Predicting Activities in Aqueous Electrolyte Solutions with Hybrid Machine Learning Zeno Romero, Maximilian Kohns, and Fabian Jirasek∗ Laboratory of Engineering Thermodynamics, RPTU Kaiserslautern, Erwin-Schrödinger-Str.
arXiv:2607.19114v1 [cs.LG] 21 Jul 2026
44, 67663 Kaiserslautern, Germany E-mail: [email protected] Abstract Activities in aqueous electrolyte solutions, usually described by ionic activity and osmotic coefficients, are important properties for modeling many processes in industry and nature. Established activity models, such as those of Pitzer or Bromley, require fitting to experimental data for each electrolyte of interest and thus cannot predict properties for unstudied systems. While some predictive approaches exist, they are typically limited in scope and rely on additional ion-specific descriptors. In this work, we introduce a new hybrid model that combines the physics-based Bromley model with a matrix completion method (MCM) from machine learning. The MCM is employed to predict the electrolyte-specific parameters of the Bromley model, exploiting the fact that these parameters can be arranged in a matrix with cations and anions as rows and columns, respectively. Due to the lack of experimental data for many electrolytes, the initial parameter matrix is sparsely populated, making the prediction of the Bromley parameters for unstudied electrolytes a matrix completion problem. The hybrid model, Bromley-MCM, was trained end-to-end on experimental data for mean ionic activity coefficients and osmotic coefficients of aqueous solutions of 478 electrolytes at 298 K from the Dortmund Data Bank. As output, we obtain a completed matrix of
1
Bromley parameters for 83 cations and 112 anions, enabling consistent prediction of concentration-dependent activities in aqueous solutions of 9,296 electrolytes at 298 K. This substantially extends the applicability of the Bromley model while maintaining high predictive accuracy, as demonstrated through evaluations on electrolytes excluded from model training.
Introduction Activities in electrolyte solutions play an important role in addressing many critical engineering challenges, e.g., electrochemical energy storage 1 and desalination applications. 2 Here, the term ”activities” is used collectively for different properties that essentially describe the chemical potentials of the ions, the electrolyte, and the solvent in electrolyte solutions. Most established experimental methods for determining such properties focus on solvent activity, often expressed as the osmotic coefficient. These approaches include the isopiestic method 3 and the measurement of the vapor pressure, 4 the freezing-point depression, 5 and the osmotic pressure. 6 Having conducted a measurement series with either of these methods, the activity of the electrolyte in solution, usually represented in terms of the mean ionic activity coefficient, can be inferred by evaluating the Gibbs-Duhem equation. A notable direct method for determining the mean ionic activity coefficient is the measurement of open-circuit voltages (formerly also called electromotive force measurements). 7 Many of the aforementioned methods suffer from slow equilibration and reliance on precise references, standards, and/or calibrations. 8 Moreover, these experiments are generally time-consuming and cost-intensive. Thus, it is unfeasible to determine the activity coefficients in all relevant systems experimentally, making predictive methods essential for engineering practice. Several physical models for activities in electrolyte solutions, such as the excess Gibbs energy (GE ) models of Pitzer 9 and Bromley, 10 have been proposed in the literature. These models can reliably generalize across concentration, but their adjustable parameters must be determined by fitting to experimental data for the specific electrolyte, solvent, and 2
temperature of interest. As a consequence, these models cannot generalize over electrolytes, i.e., they cannot be applied to electrolytes for which no activity data are available. This limitation has partially been addressed with the development of LIQUAC and LIFAC, 11–13 which enable extrapolations to unstudied temperatures and (mixed) solvents, but still require electrolyte-specific cation-anion interaction parameters that have been fit to data for each electrolyte of interest. Only a few models that enable predictions for solutions of completely unstudied electrolytes have been proposed in the literature. Two examples are the ion-specific Bromley model (IBM), 10 which targets the prediction of electrolyte-specific parameters of the Bromley model based on ion-specific parameters, and the Simoes model, 14 which, similarly, aims at predicting the electrolyte-specific parameters of the Pitzer model based on ion-specific parameters. However, these approaches have limited predictive accuracy and, in the case of the Simoes model, 14 limited applicability, as the ionic radii required as descriptors of the model are not comprehensively available for every ion. In recent years, machine learning (ML) has emerged as a powerful alternative to and complement of established, physics-based thermodynamic modeling approaches. 15,16 While purely data-driven ML approaches often lack physical interpretability and robustness, hybrid models that combine established thermodynamic frameworks with data-driven ML models enable generalization across chemical spaces inaccessible to traditional models, effectively closing gaps in established physical models, while retaining thermodynamic consistency. 17–21 One interesting class of ML methods in thermodnyamics are matrix completion methods (MCMs), 22 and, by extension, tensor completion methods, 23 which have also been combined with physical models and knowledge to yield powerful hybrid approaches for predicting various thermodynamic properties, including activity coefficients, 24–29 Henry’s-law constants, 30,31 diffusion coefficients, 23,32,33 and densities of aqueous electrolyte solutions. 34 Furthermore, MCMs have been embedded in physical modeling frameworks, enabling prediction of model parameters and thereby significantly enhancing their applicability. 17,35,36,36–38
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In this work, we present a novel hybrid model for predicting activities in aqueous electrolyte solutions by combining the physics-based Bromley model 10 with an MCM. Within this new hybrid model, which we call Bromley-MCM, the MCM is trained to predict the electrolytespecific adjustable parameters of the Bromley equation, BCA , by exploiting the fact that these Bromley parameters for a set of electrolytes can be conveniently arranged in the shape of a matrix, where the rows represent the different cations C, the columns represent the different anions A, and the entries of the matrix contain the numeric values of BCA . Since experimental data to which BCA can be directly fitted exist only for some electrolytes, i.e., only a small fraction of all possible cation-anion pairs CA, the prediction of the missing Bromley parameters becomes a matrix completion problem. To benchmark the developed Bromley-MCM, we use a consolidated database of mean ionic activity coefficients and osmotic coefficients in aqueous electrolyte solutions at 298 ± 1 K, obtained by careful evaluation of data from the Dortmund Data Bank (DDB). 39 We thereby systematically compare its results with the predictive approaches in the literature, namely, the IBM 10 and the model of Simoes, 14 each within its respective scope.
Methodology Electrolyte Activities In this work, we address the prediction of activities in aqueous solutions containing a single electrolyte. We thereby assume complete dissociation of the electrolytes in solution following, for an electrolyte CνC AνA that consists of cation C and anion A, the reaction CνC AνA → νC C zC + νA AzA ,
(1)
where νi is the stoichiometric coefficient and zi is the charge number of ion i. For brevity, we use CA to denote the electrolyte in the following, omitting the stoichiometric coefficients. To
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denote solution composition, we use the overall molality m̃ of the electrolyte, which is the number of moles of electrolyte per kilogram of pure water W as the solvent. For complete dissociation, the molality of an ion mi in the solutions is related to the overall molality m̃ via the mass balance mi = νi m̃.
(2)
We furthermore use the ionic strength I of the solution defined as
I=
zC2 mC + zA2 mA , 2m0
(3)
where the division by m0 = 1 mol kg−1 is introduced to render the ionic strength dimensionless. As is common for electrolyte solutions, we consider the chemical potentials of the ions in solution normalized according to Henry’s law on the molality scale
µi (T, p, m̃) = µref i (T, p, m̃ → 0) + RT ln
mi + RT ln γi∗ (T, p, m̃), m0
(4)
where µref i (T, p, m̃ → 0) is the chemical potential of ion i in the reference state (infinite dilution of the ion), T is the thermodynamic temperature, p is the pressure, R is the universal gas constant, and γi∗ is the activity coefficient of i on the molality scale. Combining these equations for both respective ions yields the overall chemical potential of the electrolyte CA, which in case of complete dissociation can be written as
νC νA µ̃CA (T, p, m̃) = µ̃ref CA (T, p, m̃ → 0) + RT ln νC νA
m̃ m0
ν
+ νRT ln γ̃±∗ (T, p, m̃),
(5)
where 1
γ̃±∗ = ((γC∗ )νC (γA∗ )νA ) ν
5
(6)
is the so-called mean ionic activity coefficient of the electrolyte and
ν = νC + νA .
(7)
For the chemical potential of the solvent water W, we use the normalization according to Raoult liq µ̃W (T, p, m̃) = µ̃pure (T, p) − νMW m̃RT ϕ(T, p, m̃), W
(8)
liq where µ̃pure (T, p) is the chemical potential of pure water W, MW is the molar mass of W
water, and ϕ is the osmotic coefficient. Again, equation (8) holds for the case of complete dissociation of the electrolyte in the solvent. In the following, we use γ̃±∗ and ϕ as measures for the activity of the electrolyte and the solvent water in aqueous electrolyte solutions, respectively.
Database Raw activity data for γ̃± and ϕ were taken from the DDB 2026 ”ELE” database (Vapor-Liquid Equilibria of Electrolyte Systems) 39 for binary systems containing exactly one electrolyte and the solvent water. We restricted the scope to these systems because the vast majority of available activity data for aqueous solutions is at 298 K, with very little data at other temperatures. Specifically, we found 92,254 data points for solutions of single electrolytes in water, whereas only 4,180 data points were available for the solvent methanol, and even fewer for other solvents. For the aqueous systems, 43,920 data points are reported at 298 ± 1 K, whereas substantially fewer data points are available at any other temperature. Further details on the data availability are reported in Figures S1 and S2 in the Supporting Information. Since the effect of pressure on liquid-phase properties is generally small, especially at low to moderate pressures, and data at elevated pressures are extremely scarce, the influence of pressure was neglected in this study. All data in this study were either taken at ambient pressure or at the solvent’s vapor pressure. 6
The following data consolidation and filtering procedure was applied to the raw data from the DDB; further details are reported in the Supporting Information: • Only aqueous systems, i.e., systems with water as the only solvent, were considered. • Electrolytes that could not be split into 1 type of cation and 1 type of anion were excluded. One example of an excluded electrolyte is magnesium ammonium phosphate (MgNH4 PO4 ), which has two types of cations. On the other hand, ionic coordination complexes, which do not further dissociate in solution, such as tris(ethylenediamine)cobalt(III) ([Co(en)3 ]3+ ), are included and are treated as single ions. • Only data reported at 298 ± 1 K were considered. • Only data for which the electrolyte concentration is reported in terms of molality m̃ were used. We also checked the availability of data reported in other concentration measures, but did not find additional relevant systems. • Data points at I > 6 were excluded (which is reported as the upper bound for the applicability of the Bromley model in the literature 10 ). • The database was divided into one database for γ̃± and one for ϕ, and each was checked for internal consistency. For this purpose, for a given electrolyte (e.g., NaCl), all available data sets (i.e., data reported by individual sources) in the database were compared. If two data points were reported at the same (to the second decimal) molality, the mean of these values was calculated. However, whenever one of these data points deviated by more than 5 % from this mean, the entire data set containing this data point was excluded. This procedure was done for ϕ and γ̃± individually. • For each electrolyte, the Bromley parameter BCA was fitted individually to the considered ϕ and γ̃± data for this electrolyte using a least-squares procedure. If the relative mean residual of the fit exceeded 5 % for either ϕ or γ̃± , the respective electrolyte was excluded from the dataset. This procedure was used since weak electrolytes, for 7
which the Bromley model is not suited, were not excluded a priori based on their respective solubility products. Instead, only those electrolytes for which the Bromley model reproduced the available experimental data with sufficient accuracy (as specified above) were retained. • Finally, for the retained electrolytes for which both ϕ and γ̃± were available, the Bromley model was fit to ϕ − 1 and ln γ̃± simultaneously, and electrolytes with residuals greater than 5% were excluded. This procedure was used to numerically verify Gibbs-Duhem consistency of the data for these electrolytes (within the specified tolerance), since the Bromley model itself is Gibbs-Duhem consistent. We call the database containing all data retained after applying the above-described criteria the full database in the following, which can be represented as a matrix, where rows correspond to the different cations C and columns to the different anions A. This matrix is visualized in Figure 1 (left). The numbers of cations, anions, available electrolytes, and data points in our full database are listed in Table 1, column ”Full”. The names of all cations and anions are listed in Table S1 in the Supporting Information. The full database contains data for 83 unique cations and 112 unique anions, with experimental mean ionic activity coefficients γ̃± and/or osmotic coefficients ϕ existing for 478 of those cation-anion pairs (5.1 % of all possible pairs) with a total of 11,533 experimental data points for γ̃± and 13,296 data points for ϕ. To evaluate the predictive accuracy of the hybrid model developed in this work, we use a leave-one-electrolyte-out strategy, in which the model is repeatedly trained on data from all electrolytes except one, whose data are used as test data and compared against the model predictions. This procedure requires data on at least two cation–anion pairs (i.e., different electrolytes) per ion, so we further restricted the full database to include only cations and anions with at least two experimentally available pairs for this analysis. The resulting reduced database is shown in Figure 1 (right) in matrix form. The number of cations, anions, and the number of available electrolytes and data points are listed in Table 1, column ”Reduced”. 8
The names of the cations and anions included in the reduced dataset are listed in Table S2 in the Supporting Information. Full database
Reduced database