Generating isocoulombic reactions as a tool for systematic evaluation of temperature trends of thermodynamic properties: Application to aquocomplexes of lanthanides and actinides

Most of the available thermodynamic data concerning radioactive waste disposal are restricted to values of reaction equilibrium constants (logK 298) at 25 C and 1 bar. Simple estimation methods such as isocoulombic reactions can be used for extrapolating the properties of reactions involving aqueous species and minerals to elevated temperatures. The aim of this study was to validate the applicability of various alternative isocoulombic reactions to estimate logK T values of aqueous complexation reactions for lanthanides and actinides to elevated temperatures while taking advantage of new additional literature data, and to identify criteria for choosing the ‘‘best” reactions. For each chemical species of interest, a systematic approach using dedicated software and database allowed us to identify the isocoulombic reactions and types of extrapolation that yield the best estimates of standard thermodynamic properties at elevated temperatures, when very limited or no experimental data are available. We have tested aqueous complexation reactions for selected lanthanides and actinides of different valences with chloride, fluoride, sulfate, carbonate, nitrate, phosphate and silicate ligands. ‘‘Model” complexation reactions, having known temperature trends, were systematically combined with complex formation reactions of interest whose temperature trends are unknown, into many alternative isocoulombic reactions. For each ion, we investigated which of the generated isocoulombic reactions provide the best estimates for logK T of the reaction of interest at elevated temperatures in order to compile the guidelines for choosing the optimal ones, then applying these guidelines to ‘‘prediction” subsets. In most cases, knowing only logK T at 25 C (for the reaction of interest), it was possible to obtain rather accurate estimates of log K T values at elevated temperatures using isocoulombic reactions that exchange ions with similar charge and hydration properties (hydrated ionic radius and structure of the hydration shell) and known logK T of model reactions. These ions and their complexes interact with the solvent in comparable ways, so that their similar heat capacity and entropy effects largely cancel out on both sides of an ‘‘optimal” isocoulombic reaction. 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/ licenses/by/4.0/).


INTRODUCTION
A common problem with thermodynamic modeling related to radioactive waste disposal is the lack of thermodynamic data for some relevant aqueous species and minerals, especially for the extrapolation of equilibrium constants to elevated temperature conditions. Although various extrapolation methods exist, they require a certain amount of data to be available for retrieving the necessary model parameters. Even though new experimental data eventually become available, they are by far not enough to cover the large number of aquocomplexes and minerals in broad temperature ranges and for direct evaluation of the required standard-state thermodynamic data. In such cases, correlation and prediction methods, such as the isocoulombic method, can provide reasonable estimates of the missing thermodynamic properties before these can be extracted from fitting the models to new experimental data.
It has long been recognized that isocoulombic and isoelectric reactions may be useful to estimate the standard thermodynamic properties beyond the experimental conditions when limited data are available (Gu et al., 1994;Izatt et al., 1991;Lindsay, 1989Lindsay, , 1980Machesky et al., 1994;Mesmer et al., 1988;Mesmer and Baes, 1974;Wood and Samson, 1998). An isocoulombic reaction has the same number of like-charged species on the reactant and the product side, while an isoelectric reaction has the same total numbers of positive and negative charges on each side. Such reactions are adopted in the MULTEQ computer program (Alexander and Luu, 1989) for providing high-temperature estimates of thermodynamic properties with applications in vapor-liquid partitioning and precipitation. Isocoulombic reactions have also been used with various degrees of success to extrapolate several thermodynamic properties such as equilibrium constants (Luo and Byrne, 2004;Ruaya, 1988;Tremaine et al., 2004), volume (Tagirov and Schott, 2001) and heat capacity (Schrö dle et al., 2010) effects, and kinetic rate constants (Aki et al., 2001) to different temperatures, pressures and ionic strengths. Usually there can be many ways in which a reaction of interest can be converted into an isocoulombic reaction; this leads to temperatureand other extrapolations with varying degrees of accuracy.
The aim of the present study was to systematically investigate the applicability of various alternative isocoulombic reactions for estimating the thermodynamic properties of aqueous complexation reactions for lanthanides and actinides at temperatures above the reference temperature of 25°C using the one-, two-, and three-term logK T temperature extrapolation function. As such, the isocoulombic reaction method is not new, but we can now take advantage of a dedicated database and software for fast and accurate generation of reactions and calculation of their effects, as well as of the new experimental data at elevated temperatures (previously not available for lanthanide and actinide systems) to validate the isocoulombic reactions and extrapolation types. After all, we should be able to recommend ''optimal" reactions and extrapolation types that produce the best estimates of standard properties of substances and reactions at elevated temperatures in cases when only very limited thermodynamic data are available.
We systematically tested the isocoulombic method against literature data on the temperature dependence of aqueous complexation and solubility reactions of lanthanides and actinides. We focused on alternative isocoulombic exchange reactions using the one-term logK T extrapolation type, which requires the minimum amount of available input thermodynamic data (logK 298 only). On this basis, we show how similarities between the properties of ions (charge, hydrated radius, hydration shell configuration) can be used in selecting the most appropriate analogous ion to construct an isocoulombic reaction and model the unknown temperature dependence of the complexation reaction of interest.

Temperature extrapolations
There are several different methods for extrapolating thermodynamic properties of substances and reactions from a reference temperature to the temperature of interest (for a review see Dolejš, 2013), e.g. the Helgeson Kirkham Flowers (HKF) equation of state (EoS) model (Tanger and Helgeson, 1988) or the density models (Anderson et al., 1991;Dolejš and Manning, 2010). These methods lead to accurate extrapolations with temperature and pressure if the experimental data at elevated temperatures are available to fit their parameters (Miron et al., 2017). In the case of the HKF EoS model, parameters for many aqueous species have been estimated from correlations between standard partial molal properties (Shock and Helgeson, 1988). However, some previous estimates of HKF EoS coefficients from correlations for rare earth elements (Haas et al., 1995) were shown to produce large disagreements with the new experimental data (Migdisov et al., 2009). The discrepancies could be resolved by fits to the new measured data, suggesting that for some systems, the existing correlations may need to be revised. When no or little experimental data are available, as in the case of many lanthanide and actinide complexes where only logK 298 is known, other approaches, such as simple logK T temperature extrapolations using isocoulombic reactions, can and should be considered.

Possible types of logK T temperature extrapolations
The standard Gibbs energy of a reaction (D r G T ) is related to the equilibrium constant and logK T ¼ À where R = 8.31451 J/K/mol is the universal gas constant, and K T is the thermodynamic equilibrium constant of reaction. At pressure P ref = 1 bar, the temperature T dependence of logK T is given by the standard enthalpy H (or entropy S ) and the standard heat capacity C P of reaction, as follows: where Eqs. (3a) and (3b) are related by Eq.
(2) and Depending on the available thermodynamic properties and on the underlying assumptions, the following simplifications of Eqs. (3a) and (3b) can be made: (1) Eqs. (5a) and (5b), called the ''three-term" extrapolations, assume that the heat capacity of reaction D r C P ;T ref does not depend on temperature, hence three terms must be known: the equilibrium constant, the enthalpy (or entropy) of reaction, and the heat capacity of reaction, all taken at the reference temperature T ref (usually 298.15 K or 25°C).
(2) Another simplification is to assume that D r C P ;T ¼ 0 at any temperature. This reduces Eqs. (3a) and (3b) (or (5a) and (5b)) to: Eqs. (6a) and (6b) are known as the ''two-term" extrapolations, since they require two terms to be known: the equilibrium constant and the enthalpy (or entropy) of reaction, all at the reference temperature. Eq. (6a) can also be written in the form broadly known as the "van't Hoff equation'': which yields a linear plot of the natural logarithm of the equilibrium constant against 1 T , where the slope is equal to (3) When both D r C P ;T ¼ 0 and D r S T ¼ 0 are assumed, Eq. (6b) further simplifies to: From Eq. (4), it then follows that the Gibbs energy of reaction is constant: Eq. (7) is the expression for the ''one-term" extrapolation (called A-type), as it only requires one term to be known: the equilibrium constant at T ref . However, if D r C P ;T ¼ 0 and D r H which is another variant of the one-term extrapolation (Btype) where the equilibrium constant is independent of temperature. In this case, These logK T extrapolations can be used for any type of reaction (not only isocoulombic or isoelectric) and from any reference temperature T ref (not only from 25°C and 1 bar, normally used as reference in geochemical thermodynamic databases). Their accuracy is strongly dependent on the validity of the assumptions made about the reaction properties in cases (1), (2) and (3) above. (1) Constant and large heat capacity; (2) Heat capacity close or equal to zero; (3) Values of both heat capacity and entropy/ enthalpy close or equal to zero.
In order to evaluate the impact of making wrong assumptions about the heat capacities and entropies of reaction, one can use simple error propagation schemes. For example, at Common aquatic reactions such as mineral dissolution, complex formation or dissociation, often have large entropy and heat capacity effects (Table 1). For example, the values of D r Cp 298 and D r S 298 for the dissolution reaction CeF 3 (cr) = Ce 3+ + 3F À (Migdisov et al., 2009) are À631 and À360 J•K À1 •mol À1 ; for the bicarbonate dissociation  (Thoenen et al., 2004), they are À291 and À148 J•K À1 •mol À1 , respectively. For such reactions, the assumption that D r Cp 298 and D r S 298 values are zero will result in large errors in logK T even for extrapolations over small temperature intervals. On the other hand, isocoulombic and isoelectric reactions have rather small and often negligible entropy and heat capacity effects, with similar values on both sides of the reactions that cancel out to a large extent. This can dramatically reduce the error accumulated in logK T temperature extrapolations. For the same range of temperatures, when only log i K T ref is known, using any isocoulombic reaction guarantees that the error in log i K T is much smaller, compared with the direct complexation or dissolution reaction.
Three-term (constant heat capacity of reaction) temperature extrapolations are best suited for isoelectric reactions (Puigdomenech et al., 1997); they also can yield reasonable results for non-isoelectric reactions up to temperatures of about 100-150°C (Kulik, 2002). The two-term extrapolation is well suited for isocoulombic reactions (Wood and Samson, 1998), and can provide reasonable estimates for isoelectric reactions, but is not suitable for reactions that have a large heat capacity effect due to ion hydration, such as mineral dissolution reactions. One-term extrapolation works best for isocoulombic reactions, for which the assumption that the entropies and heat capacities of reaction cancel out or are close to zero, was found to be valid in many cases (Gu et al., 1994;Puigdomenech et al., 1997;Wood and Samson, 1998). Gu et al. (1994) provided numerous applications of the one-term extrapolation (Eq. (7)) to isocoulombic acid-base neutralization, hydrolysis, redox and exchange reactions, which in some cases showed comparable or even better results than those obtained with two-term extrapolations.

Isocoulombic reactions
An isocoulombic reaction has the same number of likecharged species on the reactant and the product side. For example: CaCO 3ðaqÞ + UO 2 2þ = Ca 2þ + UO 2 CO 3ðaqÞ ð11Þ is an isocoulombic reaction, while: is not. Isocoulombic reactions are a subset of isoelectric reactions. The latter have the same number of positive and negative charges on each side, though the species are not like-charged, as in this example for a cation M: The main assumption in the use of isocoulombic reactions for temperature extrapolation is that similar aqueous species that are present on both sides of the reaction have standard thermodynamic properties (e.g. heat capacity, entropy, or volume) of comparable magnitudes that largely cancel out. The same principle can be applied when extrapolating thermodynamic properties not only to elevated temperatures, but also to elevated pressures or ionic strengths; The values were collected from several literature sources and used for testing the isocoulombic estimates. Direct use of these values to compute logK T is discouraged because this can result in large deviations above 100-150°C due to the unknown large values of  it is known that equilibrium constants of isocoulombic reactions are only weakly dependent on ionic strength (Luo and Byrne, 2004). This assumption might not always be valid for an isocoulombic reaction, but it is presumably valid for reactions that have minimal energetic, electrostatic, volumetric and structural differences between reactants and products (Gu et al., 1994). Given a generic reaction whose log e K T ref is known, but the temperature trend of log e K T is unknown and must be evaluated (hence the superscript 'e'), we can use the isocoulombic method for making reliable estimates, provided that a suitable ''model reaction" (superscript 'm') with known log m K T temperature dependence exists. This model reaction can be used to convert the reaction of interest into an isocoulombic reaction and thus serves as a ''model" for the effects of temperature on the thermodynamic properties of the reaction of interest. An essential requirement for producing accurate estimates for the properties of the reaction of interest is that the properties of the model reaction as a function of temperature are well known (at least up to the temperature of interest).

Workflow and systematic evaluation
The workflow of estimating the log e K T temperature trends using isocoulombic reactions is outlined in Fig. 1. In order to validate each extrapolation type, the estimated values of log e K T need to be compared with the known experimental data.
Consider, for example, the reaction of interest which is not isocoulombic, and its log e K T temperature dependence is assumed to be unknown.
As a first step, we construct an isocoulombic reaction by combining reaction (14) with a suitable model reaction ( Fig. 1, row 1), for instance Subtracting the model reaction (15) from the reaction of interest (14) (Fig. 1, row 2) leads to an isocoulombic reaction: whose properties are calculated as follows: where N represents any standard state property of reaction, e.g. logK At the second step, the one-, two-, or three-term extrapolation is performed alternatively in order to compute temperature functions of log i K T for this isocoulombic reaction ( Fig. 1, rows 3 and 4). In the third step, we use these temperature functions, combined with the well-known log m K T temperature function for the model reaction, to estimate and plot the corresponding log e K T temperature function for the reaction of interest (Fig. 1, rows 5 and 6) by simple addition at each temperature T: Fig. 2. Estimated log e K T for the reaction of interest Cu + + 2-HS À = Cu(HS) 2 À using the one-term A, one-term B, two-, and three-term extrapolations for log i K T of the isocoulombic reaction CuCl 2 À + 2HS À = 2Cl À + Cu(HS) 2 À and the log e K T of the model reaction Cu + + 2Cl À = CuCl 2 À determined from standard Gibbs energies of the reactants, CuCl 2 À and Cu + , Cl À , calculated with the HKF model. Values for the log m K T of the model reaction are retrieved from the standard Gibbs energies of the reactants, CuCl 2 À and Cu + , Cl À , calculated with the HKF EoS model (Tanger and Helgeson, 1988). Finally, we validate each extrapolation type (Fig. 1, row 4) by comparing the estimated log e K T with the known experimental data (Fig. 2). In this example, the equilibrium constants of the reaction of interest (14) have been experimentally determined from the solubility of copper sulfide minerals from ambient temperature up to 350°C at saturated water vapor pressure (SWVP) by Crerar and Barnes (1976) and Mountain and Seward (2003). The dissociation constants and thermodynamic properties for model reaction (15) were taken from the study of Liu and McPhail (2005), who reviewed the existing experimental data on copper chloride complexation from ambient conditions up to 350°C at the SWVP. As seen in Fig. 2, the one-term A-type extrapolation leads to the best agreement of estimated values with the experimental data. In this case, it was assumed that only the log e K 298 value is known for the reaction of interest. The one-term Btype, two-and three-term extrapolations yield worse estimates.
The challenge of applying the isocoulombic method for temperature extrapolations consists of finding the bestsuited model reaction whose reacting species have properties (size, charge, structure, etc.) closest to those of the species involved in the reaction of interest. This would result in a cation or ligand exchange reaction with perfectly balanced electrostatic and structural properties of the species.
In order to validate the log e K T temperature extrapolations obtained from isocoulombic reactions, and to identify the criteria for choosing the best model reactions, we carried out a systematic evaluation study by considering different reactions for various ions and ligands. The proposed workflow, aimed at finding the best-suited model reaction and performing the best possible temperature extrapolation, is shown in Fig. 3. From a dataset of selected reactions, the isocoulombic reactions are automatically generated by considering all possible combinations. These are then used for estimations using the one-, two-, or three-term extrapolations, as if we had only a limited amount of data ( Fig. 1). The quality of the estimates is determined by comparing them with a set of independent experimental log e K T values for the reaction of interest. Extensive, tedious and error-prone data management and thermodynamic calculations required for this systematic study were greatly simplified and accelerated by using our new ThermoHub database (https://thermohub.org) and the related software tools. All reactions were automatically generated using the Reactions Generator module of the ThermoMatch code tool for thermodynamic database management and consistency improvement (https:// thermohub.org/thermomatch/reacgen/). For a given list of substances and selected master species, a set of all possible independent reactions was generated by another Thermo-Match module. The properties of reactants and reactions at elevated temperatures were calculated and plotted using the ThermoFun client code that provides a large collection of temperature and pressure extrapolation models, such as the methods discussed above for reactions, or the HKF model for aqueous species (https://thermohub.org/thermofun/ methods/).

APPLICATION TO LANTHANIDE COMPLEXATION
The lanthanide system, which is better characterized in comparison to actinides, was chosen for evaluating the criteria for selecting the reactions to produce reliable temperature extrapolations. Consistent thermodynamic datasets, such as the speciation of lanthanides with fluoride from Migdisov et al. (2009), were used to check whether the similarities between the properties of ions (hydrated ionic radius etc.), can be utilized for making reliable temperature extrapolations. The same principles were then applied and tested for the actinide-bearing species.

Trivalent lanthanide fluoride and chloride complexation
The stabilities of trivalent lanthanide La(III) fluoride and chloride aqueous complexes (LaF 2+ , LaCl 2+ , LaCl 2 + ) at elevated temperatures have been evaluated using the HKF EoS-based model by Migdisov et al. (2009). This model is useful for extrapolating thermodynamic properties of ions and complexes to elevated temperatures and pressures, although for calibration, it requires several experimental data points measured at different T, P conditions. Standard partial molal thermodynamic properties and HKF EoS model coefficients have been retrieved on the basis of solubility experiments of La(III) fluoride solids at 150, 200, 250°C in in fluoride and chloride solutions at SWVP and from the additional low temperature data (Luo and Byrne, 2007;Luo and Millero, 2004;Migdisov and Williams-Jones, 2007).
We used both chloride and fluoride complexation datasets to test the relationship between the size of the hydrated ionic radius of the lanthanide ions and the quality of log e K T estimates using isocoulombic Ln(III) exchange reactions. As in the isocoulombic reaction example before, we assumed that for the reaction of interest, only a minimal set of parameters is available, and tested each extrapolation type (each case in Fig. 1, row 3) by comparing estimated log e K T values with those calculated from the HKF EoS model that provides an accurate description of experimental log e K T values (Fig. 1). For a given lanthanide element Ln(III), we considered all complexation reactions (Fig. 3, list of N reactions) of the form: The standard properties of reaction were calculated from standard properties of reactants and products. Next, we considered all possible combinations between the fluoride complexation reactions to generate isocoulombic reac-tions ( Fig. 3, row 3). For each Ln(III) reaction with fluoride (designated as the reaction of interest, for which the temperature dependence had to be estimated), we subtracted each of the remaining reactions, i.e. a designated model reaction with known temperature dependence. This led to isocoulombic reactions of the following form: where A and B designate any two different trivalent lanthanide Ln(III) ions. To compare the quality of different estimations, calculations were conducted for temperatures from 0 to 250°C in 5°steps (Fig. 4). For this discretized temperature interval (all 50 steps indexed with j), we computed the mean hd T i and standard deviation r d T ð Þ of the differences between log e K T j (HKF) values calculated using the HKF model and of log e K T j (estimated) values obtained from the log m K T j values of the model reactions (of another lanthanide ion) and the log i K T values from the isocoulombic reactions calculated using the one-term A-type, one-term B-type, two-, and three-term extrapolations. We compared Fig. 4. Mean values hd T i of the differences d T j (Eq. (22)) for the estimated log e K T values using one-term A-type, one-term B-type, two-, and three-term extrapolations in the temperature interval from 0 to 250°C with 5°C step, plotted against the absolute difference in the hydrated ion radius between the Ln(III) ion in the reaction of interest (shown in the upper-left corner) and the Ln(III) ion in the model reaction (marked on each data-point with the corresponding lanthanide). For the one-term A-type extrapolation, the standard deviation r d T ð Þ is also shown as error bars. Hydrated Ionic radii values from D' Angelo et al. (2011). our estimated values with those calculated using the HKF model. The HKF parameters were fitted to experimental data (25-250°C) by Migdisov et al. (2009), and provide an accurate description of the experimental values and can be used to calculate predicted values at any temperature in the chosen interval. The HKF model is also used to calculate the log m K T j values of the model reaction, which represents a different lanthanide ion than the estimated one. In this way, we can see how similar the temperature trend of the model reaction is to that of the estimated reaction.
The obtained trends of the differences (Fig. 4) point to the fact that the estimate is the better the smaller is the difference in the hydrated ionic radius between the Ln(III) ion present in the reaction of interest and the Ln(III) ion in the model reaction (i.e. the differences are close to zero). The three-term isocoulombic extrapolation yields the best estimates over the investigated temperature interval, followed by the one-term A-type, one-term B-type, and two-term extrapolations. Good estimates from the three-term extrapolation suggest that, overall, there is no large change in the heat capacity as a function of temperature over the investigated interval, and the reliability of the estimate decreases only at much more elevated temperatures. The two-term extrapolation gives the worst estimates for all combinations. This signifies that either the heat capacity of reaction is not close to zero or the entropies and heat capacities of reaction largely compensate each other. The latter conclusion is also supported by better estimates using the oneterm A-type and one-term B-type extrapolations (i.e., setting both D r S 298 and D r Cp 298 to zero produces better estimates than independently setting either one to zero).
For the complexation of lanthanides with fluoride, four groups can be distinguished, (1) La to Nd, (2) Nd to Gd, (3) Gd to Ho, and (4) Er to Lu. Several studies have identified a 'tetrad effect' in the geochemical properties of lanthanides (Ekberg et al., 2012;Kawabe and Masuda, 2001). The same effect seems to be valid also for the temperature dependence of lanthanides complexation with fluoride. The estimation of reaction properties at elevated temperatures with the one-term A-type extrapolation and isocoulombic reactions generated using model reactions with Ln(III) ions belonging to the same group results in mean differences smaller than 0.2 log units over the whole temperature interval. The groups are only weakly distinguishable in the case of chloride complexation (Fig. 4). Lanthanides are chemically quite similar; EXAFS studies on the hydrated lanthanide ions have shown that there are no sudden structural changes, but rather gradual ones along the series with decreasing ionic radii (Persson et al., 2008). Overall, the ionic radius and metal oxygen distance variation is small, 1.25-0.995 Å and 2.45-2.31 Å from La to Lu, respectively, and the same trigonal prism configuration characterizes the hydration shell throughout the La(III) group (D'Angelo et al., 2011;Persson, 2010). This is reflected in aqueous complexes (with a common ligand), and explains why the effect of temperature on the properties of isocoulombic reactions exchanging lanthanide ions with similar ionic radii is minimal, resulting in good estimates using the one-term A-type extrapolation (D r S 298 ¼ 0 and D r Cp Analogous to aqueous complexation, mineral dissolution reactions such as LnF 3 = Ln 3+ + 3F À commonly have large entropy and very large heat capacity effects due to the ion hydration in water. However, for isocoulombic reactions involving solids where two similar aqueous ions are exchanged, thermodynamic contributions of ion solvation and electrostriction should mostly cancel out. Hence, we can expect that temperature trends of logK T in dissolution reactions for two minerals with the same structure involving similar ions will also be similar. Migdisov et al. (2009) evaluated temperature trends of solubility products of Ln(III) fluoride solids from their high temperature solubility measurements and reported calorimetric data (Konings and Kovács, 2003) for the solids. We used the dataset for fluoride solids to do the same exercise as in the case of the aqueous complexation reactions. The resulting isocoulombic reactions are aqueous ion exchange reactions between REE(III) fluoride solids.  22)) for the estimated log e K T values using one-term A-type, one-term B-type, two-, and three-term extrapolations in the temperature interval from 0 to 250°C with 5°C step, plotted against the difference in the hydrated ionic radius between the Ln(III) ion in the reaction of interest (shown in the upper-left corner) and the ion in the model reaction (marked on each data point). For the one-term A extrapolation, the standard deviation r d T ð Þ is also shown as error bars. Hydrated ionic radii values from D'Angelo et al. (2011). As in previous cases, we can see improved estimates using the one-term extrapolation for log e K T when exchanging ions with similar ionic radii (Fig. 5). Over the entire temperature range, the average deviation hd T i does not exceed 0.5 log units when using either of the two variants of the one-term extrapolation. Compared to the one-term A-type, larger deviations arise in the case of the one-term B-type extrapolation, suggesting that the resulting isocoulombic exchange reactions are rather enthalpy-than entropy-driven (D r S . Two-and three-term extrapolations produce almost perfect estimates for the entire lanthanide group, independent of the ionic radius. This means that knowing D r Cp 298 for the solubility reaction of one lanthanide solid, the temperature dependence for the solubility constants of all other lanthanide solids with similar crystal structures can be estimated using only logK 298 and D r H 298 . Even if D r H 298 is not known, the error in log e K T will still be less than 0.5 when using an isocoulombic reaction in this temperature interval (250°), but will exceed several log 10 units if the direct dissolution reactions are used with D r H 298 ¼ 0. Experimental data on aqueous lanthanide hydroxide, carbonate, sulfate, and phosphate complexes at elevated temperatures are scarce (Migdisov et al., 2016), with only some measurements available for sulfate complexation (Nd, Sm, and Er; Migdisov and Williams-Jones (2008)) and Nd 3+ hydrolysis (Wood et al., 2000). Therefore, a similar type of evaluation was not possible. However, we expect similar results as in the case of fluoride and chloride complexation, with comparable temperature effects for complexation reactions of lanthanides with similar ionic radii that should largely cancel out in isocoulombic ion exchange reactions.

ACTINIDE COMPLEXATION
The assessment of estimates produced using the isocoulombic method and the temperature interval for the comparisons is limited by the available thermodynamic data on the temperature dependence of the model reactions and the existence of experimental data to validate the estimates against. In cases when for model reactions only the entropy of reaction was available, we limited the assessment to an upper value of 150°C, in cases when the heat capacity was also available the assessment was done up to 250°C. In the best-case scenario, a well-defined temperature dependence for the model reaction is available (at least 0-250°C).
Most of the available thermodynamic data for aqueous actinide complexation are restricted to values of equilibrium constants logK 298 at 25°C (298 K) and 1 bar, with very few data available on the standard entropy (or enthalpy) of reaction, and almost no data available on the standard heat capacity of reaction (Guillaumont et al., 2003;Lemire et al., 2001;Rao, 2007;Thoenen et al., 2004). For the following evaluation of the reliability of estimated data using isocoulombic reactions, we focused on the one-term A-type extrapolation that requires only a value of logK 298 to be known. For model reactions (unless otherwise stated), the available entropies (or enthalpies) of reaction were used to calculate log m K T at different temperatures using the two-term extrapolation. In this case, the precision of the calculated log m K T for model reactions is getting worse with increasing temperature and at temperatures far from the experimental conditions (>100°C). Temperatures in the range from 0 to 150°C are commonly relevant for safety assessments of radioactive waste repositories. Temperatures to be considered are usually below 80°C (see, e.g., Giffaut et al. (2014)), although higher temperatures may be expected, e.g., in the case of a canister failure scenario (up to about 160°C, Johnson et al. (2002)).
For the evaluation of the isocoulombic method, we needed data for reaction enthalpies for more than one actinide with the same ligand. Table 1 shows complexation reactions, collected from the literature, with the available reaction properties (as reported in their original reference) that were used as model reactions (lightface), and as designated reactions of interest (boldface).
To test the quality of the estimates, we assumed that only log e K 298 is available for the reactions of interest (boldface). For each group of actinide complexation, isocoulombic reactions were generated by subtracting the properties of the model reactions (lightface) from the reactions of interest (boldface). For a limited temperature interval (see above), we calculated log i K T values for the isocoulombic reactions using the one-term A-type and in some cases the one-term B-type extrapolation. Finally, we retrieved esti- , An(VI)O 2 (SO 4 ) n (2À2n) and An(VI)O 2 (CO 3 ) 3 4À complexes were taken from the NEA-TDB reviews (Guillaumont et al., 2003;Lemire et al., 2001;Rand et al., 2008). Available independent log e K T values were used to test the quality of the estimates. Xia et al. (2010) report log e K T values for the complexation of Pu(IV) with fluoride (n = 1, 2) at 25, 40 and 55°C based on solvent extraction, and Di Bernardo et al. (2018) report log e K T values for the complexation of Th(IV) with sulfate (n = 1, 2) for 10-70°C temperature interval based on spectrophotometry or H + -potentiometry. Recent experimental data on the uranyl sulfate system, reported by Kalintsev et al. (2019), were not used in the assessment due to the lack of high temperature data for the analogue hexavalent actinides (Pu, Np) to compare the estimates with. In the case of Th(IV) sulfate complexation, we limited our comparisons for temperatures up to 150°C because the analogue model reactions for tetravalent Pu, Np, U used for calculating estimates have only data for the entropy of reaction. The experimental data above 150°C for Th(IV) sulfate complexation reported by Nisbet et al. (2019) are used in the discussion section below as an example of estimating log e K T for the analogue U(IV) reaction.
Results of the estimations are shown in Fig. 6, compared with independent log e K T values. The one-term A-type extrapolation used for calculating log m K T values of isocoulombic ion exchange reactions combined with the temperature dependence of model reactions, results in estimated log e K T values for the tested reactions of interest that compare well with the independently measured values. The one-term B-type extrapolation, although not shown in the plots, gives very similar estimates, because log m K 298 of the resulting isocoulombic reactions is close to zero. There is no clearly visible difference in the quality of the estimates when exchanging either one of the considered actinide ions. Persson (2010) reports a decrease of the hydrated ionic radii for Th(IV) to U(IV) from 1.11 to 1.05 Å , a 9-fold coordination for U(IV), Np(IV), Pu(IV), and a trigonal prism for Th (IV) (configuration of the hydration shell). This could suggest that at temperatures > 100°C increasing differences between Th(IV) and the other three tetravalent actinides (U, Np, Pu) should be observed. Within the errors of log e K T measurements and in view of the limited high temperature data for the model reactions, no clear distinction can be made between estimates using isocoulombic reactions of either actinide ion of the same valence (Th, U, Np, and Pu). When high temperature complexation data  . The log i K T values of isocoulombic reactions were calculated at different temperatures using the one-term A extrapolation, and the log m K T values of the model reactions were calculated using the two-term extrapolation. As limited data (i.e. just D r S 298 ) were available for the model reactions, the estimated values above 100°C may be unreliable.
are available for any of these actinide ions, they can be used to estimate the temperature dependence of the other ions, although neighboring ions in the actinide series should be preferred, due to the likelihood of having a more comparable hydration behavior and, hence, temperature trend.
A clear difference can be seen in the case of estimating the Th(IV) sulfate complexation using the analogous U (IV) sulfate complexation as a model reaction (Fig. 6B). This suggests that there is either a significant difference in the temperature behavior between U(IV) and Th(IV) or that the D r H 298 value selected by Guillaumont et al. (2003) needs to be revised. The deviation in temperature trends could be due to the different configuration of the hydration shell for the Th(IV) trigonal prism compared to the 9-fold coordination for U(IV), Np(IV), and Pu(IV). Another possibility is that there are differences in the electronic configuration of the ions. Chaudhuri et al. (1999) suggest that U(IV) may retain d electrons which does not happen to Th(IV). However, the same poor quality of estimates should result when using Np(IV) and Pu(IV) sulfate complexation as model reactions. Di Bernardo et al. (2018) suggested a revised D r H 298 value (33.4 instead of 8 ± 2.7 kJ•mol À1 ) for the 1:1 complexation reaction. Using this value for the U(IV) sulfate complexation model reaction leads to excellent estimates for the analogous Th(IV) reaction. Although not shown by Di Bernardo et al. (2018), if a correction of the same magnitude to the D r H 298 value for the 1:2 U(IV) complexation reaction is made (58.1 instead of 32.7 ± 2.8 kJ•mol À1 ), this will lead to good estimates for the analogous Th(IV) 1:2 sulfate complexation reaction. In the case of the U(VI) 1:2 sulfate complex (Fig. 6C), the logK T values are underestimated when using the analogous Np(VI) model reaction, signifying that the D r H 298 for the latter needs reconsideration. A significant disagreement can be seen between the log e K T values for the 1:3 Np(VI) carbonate complex calculated using the van't Hoff (two-term) extrapolation (with D r H 298 from Lemire et al. (2001)) and the estimated values using the analogous U(VI) model reaction (Fig. 6D). At the same time, using the Pu(VI) model reaction produces satisfactory results, as expected from the similarities between Pu(VI) and Np (VI). All this suggests that D r H 298 for the reaction involving U(VI) needs reconsideration. There is a good agreement between the temperature dependence of the considered tetravalent actinides complexation with fluoride. In the case of the complexation with sulfate, the available stability data for U(IV) and Np(VI) (just for the 1:1 complex) show a different trend from the other actinides, while in the case of the complexation with carbonate, the stability data for U (VI) shows a different temperature trend compared to Np (VI) and Pu(VI). This makes it difficult to discern possible structure differences from errors in the available temperature dependence data.

LANTHANIDE AND ACTINIDE EXCHANGE REACTIONS
Studies on the hydrated ionic radii of actinide(III) and lanthanide(III) elements in aqueous solution (D'Angelo et al., 2013(D'Angelo et al., , 2011 have shown that these two groups have almost identical ionic radii and a similar water structure in the hydration shell (Persson, 2010), suggesting that they should have very similar hydration properties and electrostatic interactions. A close resemblance between their behavior in water advocates for comparable temperature trends of thermodynamic properties. This means that complexation reactions sharing the same ligand could be used as analogues concerning the temperature effect. If a complexation reaction of one actinide or lanthanide ion (e.g. Eu(III)) has a complete set of parameters describing the dependence of its properties on temperature then it can be used as a model reaction to estimate reaction properties for a similar trivalent actinide or lanthanide ion (e.g. Cm(III)).
To test this idea, we used the available high temperature data on trivalent actinide and lanthanide complexation with fluoride, chloride, nitrate and silicate (for Eu, Cm, and Nd, Table 1). If the available reaction properties are restricted to log m K 298 , D r H 298 and D r S 298 , (and D r Cp 298 set to zero), the log m K T of the model reaction can be calculated at T > 25°C using the two-term (van't Hoff) extrapolation, which becomes increasingly uncertain at T > 100°C. However, for the Eu(III) fluoride complexation reaction, the log m K T values were calculated using standard Gibbs energies of reactants and HKF EoS parameters derived in Migdisov et al. (2009). For the Cm(III) complexation reaction with nitrate, D r Cp 298 is also available from Skerencak et al. (2009), and its properties, when used as a model reaction, can be calculated using the three-term extrapolation. In these cases, the estimations were also done at temperatures exceeding 150°C.
Comparisons of estimates with independent log e K T values from experiments are shown in Fig. 7. There is a good agreement between experiments and estimates, suggesting that high-temperature data on the complexation of trivalent lanthanides can be used to model the temperature dependence for trivalent actinides and vice versa. The differences that can be seen for the estimation of Cm(III) fluoride complexation from Eu(III) (Fig. 7A) are probably due to the limited low temperature data (0-45°C from Luo and Millero (2004)) used by Migdisov et al. (2009) when fitting the HKF model for the EuF 2+ aqueous complex. We can now better constrain the EuF 2+ aqueous complex at low temperatures by adding log e K T values estimated using the iscoulombic method from Cm(III) (Skerencak et al., 2010), and the measured high temperature log m K T values from Migdisov et al. (2009) (see Section 7). We adjusted the temperature dependence of EuF 2+ and now the estimates for Cm(III) (Fig. 7A adjusted) appear improved over the entire temperature interval.
For complexation reactions with the silicate ligand (Fig. 7D), the reaction equilibrium constants were not corrected to infinite dilution. As expected, both the temperature and the ionic strength effects cancel out, meaning that the interaction parameters for Cm(III) and Eu(III) with another anion can be considered as equal. When doing ionic strength corrections with no interaction parameters available for the investigated system, their values can be approximated to be equal to the parameters from systems for similar ions (Guillaumont et al., 2003). This has been demonstrated from experiments for a number of isocoulombic reactions (Gu et al., 1994) and is based on the same principle as in the case of the temperature dependence.

REDOX REACTIONS
Actinide ions have several oxidation states commonly ranging from III to VI. We have tested the isocoulombic method in estimating the temperature dependence for redox reactions of Np, assuming that only log e K 298 is available. The temperature dependence of the reduction of Np(VI) to Np(IV) was estimated using the analogous reactions for the reduction of U and Pu VI to IV. Fig. 8A shows that log e K T for Np(VI) to Np(IV) reduction reaction is very well estimated using the one-term A-type extrapolation for the isocoulombic reaction and U and Pu analogous model reactions. In addition, a very good estimate is obtained when using the data for U reduction from Shock et al. (1997) and the HKF model, which, besides the entropy and heat capacity, contains an additional D r C P dT contribution to the temperature dependence. We have also estimated the Np(VI) to Np(V) and Np(VI) to Np(III) reduction reactions temperature dependence using the Pu analogue reduction as model reactions and one-term A extrapolation. Fig. 8B shows that the log e K T values of the investigated reactions are well reproduced by the estimated values.

EXCHANGE OF LIGANDS
Isocoulombic ligand exchange reactions that involve two complexation reactions sharing the same cation can also Fig. 7. Independent log e K T values (symbols) derived from experimental measurements (Jordan et al., 2018;Pathak and Choppin, 2006;Rao and Tian, 2009;Skerencak et al., 2010;Tian and Shuh, 2014) for the displayed reactions of interest, and log e K T values (curves) estimated using isocoulombic reactions generated by subtracting analogous model reactions for ions shown in the legend. Isocoulombic reactions are of the form: (A) Cm 3+ + EuF 2+ = CmF 2+ + Eu 3+ , the log e K T data points at 150, 200, and 250°C for Cm were estimated using the one-term A-type extrapolated log m K T values for the isocoulombic reaction and values reported for Eu(III) (Migdisov et al., 2009 produce accurate estimates, based only on log e K 298 and log m K T . This can be seen in the example where we estimated the log e K T values of Cu(I) hydrosulfide complexes using the Cu(I) chloride complexation reaction as a model reaction (Fig. 2) and the one-term A-type extrapolation. Ligands such as F À and OH À are known to have very similar solvation characteristics in non-aqueous and aqueous solution (Majer et al., 1997;Salomon and Hefter, 1993). The temperature trends of the association constants for NaOH (aq) and NaF (aq) complexes that share the same ion are similar (Majer et al., 1997). This suggests that the data on high temperature fluoride complexation can be used to estimate the temperature dependence of analogous hydrolysis reactions, and vice versa. We tested this hypothesis using the available temperature dependence data on U(IV), U(VI) and Th(IV) fluoride complexation (Table 1) as model reactions to estimate the U(IV), U(VI) and Th(IV) hydrolysis constants at elevated temperature. The isocoulombic reactions were generated by first subtracting from the investigated reaction the water dissociation constant and then the corresponding fluoride complexation reaction, as follows (for An(IV)): , the model reaction for Pu had only entropy data available (still produces comparable estimates with U > 150°C), while for U entropy and heat capacity were available; (B) NpO 2 2+ + An 3+ = Np 3+ + AnO 2 2+ (top) and NpO 2 2+ + AnO 2 + = NpO 2 + + AnO 2 2+ (bottom), only D r S 298 was available for model reactions; therefore, estimated values above 100°C may be unreliable. Fig. 9. Exchange of ligands log e K T values (symbols) derived using van't Hoff (two-term) extrapolation using the reported value for D r S 298 and log e K 298 values (curves) estimated using isocoulombic reactions and one-term A and B extrapolations. Isocoulombic reactions are of the form: (A) An(IV)F n (4Àn) + nOH À = An(IV)(OH) n (4Àn) + nF À ; (B) UO 2 F + + OH À = UO 2 OH + + F À ; and were generated by subtracting the water dissociation reaction and the analogous fluoride complexation reaction. Limited data (i.e. just D r S Estimated log e K T values (one-term A-and B-type for calculating log i K T ) are compared with independent log e K T values for the reaction of interest in Fig. 9. The 1:1 U(IV) hydrolysis was measured at elevated temperatures by Nikolaeva (1978). Selected values for the enthalpies of the 1:1 and 1:2 hydrolysis of Th(IV) by Brown and Ekberg (2016) are consistent with those from Rand et al. (2008) and in agreement with the isocoulombic estimates. High temperature reaction constants for the formation of Th (OH) 2 2+ are reported by Nisbet et al. (2018) between 175 and 250°C, but the iscoulombic estimates using the fluoride complexation as model reaction, with no information on the reaction heat capacity effect, are not suitable for extrapolations above 100°C (see Section 7.1 U(IV) sulfate complexation example). In the estimation process, the water dissociation reaction was also used as a model reaction in addition to the fluoride complexation. Compared to the previous examples, in this case the one-term B-type extrapolation produces better estimates than the one-term A-type extrapolation. This suggests that the resulting isocoulombic reactions exchanging OH À with F À are entropy driven and should have relatively small enthalpy effects.

DISCUSSION
The temperature trend of the equilibrium constant of a complexation reaction (at infinite dilution) is a function of its standard entropy and heat capacity effects, which both can be related to interactions among the reactants (ions and complexes) and the water solvent, which affects the structure of water. These interactions can be electrostatic and non-electrostatic and may depend on the charge (charge localization), size, and electronic configuration of the ions, with additional contributions from structural factors in the case of complex species (Abraham and Marcus, 1986;Betts and Dahlinger, 2006;Cobble, 1953a,b;Tanger and Helgeson, 1988). Abraham andMarcus (1986, 2007) show that upon complexation, due to the lower charge on the product side, part of the solvent is released from the solvation shell. This results in a change in entropy (desolvation entropy) that Abraham and Marcus (1986) relates to the number of solvent molecules released during the ion association. Reactions between ions and ligands that show similar interactions with water (e.g. hydration properties, hydration shell structure) produce complexes that have similar electrostatic and structural properties and lead to the same number of solvent molecules released from the hydration shell. Therefore, having on both sides of an exchange reaction ions and complexes interacting with the solvent in a comparable way will result in cancelation of most temperature effects (due to similar contributions to the entropy and heat capacity), thus making the Gibbs energy effect of the reaction almost independent of temperature (Eq. (7)).
The isocoulombic method is well suited to fill out the gaps when no or limited elevated temperature data are available. Only when there are sufficient data, other extrapolation methods such as the HKF model or density models (logK can provide more accurate values over a wider range of temperature and, thus, can complement the isocoulombic method by providing logK T values for model reactions. Reactions in isocoulombic form can replace conventional reactions that still largely lack temperature dependence parameters in law of mass action (LMA) type databases (unless there are some technical limitations). A pre-condition is that the temperature dependence of other secondary species (comprising the model reaction) is well known at the desired conditions. For Gibbs energy minimization (GEM) type databases, where the standard Gibbs energy of all substances must be provided, the isocoulombic method can be used to calculate the unknown standard Gibbs energy values at elevated temperatures for a substance of interest from the estimated log e K T (D r G T ), knowing the standard Gibbs energies of other reactants (from their own extrapolation method, e.g. the HKF model). Gu et al. (1994) analyzed different types of reactions and showed that the one-term A-type extrapolation can produce accurate results when applied to isocoulombic reactions containing reactants and products with the smallest electrostatic and structural differences. The success of the isocoulombic method using the one-term Atype extrapolation relies on the choice of the model reaction. There are many ions and aqueous complexes that have the same charge and that can be used for constructing isocoulombic reactions, but in many cases, the effects of temperature may be different, and the estimations may produce incorrect values. Wood (1990) combined the isocoulombic method with an electrostatic method (similar to the HKF) to extend estimated log e K T of rare-earth elements species up to 350°C, with subsequent experimental data showing that the log e K T estimates are inaccurate (Migdisov et al., 2009).
To identify the criteria for choosing the best model reactions, we conducted a systematic study into the lanthanide and actinide systems with isocoulombic estimates compared against experimental data. The main focus is on cases where the thermodynamic properties for both model reaction and the reaction of interest are known, and experimental values for log e K T are also available in order to validate the isocoulombic method and to provide guidelines for choosing the model reactions that are expected to result in the best estimates. The methodology (Figs. 1 and 2) presented is put into practice using modern technology in the form of a flexible database (ThermoHub) and related code tools (ThermoMatch, ThermoFun), https://thermohub.org. Together, they allow for efficient and fast processing and generation of all possible reactions (with calculation of their properties). These reactions are then automatically combined into isocoulombic reactions, to be further used for validating temperature extrapolations against independent measurements (Figs. 4-9).
Based on observations from this study, we arrive at recommending a three-step methodology for the reliable prediction of standard thermodynamic properties of complexation reactions of interest with unknown temperature dependence: (Step 1) seek for the ''best" available model reaction in order to construct an isocoulombic exchange reaction; (Step 2) extrapolate log i K T of the isocoulombic reaction using the appropriate one-, two-, or three-term extrapolation method depending on the availability of thermodynamic data; and (Step 3) estimate log e K T of the reaction of interest as a function of temperature, or retrieve the standard-state thermodynamic properties of the substance of interest at T and T 298 .
Step 1: Search for a suitable model reaction If only log e K 298 is known for the reaction of interest, one needs to look for an analogous model reaction that has an exchange ion with the hydrated ionic radius, structure of the hydration shell, and solvation properties closest to those of the aqueous ion in the reaction of interest with unknown temperature dependence. This task can be facilitated by considering the available information such as hydrated radii of aquo-ions (D'Angelo et al., 2013(D'Angelo et al., , 2011Persson et al., 2008), bond distances in the complexes and structure of the hydration shell (Persson, 2010). For example, the ionic radius, the mean metal oxygen bond distance, and the configuration in the hydration shell for Fe 3+ and Ti 3+ , reported by Persson (2010), are 2.00 Å , 0.66 Å , octahedron, and 2.03 Å , 0.69 Å , octahedron, respectively. Such closely similar properties of two hydrated ions suggest that the data on complexation at elevated temperature of either ion can be used to estimate the complexation of the other. Similar results can also be expected in non-aqueous solutions because the solute-solvent interactions should be alike for similar ions and complexes. If the available model reaction has the same charge pattern as the reaction of interest, but the size and structure of the reactants are different, knowledge of both entropy and heat capacity effects of the reactions is of advantage for producing reasonable two-and three-term extrapolations to elevated temperatures for the isocoulombic reaction.
For a Ln(III) or a An(III) complexation reaction with unknown temperature dependence, the best suited model reaction would be a reaction between the same ligand and another Ln(III) or a An(III) cation with the closest hydrated ionic radius. Complexation reactions of Th, U, Np, and Pu actinide ions in the same oxidation state seem to have similar temperature dependence (Section 3) and can be used as model reactions for generating isocoulombic reactions exchanging any of the four actinide ions. It is expected that exchanging U with Np whose difference in ionic radius is small will yield better estimates over a wider temperature interval than exchanging Th and U aquoions with a larger difference in ionic radius.
Step 2: Extrapolate log i K T of the isocoulombic reaction to elevated temperatures Once we have found a suitable reaction of interest and constructed the isocoulombic reaction, one possibility is to use it directly to get log i K T values at elevated temperatures and further use them in modeling calculations, or to retrieve the temperature dependence of standard partial molal properties of the aqueous complex of interest, provided that these are known for all other reactants.
If only log i K 298 is available, the log i K T values can be extrapolated to elevated temperatures, in some cases even up to 300°C or higher (Gu et al., 1994) using the oneterm extrapolation. If values for the reaction entropy and heat capacity are available, the two-or three-term extrapolation methods can be used to extrapolate log i K T values at different temperatures. Two-or three-term extrapolation methods can produce better extrapolations when there are larger differences between the ions on the reactants and the products side of the isocoulombic reaction. Our assessment of Ln(III) chloride and fluoride complexation has shown that one-term extrapolations produce better results than two-term extrapolations. This is probably due to the optimal cancelation of 3-term temperature dependence effects that takes place only when setting both entropy and heat capacity effects of the isocoulombic reaction to zero.
Step 3: Estimate log e K T values for the reaction of interest at different temperatures Another possibility is to use the ''best" suitable isocoulombic reaction along with the model reaction to estimate log e K T values for the reaction of interest with unknown temperature dependence. In this case, the model reaction must have well-defined temperature dependence parameters. The valid temperature interval for the properties of the model reaction will also apply to the estimated properties of the reaction of interest. In the simplest case, if only log e K 298 is available for the reaction of interest and only log m K 298 and D r S 298 are available for the model reaction, the entropy effect of the reaction of interest can be assumed to be equal to that of the model reaction. In such a case, estimations can be done for a relatively narrow temperature range. For accurate estimates over a wide temperature interval, it is essential to have a well-studied model reaction with values not only for D r S 298 but also for D r Cp 298 , and perhaps even for the dependence of the heat capacity on temperature (dD r Cp dT ). Preferably, the temperature dependence of the model reaction and its species should be extrapolated using more elaborate methods such as the HKF equation of state or density models.
Examples of different application cases for the isocoulombic method using the methodology and the criteria summarized above are presented in the following section.
7.1. Using the isocoulombic method to estimate temperature trends Some high temperature experimental data on the complexation of lanthanide and actinide ions (e.g. uranyl complexation with chloride or sulfate (Dargent et al., 2014;Kalintsev et al., 2019;Migdisov et al., 2018)) do not have a corresponding analogue ion to validate the estimates (i.e. no high temperature data on chloride complexation of another hexavalent actinide). Such data cannot be used in the validation process, but are appropriate for being used as model reactions for estimating standard properties for complexation reaction involving analogue ions (e.g. Pu, Np chloride or sulfate complexes).
For example, the available data for U(IV) 1:2 sulfate complexation are limited to log e K 298 and D r S 298 (Table 1). The isocoulombic method can be used to estimate log e K T up to 300°C for this reaction by choosing an appropriate model reaction. Based on the available data on the temperature dependence (0-250°C) and similarities in the properties of hydrated ions (Table 1 in Persson, 2010), we select the analogous Th reaction as a suitable model reaction (hydrated ionic radii: Th 4+ 1.11 Å , U 4+ 1.08 Å ). An isocoulombic reaction is generated by subtracting the Th(IV) sulfate complexation model reaction from the U(IV) reaction of interest, resulting in: The log i K 298 of the isocoulombic reaction is 10:51 À 9:99 ¼ 0:52 (Table 2). We use this value and the one-term type-A extrapolation (Eq. (7)) to calculate log i K T for the isocoulombic reaction from 25 to 300°C (Table 1, column 2). For the model reaction, we fit the three-term equation (Eq. (5b)) against the experimental data at low temperatures from Di Bernardo et al. (2018) and at elevated temperatures from Nisbet et al. (2019).
The resulting values for the entropy and heat capacity of the model reaction at 25°C are 387 ± 25 and 1200 ± 100 J•mol À1 •K À1 , respectively. Using these values and the three-term extrapolation (Eq. (5b)), we calculate log m K T for the model reaction from 25 to 300°C (Table 2, column 3). Finally, we retrieve the estimated log e K T values for the U(IV) 1:2 complexation reaction from 25 to 300°C (Table 2, column 4, Fig. 10) by adding the log m K T values of the model reaction (Th(IV)) to the values obtained from the isocoulombic reaction. By doing so we assume that the entropy and heat capacity of the U(IV) sulfate complexation reaction are very similar to those of the Th(IV) model reaction. Note that a different configuration of the hydration shell, 6-fold coordinated trigonal prism for the Th (IV), compared to the 9-fold coordination for U(IV), could lead to some level of discrepancies at elevated temperatures. Such possible discrepancies can only be identified when the experimental data for U(IV) become available above 70°C.   Table 1 (Guillaumont et al., 2003). Shaded area represents the estimated error propagated from the error in the standard state properties.
As can be seen in Fig. 10, a van't Hoff (two-term) extrapolation is linear in a plot against the inverse temperature. When used over large temperature intervals, this will result in large discrepancies, if the heat capacity of the reaction is not considered. The discrepancies will increase with the difference in the charge distribution between the reactant and the products. For example, in Fig. 10 we, have one ''4+" and two ''2À" charges on the reactants side and no charge on the products side. This explains the large differences between the estimated values and the two-term extrapolation.
Commonly, the isocoulombic method is used for making log e K T extrapolations and estimates starting from the reference temperature of 25°C (298.15 K) to elevated temperatures. This is because most of the thermodynamic data are available for this temperature, which is also used as a reference point in many thermodynamic databases. There are cases when no equilibrium constant values are reported for low temperatures. The stabilities of some species can be so low at room conditions that they cannot be reasonably detected, and their thermodynamic properties extracted. Such complexes can become more stable with increasing temperature, appearing in significant amounts, so that their stability constants can be estimated reliably. In this case, the isocoulombic method can be used to extrapolate the log e K T from elevated to low temperatures (e.g. estimate log e K 298 at 25°C), by setting the reference temperature to one of the values of the experiments.
Experimental data on the hydrolysis of lanthanide ions as a function of temperature are mostly available below 100°C (Brown and Ekberg, 2016;Migdisov et al., 2016). Values for log e K 298 based on experimental data and free energy relationships have been reported (Baes and Mesmer, 1981;Brown and Ekberg, 2016;Klungness and Byrne, 2000;Lee and Byrne, 1992). The temperature dependence can be estimated based on HKF model correlations reported by Haas et al. (1995) or based on the van't Hoff extrapolation using the enthalpy data reported by Klungness and Byrne (2000). Wood et al. (2002) report the only experimental log e K T at temperatures above 100°C for the Nd 3+ 1:1 hydrolysis (250 and 290°C). Although their solubility experiments ranged from 30 to 290°C, the NdOH 2+ species was present in significant amounts only in the 250 and 290°C experiments. We can use these data and the isocoulombic method to obtain estimates for log e K T for the 1:1 hydrolysis reaction from 300 to 25°C.
In previous examples (Section 6), we have seen that isocoulombic reactions exchanging OH À with F À work well with the one-term B-type extrapolation. As shown in Eqs. (23)-(26), we subtract from the Nd 1:1 hydrolysis reaction  Wood et al. (2000). b Lee and Byrne (1992). Fig. 11. Estimated log e K T values (continuous line) using isocoulombic reaction Eq. (29) and the one-term type-B extrapolation and the Nd 3+ fluoride complexation model reaction from Migdisov et al. (2009) with the water dissociation reaction (Tanger and Helgeson, 1988). Dash dotted black line: using the van't Hoff extrapolation with D r H 298 58400 J•mol À1 . Dotted black line: using the three-term extrapolation with D r H 298 41500 J•mol À1 and D r Cp 298 170 J•mol À1 •K À1 . Dotted gray line: using the data for NdOH 2+ reported by Brown and Ekberg (2016). Circle: data from Lee and Byrne (1992), squares: data from Wood et al. (2002), and diamonds: data recalculated by Brown and Ekberg (2016). the reactions of the water dissociation and the Nd 1:1 fluoride complexation reaction to obtain the isocoulombic reaction: We select the reference temperature to be 250°C (523 K). The log i K 523 for the resulting isocoulombic reaction (eq. (29)) is 0.35. The estimated log e K T values for the Nd hydrolysis are calculated as follows (Table 3): for each temperature, we add the log m K T for the water dissociation (also a model reaction) and the log m K T for Nd fluoride complexation reaction (calculated with the HKF model and parameters in Migdisov et al. (2009)) to the log i K 523 of the isocoulombic reaction ( Fig. 11 and Table 3) (oneterm B-type, assumes constant logK T ). The log e K T values of reaction (28) plot on an almost straight line against the inverse of temperature. This is expected since the reaction is isoelectric, with three positive charges on both sides of the reaction. When plotted against 1/T, both the estimated log e K T using the isocoulombic method and the values calculated using the data of Haas et al. (1995) show a linear slope. The slope and shape of the lines are almost identical, but the estimated values are systematically off by 1.7 to 2 log units from the values of Haas et al. (1995), and the log e K T estimated using the isocoulombic method at 25°C is $1.7 log units more negative than the values reported in the literature (Brown and Ekberg, 2016;Klungness and Byrne, 2000;Lee and Byrne, 1992).
There are several possible reasons for this discrepancy: errors in low-and high temperature experimental values; errors in the data on fluoride complexation; errors in log e K T values estimated at low temperatures that are far away from the starting temperature of the extrapolation (250°C); or the possibility that the temperature dependence of the chosen model reaction (Nd 1:1 fluoride complexation) does not reproduce the temperature dependence of the hydrolysis reaction sufficiently well. The choice of the model reaction was based on the comparisons in section 6 and other studies, that suggest a similar temperature behavior for hydrolysis and fluoride complexation (for Na, Majer et al., 1997;for Al, Tagirov and Schott, 2001). Deberdt et al. (1998), based on solubility experiments, suggest that the values of the 1:1 hydrolysis constants for La and Gd at 25°C should be 1-2 log units smaller than the values proposed by Mesmer (1977, 1981) of À8.5 and À8.0, which are similar to À8.89 and À7.87 reported by Brown and Ekberg (2016). This is consistent with our estimate for the Nd 1:1 hydrolysis constant at low temperatures (a constant at 25°C that is 1.7 log units more negative). Nevertheless, several studies on Ln hydrolysis, using different methods, produced consistent results (see review by Brown and Ekberg, 2016). The extensive solubility study at 25°C by Neck et al. (2009) agrees with the values for NdOH 2+ and other hydrolysis species accepted by Brown and Ekberg (2016). Because the low and high stability constants produce enthalpy values consistent with Ln neighbors of Nd, the most plausible source for the discrepancy may lie in the particular choice of the model reaction or its temperature dependence. As shown in section 5 in the case of exchanging ligands the isocoulombic method can produce good estimates over short temperature intervals when the heat capacity effect is not significant. The fluoride complexation reaction has D r Cp 298 396 J•mol À1 •K À1 (Migdisov et al., 2009), assuming that for reaction (28) the heat capacity is 0 and subtracting the water dissociation reaction heat capacity we get a value of 211 J•mol À1 •K À1 . This shows a large difference of 184 J•mol À1 •K À1 between the heat capacities of the model and of the estimated reaction. This difference alone produces a deviation of 1.3 log units in the extrapolation of the reaction constant and explains large part of the resulted discrepancy. This suggest that the assumption that D r Cp 298 ¼ 0, for the isocoulombic OH-, F-exchange reactions might not be valid also in the case of other cations.
The isocoulombic estimation method can also be useful for combining temperature dependent experimental data of reactions that involve similar ions or ligands but were measured within different temperature intervals. We can combine the reactions into the isocoulombic form and use the one-term extrapolation to calculate log i K T values at experimental temperatures. These values can then be used to convert experimental data points from one reaction into the other. In the case of the CmF 2+ complexation reaction, the available experimental data are limited to 90°C (Skerencak et al., 2010). On the other hand, for EuF 2+ at low temperatures, there are only data around 25°C, but at high temperatures, values at 150, 200, and 250°C (Migdisov et al., 2009) are available. By adding log i K T values of the isocoulombic reaction, calculated using the oneterm A-type extrapolation, to the experimental log m K T values for the EuF 2+ complex, we obtain what would be the experimental data points for CmF 2+ above 100°C ( Fig. 7A and Table 4). Now, using the experimental data for CmF 2+ at low temperatures and the data at elevated temperatures converted from the EuF 2+ trend, the temperature dependence can be better constrained by more experimental data points, and thermodynamic properties needed to calculate log e K T , D r S 298 and D r Cp 298 can then be evalu- À0.38 6.52 ± 0.21 6.14 ± 0.21 ated. The same procedure was used to convert experimental data points for Th(IV) 1:2 sulfate complexation to the analogue U(IV) complexation (Fig. 10). This is all based on the assumption of a similar temperature dependence, as suggested from the evaluation of the isocoulombic method. For a chemical subsystem consisting of similar ions (or ligands), one can construct a reaction-based thermodynamic dataset (for use in speciation codes such as PHREEQC) with temperature dependences constrained by a few selected model reactions (with well-defined standard properties). The remaining reactions in the dataset will be in the form of isocoulombic reactions, constructed using the selected model reactions. Take for example the system Th(IV)-U(IV)-Np(IV)-Pu(IV)-S(VI)-O, having the following master species: Th 4+ , U 4+ , Np 4+ , Pu 4+ , and SO 4 2À , and the following product species: Th(SO 4 ) 2(aq) , U(SO 4 ) 2 (aq) , Np(SO 4 ) 2(aq) , Pu(SO 4 ) 2(aq) . To add them to a reaction-based dataset, one can construct complexation reactions between the ions and the sulfate ligand that will result in the product species. To use these reactions for calculations at elevated temperatures, one needs a way to calculate the log e K T for each reaction. Having mostly only data on the entropy/enthalpy of reaction, extrapolations above 100°C will result in large errors. Until high quality data will be available, one can benefit from the isocoulombic method and the fact that these tetravalent actinide ions have fairly similar hydrated ionic radii, so that the temperature dependences between Th(IV) complexation and that of other tetravalent actinides largely cancel out. The reactions can be set up as follows.
serves as a model reaction with a well-defined temperature trend based on experimental data (see above), and for the remaining dependent species, using the following isocoulombic reactions (obtained by subtracting the model reaction from each remaining complexation reaction): In this case, one can calculate the logK T for reaction (30), which has a known temperature dependence, and the log i K T of the isocoulombic reactions (31)-(33) can be calculated just by using their log i K 298 and the one-term extrapo-lation (Eq. (7)), or using the enthalpy of reaction which in this case is equal to the Gibbs energy (Eq. (8)) ( Table 5). Considering the selected master and product species, the resulting four reactions (30)-(33) can be used for calculations from ambient temperature to 300°C, assuming that the temperature trends of properties of similar tetravalent actinides cancel out to a large extent. There is a considerable improvement compared to just using direct complexation reactions that have limited data with large errors when extrapolated above 100°C.
The uncertainties in calculated logK T from the different temperature extrapolations can be evaluated by simple error propagation in their respective Eqs. ((5a), (5b), (6a), (6b), (7), and (8)) (Kulik, 2002). When estimating log e K T for a reaction with unknown temperature dependence, two sources of uncertainties need to be considered, namely the uncertainity of parameters for the reaction of interest, and the uncertainity of parameters for the model reaction. For example, when using the isocoulombic method and the one-term A-type extrapolation, the uncertainty of the log i K T for the isocoulombic reaction is calculated by using the reported errors for log e K 298 of the investigated reaction and for log m K 298 of the model reaction. The total error for the estimated log e K T is then the sum of the reported error in log e K 298 and the uncertainty for the temperature dependence of the model reaction based on reported errors for D r S 298 , D r Cp 298 , and other possible temperature dependent parameters.

CONCLUSIONS
In this study, greatly facilitated by using dedicated databases and code tools, the isocoulombic method for estimating logK T temperature trends of lanthanide and actinide complexation has been systematically tested for different cations and ligands. Due to ion hydration, aqueous complexation and mineral ionic dissolution reactions usually have large entropy and large heat capacity effects. Hence, the direct usage of one-or two-term extrapolations, which set the entropy (or the heat capacity) of reaction or both to zero (e.g. as parameters from Table 1), will result in large deviations from correct trends even over moderate temperature intervals. In comparison, isocoulombic reactions can greatly reduce errors in log i K T extrapolations, especially when very limited thermodynamic data are available, For an aqueous complexation reaction with unknown temperature dependence, a suitable model reaction can be used to generate a well-balanced isocoulombic reaction having ions with the same oxidation state, coordination number, and similar ionic radii, hydration properties, and hydration shell structures of ionic species on both sides of the reaction. The one-term A-type extrapolation (with only log i K 298 known) can often be used to calculate log i K T values of the isocoulombic reactions at elevated temperatures, and, if the temperature dependence of the model reaction is available, the isocoulombic method can be used to estimate log e K T values of the unknown participating reactions requiring only their values for log e K 298 . In the latter case, the validity temperature range of the estimated values cannot exceed that of the model reaction. For a system with similar ions, a thermodynamic dataset of reactions for calculations at elevated temperatures can be constructed using a few model reactions having well known temperature dependence and the remaining reactions in isocoulombic form.
The systematic tests showed that the temperature dependence of complexation reactions of trivalent lanthanide ions can be used to estimate the properties of trivalent actinide ions and vice versa, as a result of almost identical hydrated ionic radii within the two groups. The one-term A-type extrapolation (D r S 298 , D r Cp 298 ¼ 0) was found to be useful in the case of isocoulombic reactions which exchange similar cations, whereas for the exchange of OH À and F À ligands, the isocoulombic reactions produced better estimates using the one-term B-type extrapolation (D r H 298 , D r Cp 298 ¼ 0) within in small temperature intervals (50-100°C) with small heat capacity effects. Similar results can be expected for other complexation reactions with ligands (e.g. OH À , Br À , I À , organic ligands), for which the data at elevated temperatures are not yet available and could so far not be tested, and possibly also for nonaqueous solutions.
The possibility of using isocoulombic reactions and analogous model reactions to estimate thermodynamic data at elevated temperatures is of great importance, since most of the available data for lanthanide and actinide complexation are restricted to 25°C. On the other hand, the method can also be used to estimate log e K 298 for complexes whose stabilities can only be measured at elevated temperatures, and can reveal possible inconsistencies in the available data. The isocoulombic method can be used to cover the existing gaps and to enhance the capabilities of thermodynamic calculations of chemical equilibria, until new experimental data on the temperature trends of complexation reactions become available.

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. : standard molar enthalpy of the substance (partial molal for aqueous species) S : standard molar entropy of the substance (partial molal for aqueous species) C P : standard molar heat capacity of the substance (partial molal for aqueous species) D r : reaction property prefix, for example D r S T is a standard entropy effect of the reaction at given temperature T N: any standard state molar property (partial molal property for aqueous species) three-term extrapolation: D r C P ;T ¼ const; D r S T ¼ const; D r G T ¼ const two-term extrapolation: D r C P ;T ¼ 0; D r S T ¼ const; D r G T ¼ const one-term A-type extrapolation: D r C P ;T ¼ 0; D r S T ¼ 0; D r G T ¼ const one-term B-type extrapolation: D r C P ;T ¼ 0; D r H T ¼ 0; D r S T ¼ const Ln: any given lanthanide element An: any given actinide element hd T i: mean of differences r d T ð Þ: standard deviation of the differences Associate editor: Robert H. Byrne