# Thermodynamic equilibrium and kinetic study of lanthanum chloride heptahydrate dehydration for thermal energy storage

Larysa Okhrimenko, Julie Dussouillez, Kévyn Johannes, Frédéric Kuznik

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Larysa Okhrimenko, Julie Dussouillez, Kévyn Johannes, Frédéric Kuznik. Thermodynamic equilibrium and kinetic study of lanthanum chloride heptahydrate dehydration for thermal energy storage. Journal of Energy Storage, 2022, 48, pp.103562. 10.1016/j.est.2021.103562. hal-04918189

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Submitted on 31 Mar 2025

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Thermodynamic equilibrium and kinetic study of
lanthanum chloride heptahydrate dehydration for
thermal energy storage

Larysa Okhrimenkoª, Julie Dussouillezª, Kévyn Johannesª,*, Frédéric Kuznikª

ªUniversité de Lyon, CNRS

INSA-Lyon, CETHIL, UMR5008, F-69621, Villeurbanne Cedex, France

Université Lyon 1, F-69622, Villeurbanne Cedex, France

**Abstract**

In the past decade, thermochemical materials have gained attention as promising options for heat storage. Compared to sensible heat storage and phase change materials, thermochemical materials offer certain important advantages such as high energy density and low heat loss during the storage process. Lanthanum chloride hydrates are interesting and promising materials owing to their high energy density and application at low temperature and water-vapor pressure. The thermodynamic properties of the LaCl₃–H₂O system and dehydration kinetic of LaCl₃·7H₂O are investigated using coupled thermogravimetry and differential scanning calorimetry under a controlled humidity atmosphere and X-ray diffraction. As the literature on the LaCl₃ thermodynamic equilibrium is inconsistent, the phase diagram is experimentally determined. The findings of the first part of the study are presented in the form of a reliable phase diagram. From a kinetic point of view, it has been determined that the dehydration of lanthanum sulfate heptahydrate can be analyzed using the shrinking core model, where the growth of a new phase

<footnote>*Corresponding author: Phone: +33 472 438 461; Fax: +33 472 438 522
Email address: kevyn.johannes@insa-lyon.fr (Kévyn Johannes)</footnote>
25 alone is the rate-determining step. The areic reactivity of growth decreases 26 with the increase in water-vapor pressure and increases with the increase in 27 temperature. The mechanism of the dehydration reaction is presented, in 28 which the desorption of water molecules is the rate-determining step. 29

30 *Keywords:* Reaction kinetic, Dehydration, Lanthanum chloride, Kinetic model, Thermogravimetry

31 **1. Introduction**

32 In EU-27, 27.3% of the final total energy consumption is attributed to 33 the energy systems in households [1]. It's worth mentioning that 78.9% of 34 this energy is required for heating purposes [2] (i.e. space heating and do- 35 mestic hot-water production); hence, 21.4% of the total energy consumption 36 is for domestic heating alone. Thus, reducing domestic heating is the key to 37 decreasing global energy consumption and greenhouse gas emissions. In this 38 context, thermal energy storage can play a fundamental role in rationalizing 39 and increasing the use of solar energy as well as in more sustainable energy 40 management. Among the possible solutions for heat storage, chemisorption 41 is the most promising [3], compared to sensible heat, latent heat, and phys- 42 ical sorption. However, as heat storage by chemical reaction is a less studied 43 technology up to now, fundamental research is required to develop innovative 44 systems.

45 In thermochemical energy storage, sorption/chemical process is based on 46 a reversible reaction allowing energy to be stored and then released. Each 47 gas-solid reaction has a unique reaction equilibrium characteristic and kinetic 48 behavior. Salt hydrates are commonly used as chemisorption heat storage
materials owing to their high-potential for storing heat (at least 1 GJ·m⁻³ of the energy density depending on the operating conditions) and low heat losses during the storage period [4–10]. Most, but not all, of the potential salt hydrates have been studied in detail. In this study, we selected LaCl₃ salt from 25 of the most suitable thermochemical reactions [9] and three of the most promising candidates for low-temperature thermochemical heat storage [6]. Lanthanum chloride is used in various water treatment applications, bio-chemical research, inorganic synthesis and catalysis. But LaCl₃·7H₂O still is an expensive type of chemical compound because of a rare earth metal. However, it is non-toxic, non-corrosive potential candidate with high storage energy density, with reversible and cycle-stable reaction in the relevant temperature range. Moreover, proposed studies concerned the thermodynamic properties are not consistent and the kinetic behavior stays unknown up to now. LaCl₃·7H₂O was chosen in order to enrich knowledge about salts and test as a promising candidate for storage applications according to the literature [6, 9].

The entire dehydration reaction of LaCl₃·7H₂O offers a theoretical storage energy density of 2.41 GJ·m⁻³, while the partial dehydration of LaCl₃·7H₂O to LaCl₃·3H₂O offers 1.48 GJ·m⁻³ and 1.03 GJ·m⁻³, respectively, for open and closed storage systems [9]. These data appear to encourage work at low-temperature and low water-vapor pressure, which is recommended for heat-storage applications [11–13]. Nevertheless, the application of lanthanum chloride heptahydrate for thermochemical storage remains difficult. There are few studies on the dehydration mechanism of LaCl₃·7H₂O [14–16] and the thermodynamic equilibrium of lanthanum chloride hydrates [17, 18]. More-
over, kinetic study and investigation of the reaction mechanism have not yet been performed.

In view of the above, in this study, we investigate the thermodynamic equilibrium of LaCl₃ hydrates to define the phase diagram, and develop a kinetic model for the dehydration reaction of LaCl₃·7H₂O according to the following equation:

$$LaCl_3 \cdot 7 H_2O \rightleftharpoons LaCl_3 \cdot 3 H_2O + 4H_2O. \quad (1)$$

This is a typical solid-gas dehydration reaction in the heat storage application domain [19–22]. For these kinds of dehydration reactions, the temperature and the water vapor pressure has an important effect on the overall reaction kinetic [23–27]. However, there are a lot of published studies in the kinetic of solid decomposition where the water vapor pressure is not taken into account. Most of the recent studies are devoted to the heterogeneous kinetic analysis of the experimental data following expression of the reaction rate:

$$\frac{d\alpha}{dt} = A \exp(-\frac{E}{RT})f(\alpha). \quad (2)$$

where A is the pre-exponential factor, E is the apparent activation energy, and f(α) is a mathematical function depending on the kinetic model [28]. Nevertheless, this approach considers the influence only one intensive thermodynamic parameter, the temperature, without any experimental verification of the Arrhenius rate equation. It is obvious that if gases are consumed and produced during a chemical transformation of solid, the kinetic may be influenced by temperature and partial pressure of gases. Thus, this Arrhenius-
type dependence drive to a misleading interpretation of the kinetic study.

Some authors introduce $h(P_{H_2O}, P_{eq}(T))$ function into Eq. 2 with a view to resolve this issue with dehydration reaction [29–33]. This function $h(P_{H_2O}, P_{eq}(T))$ include the equilibrium pressure of the reaction $P_{eq}(T)$ and the water vapor pressure of the system $P_{H_2O}$. The kinetic rate with this accommodation function can be written by the following expression:

$$ \frac{d\alpha}{dt} = A \exp(-\frac{E}{RT}) f(\alpha) h(P_{H_2O}, P_{eq}(T)). \quad (3) $$

As almost all decomposition reaction is reversible, the rate of the reversible reaction can exhibit a dependence on the partial pressure of the gaseous products. In the case, the gas produced by the reaction is not removed efficiently from the reaction zone, the reaction proceeds to the equilibrium. The accommodation function can be presented as following:

$$ h(P_{H_2O}, P_{eq}(T)) = 1 - \frac{P_{H_2O}}{P_{eq}(T)} \quad (4) $$

Some authors have shown that a variety of possible laws expressing the physicochemical relationship between reaction rate and the partial pressure can be obtained by means of solid-gas reactions mechanism [29]. For some of these expression, the intensive thermodynamic parameters, the temperature and the partial pressure, are not separables as implied by the Eq.3. For some systems, the accommodation function $h(P_{H_2O}, P_{eq}(T))$ can contain two water vapor pressure component [29, 34]. Thereby, if we need to realize a thorough study of a solid-gas reaction, some aspects should be taken in account: physics of the transformation, studying of the mechanism, changes undergone by the sample. The unawareness of those parameters can be prob-
lematic for several applications one of which is heat storage application. The
combination of reaction kinetics, physics of transformation, and mechanisms
give us a complete picture of what is happening inside of solid particles.
This knowledge as well as a mass transfer are required in the application
and thermo-chemical system design. In this paper, we apply a methodology
of step-by-step studying of a chemical reaction and mechanism for the heat
storage application. The kinetic model has been developed based on previous
works using TGA analysis [35–38]. Furthermore, the kinetic study for the
dehydration reaction of LaCl₃·7H₂O has been presented for the first time
as well as the importance of reaction mechanism and pressure influence.

The reaction is studied through thermogravimetric experiments (1) with
constant heating rates for the thermodynamic equilibrium study, (2) under
isobaric and isothermal conditions to establish a kinetic rate model. The
influence of the experimental conditions on the dehydration of lanthanum
sulphate heptahydrate is studied. Further, the reaction mechanism and
mathematical expressions are presented. Finally, the experimental results
and kinetic model are compared. Moreover, the variations in the areic reac-
tivity of growth as a function of the temperature and water-vapor pressure
are obtained.

## 2. Material and methods

### 2.1. *Material and characterisation*

High-purity (> 99.9%) lanthanum chloride heptahydrate powder (Sigma Aldrich) was used as the starting material in this study. For all the experiments, the powder sample was crushed beforehand.
The powder X-ray diffraction (XRD) patterns were recorded using a Bruker D8 advanced diffractometer, operating with monochromatic Cu-Kα radiation (λ = 1.5406 Å). The XRD patterns were obtained in the 20° ≤ 2θ ≤ 40° range at an angular step-size of 0.02 for a scan time of 2 s. The diffractograms of four samples S0, S1, S2, and S3 were recorded to check the different equilibrium states. Sample S0 comprised the initial lanthanum chloride heptahydrate commercial powder, whereas the other samples were obtained by heating the initial solid using a TGA-DSC apparatus to 55 °C, 90 °C, and 120 °C, respectively, under a water-vapor pressure of 10 hPa. For each dehydration step, mass stabilisation was maintained for 2 h. The experimental diffraction patterns were then compared with known patterns of lanthanum chloride heptahydrate (JCPDS 03-0069), lanthanum chloride trihydrate (JCPDS 082-1200), and lanthanum chloride anhydrous (JCPDS 073-0725). Lanthanum chloride monohydrate is unknown in the diffraction pattern database.

The material-specific surface area was estimated through nitrogen adsorption isotherm analysis, measured at 77 K using a BelSorp-Mini II (Microtrac Bel). The specific surface area of the powder was calculated according to the BET method (BrunauerEmmettTeller). The pore size distribution was estimated by Barret-Joyner-Halenda (BJH) method. The particle size distribution was measured using the Mastersizer 2000 laser-granulometer with dry-powder measurement.

## 2.2. TGA protocol

A symmetrical suspension-type thermo-balance Sensys Evo TG-DSC (Setaram) was employed for investigating the thermodynamic equilibrium of the LaCl₃-
H₂O system and the kinetics of the dehydration reaction of LaCl₃·7H₂O. This apparatus enables simultaneous measurement of the mass variation and heat flow as a function of time. Experiments were performed under atmospheric pressure with dry nitrogen mixed with pure vapor-pressure at a flow rate of 50 mL·min⁻¹. The humidity of the gas was controlled using the Wet-sys water-vapor generator (Setaram). The specified measurement accuracy was ± 1 °C, ± 0.1 µg, and ± 0.3 hPa, respectively, for the thermocouples, balance, and water-vapor pressure. A 125 µL-sample quartz crucible with a fritted glass bottom was used to provide better gas transport through the sample. As the crucible size limits the sample size, preliminary study of the influence of mass was performed. Therefore, an initial sample mass of approximately 2–3 mg was used for thermogravimetric analysis to minimise the impact of heat and mass transfer on the kinetic characterisation [39]. All experiments have been repeated in the same conditions (temperature and pressure) in order to ensure the accuracy of the obtained results.

Two different experimental protocols were applied: the first for thermodynamic equilibrium study and the second for studying the dehydration kinetics.

### 2.2.1. *Thermodynamic equilibrium*

Thermodynamic analysis was performed for the dehydration process of LaCl₃ from the heptahydrate to anhydrous form. Each sample has been initially stabilised at given temperature and given water pressure and then dehydrated at constant pressure by increasing temperature to 200 °C. During the phase of stabilisation, the sample was maintained for 3h at 30 °C and 5 hPa, 40 °C and 10 hPa, 50 °C and 25 hPa. Then the sample temperature was
increased from the stabilisation-phase temperature to 200 °C at a constant heating rate. Heating rates 0.05, 0.2, 0.5 and 10 K·min⁻¹ have been tested. During this phase, the LaCl₃ sample was dehydrated from the heptahydrate to anhydrous form.

To obtain the onset temperature of the decomposition of each hydrate of lanthanum chloride, the sample mass variations along with the heat flow evolution were studied. As shown in Fig. 1, the endothermic dehydration reaction is characterised by sudden mass loss and endothermic heat flow. The decomposition activation temperature is defined as the temperature where the reaction commences, and it corresponds to the temperature where the mass loss and heat-rate curves begin to change. Fig. 1 shows the determination of this temperature using the two above-mentioned characteristics, as the point where the mass loss and heat-rate curves exhibit a knee-point.

### 2.2.2. Dehydration kinetics

In the second protocol, all the experiments were performed under isothermal and isobaric conditions as fast as possible. Initially, the sample was maintained at 25 °C and 10 hPa for 30 min. The temperature was then increased to a given value between 30 °C and 60 °C, and the water-vapor pressure was changed to a given value between 5 hPa and 20 hPa, depending on the temperature to maintain lanthanum chloride heptahydrate equilibrium and prevent the commencement of the dehydration reaction. In the second step, the sample was heated at different water-vapor pressures at a heating rate of 10 K·min⁻¹, to various temperatures ranging from 45–90 °C.
## 3. Kinetic modelling background

The general equation of the thermal decomposition of salt hydrates is as follows:

$$A(s) = B(s) + nH_2O(g). \qquad (5)$$

In heterogeneous kinetic modelling, the chemical transformation of one solid to another involves two processes: the nucleation of the new phase, B, and the growth of phase B by consuming the reactant solid, A. The kinetics of this type of reaction can be controlled by both processes (nucleation-growth model) or by one, if one process is faster than the other. Nucleation is often instantaneous and the reaction kinetic is growth controlled. However, at times, the growth of the nuclei can be considered instantaneous and therefore, the nucleation process determines the rate of the reaction. The growth process involves a set of successive elementary steps, and in a simple case, one of them can be the rate-determining step. The general expression of the solid-gas reaction rate is as follows [35]:

$$\frac{d\alpha}{dt} = \phi(T, P_{H_2O})S_m(t), \qquad (6)$$

where $\phi(T, P_{H_2O})$ is the areic reactivity of growth expressed in mol·m⁻²·s⁻¹ and $S_m(t)$ is a space function expressed in m²·mol⁻¹. The areic reactivity of growth is a kinetic constant depending on the temperature and pressure alone. Function $S_m(t)$ is related to the extent reaction area where the rate-determining step occurs, and varies with time. Thus, kinetic modelling is based on the search of both geometrical and kinetic transformation to express
function $S_m(t)$, as well as the mechanism and nature of the rate-determining step of the growth involved in the gas-solid reaction leading to the expression of $\phi$.

The kinetic equation, Eq. 6, can be applied if, (1) the reaction occurs at steady-state conditions and (2) the reaction rate is controlled by one elementary rate-determining step in the entire range of $\alpha$ of the gas-solid reaction. It is also possible to verify, (3) whether the kinetic rate ($d\alpha/dt$) can be represented by a function of the fractional conversion $\alpha$. This indicates that one of the two processes, nucleation or growth, is rapid compared to the other. All these assumptions can be easily validated through experimental tests [36, 37]. Finally, (4) certain parameters such as the size and shape of the particle, size of development of new phase, and nature of the rate-determining step should be considered in the expression of the reaction rate.

### 3.1. Steady-state assumption

In heterogeneous kinetics, the reaction mechanism comprises elementary steps and involves reactants, various intermediates, and products. To interpret the kinetic curves using a realistic model and simplify the resolution of complex equations, certain assumptions must be verified [39]. The first assumption is the steady-state approximation.

To ensure that the concentrations of the intermediate species are constant with time, and there are no accumulations of the species in any part of the reacting solid, the pseudo-steady state assumption can be verified experimentally [37]. This test is based on the measurement of the reaction-rate evolution as a function of time, in two different ways. In general, for solid-gas reaction, thermogravimetry is used along with calorimetric measurement.
Thermogravimetry facilitates the measurement of the mass variation (Δm) as a function of time, enabling the calculation of the reaction rate. The heat flux is the kinetic rate measured directly through calorimetry. The two reaction rates are then compared. If the pseudo-steady-state condition of the system is satisfied, the rates are equivalent, and the measured heat flux can be expressed as follows:

$$ \frac{dQ}{dt} = \frac{\Delta_r H^\circ}{M_{H_2O} v_{H_2O}} \frac{d(\Delta m)}{dt}, \qquad (7) $$

where $dQ/dt$ is the heat flux, $d(\Delta m)/dt$ is the mass change rate, $\Delta_r H^\circ$ is the standard enthalpy variation of the reaction, $M_{H_2O}$ is the molar mass of water involved in the reaction ($M_{H_2O} = 18 \text{ g} \cdot \text{mol}^{-1}$), and $v_{H_2O}$ is the stoichiometric number of water molecules in the reaction (in our case $v_{H_2O} = 4$, according to reaction Eq. 1).

Besides, the enthalpy of the reaction $\Delta_r H^\circ$ can be obtained from the slope of the linear plot of $dQ/dt$ versus $d(\Delta m)/dt$, according to Eq. 7.

## 3.2. Expression of the kinetic-rate equation

After validating the steady-state assumption, the next step is to verify whether the rate of the reaction can be expressed by Eq. 6. This indicates that the reaction rate is controlled by one elementary rate-determining step for the entire range of $\alpha$ in the gas-solid reaction.

An experimental test, called «$\phi S_m$» test, was proposed to validate the assumption of the rate-determining step [37]. Its principle is based on the sudden change in temperature (or water-vapor pressure) from value $T_1$ to value $T_2$ at constant water-vapor pressure at a given time $t_i$ (or given fractional conversion $\alpha_i$). As the areic reactivity of growth $\phi$ is independent of
time under isothermal and isobaric conditions, and the space function $S_m$ is dependent on time, the ratios of the kinetic rate after and before the sudden change in temperature are determined by an expression in which term $S_m(t)$ is eliminated:

$$r = \frac{\phi(T_2, P_{H_2O})S_m(t)}{\phi(T_1, P_{H_2O})S_m(t)} = \frac{\phi(T_2, P_{H_2O})}{\phi(T_1, P_{H_2O})} \quad (8)$$

The variation of function $S_m(t)$ with time must be limited to be negligible. Hence, a thermodynamic variable (temperature or gas pressure) should vary rapidly. The assumption of the rate-determining step is validated, when ratio $r$ remains constant at all values of the fractional conversion $\alpha_i$.

### 3.3. One-process model assumption

As previously mentioned, chemical transformation involves the nucleation as well as growth process. Nucleation can be considered instantaneous at times, compared to growth. Therefore, the growth of the nuclei is the only rate-controlled process, and is included in kinetic modelling. Test «$f(\alpha)$» was proposed to determine whether the nucleation process can be supposed as instantaneous, and the transformation can be described by a one-process model [36]. In addition, with this test, it can be verified whether the reaction rate can be described by a single function $f(\alpha)$. In several articles on the kinetic modelling of reactions, this is neglected; however, it is not a general case for all transformation reactions [35, 40].

Test «$f(\alpha)$» comprises two experiments. The first is performed under constant conditions of temperature $T_1$ and water-vapor pressure $P_1$ for the entire reaction. The second experiment is performed under different condi-
tions: either at temperature T₂ and water-vapor pressure P₁ or at temperature T₁ and water-vapor pressure P₂; the condition is then modified by a sudden change to return to the same conditions as in the first experiment (temperature T₁ and water-vapor pressure P₁). If the rates of both experiments become superimposed after the sudden change, it is indicated that the rate follows the same function f(α). In this case, the reaction is controlled by a single process (growth or nucleation) and the rate can be expressed as a function of the fractional conversion.

$$ \frac{d\alpha}{dt} = \phi(T, P_{H_2O})S_m(\alpha). \qquad (9) $$

On the other hand, the rates may differ after the sudden change; hence, with this test, it can be verified whether the nucleation process is continuously followed by growth in the reaction. Thus, the sample history influences the rate of the reaction. In this case, function Sₘ(t) cannot be represented as a function of the fractional conversion.

### 3.4. Reaction mechanism

The easiest way to solve a chemical kinetic problem is by considering that the reaction rate is governed by one elementary step in the overall mechanism, while the other elementary steps are at equilibrium [41]. The areic reactivity of growth φ represents a kinetic constant depending on the operating conditions alone, i.e. the temperature and water-vapor pressure. The nature of the growth rate-determining step and the expression for the areic reactivity of growth can be deduced from the growth mechanism. Using the Kröger-Vink notation [42]. The various elementary steps involved in the mechanism are detailed as follows:
(I) Internal interface reaction with the formation of water-molecule vacancy:

$$[\text{H}_2\text{O}_{\text{H}_2\text{O}}]_h + s \rightleftharpoons \text{H}_2\text{O} - s + [\text{V}_{\text{H}_2\text{O}}]_h.$$

(II) Desorption of the water molecules generated from the initial solid at the internal interface:

$$\text{H}_2\text{O} - s \rightleftharpoons s + \text{H}_2\text{O}_g.$$

(III) New-phase formation by the annihilation of vacancies at the internal interface:

$$(\text{La}_{\text{La}} + 3\text{Cl}_{\text{Cl}} + 3\text{H}_2\text{O}_{\text{H}_2\text{O}} + 4\text{V}_{\text{H}_2\text{O}})_h \rightleftharpoons (\text{La}_{\text{La}} + 3\text{Cl}_{\text{Cl}} + 3\text{H}_2\text{O}_{\text{H}_2\text{O}})_t,$$

where "s" is the adsorption site, $V_{H_2O}$ is the vacancy of the water molecule, $H_2O_{H_2O}$, $La_{La}$, and $Cl_{Cl}$ are the water molecule, lanthanum ion, and chloride ion, respectively, in the normal crystal position in lanthanum chloride hydrate; subscripts "h" and "t" denote the correspondence of the species with the heptahydrate or trihydrate of lanthanum chloride, respectively.

The obtained lanthanum chloride trihydrate is porous with an average pore diameter of 14 nm. Thus, the diffusion of gaseous water molecules, produced by the reaction, through the layer of the formed phase of trihydrate is rapid, compared to the interfacial steps. Therefore, the interfacial growth steps alone are considered in the growth mechanism. Furthermore, as there is no diffusion through the formed solid, the development is internal.
Applying the mass action law to all the elementary steps, a system of 343 equations can be generated. 344

$$K_1 = \frac{[\text{H}_2\text{O} - s][V_{\text{H}_2\text{O}}]}{[s]}, \qquad (10)$$

$$K_2 = \frac{[s]P_{\text{H}_2\text{O}}}{[\text{H}_2\text{O} - s]}, \qquad (11)$$

$$K_3 = \frac{1}{[V_{\text{H}_2\text{O}}]^4}, \qquad (12)$$

The overall equilibrium constant $K_{eq}$ for the reaction in Eq. 1 can be expressed as following: 345 346

$$K_{eq} = \prod_{i} K_{i}^{\lambda} = K_{1}^{4} K_{2}^{4} K_{3}, \qquad (13)$$

where $\lambda_i$ is a positive number corresponding to the $i$th elementary steps of 347 the mechanism and a linear combination of them should result the global 348 dehydration reaction balance Eq.1. 349

Considering that one of the steps is rate-determining, and solving this 350 system, the expression for the rate equation of the growth process can be 351 obtained. The expressions for $\phi$, for elementary steps (I)-(III) are as follows 352 [41]: 353

$$\phi_1 = k_1 \frac{K_2}{K_2 + P_{\text{H}_2\text{O}}} \left(1 - \frac{P_{\text{H}_2\text{O}}}{P_{eq}}\right), \qquad (14)$$

$$\phi_2 = k_2 \frac{K_1 \sqrt[K_3]{K_3}}{K_1 \sqrt[K_3]{K_3} + 1} \left(1 - \frac{P_{\text{H}_2\text{O}}}{P_{eq}}\right), \qquad (15)$$

$$\phi_3 = k_3 \frac{K_1 K_2}{P_{\text{H}_2\text{O}}} \left(1 - \frac{P_{\text{H}_2\text{O}}}{P_{eq}}\right), \qquad (16)$$
354 where kᵢ is the kinetic constant, Kᵢ is the equilibrium constant of the corre-
sponding step, and Pₐq is the equilibrium pressure. The term (1-P<sub>H₂O</sub>/Pₐq)
355 represents the deviation from equilibrium conditions which is an accommo-
dation function presented in Eq. 3. But from these expressions, it is obvious
356 that simple introduction of the accommodation term do not allow to obtain a
357 good agreement between experimental and calculated data. For φ₃, the term
358 preceding the accommodation unction is a function of P<sub>H₂O</sub>⁻¹, whereas for
359 φ₁, the dependence of water vapor pressure is presented as (a/(b+P<sub>H₂O</sub>)).
360
361

362
4. Results and discussion

363
4.1. *Sample characterisation*

364
Fig. 2 depicts the results of the experimental diffraction patterns of the
365
commercial lanthanum chloride heptahydrate powder and the hydrates ob-
tained with in-situ TGA dehydration as the final step. The crystalline phase
366
compositions were compared with known patterns. In the X-ray pattern of
367
the initial solid (black spectrum), LaCl₃·7H₂O alone was detected. After de-
368
hydration at 55 °C with a water-vapor pressure of 10 hPa, the diffractograms
369
of samples S1 (red spectrum) and S3 (orange spectrum) displayed profiles cor-
370
responding to well definite lanthanum chloride trihydrate (JCPDS 082-1200)
371
and lanthanum chloride anhydrous (JCPDS 073-0725). The diffractogram in
372
the blue spectrum (sample S2) appeared similar to the experimental diffrac-
tion pattern obtained at 105 °C by Soho et al. [43], but no reported diffraction
373
pattern is available for comparison.
374
375

376
The particle-size distribution of the initial crushed powder was measured
377
with a laser granulometer in a dry process. Fig. 3 shows the obtained re-
sults of the number and volume distribution of the particles. The grain-size
distribution ranges from 1–700 µm; large particles, corresponding to the ag-
gglomerated particles, range from 100 µm to 700 µm. The average particle size
in volume was estimated to be 190 µm, whereas the average particle size in
number was 3.5 µm.

The average particle size was also calculated from the specific surface area
S<sub>BET</sub> (assuming that the particle is spherical and nonporous). The particle
diameter d was calculated according to equation d=6/ρS<sub>BET</sub>, where ρ is the
density of lanthanum chloride heptahydrate, which is 2.23 g·cm<sup>-3</sup>, and S<sub>BET</sub>
is the specific surface, which is 0.89 m<sup>2</sup>·g<sup>-1</sup>. The particle diameter d was
found to be 3 µm. This value agrees with the average particle size in number
obtained through laser granulometric analysis.

## 4.2. Thermodynamic equilibrium

Fig. 4 displays the TG plot of the dehydration of LaCl₃, comparing the mass loss with the heating rate. Three consecutive mass loss corresponding to the three chemical reactions can be identified. Further, by observing the heat flow, three endothermic peaks corresponding to the stored heat can be observed. To quantify this stored heat, the enthalpy of the reaction is obtained by calculating the integral of the curve. The results are presented in Table 1.

The slower the dehydration rate, the more stable and easily identifiable
are the three-step reactions. As 10 K·min⁻¹ is rapid, the reactions occur
simultaneously (i.e. the second reaction commences before the completion
of the first), and the three consecutive dehydration steps cannot be clearly
identified. Both 0.2 K·min⁻¹ and 0.5 K·min⁻¹ achieve the established tar-
get of 34% theoretical final mass loss. The first reaction at 10 K·min⁻¹ is
incomplete, and it can be observed that the mass loss of the final reaction at
0.05 K·min⁻¹ exceeds the theoretical mass loss of LaCl₃. This is owing to
the formation of HCl because of the very slow heating rate [43, 44].

Based on these results, it can be considered that the heating rate in-
fluences the read onset temperature. At higher dehydration rate, the start
of hydration reaction of each hydrate is shifted to higher temperature. To
evaluate the impact of the heating rate on the accuracy of thermodynamic
reactions, an experimental study was conducted at different heating rates
with various water-vapor pressure. Fig. 5 shows the obtained results of
the onset temperatures with the dehydration rate for the three-step reac-
tions: first from LaCl₃ heptahydrate to trihydrate, second from trihydrate
to monohydrate, and the last from monohydrate to anhydrous form. Three
heating rates (0.05, 0.2, and 0.5°C·min⁻¹) were first studied at 10hPa. The
results demonstrated that the obtained onset temperatures were linear with
the heating rates. This indicates that the ‘real’ onset temperature can be
found at 0°C·min⁻¹, corresponding to the intersection between the linear
function of the onset temperature and the Y-axis of the graph. The ‘real’
onset temperatures are represented by arrows in Fig. 5. Moreover, it is
important to note that the slope is not identical for the three chemical re-
actions. For the first two dehydration reactions, the heating rate has low
impact on the onset temperature (the slope is gentler), whereas it has higher
impact for the third reaction. There is more than 10 °C difference in the
last step, between the minimum and maximum heating rate, which can be
explained based on the operation of the TG instrument. The measured tem-
perature is not the exact sample temperature (salt crystals), but the crucible temperature. This is a well-known phenomenon in phase change materials; in the TGA instrument, the temperature of the crucible differs from those of the salt crystals within because of the thermal transfer resistance (i.e. conduction) [45]. The greater the increase in furnace temperature (and hence, the crucible), the greater is the difference with the salt-crystal temperature. Moreover, for dehydration, the heat flow is endothermic, increasing the temperature difference. To confirm and validate these results, a second experimental study was performed at 5hPa, at two heating rates of 0.2 and 0.5 °C·min⁻¹. The results were conclusive and close to those at 10hPa with different heating rates. The leading coefficient of the straight line connecting the onset temperature with the dehydration rate was the same as that at 10hPa, for each of the three LaCl₃ dehydration step reactions, respectively. The ‘real’ onset temperatures (at 0 °C·min⁻¹) were then deduced for 5hPa from this second experimental study. Finally, the results obtained at 25hPa for a heating rate of 0.5 °C·min⁻¹ were extrapolated using the slopes of the other two studies at 5 and 10hPa, and the final ‘real’ onset temperatures were defined. The ‘real’ onset temperatures as per the three experimental points of each reaction step are represented in Fig. 6. They were then fitted with the experimental enthalpy of the reaction (discussed previously) using Gibbs equation to determine the three entropies of the reaction and obtain the three thermodynamic equilibria of LaCl₃. Our results are in good agreement with experimental data obtained by Polyachenok [17, 18] for LaCl₃·7H₂O decomposition reactions and very closed for second and third steps of decomposition reaction.
Fig. 7 depicts the three-step dehydration reaction (mass loss) of LaCl₃ for the three water-vapor pressures studied previously, at heating rates of 0.5°C·min⁻¹: 5, 10, and 25hPa, respectively. The three experimental mass losses converge with the theoretical values (34wt.%).

### 4.3. Kinetic behavior

To ensure that the reaction is carried out at the same temperature and water vapor pressure conditions, the influence of various initial sample mass was studied. Fig. 8 displays the TG dehydration curves of the sample mass loss as a function of time, obtained under isothermal-isobaric conditions (temperature of 60 °C and water-vapor pressure of 10 hPa). Experiments performed with initial sample mass in the range 1.5-3 mg present similarly shaped curves during dehydration. The mass loss curve obtained with 5 mg of sample shows longer reaction time compared to samples with smaller mass. Those with initial sample masses of 1.5, 2.4, 2.5, 2.7 and 3 mg exhibit closed mass-loss curves and satisfactory reproducibility for all the experiments at these conditions. These results highlight that there are no heat and mass gradients in the powder layer during the reaction. Therefore, an initial sample mass of approximately 2.5 mg was selected for further experimentation. The measured percentage mass loss was 19.5 ±0.2 %, which is close to the stoichiometric value of 19.4 % for the loss of four water molecules during the dehydration of LaCl₃·7H₂O, according to Eq. 1.

Fig. 9 indicates that the rate of temperature change (10 K·min⁻¹) is sufficiently fast to study the dehydration kinetic. For both experiments performed at 60 °C and 70 °C, the reaction commences under isothermal and isobaric conditions.
Experiments at various temperatures were performed to study the tem-
perature influence on the dehydration kinetic of lanthanum chloride heptahy-
drate. All the experiments were conducted under a constant water-vapor
pressure of 10 hPa, and at a temperature range of 55 °C–80 °C. Fig. 10 dis-
plays the α(t) curves as a function of time, as well as those of the reaction
rate as a function of the fractional conversion α. It is evident that tem-
perature increase accelerates dehydration. Although the heating rate of the
device is limited at 10 K · min⁻¹, the dehydration reaction commences un-
der isothermal-isobaric conditions for all the performed experiments (see the
example in the magnified area for α between 0 and 0.1 for the experiments
performed from 60 to 70 °C in Fig. 10).

Furthermore, the influence of water-vapor pressure on the dehydration
kinetic of lanthanum chloride heptahydrate was studied. Fig. 11 shows the
corresponding results of the α(t) curves as a function of time and the reac-
tion rate as a function of the fractional conversion α obtained at a constant
temperature of 60 °C for various water- vapor pressures (from 5–30 hPa).
Similar to the experiments with fixed temperatures, the curves monotonously
decrease under isothermal-isobaric conditions. However, these results show
that the increase in water- vapor pressure decreases the kinetic.

4.4. Verification of the assumptions

A pseudo-steady test of the dehydration of lanthanum chloride heptahy-
drate was conducted at 60 °C and 10 hPa water-vapor pressure. Fig. 12a
displays the experimental curves of the heat flux and mass change rate as a
function of time during the reaction. The superimposition of the two curves
establishes the validity of the pseudo-steady state assumption of the dehydra-
tion reaction. The heat flux is plotted as a function of the mass change rate in Fig. 12b, in which the longest part is a straight line. Fig. 12b indicates that Eq. (7) is followed and therefore, the standard enthalpy of the reaction can be obtained from the slope, which is equal to 58.5 kJ·mol⁻¹ of water. This result is proximate to the standard enthalpy of the reaction specified in literature (61.1 kJ·mol⁻¹) [9].

The verification of the rate-determining step described in Section 3.1 was also validated. Fig. 13 depicts the ratios of the rates at various values of α obtained through the «$\phi S_m$» test performed under a water-vapor pressure of 10 hPa, with temperature jumps from 60 °C to 70 °C (a heating rate of 10 K·min⁻¹ was applied). The ratios in the entire range of α from 0.1–0.9 remained constant at a value of 2.58 ± 0.12. Therefore, from these results, it can be concluded that the growth reaction is controlled by one of the elementary steps of the overall reaction mechanism. Thus, the assumption is validated, and the reaction rate can be expressed by Eq. 6.

As previously mentioned, it is recommended to verify whether the reaction rate follows the one- or two-process model for determining if it can be expressed by a single function $f(\alpha)$. Fig. 14 shows the curves of the dehydration rate $d\alpha/dt$ as a function of the fractional conversion $\alpha$. The two solid lines represent the experiments performed at 60 °C (blue curve) and 70 °C (green curve) under the same water-vapor pressure of 10 hPa. The dotted lines indicate the first dehydration rate at 60 °C, after which the temperature is suddenly increased to 70 °C under the same water-vapor pressure of 10 hPa. This test was performed at two different fractional conversions: 0.5 and 0.6, respectively, represented by red and orange dotted lines. We can
observe that the reaction rate after the jump is perfectly superimposed with
the rate of the reference experiment. Thereby, we can conclude that this
test is validated, and the reaction rate can be described by the one-process
model.

### 4.5. Reaction mechanism, $\phi(T, P_{H_2O})$ and $S_m(\alpha)$ expressions

According to the previous results, the following characteristics of LaCl₃·7H₂O
dehydration have been determined:

* A pseudo-steady state is maintained for all dehydration reaction.

* The reaction rate is controlled by the rate-determining step, the «$\phi S_m$» test is confirmed.

* The $f(\alpha)$ is validated which means the reaction kinetic is governed only by growth process

* Development is inward.

* The rate-determining step is located at the internal interface.

The last step in the physical modelling of the decomposition reaction
involves the determination of the expression for space function $S_m$, which de-
pends only on the area where the growth rate-determined step occurs. The
calculation of the $S_m(\alpha)$ function is based on certain geometrical consider-
ations: the particles are spherical with the same initial radius of 1.5 µm,
the rate-determined step is located at the internal interface, the direction
of growth is inward as previously discussed, and the growth of the nuclei is
considered anisotropic. This indicates that the rate of tangential growth at
the surface of the particle is rapid, compared to the perpendicular growth at
the surface [35]. Considering all these assumptions, the S<sub>m</sub>(α) function can
be expressed by Eq.17 (see detailed calculation in [35]):

$$
S_m = \frac{3V_{mA}}{r_0} (1 - \alpha)^{2/3}, \qquad (17)
$$

where V<sub>mA</sub> is the molar volume of the initial sample and r<sub>0</sub> is the initial radius of the particle.

Using the general equation of solid-gas transformation, Eq.6, the reaction
rate can be expressed as follows:

$$
\frac{d\alpha}{dt} = \frac{3V_{mA}\phi}{r_0}(1-\alpha)^{2/3}, \qquad (18)
$$

Integrating Eq. 18, the expression for the fraction conversion α can be
obtained as a function of time:

$$
\alpha = 1 - \left(1 - \frac{3V_{mA}\phi}{r_0}t\right)^3. \qquad (19)
$$

This is the shrinking core model for spherical particles known as the "R3
law" [46]. Comparing this law, which is the general rate equation $d\alpha/dt =$
$k(T)$ $f(\alpha)$, with $f(\alpha) = k(1-\alpha)^{2/3}$ in Eq. 18, the rate is linked to the particle
size and molar volume of the initial reactant; however, the kinetic constant
$A$ in the "R3 law", the rate constant $k$ is represented by the Arrhenius
equation $k = A\exp(-E_A/RT)$, cannot be considered constant and is a variable
parameter [38]. The main objective of this article is to not only develop
a mathematical expression for the dehydration kinetic, but also provide a
physical interpretation for the measurements.
568 4.6. Kinetic modelling

569 Optimisation was realised using the φ parameter, applying the least-570 squares method to obtain agreement between the experimental and calcu-571 lated results. The agreement of the kinetic model with the fraction conver-572 sion experimental data is shown in Fig. 15 for the experiments conducted at573 10 hPa and 55, 60, 70 and 80 °C, and in Fig. 16 for the experiments conducted574 at 60 °C and 5, 10, 15, 20, and 30 hPa.

575 Figs. 15 and 16 demonstrate that for each water-vapor pressure or tem-
576 perature, the experimental data are correctly described by the anisotropic
577 growth model. Table 2 lists the values of the kinetic parameter φ for each
578 experimental condition. The effect of the water-vapor pressure on the areic
579 reactivity of growth is depicted in Fig. 17a at 60 °C; the areic reactivity
580 of growth monotonously decreases, when the water-vapor pressure increases.
581 After establishing the expressions for the areic growth reactivity (φᵢ), the
582 compare of these expressions allow to determine the rate-determining step.
583 The three Eqs. 14, 15 and 16 were matched with the values of φ as a function
584 of the water-vapor pressure to identify the rate-determining step of growth.
585 The fitting procedure was performed using the least-squares method. As
586 seen in Fig. 17a, Eq. 14 is better fitted to the experimental values of φ with
587 a correlation factor of 0.9946. It is to be noted that the temperature and
588 water-vapor pressure are not separable and the φ function does not follow
589 Arrhenius law. Therefore, the application of Eq. 6 is more appropriate than
590 the usual one dα/dt = k(T) f(α) f(P). Fig.17b shows that the areic reactiv-
591 ity of growth increases with the increase in temperature for the experiments
592 performed at 10 hPa.
As the rate coefficient $k_1$ obeys Arrhenius law, and the equilibrium constant $K_2$ must vary with temperature according to Van't Hoff's law, the areic reactivity of growth $\phi_1$ can be represented as follows:

$$ \phi_1 = \frac{k_1^0 K_2^0 \exp(-\frac{E_{a,1}+\Delta H_2}{RT})}{K_2^0 \exp(-\frac{\Delta H_2}{RT}) + P_{H_2O}} (1 - \frac{P_{H_2O}}{P_{eq}}), \quad (20) $$

where $k_1^0$ and $K_2^0$ are the pre-exponential factors, $E_{a,1}$ is the real activation energy of step (I), and $\Delta H_2$ is the enthalpy variation of step (II).

The results of the $\phi$ function obtained through kinetic modelling (Table 2) were fitted with Eq. 20. The kinetic parameters obtained from this fit were $k_1^0 K_2^0 = 6.9 \times 10^6 \text{ mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1} \cdot \text{Pa}$, $\Delta H_2 = -55 \text{ kJ} \cdot \text{mol}^{-1}$, $E_{a,1} = 106 \text{ kJ} \cdot \text{mol}^{-1}$, $K_2^0 = 3 \times 10^{-6} \text{Pa}$. Fig. 18 shows good agreement between the values obtained through kinetic modelling and the experimental data, and the theoretical $\phi$ function obtained from the growth mechanism.

## 5. Conclusions

In this study, the thermodynamic properties of the LaCl$_3$–H$_2$O system and the kinetics of the dehydration reaction of lanthanum sulphate heptahydrate to lanthanum sulphate trihydrate were investigated. The thermodynamic equilibrium was studied to generate a reliable phase diagram. Further, the dehydration reaction of LaCl$_3$·7 H$_2$O was performed at grain scale under isothermal and isobaric conditions in the temperature range of 55-80 °C and water-vapor pressure range of 5–30 hPa. The kinetic modelling assumptions were validated to ensure steady-state maintenance during the entire dehydration reaction. The reaction rate is controlled by the rate-determining step; the rate expression can be represented by a single function $f(\alpha)$, indicating
that one of two processes, the nucleation, is faster and negligible. The kinetic model of the dehydration of LaCl₃·7H₂O is based on surface nucleation and the inward anisotropic growth of the formed nuclei. It was demonstrated that the thermodynamic and morphological variables can be separated, and the areic reactivity of growth was defined as a function of the temperature and water-vapor pressure. The rate-determining step was the desorption of water molecules generated at the internal interface. Even if the areic reactivity of growth follows Arrhenius law, the apparent activation energy cannot be defined because the thermodynamic variables T and PₕO are not separable.

## Acknowledgement

The authors would like to thank the French National Research Agency (ANR-16-CE22-0006-01) for the funding.

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LIST OF SYMBOLS

<table><tr><td>d</td><td>particle diameter</td><td>m</td></tr><tr><td>E<sub>a</sub></td><td>activation energy</td><td>J &middot; mol<sup>-1</sup></td></tr><tr><td>&Delta; H</td><td>enthalpy of the reaction</td><td>kJ &middot; mol<sup>-1</sup></td></tr><tr><td>k</td><td>kinetic constant</td><td></td></tr><tr><td>K</td><td>equilibrium constant</td><td></td></tr><tr><td>m</td><td>sample mass</td><td>mg</td></tr><tr><td>M</td><td>molar mass</td><td>g &middot; mol<sup>-1</sup></td></tr><tr><td>P<sub>H2O</sub></td><td>water-vapor pressure</td><td>Pa</td></tr><tr><td>Q</td><td>energy</td><td>J</td></tr><tr><td>r</td><td>radius of particle</td><td>m</td></tr><tr><td>R</td><td>universal gas constant</td><td>J &middot; K<sup>-1</sup> &middot; mol<sup>-1</sup></td></tr><tr><td>S<sub>BET</sub></td><td>specific surface area</td><td>m<sup>2</sup> &middot; g<sup>-1</sup></td></tr><tr><td>S<sub>m</sub></td><td>space function</td><td>m<sup>2</sup> &middot; mol<sup>-1</sup></td></tr><tr><td>t</td><td>time</td><td>s, min or h</td></tr><tr><td>T</td><td>temperature</td><td>K, &deg;C</td></tr><tr><td>V<sub>m</sub></td><td>molar volume of the solid</td><td>m<sup>3</sup> &middot; mol<sup>-1</sup></td></tr><tr><td>Cl<sub>Cl</sub></td><td>chloride-ion at the normal position in lanthanum chloride hydrate</td><td></td></tr><tr><td>H<sub>2</sub>O<sub>g</sub></td><td>water molecules in the vapor phase</td><td></td></tr><tr><td>H<sub>2</sub>O<sub>H2O</sub></td><td>Water molecule at the normal position in the lattice</td><td></td></tr><tr><td>La<sub>La</sub></td><td>lanthanum-ion at the normal position in lanthanum chloride hydrate</td><td></td></tr><tr><td>V<sub>H2O</sub></td><td>vacancy of the water molecule</td><td></td></tr><tr><td colspan="3"><em>Greek symbols</em></td></tr><tr><td>&alpha;</td><td>fractional conversion</td><td>36</td></tr><tr><td>&rho;</td><td>density</td><td>g &middot; cm<sup>-3</sup></td></tr><tr><td>&phi;</td><td>areic reactivity of growth</td><td>mol &middot; m<sup>-2</sup> &middot; s<sup>-1</sup></td></tr></table>
<figure><img src="image_1.png" /><figcaption>Figure 1: Graphical representation of dehydration-temperature determination</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Figure 2: XRD diffractograms of the initial lanthanum chloride heptahydrate powder sample and the hydrates obtained by in-situ TGA</figcaption></figure>

<table><caption>Table 1: LaCl₃ intrinsic characteristics: enthalpy and entropy of the reaction</caption><thead><tr><th>Reaction step</th><th>Mass loss, wt%</th><th>Δh<sub>r</sub><sup>0</sup>, J/mol – H<sub>2</sub>O</th><th>Δs<sub>r</sub><sup>0</sup>, J·K<sup>-1</sup></th></tr></thead><tbody><tr><td>LaCl<sub>3</sub>.7H<sub>2</sub>O → LaCl<sub>3</sub>.3H<sub>2</sub>O + 4H<sub>2</sub>O</td><td>19.4</td><td>57 000</td><td>142.6</td></tr><tr><td>LaCl<sub>3</sub>.3H<sub>2</sub>O → LaCl<sub>3</sub>.H<sub>2</sub>O + 2H<sub>2</sub>O</td><td>12.0</td><td>62 000</td><td>137.5</td></tr><tr><td>LaCl<sub>3</sub>.H<sub>2</sub>O → LaCl<sub>3</sub> + H<sub>2</sub>O</td><td>6.8</td><td>73 000</td><td>149.8</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Figure 3: Particle-size distribution</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 4: Plot of the mass loss in wt% with the dehydration rate in K·min⁻¹</figcaption></figure>
<figure><img src="image_5.png" /></figure>

Figure 5: Dehydration temperature in [°C] with the dehydration rate in [K·min⁻¹] for the three-step reactions of LaCl₃·7H₂O: decomposition at 25 hPa (solid line), 10 hPa (dash), and 5 hPa (dash-dot). The theoretical onset temperatures (at 0 °C·min⁻¹) are represented by arrows on the Y-axis.

<table><caption>Table 2: Values of the kinetic parameter φ determined by adjustment</caption><thead><tr><th>T, °C</th><th>P<sub>H₂O</sub>, hPa</th><th>φ, mol·m<sup>-2</sup>·s<sup>-1</sup></th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>55</td><td>10</td><td>1.0×10<sup>-5</sup></td><td>0.9923</td></tr><tr><td rowspan="6">60</td><td>5</td><td>2.9×10<sup>-5</sup></td><td>0.9901</td></tr><tr><td>10</td><td>2.2×10<sup>-5</sup></td><td>0.9868</td></tr><tr><td>15</td><td>1.6×10<sup>-5</sup></td><td>0.9907</td></tr><tr><td>20</td><td>1.4×10<sup>-5</sup></td><td>0.9911</td></tr><tr><td>30</td><td>1.0×10<sup>-5</sup></td><td>0.9901</td></tr><tr><td>70</td><td>10</td><td>7.5×10<sup>-5</sup></td><td>0.9867</td></tr><tr><td>80</td><td>10</td><td>1.3×10<sup>-4</sup></td><td>0.9569</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 6: Thermodynamic diagram: literature and experimental points</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 7: Plot of the desorption of LaCl₃ from heptahydrate to anhydrous form for 5, 10, and 25hPa at 0,5°C·min⁻¹ as a function of the temperature</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 8: TG dehydration curves for different initial sample mass at 60°C and 10 hPa.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 9: Kinetic curves during temperature change.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Kinetic curves of lanthanum chloride heptahydrate dehydration as a function of time at 10 hPa: (a) $\alpha(t)$ curves and (b) $d\alpha/dt(\alpha)$ rate curves.</figcaption></figure>
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Figure 11: Kinetic curves of lanthanum chloride heptahydrate dehydration as a function of time at 60 °C: (a) $\alpha(t)$ curves and (b) $d\alpha/dt(\alpha)$ rate curves.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 12: Steady state test for the dehydration of LaCl₃·7H₂O at T = 60°C and P_H₂O = 10 hPa: (a) Heat flux and mass change rate as a function of time and (b) heat flux versus mass change rate</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 13: «$\phi S_m$» test with temperature jumps from 60 °C to 70 °C at a constant water-vapor pressure of 10 hPa.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 14: «$f(\alpha)$» test with the temperature jumps from 60 °C to 70 °C at constant water vapor pressure of 10 hPa.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 15: Numerical optimisation results and the experimental kinetic curves obtained at 10 hPa, and 55, 60, 70, and 80 °C: $\alpha(t)$.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 16: Numerical optimisation results and the experimental kinetic curves obtained at 60 °C, and 5, 10, 15, 20, and 30 hPa: $\alpha(t)$.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 17: Plot of areic growth reactivity $\phi$ with (a) the water vapor pressure and (b) the temperature.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 18: 3D plot of areic growth reactivity $\phi$ with the water vapor pressure and the temperature.</figcaption></figure>