures.
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Lactate purity (a) and pH (b) in the diluate and acetate purity (c) and pH (d) in the concentrate as a function of the recovered fraction for different initial pH values in the diluate. C<sub>Ac</sub> = 30 g/L, Experiment 10: U = 20 V, pH<sub>D</sub> = 7; Experiment 11: U = 20 V, pH<sub>D</sub> = 4.2; Experiment 14: U = 20 V, pH<sub>D</sub> = 4.7.</figcaption></figure>

### 4.2.3. Acetate/butyrate mixture

The same methodology was applied to acetate-butyrate mixtures. The initial proportions were set at 50/50 (mol%), and the studied pH range varied between 4.8 and 8. Fig. 7 illustrates the variation in pH and butyrate purity within the diluate as well as the variation in pH and acetate purity in the concentrate as function of the recovered fraction.

Fig. 7 shows that the initial pH has little effect on the purity of the solutes. This can be explained by the similar pKa values of acetate and butyrate, leading to comparable charged fractions regardless of the pH (Fig. 1).

In this section, we studied the influence of pH. First, it was shown that selectivity depends on the acid dissociation constants and on pH. In the formate/acetate mixture, lowering the pH was shown to enhance electrodialysis selectivity. For the acetate/lactate mixture, lowering the pH reduces selectivity. This reduction is sufficient to induce a preferential transfer of lactate over acetate at pH < 4.2. Finally, it has been shown that the pH has little effect on the selectivity in the case of acetate-butyrate mixture, due to their similar pKa.

## 4.3. Modeling of selective extraction

### 4.3.1. Experimental determination of model parameters

As previously explained, the model requires to determine some parameters that can be obtained experimentally. These include the hydration numbers $n^H$ of migrating ions, the permselectivity $P_A/B$ between anions, and the permselectivity between cations $P_{H^+/Na^+}$. The determination of ion hydration number was realized using the methodology developed in a previous paper [25]. Values obtained for the hydration numbers are presented in Table 7.

One can notice that the sodium hydration number $n_{Na^+}^H = 6.1$ (CMX) determined is very close to that, $n_{Na^+}^H = 6$, reported in earlier studies with same membranes [25].

It has been shown in a previous paper that the permselectivity between ions depends only on the membrane used and the specific couple of ions investigated [23]. A methodology was also developed to determine the permselectivity of couples of ions. This methodology is used to get the permselectivity of acetate/formate and acetate/lactate couples at pH 7, as illustrated in Fig. 8.

While the model and methodology are applicable to various acid couples, the acetate-butyrate system is not considered here, as previous results showed that its selectivity and purity variations remain almost unaffected by pH.

As expected, the plots of $\ln(n_A/n_A^0)$ versus $\ln(n_B/n_B^0)$ are linear for both mixtures, with regression coefficients $R^2 > 0.99$. The slopes of these linear fits give the permselectivity values. For the formate/acetate mixture, a permselectivity of $P_{F_0/A_c} = 3.4 \pm 0.3$ is obtained, while the acetate/lactate mixture yields a value of $P_{Ac/Lac} = 2.0 \pm 0.3$.

This methodology is based on mass balances to determine the permselectivity, where the quantities of each ion, calculated from concentrations and volumes in the system [23].

But it is inadequate for systems involving protons. Indeed, the concentration of $H^+$ is not only governed by transport but also tightly linked to the acid-base equilibrium of weak acids in solution. As a result, the proton flux $J_{H^+}$ cannot be reliably calculated using concentration measurements alone. To address this limitation, an alternative approach was adopted in this study where the cation permselectivity was determined by fitting the model to the experimental pH using a least-squares method.

For this purpose, the experiment previously described (experiment 1), using a sodium acetate solution at pH = 4.8, is employed. Fig. 9 shows the comparison between experimental and simulated pH and sodium concentration variations in the diluate. The simulations were performed using the initial experimental conditions for concentration, volume, and pH, along with previously determined hydration numbers.
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Butyrate purity (a) and pH (b) in the diluate and acetate purity (c) and pH (d) in the concentrate as a function of the recovered fraction for different initial values in the diluate. Experiment 15: *I* = 1 A, *pH*<sub>D</sub> = 7; Experiment 16: *U* = 20 V, *pH*<sub>D</sub> = 4.8, *C* = 20 g/L.</figcaption></figure>

<table><caption>**Table 7**
Determined ion hydration number with ACS and CMX membranes.</caption><thead><tr><th colspan="3">Hydratation number n<sup>H</sup></th></tr><tr><th>Solute</th><th>ACS membrane</th><th>CMX membrane</th></tr></thead><tbody><tr><td>Acetate</td><td>5.4 ± 0.6</td><td></td></tr><tr><td>Lactate</td><td>13.2 ± 3.9</td><td></td></tr><tr><td>Formate</td><td>5.8 ± 1.2</td><td></td></tr><tr><td>Sodium</td><td></td><td>6.1 ± 0.2</td></tr><tr><td>Hydronium H<sup>+</sup></td><td></td><td>4.3 ± 2.3</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Formate-Acetate and Acetate-Lactate permselectivity determination (method from [23]). *C*<sub>Ac</sub> = 30 g/L, Experiment 2: *I* = 2 A, *pH*<sub>D</sub> = 7; Experiment 10: *U* = 20 V, *pH*<sub>D</sub> = 7.</figcaption></figure>

The shaded area represents the limit of the validity domain of the model which has been fixed at Λ = 95%. The cation permselectivity obtained by fitting is *P*<sub>H⁺/Na⁺</sub> = 18. This value enables an accurate description of both the sodium concentration (R² > 0.998) and pH (R² > 0.989) evolution during electrodialysis. This value *P*<sub>H⁺/Na⁺</sub> = 18 will be used for further modeling.

### 4.3.2. Comparison between calculated and experimental results

This section introduces the modeling approach developed to simulate the influence of pH on concentration variations and selectivity during the electrodialysis of organic acid mixtures to compare it to experimental values.

Fig. 10 and Fig. 11 plot the variation of purity as a function of recovered fraction for formate/acetate and acetate/lactate, respectively.

It is confirmed that the model accurately describes composition changes at pH 7 [23]. However, for pH values below 7, a significant discrepancy is observed between the model predictions and experimental data.

These deviations are particularly evident in the model's overestimation of selectivity. In the case of the formate/acetate mixture, the calculated purities are much higher than the experimental values at a given recovered fraction, indicating that the model overpredicts selective formate transport. A similar observation can be made for the acetate/lactate system, where the model underestimates lactate purity in the diluate and acetate purity in the concentrate. From these observations, it is assumed that this deviation comes from the charged fraction ratio, which would be lower in practice, for Fo/Ac as well as Lac/Ac mixtures, than the model predicts.

This discrepancy can be attributed to the acid dissociation mechanism at the membrane interface. As discussed in the theoretical background, two main theories account for this phenomenon: one proposes a shift in local pH near the membrane, while the other suggests a field-
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Variation of sodium concentration and pH in the diluate as a function of the recovered fraction. Experimental data (points) and calculated data (curves). Experiment 1: U = 20 V, C<sub>Ac</sub> = 30 g/L.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Acetate purity in the diluate and formate purity in the concentrate as a function of recovered fraction for different initial pH formate/acetate mixture. Experimental points (symbols) vs. calculated curves. C<sub>Ac</sub> = 30 g/L, Experiment 2: I = 2 A, pH<sub>D</sub> = 7; Experiment 3: U = 20 V, pH<sub>D</sub> = 4.7; Experiment 7: U = 20 V, pH<sub>D</sub> = 3.7.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Lactate purity in the diluate and acetate purity in the concentrate as a function of recovered fraction for different initial pH values acetate/lactate mixture. Experimental points (symbols) vs calculated curves. The straight dashed line represents 50% purity as an indicator. C<sub>Ac</sub> = 30 g/L, Experiment 10: U = 20 V, pH<sub>D</sub> = 7; Experiment 11: U = 20 V, pH<sub>D</sub> = 4.2; Experiment 14: U = 20 V, pH<sub>D</sub> = 4.7.</figcaption></figure>
induced modification of the acid dissociation constant $Ka$.

To account for this mechanism, a corrective coefficient $\sigma$ is introduced into the model. Depending on the underlying mechanism, $\sigma$ is defined either as the difference between the bulk solution pH and the pH near the membrane interface ($pH_m$) (Eq. (16)), membrane-catalyzed water-splitting), or as the difference between the dissociation constant in solution and the dissociation constant under a strong local electric field $pKa(E_V)$ (Eq. (17)), second Wien effect).

$$ \sigma = pH_m - pH \qquad (16) $$

$$ \sigma = pKa(0) - pKa(E_V) \qquad (17) $$

The Eq. (2), which is used to calculate the charged fraction $X_A$ of an acid A, is thus modified to yield Eq. (18).

$$ X_A = \frac{1}{1 + 10^{(pKa_A(0) - pH) - \sigma}} \qquad (18) $$

The value of $\sigma$ can be determined for each experiment by fitting experimental values with calculated ones (least squares method).

Fig. 12 shows the concentration variations of acetate and formate in the diluate for two initial pH values (7.0 and 3.7). Experimental points correspond to total concentration ($HA + A^-$) of the acids, as measured by HPLC. Experimental data are compared with model predictions, with and without the corrective coefficient $\sigma$. The value obtained to fit experimental and calculated value is $\sigma_{Fo/Ac} = 1.6$.

Fig. 13 displays as an example the calculated and experimental variations of acetate purity in the diluate and formate purity in the concentrate, as function of their respective yield. Calculated values have been obtained using $\sigma_{Fo/Ac} = 1.6$.

A very good match between the experimental data and the model predictions is observed. Same approach was used for every experiment. The fitted values of $\sigma$ obtained are provided in Table 8.

For both formate/acetate and acetate/lactate systems, $\sigma$ values range from 1.3 to 1.6, with an average of $\sigma_{Fo/Ac} = 1.4$ and $\sigma_{Ac/Lac} = 1.5$, respectively. These values being very close, a Student t-test was conducted to assess whether the difference is statistically significant. The result ($p = 0.71$, $\alpha = 0.05$) indicates no significant difference. Therefore, a single constant value of $\sigma = 1.4 \pm 0.1$ is further used for both mixtures.

To evaluate the physical meaning of the fitted coefficient $\sigma$, the theoretical framework of the Second Wien Effect can be used. This approach allows the estimation of $E_V$ from changes in the dissociation constant $Ka$ and the relative permittivity $\epsilon_R$ of the medium. Detailed calculation is developed in Appendix B.

Assuming that the relative permittivity lies within $20 < \epsilon_R < 80$, the calculated local electric field $E_V$ lies between $0.9 \times 10^8 V/m$ and $3.8 \times 10^8 V/m$ with a coefficient $\sigma = 1.4$. This value can be compared to those reported previously.

According to the literature, local electric fields to observe the Second Wien Effect typically range from $10^7$ to $10^8 V/m$ [28,29]. For example, a commonly cited value is the average electric field $E_V = 2.5 \times 10^8 V/m$ obtained with a relative permittivity $\epsilon_R = 20$ by Simons [28] through integration of the Poisson equation. This value supports the introduction of the coefficient $\sigma$ in our model to account for dissociation mechanisms since the calculated electric field is in the range of the ones reported in previous studies.

This section shows that the membrane permselectivity $P_{A/B}$, determined under conditions where both acids are fully dissociated, is a necessary but insufficient parameter to predict the overall separation performance across a wide pH range. The effective selectivity between weak acids varies strongly with pH through its effect on acid dissociation (pKa) and on the reaction-transport mechanism at the membrane interface.

Moreover, it was observed that the influence of pH on selectivity

<figure><img src="image_14.png" /><figcaption>Fig. 12. Evolution of acetate and formate concentrations in the diluate as a function of the recovered fraction, formate/acetate mixtures at pH 7 and 3.7. Experimental points and calculated curves with $\sigma = 0$ and $\sigma = 1.6$. $C_{Ac} = 30$ g/L, Experiment 2: $I = 2$ A, $pH_D = 7$; Experiment 7: $U = 20$ V, $pH_D = 3.7$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Variation in acetate and formate purity in the dilute and concentrate, respectively, versus their respective yield. Experimental points and calculated curves with $\sigma = 1.6$. $C_{Ac} = 30$ g/L, Experiment 2: $I = 2$ A, $pH_D = 7$; Experiment 7: $U = 20$ V, $pH_D = 3.7$.</figcaption></figure>

<table><caption>**Table 8**
Values of $\sigma$ obtained for all experiments.</caption><thead><tr><th>Mixture</th><th>Experiment</th><th>Initial pH</th><th>σ</th></tr></thead><tbody><tr><td rowspan="7">Formate/acetate</td><td>3</td><td>4.8</td><td>1.2</td></tr><tr><td>4</td><td>4.8</td><td>1.2</td></tr><tr><td>5</td><td>4.2</td><td>1.3</td></tr><tr><td>6</td><td>3.7</td><td>1.6</td></tr><tr><td>7</td><td>2.5</td><td>1.4</td></tr><tr><td>8</td><td>4</td><td>1.5</td></tr><tr><td>9</td><td>3.7</td><td>1.6</td></tr><tr><td rowspan="4">Acetate-lactate</td><td>11</td><td>4.2</td><td>1.5</td></tr><tr><td>12</td><td>5.6</td><td>1.5</td></tr><tr><td>13</td><td>4.2</td><td>1.3</td></tr><tr><td>14</td><td>4.7</td><td>1.6</td></tr></tbody></table>

varies depending on the acid pair considered. To capture these effects, the model was extended to account for composition changes induced by pH during extraction. This required the determination of not only the permselectivity between the organic acids, but also between sodium and $H^+$ ions, as well as that of ion hydration numbers.

Additionally, the presence of the dissociation mechanism is represented by a constant coefficient $\sigma = 1.4 \pm 0.1$, which was determined by fitting the calculated and experimental data. Incorporating this coefficient into the model enables the accurate prediction of organic acid composition during electrodialysis under any operating condition.

### 4.3.3. Modeling of organic acids extraction

In the previous section, the model was validated against experimental data for selected pH values and couples of acids. The current section aims to use the model to simulate and visualize how selectivity evolves across a wide range of initial pH values and recovered fractions. This view offers a better understanding of pH-dependent separation trends and the role of sodium as a counter-ion in determining the process selectivity.

In particular, 3D and contour plots are used to provide a continuous representation of solute purity evolution in both dilute and concentrate compartments. This graphical approach highlights key transition zones, specifically where proton transport begins to dominate over that of sodium, i.e. where the dissociation mechanism takes places, and shows how these transitions influence selectivity.

The model was used to compute the purity obtained with equimolar formate/acetate and acetate/lactate mixtures over initial pH ranges from 3.7 to 7 and from 4.2 to 7, respectively and for a total concentration of $C = 30$ g/L.

**Fig. 14** presents, from two different perspectives, the variation of acetate purity in the dilute as a function of the initial pH and recovered fraction for a 50/50 (mol%) mixture of formate/acetate.

In the same manner, Fig. 15 illustrates the variation in formate purity in the concentrate with respect to the initial pH and recovered fraction.

The results show that acetate purity in the dilute steadily increases during extraction. This trend is directly related to the permselectivity between acetate and formate ($P_{fo/Ac}=3.4$), with formate migrating preferentially to the concentrate, as seen in Fig. 14 and Fig. 15.

However, the purity changes differently depending on the initial pH. Indeed, both figures indicate that for pH values below 5, the acetate purity in the dilute and the formate purity in the concentrate increase significantly compared to pH 7. This increase is directly correlated with the sodium concentration in the dilute. The dashed curve represents the recovered fraction at which the sodium flux becomes less than or equal to the proton flux. In other words, this curve marks the boundary of the domain where the acid dissociation mechanism catalyzed at the membrane becomes significant. Beyond that recovered fraction, the sodium concentration still decreases and becomes negligible, and the current is carried by $H^+$ ions generated at the membrane.

Earlier in this paper, it was shown that transferring organic acids changes the ratio $A^-/AH$ in the dilute and decreases the pH. Additionally, maintaining electroneutrality requires that the concentration of the dissociated form ($A^-$) be balanced by counter-ions, making it dependent on sodium concentration. Finally, the results have demonstrated that selectivity is pH-dependent. Since selectivity was also shown to be pH-dependent, it follows that the selectivity between two weak acids is indirectly governed by the concentration of the counter-ion, sodium.

The results obtained with the acetate/lactate mixture are shown in Fig. 16 and Fig. 17.

Fig. 16 presents lactate purity in the dilute, while Fig. 17 shows acetate purity in the concentrate, both as functions of initial pH and recovered fraction for the acetate-lactate system.

At pH values above 5, the behavior is similar to that observed for the acetate/formate mixture: the most mobile species (acetate, in that case) transfer preferentially in the concentrate as extraction proceeds, increasing lactate purity in the dilute. However, with a lower pH, this trend reverses as the lactate purity tends to decrease in the dilute.

As in the previous case, the dashed line in Fig. 16 marks the boundary where the fluxes of $Na^+$ and $H^+$ are equal, distinguishing the two operational regimes. At first, the sodium is the dominant cation to carry the current to equilibrate the initial charged fraction of organic acid. Then, due to the decrease of sodium concentration, the current is mainly
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Acetate purity in the diluate as a function of the initial pH and recovered fraction. 3D and contour plots. The dashed curve indicates the recovered fraction at which $J_{Na^+} = J_{H^+}$. Initial 50/50 (mol%) formate/acetate mixture in the diluate.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Formate purity in the concentrate as a function of the diluate initial pH and the recovered fraction. 3D and contour plots. The dashed curve marks the recovered fraction at which $J_{Na^+} = J_{H^+}$. Initial 50/50 (mol%) formate/acetate mixture in the diluate.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Lactate purity in the diluate as a function of the diluate initial pH and the recovered fraction. 3D and contour plots. Dashed curve: recovered fraction at which $J_{Na^+} = J_{H^+}$. Initial 50/50 (mol%) acetate/lactate mixture in the diluate. Validity limit of the model: recovered fraction <95%.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Fig. 17.** Acetate purity in the concentrate as a function of the diluate initial pH and the recovered fraction. 3D and contour plots. Dashed curve: recovered fraction at which $J_{Na^+} = J_{H^+}$. Initial 50/50 (mol%) acetate/lactate mixture in the diluate. Validity limit of the model: recovered fraction <95%.</figcaption></figure>

carried by the protons released by the acid dissociation mechanism.

These results show that in order to realize an efficient separation between lactate and acetate, the process must start with an initial pH solution of pH > 5.5 or pH < 4.2. If pH > 5.5, full organic acids are totally dissociated leading to preferential acetate transfer during the whole extraction. For pH < 4.2, membrane-catalyzed dissociation and the charged fraction of the acids lead to a preferential lactate transport, due to its lower pKA and higher dissociation degree, compared to acetate. For an initial pH between 4.2 and 5.5, the process exhibits minimal selectivity, with purities remaining near 50%.

Similar observations can be made from Fig. 17. Indeed, working with an acetate/lactate mixture enriches acetate in the concentrate compartment, as found experimentally. However, lowering the initial pH leads to a loss of selectivity. Thus, for lower pH, lactate becomes the preferentially transferred species relative to acetate.

These model-based visualizations highlight how the transition in charge carrier, from sodium to protons, can shape the electrodialysis selectivity. By capturing the interplay between pH, acid dissociation (pKA) and mechanisms at the membrane, the model reveals optimal operating conditions for selective separation of organic acid.

## 5. Conclusion

In this study, we investigated the influence of pH on the selective transport of organic acids in electrodialysis, with a particular focus on the role of a reaction-transport mechanism that promotes weak acid dissociation at the membrane interface. Electrodialysis experiments using a sodium acetate solution at pH 4.8 provided direct evidence of this mechanism. Specifically, a decrease in acetic acid concentration in the diluate appears when the sodium concentration becomes negligible, accompanied by an increase in acetic acid concentration in the concentrate. This behavior indicates that initially neutral acids undergo dissociation in the vicinity of the membrane, enabling both the released proton and the resulting anion to participate in ion transport.

Further experiments were carried out with equimolar mixtures of acetate with formate, lactate, and butyrate for various initial pH. The results showed that selectivity is strongly influenced by the pH and the pKa values of the acids, which determine the proportions of dissociated species.

In formate/acetate systems, formate was preferentially transferred, and this selectivity increased with decreasing pH due to the greater dissociation of formate compared to acetate ($pKa_{Fo} < pKa_{Ac}$). In contrast, in acetate/lactate mixtures, acetate was more selectively

transferred at higher pH, but selectivity declined at lower pH due to the lower dissociation degree of acetate over lactate ($pKa_{Ac} > pKa_{Lac}$). Finally, with acetate/butyrate mixtures, the selectivity remains constant whatever the pH due to their similar pKA.

The model previously developed for charged ions was extended to include pH-dependent ion transfer and solvent flux, enabling the calculation of both concentration and pH variations during the process. To supply the model with the necessary parameters, dedicated experiments were conducted to determine the ion hydration numbers and the anion permselectivities. Specifically, permselectivities between formate and acetate $P_{Fo/Ac} = 3.4 \pm 0.3$ and between acetate and lactate $P_{Ac/Lac} = 2.0 \pm 0.3$ were measured using established analytical methodologies. Additionally, the cation permselectivity between sodium and protons was derived through least-squares fitting of simulated and experimental pH variations, yielding a value of $P_{H^+/Na^+} = 18$.

To fit experimental results with model calculations, it was necessary to introduce a corrective coefficient $\sigma$ into the equation giving the charged fraction of the organic acids. An average value of $\sigma = 1.4 \pm 0.1$ was determined by fitting experimental and calculated concentrations as function of the recovered fraction. No significant influence of the operating parameters was observed. It reflects either a lowering of acid pKa under a strong local electric field (the “electric field” theory, based on the SWE) or enhanced, membrane-catalyzed acid dissociation (the “catalytic” theory). Considering the SWE, the resulting local electric field was calculated and the value was found to be in the range of published data, suggesting that $\sigma$ indeed captures the reaction-transport mechanism.

Finally, incorporating the reaction-transport mechanism into the model allows for accurate simulation of both solute concentrations and pH variations across a wide range of conditions, including the typical pH conditions of fermentation-based processes. By enabling the visualization of key variables such as concentration and pH evolution, as functions of initial pH and recovered fraction, the model offers valuable insight into the influence of charge carrier transitions, from sodium to protons, on process selectivity. These capabilities extend beyond validating experimental trends; they allow for the identification of optimal operating conditions for the selective recovered of organic acids. This makes the model particularly useful for optimizing pH-dependent separation strategies in biotechnological applications. Overall, the results demonstrate that integrating both pH effects and membrane-driven dissociation mechanisms is essential for the reliable prediction and optimization of electrodialysis performance in the selective separation of weak acids. In this sense, the present process-level analysis of volatile
fatty acid mixtures bridges the gap between mechanistic models of weak-electrolyte transport and the practical design of electrodialysis conditions for selective recovery of organic acids.

### CRediT authorship contribution statement

**R. Caveriviere:** Writing – original draft, Methodology, Investiga- tion, Formal analysis. **S. Galier:** Writing – review & editing, Validation, Supervision, Conceptualization. **H. Roux-de Balmann:** Writing – re- view & editing, Supervision, Resources, Funding acquisition, Conceptualization.

### Funding

This research was co-funded by Toulouse INP and the Région

Occitanie through the PhD grant of Robin Caverivière 2021–09 COMUE/Région “Projet BioCO2Organics”. The authors acknowledge also the financial support provided by the French state aid managed by the National Research Agency under the Investments for the Future program bearing the reference ANR-18-EURE-0021.

### 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.

## Appendix A. Concentrations and pH model calculation

All equations are developed for the calculation of the concentrations and the pH in the dilute compartment during ED.

First, the charged fraction of an organic acid is obtained using Eq. (A.1).

$$X_i = \frac{1}{1 + 10^{(pKa_i - pH)}} \qquad (A.1)$$

By definition, permselectivity depends on the concentration of ionized species. The charged fraction is therefore used to convert the total con- centration of an acid into its ionic concentration. Hence, permselectivity is expressed in terms of the total concentration of A and B Eq. (A.2).

$$P_{A/B^{-}} = \frac{J_{A^{-}}/J_{B^{-}}}{C_{A}X_{A}/C_{B}X_{B}} \qquad (A.2)$$

Solving the system composed of Eqs. (A.2) and (A.3) allows us to find the fluxes of ions A⁻ and B⁻. Combining these two equations yields Eqs. (A.4) and (A.5).

$$\sum_{i} J_{i} = \eta \frac{N}{F} I \qquad (A.3)$$

$$J_{B^{-}} = \frac{N_{F}^{I}}{1 + P_{A^{-}/B^{-}} \times \frac{C_{A} X_{A}}{C_{B} X_{B}}} \qquad (A.4)$$

$$J_{A^{-}} = N_{F}^{I} - J_{B^{-}} \qquad (A.5)$$

Similarly, cation fluxes are calculated using the definition of permselectivity between Na⁺ and H⁺. This is given by Eq. (A.6):

$$P_{H^{+}/Na^{+}} = \frac{J_{H^{+}}/J_{Na^{+}}}{C_{H^{+}}/C_{Na^{+}}} \qquad (A.6)$$

Solving the flux system for cations follows the same approach as for anions. The permselectivity definition is combined with Eq. (A.3) to obtain J<sub>H⁺</sub> and J<sub>Na⁺</sub>. The combination of the two equations leads to Eq. (A.7), expressed as function of the concentration of H⁺ (pH) and Eq. (A.8):

$$J_{Na^{+}} = \frac{N_{F}^{I}}{1 + P_{H^{+}/Na^{+}} \times \frac{C_{H^{+}}}{C_{Na^{+}}}} \qquad (A.7)$$

$$J_{H^{+}} = N_{F}^{I} - J_{Na^{+}} \qquad (A.8)$$

To take into account the variation of the volume of the compartment, solvent transfer must be considered. To calculate the solvent flux associated with ion transport, Eq. (A.9) is used.

$$J_{solvent} = V_{m}^{H_{2}O} \sum_{i} J_{i} \times n_{i}^{H} \qquad (A.9)$$

Water molar volume is V<sub>m</sub><sup>H₂O</sup> = 18 × 10⁻⁶ m³/mol at 25 °C and the hydration number n<sub>i</sub><sup>H</sup> is in mol<sub>H₂O</sub>/mol<sub>i</sub>. One must know the fluxes of each anion and cation J<sub>i</sub>, as well as their respective hydration numbers. In the case of two anions A⁻ and B⁻ and two cations Na⁺ and H⁺, the solvent flux is given by Eq. (A.10):

$$J_{solvent} = V_{m}^{H_{2}O} \times (J_{A^{-}} \times n_{A}^{H} + J_{B^{-}} \times n_{B}^{H} + J_{Na^{+}} \times n_{Na^{+}}^{H} + J_{H^{+}} \times n_{H^{+}}^{H}) \qquad (A.10)$$

The time required for a complete extraction, denoted t(n<sup>i</sup>), is calculated via Eq. (A.11), which is derived from Eq. (A.3):

$$t(n^i) = \frac{F}{N \eta I} (n_A^0 + n_B^0) \qquad (A.11)$$

The time interval Δt between two iterations is given by Eq. (A.12), with n<sup>i</sup> being the total number of iterations in the model:
$$ \Delta t = \frac{t(n^i)}{n^i} \qquad (A.12) $$

The time interval $\Delta t$ is chosen sufficiently small so that $J_i$ may be considered constant (Eq. (A.13)):

$$ \int_{t}^{t+\Delta t} J_i dt \approx J_i \times \Delta t \qquad (A.13) $$

The previously determined fluxes are then used to compute the quantities $n_{A^-}$, $n_{B^-}$ and $n_{Na^+}$ using Eq. (A.14). Because the calculations are performed for the dilute compartment, the fluxes calculated are outfluxes (transfers from the dilute to the concentrate):

$$ n_i(t + \Delta t) = n_i(t) - J_i(t) \times \Delta t \qquad (A.14) $$

The recovered fraction $\Lambda$ is defined by Eq. (A.15). Its expression is given by Eq. (A.16) in our conditions:

$$ \Lambda = 1 - \frac{\sum_{a} n_{a}^{D}}{\sum_{a} n_{a}^{0D}} \qquad (A.15) $$

$$ \Lambda (t + \Delta t) = 1 - \frac{n_A(t + \Delta t) + n_B(t + \Delta t)}{n_A(0) + n_B(0)} \qquad (A.16) $$

Similar to solute fluxes, the solvent flux is assumed constant over the time interval $\Delta t$ and is expressed by Eq. (A.17). In the dilute, solvent transfer is an outflux as it is associated with ion transport:

$$ \int_{t}^{t+\Delta t} J_{\text{solvent}} dt \approx J_{\text{solvent}} \times \Delta t \qquad (A.17) $$

The volume is then calculated using Eq. (A.18).

$$ V(t + \Delta t) = V(t) - J_{\text{solvent}}(t) \times \Delta t \qquad (A.18) $$

Finally, the concentration of each ion (except $H^+$) is determined using Eq. (A.19):

$$ C_i(t + \Delta t) = \frac{n_i(t + \Delta t)}{V(t + \Delta t)} \qquad (A.19) $$

The pH ($H^+$ ion concentration) value is obtained from the electroneutrality equation between anions $a^-$ and cations $c^+$, as stated in Eq. (A.20):

$$ \sum_{a} z_{a} \cdot C_{a^{-}} = \sum_{c^{+}} z_{c^{+}} \cdot C_{c^{+}} \qquad (A.20) $$

Using Eq. (A.20) in the case of either one or two organic acids A and B in solution with sodium $Na^+$ gives Eqs. (A.21) and (A.22), respectively:

$$ C_{A^{-}} + C_{OH^{-}} = C_{Na^{+}} + C_{H^{+}} \qquad (A.21) $$

$$ C_{A^{-}} + C_{B^{-}} + C_{OH^{-}} = C_{Na^{+}} + C_{H^{+}} \qquad (A.22) $$

In our work, the pH range of interest lies between 2.5 and 8. Hence, the concentration of hydroxide ions $OH^{-}$ is considered negligible relative to other anions. The derivation for solving these equations yields a quadratic equation in the case of a single weak acid in solution. Calculating the pH thus involves solving the quadratic roots, which analytically gives us Eq. (A.23).

$$ pH = -\log\left(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right) \qquad (A.23) $$

With the following coefficients:

$$ \begin{aligned} a &= 1 \\ b &= C_{Na^{+}} + K a_{A} \\ c &= K a_{A} (C_{Na^{+}} - C_{A}) \end{aligned} $$

In the second case, where two organic acids are present in solution, we obtain Eq. (A.24):

$$ a C_{H^{+}}^{3} + b C_{H^{+}}^{2} + c C_{H^{+}} + d = 0 \qquad (A.24) $$

The coefficients in this case are:

$$ \begin{aligned} a &= 1 \\ b &= K a_{A} + K a_{B} + C_{Na^{+}} \\ c &= (K a_{A} + K a_{B}) C_{Na^{+}} + K a_{A} K a_{B} - K a_{A} C_{A} - K a_{B} C_{B} \\ d &= K a_{A} K a_{B} (C_{Na^{+}} - (C_{A} + C_{B})) \end{aligned} $$

Hence, a cubic equation is obtained, which determines the concentration of $H^+$ ions and thus the pH. The concentration $C_{H^+}$ and the resulting pH are found numerically (Python 3.9.13).
## Appendix B. Local electric field calculation (SWE)

A theoretical calculation of the variation in the dissociation constant was first proposed by Lars Onsager in 1934 [17,18]. The solution derived from his theory includes a first-order Bessel function $Γ₁$ as described by Eq. (B.1). It expresses the ratio between the dissociation constant in the presence of an electric field $Ka(E_V)$ and that in the absence of an electric field $Ka(0)$.

$$ \frac{Ka(E_V)}{Ka(0)} = \frac{Γ_1(4\sqrt{-\frac{b}{2}})}{2\sqrt{-\frac{b}{2}}} = 1 + b + \frac{b^2}{3} + \frac{b^3}{18} + \frac{b^4}{180} + \frac{b^5}{2700} + \frac{b^6}{56700} + \dots \quad (B.1) $$

The coefficient b is defined in Eq. (B.2).

$$ b = \frac{e^3}{8 \pi \epsilon_0 (k_B T)^2} \times \frac{E_V}{\epsilon_R} \quad (B.2) $$

In this equation, $ε_0$ (F/m) and $ε_R$ represent the vacuum and relative permittivity, respectively, $k_B$ (J/K) is the Boltzmann constant, e (C) is the elementary charge, $E_V$ (V/m) is the local electric field, and T (K) is the temperature.

Furthermore, Eq. (17) can be rewritten into Eq. (B.3).

$$ \sigma = \log \left( \frac{Ka(E_V)}{Ka(0)} \right) \quad (B.3) $$

By then combining Eq. (B.1) with Eq. (B.3), one arrives at Eq. (B.4).

$$ \sigma = \log \left( \frac{\Gamma_1 \left( 4 \sqrt{-\frac{b}{2}} \right)}{2 \sqrt{-\frac{b}{2}}} \right) \quad (B.4) $$

Calculating the electric field requires knowledge of the relative permittivity $ε_R$ near the membrane, which can range from $ε_R = 20$ to $ε_R = 80$ for aqueous solutions [21,28]. Assuming the relative permittivity lies within this interval, solving the equation gives a local electric field $E_V$ between $0.9 \times 10^8$ V/m and $3.8 \times 10^8$ V/m.

## Data availability

Data will be made available on request.

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