Anomalous Correlations Between Ionic Conductivity Isotherms and Phase Diagrams in Hydrofluoric and Acetic Acid Aqueous Systems

Hilal Al-Salih 1

Asefeh Poshtchioskuei 1

Yaser Abu-Lebdeh 1✉ EmailYaser.Abu-Lebdeh@nrc-cnrc.gc.ca

1 Energy, Mining, and Environment Research Centre National Research Council of Canada 1200 Montreal Road K1A 0R6 Ottawa Ontario Canada

Hilal Al-Salih, Asefeh Poshtchioskuei, Yaser Abu-Lebdeh^*

Energy, Mining, and Environment Research Centre, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario K1A 0R6, Canada

* Corresponding author: Yaser.Abu-Lebdeh@nrc-cnrc.gc.ca

Abstract

There is an empirically established correlation between ionic conductivity isotherms and solid–liquid phase diagrams in most electrolyte solutions, wherein the composition of maximum conductivity (x_max) typically coincides with the eutectic composition (x_eutectic). In this work, we investigate pronounced deviations from this correlation observed in hydrofluoric acid (HF) and acetic acid (AA) aqueous solutions, two systems that behave anomalously compared to other common acids. Ionic conductivity data across multiple temperatures and concentrations were compiled from literature and supplemented with new experimental measurements. Hydrofluoric acid exhibits nearly linear isotherms that bypass all eutectic features without showing any conductivity maximum, whereas acetic acid displays a parabolic isotherm with an x_max occurring far earlier than its eutectic point, representing an extreme case of x_max ≪ x_eutectic. We attribute these deviations to peculiar chemical equilibria and a less-understood liquid microstructure that likely give rise to distinct microdomain organization and modified ion-transport dynamics. To further probe these anomalies, activation energy and pre-exponential factor values were extracted as functions of molar fraction and examined for both acids. These findings highlight the need for refined theoretical frameworks capable of capturing transport behaviour in systems where traditional conductivity–phase diagram correlations break down.

Keywords:

acids

aqueous electrolyte solutions

liquid solution structure

phase diagram

ionic conductivity

hydrofluoric acid

acetic acid

eutectics

activity coefficients

Introduction

The connection between the microstructure of liquid electrolyte solutions at different concentrations and their corresponding conductivity isotherms is quite an intricate one.¹ Abu-Lebdeh’s research group delved deep into examining this connection, and managed to establish relationships between phase diagrams and conductivity isotherms for various organic/inorganic salts,^2,3 aqueous nitrates,⁴ and aqueous acid solutions,⁵ thereby enriching the understanding of the microstructure beyond the infinitely dilute concentration, which is often explained by the Debye–Hückel theory and its extensions.⁶ One solid established correlation is that the first eutectic point (x_eutectic) in the binary salt/water phase diagram usually corresponds to the peak point in the corresponding conductivity isotherm; the x_max. This has been observed in the cases of different aqueous and non aqueous electrolyte solutions. ^2–5,7,8

Fig. 1

(a) Schematic illustration of Abu-Lebdeh's model for liquid electrolytes' microstructure at various concentrations. The figure is generalized to cover all aqueous solutions. (b) Illustration showing the typical close convergence of the point of maximum ionic conductivity (x_max) and the eutectic composition point (x_e). The top panel depicts the ionic conductivity (κ) as a function of composition (x), while the bottom panel illustrates the phase diagram with temperature (T) versus composition (x).

The correlation between x_max and x_eutectic can be understood through the structural changes in the electrolyte solution at the eutectic composition, where optimal solute-solvent interactions and structural stability occur.² Approaching this composition, the differential increase in charge carrier concentration (

$\:\frac{d{n}_{ion}}{{d}_{x}}$

) rises rapidly and then at a slower rate, balancing with the differential change in viscosity (

$\:\frac{d\eta\:}{{d}_{x}}$

), which initially increases gradually (linearly) but then more aggressively (exponentially) at higher concentrations. This balance results in the peak ion mobility and conductivity at the eutectic point as seen in Fig. 1 (b). The model proposed by Abu-Lebdeh et al.⁴ indicates that the microstructure at the eutectic composition features fewer distinct microdomains, with the liquid and solid phases forming laminar-like structures. These uniform, laminar structures contribute to optimal conductivity by minimizing disruptive microdomains and reducing shear, thereby facilitating smoother ion transport. There is a drop in conductivity beyond the eutectic that tends to be more aggressive if the phase diagram contains a solvate composition. Abu-Lebdeh et al.⁴ proposed a model to explain the microstructural domains that exist at all concentrations. According to this model, above the liquidus line, the liquid structure is a heterogeneous mixture similar to the solid state observed below the solidus line. This mixture comprises distinct molten sub-micron domains, each populated by charge carriers originating from the fragmentation of the bulk structure into components such as ion pairs (IPs), ion clusters (ICs), or solvent aggregates. The number and mobility (bulk or local) of the ions determine the ionic conductivity. The dominant microdomain dictates the bulk transport properties of the liquid. For example, the molten solvent domain is dominant before the eutectic composition and hence viscosity is low and ionic conductivity is higher compared to beyond the eutectic composition where the molten-solvate or salt is dominant and causes overall decrease in bulk ionic transport. However, equal conductivity can be achieved before and beyond concentration due to change in mechanisms and local properties play a role. e.g., decoupling of ionic conductivity from viscosity. It is important to note that the presence of microdomains in aqueous electrolyte solutions, including HF and AA, remains a working hypothesis, as direct visualization of molecular-scale heterogeneities in concentrated liquid solutions is inherently difficult. Advanced structural characterization methods, such as neutron scattering or ultrafast spectroscopy, could provide deeper insight into these transitions, but current techniques rely on indirect spectroscopic and computational interpretations. Nevertheless, deviations in ionic conductivity trends, activation energy shifts, and phase behavior observed in this study provide meaningful experimental evidence that supports the relevance of microstructural domains in concentrated HF and AA solutions.

Figure 1 provides a schematic illustration of this model. The plot in the figure depicts a generic, simple eutectic phase diagram with temperature (T) on the y-axis and composition (x) on the x-axis, showcasing the liquidus and solidus lines. Surrounding this phase diagram are various microstructural snapshots at different concentrations and temperatures, helping to visualize the proposed model. These snapshots illustrate the complex microstructural changes, offering a comprehensive understanding of the interplay between phase behaviour and ionic conductivity in electrolyte solutions, thereby bridging the gap in our understanding beyond the dilute solution regime.

As with any new scientific effort, exceptions and outliers are bound to emerge. These exceptions, rather than undermining established relationships, offer an opportunity to test their limits and motivate refinement. By understanding the reasons behind the deviations, we can provide a more robust justification for the established norms. In this context, two electrolyte solutions, acetic acid (AA) and hydrofluoric acid (HF) in water, have shown behaviours that do not conform to the previously established relationships. While prior studies such as Broderick et al.⁹ have reported conductivity data for HF, these anomalies have not been analyzed through the lens of the broader empirical correlation between conductivity maxima and eutectic compositions that is observed across many aqueous systems. In this work, we reinterpret HF and AA as meaningful exceptions to that trend and use them as case studies to explore potential breakdowns in the established model. The importance of addressing such outliers lies in the potential to fortify the overarching theory and provide a better-rounded understanding of the subject. Herein, we aim to delve into these exceptions, exploring the unique behaviours of acetic acid and hydrofluoric acid in water. Through a combination of theoretical reasoning and empirical data, we seek to shed light on the reasons behind their deviations from the established norms. In doing so, we aim to provide clarity on these specific systems while identifying the limits of the foundational

relationships established in our previous work. Moreover, our findings demonstrate that a previously introduced semi-empirical equation for ionic conductivity remains adaptable and useful even in anomalous aqueous electrolyte solutions—highlighting its flexibility in capturing trends where traditional models fall short.

Fig. 2

(a) Acetic acid/water phase diagram with experimental 25°C conductivity isotherm (b) Hydrofluoric acid/water phase diagram with 0 ℃ conductivity isotherm.¹² Hydrates at specific molar ratios are labelled as Hx where x is the ratio of water moles to acetic acid

Methodology

We base our investigation and analysis in this paper on two main features: Phase diagrams and conductivity isotherms. Phase diagrams for acetic acid and hydrofluoric acid solutions in water were reproduced from Barr et al.¹⁰, and Giguère et al.¹¹, respectively. Data from the two binary phase diagrams were digitized using WebPlotDigitizer software. The units used were unified to be K for temperature (T) on the y-axis and molar fraction, given by (x) on the x-axis.

The ionic conductivity for hydrofluoric acid in water over the whole concentration and temperature ranges were reproduced from Sirkar et al.¹² and Broderik et al.⁹ For acetic acid in water, we found contrasting data in the literature (~ 23% difference between 25°C temperature κ_max values).^13,14 Therefore, we measured ionic conductivity. The measurements were conducted using an Oakton Waterproof pH/Con 300 Meter from Cole-Parmer,

Results and Discussion

Experimental Trends and Microstructural Origins of Conductivity Anomalies

Fig. 2

presents the conductivity isotherms of acetic acid and hydrofluoric acid aqueous solutions at 25°C and 0°C, respectively, and their respective phase diagrams. For acetic acid/water solutions (Fig. 2a), the phase diagram is a simple one with one eutectic point and no solvate formation. The conductivity isotherm on the other hand has an early peak point in the dilute region with conductivity gradually dropping thereafter. For hydrofluoric acid/water solutions (Fig. 2b), the phase diagram has three solvate formation points and three eutectic points and a conductivity isotherm that never peaks or drops. Rather, it increases linearly with molar fraction. Table 1 lists all of the data used in the above description for phase diagrams and 25°C conductivity isotherms.

Acid	Chemical structure	x_eutectic	T_eutectic (K)	x_max	κ _max (mS cm^− 1)	x_solvate
Acetic acid	CH₃COOH	0.30	246	0.06	1.82	--
Hydrofluoric acid	HF	0.27 0.76 0.88	203 171 163	--	375*	0.5 0.67 0.8

Previously, we have studied different organic/inorganic salts in water, nitrate salts in water, and acids in water. In our studies, we have observed few outstanding features that link conductivity isotherms to the corresponding binary salt/water phase diagrams. The most apparent and common correlation being the overlap of the eutectic molar fraction (x_eutectic) with point of highest conductivity in the isotherm (x_max). Acetic acid emerges as an extreme case, with its x_max appearing much earlier at x = 0.06, diverging strongly from its x_eutectic = 0.30, a deviation far more pronounced than in other systems studied to date. The conductivity maximum suggests a structural transition and a possible change in conductivity mechanism; however, the degree of deviation from x_eutectic is far beyond what is typically captured by Abu-Lebdeh’s model.⁴ One might expect the degree of ionization in an electrolyte solution to peak at an equimolar composition (x ≈ 0.5) where solute and solvent are balanced.¹⁹

H₂O + CH₃COOH ⇌ CH₃COO⁻ + H₃O⁺ K_a =

$\:\frac{{\text{a}}_{{\text{H}}^{+}}{\text{a}}_{{\text{C}\text{H}3\text{C}\text{O}\text{O}}^{-}}}{{\text{a}}_{\text{C}\text{H}3\text{C}\text{O}\text{O}\text{H}}}$

[1]

However, because the acid dissociation constant (Ka) is highly sensitive to the local dielectric environment, the actual ionization maximum occurs at much lower solute fractions (around x ≈ 0.1).¹⁹ For acetic acid, this peak shifts even further to x ≈ 0.06, coinciding with a sharp drop in the mixture’s dielectric constant and the extensive formation of hydrogen-bonded acetic acid dimers that sequester protons and hinder long-range ionic transport. Previous theoretical and experimental investigations of aqueous acetic acid solutions—using X-ray and neutron scattering, IR, Raman, and NMR spectroscopy, combined with DFT and molecular dynamics simulations—have demonstrated that liquid acetic acid solutions form extensive hydrogen-bond networks, including possible cyclic hydrogen-bonded rings and other O–H…O associations., but the exact structure still not known.^15–18 As the mixture becomes richer in acetic acid, structural heterogeneity diminishes and distinct solvation shells disappear, leading to a homogeneous, low-permittivity environment that further suppresses ionization.¹⁹ This behavior likely arises from a coupling between the ionization equilibrium and local dielectric inhomogeneities, although direct structural or spectroscopic evidence for this mechanism is still needed. Delving into this behaviour, recent studies illuminate the intricate interplay between ionization dynamics and the mixture's effective dielectric constant.¹⁹ In an elementary perspective, one would anticipate the degree of ionization to peak at x = 0.5. However, the dependence of the acid dissociation constant (K_a) on the dielectric constant primarily shifts this ionization maximum from x = 0.5 to x = 0.1 and the shift is further accentuated to x = 0.06. At the molecular level, this premature onset might originate from poor dissociation of the acid and its dimeric molecular nature that leads to poor heterogeneity when a solution structure change from molten H₂O/molten eutectic sub-micro domain to a molten eutectic/molten CH₃COOH sub-micro domain as per the phase diagram in Fig. 2a. While this behavior may stem from a coupling between ionization equilibria and local dielectric inhomogeneity, this remains a working hypothesis in the absence of direct structural validation. Contrasting this behaviour with nitric acid (HNO₃) for instance, a strong acid that undergoes near-complete dissociation in water, the uniqueness of acetic acid's conductivity profile becomes evident. HNO₃, lacking an organic moiety and exhibiting full dissociation, has its conductivity maximum closer to x = 0.10. Previous studies of liquids acetic acid and its aqueous solutions confirmed the presence of double hydrogen bonded cyclic dimers similar to the gas phase in hydrogen bonded linear chains similar to the solid phase corroborating the heterogeneity of the liquid structure.²⁰

The aqueous hydrofluoric acid solution exhibits one of the most anomalous conductivity trends observed, with a nearly linear increase and no x_max up to very high concentrations (x = 0.7, as reported by Hill and Sirkar at 0°C).¹² Unlike typical electrolyte solutions, HF does not show a conductivity peak at x_eutectic or at solvate-forming compositions (x_solvate = 0.5, 0.67), indicating a fundamentally different transport mechanism. Hydrofluoric acid's distinctive interaction with water, characterized by robust hydrogen bonding and the plethora of complexes, is instrumental in shaping its conductivity profile. Moreover, the unique hydration dynamics of the fluoride ion and the complex dissociation behavior of hydrofluoric acid serve to amplify its conductivity. The ionization of hydrofluoric acid (HF) in water has long been a subject of intrigue and debate among chemists, but was well investigated by Kolasinski in 2005.²¹ The peculiarities of hydrofluoric acid ionization were first hinted at in 1912 when Pick²² proposed that aqueous hydrofluoric acid solutions’ conductivity is consistent with two equilibria.

HF ⇌ H⁺ + F⁻ K_a =

$\:\frac{{\text{a}}_{{\text{H}}^{+}}{\text{a}}_{{\text{F}}^{-}}}{{\text{a}}_{\text{H}\text{F}}}$

[1]

HF + F⁻ ⇌ HF₂⁻ K₁ =

$\:\frac{{\text{a}}_{{\text{H}{F}_{2}}^{-}}}{{a}_{HF}{a}_{{F}^{-}}}$

[2]

Where a is the activity of species. Kolasinski²¹ suggests that these two equilibria are ‘largely suffice’ only in the x < 0.005 compositional range. Above this molar fraction and up to x = 0.1, the system is better described by a third equilibrium equation given first by Mctigue et. al.^23,24

HF + HF₂⁻ ⇌ H₂F₃⁻ K₂ =

$\:\frac{{\text{a}}_{{{H}_{2}{F}_{3}}^{-}}}{{a}_{HF}{a}_{H{{F}_{2}}^{-}}}$

[3]

Warren²⁵ suggested that the dimer (HF)₂ must be considered but Kolasinski states there is no additional supporting evidence from the literature for this species up to x = 0.1 but it could be present in very concentrated solutions.²³ In fact, they have been detected spectroscopically at > 50 wt. % hydrofluoric acid. Giguère and Turrell²⁶ theorize that hydrofluoric acid does actually dissociate strongly but it interacts with water and gets bound in a contact ion pair that is ionic, yet neutral electronically and that is the H₃O⁺.F⁻. The existence of this complex is now spectroscopically established as the major species in aqueous hydrofluoric acid solutions. The ionization process is represented by this double equilibrium

H₂O + HF ⇌ [H₃O⁺. F⁻] ⇌ H₃O⁺ + F⁻ [4]

While the exact concentrations of these species remain undetermined from spectroscopic data, it's evident that the first equilibrium is shifted significantly to the right, with cryoscopic data suggesting a dissociation of the complex around 15%.

The unique interplay of molecular interactions, particularly involving HF₂⁻ and H₃O⁺ ions, enhances proton transport in HF solutions. The observed increase in conductivity suggests a Grotthuss-like hopping mechanism facilitated by hydrogen-bonded HF₂⁻ networks, consistent with prior spectroscopic studies.⁵ While direct experimental confirmation of proton hopping in HF solutions remains limited, previous spectroscopic studies have identified structural features consistent with this transport mechanism. This behavior, starkly contrasting with other salt/solvent electrolyte solutions, emphasizes the profound impact of these molecular dynamics on the macroscopic property of conductivity. Interestingly, the absence of a conductivity maximum, as typically expected in Abu-Lebdeh’s model, suggests that the solution's structure remains unaltered and is not heterogeneous which contradicts the presence of the eutectic in the phase diagram. The answer might lie in the formation of unique microdomains with enhanced stability that is likely due to hydrofluoric acid's small size, which seamlessly integrates into the hydrogen-bonded chains of water, forming (HF)_x(H₂O)_y clusters akin and intertwined to the (H₂O)_x clusters. These extended structures or clusters have been theoretically predicted where the larger clusters with equimolar amounts of HF and H₂O are the most stable due the very strong H₂O --- HF interactions.²⁷ Such a structural formation facilitates a Grotthus-type hopping mechanism that prevails across varying concentrations, further elucidating hydrofluoric acid's distinctive conductivity attributes in contrast to conventional solvent/electrolyte solutions as depicted in the phase diagram of Fig. 2b.

Modeling the behavior of HF in water at high concentrations is challenging due to several limiting factors. First, the ionic strength of concentrated HF solutions lies outside the range where simple electrolyte solution models (e.g., the Debye–Hückel theory) are accurate. Early researchers addressed this issue by introducing a concentration-dependent dissociation degree or “stoichiometric” equilibrium constant for HF to fit experimental data.²⁸ This approach acknowledged that the apparent extent of HF dissociation decreases as total HF concentration increases. Modern thermodynamic models (such as those based on Pitzer ion-interaction equations or specific ion-pairing schemes) improve upon simple theories and have been applied to HF–H₂O systems.²⁹ However, these models still require reliable experimental data for calibration, and beyond a certain concentration (above roughly 6–10 M HF) such data become scarce or uncertain. Another limitation is that conventional speciation models may not include all relevant fluoride species present at high acid concentrations. In reality, at high [HF] the formation of HF₂⁻ (and potentially higher-order fluoride complexes) significantly affects the solution chemistry.³⁰ If a model neglects such associations and assumes that HF dissociates only into H⁺ and F⁻, it will underestimate the true acidity and mis predict the species distribution. In practice, treating HF with a fixed dissociation constant or as fully dissociated leads to errors when predicting thermodynamic properties of concentrated solutions.³¹ This is because effective equilibrium constants themselves change with concentration due to medium effects and ion pairing that are not captured by simplistic models.

Modeling Ionic Transport: Speciation, Mobility, and κ(T,x) Analysis

Ionic conductivity is a product of the number of charge carriers, by their charge, and charge mobility as represented in Eq. 1

$\:\kappa\:={\sum\:}_{i}{n}_{i}\bullet\:\:{q}_{i}\:\bullet\:{\mu\:}_{i}$

where n_i is number of free charged species, q_i denotes charge number and 'µ_i' is mobility of free charged species. To perform the theoretical calculation of ionic conductivity based on Eq. 1, one must first understand speciation of ions present in the aqueous solutions of both acids, and obtain ionic mobility and activity data for each. We purposefully ignored the contribution of OH⁻, and H⁺ from water auto dissociation, as it is few orders of magnitude lower than the contributions of the ions dissociated from most acids, even weak ones. n_i which represents the effective concentration of ions in molarity, i.e. activity, a_i. For acetic acid, data for activity is only available up to 1 M from Cohn et al.³² While that for hydrofluoric acid was obtained to higher concentrations from the National Bureau of standards.²⁸ On the other hand, 25°C values of mobility at infinite dilution, µ_io, were sourced from the literature to be 349.8 Scm² for H⁺, 40.9 Scm² for CH₃COO⁻, 55.4 Scm² for F⁻ ions,³³ and 84.6 Scm² for HF₂⁻.³⁰ pH values versus concentration for acetic acid and hydrofluoric acid are tabulated in table S.1 and S.2, respectively. To the best of our knowledge, mobility values at elevated temperatures and concentrations are scarce in literature for most ions. We have used these extracted species activity and mobility data to theoretically calculate conductivity and the results of this exercise are presented graphically in Fig S.1. For acetic acid, one can see clearly the great discrepancy between the theoretically calculated line versus the experimental reported conductivity curves. This is mainly due to underestimating the reduced ionic mobility as concentration increases. Several studies have focused solely on the this effect and introduced empirical equations to correct for mobilities at higher concentration (< 0.1 M) (Eq. 2).^34,35

$\:\frac{{\mu\:}_{i}}{{\mu\:}_{oi}}=exp\left(-\text{d}\sqrt{{\text{z}}_{i}\text{I}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(2\right)$

Where d is a parameter with a value > 0.5, z is the charge number and I is ionic strength. But at higher concentration it can be directly related to x (empirically to simplify the equation due to lack of literature values above 0.1 M).³⁶

$\:\frac{{\mu\:}_{i}}{{\mu\:}_{oi}}=exp\left(-\text{D}\text{x}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(3\right)\:\:\:\:\:\:\:$

Ion mobility refers to the average speed of ions per unit electric field strength, determined by the interplay between the external electric field and resistance to ion movement. When considering the ion and its hydration layer as a single entity, the resistance encountered during migration includes ion–ion, ion–solvent, and solvent–solvent interactions. Ion–ion forces are long-range electrostatic interactions, while ion–solvent and solvent–solvent forces are short-range interactions. At low concentrations, long-range interactions dominate, allowing short-range interactions to be neglected. However, as concentration of electrolyte solution increases, the molecular spacing decreases, making short-range interactions significant and causing a rapid increase in movement resistance. Consequently, ion mobility typically decreases with higher electrolyte solution concentrations.³⁶ This helps explain the presence of a maximum in conductivity isotherms of most aqueous electrolyte solutions. For hydrofluoric acid, one expects that this is not the case. In fact, we found evidence that the mobility of the HF₂⁻ triple ion actually increases with concentration!³⁰

Our research group has introduced a semi-empirical equation (Eq. 4) that combines equations 1 and 3 and introduces temperature effect (that causes an increase in the number of free charges and their mobility) hence takes into account mobility and activity changes with temperature and concentration.⁵

$\:\kappa\:=A{T}^{B}{x}^{C}exp\left(\frac{-Dx}{T}\right)\:\:\:\:\:\text{o}\text{r}\:\:\:\:\:\kappa\:=Ax\:exp\left(-Bx\right)\:\text{a}\text{t}\:\text{c}\text{o}\text{n}\text{s}\text{t}\text{a}\text{n}\text{t}\:\text{T}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(4\right)$

where A, B, C, and D are parameters and the pre-exponential term corrects activity while the exponential term is related to mobility. Fitting both acids’ κ vs x and T data lead to perfect surface fits with both fits’ coefficient of determination (R²) being 100% as shown in Fig. 3 below.

Fig. 3

The 3D surface plot fitted using Eq. 4 showing variation of ionic conductivity (κ) with molar ratio (x) and temperature (T) for (a) acetic acid (b) hydrofluoric acid. Red colour is Data surface. Blue is fit surface

Figure 4(a) elucidates more clearly in 2D the variation of ionic conductivity (k) with molar fraction (x) at different temperatures (25°C, 45°C, and 65°C) for acetic acid solutions. In the extremely dilute region, specifically below x = 0.03, a noteworthy observation is the minimal deviation in conductivity across the three temperatures, with the curves nearly overlapping. This subtlety in temperature influence can be attributed to the pronounced effects of low molar volume and viscosity in water-rich mixtures, optimizing ion mobility and conductivity. However, as the solution approaches x_max, the disparity in conductivity between the different temperatures becomes significantly pronounced. Interestingly, the increment in conductivity observed when transitioning from 25°C to 45°C is nearly double that of the shift from 45°C to 65°C. Beyond x_max, while the temperature effect persists, it is less pronounced, with the conductivity values converging at higher concentrations. Figure 4(b) presents the ionic conductivity (k) of hydrofluoric acid (HF) solutions across varying molar fractions (x) at temperatures of 0°C, 18°C, 37.8°C, and 93.3°C. Remarkably, the isotherms depicted are linear, with a near overlap in the extremely dilute region, similar to the behavior observed in acetic acid solutions (Fig. 4(a)). However, a distinct characteristic of hydrofluoric acid solutions is the continuous and increasing influence of temperature on conductivity, with no discernible x_max. The temperature effect strengthens as x increases, and the increment in conductivity is greater between 0°C and 18°C than between subsequent temperature intervals. This pattern mirrors the non-linear temperature effect observed in acetic acid solutions, albeit in the absence of a conductivity peak. The pronounced temperature sensitivity in hydrofluoric acid solutions especially in concentrated regions, was observed previously for other solutions such as nitrates and acids, and can be attributed to the presence of a higher number of ion pairs and clusters (IPs and ICs) and the consequent increase in mobile "free" ions as temperature rises, a phenomenon previously discussed.⁴ The expansion of the liquid and the increase in free volume at higher temperatures further enhance ion mobility, contributing to the substantial increase in conductivity.

Fig. 4

The variation of ionic conductivity (κ) with molar ratio (x) and temperature (T) for (a) acetic acid and (b) hydrofluoric acid. Acetic acid data were experimentally measured in triplicate, with a standard deviation of less than ± 5%. Hydrofluoric acid data were obtained from literature sources where error bars were not reported.

Figure 5 outlines the variation of the natural logarithm of ionic conductivity (ln(k)) against the reciprocal of temperature (1000/T) for both hydrofluoric acid and acetic acid solutions. The linear trend exhibited by both acids in this plot signifies their adherence to the classical Arrhenius behavior, described by Eq. (1)

$\:\kappa\:=Aexp\left(\frac{-{E}_{a}}{RT}\right)$

(5) where E_a is the activation energy, R is the gas constant, T is the temperature in Kelvin, and “A” is the pre-exponential factor. While the original conductivity data for hydrofluoric acid displayed linearity with respect to x, this graph is valuable for extracting the variation of activation energy and the pre-exponential factor across different molar fractions.

Fig. 5

ln k of (a) acetic acid (b) hydrofluoric acid as a function of the reciprocal of the temperature

Figure 6(a) and (b) presents the variation of activation energy (E_a) with molar fraction (x). In the case of acetic acid, a notable increase in E_a is observed in the dilute region, reaching a plateau around x_max. Beyond this point, E_a exhibits a further increase until x = 0.15, followed by a sudden drop to values close to the value near x_max. Subsequently, E_a experiences another increase, stabilizing between x = 0.27 and 0.33, before rising once more at higher molar fractions. A similar trend has been observed in another weak acid (H₃PO₄),⁵ though further research is needed to determine whether this behavior is common among weak acids. This trend in the E_a vs x graph is very similar to that of aqueous phosphoric acid solutions. For hydrofluoric acid, while data in the extremely dilute region is unavailable, we hypothesize that E_a initiates at a value comparable to most acids (approximately 8.5 kJ mol^− 1) and undergoes a decrease, aligning with the first available data point at x = 0.08 where E_a is around 3.3 kJ mol ^− 1. Following this, a continuous decrease in E_a is observed until a minimum is reached at x = 0.14, after which a slight increment and subsequent decrement are noted at higher molar fractions. This E_a vs x trend is very similar to that observed in aqueous nitric acid solutions except that nitric acid’s E_a does not drop as much as that of hydrofluoric acid. The E_a drops at the minimum point and also it continues to increase beyond the minimum unlike that of hydrofluoric acid. This may be related to the unique structural features of hydrofluoric acid (e.g., HF₂⁻ formation and hydrogen bonding), though we emphasize that this remains a hypothesis due to the lack of direct structural validation. However, it is worth noting that the overall decrease in E_a is less than the thermal activation energy of 25°C at ~ 2 kJ mol ^− 1 indicating a facile transport of ions with little activation necessary.

Fig. 6

Activation energy (E_a) vs. molar fraction (x) for (a) acetic acid (b) hydrofluoric acid aqueous solutions. Pre-exponential factor (A) vs. molar fraction (x) for (c) acetic acid (d) hydrofluoric acid

Ionic conductivity in liquid electrolyte solutions shares some conceptual similarities with amorphous materials, such as glass-forming liquids or solid ionic glasses above the glass transition temperature (T_g). Regardless of the conduction mechanism, ionic conductivity can be expressed as the product of the concentration of charge carriers, their mobility, and their charge, as described in Eq. 1. While we reference the empirical Vogel-Tammann-Fulcher (VTF) equation as a useful tool to describe conductivity-temperature relationships in some materials, we clarify that we are not asserting that our data strictly follows the VTF equation. Instead, we included the VTF discussion as a conceptual framework to highlight the transition from Arrhenius to non-Arrhenius behavior in many ionic systems, especially in glassy or polymeric electrolyte. It is also important to note that direct application of the VTF equation to liquid aqueous electrolyte solutions is non-trivial because defining key parameters such as the ideal vitreous temperature (T₀) and free volume (Vf) in these systems remains challenging. While VTF theory has successfully described transport in polymeric or glassy systems, its role in liquid electrolyte solutions remains largely qualitative. Previous studies have drawn analogies between VTF-type behavior and ionic conduction in highly concentrated or supercooled liquids, but these analogies are limited by the lack of a well-defined T₀. In light of this, we treated VTF as a conceptual tool rather than an explicit fitting model, which is why we did not perform a VTF analysis on our data. To analyze these behaviours, we primarily apply the Arrhenius model, while referencing the VTF framework conceptually to discuss its potential relevance to ionic transport in certain cases focusing on the role of activation energy, ion mobility, and the potential for free volume influence in determining conductivity. In glass electrolytes, two different mechanisms govern temperature dependence: below T_g, conductivity follows the Arrhenius behavior, while above T_g, chain movement and free volume rearrangement become significant, resulting in Vogel-Tammann-Fulcher (VTF) dependence.

Nascimento et al.³⁷ summarized this behavior and provided an expression for the concentration of charge carriers (n⁺):

$\:{n}_{+}\:=\:n\:exp\:\left(\frac{-\varDelta\:{G}_{f}}{2RT}\right)\:$

where

$\:\varDelta\:{G}_{f}=\:\varDelta\:{H}_{f}\:-T\varDelta\:{S}_{f}$

is the Gibbs energy of cationic vacancy and interstitial pair formation, calculated from the corresponding formation enthalpy (

$\:\varDelta\:{H}_{f})$

and entropy

$\:(\varDelta\:{S}_{f}$

) Nascimento also described the equation to calculate the mobility (µ+) of charge carriers in the presence of an electric field at low temperatures (below T_g):

$\:{\mu\:}_{+}=\frac{F{{\uplambda\:}}^{2}{{\upnu\:}}_{0}}{6RT}\text{e}\text{x}\text{p}(-\frac{\varDelta\:{G}_{m}}{RT})$

where λ is the mean distance between cationic sites,

$\:{{\upnu\:}}_{0}$

is the jump attempt frequency and

$\:\varDelta\:{G}_{m}=\:\varDelta\:{H}_{m}-T\varDelta\:{S}_{m}$

is the migration Gibbs free energy calculated the migration enthalpy (

$\:\varDelta\:{H}_{m})$

and entropy

$\:(\varDelta\:{S}_{m}$

Using Equations 1, 6, and 7, Nascimento derived the equation to describe the cationic conductivity (σT) in the low-temperature range for glass electrolytes below T_g:

$\:{\upsigma\:}\text{T}=\text{n}\frac{{F}^{2}{{\uplambda\:}}^{2}{{\upnu\:}}_{0}}{6R}\text{exp}\left(\frac{-\frac{\varDelta\:{S}_{f}}{2}+-\varDelta\:{S}_{m}}{R}\right)\text{e}\text{x}\text{p}\left(\frac{-\frac{\varDelta\:{H}_{f}}{2}+-\varDelta\:{H}_{m}}{RT}\right)$

This is essentially the Arrhenius equation, where the experimental value of activation energy (

$\:{E}_{{\upsigma\:}}^{A}$

) and the pre-exponential factor (

$\:{A}_{{\upsigma\:}}^{low}$

) are identified as:

$\:{E}_{{\upsigma\:}}^{A}=\:\frac{\varDelta\:{H}_{f}}{2}+\varDelta\:{H}_{m}$

$\:{A}_{{\upsigma\:}}^{low}=\:\text{n}\frac{{F}^{2}{{\uplambda\:}}^{2}{{\upnu\:}}_{0}}{6R}\text{exp}\left(\frac{-\frac{\varDelta\:{S}_{f}}{2}+-\varDelta\:{S}_{m}}{R}\right)$

In hydrofluoric acid, the Arrhenius-like behavior observed at low to moderate concentrations aligns with Eq. 8, suggesting that ion mobility remains relatively unaffected by free volume changes. Instead, conductivity relies on stable hydrogen-bonded clusters and proton transfer, allowing for a unique linear increase without transitioning to VTF behavior.

Above T_g, when free volume is activated, another conduction mechanism is observed which involve the transfer of defects to neighboring positions because of the local deformations of the macromolecular chains due to fluctuations in the free volume of the chain segments.³⁸ However, in aqueous solutions, the concept of free volume is not as well-defined as in polymers or glasses, making the direct application of VTF theory challenging. We have therefore avoided a formal VTF analysis, as it would require detailed viscosity and relaxation data not presently available. Moreover, the conceptual application of VTF theory to aqueous systems remains nontrivial due to the limited understanding of free volume behaviour in liquids. Instead, we focus on the conceptual aspects of ion transport and their implications for observed conductivity trends. To arrive to an equation equivalent to Eq. 8 but now accounting for this new conduction mechanism, one must first define a couple of parameters. First, this second mechanism only appears above the ideal vitreous temperature (T₀), where at T > T₀, the cationic conduction can occur by either a successful activated jump or by the entropic free volume mechanism. The probability of an activated jump is given by P₁ which is related the enthalpy of migration by the following equation:³⁸

$\:{P}_{1}=\text{e}\text{x}\text{p}(-\frac{\varDelta\:{H}_{m}}{RT})$

and P₂ is the probability that minimum free volume is available for the second mechanism: ³⁸

$\:{P}_{2}=\text{e}\text{x}\text{p}(-\frac{{V}_{f}*}{{V}^{f}})$

where Vf^* is the minimum required free volume for conduction and V_f is the chain segment mean free volume. V_f temperature dependence is given by the following equation: ³⁸

$\:{V}_{f}={V}_{0}{\alpha\:}_{f}(T-{T}_{0})$

where

$\:{\alpha\:}_{f}$

is the free volume thermal expansion coefficient, P₂ can now be re-arranged to be: ³⁸

$\:{P}_{2}=\text{e}\text{x}\text{p}(-\frac{{V}_{f}*}{{V}_{0}{\alpha\:}_{f}(T-{T}_{0})})$

This equation confirms that conduction due to defect transfer can happen only at T > T₀. At this high T range, cationic displacement can occur by either mechanisms and the total probability of a successful would be statistically³⁸

$\:P=\:{P}_{1}+{P}_{2}-{P}_{1}{P}_{2}$

and we can now arrive to a new equation for mobility:

$\:{\mu\:}_{+}=\frac{F{\text{l}}^{2}{{\upnu\:}}_{0}}{6RT}{(P}_{1}+{P}_{2}-{P}_{1}{P}_{2}$

) (16)

meaning that mobility at the high temperatures depends on the probabilities P₁ and P₂ and a free volume migration is only possible above T₀. However, it is only noticeable for T > T_g. Souquet et al.³⁸ suggest that at the following temperature range T₀ < T < T_g, P₂ remains very small compared to P₁ and the mobility equation remains the same as Eq. 7. At temperatures above T_g, P₂ is not negligible and the conductivity equation now becomes: ³⁸

$\:{\upsigma\:}\text{T}=\text{A}\text{exp}\left(\frac{-\varDelta\:{H}_{f}}{2RT}\right)\text{e}\text{x}\text{p}(-\frac{{V}_{f}*}{{V}_{0}{\alpha\:}_{f}(T-{T}_{0})})$

This equation is the product of two exponentials where the first represents the activated mechanism and the second is associated with the VTF behavior. Notably, the pre-exponential factor is not changed in this equation which means that the limit of the conductivity-temperature product at infinite T does not depend on any conduction mechanism. VTF behavior typically requires sufficient free volume, enabling large-scale ion migration facilitated by chain motion or defect movement. In acetic acid, we observe features that may suggest a transition beyond x_max where ion mobility is restricted—possibly due to viscosity and ion pairing—which is conceptually consistent with VTF-type behavior described in Eq. 17. However, hydrofluoric acid does not exhibit VTF-like behavior due to its unique proton hopping mechanism along hydrogen-bonded networks, a Grotthuss-like process that sustains high mobility without the need for free volume reorganization. Thus, while acetic acid’s conductivity behavior transitions as expected with increasing free volume, hydrofluoric acid’s proton-hopping mechanism maintains a stable structure that bypasses typical VTF dependencies, even at higher concentrations

Figure 6(c) and (d) shows the variation of the pre-exponential factor (A) with molar fraction (x). A can also be described as the conductivity when charge carriers encounter no energy barriers, meaning the activation energy (E_a) is non-existent.^39,40 For acetic acid, (A) exhibits an initial increase with increasing x, reflecting the enhancement in ion jump frequency and available conductive sites due to higher ion concentrations. This trend continues until reaching a plateau around x_max, where the solution's structural constraints begin to counterbalance the benefits of increased ion concentration. Beyond x = 0.15, a notable sudden drop followed by a consistent decline in (A) is observed, that could be attributed to the increased formation of non-conductive ion pairs or clusters and a significant rise in viscosity, which collectively impede ion mobility. For hydrofluoric acid, (A) demonstrates a slow and gradual continuous increase, suggesting that the solution's structural changes facilitate better total ion mobility across all concentrations. We hypothesize that this behaviour may arise from the formation of complex ions that facilitate charge transport more efficiently, without strong hindrance from ion pairing or aggregation. However, this remains a proposed explanation based on indirect inference.

Lastly, it is important to note that we have based our calculations of E_a and A on the simple Arrhenius equation, assuming its validity in the κ vs T data range despite dealing with aqueous solutions that actually behave like glasses or polymers that are well above their T_g. However, one can say that the isotherms follow pseudo-Arrhenius behavior due to the narrowness of the temperature range we are discussing especially for acetic acid, where the difference between the highest temperature and the lowest is just 40°C. if the temperature range was to be larger, it would be wiser to have considered the VTF behavior when linearizing the data.

We emphasize that many of the mechanistic suggestions made in this section—particularly those involving ion pairing, complex formation, or structural transport modes—are working hypotheses motivated by literature and consistent with our data trends, but they are not experimentally verified. This reflects a broader challenge in the field, as probing the microstructure of highly concentrated or structurally unconventional liquid systems remains a major experimental and theoretical difficulty.

Conclusion

In this study, we delved into the unique conductivity behaviors of acetic acid (AA) and hydrofluoric acid (HF) in water, two electrolyte solutions that present intriguing deviations from established correlations between conductivity isotherms and phase diagrams. Our exploration has highlighted the intricate interplay of molecular interactions, ionization dynamics, and solvation that, in these two cases, give rise to conductivity behaviours that deviate significantly from the typical correlation between x_max and x_eutectic. For acetic acid, the early manifestation of x_max at x = 0.06, diverging from its x_eutectic at x = 0.30, has been attributed to the unique dissociation into acetate dimers. Conductivity at elevated temperatures is impacted by the reduced water viscosity, and reduced dielectric constant as is this case with most aqueous solution. Isothermally, we suggest that the dominance of the conductive microdomains in acetic acid decreases as the concentration increases way before reaching the eutectic point. hydrofluoric acid, on the other hand, exhibits a linear increase in conductivity with concentration without a maximum, a behavior unseen in any other salt in water solution. The formation of the H₃O⁺.F⁻ ion pair or proton-transfer complex and the subsequent ionization processes elucidate the near-complete ionization of hydrofluoric acid in water. The observed sharp rise in conductivity with concentration is attributed to the participation of both the HF₂⁻ ion and the separate H₃O⁺ ions in the proton-migration process through (HF)_x(H₂O)_y clusters akin to the (H₂O)_x clusters. This led us to suggest that the dominant microdomains in hydrofluoric acid remain highly conductive throughout the concentration range.

In conclusion, the exploration of acetic acid and hydrofluoric acid in water has provided clarity on these specific systems while revealing the limitations of previously established relationships and highlighting the need for expanded theoretical models. In both acids, these deviations appear to arise from their peculiar chemical equilibria and a liquid microstructure that remains less well understood, which may be interpreted as giving rise to different microdomain organization and transport dynamics compared to typical aqueous electrolyte solutions. This study, by addressing the exceptions and refining the overarching theory, contributes to a better-rounded understanding of the subject and lays the groundwork for future research in unraveling the complexities of liquid electrolyte solutions’ microstructure and conductivity.

These findings contribute to a broader effort to identify and interpret outlier systems within aqueous electrolyte solutions. Although our theoretical interpretations remain preliminary, they are grounded in consistent thermodynamic reasoning and comparative analysis across many systems. We hope this work stimulates further investigation, particularly in developing experimental methods capable of probing concentrated liquid microstructure.

Data Availability

All data except acetic acid conductivity was obtained exclusively from literature. Data for acetic acid conductivity was obtained in lab and in literature, as outline in the methodology section. all data points were further processed and analyzed in excel sheets that are available from the corresponding author on reasonable request.

Acknowledgments

The authors would like to acknowledge funding from NRC’s Critical Battery Materials Initiative

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