# Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art on the U–Te System and Selection of the Data for the Assessment

#### 2.1. Crystal Structure Data

#### 2.2. Phase Diagram Data

#### 2.3. Thermodynamic Data

## 3. Thermodynamic Modeling with the CALPHAD Method

#### 3.1. Gibbs Energy Models

#### 3.2. Optimization Procedure

## 4. Use of Conjugate Prior Distribution for CALPHAD Modeling

#### 4.1. Bayesian Inference

#### 4.2. Conjugate Prior Probability Distribution

#### 4.3. Uncertainty on the Calculated Thermodynamic Quantities

#### 4.3.1. Enthalpy of Formation

#### 4.3.2. Entropy

#### 4.3.3. Partial Pressures for the Two-Phase Regions

#### 4.4. Uncertatinty on the Calculated Phase Diagram

## 5. Results and Discussion

#### 5.1. CALPHAD Assessment

#### 5.2. Uncertainty Propagation with the Bayesian Approach

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Reduction of the Number of Parameters

#### Appendix A.1. Degree of Freedom and System Dimension Reduction

#### Appendix A.2. Reduced Prior Probability Distribution and Relations with the Initial Set of Parameters

## Appendix B. Additional Table: Correlation Values for the Posterior Probability Distribution

${\mathit{L}}_{0}\mathit{B}$ | ${\mathit{L}}_{0}\mathit{C}$ | ${\mathit{L}}_{1}\mathit{B}$ | ${\mathit{L}}_{1}\mathit{C}$ | ${\mathit{L}}_{2}\mathit{B}$ | ${\mathit{L}}_{2}\mathit{C}$ | ${\mathit{UTe}}_{0}$ | ${\mathit{UTe}}_{1}$ | $\mathit{U}3\mathit{Te}{4}_{0}$ | $\mathit{U}3\mathit{Te}{4}_{1}$ | $\mathit{U}2\mathit{Te}{3}_{0}$ | $\mathit{U}2\mathit{Te}{3}_{1}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${L}_{0}B$ | 1 | −0.015 | −0.010 | −0.012 | −0.046 | 0.030 | 0.034 | 0.028 | −0.010 | 0.034 | 0.052 | 0.031 |

${L}_{0}C$ | 1 | −0.009 | 0.036 | −0.078 | −0.033 | −0.002 | −0.688 | −0.012 | −0.368 | −0.018 | −0.100 | |

${L}_{1}B$ | 1 | 0.021 | −0.006 | 0.002 | 0.039 | −0.007 | 0.020 | −0.006 | 0.055 | −0.006 | ||

${L}_{1}C$ | 1 | −0.033 | −0.040 | 0.018 | −0.296 | −0.024 | −0.002 | 0.061 | 0.137 | |||

${L}_{2}B$ | 1 | 0.012 | 0.074 | 0.063 | 0.055 | 0.032 | −0.018 | 0.008 | ||||

${L}_{2}C$ | 1 | 0.030 | −0.143 | 0.019 | 0.013 | −0.010 | 0.047 | |||||

$UT{e}_{0}$ | 1 | −0.014 | 0.036 | −0.003 | 0.020 | 0.000 | ||||||

$UT{e}_{1}$ | 1 | 0.008 | 0.854 | −0.010 | 0.661 | |||||||

$U3Te{4}_{0}$ | 1 | −0.002 | −0.011 | −0.006 | ||||||||

$U3Te{4}_{1}$ | 1 | −0.003 | 0.953 | |||||||||

$U2Te{3}_{0}$ | 1 | −0.003 | ||||||||||

$U2Te{3}_{1}$ | 1 | |||||||||||

$\mathit{U}\mathbf{3}\mathit{Te}{\mathbf{5}}_{\mathbf{0}}$ | $\mathit{U}\mathbf{3}\mathit{Te}{\mathbf{5}}_{\mathbf{1}}$ | $\mathit{UTe}{\mathbf{2}}_{\mathbf{0}}$ | $\mathit{UTe}{\mathbf{2}}_{\mathbf{1}}$ | $\mathit{U}\mathbf{2}\mathit{Te}{\mathbf{5}}_{\mathbf{0}}$ | $\mathit{U}\mathbf{2}\mathit{Te}{\mathbf{5}}_{\mathbf{1}}$ | $\mathit{UTe}{\mathbf{3}}_{\mathbf{0}}$ | $\mathit{UTe}{\mathbf{3}}_{\mathbf{1}}$ | $\mathit{UTe}{\mathbf{5}}_{\mathbf{0}}$ | $\mathit{UTe}{\mathbf{5}}_{\mathbf{1}}$ | |||

${L}_{0}B$ | 0.020 | 0.027 | −0.015 | 0.017 | −0.044 | 0.016 | −0.018 | 0.014 | 0.020 | 0.017 | ||

${L}_{0}C$ | −0.011 | 0.121 | −0.034 | 0.463 | 0.021 | 0.615 | −0.007 | 0.687 | 0.004 | 0.682 | ||

${L}_{1}B$ | 0.001 | −0.008 | 0.037 | −0.012 | 0.005 | −0.017 | −0.014 | −0.020 | 0.027 | −0.022 | ||

${L}_{1}C$ | 0.010 | 0.147 | −0.023 | 0.093 | 0.017 | −0.133 | −0.055 | −0.292 | 0.022 | −0.389 | ||

${L}_{2}B$ | 0.015 | −0.010 | 0.013 | −0.036 | −0.031 | −0.041 | −0.011 | −0.041 | −0.009 | −0.037 | ||

${L}_{2}C$ | 0.020 | −0.001 | −0.015 | −0.094 | 0.001 | −0.052 | 0.038 | −0.009 | 0.010 | 0.104 | ||

$UT{e}_{0}$ | −0.008 | −0.002 | 0.013 | −0.005 | 0.035 | −0.008 | −0.004 | −0.009 | 0.083 | −0.008 | ||

$UT{e}_{1}$ | 0.030 | 0.514 | 0.067 | 0.245 | −0.055 | 0.126 | 0.010 | 0.037 | 0.003 | 0.008 | ||

$U3Te{4}_{0}$ | −0.009 | −0.010 | 0.003 | −0.013 | −0.005 | −0.008 | −0.006 | −0.004 | −0.009 | 0.000 | ||

$U3Te{4}_{1}$ | 0.045 | 0.870 | 0.063 | 0.648 | −0.059 | 0.489 | −0.003 | 0.351 | 0.018 | 0.295 | ||

$U2Te{3}_{0}$ | −0.020 | −0.004 | −0.020 | −0.013 | −0.036 | −0.029 | 0.037 | −0.039 | 0.001 | −0.045 | ||

$U2Te{3}_{1}$ | 0.047 | 0.975 | 0.053 | 0.826 | −0.054 | 0.674 | −0.011 | 0.530 | 0.023 | 0.462 | ||

$U3Te{5}_{0}$ | 1 | 0.043 | −0.039 | 0.034 | 0.002 | 0.025 | 0.040 | 0.018 | 0.021 | 0.017 | ||

$U3Te{5}_{1}$ | 1 | 0.046 | 0.931 | −0.049 | 0.810 | −0.014 | 0.680 | 0.024 | 0.606 | |||

$UTe{2}_{0}$ | 1 | 0.029 | −0.009 | 0.026 | 0.003 | 0.020 | −0.016 | 0.017 | ||||

$UTe{2}_{1}$ | 1 | −0.038 | 0.955 | −0.015 | 0.869 | 0.021 | 0.797 | |||||

$U2Te{5}_{0}$ | 1 | −0.034 | −0.018 | −0.027 | 0.033 | −0.025 | ||||||

$U2Te{5}_{1}$ | 1 | −0.001 | 0.977 | 0.014 | 0.937 | |||||||

$UTe{3}_{0}$ | 1 | 0.006 | −0.070 | 0.021 | ||||||||

$UTe{3}_{1}$ | 1 | 0.009 | 0.987 | |||||||||

$UTe{5}_{0}$ | 1 | 0.003 | ||||||||||

$UTe{5}_{1}$ | 1 |

## References

- Samuelsson, K.; Dumas, J.C.; Sundman, B.; Lainet, M. An improved method to evaluate the “Joint Oxyde-Gaine” formation in (U,Pu)O
_{2}irradiated fuels using the GERMINAL V2 code coupled to Calphad thermodynamic computations. EPJ Nucl. Sci. Technol.**2020**, 6, 47. [Google Scholar] [CrossRef] - Samuelsson, K.; Dumas, J.C.; Sundman, B.; Lamontagne, J.; Guéneau, C. Simulation of the chemical state of high burnup (U,Pu)O
_{2}fuel in fast reactors based on thermodynamic calculations. J. Nucl. Mater.**2020**, 532, 151969. [Google Scholar] [CrossRef] - Guéneau, C.; Dupin, N.; Kjellqvist, L.; Geiger, E.; Kurata, M.; Gossé, S.; Corcoran, E.; Quaini, A.; Hania, R.; Smith, A.L.; et al. TAF-ID: An international thermodynamic database for nuclear fuels applications. Calphad
**2021**, 72, 102212. [Google Scholar] [CrossRef] - Lukas, H.L.; Fries, S.G.; Sundman, B. Computational Thermodynamics: The Calphad Method; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Paulson, N.H.; Bocklund, B.J.; Otis, R.A.; Liu, Z.-K.; Stan, M. Quantified uncertainty in thermodynamic modeling for materials design. Acta Mater.
**2019**, 174, 9–15. [Google Scholar] [CrossRef] [Green Version] - Paulson, N.H.; Jennings, E.; Stan, M. Bayesian strategies for uncertainty quantification of the thermodynamic properties of materials. Int. J. Eng. Sci.
**2019**, 142, 74–93. [Google Scholar] [CrossRef] [Green Version] - Paulson, N.H.; Zomorodpoosh, S.; Roslyakova, I.; Stan, M. Comparison of statistically-based methods for automated weighting of experimental data in CALPHAD-type assessment. Calphad
**2020**, 68, 101728. [Google Scholar] [CrossRef] - Thermo-Calc Software, Version 2021a. Available online: https://thermocalc.com/ (accessed on 15 January 2021).
- Okamoto, H. Te-U (Tellurium-Uranium). J. Phase Equilibria
**1993**, 14, 129–130. [Google Scholar] [CrossRef] - Herrmannsdörfer, T.; Fischer, P.; Mattenberger, K.; Vogt, O. Temperature dependences of rhombohedral lattice distortion and of ferromagnetic uranium ordering in the uranium monochalcogenides. J. Alloys Compd.
**2006**, 414, 14–19. [Google Scholar] [CrossRef] - Tougait, O.; André, G.; Bourée, F.; Noël, H. Neutron diffraction study of magnetic ordering of two binary uranium tellurides U
_{3}Te_{5}and U_{2}Te_{3}. J. Alloys Compd.**2001**, 317–318, 227–232. [Google Scholar] [CrossRef] - Stöwe, K. Contributions to the Crystal Chemistry of Uranium Tellurides. III. Temperature-Dependent Structural Investigations on Uranium Ditelluride. J. Solid State Chem.
**1996**, 127, 202–210. [Google Scholar] [CrossRef] - Solvyanskikh, V.K.; Yarembash, E.I.; Ellert, G.V.; Eliseev, A.A. On the system U-Te. Izv. Akad. SSSR Ser. Neor. Materialy
**1968**, 4, 543–545. [Google Scholar] - Solvyanskikh, V.K.; Rozanov, I.A.; Gracheva, N.V. The S-Te-U. Russ. J. Inorg. Chem.
**1977**, 22, 893–896. [Google Scholar] - Tougait, O.; Potel, M.; Noël, H. Characterization of the Binary Uranium and Thorium Tellurides U
_{7}Te_{12}and Th_{7}Te_{12}. Inorg. Chem.**1998**, 37, 5088–5091. [Google Scholar] [CrossRef] - Tougait, O.; Potel, M.; Padiou, J.; Noël, H. Crystal structure and properties of the binary uranium telluride U
_{2}Te_{5}. J. Alloys Compd.**1997**, 262–263, 320–324. [Google Scholar] [CrossRef] - Boehme, D.R.; Nichols, M.C.; Snyder, R.L.; Matheis, D.P. An investigation of the tellutium-rich uranium tellurides using X-ray powder diffraction. J. Alloys Compd.
**1992**, 179, 37–59. [Google Scholar] [CrossRef] - Ellert, G.V.; Sevast’yanov, V.G.; Solvyanskikh, V.K. The Se-U and Te-U Systems. Russ. J. Inorg. Chem.
**1975**, 20, 120–124. [Google Scholar] - Czechowicz, D.G. Combustion Synthesis and Characterization of Uranium and Thorium Tellurides: LA-10559-T. Master’s Thesis, Los Alamos National Laboratory, Los Alamos, NM, USA, 1985. [Google Scholar]
- Wolf, A. Modellierungen zur Kristallzüchtung von CrSb2 und UPTe, Ein Beitrag zur Rationale Syntheseplanung; Springer: Wiesbaden, Germany, 2017. [Google Scholar] [CrossRef]
- Westrum, E.F.; Gronvold, F. Chemical Thermodynamics of the Actinide Element Chalcogenides. In Proceedings of the Symposium on Thermodynamics of Nuclear Materials (IAEA), Vienna, Austria, 21–25 May 1962. [Google Scholar]
- Czechowicz, D.G. A Study of vaporization Thermodynamics in the Uranium-Tellurium System: LA-10621-T. Master’s Thesis, Los Alamos National Laboratory, Los Alamos, NM, USA, 1986. [Google Scholar]
- Baskin, Y.; Smith, S.D. Enthalpy of formation data on compounds of uranium with groups VA and VIA elements. J. Nucl. Mater.
**1970**, 37, 209–222. [Google Scholar] [CrossRef] - Mills, K.C. Thermodynamic Data for Inorganic Sulphides, Selenides and Tellurides; Butterworth & Co.: London, UK, 1974. [Google Scholar]
- Qian, S.; Qiu, R.; Tang, J.; Chen, J.; Liu, P.; Ao, B. Theoretical Assignment of Oxidation State of Uranium in Binary, Ternary, and Quaternary Tellurides. J. Phys. Chem.
**2021**, 125, 1029–1040. [Google Scholar] [CrossRef] - Saal, J.E.; Kirklin, S.; Aykol, M.; Meredig, B.; Wolverton, C. Materials Design and Discovery with High-Throughput Density Functional Theory: The Open Quantum Materials Database (OQMD). JOM
**2013**, 65, 1501–1509. [Google Scholar] [CrossRef] - Solvyanskikh, V.K.; Ellert, G.V.; Yarembash, E.S. Transport mechanism and kinetics in Uranium chalcogenides. Izv. Akad. SSSR Ser. Neor. Mater.
**1967**, 3, 1133. [Google Scholar] - Sevast’yanov, V.G.; Solvyanskikh, V.K.; Ellert, G.V. Equilibria in the US
_{x}–Br_{2}and USe_{x}–Br_{2}system. Zh. Neorg. Khim.**1971**, 16, 3357. [Google Scholar] - Dinsdale, A.T. SGTE data for pure elements. Calphad
**1991**, 15, 317–425. [Google Scholar] [CrossRef] - Hillert, M.; Jansson, B.; Sundman, B.; Agren, J. A two-sublattice model for molten solutions with different tendency for ionization. Metall. Trans.
**1985**, 16, 661. [Google Scholar] [CrossRef] - SSUB5: SGTE Substances Database, Version 5.2. Available online: https://thermocalc.com/products/databases/general-alloys-and-pure-substances/ (accessed on 15 January 2016).
- Guillaumont, R. Update on the Chemical Thermodynamics of Uranium, Neptunium, Plutonium, Americium and Technetium; OECD Nuclear Energy Agency, Data Bank: Issy-les-Moulineaux, France, 2003. [Google Scholar]
- Chatterjee, N.D.; Krüger, R.; Haller, G.; Olbricht, W. The bayesian approach to an internally consistent thermodynamic database: Theory, database, and generation of phasediagrams. Contrib. Mineral. Petrol.
**1998**, 133, 149–168. [Google Scholar] [CrossRef] - Duong, T.C.; Hackenberg, R.E.; Landa, A.; Honarmandi, P.; Talapatra, A.; Volz, H.M.; Llobet, A.; Smith, A.I.; King, G.; Bajaj, S.; et al. Revisiting thermodynamics and kinetic diffusivities of uranium–niobium with bayesian uncertainty analysis. Calphad
**2016**, 55, 219–230. [Google Scholar] [CrossRef] [Green Version] - Lawrence, E. Reconstruction Fonctionnelle et Analyse D’incertitude dans le Cadre d’un Problème Inverse de Thermodynamique Chimique. Ph.D. Thesis, Université Paul Sabatier Toulouse 3, Toulouse, France, 2020. [Google Scholar]
- Robert, C.P. The Bayesian Choice: A Decision-Theoretic Motivation; Springer: New York, NY, USA, 1994. [Google Scholar]
- Azaïs, J.M.; Bardet, J.M. Le Modèle Linéaire par L’exemple. Régression, Analyse de la Variance et Plans d’Expériences. Illustrations Numériques avec les Logiciels R, SAS et Splus; Dunod: Paris, France, 2006. [Google Scholar]
- Sundman, B.; Lu, X.G.; Ohtani, H. The implementation of an algorithm to calculate thermodynamic equilibria for multi-component systems with non-ideal phases in a free software. Comput. Mater. Sci.
**2015**, 101, 127–137. [Google Scholar] [CrossRef] - The OpenCalphad Repository. Available online: http://github.com/sundmanbo/opencalphad (accessed on 12 December 2018).

**Figure 1.**Enthalpy of formation data of the U${}_{x}$Te${}_{y}$ compounds coming from the literature. The uncertainty bars on the experimental data measured by calorimetry (in blue) are reported.

**Figure 4.**Calculated enthalpy of formation of UTe, U${}_{3}$Te${}_{4}$, U${}_{2}$Te${}_{3}$, U${}_{3}$Te${}_{5}$, UTe${}_{2}$, U${}_{2}$Te${}_{5}$, UTe${}_{3}$, and UTe${}_{5}$. Comparison with the experimental data from Baskin et al. [23], the models from Czechowitz [19] and Wolf [20], and the DFT calculations from Qian [25] and the OQMD database [26].

**Figure 7.**Parameter posterior probability distributions and correlations matrix plot for the liquid phase on UTe–Te side and UTe. The posterior correlation values can be found in Appendix B.

**Figure 8.**Parameter posterior probability distributions and correlations matrix plot for the compounds entropic parameters. The posterior correlation values can be found in Appendix B.

**Figure 9.**Uncertainty estimation given by the 95% credible interval of the posterior probability distribution for the calculated enthalpies of formation for UTe, U${}_{3}$Te${}_{4}$, U${}_{2}$Te${}_{3}$, U${}_{3}$Te${}_{5}$, UTe${}_{2}$, U${}_{2}$Te${}_{5}$, UTe${}_{3}$, and UTe${}_{5}$. The experimental error bars are in dotted black lines. The propagated uncertainties (95% credible interval) for each computed quantity are in bold red lines.

**Figure 10.**Uncertainty estimation given by the 95% credible interval of the posterior probability distribution for the calculated entropies for UTe, U${}_{3}$Te${}_{4}$, U${}_{2}$Te${}_{3}$, UTe${}_{2}$, U${}_{2}$Te${}_{5}$, and UTe${}_{3}$. The experimental error bars are in doted black lines. The propagated uncertainties (95% credible interval) for each computed quantity are in bold red lines.

**Figure 11.**Uncertainty estimation given by the 95% credible interval of the posterior probability distribution for the calculated partial pressure. Plain lines are the posterior means. Dotted lines correspond to the 95% upper and lower bounds of the credible interval for each temperature value.

**Figure 12.**Simulation of $N=100$ phase diagram from the parameters posterior probability distribution.

**Table 1.**Crystal structure data on the phases of the U–Te system. The compounds in

**bold**are considered in the present work. The compounds in italic gray${}^{\star}$ are not taken into account.

Compounds | Composition At. % Te | Space Group | Reference |
---|---|---|---|

$\gamma $-U | 0 | $\mathbf{Im}\overline{\mathbf{3}}\mathbf{m}$ | Okamoto (1993) [9] |

$\beta $-U | 0 | $\mathbf{P}{\mathbf{4}}_{\mathbf{2}}/\mathbf{mmm}$ | Okamoto (1993) [9] |

$\alpha $-U | 0 | Clcl | Okamoto (1993) [9] |

U${}_{\mathit{10}}$Te${}^{\star}$ | 9 | $Fm-3m$ | Okamoto (1993) [9] |

UTe | 50 | Fm-3m | Hermannsdörfer et al. (2006) [10] |

U${}_{\mathbf{3}}$Te${}_{\mathbf{4}}$ | 57.1 | I-43d | Solvyanskikh et al. (1977) [14] |

U${}_{\mathbf{2}}$Te${}_{\mathbf{3}}$ | 60 | Pnma | Tougait et al. (2001) [11] |

U${}_{\mathbf{3}}$Te${}_{\mathbf{5}}$ | 62.5 | Pnma | Tougait et al. (2001) [11] |

U${}_{\mathit{7}}$Te${{}_{\mathit{12}}}^{\star}$ | 63.2 | $P-6$ | Tougait et al. (1998) [15] |

$\beta $-UTe${}_{\mathbf{2}}$ | 66.7 | Unknown | Okamoto (1993) [9] |

$\alpha $-UTe${}_{\mathbf{2}}$ | 2 | 2 | Stöwe (1996) [12] |

U${}_{\mathbf{2}}$Te${}_{\mathbf{5}}$ | 71.4 | C12m1 | Tougait et al. (1997) [16] |

UTe${}_{\mathbf{3}}$ | 75 | $\mathbf{P}{\mathbf{12}}_{\mathbf{1}}/\mathbf{m}\mathbf{1}$ | Stöwe (1996) [12] |

U${}_{2}$Te${{}_{7}}^{\star}$ | 77.8 | Unknown | Okamoto (1993) [9] |

UTe${}_{\mathbf{5}}$ | 83.3 | Pnma | Boehme et al. (1992) [17] |

Te | 100 | $\mathbf{P}{\mathbf{3}}_{\mathbf{1}}\mathbf{21}$ | Okamoto (1993) [9] |

Composition Range At. % Te | Method | Reference | Comments |
---|---|---|---|

58–100 | DTA, XRD | Slovyanskikh et al. (1968) [13] | Selected |

0–65 | DTA, XRD | Ellert et al. (1975) [18] | Selected |

0–100 | Review | Czechowitz (1985) [19] | Not selected |

0–100 | Review | Okamoto (1993) [9] | Not selected |

100 | Calphad model | Wolf (2017) [20] | Not selected |

**Table 3.**Thermodynamic data on the phases of the U–Te system. *: the condensed phases were not characterized. The reported phases by Czechowicz 1986 [22] were proposed on the basis of the phase diagram. The method DFT+U refers to the density functional theory + Hubbard model.

Thermodynamic Data | Phases | Method | Reference | Comments |
---|---|---|---|---|

${S}_{298.15K}^{0}$ | $\begin{array}{c}UTe,{U}_{3}T{e}_{4},{U}_{2}T{e}_{3}\\ UT{e}_{2},{U}_{2}T{e}_{5},UT{e}_{3}\end{array}$ | Estimation | Westrum 1962 [21] | Selected |

$\Delta {H}_{f,298.15K}^{0}$ | $UTe$, ${U}_{3}T{e}_{4}$ | $\begin{array}{c}\mathrm{Direct}\phantom{\rule{4.pt}{0ex}}\mathrm{reaction}\\ \mathrm{calorimetry}\end{array}$ | Baskin and Smith 1970 [23] | Selected |

$\begin{array}{c}\Delta {H}_{f,298.15K}^{0},{S}_{298.15K}^{0}\\ {C}_{p}\left(T\right),\Delta {H}_{fusion}\end{array}$ | $\begin{array}{c}UTe,{U}_{3}T{e}_{4},{U}_{2}T{e}_{3},{U}_{3}T{e}_{5}\\ UT{e}_{2},{U}_{2}T{e}_{5},UT{e}_{3},UT{e}_{5}\end{array}$ | Estimation Solgasmix | Czechowicz 1985 [19] | $\begin{array}{c}\Delta {H}_{f,298.15K}^{0}\phantom{\rule{4.pt}{0ex}}\mathrm{selected}\\ \mathrm{except}\phantom{\rule{4.pt}{0ex}}UTe\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{U}_{3}T{e}_{4}\end{array}$ |

${p}_{T{e}_{2}}$ | *$UT{e}_{3}$/${U}_{2}T{e}_{5}$ | $\begin{array}{c}\mathrm{Quartz}\phantom{\rule{4.pt}{0ex}}\mathrm{membrane}\\ \mathrm{null}\phantom{\rule{4.pt}{0ex}}\mathrm{manometer}\\ \mathrm{Chemical}\phantom{\rule{4.pt}{0ex}}\mathrm{transport}\\ \mathrm{reaction}\end{array}$ | Slovyanskikh et al., 1967 [27] | Selected |

${p}_{T{e}_{2}}$ | *$UT{e}_{2}$/${U}_{3}T{e}_{5}$ | $\begin{array}{c}\mathrm{Bourdon}\phantom{\rule{4.pt}{0ex}}\mathrm{null}\\ \mathrm{manometer}\\ \mathrm{Chemical}\phantom{\rule{4.pt}{0ex}}\mathrm{transport}\\ \mathrm{reaction}\end{array}$ | Sevast’yanov et al., 1971 [28] | Selected |

${p}_{Te}$, ${p}_{T{e}_{2}}$ | ${U}_{3}T{e}_{4}$/$UTe$ | KEMS | Czechowicz 1986 [22] | Selected |

$\begin{array}{c}\Delta {H}_{f,298.15K}^{0},{S}_{298.15K}^{0}\\ {C}_{p}\left(T\right)\end{array}$ | $\begin{array}{c}UTe,{U}_{3}T{e}_{4},{U}_{2}T{e}_{3},{U}_{3}T{e}_{5}\\ UT{e}_{2},{U}_{2}T{e}_{5},UT{e}_{3},UT{e}_{5}\end{array}$ | Calphad model | Wolf 2017 [20] | Only ${C}_{p}\left(T\right)$ selected |

$\Delta {H}_{f,0}^{0}$ | $\begin{array}{c}UTe,{U}_{3}T{e}_{4},{U}_{2}T{e}_{3}\\ {U}_{3}T{e}_{5},{U}_{7}T{e}_{12},UT{e}_{2}\\ {U}_{2}T{e}_{5},UT{e}_{3},UT{e}_{5}\end{array}$ | DFT+U | Qian et al., 2021 [25] | Not selected |

$\Delta {H}_{f,0}^{0}$ | $\begin{array}{c}UTe,{U}_{3}T{e}_{4},{U}_{2}T{e}_{3},{U}_{3}T{e}_{5}\\ UT{e}_{2},{U}_{2}T{e}_{5},UT{e}_{3},UT{e}_{5}\end{array}$ | DFT | OQMD [26] | Not selected |

**Table 4.**Posterior mean and standard deviation of the U–Te system parameters. The correlation matrix of the system parameters can be found in Appendix B.

Phase | Parameter Name | Assessment Value | Posterior Mean Value | Posterior Standard Deviation |
---|---|---|---|---|

Liquid | ${L}_{(UTe,Te)}^{0}$B | −171,903.278 | −171,903.269 | 1.000 |

${L}_{(UTe,Te)}^{0}$C | 81.899 | 81.836 | 0.840 | |

${L}_{(UTe,Te)}^{1}$B | 111,518.216 | 111,518.211 | 1.000 | |

${L}_{(UTe,Te)}^{1}$C | −55.379 | −55.635 | 0.965 | |

${L}_{(UTe,Te)}^{2}$B | −69,986.365 | −69,986.364 | 1.000 | |

${L}_{(UTe,Te)}^{2}$C | 46.920 | 47.115 | 0.994 | |

UTe | ${A}_{0}$ | −221,461.935 | −221,461.944 | 1.000 |

${A}_{1}$ | 198.439 | 198.314 | 0.236 | |

U${}_{3}$Te${}_{4}$ | ${A}_{0}$ | −736,731.990 | −736,731.991 | 1.000 |

${A}_{1}$ | 689.583 | 689.274 | 0.478 | |

U${}_{2}$Te${}_{3}$ | ${A}_{0}$ | −514,925.773 | −514,925.777 | 1.000 |

${A}_{1}$ | 494.721 | 494.781 | 0.379 | |

U${}_{3}$Te${}_{5}$ | ${A}_{0}$ | −808,509.276 | −808,509.286 | 1.000 |

${A}_{1}$ | 796.312 | 796.470 | 0.686 | |

UTe${}_{2}$ | ${A}_{0}$ | −272,474.416 | −272,474.408 | 1.000 |

${A}_{1}$ | 288.876 | 288.996 | 0.160 | |

U${}_{2}$Te${}_{5}$ | ${A}_{0}$ | −556,197.559 | −556,197.562 | 1.000 |

${A}_{1}$ | 656.416 | 656.504 | 0.402 | |

UTe${}_{3}$ | ${A}_{0}$ | −283,794.079 | −283,794.075 | 1.000 |

${A}_{1}$ | 367.797 | 367.730 | 0.250 | |

UTe${}_{5}$ | ${A}_{0}$ | −312,579.609 | −312,579.607 | 1.000 |

${A}_{1}$ | 557.353 | 557.329 | 0.269 |

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**MDPI and ACS Style**

Guéneau, C.; Lawrence, E.; Klein, T.; Gamboa, F.
Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach. *Thermo* **2022**, *2*, 15-38.
https://doi.org/10.3390/thermo2010003

**AMA Style**

Guéneau C, Lawrence E, Klein T, Gamboa F.
Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach. *Thermo*. 2022; 2(1):15-38.
https://doi.org/10.3390/thermo2010003

**Chicago/Turabian Style**

Guéneau, Christine, Eva Lawrence, Thierry Klein, and Fabrice Gamboa.
2022. "Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach" *Thermo* 2, no. 1: 15-38.
https://doi.org/10.3390/thermo2010003