Lauriane Huguet

Domitile Lourdeaux

Nicolas Sabouret

Marie-Hélène Ferrer

Perturbed Communication in a Virtual Environment to Train Medical Team Leaders

Keywords: Virtual environment, medical team, non-technical skills, verbal communication

The VICTEAMS project aims at designing a virtual environment for training medical team leaders to non-technical skills. The virtual environment is populated with autonomous virtual agents who are able to make mistakes (in action or communication) in order to train rescue team leaders and to make them adaptive with all kinds of situations or teams.

Introduction and Context of the Research

Health care delivery in military conflict, in peacekeeping missions or in the aftermath of disaster, implies high stress environments with danger exposures, life-threatening events and high levels of work demand. Crisis and emergency risk communication remains a real challenge in these situations [START_REF] Glik | Risk communication for public health emergencies[END_REF]. The rapid response of emergency medical teams has an important role to play in preventing serious adverse events [START_REF] Jones | Effectiveness of the medical emergency team: the importance of dose[END_REF] but this need for rapid actions can also increase stress level. Elevated stress levels can reduce performance on tasks that require divided attention, working memory, retrieval of information from memory and decision making.

During critical events, medical errors can be related to human or system factors, including ineffective team leadership, non-standardized team communication, a lack of global situation awareness, poor use of resources and inappropriate triage and prioritization [START_REF] Petrosoniak | Beyond crisis resource management: new frontiers in human factors training for acute care medicine[END_REF]. Team training optimize the non-technical skills [START_REF] Petrosoniak | Beyond crisis resource management: new frontiers in human factors training for acute care medicine[END_REF] and team feedback influence behaviour and changing practices [START_REF] Frykman | Functions of behavior change interventions when implementing multi-professional teamwork at an emergency department: a comparative case study[END_REF]. The VICTEAMS (Virtual Characters for Team Training: Emotional, Adaptive, Motivated and Social) project aims at building a virtual environment for training non-technical skills of rescue team leaders. In this interactive learning environment, which is a work in progress, virtual characters play the role of the team members and interact with each other in a natural manner.

Virtual Reality provides compelling, engaging learning experiences that have been shown to reduce training time and increase training persistence.

In opposition to existing systems [START_REF] Gordon | Branching storylines in virtual reality environments for leadership development[END_REF][START_REF] Magerko | Improving interactive training through individualized content and increased engagement[END_REF][START_REF] Hill | Virtual Humans in the Mission Rehearsal Exercise System[END_REF], our idea by proposing virtual characters with varied behaviours, is not to train a team of technical experts but an expert and adaptable team. The parameters influencing collective decision aspects can then be controlled by a pedagogical director to tailor collective behaviours according to the dynamic profile of the learner. In the VICTEAMS project, we have to face the challenge of interpersonal verbal exchanges in the virtual environment and with the learner who uses the interface. In this article, we focus on communications and how aspects of autonomous characters can adapt and create consistent and relevant communications (i.e. ideal or erroneous communications).

The following section describes the technical setup and the communication architecture.

Technical Setup

The VICTEAMS platform is based on multi-agent system (MAS): each team member is represented by an autonomous agent that makes decision based on local information and communicates with other agents to synchronise. This MAS is coupled with the virtual environment in which the scene is played, so that the team leader -the only human interacting with autonomous agents in the scene -is immersed in the simulation.

Virtual Environment

The virtual environment reproduces a Field Hospital, located in a tent, with regular/standard medical material in cupboards. The victims are already placed on beds and the medical staff is scattered in the room, taking care of the victims. The profiles of the victims and medical staff can be changed by the trainer between training sessions to optimize adaptive skills of the learner. Figure 1 gives an overview of this virtual environment. The team leader can interact and move in the room like in a first-person game using a joy pad. A wheel is used to displays the different communication options if the user is facing a character, or actions if the player is facing objects.

In this paper, we focus on the communication between the team members: [START_REF] Driskell | Collective behavior and team performance[END_REF] has proven that it is a very important skill for teamwork.

Communication

Research in MAS and Artificial Intelligence (AI) has proposed several models for communication. The FIPA-ACL model [START_REF]Specification part 2: Agent communication language, Technical report[END_REF], is now widely used for agent coordination. One important property of AI model is that the illocutionary act (what is said in the message) is equal to the perlocutionary act (what is understood by the receiver), which ensures a correct and reliable global behaviour in the MAS. The originality of our research is to consider situations in which agents do not understand each other correctly, because of the stressful condition of their activity.

We have designed a new communication architecture to reproduce communication errors in an agent team. This model is based on both field observation (conducted by ergonomist) and on psychological study about stress, behaviour, critical situation, leadership and teamwork [START_REF] Flin | Safety at the sharp end: a guide to non-technical skills[END_REF][START_REF] St Pierre | Crisis Management in Acute Care Settings[END_REF]. When the receiver intercepts the message, he might not select the initial meaning.

Second, communication errors can appear at four different stages of the process. Each of the 4 stages of message transport correspond to different processing functions. While we cannot present all of them here, we can briefly show the difficulty they arise:

1. The alternative meanings have to be built automatically, based on a domain ontology and expert rules for message transformation. 2. The message has to be send to all agents within range of the sender, and altered differently for each agents, depending on the state of the channel between receiver and sender. 3. Agents can ignore a message or ask for a clearer one, depending on their stress and the degree of alteration of the message. We use the following polynomial to compute the probability that a message is accepted: 0.625s 2 -1.55d + 0.125s + 0.95d + 0.04, where s is the receiver's stress level (between 0 and 1) and d is the message alteration. 4. The receiver must correct the message using its domain ontology and its current task model. This part is still worked on.

This communication model has been implemented and a preliminary experiment has been conducted to evaluate the impact on the agent's behaviour. Another experiment with the complete simulator is to be conducted soon.

Figure 1 .

 1 Figure 1. The Virtual Field Hospital -designed by Reviatech

M

  = f (d, M, {channel}) M = ({P, {(C, channel)}}, S, {D})

Figure 2 .

 2 Figure 2. Overview of the communication model This model, presented on figure 2, relies on two major ideas: First, to support misinterpretation, and based on the idea proposed by [1,2], each message is composed of several (Per f ormative,Content) couples -instead of one in classical agent communication languages: Message = ({ Meaning }, Sender, { Addressees }), where Meaning = (Performative, { (content, channel) }).

Corresponding Author: Huguet Lauriane, Heudiasyc; E-mail: lauriane.huguet@hds.utc.fr

Summary and Future Work

The VICTEAMS project's system is a virtual environment for training non-technical skills, based on multi-agent systems. Our work focused on the communication in the team, which is a very important skill for teamwork [4]. Our next steps are to evaluate this communication mechanism in the complete simulation, and especially the impact of the communication behaviour on the team leader's non-technical skills training. We also want to integrate collective tasks in our systems, by developing a decision model for collective behaviour and teamwork supporting erroneous communications and partial knowledge of the situation by each agent.

The VICTEAMS project (ANR-14-CE24-0027) has received funding from French National Research Agency (ANR), the French Defence Procurement Agency (DGA).The authors would like to thank the Picardie region and the European Regional Development Fund (ERDF) 2014/2020 for the funding of this work. The institution involved are : Heudiasyc, Limsi, IRBA, EVDG, BSPP, Reviatech. The opinions or assertions expressed here in are the private views of the authors and are not to be considered as official or as reflecting the views of our organizations.


Adeline Lach
email: a.lach@brgm.fr

Laurent André
email: l.andre@brgm.fr

Arnault Lassin
email: a.lassin@brgm.fr

Mohamed Azaroual
email: m.azaroual@brgm.fr

Jean-Paul Serin
email: jean-paul.serin@univ-pau.fr

Pierre Cézac
email: pierre.cezac@univ-pau.fr

A new

Keywords: sodium hydroxide, Pitzer model, high salinities, thermodynamic excess properties

Introduction

Understanding the physico-chemical behaviour of natural and synthetic saline aqueous systems is very important for environmental and industrial purposes. Various models have been developed to describe such systems [START_REF] Li | A gE model for single and mixed solvent electrolyte systems: 1. Model and results for strong electrolytes[END_REF][START_REF] Chen | Local composition model for excess Gibbs energy of electrolyte systems. Part I: Single solvent, single completely dissociated electrolyte systems[END_REF][START_REF] Abovsky | Representation of nonideality in concentrated electrolyte solutions using the Electrolyte NRTL model with concentration-dependent parameters[END_REF][START_REF] Thomsen | Aqueous Electrolytes Model Parameters and Process Simulation[END_REF][START_REF] Zhao | Extension of the Wilson model to electrolyte solutions[END_REF]. Pitzer [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF] developed one of the most promising and frequently used models. Indeed, researchers have shown that it can represent the physico-chemical solution properties of very low to very high salinities [START_REF] Greenberg | The prediction of mineral solubilities in natural waters : A chemical equilibrium model for the system to high concentration from 0 to 250 ° C. Geochim[END_REF] in complex systems [START_REF] Harvie | The prediction of mineral solubilities in natural waters: The Na-K-Mg-Ca-H-Cl-SO 4 -OH-HCO 3 -CO 3 -CO 2 -H 2 O system to high ionic strengths at 25°C[END_REF], and at high temperatures [START_REF] Greenberg | The prediction of mineral solubilities in natural waters : A chemical equilibrium model for the system to high concentration from 0 to 250 ° C. Geochim[END_REF], and it has been applied to many different chemical systems ( [START_REF] Christov | Thermodynamics of formation of double salts and mixed crystals from aqueous solutions[END_REF], and references therein). The semi-empirical Pitzer approach considers specific interactions between dissolved species and uses interaction parameters, thermodynamic constants for aqueous complexation reactions, and mineral (respectively gas) solubility products for dissolution/precipitation (respectively dissolution/degassing) reactions. All of these parameters are included in thermodynamic databases that can be used by geochemical codes. However, extending these databases to include new species might be problematic in terms of numerical stability and/or model consistency, especially when the previously tabulated parameters are limited by their salinity range of validity. Fundamental systems should therefore be reviewed and updated to extend their thermodynamic characterisation to very high salinities and high temperatures.

Here, we consider the NaOH-H 2 O binary system because of the very high solubility of NaOH salts [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF]. This electrolyte has been widely studied both in the binary system NaOH-H 2 O and in mixtures. Pitzer and Mayorga [START_REF] Pitzer | Thermodynamics of electrolytes. II. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent[END_REF] worked on this binary system up to 6 mol•kgw -1 at 298.15 K. Harvie et al. [START_REF] Harvie | The prediction of mineral solubilities in natural waters: The Na-K-Mg-Ca-H-Cl-SO 4 -OH-HCO 3 -CO 3 -CO 2 -H 2 O system to high ionic strengths at 25°C[END_REF] predicted solid-liquid equilibria in the Na-K-Mg-Ca-H-Cl-SO 4 -OH-HCO 3 -CO 3 -CO 2 -H 2 O systems at 298.15 K up to 6 mol•kgw -1 . Pabalan and Pitzer [START_REF] Pabalan | Thermodynamics of concentrated electrolyte mixtures and the prediction of mineral solubilities to high temperatures for mixtures in the system Na-K-Mg-Cl-SO 4 -OH-H 2 O. Geochim[END_REF] predicted solid-liquid equilibria in the Na-K-Mg-Cl-SO 4 -OH-H 2 O at high temperature.

Pabalan and Pitzer [START_REF] Pabalan | Thermodynamics of NaOH(aq) in hydrothermal solutions[END_REF] provided parameters for the binary system NaOH-H 2 O between 273.15-623.15 K and up to 10 mol•kgw -1 using both osmotic coefficient data and volumetric data such as apparent molar volumes. Simonson et al. [START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF] deduced, from heat capacity and dilution enthalpy measurements in the NaOH-H 2 O system, the osmotic coefficient between 273.15 K and 523.15 K and up to 6 mol•kgw -1 . They represented these data with Pitzer's model and introduced the interaction parameter β (2) in order to improve their fit. Petrenko and Pitzer [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF] measured vapor pressure and deduced water activity values. Using a modified version of the original Pitzer model, based on a mole fraction scale instead of a molality scale, they modeled these data between 273.15 and 523.15 K and for molalities ranging from 0 to 249.786 mol•kgw -1 . Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF] worked on the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system at high temperature (523.15 K). The parameters for the NaOH-H 2 O system are valid up to 6 mol•kgw -1 . The excess properties of NaOH solutions are described with accuracy up to this concentration limit. In this paper, the authors also underline the non validity of their parameters at higher concentration. However these authors tested with success their parameters to determine the solubility of some hydrated salts up to about 60°C and 50 mol•kgw -1 .

The work described here goes further and aims to develop a set of parameters able to represent both the excess properties (e.g. osmotic coefficient, activity coefficient) and the mineral solubility in the NaOH-H 2 O binary system to very high salinities, from 273.15 to 523.15 K, using the Pitzer approach. The originality of this study is to propose a model valid up to salt saturation between 273.15 and 523.15 K. Up to our knowledge, no Pitzer model exists on a such system up to so high concentrations (higher than 30 mol•kgw -1 ). Its compatibility with more complex systems is checked on the LiOH-NaOH-H 2 O and NaOH-NaCl-H 2 O ternary systems.

Theoretical background

Pitzer [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF] developed a semi-empirical thermodynamic model to represent the excess properties of electrolyte solutions. He improved on the work of Debye-Hückel [START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF], whose equation is valid for a low range of molalities, to extend its field of application. Pitzer [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF] and Pitzer and Mayorga [START_REF] Pitzer | Thermodynamics of electrolytes. II. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent[END_REF] provided specific interaction parameters (3 for binary systems:  0) ,  1) and C  ) for 1-1, 2-1, 3-1, 4-1 and 5-1 type electrolytes for maximum molalities of 6 mol•kgw -1 , assuming total dissociation of the electrolytes. Pitzer and Mayorga [START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF] then showed that an additional parameter ( 2) ) is needed to represent the behaviour of the 2-2 type electrolytes above 0.1 mol•kgw -1 . Pitzer and Kim [START_REF] Pitzer | Thermodynamics of electrolytes. IV. Activity and osmotic coefficients for mixed electrolytes[END_REF] extended the model to include mixed electrolytes. A few years later, Kim and Frederick [START_REF] Kim | Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25.degree.C. 1. Single salt parameters[END_REF] provided ion interaction parameters for 304 single salts and for all types of electrolytes. These parameters apply at 298.15 K and generally up to saturation (for instance for NaOH up to 29 mol•kgw -1 ). They could give a similar result as the one obtained in case 1 of this study (see below). In order to improve Pitzer's model and its ability to accurately describe experimental data, authors have included virial coefficients of higher orders than C φ ( [START_REF] Ananthaswamy | Thermodynamics of concentrated electrolyte mixtures. 5. A review of the thermodynamic properties of aqueous calcium chloride in the temperature range 273.15[END_REF][START_REF] Anstiss | Thermodynamics of very concentrated aqueous electrolytes: LiCl, ZnCl 2 , and ZnCl 2 -NaCl at 25°C[END_REF][START_REF] Pitzer | Thermodynamics of Electrolytes. 13. Ionic Strength Dependence of Higher-Order Terms; Equations for CaCl 2 and MgCl 2[END_REF]). Ananthaswamy and Atkinson [START_REF] Ananthaswamy | Thermodynamics of concentrated electrolyte mixtures. 5. A review of the thermodynamic properties of aqueous calcium chloride in the temperature range 273.15[END_REF] introduced three higher order virial coefficients (D φ , E φ and F φ ) to represent the excess properties (osmotic coefficient, apparent molal enthalpy and apparent molal heat capacity) of CaCl 2 between 273.15 and 373.15 K up to 9 mol•kgw -1 . Anstiss and Pitzer [START_REF] Anstiss | Thermodynamics of very concentrated aqueous electrolytes: LiCl, ZnCl 2 , and ZnCl 2 -NaCl at 25°C[END_REF] determined interaction parameters for LiCl up to 20 mol•kgw -1 by including D φ and E φ parameters. Pitzer et al. [START_REF] Pitzer | Thermodynamics of Electrolytes. 13. Ionic Strength Dependence of Higher-Order Terms; Equations for CaCl 2 and MgCl 2[END_REF] also introduced an ionic strength dependence in the higher order virial coefficients for CaCl 2 and MgCl 2 systems. But when pushed to more than 27 mol•kgw -1 , these models which rely on the full dissociation hypothesis for the electrolyte, the number of charged species exceeds that of water. The paradoxical situation takes place where charged species are in direct contact with each other but do not form a complex because of the initial hypothesis of full dissociation. Physically, this is hardly justifiable. Thus, some electrolytes cannot be described if we only consider total dissociation. Indeed, rather than high order virial coefficients, Pitzer and Silvester [START_REF] Pitzer | Thermodynamics of electrolytes. VI. Weak electrolytes including H 3 PO 4[END_REF] introduced a neutral species that comes from the partial dissociation of phosphoric acid. Later, several authors successfully introduced the H 3 PO 4 0 (aq) species [START_REF] Holmes | Isopiestic Studies of H 3 PO 4 (aq) at Elevated Temperatures[END_REF][START_REF] Jiang | Thermodynamics of aqueous phosphoric acid solution at 25°C[END_REF][START_REF] Cherif | Modelling of the equilibrium properties of the system H 3 PO 4 -H 2 O: representation of VLE and liquid phase composition[END_REF] to describe the behaviour of the phosphoric acid system up to 24 mol•kgw -1 using either the original Pitzer's model or the modified Pitzer's model [START_REF] Lassin | A thermodynamic model of aqueous electrolyte solution behavior and solid-liquid equilibrium in the Li-H-Na-K-Cl-OH-LiCl 0 (aq)-H2O system to very high concentrations (40 molal) and from 0 to 250[END_REF]. In our model for NaOH-H 2 O at high temperature and high salinity, it was necessary to introduce the neutral species NaOH 0 (aq).

However, considering the existence of a neutral species in solution is not inconsequential and particular attention must be paid to the meaning of the available experimental data, especially when these have been acquired with isopiestic or electromotive force (e.m.f.) methods. For example, osmotic coefficient data given in the literature (𝜙 𝑙𝑖𝑡  generally consider full dissociation of the electrolytes. But when using a geochemical calculation code, the osmotic coefficients of a solution (𝜙) and other excess properties (such as mean activity coefficients) are calculated by considering the detailed speciation of the solution. This is the case of PHREEQC v2.18 [START_REF] Parkhurst | User's guide to PHREEQC (Version 2) : a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF], which is the geochemical calculation code we used in this study. In order to be able to compare the modelled values with literature data, we had to correct the calculated osmotic coefficient by taking the partial dissociation into account. The osmotic coefficient (𝜙) was then calculated using the equation of Clegg and Whitfield [START_REF] Clegg | Activity coefficient in natural waters[END_REF] who developed the Pitzer's model equations for a system containing an indefinite number of both neutral and ionic solutes (see the Online Resource 1). The correction is as follows:

𝜙 ′ = 𝜙 ∑ 𝑚 𝑖 𝑖 ∑ 𝜈 𝑖 𝑚 𝑖 𝑖 ⁄ (1a)
where 𝜈 𝑖 is the number of ions formed by the complete dissociation of one molecule of solute and 𝑚 𝑖 is the molality of solute in mol•kgw -1 . In the case of the NaOH-H 2 O system:

𝜙 ′ = 𝜙 (𝑚 𝑁𝑎+ + 𝑚 𝑂𝐻-+ 𝑚 𝑁𝑎𝑂𝐻 ) (𝑚 𝑁𝑎+ + 𝑚 𝑂𝐻-+ 2 * 𝑚 𝑁𝑎𝑂𝐻 ) ⁄ (1b)
We can now compare 𝜙 ′ with the literature data (𝜙 𝑙𝑖𝑡 . The mean activity coefficient also had to be corrected. Indeed, like the osmotic coefficient, the activity coefficient provided in the literature ( lit ) is based on the total dissociation of the electrolyte. In order to compare literature data and PHREEQC calculations (𝛾 ± ), we calculated the corrected activity coefficient (𝛾 ± ′ ) using the following equation [START_REF] Holmes | Isopiestic Studies of H 3 PO 4 (aq) at Elevated Temperatures[END_REF]:

ln 𝛾 ± ′ = ( 𝑚 𝑁𝑎+ 𝑚 𝑁𝑎+ + 𝑚 𝑁𝑎𝑂𝐻 ) ln 𝛾 ± (2a) ln 𝛾 ± = (ln 𝛾 𝑁𝑎+ + ln 𝛾 𝑂𝐻-) 2 ⁄ (2b)
The activity of water does not need to be corrected. It is calculated in coherence with the aqueous speciation using the following equation:

ln 𝑎 𝑤 = (-𝜙𝑀 𝑤 ∑ 𝑚 𝑖 𝑖 ) 1000 ⁄ (3)
where 𝑀 𝑤 is the molar mass of water (18.0153 g•mol -1 ).

Model parameterization and results

3.1.Methodology

We used the theories described above to develop a two-step procedure. We first developed the simplest model and then increased its complexity until the model results matched the available experimental data (e.g. for osmotic coefficient or water activity). We started parameter fitting using the fully dissociated approach considering only 3 parameters: β (0) , β (1) (with its associated parameter α 1 equal to 2) and C  . If this model was not able to correctly reproduce the experimental thermodynamic properties, we added another adjustable interaction parameter β (2) and its adjusted associated parameter, α 2 . In this last case, the parameter α 1 is equal to 1.4 (see the Online Resource 1 which describes the Pitzer equations) [START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF]. Several authors reported that when they are unable to reproduce excess properties on the concentration and temperature ranges under consideration, they introduce the binary interaction parameter β (2) and the associated parameter α 2 rather than considering partial dissociation [START_REF] Christov | Thermodynamics of formation of double salts and mixed crystals from aqueous solutions[END_REF][START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF]. The β (2) parameter was first introduced by Pitzer and Mayorga [START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF] to describe the aqueous properties of 2-2 electrolytes. In Pitzer's classic theory, α 1 = 1.4 and α 2 = 12 [START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF]. However, to obtain a better fit, Filippov [START_REF] Filippov | Thermodynamics of the systems Ni//Cl, SO 4 -H 2 O and Co//Cl, SO 4 -H 2 O at 25[END_REF] proposed a value of 1 or -1, and Christov [START_REF] Christov | Thermodynamics of formation of double salts and mixed crystals from aqueous solutions[END_REF] considered it to be an adjustable parameter. If these α variations cannot reproduce the experimental results over the entire range of concentrations and temperatures under consideration, the alternative used by Pitzer and Silvester [START_REF] Pitzer | Thermodynamics of electrolytes. VI. Weak electrolytes including H 3 PO 4[END_REF] consists in introducing a neutral and/or aqueous complex species together with the associated binary and ternary interaction parameters, and its thermodynamic equilibrium constant (K) of formation. In this case, the β (2) and associated  2 parameters for the charged species are redundant and must be removed from the model.

We coupled an optimization software with PHREEQC v2.18 [START_REF] Parkhurst | User's guide to PHREEQC (Version 2) : a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] to estimate the parameters after modifying the temperature dependence of interaction parameters (β (0) , β (1) ,C  , λ NaOH/NaOH and ζ NaOH/Na+/OH-) so that the sets of parameters of both Møller [START_REF] Møller | The prediction of mineral solubilities in natural waters: A chemical equilibrium model for the Na-Ca-Cl-SO 4 -H 2 O system, to high temperature and concentration[END_REF] and Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF]

where 𝑇 𝑘 is the temperature in Kelvin and 𝑎 𝑖 are adjustable parameters.

The equilibrium constants of the reactions are also dependant on temperature according to the following equation: (5)

Application to the binary NaOH-H 2 O system

Data selection

The data used for the parameterization of the NaOH-H 2 O system are osmotic coefficients and water activity measurements issued from literature (Fig. 1 and Table 1). Osmotic coefficient data for NaOH solutions at 298.15 K are taken from Hamer and Wu [START_REF] Hamer | Osmotic Coefficients and Mean Activity Coefficients of Uniunivalent Electrolytes in Water at 25°C[END_REF] for molalities ranging from 0.001 to 29 mol•kgw -1 . These authors compiled isopiestic data from Robinson and Stokes [START_REF] Robinson | Tables of osmotic and activity coefficients of electrolytes in aqueous solution at 25°C[END_REF], and vapour pressure data from Kangro and Groeneveld [START_REF] Kangro | Konzentrierte wäßrige Lösungen, I[END_REF]. Simonson et al.

[14] also provided osmotic coefficients for NaOH molalities ranging from 0.01 to 6 mol•kgw - 1 and temperatures from 273.15 to 523.15 K, calculated using Pitzer interaction parameters optimized from heat capacity and heat of dilution measurements. Water activity data are taken from Petrenko and Pitzer [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF] for molalities ranging from 0.28 to 249.786 mol•kgw -1 between 273.15 and 523.15 K. Petrenko and Pitzer [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF] computed water activity data from measured vapour pressures [START_REF] Kangro | Konzentrierte wäßrige Lösungen, I[END_REF][START_REF] Dibrov | Saturated vapor pressure of caustic soda and sodium aluminate solutions in the 25-350°C temperature range over a wide range of concentrations[END_REF][START_REF] Campbell | Osmotic and activity coefficients of sodium hydroxide in water from 150 to 250°C[END_REF][START_REF] Krumgalz | Vapor pressure of NaOH solutions (of over 45% concentration) at temperatures up to 400[END_REF].

Holmes and Mesmer [START_REF] Holmes | Isopiestic molalities for aqueous solutions of the alkali metal hydroxides at elevated temperatures[END_REF] also gave osmotic coefficient data for sodium hydroxide systems up to 7 mol•kgw -1 at three temperatures (383.25, 413.12 and 443.09 K). These authors measured osmotic coefficients directly using the isopiestic method, with NaCl as the reference electrolyte. Because of the limited range of concentrations (up to 7 mol.kgw -1 ) of this latter set of experimental data, we decided not to use it. Indeed, in the optimization process, we worked temperature by temperature. Consequently, if we had optimized parameters on these data at one temperature, the resulting interaction parameters would have been valid up to 7 mol.kgw -1 only. Therefore, this set of parameters would have not been coherent with the sets of parameters determined for other temperatures. The activity coefficient data are not used for the optimisation. But the values obtained with our model are compared with values provided by literature [START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF][START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF][START_REF] Hamer | Osmotic Coefficients and Mean Activity Coefficients of Uniunivalent Electrolytes in Water at 25°C[END_REF][START_REF] Robinson | Tables of osmotic and activity coefficients of electrolytes in aqueous solution at 25°C[END_REF][START_REF] Holmes | Isopiestic molalities for aqueous solutions of the alkali metal hydroxides at elevated temperatures[END_REF].

Solubility data for NaOH.nH 2 O minerals were from Pickering [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF], who identified anhydrous NaOH(cr) and eight hydrates of sodium hydroxide (1-2-3.11-3.5-4α-4β-5-7). For temperatures above 273.15 K only anhydrous NaOH(cr) and five hydrates can exist (1-2-3.11-3.5-4α.

Pickering's [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF] measurements were verified by Cohen-Adad et al. [START_REF] Cohen-Adad | Le système eauhydroxyde de sodium[END_REF]. Experimental invariant points are reported in Table 2 for comparison. The principal difference is for NaOH:3.5 H 2 O + NaOH:H 2 O. These data from Pickering [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF] and Cohen-Adad et al. [START_REF] Cohen-Adad | Le système eauhydroxyde de sodium[END_REF] are compared with those compiled by Zdanovskii [START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF] and Linke [START_REF] Linke | Solubilities inorganic and metal-organic compounds[END_REF] for ternary systems. The sets of data are coherent. For anhydrous NaOH(cr) we used those provided by Cohen-Adad et al. [START_REF] Cohen-Adad | Le système eauhydroxyde de sodium[END_REF], Linke [START_REF] Linke | Solubilities inorganic and metal-organic compounds[END_REF] and Zdanovskii [START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF].

Model parameterization

Our objective was to determine the excess properties of NaOH solutions up to 523.15 K and from pure water to saturation with respect to the most soluble phase of NaOH.nH 2 O hydrate minerals. In a first step, we began with the simplest model at 298.15 K only, simply considering the three interaction parameters β (0) Na+/OH-, β (1) Na+/OH-(α 1 =2) and C  Na+/OH-(Case 1). In this first case, the calculated standard deviation, σ (calculated on 76 measured data from [START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF], [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF], [START_REF] Hamer | Osmotic Coefficients and Mean Activity Coefficients of Uniunivalent Electrolytes in Water at 25°C[END_REF]-Table 1), on the osmotic coefficient is 0.0808. This model did not correctly represent osmotic coefficient data throughout the entire domain of molalities at 298.15 K. The adjustable interaction parameter β (2) Na+/OH-was therefore added with α 1 =1.4 and α 2 =12 (Case 2, σ=0.0805). As described in the methodology part the α 2 value was also tentatively optimized whereas α 1 is kept equal to 1.4. The best fit is obtained with α 2 =0.695 (Case 3, σ=0.0285). Finally, we removed the β (2) term and we included the neutral species NaOH 0 (aq) in the speciation calculation (Case 4, σ=0.00354). The equilibrium constant (K NaOH ) and two additional ternary (ζ Na+/OH-/NaOH ) and binary (λ NaOH/NaOH ) interaction parameters were added to β (0) Na+/OH-, β (1) Na+/OH-(α 1 =2) and C ϕ Na+/OH-. Several other studies [START_REF] Lindsay | Chemistry of steam cycle solutions: principles[END_REF][START_REF] Chen | Enthalpy of dissociation of water at 325°C and LogK, ΔH, ΔS, and ΔCp values for the formation of NaOH(aq) from 250 to 325°C[END_REF][START_REF] Ho | Ion association of dilute aqueous sodium hydroxide solutions to 600°C and 300 MPa by conductance measurements[END_REF][START_REF] Ho | Conductivity measurements of dilute aqueous LiOH, NaOH, and KOH solutions to high temperatures and pressures using a flow-through cell[END_REF][START_REF] Fuangswasdi | Enthalpies of dilution of NaOH, KOH, and HCl and thermodynamic quantities for the formation of these species from their constituent ions in aqueous solution from 300[END_REF][START_REF] Bianchi | Electrical conductivity of aqueous sodium hydroxide solutions at high temperatures[END_REF][START_REF] Gimblett | Emf studies of electrolytic dissociation. Part 7.-Some alkali and alkaline earth metal hydroxides in water[END_REF][START_REF] Sharma | Ionic interaction in alkali metal hydroxide solutions -A raman spectral investigation[END_REF] consider the neutral species NaOH 0 (aq). In particular, Sharma and Kashyap [START_REF] Sharma | Ionic interaction in alkali metal hydroxide solutions -A raman spectral investigation[END_REF] have experimentally identified this ion pair in concentrated aqueous solutions of sodium hydroxide, using a Raman spectrometer. The lowest standard deviation is obtained for Case 4 when the partial dissociation of NaOH is considered. Moreover, at 298.15 K, this approach also enables us to accurately represent the mean activity coefficient of 52 data points ([33]-Table 3).

In the following step, we therefore used the same optimization approach at temperatures other than 298.15 K. The interaction parameters were adjusted using both the osmotic coefficients given by Simonson et al. [START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF] up to 6 mol•kgw -1 and the water activity data given by Petrenko and Pitzer [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF] up to saturation with respect to hydrated minerals. The results cover a wide range of temperatures, from 273.15 to 523.15 K.

The osmotic coefficient and water activity can be represented up to x 2 =0.82 (≈ 250 mol•kgw - 1 ) (Fig. 2), where x 2 is the molar fraction of total of Na in the aqueous solution, defined by:

𝑥 2 = 𝑛 𝑡𝑜𝑡 𝑁𝑎 (𝑛 𝑡𝑜𝑡 𝑁𝑎 + 𝑛 𝑤 ) ⁄ (6)
where 𝑛 𝑤 is the number of moles of water.

The osmotic coefficients from Petrenko and Pitzer [START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF] were calculated from water activity data using a rearrangement of Eq. 3. Here, the osmotic coefficient is calculated assuming total dissociation and we could compare these values with model results. They are consistent with Simonson et al.'s data [START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF].

The standard deviation (σ) obtained with our model for each experimental data set of osmotic coefficients is given in Table 1. Values of σ illustrate the good match both for the data used for the parameterization and for the data used from Holmes and Mesmer [START_REF] Holmes | Isopiestic molalities for aqueous solutions of the alkali metal hydroxides at elevated temperatures[END_REF]. The regression coefficients of Eq. 4 are given in Table 4.

The mean activity coefficient of NaOH was also calculated according to temperature. The speciation of the aqueous solution and the partial dissociation were taken into account according to Eq. 2. The mean activity coefficients calculated with this equation are given in

NaOH.nH 2 O(cr)  Na + + OH -+ nH 2 O
The temperature functions of the equilibrium constants are plotted in Fig. 4 and the corresponding coefficients (see Eq. 5) are given in Table 5. On Fig. 4, the lines are the values determined in this study. The white symbols are the values determined by Christov and

Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF] and the grey symbols were calculated using the standard thermodynamic properties of formation (ΔH°f, S° and C p ) of NaOH.H 2 O provided by Wagman et al. [START_REF] Wagman | The NBS tables of chemical thermodynamic properties: Selected values for inorganic and C1 and C2 organic substances in SI units[END_REF]. The solubility of the different hydrated NaOH salts in water is plotted against temperature on Fig. 5. Model calculations (lines) are consistent with experimental data (symbols) from Pickering [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF] and the invariant points (black symbols) are well reproduced by calculations (see also

Testing against boiling point, ternary systems and discussion

4.1.Boiling point elevation

The parameterization of the NaOH-H 2 O binary system was tested by calculating an independent property that is the boiling point elevation of this system depending on the NaOH concentration. The boiling point elevation,Θ 𝐵 , is equal to the difference between the boiling temperature (𝑇 𝐵 ) of a given solution and that of the pure solvent at the same pressure. Ge and Wang [START_REF] Ge | Estimation of Freezing Point Depression, Boiling Point Elevation, and Vaporization Enthalpies of Electrolyte Solutions[END_REF] proposed an equation in coherence with the Pitzer theory. They demonstrated that the boiling point elevation can be calculated with the following equation:

Θ 𝐵 = -∆𝐻 0,𝑇 𝐵 𝑣𝑎𝑝 -2𝑅𝑇 𝐵 ln 𝑎 𝑙𝑖𝑞 + √ (∆𝐻 0,𝑇 𝐵 𝑣𝑎𝑝 ) 2 -2Δ𝐶 𝑝 𝑣𝑎𝑝 𝑇 𝐵 2 𝑅 ln 𝑎 𝑙𝑖𝑞 2 ( ∆𝐻 0,𝑇 𝐵 𝑣𝑎𝑝 𝑇 𝐵 + 0.5Δ𝐶 𝑝 𝑣𝑎𝑝 + 𝑅 ln 𝑎 𝑙𝑖𝑞 ) (7)
where ∆𝐻 0,𝑇 𝐵 𝑣𝑎𝑝 is the enthalpy of vaporization of a pure solvent at its normal boiling point 𝑇 𝐵 and Δ𝐶 𝑝 𝑣𝑎𝑝 is the difference in heat capacity between the vapor (C p The boiling point elevation was determined by iteration. We set an arbitrary initial value for Θ 𝐵 and the corresponding water activity was calculated with PHREEQC. By applying Eq. 8,

we obtained a new value for Θ 𝐵 . If this value was different from the previous one, it was used to recalculate the water activity. This iterative procedure was continued as long as the boiling point rise obtained at two successive steps were different.

The results of these calculations are given in Fig. 6 and compared with the experimental data.

Kirschstein [START_REF] Kirschstein | Physikalische Eigenschaften der festen Salze[END_REF] provided boiling point values for molalities ranging from 1.31 to 141.7 mol•kgw -1 . The range of molalities studied by Bialik et al. [START_REF] Bialik | Boiling Point Rise Calculations in Sodium Salt Solutions[END_REF] is from 0.9876 to 9.77 mol•kgw -1 . We also plotted the values predicted with the Dühring equation [START_REF] Haltenberger | Enthalpy-Concentration Charts from Vapor Pressure Data[END_REF]. This empirical law assumes that the boiling point of a given solution is a linear function of the boiling point of pure water at the same pressure. We then calculated the boiling point elevation with the interaction parameters of Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF]. The various boiling point elevation values are given in Online Resource 2. The interaction parameters defined in this study can accurately predict the boiling point elevation up to 75 mol•kgw -1 without any further optimization.

4.2.Ternary system NaOH-NaCl-H 2 O

To check the validity of our interaction parameters for the NaOH binary system (Table 4), we studied the ternary system NaOH-NaCl-H 2 O and the solubility of halite in NaOH solutions as a function of temperature. The interaction parameters of the binary system NaCl-H 2 O are provided by Møller [START_REF] Møller | The prediction of mineral solubilities in natural waters: A chemical equilibrium model for the Na-Ca-Cl-SO 4 -H 2 O system, to high temperature and concentration[END_REF] who considers NaCl to be completely dissociated (Table 6). The binary parameter interaction θ OH-/Cl-is given by Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF]. The equilibrium constant of halite is provided by Chase [START_REF] Chase | NIST-JANAF Thermochemical Tables[END_REF], consistent with the Thermoddem database [START_REF] Blanc | Thermoddem a database devoted to waste minerals[END_REF].

Only two additional interaction parameters (ψ Na+/Cl-/OH-and ζ Na+/Cl-/NaOH ) were optimized. The results are shown in Fig. 7. Experimental solubility data come from several authors and were compiled by Linke [START_REF] Linke | Solubilities inorganic and metal-organic compounds[END_REF] and Zdanovskii et al. [START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF]. By optimizing the ternary parameters With this full parameterization, we calculated osmotic coefficients for the ternary system NaCl-NaOH-H 2 O at three temperatures (298.15, 323.15 and 373.15 K) (Fig. 8). They are similar to the isopiestic measurements of Königsberger et al. [START_REF] Königsberger | Zdanovskii's Rule and Isopiestic Measurements Applied to Synthetic Bayer Liquors[END_REF]. The standard deviations are 0.0113, 0.0069 and 0.0202 for 298.15, 323.15 and 373.15 K, respectively. Harned and Harris [START_REF] Harned | The activity coefficients of sodium and potassium hydroxides in their corresponding chloride solutions at high constant total molality[END_REF] and Harned and Cook [START_REF] Harned | The Activity and Osmotic Coefficients of Some Hydroxide-Chloride Mixtures in Aqueous Solution[END_REF] recommended activity coefficient values in ternary solutions and for four total constant molalities (m NaOH + m NaCl ) (Fig. 9). Our calculations give standard deviations for the mean activity coefficients of NaOH of 0.00489, 0.00544, 0.0112 and 0.0429 for the total molalities of 0.5, 1, 3 and 5 mol•kgw -1 , respectively. For the mean activity coefficients of NaCl, we obtained standard deviations of 0.008 and 0.00609 at total molalities of 0.5 and 1 mol•kgw -1 , respectively.

4.3.Ternary system NaOH-LiOH-H 2 O

We also studied the ternary system NaOH-LiOH-H 2 O. This system was studied by Lassin et al. [START_REF] Lassin | A thermodynamic model of aqueous electrolyte solution behavior and solid-liquid equilibrium in the Li-H-Na-K-Cl-OH-LiCl 0 (aq)-H2O system to very high concentrations (40 molal) and from 0 to 250[END_REF] using the interaction parameters provided by Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF] for the NaOH-H 2 O system. Because of the concentration limits of this set of parameters, we reconsidered the system and used the NaOH parameterisation determined in this study. The interaction parameters of the binary system LiOH-H 2 O come from Lassin et al. [START_REF] Lassin | A thermodynamic model of aqueous electrolyte solution behavior and solid-liquid equilibrium in the Li-H-Na-K-Cl-OH-LiCl 0 (aq)-H2O system to very high concentrations (40 molal) and from 0 to 250[END_REF] who consider dissolved LiOH to be completely dissociated (additional data are given in Online Resource 2).

Few experimental data exist on this system [START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF][START_REF] Zdanovskii | Experimental solubility data on salt -water systems. Vol. 2, four component and more complex systems[END_REF] and these authors publish inconsistent values. Some of these authors gave solubility data for the NaOH-H 2 O system also. This enabled us to compare their values with the data from Pickering [START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF] and to choose the set of data which seems to be the most coherent namely Evteeva (1965) and Itkina et al. (1963,1966,1968) compiled by Zdanovskii et al. [START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF].

Two mixing parameters (ψ Li+/Na+/OH-and λ Li+/NaOH ) were added to the model and were optimised on solubility data of LiOH in NaOH solutions as a function of temperature (303.15, 323.15, 373.15 and 423.15 K) (Table 8). At moderately high molalities (around NaOH = 10-15 mol•kgw -1 ), only the ternary interaction parameter ψ Li+/Na+/OH-has an influence on the calculated results since the concentration of the NaOH 0 (aq) ion pair is small (see previous sections). We therefore first optimised this parameter before adjusting λ Li+/NaOH to tentatively extend the representation of the solubility of LiOH in NaOH solutions to the entire molality range (Fig. 10). Calculation results are consistent with experimental data except at 373.15 K where the predicted LiOH solubility is lower than measured data. However, there is too little experimental data to draw meaningful conclusions.

This new parameterization greatly improves the thermodynamic description of the LiOH-NaOH-H 2 O system by Lassin et al. [START_REF] Lassin | A thermodynamic model of aqueous electrolyte solution behavior and solid-liquid equilibrium in the Li-H-Na-K-Cl-OH-LiCl 0 (aq)-H2O system to very high concentrations (40 molal) and from 0 to 250[END_REF]. However, due to the scarcity of measured data, we are unable to further improve the model developed in this study. Additional experimental data are needed.

4.4.Discussion

The results obtained on the NaOH-H 2 O system show that the proposed set of parameters is able to represent most of the excess properties up to saturation between 273.15 and 523.15 K.

Using the classical Pitzer equations, a good match between numerical results and measurements cannot be achieved without the use of the neutral ion pair NaOH 0 (aq) in the speciation calculations. Pokrovskii and Helgeson [START_REF] Pokrovskii | Thermodynamic properties of aqueous species and the solubilities of minerals at high pressures and temperatures : the system Al 2 O 3 -H 2 O-NaCl[END_REF] It is questionable whether or not such complex species (NaOH 0 (aq)) can be used in the Pitzer theory since Pitzer [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF] originally developed his model considering the electrolyte to be fully dissociated. This is valid for many salts and strong acids and bases. However, a few years later Pitzer, extended his model [START_REF] Pitzer | Thermodynamics of electrolytes. VI. Weak electrolytes including H 3 PO 4[END_REF] to solutions using a dissociation equilibrium. In some cases, Pitzer and Silvester [START_REF] Pitzer | Thermodynamics of electrolytes. VI. Weak electrolytes including H 3 PO 4[END_REF] showed that the partial dissociation of the electrolyte needed to be taken into account and that neutral species (and ionic aqueous complexes) could be added to the speciation calculations. To do this, they took the example of phosphoric acid, which they considered to be partially dissociated. The neutral aqueous species H 3 PO 4 0 (aq) was added to the model together with the equilibrium constant of the corresponding dissociation reaction, and parameters for the interaction between H 3 PO 4 0 (aq) and the other species in solution. The equations are similar to the original model. They simply include a correction of the osmotic coefficient to integrate the partial dissociation. This model was improved by Cherif et al. [START_REF] Cherif | Modelling of the equilibrium properties of the system H 3 PO 4 -H 2 O: representation of VLE and liquid phase composition[END_REF], who introduced additional complex ions based on Raman spectroscopy measurements [START_REF] Cherif | A New Investigation of Aqueous Orthophosphoric Acid Speciation Using Raman Spectroscopy[END_REF].

Pitzer et al. [START_REF] Pitzer | Thermodynamics of electrolytes. 7. Sulfuric acid[END_REF] also used sulphuric acid as another example of the integration of ionic association. They assumed that the second dissociation was partial and they added the aqueous complex HSO 4 -(aq). This approach was used later by Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF].

Felmy et al. [START_REF] Felmy | An aqueous thermodynamic model for polymerized silica species to high ionic strength[END_REF] successfully developed a model for polymerized silica species at high ionic strength based on the classical Pitzer equations. Møller et al. [START_REF] Møller | Thermodynamic model for predicting interactions of geothermal brines with hydrothermal aluminium silicate minerals[END_REF] added the neutral aqueous species Al(OH) 3 0 (aq) to describe the solubility of gibbsite and boehmite over a wide pH range.

On the other hand, Møller [START_REF] Møller | The prediction of mineral solubilities in natural waters: A chemical equilibrium model for the Na-Ca-Cl-SO 4 -H 2 O system, to high temperature and concentration[END_REF] studied the Na-Ca-Cl-SO 4 -H 2 O system between 298.15 and 523.15 K and up to 8 mol•kgw -1 . She introduced the species CaSO 4 0 (aq) to reproduce the solubility data. She determined the dissociation constant but fixed the interaction parameter with this neutral species equal to zero. When Greenberg and Møller [START_REF] Greenberg | The prediction of mineral solubilities in natural waters : A chemical equilibrium model for the system to high concentration from 0 to 250 ° C. Geochim[END_REF] extended the model of Møller [START_REF] Møller | The prediction of mineral solubilities in natural waters: A chemical equilibrium model for the Na-Ca-Cl-SO 4 -H 2 O system, to high temperature and concentration[END_REF], they retained the species CaSO 4 0 (aq). With such an approach, the extension of the chemical system to other elements became very difficult. This led Christov [START_REF] Christov | Temperature variable chemical model of bromide-sulfate solution interaction parameters and solid-liquid equilibria in the Na-K-Ca-Br-SO 4 -H 2 O system[END_REF] to Finally, the Pitzer model can combine an association equilibrium with solute interactions treated by using virial coefficients [START_REF] Pitzer | Thermodynamics of electrolytes. VI. Weak electrolytes including H 3 PO 4[END_REF]. When the classical approach fails at high molality assuming full dissociation, introducing the ion association can significantly improve the results. Nevertheless, for the sake of physical significance, the species added to the model should be compatible with the actual composition of the aqueous solution and should therefore rely on its characterization. Experience suggests that, for a given element, the aqueous complexes introduced in the model should be, at least, present in significant amounts if not predominant, in the solution.

Concerning the fraction of NaOH 0 (aq) present in solution, at total NaOH concentrations above about 10 mol•kgw -1 , the concentration of NaOH 0 (aq) is higher at low temperatures (273.15 and 298.15 K) (Fig. 4). This is surprising since the ionic association should be stronger at high temperatures because of the decrease in the dielectric constant of water. A possible explanation might be a competitive influence of both the dielectric constant and the equilibrium constant of the reaction of formation of the aqueous complex NaOH 0 (aq). Indeed, in the model presented here, these equilibrium constant increases at low temperature, and therefore stabilizes the aqueous complex, with a minimum at 298.15 K. The real weight of physical and thermodynamic contributions is hard to evaluate, mainly because there are few available data on aqueous speciation. New data are needed to clarify this point. Currently, the parameters retained in this study provide the best fit with the available excess properties (osmotic coefficient, activity of water).

We have seen that we can represent thermodynamic excess properties up to saturation. At 523.15 K, for instance, we can represent the excess properties up to 250 mol•kgw -1 , at which point the water activity is very low (0.012356). We are aware that an "aqueous solution" with such a low water activity is very unusual, but the aim of our study was not to provide new information concerning the structure and behavior of water under such extreme conditions.

We show here, for the first time to our best knowledge, the robustness of the Pitzer's approach and the ability of the model to represent these aqueous properties at very high molality and we provide a comprehensive set of parameters that can be used with confidence and be extended to others chemical systems. This is illustrated by the extension of the binary NaOH-H 2 O system to two ternary systems and the calculation of independent data such as the boiling point rise. At this stage, we consider that the proposed set of parameters constitutes a sound basis for describing the quaternary system NaOH-LiOH-NaCl-LiCl-H 2 O up to mineral saturation and to 423.15 K, in the absence of any double salt.

Conclusion

In

that we compared to values obtain by Christov and Møller [START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF]. For NaOH.H 2 O, our equilibrium constant is roughly consistent with thermodynamic data. These new parameter values were successfully tested on the boiling point elevation, which was accurately predicted, and on the prediction of salt solubility in two ternary systems (NaOH-NaCl-H 2 O and NaOH-LiOH-H 2 O), which was improved at high molalities.

This study is part of on-going work to redefine basic binary systems by extending them to high salinities and temperatures in order to ensure the consistency of an extension to complex systems similar to real and industrial brines.

Online Resource 1: Description of Pitzer's equations

Pitzer's model is described in detail elsewhere [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF][START_REF] Pitzer | Thermodynamics of electrolytes. II. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent[END_REF][START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF][START_REF] Pitzer | Thermodynamics of electrolytes. IV. Activity and osmotic coefficients for mixed electrolytes[END_REF][START_REF] Pitzer | Activity coefficients in electrolyte solutions[END_REF][START_REF] Zemaitis | Handbook of aqueous electrolyte thermodynamics[END_REF]. We give here the equations for the osmotic coefficient (𝜙) and activity coefficient for aqueous (𝛾 𝑀 and 𝛾 𝑋 ) and neutral species (𝛾 𝑁 ). These are developed for a system that contains c cations, a anions and n neutral species [START_REF] Clegg | Activity coefficient in natural waters[END_REF] (𝜙 -1) = This study

( 2 ∑ 𝑚 𝑖 𝑖 ) { -𝐴 𝜙 𝐼 3 2 ⁄ 1+1.2𝐼 1 2 ⁄ + ∑ ∑ 𝑚 𝑐 𝑚 𝑎 (𝐵 𝑐𝑎 𝜙 + 𝑧𝐶 𝑐𝑎 ) 𝑎 𝑐 (i)≥1 salt + ∑ ∑ 𝑚 𝑐 𝑚 𝑐 ′ (𝛷 𝑐𝑐 ′ 𝜙 + ∑ 𝑚 𝑎 𝜓 𝑐𝑐 ′ 𝑎 𝑎 ) <𝑐 ′ 𝑐 (ii)≥2 cations + ∑ ∑ 𝑚 𝑎 𝑚 𝑎 ′ (𝛷 𝑎𝑎 ′ 𝜙 + ∑ 𝑚 𝑐 𝜓 𝑎𝑎 ′ 𝑐 𝑐 ) <𝑎 ′ 𝑎 (iii)≥2 anions + 1 2 ∑ 𝑚 𝑛 2 𝜆 𝑛𝑛 𝑛 + ∑ 𝑚 𝑛 3 𝜇 𝑛𝑛𝑛 𝑛 (iv)≥1 neutral + ∑ ∑ 𝑚 𝑛 𝑚 𝑛 ′ 𝜆 𝑛𝑛 ′ <𝑛 ′ 𝑛 + 3 ∑ ∑ 𝑚 𝑛 2 𝑚 𝑛 ′ 𝜇 𝑛𝑛𝑛 ′ <𝑛 ′ 𝑛 (v)≥2 neutral +6 ∑ ∑ ∑ 𝑚 𝑛 𝑚 𝑛 ′ 𝑚 𝑛 " 𝜇 𝑛𝑛′𝑛" 𝑛" 𝑛 ′ < 𝑛< (vi)≥3 neutral + ∑ ∑ 𝑚 𝑛 𝑚 𝑐 𝜆 𝑛𝑐 𝑐 𝑛 + 3 ∑ ∑ 𝑚 𝑛 2 𝑚 𝑐 𝜇 𝑛𝑛𝑐 𝑐 𝑛 (vii)≥1 cation, ≥1 neutral + ∑ ∑ 𝑚 𝑛 𝑚 𝑎 𝜆 𝑛𝑎 𝑎 𝑛 + 3 ∑ ∑ 𝑚 𝑛 2 𝑚 𝑎 𝜇 𝑛𝑛𝑎 𝑎 𝑛 (viii)≥1 anion, ≥1 neutral + ∑ ∑ ∑ 𝑚 𝑛 𝑚 𝑐 𝑚 𝑎 𝜁 𝑛𝑐𝑎 𝑎 𝑐 𝑛 (ix)≥1 cation, ≥1 anion, ≥1 neutral + ∑ ∑ ∑ 𝑚 𝑛 𝑚 𝑐 𝑚 𝑐 ′ 𝜂 𝑛𝑐𝑐 ′ <𝑐 ′ 𝑐 𝑛 (x)≥2 cations, ≥1 neutral + ∑ ∑ ∑ 𝑚 𝑛 𝑚 𝑐 𝑚 𝑎 ′ 𝜂 𝑛𝑎𝑎 ′ <𝑎 ′ 𝑎 𝑛 (xi)≥2 anions, ≥1 neutral +6 ∑ ∑ ∑ 𝑚 𝑐 𝑚 𝑛 𝑚 𝑛 ′ 𝜇 𝑐𝑛𝑛 ′ <𝑛 ′ 𝑛 𝑐 (xii)≥1 cation, ≥2 neutral +6 ∑ ∑ ∑ 𝑚 𝑎 𝑚 𝑛 𝑚 𝑛 ′ 𝜇 𝑎𝑛𝑛 ′ <𝑛 ′ 𝑛 𝑎 (xii)≥1 anion, ≥2 neutral } (A-1) 𝑙𝑛 𝛾 𝑀 = [ 𝑧 𝑀 2 𝐹 + ∑ 𝑚 𝑎 (2𝐵 𝑀𝑎 + 𝑧𝐶 𝑀𝑎 ) 𝑎 + |𝑧 𝑀 | ∑ ∑ 𝑚 𝑎 𝐶 𝑐𝑎 𝑎 𝑐 (i)≥1 sel + ∑ 𝑚 𝑐 (2𝛷 𝑀𝐶 + ∑ 𝑚 𝑎 𝜓 𝑀𝑐𝑎 𝑎 ) 𝑐 (ii)≥2 cations + ∑ ∑ 𝑚 𝑎 𝑚 𝑎 ′ 𝜓 𝑀𝑎𝑎 ′ <𝑎 ′ 𝑎 (iii)≥2 anions +2 ∑ 𝑚 𝑛 𝜆 𝑀𝑛 𝑛 + 3 ∑ 𝑚 𝑛 2 𝜇 𝑀𝑛𝑛 𝑛 (iv)≥1 neutral +6 ∑ ∑ 𝑚 𝑛 𝑚 𝑛 ′ 𝜇 𝑀𝑛𝑛′ 𝑛 ′ 𝑛< (v)≥2 neutral + ∑ ∑ 𝑚 𝑛 𝑚 𝑎 𝜁 𝑀𝑛𝑎 𝑎 𝑛 (vi)≥1 anion, ≥1 neutral +6 ∑ ∑ 𝑚 𝑛 𝑚 𝑐 𝜂 𝑀𝑛𝑐 𝑐 𝑛 (vii)≥2 cations, ≥1 neutral ] (A-2) 𝑙𝑛 𝛾 𝑋 = (A-3)
mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1
•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1 mol•kgw -1
Dühring's law Kirschstein [START_REF] Kirschstein | Physikalische Eigenschaften der festen Salze[END_REF] Bialik et al. [START_REF] Bialik | Boiling Point Rise Calculations in Sodium Salt Solutions[END_REF] Dühring's law [START_REF] Haltenberger | Enthalpy-Concentration Charts from Vapor Pressure Data[END_REF] Figure 7 c)

  the normal boiling point of the pure solvent. The values of these parameters are given in Online Resource 2 and are provide by Lide[START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF]. 𝑎 𝑙𝑖𝑞 is the water activity in the aqueous solution at 𝑇 𝐵 + Θ 𝐵 .

ψ

  Na+/Cl-/OH-and ζ Na+/Cl-/NaOH , the halite solubility can be satisfactorily reproduced up to 423.15 K and up to saturation of NaOH. Except at 423.15 K, where we represented solubility up to 55 mol•kgw -1 . The regression coefficients of Eq. 4 are given in Table7for the interaction parameters (ψ Na+/Cl-/OH-and ζ Na+/Cl-/NaOH ).

3 -

 3 propose a new parameterization without CaSO 4 0 (aq). Harvie et al.[START_REF] Harvie | The prediction of mineral solubilities in natural waters: The Na-K-Mg-Ca-H-Cl-SO 4 -OH-HCO 3 -CO 3 -CO 2 -H 2 O system to high ionic strengths at 25°C[END_REF] studied the solubility of the Na-K-Mg-Ca-H-Cl-SO 4 -OH-HCO 3 -CO 3 -CO 2 -H 2 O system and introduced several complex ions (MgOH + , HCO )). But it is questionable whether these aqueous complexes will enable the extension to larger chemical systems since parameters should, theoretically, be estimated for specific interactions between parameters and for every new species added to the chemical system. Moreover, the compatibility of parameter sets based on ion association and parameter sets based on 4-parameter descriptions is an open debate.

Figure captions Fig. 1

 captions1 Figure captions

Fig. 2

 2 Fig. 2 Calculated (lines) and experimental (open diamonds) osmotic coefficient, , for the

Fig. 3

 3 Fig. 3 Evolution of concentration of NaOH 0 (aq) as a function on a total molality of Na.

Fig. 4 Fig. 5

 45 Fig.[START_REF] Thomsen | Aqueous Electrolytes Model Parameters and Process Simulation[END_REF] Comparison of thermodynamics solubility product of hydroxide sodium solids

Fig. 6 Fig. 7

 67 Fig. 6 Calculated (lines) and experimental (open symbols) boiling point elevation as a

Fig. 8 Fig. 9 Fig. 10

 8910 Fig. 8 Calculated (lines) and experimental (open diamonds) osmotic coefficient, , for the

  Figure 4

  Figure 8

  could be used without further adjustment :

	𝑃𝑎𝑟𝑎(𝑇 𝑘 ) = 𝑎 1 + 𝑎 2 𝑇 𝑘 + 𝑎 3 𝑇 𝑘 2 + 𝑎 4 𝑇 𝑘 3 + 𝑎 5 𝑇 𝑘 ⁄ + 𝑎 6 ln 𝑇 𝑘 + 𝑎 7 (𝑇 𝑘 -263) ⁄	+ 𝑎 8 (680 -𝑇 𝑘 ) ⁄
	+ 𝑎 9 (𝑇 𝑘 -227) ⁄

Table 3

 3 together with literature values given for comparison. Although the values are extremely variable (=42.8 at 22 mol•kgw -1 and 273.15 K and <1 at 523.15 K for the same molality), our values are similar to those recommended by several authors[START_REF] Simonson | The enthalpy of dilution and apparent molar heat capacity of NaOH(aq) to 523 K and 40 MPa[END_REF][START_REF] Petrenko | Thermodynamics of Aqueous NaOH over the Complete Composition Range and to 523 K and 400 MPa[END_REF][START_REF] Hamer | Osmotic Coefficients and Mean Activity Coefficients of Uniunivalent Electrolytes in Water at 25°C[END_REF][START_REF] Robinson | Tables of osmotic and activity coefficients of electrolytes in aqueous solution at 25°C[END_REF][START_REF] Holmes | Isopiestic molalities for aqueous solutions of the alkali metal hydroxides at elevated temperatures[END_REF]. NaOH 0 (aq) appears when the total NaOH molality is around 10 mol•kgw -1 . It becomes increasingly predominant when the concentration of total NaOH increases. For instance, at 30 mol•kgw -1 , the proportion of NaOH 0 (aq) ion pair reaches 35 %, except at 273.15 and 298.15 where 50 % only of the total dissolved NaOH is dissociated. This suggests

	For instance, the comparison with the experimental values compiled by Hamer and Wu [33] at
	298.15 K, shows that our model better represents the data than the other models up to 22
	mol.kgw -1 .

The speciation of dissolved NaOH is plotted on Fig.

3

in terms of molar percentage of NaOH 0 (aq) as a function of the total NaOH molality at various temperatures. Up to 423.15 K, the species solubility could be studied and we estimated the temperature functions of equilibrium constants for the dissolution reaction of anhydrous NaOH(cr) and hydrated NaOH.nH 2 O(cr) salts [n=1, 2, 3.11, 3.5 or 4α]. The irreversible general dissolution reaction is:

Table 1

 1

	). Although NaOH.nH 2 O [with n=2, 3.5 or 4] are not stable at 298.15 K, the log K of
	those three salts can be calculated which gives 3.95, 2.5 and 2.11, respectively. These values

are close to the values proposed by Christov and Moller

[START_REF] Christov | Chemical equilibrium model of solution behavior and solubility in the H-Na-K-OH-Cl-HSO 4 -SO 4 -H 2 O system to high concentration and temperature[END_REF]

who found 3.56, 2.23 and 1.93 for NaOH.2H 2 O, NaOH.3.5H 2 O and NaOH.4H 2 O, respectively.

  NaOH 0 (aq) species to represent the excess properties over the entire range of concentrations, without losing accuracy at low molalities. At 298.15 K, 373.15 K and 473.15 K, we found -3.5, -3.16 and -1.35, respectively, which is consistent with the calorimetric data of Chen et al.

	[44] for temperatures above 523.15 K. New experimental data concerning speciation is
	needed to enable us to reduce this uncertainty.
	determined log K values between
	273.15 and 623.15 K with data up to 6 mol•kgw -1 . They found -0.75, -0.66 and -0.21 at
	298.15, 373.15 and 473.15 K, respectively, for the reaction Na + + OH -= NaOH 0 (aq). We
	estimated Pitzer interaction parameters by first using the log K values proposed by Pokrovskii
	and Helgeson [63]. The resulting set of parameters was not able to reproduce excess

properties over the entire range of concentrations. It could represent the data only up to 14 mol•kgw -1 . We therefore had to propose new values for the formation constant of the

  this study we proposed a new Pitzer parameterization for the binary NaOH-H 2 O system and for ternary NaOH-NaCl-H 2 O and NaOH-LiOH-H 2 O systems. We represent the excess solution properties (osmotic coefficient, water activity, etc.) up to saturation and between 273.15 and 523.15 K (i.e., 30 mol•kgw -1 at 298.15 K and up to 250 mol•kgw -1 at 523.15 K).To do this, it is necessary to consider the partial dissociation of NaOH and add the neutral ion pair NaOH 0 (aq), which has been considered by other authors. In this case, several parameters must be added to the Pitzer model (𝜇 𝑛𝑛𝑐 , 𝜆 𝑛𝑐 , 𝜆 𝑛𝑎 … where n, c, a stand for neutral, cation and anion species respectively). After trial and error testing, we selected 𝜁 𝑛𝑐𝑎 and 𝜆 𝑛𝑛 . The approach is fully consistent with the classical Pitzer model and can be easily applied to

common geochemical calculation codes. We also determined the solubility products of anhydrous NaOH(cr) and five hydrated NaOHnH 2 O(cr) salts (where n = 1,

Table B -

 B 2. Boiling point raise (θ B ) of NaOH-H 2 O system

	Recommended values of :			θ B Boiling point raise (°C)
	0.98760	1.90790	3.50980	6.25048	9.76640	13.46258 16.66796 20.45613

Table B -

 B [START_REF] Abovsky | Representation of nonideality in concentrated electrolyte solutions using the Electrolyte NRTL model with concentration-dependent parameters[END_REF]. Values of the fitting constants (Eq. (4)) for the interaction parameters of the LiOH-H 2 O system

			T range				Constants (T in K)
	Parameters										Ref.
			(K)	a 1 (-)	a 2 (T)	a 3 (T²)	a 5 (1/T)	a 6 (ln(T))	a 7 (1/(T-263))
	This study β 0 Li+/OH-	46.93 273.15-473.15	54.95 0.121114	63.88 -3.45E-04	74.43 0	86.96 3.917769 100.65	114.19 0	127.76 0	[2]
	Kirschstein [54] β 1 Li+/OH-	42.5 273.15-473.15	50.1 1.235033	58.8 -3.984E-03	69	0	81.5	97.1 101.9411		117.5 0	144.9 0	[2]
	C φ	Li+/OH-	273.15-473.15	1.00E-10	2.51E-04		0		-0.51131		0	0	[2]

Table 1 .

 1 Standard deviation (σ) values of osmotic coefficient in the NaOH-H 2 O binary system (n is the number of experimental data points)

	Zdanovskii et al (1975)
	Zdanovskii et al (1973)
	LiOH:H2O
	LiOH
	423.15 K

Table 2 .

 2 Molality and temperature of invariant point in the NaOH-H 2 O binary system Experimental values of Pickering[START_REF] Pickering | LXI.-The hydrates of sodium, potassium, and lithium hydroxides[END_REF]; b Experimental values of Cohen-Adad et al.[START_REF] Holmes | Isopiestic molalities for aqueous solutions of the alkali metal hydroxides at elevated temperatures[END_REF]; c Experimental values of Linke[START_REF] Zdanovskii | Experimental solubility data on salt-water systems[END_REF]

	m S /
	(mol.kg -1 )

Table 3 .

 3 Mean activity coefficient (γ ± ) of NaOH in binary solutions as a function of molality and temperature

							γ ± (NaOH)
	Recommended values of:	T/K	0.1	0.5	1	4	6	10	14	18	22
							mol•kgw -1
	This study	273.15 0.784	0.686	0.662	0.884	1.363	4.289	11.232	22.927	42.836
	Simonson et al. [14]		0.782	0.682	0.656	0.864	1.320	-	-	-	-
	Holmes and Mesmer [39]		0.782	0.682	0.656	0.864	-	-	-	-	-
	Petrenko and Pitzer [15]		-	-	-	-	1.514	4.520	11.694	24.332	41.517
	Christov and Møller [16]		0.773	0.672	0.658	0.970	1.491	-	-	-	-
	Pabalan et Pitzer [13]		0.778	0.675	0.652	0.877	1.312	4.084
	This study	298.15 0.778	0.687	0.673	0.918	1.315	3.234	8.208	15.792	22.989
	Hamer and Wu [33]		0.775	0.685	0.674	0.911	1.302	3.258	8.083	15.570	23.030
	Simonson et al. [14]		0.781	0.692	0.676	0.902	1.295	-	-	-	-
	Holmes and Mesmer [39]		0.781	0.691	0.674	0.901	-	-	-	-	-
	Petrenko and Pitzer [15]		-	-	-	-	1.463	3.683	8.064	14.648	22.565
	Christov and Møller [16]		0.768	0.667	0.652	0.925	1.342	-	-	-	-
	Pabalan et Pitzer [13]		0.777	0.684	0.668	0.900	1.287	3.293
	This study	323.15 0.769	0.676	0.661	0.875	1.185	2.399	4.848	8.251	11.560
	Simonson et al. [14]		0.772	0.682	0.666	0.865	1.176	-	-	-	-
	Holmes and Mesmer [39]		0.773	0.683	0.667	0.867	-	-	-	-	-
	Petrenko and Pitzer [15]		-	-	-	-	1.319	2.855	5.430	8.822	12.506
	Christov and Møller [16]		0.759	0.654	0.635	0.856	1.177	-	-	-	-
	Pabalan et Pitzer [13]		0.772	0.682	0.668	0.887	1.219	2.666
	This study	373.15 0.743	0.633	0.604	0.700	0.865	1.422	2.308	3.310	4.112
	Simonson et al. [14]		0.744	0.637	0.607	0.704	0.865	-	-	-	-
	Holmes and Mesmer [39]		0.746	0.639	0.610	0.703	-	-	-	-	-
	Petrenko and Pitzer [15]		-	-	-	-	0.944	1.587	2.419	3.310	4.139
	Christov and Møller [16]		0.730	0.606	0.572	0.671	0.838	-	-	-	-

Table 4 .

 4 Values of the fitting constants (Eq. 4) for the interaction parameters of the NaOH-H 2 O system

	Parameters	T range (K)

Table 5 .

 5 Values of fitting constants (Eq. 5) for log K of formation of aqueous complex and dissolution of solid phases.

	Compound	T range
		(K)

Table 6 .

 6 Values of the constants (Eq. 4) for the interaction parameters of the NaCl-H 2 O system

	Parameters	T range (K)

Table 7 .

 7 Values of the fitting constants (Eq. (4)) for the interaction parameters of the NaOH-NaCl-H 2 O system

	Parameters	T range (K)

Table 8 .

 8 Values of the fitting constants (Eq. 4) for the interaction parameters of the NaOH-LiOH-H 2 O system

	Parameters	T range (K)	a 1 (-)	a 2 (T)	Constants (T in K) a 3 (T²)	a 5 (1/T)	a 7 (1/(T-263))	Ref.
	θ Li+/Na+	303.15-423.15 -2.304375225	9.185919E-3 -11.382562E-6 195.8054794	0	[16]
	ψ Na+/Li+/OH-	303.15-423.15 -4.254512E-2	9.112903E-5	0	0	0	This study
	λ Li+/NaOH	303.15-423.15 4.0637268125	-4.6714348E-03	0	-9.344074E+02	17.856540491	This study

Acknowledgements

This work was funded by BRGM, the French Geological Survey. The authors thank the LABEX Voltaire (ANR-10-LABX-100-01). The authors warmly thank Christomir Christov and an anonymous reviewer for their reviews of this manuscript.

Functions 𝑓 𝜙 (extended form of the Debye-Hückel equation) and 𝐵 𝑐𝑎 𝜙 (form of the ionic strength dependence) are determined empirically. In his first paper, Pitzer [START_REF] Pitzer | Thermodynamics of electrolytes. I. Theoretical basis and general equations[END_REF] tried several forms for these functions and the best results for type 1-1, 2-1 and 1-2 electrolytes at 298.15 K were obtained with:

where I is the ionic strength defined by:

In these equations, 𝑏 and 𝛼 are universal parameters whose values are 1.2 kg 1/2 •mol -1/2 and 2.0 kg 1/2 •mol -1/2 , respectively.

Pitzer and Mayorga [START_REF] Pitzer | Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2-2 electrolytes[END_REF] studied the type 2-2 electrolyte and explained that for this electrolyte another form could be used for the second virial coefficient in order to reproduce the behaviour of this electrolyte below 0.1 mol•kgw -1 . They added 𝛽 𝑐𝑎 (2) and 𝛼 2 parameters.

𝐵 𝑐𝑎 𝜙 = 𝛽 𝑐𝑎 (0) + 𝛽 𝑐𝑎 (1) exp(-𝛼 1 𝐼 1 2 ⁄ ) + 𝛽 𝑐𝑎 (2) exp (-𝛼 2 𝐼 1 2 ⁄ )

with values of 1.4 kg 1/2 •mol -1/2 and 12 kg 1/2 •mol -1/2 for 𝛼 1 and 𝛼 2 , respectively.

The equation for the Debye-Hückel parameter 𝐴 𝜙 is:

The 𝐶 𝑐𝑎 third virial coefficients are independent of ionic strength:

But this term is coupled with the parameter 𝑧, which depends on the concentration:

𝐶 𝑐𝑎 𝜙 is the adjustable parameter.

The other parameters in the activity coefficient equation are defined by:

where:

𝐵 𝑐𝑎 ′ = [𝛽 𝑐𝑎 (1) g ′(𝛼 1 𝐼 1 2 ⁄ ) + 𝛽 𝑐𝑎 (2) g′(𝛼 2 𝐼 1 2 ⁄ )] 𝐼 ⁄ (A-15)

)

The complete equation for Φ 𝑖𝑗 , which takes into account the interaction between two ions of the same sign, is:

where 𝜃 𝑖𝑗 𝐸 and 𝜃 𝑖𝑗 ′ 𝐸 take into account electrostatic unsymmetric mixing effects and only depend on ion charges, total ionic strength and solvent properties. 𝜃 𝑖𝑗 is an adjustable parameter. The third virial coefficient ψ ijk is a mixed electrolyte parameter for each cationcation-anion or anion-anion-cation triplet in a mixed electrolyte solution. This term is assumed to be independent of ionic strength.

Online Resource 2: Supplementary data