# Thermodynamic modelling assessment of the ternary system Cs-Mo-O

A.L. Smith, T.N. Pham Thi, C. Guéneau, J.-C. Dumas, E. Epifano, W. van Burik, N. Dupin

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A.L. Smith, T.N. Pham Thi, C. Guéneau, J.-C. Dumas, E. Epifano, et al. Thermodynamic modelling assessment of the ternary system Cs-Mo-O. Calphad, 2021, 75, pp.102350. 10.1016/j.calphad.2021.102350. hal-03852595

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# Thermodynamic modelling assessment of the ternary system Cs-Mo-O

A.L. Smith<sup>a,*</sup>, T.N. Pham Thi<sup>b</sup>, C. Guéneau<sup>c</sup>, J.-C. Dumas<sup>b</sup>, E. Epifano<sup>a,e</sup>, W. van Burik<sup>a</sup>, N. Dupin<sup>d</sup>

## ARTICLE INFO

**Keywords:**

Joint oxyde-gaine

Cesium polymolybdates

CALPHAD

Fast neutron reactors

## ABSTRACT

This work reports the thermodynamic modelling assessment of the rather complex Cs-Mo-O system, which is key for the understanding of fission products chemistry in oxide fuelled Light Water Reactors (LWRs) and next generation Sodium-cooled and Lead-cooled Fast Reactors (SFRs and LFRs). The model accounts for the existence of the ternary molybdates $Cs_2MoO_4$ ($\alpha$ and $\beta$), $Cs_2Mo_2O_7$ ($\alpha$ and $\beta$), $Cs_2Mo_3O_{10}$, $Cs_2Mo_4O_{13}$, $Cs_2Mo_5O_{16}$, and $Cs_2Mo_7O_{22}$, for which sufficient structural and thermodynamic information are available in the literature. These phases are treated as stoichiometric in the model. The liquid phase is described with an ionic two-sublattice model, and the gas phase as an ideal mixture. The optimized Gibbs energies are assessed with respect to the known thermodynamic and phase equilibrium data in the $Cs_2MoO_4-MoO_3$ pseudo-binary section. A good agreement is generally obtained within experimental uncertainties. The calculated vapour pressures above $Cs_2MoO_4$ (solid and liquid) are also compared to the available experimental data. Finally, isotherms of the Cs-Mo-O ternary phase diagram are calculated at relevant temperatures for the assessment of the fuel pin behaviour in LWRs, SFRs and LFRs.

## 1. Introduction

The safety assessment of nuclear fuel behaviour in a nuclear fission reactor requires the development of models and simulation codes of the complex chemistry of the numerous fission products generated during irradiation. The chemical elements generated inside the oxide fuel matrix (U, Pu)$O_2$ of Light Water Reactors (LWRs) and next generation Sodium-cooled and Lead-cooled Fast Reactors (SFRs and LFRs) form gaseous or volatile species, metallic precipitates, oxide precipitates, or constitute soluble species according to the classification of Kleykamp [1]. The exact speciation depends on the specific physico-chemical properties of the elements, and the conditions on burnup (as well as isotopic composition of the fuel and neutron flux spectrum), temperature, and oxygen potential under normal operation and accidental conditions. Among those fission products, the speciation of cesium and molybdenum is of particular interest. Cs and Mo are firstly generated with a high fission yield (e.g. ~20 and 21% in the fast reactor Phénix fuel pin ($U_{0.77}Pu_{0.23}O_{2-x}$) at 10 at. % burnup [2]). Moreover, cesium belongs to the class of volatile products [3], and the potential release of <sup>135</sup>Cs and <sup>137</sup>Cs to the coolant and subsequently

to the environment in a severe accident scenario should be assessed carefully for the protection of the general public.

The chemical form of cesium varies with the value of the oxygen potential that increases during irradiation. For low oxygen potential values, e.g. around – 500 kJ·mol<sup>–1</sup> in hypostoichiometric oxide fuels, the formation of CsI, $Cs_2Te$, $Cs_2UO_4$ and $Cs_2MoO_4$ is expected from thermodynamic calculations in increasing order of abundance [2]. For higher oxygen potential values, e.g. above – 400 kJ·mol<sup>–1</sup>, CsI and $Cs_2MoO_4$ become the most stable compounds (again in this increasing order of abundance as the yield of iodine, ~ 9%, is much lower than that of molybdenum) [2]. The formation of $Cs_2MoO_4$ is thus expected to be rather significant, especially at high burnups, as is the case in fast neutron reactors such as in SFRs and LFRs.

A phenomenon very specific to fast reactor oxide fuel pin is in fact observed at the (U, Pu)$O_2$ fuel periphery, namely the formation of a so-called JOG (Joint-Oxyde-Gaine, a French term that corresponds to the oxide-cladding joint) layer, at high burn-up (above 7%–8% fission per initial metal atom), in between fuel and cladding. The volatile fission products mentioned earlier (Cs but also I, Te, Mo) migrate radially due to the high thermal gradients in this type of fuel (~973 K at the

<footnote><sup>a</sup> Delft University of Technology, Faculty of Applied Sciences, Radiation Science & Technology Department, Mekelweg 15 2629 JB Delft, The Netherlands</footnote>

<footnote><sup>b</sup> CEA, DES, IRESNE, DEC, Cadarache F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote><sup>c</sup> Université Paris-Saclay, CEA, Service de la Corrosion et du Comportement des Matériaux dans leur Environnement, 91191, Gif-sur-Yvette, France</footnote>

<footnote><sup>d</sup> Calcul Thermodynamique, 3 rue de l'avenir, 63670 Orcet, France</footnote>

<footnote><sup>e</sup> CIRIMAT laboratory, University of Toulouse, CNRS, INPT, UPS, ENSIACET 4 allée Emile Monso, BP-44362, 31030 Toulouse Cedex 4, France</footnote>

<footnote>* Corresponding author.
E-mail address: a.l.smith@tudelft.nl (A.L. Smith).</footnote>
periphery and ~2273 K at the centre), and build up in the cooler rim region in the JOG layer of thickness 150–300 µm [2].

The JOG layer was found to have a porous and highly heterogeneous structure, and was reported to include the fission products (Cs, Mo, Te, I, Zr, Ba) and cladding components (Fe, Cr) [4]. From the combination of post-irradiation examinations (PIE) and thermodynamic calculations, the JOG layer was shown to be made of Cs₂MoO₄ (main constituent), CsI, Cs₂Te and Cs₂UO₄ [4–10]. The presence of higher polymolybdates was also reported in localized areas by Cappia et al. [4], in particular Cs₂Mo₃O₁₀. The authors also reported the signature of a structure with monoclinic P2₁/c symmetry, which they attributed to Cs₂Ba₂O₃ (following Cs decay into Ba), but could also be interpreted as α-Cs₂Mo₂O₇. From these findings, it is clear that a thorough knowledge of the thermochemistry of the Cs–Mo–O system, and of the thermochemical and thermophysical properties of its constituting cesium polymolybdates is essential for the assessment of the fuel behaviour. It is in particular worth mentioning that Cs₂MoO₄ has a thermal conductivity lower than the fuel by about one order of magnitude [11,12]. It also shows a strong and anisotropic thermal expansion upon increasing temperature at the expected operating temperatures [13]. All these data are necessary to feed so-called Fuel Performance Codes (FPCs) that are able to predict fuel pin behaviour at the engineering scale.

This work reports a thermodynamic modelling assessment of the key Cs–Mo–O system, based on the CALPHAD methodology, and with descriptions compatible with the formalisms used in the TAF-ID database (Thermodynamics of Advanced Fuels – International Database) of the OECD/NEA [14]. The available literature data, including structural, thermodynamic, phase diagram and vapour pressure data, are first critically reviewed, after which the models used are described in detail, and benchmarked against the experimental information.

## 2. Review of literature data

Cs–Mo–O is a rather complex system, with a number of ternary cesium molybdate phases reported: Cs₂MoO₄, Cs₂Mo₂O₇, Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, Cs₂Mo₇O₂₂, Cs₆Mo₂O₉, Cs₀.₁₄MoO₃, Cs₀.₂₅MoO₃, Cs₀.₃₃MoO₃, Cs₀.₂₈MoO₃, and CsMo₄⁻ₓO₁₂ (x = 0.13) [15]. A comprehensive literature review of this system has been reported by Fabrichnaya [16] and more recently by Smith et al. [17]. Its key features are detailed hereafter, including reported phase diagram data, thermodynamic and vapour pressure data. The thermodynamic assessment reported herein accounts for the following ternary phases, for which sufficient structural and thermodynamic information is available: Cs₂MoO₄ (α and β), Cs₂Mo₂O₇ (α and β), Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, and Cs₂Mo₇O₂₂. The knowledge on the other phases is too limited to this date to include them in the thermodynamic model.

### 2.1. Constituting binary systems

The phase diagrams of the three binary sub-systems constituting this ternary system are shown in Figs. 1(a)–1(c). They are already included in the TAF-ID database of the OECD/NEA [14], and were used herein without further modification. The Cs–O system shows eight stable binary oxides Cs₇O, Cs₄O, Cs₇O₂, Cs₃O, Cs₂O, CsO, CsO₂ and CsO₃, all considered stoichiometric in the TAF-ID. The optimized parameters of the corresponding model can be found in Guéneau et al. [18]. The Mo–O system shows five stable binary oxide compounds MoO₂, MoO₃, Mo₄O₁₁, Mo₈O₂₃, Mo₉O₂₆, also treated as stoichiometric. The corresponding model was optimized by Corcoran et al. [19], and the liquid phase parameters were re-assessed by Kauric [20] so as to suppress the appearance of a miscibility gap towards high MoO₃ content in several ternary systems such as Na–Mo–O and Ba–Mo–O [21]. The full list of optimized parameters are given in [21]. The Cs–Mo system is characterized by a low reciprocal solubility of the elements. The available experimental data are scarce. The model for this binary system was optimized in the framework of the Fuelbase project (2011) [22]. The assessed parameters are listed in Appendix.

### 2.2. Structural data

The crystal structures of the aforementioned ternary phases is summarized in Table 1. Note that two polymorphs have been reported for the Cs₂MoO₄ and Cs₂Mo₂O₇ intermediates, with transition temperatures at Tᵢₜ (Cs₂MoO₄, α-β) = (841.3 ± 1.0) K [23] and Tᵢₜ (Cs₂Mo₂O₇, α-β) = (650 ± 5) K [17], respectively. Two structures have also been reported for Cs₂Mo₄O₁₃, one triclinic, in space group P̅−1 (supposedly the phase stable at room temperature) [24], and one monoclinic, in space group C2/c (supposedly a phase stable at high temperatures). However, the exact transition temperature and nature of the transition between these two modifications is not known with certainty. The phase diagram studies of Hoekstra [25] and Spitsyn and Kuleshov [26], discussed in the next section, suggested the existence of a metastable phase, formed when quenching the melt at room temperature. The authors observed exothermic events in their thermal analysis measurements around 608 K and 643 K, respectively, which Hoekstra interpreted as the transformation from the metastable form to the stable phase of Cs₂Mo₄O₁₃ upon heating. The polymorphism of Cs₂Mo₄O₁₃ thus still needs to be clarified. We have included only one modification in the present thermodynamic model.

### 2.3. Phase diagram data

A number of phase equilibria studies have been reported in the pseudo-binary section between Cs₂MoO₄ and MoO₃: by Spitsyn and Kuleshov<fnref n="1" /> [26], Salmon and Caillet<fnref n="2" /> [29], Hoekstra<fnref n="3" /> [25], and Bazarova et al.<fnref n="4" /> [30]. Some discrepancies exist between those studies, however. According to the review of Fabrichnaya in 2007 [16], the data of Hoekstra [25] are believed to be most reliable. Smith et al. [17] very recently re-investigated the Cs₂MoO₄–MoO₃ pseudo-binary phase diagram using Differential Scanning Calorimetry (DSC) and coupled Thermogravimetry-Differential Scanning Calorimetry (TG-DSC) to solve some of these discrepancies. The most important outcomes are summarized below:

* the existence of a phase transition in the Cs₂Mo₂O₇ compound was confirmed, and measured at Tᵢₜ = (650 ± 5) K.

* the melting temperature of Cs₂Mo₂O₇ was measured at Tᵢₜ = (725 ± 5) K, in good agreement with the data reported previously by Smith et al. [27] (720.2 ± 5.0 K). It should be noted that the other sources report higher values: 737 K for Hoekstra [25], 749 K for Salmon and Caillet [29], and 767 K for Spitsyn and Kuleshov [26].

* the (congruent) melting temperature of Cs₂Mo₃O₁₀ was found at Tᵢₜ = (806 ± 5) K, which is ~15 K lower than reported by Hoekstra [25] (820 K), Salmon and Caillet [29] (823 K), and Spitsyn and Kuleshov [26] (818 K). It should be noted that the onset temperature of the heat flow curve was selected for the analysis. The extremum temperature of the same heat flow signal yielded (825 ± 5) K, in closer agreement with the other data.

* the peritectic decomposition of Cs₂Mo₅O₁₆ was found at Tᵢₜ = (813 ± 5) K. Hoekstra, Spitsyn and Kuleshov, and Salmon and Caillet reported 823 K, 823 K and 818 K, respectively.

* the peritectic decomposition of Cs₂Mo₇O₂₂ was found at Tᵢₜ = (836 ± 5) K, in good agreement with Salmon and Caillet (837 K), but again lower than the data of Hoekstra (846 K). It should be noted that in the work of Salmon and Caillet the latter data point was interpreted as the polymorphic transition of a solid solution around the Cs₂Mo₉O₂₈ composition, however.

<footnote n="1">¹ Thermal analysis by a visual-polythermal technique.</footnote>

<footnote n="2">² Solid state reactions coupled with X-ray diffraction, Differential Thermal Analysis and infrared spectroscopy.</footnote>

<footnote n="3">³ X-ray diffraction analyses, simultaneous Differential Thermal Analysis-T thermogravimetry Analysis, infrared and Raman spectroscopy.</footnote>

<footnote n="4">⁴ X-ray diffraction, Differential Thermal Analysis, infrared spectroscopy and electric conductivity measurements.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Phase diagrams of the (a) Cs–O (b) Mo–O and (c) Cs–Mo sub-systems calculated using the TAF-ID models [14].</figcaption></figure>

<table><caption>**Table 1**
Structural data for the ternary phases in the Cs–Mo–O system (1 Å = 0.1 nm). Uncertainties relative to the last digits are given in parenthesis.</caption><thead><tr><th rowspan="2">Phase</th><th rowspan="2">Symmetry</th><th rowspan="2">Space group</th><th colspan="4">Lattice parameters</th><th rowspan="2">Ref.</th></tr><tr><th>a/nm</th><th>b/nm</th><th>c/nm</th><th>β/°</th></tr></thead><tbody><tr><td>α-Cs<sub>2</sub>MoO<sub>4</sub></td><td>Orthorhombic</td><td>Pnma</td><td>1.15623(8)</td><td>0.65406(4)</td><td>0.84906(5)</td><td>90</td><td>[27]</td></tr><tr><td>β-Cs<sub>2</sub>MoO<sub>4</sub><sup>a</sup></td><td>Hexagonal</td><td>P6<sub>3</sub>/mmc</td><td>0.6840(1)</td><td>0.6840(1)</td><td>0.8977(2)</td><td>90</td><td>[13]</td></tr><tr><td>α-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>Monoclinic</td><td>P2<sub>1</sub>/c</td><td>1.5560(2)</td><td>1.5184(1)</td><td>0.72244(7)</td><td>90.120(7)</td><td>[17]</td></tr><tr><td>β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub><sup>b</sup></td><td>Orthorhombic</td><td>Pcm</td><td>0.799136(11)</td><td>1.548597(21)</td><td>0.724613(11)</td><td>90</td><td>[27]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub></td><td>Monoclinic</td><td>C2/c</td><td>1.4472(2)</td><td>0.8404(1)</td><td>0.9464(1)</td><td>97.731(6)</td><td>[17]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>4</sub>O<sub>13</sub></td><td>Triclinic</td><td>P − 1</td><td>0.8396(5)</td><td>0.8655(5)</td><td>1.0413(5)</td><td>see<sup>c</sup></td><td>[24]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>4</sub>O<sub>13</sub></td><td>Monoclinic</td><td>C2/c</td><td>4.592(5)</td><td>1.0418(3)</td><td>0.7923(8)</td><td>92.94(5)</td><td>[28]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>4</sub>O<sub>16</sub></td><td>Monoclinic</td><td>C2/c</td><td>2.1444(2)</td><td>0.55600(4)</td><td>1.4339(1)</td><td>122.735(5)</td><td>[17]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub></td><td>Monoclinic</td><td>C2/c</td><td>2.1546(1)</td><td>0.55385(3)</td><td>1.89179(9)</td><td>122.744(3)</td><td>[17]</td></tr></tbody></table>

The collected data by Smith et al. [17] are found generally lower than Hoekstra [25]. It is not clear, however, how the thermal analysis data were treated in the work of Hoekstra with respect to temperature calibration and choice of the offset or onset temperature of the signals, as reported by [17]. The most recent phase equilibrium data by Smith et al. [17] have been selected for the re-assessment of the Cs–Mo–O system as the temperature calibration of the DSC and TG-DSC measurements was carefully checked with the measurement of the well-known three phase transitions and melting temperatures of the Na₂MoO₄ compound [17].

## 2.4. Thermodynamic data

The standard enthalpies of formation of α-Cs₂MoO₄, α-Cs₂Mo₂O₇, Cs₂Mo₃O₁₀, Cs₂Mo₅O₁₆ and Cs₂Mo₇O₂₂ were measured by O'Hare and Hoekstra [31,32], Smith et al. [17,33] and Benigni et al. [34]. The latter data are summarized in Table 2. A very good agreement is seen between the various studies.

The enthalpies of formation listed in Table 2 were derived using a thermochemical cycle involving the enthalpy of formation of MoO₃(cr), i.e. –(745.0 ± 1.0) kJ·mol⁻¹ (value selected in the review by Cordfunke and Konings [23]). The reported uncertainty ± 1 kJ·mol⁻¹ is most probably largely underestimated, however, as stressed by [17]. This value

<footnote><sup>a</sup>Data measured at *T* = 948 K.</footnote>

<footnote><sup>b</sup>Data measured at *T* = 683 K.</footnote>

<footnote><sup>c</sup>α = 106.158(5), β = 103.686(5), γ = 109.761(5).</footnote>
<table><caption>**Table 2**

Thermodynamic data reported on the cesium polymolybdates.</caption><thead><tr><th>Compound</th><th>Δ<sub>f</sub> H<sup>o</sup><sub>m</sub> (cr, 298.15 K)<br>(kJ·mol<sup>-1</sup>)</th><th>Ref.</th><th>C<sup>o</sup><sub>p,m</sub> (cr, 298.15 K)<br>(J·K<sup>-1</sup>·mol<sup>-1</sup>)</th><th>S<sup>o</sup><sub>m</sub> (cr, 298.15 K)<br>(J·K<sup>-1</sup>·mol<sup>-1</sup>)</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="3">α-Cs<sub>2</sub>MoO<sub>4</sub></td><td>-(1514.5 ± 1.0)</td><td>[23,31]</td><td rowspan="3">148.67 ± 0.15</td><td rowspan="3">248.35 ± 0.30</td><td rowspan="3">[23,35]</td></tr><tr><td>-(1514.69 ± 1.54)</td><td>[33]</td></tr><tr><td>-(1513.56 ± 1.11)</td><td>[34]</td></tr><tr><td rowspan="3">α-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>-(2302.4 ± 2.1)</td><td>[23,32]</td><td rowspan="3">211.9 ± 2.1</td><td rowspan="3">317.4 ± 4.3</td><td rowspan="3">[27]</td></tr><tr><td>-(2301.84 ± 2.37)</td><td>[34]</td></tr><tr><td>-(2301.6 ± 4.7)</td><td>[17]</td></tr><tr><td rowspan="2">Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub></td><td>-(3077.17 ± 3.47)</td><td>[34]</td><td></td><td></td><td></td></tr><tr><td>-(3075.6 ± 6.5)</td><td>[17]</td><td></td><td></td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub></td><td>-(4601.7 ± 5.96)</td><td>[34]</td><td></td><td></td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub></td><td>-(6087.16 ± 8.79)</td><td>[34]</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 3**

Re-calculated enthalpies of formation of the cesium polymolybdates based on an uncertainty of 1% on the enthalpy of formation of MoO₃(cr) involved in the thermochemical cycles of the solution calorimetry measurements. The values selected for the present assessment are marked in bold.</caption><thead><tr><th>Compound</th><th>Δ<sub>f</sub> H<sup>o</sup><sub>m</sub> (cr, 298.15 K)<br>(kJ·mol<sup>-1</sup>)</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="3">α-Cs<sub>2</sub>MoO<sub>4</sub></td><td>-(1514.5 ± 7.45)</td><td>[23,31]</td></tr><tr><td>-(1514.69 ± 7.54)</td><td>[27]</td></tr><tr><td>-(1513.56 ± 7.52)</td><td>[34]</td></tr><tr><td rowspan="5">α-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>-(1514.2 ± 4.3)</td><td>Weighted average (this work)</td></tr><tr><td>-(2302.4 ± 14.9)</td><td>[23,32]</td></tr><tr><td>-(2301.84 ± 15.07)</td><td>[34]</td></tr><tr><td>-(2301.6 ± 15.5)</td><td>[17]</td></tr><tr><td>-(2302.1 ± 8.8)</td><td>Weighted average (this work)</td></tr><tr><td rowspan="3">Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub></td><td>-(3077.17 ± 22.35)</td><td>[34]</td></tr><tr><td>-(3075.6 ± 23.1)</td><td>[17]</td></tr><tr><td>-(3076.8 ± 16.1)</td><td>Weighted average (this work)</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub></td><td>-(4601.7 ± 37.25)</td><td>[34]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub></td><td>-(6087.16 ± 52.16)</td><td>[34]</td></tr></tbody></table>

was derived from the average of the measurements of Staskiewicz et al. (–744.65 ± 0.40 kJ·mol⁻¹) [36] and Mah (–745.4 ± 0.5 kJ·mol⁻¹) [37]. However, the latter data were derived from the incomplete combustion of molybdenum and MoO₂ to MoO₃ (it reached only 70%–93%). Even if corrections were made to account for this, we believe that the quoted final uncertainties amounting to less than 0.07% are largely underestimated. The review by Cordfunke and Konings [23] assigning an uncertainty of 0.1% is already more conservative. But in the present work, the uncertainties on the enthalpies of formation have been recalculated considering an uncertainty of 1% (± 7.45 kJ·mol⁻¹) on the enthalpy of formation of MoO₃, which is believed to be more realistic as reported already in [17]. The newly assessed uncertainties are listed in Table 3. Moreover, the weighted average of the latter values has been calculated for each composition, and is selected for the present thermodynamic assessment.<fnref n="5" />

The standard entropies of α-Cs₂MoO₄ and α-Cs₂Mo₂O₇ were obtained from low-temperature heat capacity measurements using adiabatic calorimetry [35] and thermal-relaxation calorimetry [27], respectively. The measured data are listed in Table 2 and selected for the present assessment.

$$X = \frac{\sum_i (X_i / \sigma_i^2)}{\sum_i (1 / \sigma_i^2)} \qquad (1)$$

The uncertainty was calculated using the formula:

$$\sigma_X = \sqrt{\frac{1}{\sum_i (1/\sigma_i^2)}} \qquad (2)$$

Fredrickson and Chasanov [39], Konings and Cordfunke [40], and Denielou et al. [41] reported high temperature enthalpy increment measurements for Cs₂MoO₄ in the temperature ranges (845–1191 K), (415–700 K), and (1232–1500 K), respectively. The review of Cordfunke and Konings [23] discards the data of Fredrickson and Chasanov because of discrepancies with the other sets of data. Kohli and Lacom [42] reported heat capacity measurements in the range 300–800 K using differential scanning calorimetry (DSC). The high temperature heat capacity equations selected in the review by Cordfunke and Konings [23] for α-Cs₂MoO₄ (orthorhombic in space group Pbam) and β-Cs₂MoO₄ (hexagonal in space group P6₃/mmc) are those of Konings and Cordfunke [40]. The data of Denielou et al. [41] were adopted for the liquid phase in [23]. The heat capacity function of Cs₂MoO₄ was optimized as part of this assessment based on the low-temperature heat capacity data of Osborne et al. [35], high temperature heat capacity data of Kohli and Lacom [42], and enthalpy increment data of Konings and Cordfunke [40].

The high temperature heat capacity of Cs₂MoO₇ was measured in the range (310–700 K) by Kohli [43] based on DSC measurements and the step method, but found in poor agreement with the low-temperature heat capacity data of Smith et al. [27]. In their work, Smith et al. [27] suggested an estimation of the high-temperature heat capacity by combining their measured low-temperature heat capacity data in the range (248.7–313.2 K) with data estimated above T = 500 K using the Neumann–Kopp approximation applied to the heat capacity functions of Cs₂MoO₄ [23,35] and MoO₃ [23,44]. The latter function was retained for the present assessment of the Cs–Mo–O system. (See Table 4.)

The reported phase transition temperatures for the cesium polymolybdates are summarized in Table 7 together with the optimized values in the present CALPHAD model. The associated transition enthalpies were determined by Konings and Cordfunke [40], Fredrickson and Chasanov [39], Denielou et al. [41], and Smith et al. [17]. They are also listed in Table 7, where they are compared to the optimized values.

## 2.5. Vapour pressure data

Several studies have been reported on the vaporization and sublimation of Cs₂MoO₄ (see Table 5), i.e. by Johnson [45] (1070.6–1167.2 K), Tangri et al. [46] (1229–1310 K), Cordfunke et al. [47] (1116.8–1190.1 K), Yamawaki et al. [48] (1134.36–1242.16 K), Kazenas et al. [49] (1026–1148 K), and Stolyarova et al. [50] (1030–1198 K), with a rather good agreement between the various studies. The various works agree on a congruent vaporization and sublimation process with the sole formation of Cs₂MoO₄(g) in the gas phase. The reported experimental data are shown in Fig. 6 together with the calculated vapour pressure over Cs₂MoO₄(cr, l) from our thermodynamic model.

Thi-Mai-Dung Do et al. [52] investigated the vaporization of Cs₂Mo₂O₇ by combining thermogravimetry with a transpiration method. The authors suggested a congruent vaporization for Cs₂Mo₂O₇(l) and

<footnote n="5">⁵ Note that the final averaged result X was calculated as recommended in [38] for independent source of data as a weighted mean using the formula:</footnote>
<table><caption>**Table 4**
Summary of heat capacity data for $Cs_2MoO_4$.</caption><thead><tr><th rowspan="2">Phase</th><th colspan="4">C<sub>p,m</sub> = A + B·T + C·T<sup>-2</sup> + D·T<sup>2</sup>/J·K<sup>-1</sup>·mol<sup>-1</sup></th><th rowspan="2">Temp. range/K</th><th rowspan="2">Ref.</th></tr><tr><th>A</th><th>B</th><th>C</th><th>D</th></tr></thead><tbody><tr><td>Cs<sub>2</sub>MoO<sub>4</sub>(cr, α)</td><td>116.404</td><td>108.2212·10<sup>-3</sup></td><td></td><td></td><td>298.15–841.3</td><td>[23]</td></tr><tr><td>Cs<sub>2</sub>MoO<sub>4</sub>(cr, β)</td><td>122.295</td><td>97.049·10<sup>-3</sup></td><td></td><td></td><td>841.3–1229.5</td><td>[23]</td></tr><tr><td>Cs<sub>2</sub>MoO<sub>4</sub>(cr)</td><td>95.9182</td><td>18.82224·10<sup>-2</sup></td><td>-95472</td><td>-7.0314·10<sup>-5</sup></td><td>298.15–1229.5</td><td>This work</td></tr><tr><td>Cs<sub>2</sub>MoO<sub>4</sub>(l)</td><td>210.154</td><td></td><td></td><td></td><td>&gt;1229.5</td><td>[23]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub>(cr)</td><td>266.321</td><td>7.173·10<sup>-2</sup></td><td>-5.65075·10<sup>6</sup></td><td></td><td>310–700</td><td>[43]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub>(cr)</td><td>131.487</td><td>0.30519</td><td></td><td>-1.12277·10<sup>-4</sup></td><td>250–720</td><td>[27]</td></tr></tbody></table>

<table><caption>**Table 5**
Experimental data on the vaporization behaviour of $Cs_2MoO_4$. KEMS: Knudsen Effusion Mass Spectrometry.</caption><thead><tr><th>Reference</th><th>logP<sub>10</sub>(Pa)</th><th>T(K)</th><th>Method</th><th>Cell/crucible material</th></tr></thead><tbody><tr><td>Johnson [45]</td><td>(11.58 ± 0.02)–(13984 ± 600)/T(K)<sup>a</sup></td><td>1070.6–1167.2</td><td>KEMS</td><td>Alumina<sup>c</sup></td></tr><tr><td>Tangri et al. [46]</td><td>(9.37 ± 0.29)–(11452 ± 370)/T(K)<sup>b</sup></td><td>1229–1310 K</td><td>Transpiration<sup>b</sup></td><td>Alumina</td></tr><tr><td>Cordfunke et al. [47]</td><td>(12.038 ± 0.507)–(14470.5 ± 584.7)/T(K)<sup>f</sup></td><td>1116.8–1190.1 K</td><td>Transpiration<sup>b</sup></td><td>Gold<sup>d</sup></td></tr><tr><td>Cordfunke et al. [47]</td><td>(11.850 ± 0.567)–(14229.3 ± 653.6)/T(K)<sup>g</sup></td><td>1116.8–1190.1 K</td><td>Transpiration<sup>b</sup></td><td>Gold<sup>d</sup></td></tr><tr><td>Yamawaki et al. [48]</td><td>9.80–12100/T(K)</td><td>1231.56–1242.16 K</td><td>KEMS</td><td>Alumina<sup>e</sup></td></tr><tr><td>Yamawaki et al. [48]</td><td>11.02–13600/T(K)</td><td>1134.36–1206.01 K</td><td>KEMS</td><td>Alumina<sup>e</sup></td></tr><tr><td>Kazenas et al. [49]</td><td>11.47–13724/T(K)</td><td>1026–1148 K</td><td>KEMS</td><td>Platinum</td></tr><tr><td>Stolyarova et al. [50]</td><td>(11.5 ± 0.9)–(13616 ± 514)/T(K)</td><td>1030–1198 K</td><td>KEMS (twin-cells)</td><td>Platinum</td></tr></tbody></table>

<sup>a</sup>The authors report a 5% uncertainty on the vapour pressures.

<sup>b</sup>The authors report a 10% uncertainty on the vapour pressures.

<sup>c</sup>Molybdenum effusion cell with gold liner and alumina crucible inside the gold liner.

<sup>d</sup>Y<sub>2</sub>O<sub>3</sub>-stabilized zirconia condenser tube with a protective gold tube.

<sup>e</sup>Molybdenum effusion cell with alumina liner.

<sup>f</sup>Data based on cesium analyses.

<sup>g</sup>Data based on molybdenum analyses.

<sup>h</sup>Dry oxygen carrier gas.

<table><caption>**Table 6**
Thermodynamic data optimized in this work. SER refers to the phase of the element stable at 298.15 K. Optimized coefficients are shown in bold.</caption><thead><tr><th>Phase</th><th>Gibbs energy/(J·mol<sup>-1</sup>)</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="4">Liquid</td><td><sup>a</sup>G(Cs<sup>+</sup>)<sub>2</sub>(MoO<sub>4</sub><sup>2-</sup>) – 2<sup>9</sup>H<sub>Cs</sub><sup>SER</sup> – <sup>9</sup>H<sub>Mo</sub><sup>SER</sup> – 4<sup>9</sup>H<sub>O</sub><sup>SER</sup> = G<sub>Cs,MoO<sub>4</sub></sub> + 36437.799 – 31.504173T</td><td>This work</td></tr><tr><td>L<sup>0</sup>(Cs<sup>+</sup>)<sub>p</sub>(MoO<sub>4</sub><sup>2-</sup>,MoO<sub>3</sub>)<sub>Q</sub> = –96000 – 11T</td><td>This work</td></tr><tr><td>L<sup>1</sup>(Cs<sup>+</sup>)<sub>p</sub>(MoO<sub>4</sub><sup>2-</sup>,MoO<sub>3</sub>)<sub>Q</sub> = 62000</td><td>This work</td></tr><tr><td>L<sup>2</sup>(Cs<sup>+</sup>)<sub>p</sub>(MoO<sub>4</sub><sup>2-</sup>,MoO<sub>3</sub>)<sub>Q</sub> = –9000</td><td>This work</td></tr><tr><td rowspan="8">Hexavalent molybdates</td><td>G<sub>d–Cs<sub>2</sub>MoO<sub>4</sub></sub> = G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub></td><td>This work</td></tr><tr><td>G<sub>β–Cs<sub>2</sub>MoO<sub>4</sub></sub> = G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub> + 4600 – 5.4677T</td><td>This work</td></tr><tr><td>G<sub>d–Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub></td><td>This work</td></tr><tr><td>G<sub>β–Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> + 423 – 0.650769T</td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> + 2G<sub>MoO<sub>3</sub></sub> – 53000 – 12T</td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub> + 3G<sub>MoO<sub>3</sub></sub> – 76017 + 8.9T</td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub> + 4G<sub>MoO<sub>3</sub></sub> – 89000 + 17T</td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> + 6G<sub>MoO<sub>3</sub></sub> – 103000 + 26.4T</td><td>This work</td></tr><tr><td rowspan="6">Gas functions</td><td><sup>a</sup>G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub><sup>gai</sup> – 2<sup>9</sup>H<sub>Cs</sub><sup>SER</sup> – <sup>9</sup>H<sub>Mo</sub><sup>SER</sup> – 4<sup>9</sup>H<sub>O</sub><sup>SER</sup> = G<sub>Cs,MoO<sub>4</sub></sub><sup>g</sup></td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>MoO<sub>4</sub></sub> = –1554486.3 + 448.47296T–95.9182T ln(T)–0.0941112T<sup>2</sup> +1.1719·10<sup>-5</sup>T<sup>3</sup> + 47736T<sup>-1</sup></td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></sub> = –2354000+651.0T – 131.487Tln(T) – 0.152595T<sup>2</sup> +1.87128333·10<sup>-5</sup>T<sup>3</sup></td><td>This work</td></tr><tr><td>G<sub>Cs<sub>2</sub>MoO<sub>3</sub></sub> = –1246344 + 340.447912T – 110.7356Tln(T) – 0.05133275T<sup>2</sup></td><td>[51]</td></tr><tr><td>+1.09630117·10<sup>-5</sup>T<sup>3</sup> +454105.3T<sup>-1</sup> (298.15–600 K)</td><td>[51]</td></tr><tr><td>= –1263115.58 + 612.12144T – 153.3783Tln(T) – 0.002280958T<sup>2</sup></td><td>[51]</td></tr><tr><td>+2.12845167·10<sup>-7</sup>T<sup>3</sup> +1714266T<sup>-1</sup> (600–1700 K)</td><td>[51]</td></tr><tr><td>= –1258019.45 + 595.921622T – 151.5654Tln(T) – 0.001915096T<sup>2</sup></td><td>[51]</td></tr><tr><td>+1.049755·10<sup>-7</sup>T<sup>3</sup> (1700–6000 K)</td><td>[51]</td></tr></tbody></table>

the formation of $Cs_2Mo_2O_7(g)$ gaseous species. They derived the following equation for the total vapour pressure in the temperature range (1273–1573 K) based on their thermogravimetry data and the Langmuir equation for free evaporation [53]:

$$ \log_{10} P(Pa) = (8.95 \pm 0.07) – (10300 \pm 100)/T(K) \quad (3) $$

The authors suggest the formation of $Cs_2Mo_2O_7(g)$ based on Raman spectra of deposits collected on platinum sheets in the downstream of a transpiration set-up.

We have accounted for $Cs_2MoO_4(g)$ in the description of the gas phase in the present model. However, we have not included

$Cs_2Mo_2O_7(g)$ at this stage. We recommend to first confirm the formation and stability of such a heavy molecule as $Cs_2Mo_2O_7(g)$ (molar weight 569.7 g·mol<sup>-1</sup>) by analysis directly the gas phase itself using mass spectrometry, and to proceed to the determination of all necessary thermodynamic functions (enthalpy of formation and standard entropy at 298.15 K as well as heat capacity at high temperatures).

### 3. Thermodynamic model

The optimization was performed using the PARROT module of the Thermo-Calc software (Version 2016b) [54,55]. The Gibbs energy
<figure><img src="image_5.png" /><figcaption>Fig. 2. (a) Pseudo-binary Cs₂MoO₄–MoO₃ phase diagram calculated using the present model compared to the equilibrium data reported in the literature by Spitsyn and Kuleshov [26], Salmon and Caillet [29], Hoekstra [25], and Smith et al. [17]; (b) Enlarged section between x(MoO₃) = 0.5 and x(MoO₃) = 1 of the pseudo-binary Cs₂MoO₄–MoO₃ phase diagram.</figcaption></figure>

functions in the model are referred to the enthalpy of the pure elements in their stable state at room temperature 298.15 K and 1 bar (°Hᵢ<sup>SER</sup>(298.15 K)). The optimized parameters are listed in Table 6.

### 3.1. Pure elements

The Gibbs energy function of the pure element i at temperature T and in its state φ is given by:

$$G_{i}^{\phi}(T) - {}^{\circ}H_{i}^{SER}(298.15 \text{ K}) = a + b \cdot T + c \cdot T \cdot \ln T + \sum d_{n} T^{n} \quad (4)$$

where n is an integer (2, 3, -1...) and a, b, c, and d_n are parameters assessed based on experimental and theoretical information. The parameters reported by Dinsdale are used in this work for pure cesium, molybdenum, and oxygen [56].

### 3.2. Binary oxides

The binary oxides Cs₇O, Cs₄O, Cs₇O₂, Cs₃O, Cs₂O, CsO, CsO₂, CsO₃, MoO₂, MoO₃, Mo₄O₁₁, Mo₈O₂₃, and Mo₉O₂₆ are described as stoichiometric compounds. The corresponding Gibbs energy functions have the same form as in Eq. (4).

$$G^{\phi}(T) - \sum_{i} n_{i}^{\phi 0} H_{i}^{SER}(298.15 \text{ K}) = a + b \cdot T + c \cdot T \cdot \ln T + \sum d_{n} T^{n} \quad (5)$$

where nᵢ<sup>φ</sup> is the number of atoms of the element i in the oxide formula. These are taken from the TAF-ID database [14].

### 3.3. Hexavalent ternary molybdates

Considering the available structural and thermodynamic data in the literature, the present model considers only the existence of the following ternary phases: Cs₂MoO₄ (α and β), Cs₂Mo₂O₇ (α and β), Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, and Cs₂Mo₇O₂₂. They are treated as stoichiometric. Their Gibbs energies have been expressed based on the recommended enthalpies of formation, entropies and heat capacities (see literature review). The enthalpies of formation and standard entropies have been further optimized to fit the phase equilibrium data in the Cs₂MoO₄–MoO₃ pseudo-binary section. Moreover, only the heat capacity of Cs₂MoO₄ has been optimized, with a single function for the α and β modifications. The heat capacity of Cs₂Mo₂O₇ was taken from the selection made in the literature review without further optimization. The heat capacities of Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, and Cs₂Mo₇O₂₂ were calculated using the Neumann–Kopp rule applied to Cs₂MoO₄ and MoO₃, and were not further optimized.

The Gibbs energies of the ternary molybdates is given by:

$$G^{\phi}(T) - \sum_{i} n_{i}^{\phi 0} H_{i}^{SER}(298.15 \text{ K}) = a + b \cdot T + c \cdot T \cdot \ln T + \sum d_{n} T^{n} \quad (6)$$

where nᵢ<sup>φ</sup> is the number of atoms of the element i in the oxide formula.

### 3.4. Liquid phase

The liquid phase is described using an ionic two-sublattice model, with Cs⁺ and Mo⁺ cations on the first sublattice, and MoO₄⁻, O₂⁻ anions, charged vacancies VaO⁻, neutral MoO₃, and neutral oxygen O on the second sublattice:

$$(\text{Cs}^{+}, \text{Mo}^{4+})_p (\text{MoO}_4^{2-}, \text{O}^{2-}, \text{Va}^{O-}, \text{MoO}_3, \text{O})_Q \quad (7)$$

P and Q are equal to the average charge of the opposite sublattice:

$$\begin{align}
Q &= y_{\text{Cs}^{+}} + 4y_{\text{Mo}^{4+}} \nonumber \\
P &= 2y_{\text{MoO}_{4}^{2-}} + 2y_{\text{O}_{2}^{2-}} + Qy_{\text{Va}^{O-}} \tag{8}
\end{align}$$

where yᵢ is the site fraction for species i in the sublattice considered. P and Q vary with composition via the site fractions so as to keep the phase electrically neutral.

### 3.5. Gas phase

The gas phase is described as an ideal mixture of the gaseous species Cs, Cs₂, Cs₂O, Cs₂O₂, CsO, Mo, Mo₂, MoO, MoO₂, MoO₃, Mo₂O₆, Mo₃O₉, Mo₄O₁₂, Mo₅O₁₅, Cs₂MoO₄, O, O₂, and O₃. The Gibbs energy is expressed by:

$$G^{\text{gas}} = \sum_{i} y_{i} {}^{\circ}G_{i}^{\text{gas}} + RT \sum_{i} y_{i} \ln y_{i} + RT \ln P/P^{\circ} \quad (9)$$

where yᵢ is the fraction of the species i in the gas phase, °Gᵢ<sup>gas</sup> the standard Gibbs energy of the gaseous species i, and P<sup>o</sup> the standard pressure. The thermodynamic functions of the gaseous species were not optimized in this work. In particular, the Gibbs energy function of Cs₂MoO₄(g) was taken from [51].

## 4. Results and discussion

### 4.1. Cs₂MoO₄–MoO₃ pseudo-binary phase diagram

The calculated Cs₂MoO₄–MoO₃ pseudo-binary phase diagram is shown in Figs. 2(a) and 2(b). The associated invariant equilibria are
<table><caption>**Table 7**

Invariant equilibria calculated in this work compared to the literature data.</caption><thead><tr><th>Equilibrium reaction</th><th>Equilibrium</th><th>x(MoO<sub>3</sub>)</th><th>T/K</th><th>Δ<sub>tr</sub> H<sub>m</sub><sup>o</sup>/kJ·mol<sup>-1</sup></th><th>Ref.</th></tr></thead><tbody><tr><td>α-Cs<sub>2</sub>MoO<sub>4</sub> = β-Cs<sub>2</sub>MoO<sub>4</sub></td><td>Polymorphism</td><td>0.5</td><td><strong>841.3</strong><br>835<sup>a</sup>/839<sup>b</sup> ± 5<sup>a</sup></td><td><strong>4.6</strong><br>(4.6 ± 0.1)</td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>841.3 ± 1.0</td><td></td><td>[23,40]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>844</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>840.2 ± 1.1</td><td></td><td>[27]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>841 ± 1</td><td></td><td>[57]</td></tr><tr><td>β-Cs<sub>2</sub>MoO<sub>4</sub> = liq.</td><td>Congruent melting</td><td>0.5</td><td><strong>1223</strong><br>1225<sup>a</sup>/1226<sup>b</sup> ± 5<sup>a</sup></td><td><strong>31.8</strong><br>(31.8 ± 0.8)</td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>1229.5 ± 0.2</td><td></td><td>[23,41]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>1213</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>1208.4 ± 0.5</td><td></td><td>[27]</td></tr><tr><td>Liq. = α-Cs<sub>2</sub>MoO<sub>4</sub> + β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>Eutectic</td><td>0.661</td><td><strong>732.2</strong><br>731</td><td></td><td>This work<br>[25,26]</td></tr><tr><td>–</td><td>–</td><td>0.645</td><td>733</td><td></td><td>[29]</td></tr><tr><td>–</td><td>–</td><td>0.649</td><td><strong>0.6667</strong><br>650</td><td><strong>0.423</strong><br>(0.423 ± 0.191)<sup>a</sup></td><td>This work<br>[17]</td></tr><tr><td>α-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> = β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>Polymorphism</td><td>–</td><td>650<sup>a,b</sup> ± 5</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>668</td><td></td><td></td></tr><tr><td>β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> = Liq.</td><td>Congruent melting</td><td>0.667</td><td><strong>732.5</strong><br>725<sup>a,b</sup> ± 5</td><td><strong>77.7</strong><br>(81.2 ± 7.1)<sup>a</sup></td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>737</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>749</td><td></td><td>[29]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>767</td><td></td><td>[26]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>720 ± 1</td><td></td><td>[27]</td></tr><tr><td>Liq. = β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> + Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub></td><td>Eutectic</td><td>0.6674</td><td><strong>732.5</strong><br>736</td><td></td><td>This work<br>[25]</td></tr><tr><td>–</td><td>–</td><td>0.669</td><td>736</td><td></td><td>[29]</td></tr><tr><td>–</td><td>–</td><td>0.676</td><td>736</td><td></td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub> = Liq.</td><td>Congruent melting</td><td>0.75</td><td><strong>812.3</strong><br>806<sup>a,b</sup> ± 5</td><td><strong>107.9</strong><br>(81.8 ± 16.8)<sup>a</sup></td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>820</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>823</td><td></td><td>[29]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>818</td><td></td><td>[26]</td></tr><tr><td>Liq. = Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub> + Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub></td><td>Eutectic</td><td>0.788</td><td><strong>793.2</strong><br>798</td><td></td><td>This work<br>[25]</td></tr><tr><td>–</td><td>–</td><td>0.784</td><td>796</td><td></td><td>[26]</td></tr><tr><td>–</td><td>–</td><td>0.781</td><td>792</td><td></td><td>[29]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>4</sub>O<sub>13</sub> = Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub> + Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub></td><td>Peritectoid</td><td>0.8</td><td><strong>783.9</strong><br>776</td><td></td><td>This work<br>[25]</td></tr><tr><td>–</td><td>–</td><td>0.8</td><td>797</td><td></td><td>[29]</td></tr><tr><td>–</td><td>–</td><td>0.773(5)</td><td>781 ± 10<sup>b</sup></td><td></td><td>[17]</td></tr><tr><td>–</td><td>–</td><td>0.811(5)</td><td>787 ± 10<sup>b</sup></td><td></td><td>[17]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub> = Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub> + Liq.</td><td>Peritectic</td><td>0.833</td><td><strong>808.7</strong><br>810<sup>a</sup>/813<sup>b</sup> ± 5</td><td><strong>134.8</strong><br>(121.3 ± 8.2)<sup>a</sup></td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>823</td><td></td><td>[25,26]</td></tr><tr><td>–</td><td>–</td><td>0.813</td><td>818</td><td></td><td>[29]</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub> = MoO<sub>3</sub> + Liq</td><td>Peritectic</td><td>0.875</td><td><strong>836.2</strong><br>835<sup>a</sup>/836<sup>b</sup> ± 5</td><td><strong>237.9</strong><br>(216.5 ± 24.2)<sup>a</sup></td><td>This work<br>[17]</td></tr><tr><td>–</td><td>–</td><td>–</td><td>848</td><td></td><td>[26]</td></tr><tr><td>–</td><td>–</td><td>0.862</td><td>846</td><td></td><td>[25]</td></tr><tr><td>–</td><td>–</td><td>0.857</td><td>837</td><td></td><td>[29]</td></tr></tbody></table>

<sup>a</sup>Data measured by DSC.

<sup>b</sup>Data measured by TG-DSC.

<table><caption>**Table 8**

Enthalpies of formation and standard entropies of the ternary cesium molybdates optimized in this work.</caption><thead><tr><th rowspan="2">Compound</th><th colspan="2">Δ<sub>f</sub> H<sub>m</sub><sup>o</sup> (298.15 K) (kJ·mol<sup>-1</sup>)</th><th colspan="2">S<sub>m</sub><sup>o</sup> (298.15 K) (J·K<sup>-1</sup>·mol<sup>-1</sup>)</th></tr><tr><th>Optimized</th><th>Selected<sup>a</sup></th><th>Optimized</th><th>Selected<sup>a</sup></th></tr></thead><tbody><tr><td>α-Cs<sub>2</sub>MoO<sub>4</sub></td><td>-1517.8</td><td>(-1514.2 ± 4.3)</td><td>247.5</td><td>(248.35 ± 0.30)</td></tr><tr><td>α-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub></td><td>-2302.2</td><td>(-2302.1 ± 8.8)</td><td>315.6</td><td>(317.4 ± 4.3)</td></tr><tr><td>Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub></td><td>-3060.0</td><td>(-3076.8 ± 16.1)</td><td>414.8</td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>3</sub>O<sub>13</sub></td><td>-3827.6</td><td></td><td>471.6</td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>5</sub>O<sub>16</sub></td><td>-4585.2</td><td>(-4601.7 ± 37.25)</td><td>541.1</td><td></td></tr><tr><td>Cs<sub>2</sub>Mo<sub>7</sub>O<sub>22</sub></td><td>-6088.4</td><td>(-6087.16 ± 52.16)</td><td>687.0</td><td></td></tr></tbody></table>

<sup>a</sup>Values selected after critical review of the literature (see Section 2).

listed in Table 7. This pseudo-binary system includes three eutectic equilibria, calculated at {x(MoO<sub>3</sub>) =0.661 and T = 732.2 K}, {x(MoO<sub>3</sub>) =0.6674 and T = 732.5 K}, and {x(MoO<sub>3</sub>) =0.788 and T = 793.2 K}, respectively. The transition temperatures (841.3 K for the α to β transition, and 1223 K for the congruent melting) of Cs<sub>2</sub>MoO<sub>4</sub> are in

very good agreement with the literature data. The α to β transition temperature of Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> is calculated at 650 K, as reported in the work of Smith et al. [17]. The congruent melting of the same compound is calculated at 732.5 K, in between the data of Smith et al. [17] (i.e. 725 ± 5 K), and the data of Hoekstra [25] (737 K), Salmon and Caillet [29] (749 K), and Spitsyn and Kuleshov [26] (767 K). Note that the nature of the thermal decomposition of β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> (congruent melting or peritectic decomposition) is not known with certainty [17]. In the present model, the calculated congruent melting point is only 0.01 K higher than the eutectic equilibrium {Liq = β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> + Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub>} and 0.35 K higher than the eutectic equilibrium {Liq = β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> + α-Cs<sub>2</sub>MoO<sub>4</sub>}, which is not possible to distinguish experimentally. A post-characterization using microscopy would be required after thermal analysis to ascertain if β-Cs<sub>2</sub>Mo<sub>2</sub>O<sub>7</sub> shows a congruent melting or a peritectic decomposition upon heating. The congruent melting of Cs<sub>2</sub>Mo<sub>3</sub>O<sub>10</sub> is calculated at 812.3 K, which is again in between the data of Smith et al. [17] (i.e. 806 ± 5 K) and that of Hoekstra [25] (820 K), Salmon and Caillet [29] (823 K), and Spitsyn and Kuleshov [26] (818 K). The calculated peritectoid decomposition of
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a) Enthalpy increments of Cs₂MoO₄ calculated with the present description (solid line) compared to the literature data of Osborne et al. [35], Konings and Cordfunke [40], and Kohli and Lacom [42]. (b) Heat capacity of Cs₂MoO₄ calculated with the present description (solid line) compared to the literature data of Osborne et al. [35] and Kohli and Lacom [42].</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Heat capacity of Cs₂Mo₂O₇ calculated in the model (solid line) compared to the literature data of Kohli [43] and Smith et al. [27].</figcaption></figure>

Cs₂Mo₄O₁₃ (783.9 K) is in very good agreement with Smith et al. [17] (781 and 787 K) and Hoekstra [25] (776 K). By contrast with the other studies, Salmon and Caillet report a peritectic decomposition for Cs₂Mo₄O₁₃ at 797 K [29], while Spitsyn and Kuleshov [26] report a

melting at 807 K. Finally, the calculated peritectic decompositions of Cs₂Mo₅O₁₆ and Cs₂Mo₇O₂₂ (i.e. 808.7 K and 836.2 K, respectively) are in very good agreement with the data of Smith et al. [17] selected for the optimization.

## 4.2. Optimized thermodynamic data

The standard enthalpies of formation and standard entropies at 298.15 K optimized in this work are listed in Table 8 and shown in Fig. 5(a), where they are compared to the selected values (see Section 2). A very good agreement is obtained, within the experimental uncertainties.

The calculated enthalpy increments and heat capacities of Cs₂MoO₄ and Cs₂Mo₂O₇ are shown in Figs. 3(a), 3(b), and 4, where they are compared to the available literature data. As explained in more detail in the literature review, the estimation by Smith et al. [27] for the heat capacity of Cs₂Mo₂O₇ was preferred over the DSC data of Kohli [43]. The heat capacity function at high temperature shows a smooth transition with the low-temperature data.

Finally, the calculated transition enthalpies associated with the invariant reactions in the Cs₂MoO₄-MoO₃ pseudo-binary section are listed in Table 7 and shown in Fig. 5(b), where they are compared to the experimental data available. The enthalpy of fusion of β-Cs₂MoO₄ is in very good agreement with the data of Denielou et al. [41] (selected in the review by Cordfunke and Konings [23]). The calculated transition for the congruent melting of β-Cs₂Mo₂O₇ is 77.7 kJ·mol⁻¹, in good agreement with the experimental data within the measured uncertainties. The calculated enthalpy of congruent melting of Cs₂Mo₃O₁₀ in the present model is slightly higher than the experimental data,

<figure><img src="image_8.png" /><figcaption>Fig. 5. (a) Enthalpies of formation at 298.15 K calculated in the present model, and comparison with the experimental data selected in this work (see Section 2); (b) Transition enthalpies calculated in the present model at the transition temperature, and comparison with the literature data of Denielou et al. [41] and Smith et al. [17].</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 6. Calculated total vapour pressure above $Cs_2MoO_4$(cr) and $Cs_2MoO_4$(l), and comparison with the data of Johnson [45], Tangri et al. [46], Cordfunke et al. [47], Yamawaki et al. [48], Kazenas et al. [49], and Stolyarova et al. [50].</figcaption></figure>

which might suggest that experimental confirmation is required for these data. The enthalpy of the peritectic decomposition of $Cs_2Mo_5O_{16}$ (134.8 kJ·mol⁻¹) is only slightly higher than the experimental data, i.e. (121.3 ± 8.2) kJ·mol⁻¹ [17]. The enthalpy of the peritectic decomposition of $Cs_2Mo_7O_{22}$ (237.9 kJ·mol⁻¹) is in good agreement with the experimental data within the measured uncertainties, i.e. (216.5 ± 24.2) kJ·mol⁻¹ [17].

### 4.3. Vapour pressure above $Cs_2MoO_4$

The calculated (total) vapour pressure of the congruent equilibrium $Cs_2MoO_4$(cr, l) = $Cs_2MoO_4$(g) is shown in Fig. 6 and compared to the data of Johnson [45], Tangri et al. [46], Cordfunke et al. [47], Yamawaki et al. [48], Kazenas et al. [49], and Stolyarova et al. [50]. Note that the thermodynamic functions of $Cs_2MoO_4$(g) were not optimized in this work, but taken from [51]. Only the thermodynamic functions of $Cs_2MoO_4$(cr) and of the liquid solution were optimized so as to match as best as possible the known phase diagram data in the $Cs_2MoO_4$–$MoO_3$ section and thermodynamic data on $Cs_2MoO_4$(cr). The calculated vapour pressures are given by the following equations for the temperature ranges 1000–1214 K and 1232–1300 K, respectively:

$$ \log_{10} P(Pa) = 11.84 - 14176.1/T(K) \qquad (10) $$

$$ \log_{10} P(Pa) = 10.21 - 12126.9/T(K) \qquad (11) $$

The agreement with the experimental data of Johnson, Cordfunke et al. and Kazenas et al. is generally good, with deviations on the absolute pressures of the order of 19%, 13%, and 10%, respectively. The data of Tangri et al. and Yamawaki et al. are somewhat lower,

<figure><img src="image_10.png" /><figcaption>Fig. 7. Calculated isothermal sections of the Cs–Mo–O system at (a) T = 300 K, (b) T = 850 K, (c) T = 1000 K, and (d) T = 2000 K.</figcaption></figure>
<table><caption>**Table A.1**
Summary of the thermodynamic data for pure elements and oxides selected in the Cs–Mo system. SER refers to the phase of the element stable at 298.15 K.</caption><thead><tr><th>Phase</th><th>Gibbs energy/(J·mol<sup>-1</sup>)</th><th>Reference</th></tr></thead><tbody><tr><td>Liquid<br>(Cs<sup>+</sup>, Mo<sup>4+</sup>)(Va<sup>O<sup>-</sup></sup>)</td><td><sup>0</sup>G(Cs<sup>+</sup>)(Va<sup>O<sup>-</sup></sup>)<sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Cs</sub> = G<sup>Liq</sup><sub>Cs</sub><br><sup>0</sup>G(Mo<sup>4+</sup>)(Va<sup>O<sup>-</sup></sup>)<sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Mo</sub> = G<sup>Liq</sup><sub>Mo</sub><br>L<sup>0</sup>(Cs<sup>+</sup>, Mo<sup>4+</sup>)<sub>p</sub>(Va<sup>O<sup>-</sup></sup>)<sub>Q</sub> = +50000</td><td>[56]<br>[56]<br>[14]</td></tr><tr><td>(Cs, Mo) (bcc)<br>(Cs, Mo)(Va)<sub>3</sub></td><td><sup>0</sup>G(Cs)(Va)<sub>3</sub><sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Cs</sub> = G<sup>SER</sup><sub>Cs</sub><br><sup>0</sup>G(Mo)(Va)<sub>3</sub><sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Mo</sub> = G<sup>SER</sup><sub>Mo</sub><br>L<sup>0</sup>(Cs, Mo)(Va)<sub>3</sub> = +50000</td><td>[56]<br>[56]<br>[14]</td></tr><tr><td>Gas<br>(Cs, Cs<sub>2</sub>, Mo, Mo<sub>2</sub>)</td><td><sup>0</sup>G<sup>gas</sup><sub>Cs</sub> <sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Cs</sub> = G<sup>G</sup><sub>Cs</sub> + RTln (10<sup>-5</sup>P)<br><sup>0</sup>G<sup>gas</sup><sub>Cs<sub>2</sub></sub> <sup>-</sup><sup>2</sup>H<sup>SER</sup><sub>Cs</sub> = G<sup>G</sup><sub>Cs<sub>2</sub></sub> + RTln (10<sup>-5</sup>P)<br><sup>0</sup>G<sup>gas</sup><sub>Mo</sub> <sup>-</sup><sup>0</sup>H<sup>SER</sup><sub>Mo</sub> = G<sup>G</sup><sub>Mo</sub> + RTln (10<sup>-5</sup>P)<br><sup>0</sup>G<sup>gas</sup><sub>Mo<sub>2</sub></sub> <sup>-</sup><sup>2</sup>H<sup>SER</sup><sub>Mo<sub>2</sub></sub> = G<sup>G</sup><sub>Mo<sub>2</sub></sub> + RTln (10<sup>-5</sup>P) [56]</td><td>[58]<br>[58]<br>[56]</td></tr><tr><td>Functions</td><td>G<sup>SER</sup><sub>Cs</sub> = -17373.82 + 436.899787T - 90.5212584Tln (T) + 0.2029422T<sup>2</sup><br>- 1.27907669·10<sup>-4</sup>T<sup>3</sup> + 245245T<sup>-1</sup> (298 &lt; T/K &lt; 301.59 K)<br>= -13553.817 + 218.689955T - 46.7273304Tln (T) + 0.02043269T<sup>2</sup><br>- 4.074846·10<sup>-6</sup>T<sup>3</sup> + 181528T<sup>-1</sup> + 7.8016·10<sup>21</sup>T<sup>-9</sup> (301.59&lt;T/K&lt;2000 K)<br>G<sup>SER</sup><sub>Mo</sub> = -7746.302 + 131.9197T - 23.56414Tln (T) - 0.003443396T<sup>2</sup><br>+ 5.66283·10<sup>-7</sup>T<sup>3</sup> + 65812T<sup>-1</sup> - 1.30927·10<sup>-10</sup>T<sup>4</sup> (298 &lt; T/K &lt; 2896 K)<br>G<sup>Liq</sup><sub>Cs</sub> = -15282.679 + 429.968752T - 90.5212584Tln (T) + 0.2029422T<sup>2</sup><br>- 1.27907669·10<sup>-4</sup>T<sup>3</sup> + 245245T<sup>-1</sup> - 3.56867·10<sup>-18</sup>T<sup>7</sup> (298 &lt; T/K &lt; 301.59 K)<br>= -11454.038 + 211.728844T - 46.7273304Tln (T) + 0.02043269T<sup>2</sup><br>- 4.074846·10<sup>-6</sup>T<sup>3</sup> + 181528T<sup>-1</sup> (301.59 &lt; T/K &lt; 2000 K)<br>G<sup>Liq</sup><sub>Mo</sub> = +34085.045 + 117.224788T - 23.56414Tln (T)<br>- 0.003443396T<sup>2</sup> + 5.66283·10<sup>-7</sup>T<sup>3</sup> + 65812T<sup>-1</sup> - 1.30927·10<sup>-10</sup>T<sup>4</sup><br>+ 4.24519·10<sup>-22</sup>T<sup>7</sup> (298 &lt; T &lt; 2896 K)<br>= +3538.963 + 271.6697T - 42.63829Tln (T) (2896 &lt; T &lt; 5000 K)</td><td>[56]</td></tr></tbody></table>

<table><caption>**Table A.2**
Summary of the thermodynamic data for pure elements and oxides selected in the Cs–Mo system. SER refers to the phase of the element stable at 298.15 K.</caption><thead><tr><th>Phase</th><th>Gibbs energy/(J·mol<sup>-1</sup>)</th><th>Reference</th></tr></thead><tbody><tr><td>Functions</td><td>G<sup>G</sup><sub>Cs</sub> = +70221.1857 - 34.9432617T - 21.01225Tln (T)<br>+ 2.3869595·10<sup>-4</sup>T<sup>2</sup> - 4.17045667·10<sup>-8</sup>T<sup>3</sup> + 4922.448T<sup>-1</sup> (298 &lt; T &lt; 1600 K)<br>= +65030.9624 + 16.0373197T - 28.23664Tln (T)<br>+ 0.0041988735T<sup>2</sup> - 4.22771·10<sup>-7</sup>T<sup>3</sup> + 522968.5T<sup>-1</sup> (1600 &lt; T &lt; 3400 K)<br>= +366608.608 - 1031.64395T + 100.1023Tln (T)<br>- 0.020285495T<sup>2</sup> + 4.61329·10<sup>-7</sup>T<sup>3</sup> - 1.333813·10<sup>8</sup>T<sup>-1</sup> (3400 &lt; T &lt; 5300 K)<br>= +21277.4247 - 330.32731T + 20.43107ln (T) - 0.012545005T<sup>2</sup><br>+ 3.32779833·10<sup>-7</sup>T<sup>3</sup> + 1.375229·10<sup>8</sup>T<sup>-1</sup> (5300 &lt; T &lt; 7000 K)<br>= -1347367.73 + 2313.0949T - 278.8398Tln (T) + 0.016571635T<sup>2</sup><br>- 1.98924·10<sup>-7</sup>T<sup>3</sup> + 1.3125275·10<sup>9</sup>T<sup>-1</sup> (7000 &lt; T &lt; 100000 K)<br>G<sup>G</sup><sub>Cs</sub> = +100114.046 - 81.8752338T - 29.1921Tln (T)<br>- 0.01584241T<sup>2</sup> + 4.411035·10<sup>-6</sup>T<sup>3</sup> - 87527.95T<sup>-1</sup> (298&lt;T&lt;700 K)<br>= +76365.6162 + 232.868668T - 76.74226Tln (T) + 0.026454175T<sup>2</sup><br>- 2.64218333·10<sup>-6</sup>T<sup>3</sup> + 2135020T<sup>-1</sup> (700 &lt; T &lt; 1700 K)<br>= +250375.533 - 857.020867T + 68.80663Tln (T) - 0.028077265T<sup>2</sup><br>+ 1.22328533·10<sup>-6</sup>T<sup>3</sup> - 37126965T<sup>-1</sup> (1700&lt;T&lt;3000 K)<br>= -161131.901 + 641.492817T - 115.7658Tln (T) + 0.008419445T<sup>2</sup><br>- 1.1690215·10<sup>-7</sup>T<sup>3</sup> + 1.3370865·10<sup>8</sup>T<sup>-1</sup> (3000 &lt; T &lt; 4900 K)<br>= -155428.026 + 692.773744T - 122.889Tln (T) + 0.01069752T<sup>2</sup><br>- 2.11018·10<sup>-7</sup>T<sup>3</sup> + 1.1397085·10<sup>8</sup>T<sup>-1</sup> (4900 &lt; T &lt; 6000 K)<br>G<sup>G</sup><sub>Mo</sub> = +651326.983 - 41.0946081T - 21.0437Tln (T)<br>+ 2.7252545·10<sup>-4</sup>T<sup>2</sup> - 4.77857833·10<sup>-8</sup>T<sup>3</sup> + 5594.93T<sup>-1</sup> (298 &lt; T &lt; 1600 K)<br>= +633752.125 + 83.3186552T - 37.98605Tln (T) + 0.007623775T<sup>2</sup><br>-6.52545667·10<sup>-7</sup>T<sup>3</sup> + 3471277.5T<sup>-1</sup> (1600 &lt; T &lt; 3500 K)<br>= +1134689.66 - 1617.69443T + 169.6658Tln (T) - 0.03080567T<sup>2</sup><br>+ 6.87557667·10<sup>-7</sup>T<sup>3</sup> - 2.240338·10<sup>8</sup>T<sup>-1</sup> (3500 &lt; T &lt; 5000 K)<br>= +1008188.16 - 1437.30781T + 150.7395Tln (T) - 0.03086233T<sup>2</sup><br>+ 7.54569333·10<sup>-7</sup>T<sup>3</sup> - 1.0601775·10<sup>8</sup>T<sup>-1</sup> (5000 &lt; T &lt; 6200 K)<br>G<sup>G</sup><sub>Mo<sub>2</sub></sub> = +892320.138 + 5.95304372T - 37.156Tln (T)<br>+ 3.61·10<sup>-5</sup>T<sup>2</sup> - 1.10858333·10<sup>-7</sup>T<sup>3</sup> + 113615T<sup>-1</sup> (298 &lt; T &lt; 1000 K)<br>= +878307.025 + 162.383892T - 60.077Tln (T)<br>+ 0.0165846T<sup>2</sup> - 2.3378·10<sup>-6</sup>T<sup>3</sup> + 1706980T<sup>-1</sup> (1000 &lt; T &lt; 1800 K)<br>= +1105100.83 - 1223.34666T + 124.219Tln (T) - 0.05020415T<sup>2</sup><br>+ 2.21706167·10<sup>-6</sup>T<sup>3</sup> - 50788800T<sup>-1</sup> (1800 &lt; T &lt; 3100 K)<br>= +641365.422 + 619.744903T - 105.798Tln (T)<br>+ 6.5385·10<sup>-4</sup>T<sup>2</sup> + 1.02916667·10<sup>-7</sup>T<sup>3</sup> + 1.25079·10<sup>8</sup>T<sup>-1</sup> (3100 &lt; T &lt; 6000 K)</td><td>[58]<br>[59]</td></tr></tbody></table>

with deviations of the order of 51% and 53%–59%, respectively, while the data of Stolyarova et al. is higher than calculated in the model with deviations of the order of 45%. Cordfunke et al. [47] mention that the

lower vapour pressure results of Tangri et al. above Cs<sub>2</sub>MoO<sub>4</sub>(l) might come from the use of an alumina condenser tube and the reaction of Cs<sub>2</sub>MoO<sub>4</sub>(g) with alumina at high temperatures. The same might be
true for the measurements of Yamawaki et al. Thus the latter two sets of data could be less reliable. The origin of the slightly higher vapour pressures measured by Stolyarova et al. compared to the studies of Johnson, Cordfunke et al. and Kazenas et al. is unclear. Complementary measurements above liquid cesium molybdate would be valuable to compare with the present calculation.

## 4.4. Calculated Cs-Mo-O isotherms

Finally, isotherms are calculated at relevant temperatures for the safety assessment of the oxide fuel pin behaviour in a fast neutron reactor, i.e. at T = 300 K, T = 850 K, T = 1000 K and T = 2000 K. They are shown in Figs. 7(a)–7(c), and 7(d), respectively. Note that under “standard” irradiation conditions, such as in the Phénix sodium-cooled fast reactor, the expected temperatures at the pellet rim and centre range between 700 and 2400 K, respectively, as reported by Samuelsson et al. [10]. 850 K was selected as a representative temperature for the JOG layer, while 1000 K is a temperature that can be expected on the surface of the JOG layer on the fuel side. 2000 K was selected for the calculation as representative temperature for the centre of the fuel pin. In the JOG layer, the main constituent, i.e. Cs₂MoO₄, is thus expected in its hexagonal β modification. In regions of the fuel pin where the temperature is higher than 1223 K, the presence of the cesium molybdate can lead to the formation of a liquid phase.

# 5. Conclusions

This work reports a thermodynamic modelling assessment of the key fission product system Cs-Mo-O, using formalisms compatible with the TAF-ID database, namely the compound energy formalism and ionic-two sublattice model. The overall agreement with the available experimental data on thermodynamic properties and vapour pressure of ternary cesium polymolybdates, and phase diagram equilibria in the Cs₂MoO₄-MoO₃ pseudo-binary section is generally good. A few issues remain unresolved still, that would benefit from complementary investigations:

* The polymorphism of Cs₂Mo₄O₁₃ needs further investigations to determine the nature and exact temperature of the phase transition between the reported triclinic (in space group *P* − 1) and monoclinic (in space group *C2*/c) modifications.

* To determine the exact nature of the thermal decomposition of β-Cs₂Mo₂O₇, i.e. congruent melting or peritectic decomposition, post microscopic characterizations would be necessary after thermal analysis measurements.

* The low-temperature heat capacities and standard entropies at 298.15 K of Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, Cs₂Mo₇O₂₂, as well as standard enthalpy of formation at 298.15 K of Cs₂Mo₄O₁₃ have not been measured to this date.

* The high temperature heat capacities (and/or enthalpy increments) of Cs₂Mo₂O₇, Cs₂Mo₃O₁₀, Cs₂Mo₄O₁₃, Cs₂Mo₅O₁₆, Cs₂Mo₇O₂₂ should be determined experimentally.

* We recommend to confirm the congruent vaporization of Cs₂Mo₂O₇ as suggested by Do et al. [52] using mass spectrometry before introducing Cs₂Mo₂O₇(g) gaseous species in the thermodynamic model of the Cs-Mo-O system. The current data available are derived from the thermogravimetry method, which gives only the total vapour pressure. Using a technique such as Knudsen Effusion Mass Spectrometry would allow to obtain more reliable results. The thermodynamic functions of Cs₂Mo₂O₇(g) would then need to be determined too.

* The vapour pressure above liquid Cs₂MoO₄ should be scrutinized again with complementary measurements using a technique such as transpiration or Knudsen effusion Mass Spectrometry.

# Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

# Data availability statement

The data presented herein are available from the corresponding author upon request.

# Acknowledgement

Modelling of the Cs-Mo-O system was supported by the OECD/NEA TAF-ID (Thermodynamics of Advanced Fuels – International Database) project.

# Appendix. Thermodynamic modelling assessment of the Cs-Mo system

See Tables A.1 and A.2.

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