Thermodynamics of formation of solid solutions between BaZrO3 and BaPrO3 42 D O I: 1 0. 15 82 6/ ch im te ch .2 02 0. 7. 2. 01 Dmitry S. Tsvetkov, Vladimir V. Sereda, Dmitry A. Malyshkin, Anton L. Sednev-Lugovets, Andrey Yu. Zuev, Ivan L. Ivanov Chimica Techno Acta. 2020. Vol. 7, no. 2. P. 42–50. ISSN 2409–5613 Dmitry S. Tsvetkov, Vladimir V. Sereda, Dmitry A. Malyshkin, Anton L. Sednev-Lugovets, Andrey Yu. Zuev, Ivan L. Ivanov* Institute of Natural Sciences and Mathematics, Ural Federal University, 620002, 19 Mira St., Ekaterinburg, Russia *email: ivan.ivanov@urfu.ru Thermodynamics of formation of solid solutions between BaZrO3 and BaPrO3 A linear relationship between the standard enthalpy of formation from bi- nary oxides, ΔfH°ox, and the Goldschmidt tolerance factor, t, for some A IIBIVO3 (A = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce, Pr, Tb, U, Pu, Am) perovskite oxides was used for estimation of ΔfH°ox of Pr-substituted barium zirconates BaZr1–xPrxO3. A de- pendence of the relative change of the standard entropies, S°298, on the relative change of the molar volumes in the reactions of formation of AIIBIVO3 (A = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce) from binary oxides was also found to be linear. Using this dependence, a relatively precise method of estimating S°298 was proposed, and S°298 of BaPrO3 was calculated as (162.8 ± 2.8) J·mol –1·K–1. Knowing S°298 of BaPrO3 and using the literature data for S°298 of BaZrO3, the values of S°298 of BaZr1–xPrxO3 were predicted on the assumption that BaZr1–xPrxO3 is a regular or ideal solution of BaPrO3 in BaZrO3 as evidenced by the very small enthalpy of mixing calculated based on the estimated ΔfH°ox. The values of standard entropy changes, ΔfS°ox, and Gibbs energy changes, ΔfG°ox, for the reactions of formation of BaZr1–xPrxO3 from BaO, ZrO2 and PrO2 were also estimated. Substituting Pr for Zr in BaZr1–xPrxO3 results in ΔfH°ox and ΔfG°ox becoming more positive, indicating the decrease of the relative stability with respect to the corresponding binary oxides. Expanded uncertainties of the estimated values of ΔfH°ox and ΔfG°ox are equal to 14 kJ · mol–1, and those of S°298 and ΔfS°ox — less than 2.8 J · mol –1·K–1 and 3.5 J · mol–1·K–1, respectively, for BaZr1–xPrxO3 (x = 0.0–1.0). Keywords: doped barium zirconate; thermodynamics; thermodynamic properties prediction Received: 30.03.2020. Accepted: 11.05.2020. Published: 30.06.2020. © Dmitry S. Tsvetkov, Vladimir V. Sereda, Dmitry A. Malyshkin, Anton L. Sednev-Lugovets, Andrey Yu. Zuev, Ivan L. Ivanov, 2020 Introduction Partially substituted barium zir- conates, BaZr1–xMxO3–δ (M = rare-earth or alkaline-earth element), are the state-of- the-art proton-conducting electrolyte ma- terials for intermediate-temperature solid oxide fuel cells [1–3]. These complex ox- ides possess high proton conductivity upon hydration, good chemical and mechanical stability. Among their known drawbacks are high grain boundary resistance, slow 43 grain growth and, as a consequence, very high sintering temperatures (1900–2000 K) required for obtaining dense ceramics [4– 8]. Praseodymium doping was suggested as  a  possible way not only to  overcome these drawbacks [9] but also, due to po- tentially mixed-valent state of Pr, to obtain triple-conducting (electron-proton-oxide ion) and catalytically active electrode mate- rials for highly efficient proton-conducting solid oxide fuel cells (PC SOFCs) [10, 11]. In spite of the promising electrochemical properties of the BaZr1–xPrxO3 zirconates [10, 11], the influence of Pr doping on their thermodynamics of formation is still un- known. At the same time, understanding the thermodynamics of key materials for PC  SOFCs is  of  utmost importance for the assessment of the long-term behavior of the whole device. Some thermodynam- ic properties of BaZr1–xPrxO3 oxides such as enthalpy increments and constant-pres- sure heat capacities have been studied by us earlier [12]. This work continues systematic investigation of the influence of Pr doping on the thermodynamics of barium zirco- nates and was aimed to estimate the stand- ard thermodynamic functions (enthalpy, entropy and Gibbs free energy) of forma- tion of BaZr1–xPrxO3 oxides. Results and discussion Typically, when it is necessary to experi- mentally determine the standard formation enthalpy of a compound, the solution calo- rimetry is the most straightforward method of choice. However, the dissolution of zirco- nates is quite a hard task, as our preliminary experiments showed. It requires using ei- ther highly corrosive mixtures of acids such as, for example, HF and HNO3 employed by Huntelaar et al. [13], or high-tempera- ture melts [14]. Importantly, in the latter case the solvent stirring is necessary since the dissolution kinetics is slow. Unfortu- nately, neither of  the  above mentioned possibilities was available for the authors. Indeed, the measurements on MHTC 96 (Setaram, France) calorimeter, in  which the solvent stirring is not implemented, re- sulted in irreproducible solution enthalpies of BaZr1–xPrxO3. Besides, the hydrofluoric acid resistant measurement cell for the solu- tion calorimeter has to be custom-made and was not readily available. Because of these reasons, the standard formation enthalpies of BaZr1–xPrxO3 zirconates were estimated using the  well-known strong correlation between the formation enthalpy and Gold- schmidt’s tolerance factor [15–18]. This correlation was shown to allow predicting reasonably good, i.e. very close to the ex- perimental values, estimates of the forma- tion enthalpies for many perovskite oxides. The  standard enthalpy of  formation at 298.15 K, ∆fH°ox, corresponding to the re- action AO+ BO2 = ABO3 (1) calculated for a number of AIIBIVO3 perovs- kite-type oxides, is  shown in  Fig.  1 as a function of Goldschmidt’s tolerance fac tor, ( ) 0 0 . 2 A B r r t r r + = + The  va lues of the tolerance factor were calculated using the crystal radii reported by Shannon [19] with the following coordination numbers: 12 — A2+ cation, 6 — for both B4+ cation and O2– anion. The necessary thermodynamic data were taken from [20–27]. It should be noted that while the AO oxides (namely, CaO, SrO and BaO) belong to  the  same rock-salt crystal structure class, it is  not the case for BO2 and ABO3 oxides which possess different crystal structure depend- ing on the nature of the A and B cations. 44 However, the  differences in  the  crystal structure of both BO2 and ABO3 with dif- ferent cations were not taken into account. The  enthalpies of  slight distortions of the perovskite structure in ABO3 are gen- erally small and were thought to be much less than the standard deviation of the esti- mated values. In turn, even though the crys- tal structure of BO2 varies more than that of  ABO3, judging by  the  good linearity of the ∆fH°ox,(t) dependence in Fig. 1, its in- fluence should also be rather small. The  linear dependence observed in Fig. 1 was least squares fitted. The result- ing equation is the following: ( )1f ox kJ mol 793.8 907.2H t−∆ ⋅ = − ⋅ (2) with the  coefficient of  determination R2 = 0.98. The standard formation enthalp- ies of BaZr1–xPrxO3 oxides calculated accord- ing to Eq. (2) are summarized in Table 1. The  standard deviation of  the  fitted line from the points in Fig. 1 was found to be 7 kJ·mol–1; therefore, the expanded uncer- tainty (95% confidence level) of the ∆fH°ox values reported in  Table  1 is  equal to  14  kJ·mol–1. However, since the  ex- perimental points corresponding to both BaZrO3 and BaPrO3 in Fig. 1 deviate from the fitted line (i.e. from Eq. (2)) by less than 5.6 kJ·mol–1, the accuracy of our predicted ∆fH°ox values is likely to be somewhat bet- ter than this rather conservative estimate of 14 kJ·mol–1. As  follows from Fig.  1 and Table  1, the  standard formation enthalpy of  zir- conates BaZr1–xPrxO3 increases with dop- ing level, x, becoming less negative. This corresponds to  increasing distortions of  the  perovskite lattice, as  evidenced by the results of the structural studies [28, 29] and the gradual decrease of the toler- ance factor, t, from the  value of  1, char- acteristic of undoped BaZrO3 possessing ideal cubic perovskite structure, to 0.946 for BaPrO3 with orthorhombic distortions of the lattice. Similar, but significantly more pronounced trend — the decrease in ∆fH°ox with the increase in x — was also report- ed for BaZr1–xYxO3–δ (x  =  0.0–0.3) [14]. In contrast with BaZr1–xPrxO3, the struc- ture of  BaZr1–xYxO3–δ is  destabilized not only by the difference in crystal radii of Zr and Y, but also by the formation of the oxy- gen vacancies. Moreover, Ba-loss during synthesis procedure and associated Y re- distribution between A- and B-sublattice, not to mention of ordering of oxygen va- cancies, are also influencing the stability of BaZr1–xYxO3–δ. These additional factors should be responsible for more abruptly increasing ∆fH°ox of  BaZr1–xYxO3–δ with the  dopant concentration, as  compared to BaZr1–xPrxO3. It is  also of  interest that the  mixing enthalpy of  BaZr1–xPrxO3 solid solution, calculated as ( ) ( ) ( ) mix f ox 1 3 f ox 3 f ox 3 BaZr Pr O (1 ) BaZrO BaPrO , x xH H x H x H −∆ ° = ∆ − − − ⋅∆ − − ⋅∆    (3) Fig. 1. Standard enthalpy of formation from binary oxides vs tolerance factor for some AIIBIVO3 oxides. Points — calculation using the literature data [20–27], line — linear fit. The reference thermodynamic data values are also given in Supplementary 45 is slightly positive, as seen in Table 1, most probably, as a result of both the abovemen- tioned difference in the crystal structure of the end members and the size mismatch between Zr4+ and Pr4+cations. However, the absolute value of ΔmixH° is well within the estimated level of uncertainty, indicat- ing the behavior close to that of the ideal or regular (the maximum of ΔmixH° cor- responds to x = 0.5) solution. This is con- sistent with a very small positive change of the molar volume upon mixing BaZrO3 and BaPrO3 [28]. The  ideal (or regular) solution behavior opens up a  possibility to  estimate the  entropy of  BaZr1–xPrxO3 solid solution as ( ) [ ] 1 3 3 3 BaZr Pr O (BaZrO ) (BaPrO ) ln( ) (1 ) ln(1 ) (1 ) , x x S R x x x x x S x S − = = − ⋅ + − ⋅ − + + − ⋅ + ⋅    (4) where R is  the  universal gas constant, the  first term in  the  right hand side is the entropy of ideal mixing and 3(BaZrO ) S and 3(BaPrO ) S   — the  standard entropies of  BaPrO3 and BaPrO3, respectively. Table 1 Estimated standard thermodynamic functions of BaZr1 — xPrxO3 (x = 0.0–1.0) x ta ΔfH°ox b / kJ·mol–1 ΔfH°el c / kJ·mol–1 ΔmixH° d / kJ·mol–1 S°298 e / J·mol–1·K–1 ΔfS°ox f / J·mol–1·K–1 ΔfG°ox g / kJ·mol–1 0.0 1.004  — 117.0* –1762.5 0.00 125.5 5.1 –118.5 0.1 0.998 –111.4 –1742.1 0.29 131.9 8.5 –113.9 0.2 0.992 –106.0 –1721.8 0.51 137.1 10.7 –109.2 0.3 0.986 –100.5 –1701.6 0.67 141.8 12.3 –104.2 0.4 0.980 –95.2 –1681.4 0.76 146.0 13.5 –99.2 0.5 0.974 –89.9 –1661.3 0.78 149.9 14.3 –94.2 0.6 0.968 –84.7 –1641.3 0.75 153.5 14.8 –89.1 0.7 0.963 –79.5 –1621.3 0.65 156.7 15.0 –84.0 0.8 0.957 –74.4 –1601.4 0.49 159.5 14.8 –78.8 0.9 0.951 –69.4 –1581.5 0.28 161.8 14.0 –73.6 1.0 0.946  — 64.4** –1561.7 0.00 162.8 12.0 –68.0 a Goldschmidt’s tolerance factor, (crystal radii, coordination numbers: 12 — for A2+ cation, 6 — for B4+ cation and O2– anion). b Standard enthalpy of formation from binary oxides at 298.15 K, the expanded uncertainty (95% confidence level) is ±14 kJ·mol–1. c Standard enthalpy of forma- tion from elements at 298.15 K, the expanded uncertainty (95% confidence level) is ±14 kJ·mol–1. d Standard enthalpy of mixing at 298.15 K, the expanded uncertainty (95% confidence level) is ±14 kJ·mol–1. e Standard entropy at 298.15 K, the expanded uncertainty (95% confidence level) linearly scales with x from ±1 J·mol–1·K–1 for BaZrO3 (x = 0) to ±2.8 J·mol –1·K–1 for BaPrO3 (x = 1). f Standard entropy of formation from binary oxides at 298.15 K, the expanded uncertainty (95% confidence level) linearly scales with x from ±1.2 J·mol–1·K–1 for BaZrO3 (x=0) to ±3.5 J·mol –1·K–1 for BaPrO3 (x = 1). g Standard Gibbs free energy of formation from binary oxides at 298.15 K, the expanded uncertainty (95% confidence level) is ±14 kJ·mol–1. * Experimental formation enthalpy f oxH∆   = ( — 115.12 ± 3.69) kJ·mol –1 [14], (–117.44 ± 3.7) kJ·mol–1 [13] ** Experimental formation enthalpy f oxH∆   = ( — 70 ± 10) kJ·mol –1 [20], (–147 ± 8) kJ·mol–1 [21] 46 The only unknown parameter in the Eq. (4) is the standard entropy of BaPrO3, 3(BaPrO ) ,S  which has to  be estimated since no ex- perimental value has been reported so far. To do this, we, first, tried to correlate the standard entropies available for some of the AIIBIVO3 oxides with their molar vol- umes in  line with the  so-called volume- based approach introduced by Glasser and Jenkins [30]. However, it was found that much better correlation can be established using relative changes of entropy and mo- lar volume instead of  their absolute val- ues. These relative changes correspond to the formation from binary oxides (re- action (1)) and can be calculated as follows: 2 f ox AO BO ,S S S S ∆ ω = +    (5) 2 f (ox) (AO) (BO ) ,mV m m V V V ∆ ω = + (6) where ωS and ωV are the relative changes of entropy and molar volume; f oxS∆  and f (ox)mV∆ are the absolute changes of stand- ard entropy and molar volume in the for- mation reaction (1); 2AO BO ,S S  and Vm(AO), 2(BO )m V  — are standard entropies and mo- lar volumes of constituting binary oxides, respectively. ωS as a function of ωV is shown in Fig. 2 for the AIIBIVO3 oxides for which we have managed to find the literature val- ues of the absolute entropies. Surprisingly good linear correlation can be observed between ωS and ωV. The two outliers are CaHfO3 and BaTiO3. The reason for these deviations is unclear, but, taking into ac- count the  good linear trend for the  rest of  the  AIIBIVO3 oxides, it seems that one can suggest some errors in the reference data reported for BaTiO3 and CaHfO3. The observed ωS(ωV) linear dependence (see Fig. 2) was least squares fitted. The re- sulting equation is the following: 27.99 10 0.51 .S V −ω = ⋅ + ⋅ω (7) The  coefficient of  determination is  R2  =  0.97. The  standard deviation of the fitted line from the points in Fig. 2 is  0.006 (note that BaTiO3 and CaHfO3 were not taken into account). The absolute entropy of  the  perovskite oxide AIIBIVO3 can be calculated using the Eq. (7) as ( ) ( )( ) 3 2 2 ABO AO BO f ox AO BO 1.0799 0.51 .V S S S S S S = + + ∆ = = + + ⋅ω       (8) The standard deviation of the as-esti- mated entropy from the experimental ref- erence data does not exceed 0.5 J·mol–1·K–1. For the  outliers, BaTiO3 and CaHfO3, calculation according to  Eq.  (8) leads to the standard entropies 120.6 J·mol–1·K–1 and 107.0 J·mol–1·K–1, respectively, with the  deviations from the  reference ther- modynamic data [20–27,31–34] equal to  +12.7  J·mol–1·K–1 (i.e. +11.8%) and –6.4 J·mol–1·K–1 (i.e. –5.7%), respectively. The absolute entropy of BaPrO3, calcu- lated using Eq. (8), equals 162.8 J·mol–1·K–1 Fig. 2. Relative change of entropy vs relative change of molar volume in the reaction (1) for some AIIBIVO3 perovskites. Points — calculation using the literature data [20–27, 31, 32], line — linear fit. The reference thermodynamic and structural data values are also given in Supplementary 47 with probable expanded uncertainty (95% confidence level) of 2.8 J·mol–1·K–1. Here, rather high uncertainty of the  3Pr O S value (2.0 J·mol–1·K–1 [34]) has the most influence on the  expanded uncertainty of  3Ba Pr O .S With this value in hand, it is now possible to estimate the entropies of BaZr1–xPrxO3 solid solutions, 298 ,S according to Eq. (4). The  calculated values are summarized in Table 1. The combined uncertainty of  298S depends on the amount of dopant x in BaZr1–xPrxO3, and can be evaluated using the following expression: ( ) ( ) 3 3 298 2 2 BaZrO BaPrO ( ) (1 ) ( ) ( ) , S x S x S δ = = − ⋅δ + ⋅δ    (9) where δ(S) is the uncertainty of the cor- responding entropy value S. With the val- ues of  3BaZrO ( )Sδ    =  1.0  J·mol–1·K–1 [32], which is less than our predicted value of  3BaPrO ( )Sδ   = 2.8 J·mol–1·K–1, 298( )Sδ  is also less than 2.8  J·mol–1·K–1 for any possible value of x. The  entropies of  formation from ox- ides, f ox ,S∆  listed in  Table  1, obviously, also depend on the concentration of pra- seodymium: ( )f ox 298 1 3 298 298 2 298 2 BaZr Pr O (BaO) (1 ) (ZrO ) (Pr O ), x xS S S x S x S −∆ = − − − − ⋅ − − ⋅      (10) and so does their uncertainty, which in- creases with x from 1.2  J·mol–1·K–1 for BaZrO3 to 3.5 J·mol –1·K–1 for BaPrO3. The standard Gibbs free energy of for- mation at 298.15 K of BaZr1–xPrxO3 oxides, calculated as f ox f ox f ox298.15 ,G H S∆ = ∆ − ⋅∆   (11) using estimated enthalpies and entropies, is  also given in  Table  1. The  combined expanded uncertainty of  f ox ,G∆  is deter- mined by the uncertainty of  f ox ,H∆  which is much higher than that of the entropic term, and is equal to 14 kJ·mol–1. As seen, all the solid solutions studied are stable against their constituting binary oxides. However, the  relative stability of  BaZr1–xPrxO3 de- creases with the amount of Pr. Conclusions The  dependence of  the  standard en- thalpy of formation from binary oxides on the Goldschmidt tolerance factor, f ox ( ),H t∆  was shown to  be linear for a  number of  perovskite-type AIIBIVO3 (A  = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce, Pr, Tb, U, Pu, Am) oxides. This dependence was used to pre- dict the  f oxH∆  values for praseodymium- substituted barium zirconates BaZr1–xPrxO3. The increase in x results in the distortions of the crystal lattice, decreasing the toler- ance factor and making f oxH∆  more posi- tive. The values of the enthalpies of mixing, calculated regarding BaZr1–xPrxO3 as a solid solution of BaPrO3 in BaZrO3, were found to be indicative of the regular or ideal solu- tion behavior. Thus, to estimate the absolute entropy of BaZr1–xPrxO3 using the expres- sion for the entropy of ideal mixing, the ab- solute entropy of BaPrO3, not yet reported in the literature, had to be estimated first. We found that for some of  the  AIIBIVO3 (A = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce) per- ovskites, for which the entropy values are known, an almost perfectly linear relation- ship exists between the  relative changes of entropy and molar volume in the reac- tion of formation of AIIBIVO3 from AO and BO2. This relationship allowed predict- ing the  entropy of  BaPrO3 with relative uncertainty of  less than 2% of  its value, the uncertainty being virtually determined 48 by the uncertainties of the reference S°298 data for the corresponding binary oxides. With the knowledge of S°298(BaPrO3), not only the absolute entropy values, but also the standard entropies and Gibbs energies of formation of BaZr1–xPrxO3 from binary oxides were calculated. The latter, though increasing with x in BaZr1–xPrxO3, are nega- tive for all x from 0.0 to 1.0, so BaZr1–xPrxO3 should be stable with respect to BaO, ZrO2 and PrO2. The methodology employed in predict- ing the enthalpy, f ox ,H∆  and, especially, the absolute entropy of BaZr1–xPrxO3 can be applied to other similar oxides. We believe that, especially in the absence of experi- mental data, our work would be of inter- est to  the  researchers who are studying the  thermodynamics and stability issues of substituted barium zirconates, and that it could provide the  data for the  future thermodynamic assessments and phase diagram calculations in BaO–ZrO2–PrO2 and related oxide systems. Acknowledgements This work was supported by the Russian Science Foundation (project No. 18-73-00022). References 1. Kreuer KD. Proton-Conducting Oxides. Annu Rev Mater Res. 2003;33(1):333–59. DOI:10.1146/annurev.matsci.33.022802.091825 2. Norby T. Proton Conductivity in Perovskite Oxides. Boston, MA: Springer US; 2009. 217 p. (Ishihara T, editor. Perovskite Oxide for Solid Oxide Fuel Cells). DOI:10.1007/978-0-387-77708-5_11 3. Sažinas R, Einarsrud M-A, Grande T. Toughening of Y-doped BaZrO3 proton con- ducting electrolytes by hydration. J Mater Chem A. 2017;5(12):5846–57. DOI:10.1039/C6TA11022C 4. Iguchi F, Tsurui T, Sata N, Nagao Y, Yugami H. The relationship between chemical composition distributions and specific grain boundary conductivity in Y-doped BaZrO3 proton conductors. Solid State Ion. 2009;6–8(180):563–8. DOI:10.1016/j.ssi.2008.12.006 5. Babilo P, Uda T, Haile SM. Processing of yttrium-doped barium zirconate for high proton conductivity. J Mater Res. 2007;22(5):1322–30. DOI:10.1557/jmr.2007.0163 6. Ryu KH, Haile SM. Chemical stability and proton conductivity of doped BaCeO3– BaZrO3 solid solutions. Solid State Ion. 1999;125(1):355–67. DOI:10.1016/S0167-2738(99)00196–4 7. Duval SBC, Holtappels P, Vogt UF, Pomjakushina E, Conder K, Stimming U, Graule T. Electrical conductivity of the proton conductor BaZr0.9Y0.1O3−δ obtained by high temperature annealing. Solid State Ion. 2007;178(25):1437–41. DOI:10.1016/j.ssi.2007.08.006 8. Kjølseth C, Fjeld H, Prytz Ø, Dahl P, Estournès C, Haugsrud R, Norby T. Space — charge theory applied to  the  grain boundary impedance of  proton conducting BaZr0.9Y0.1O3−δ. Solid State Ion. 2010;181. DOI:10.1016/j.ssi.2010.01.014 49 9. Magrasó A, Frontera C, Gunnæs AE, Tarancón A, Marrero-López D, Norby T, Haugsrud R. Structure, chemical stability and mixed proton — electron conductivity in BaZr0.9−xPrxGd0.1O3−δ. J Power Sources. 2011;196(22):9141–7. DOI:10.1016/j.jpowsour.2011.06.076 10. Fabbri E, Markus I, Bi L, Pergolesi D, Traversa E. Tailoring mixed proton-electronic conductivity of BaZrO3 by Y and Pr co-doping for cathode application in protonic SOFCs. Solid State Ion. 2011;202(1):30–5. DOI:10.1016/j.ssi.2011.08.019 11. Fabbri E, Bi L, Tanaka H, Pergolesi D, Traversa E. Chemically Stable Pr and Y Co- Doped Barium Zirconate Electrolytes with High Proton Conductivity for Interme- diate-Temperature Solid Oxide Fuel Cells. Adv Funct Mater. 2011;21(1):158–66. DOI:10.1002/adfm.201001540 12. Tsvetkov D, Sednev-Lugovets A, Malyshkin D, Sereda V, Zuev A, Ivanov I. Crystal structure and high-temperature thermodynamic properties of Pr-doped barium zirconates, BaZr1-xPrxO3 (x = 0.1, 0.5). J Phys Chem Solids. Forthcoming 2020. 13. Huntelaar ME, Booij AS, Cordfunke EHP. The standard molar enthalpies of forma- tion of BaZrO3(s) and SrZrO3(s). J Chem Thermodyn. 1994;26(10):1095–101. DOI:10.1006/jcht.1994.1127 14. Gonçalves MD, Maram PS, Muccillo R, Navrotsky A. Enthalpy of formation and ther- modynamic insights into yttrium doped BaZrO3. J Mater Chem A. 2014;2(42):17840–7. DOI:10.1039/C4TA03487B 15. Katsura T, Kitayama K, Sugihara T, Kimizuka N. Thermochemical Properties of Lanth- anoid-Iron-Perovskite at High Temperatures. Bull Chem Soc Jpn. 1975;48(6):1809–11. DOI:10.1246/bcsj.48.1809 16. Katsura T, Sekine T, Kitayama K, Sugihara T, Kimizuka N. Thermodynamic prop- erties of  Fe-lathanoid-O compounds at  high temperatures. J Solid State Chem. 1978;23(1):43–57. DOI:10.1016/0022–4596(78)90052-X 17. Petrov AN, Kropanev AY, Zhukovskij BM. Thermodynamic properties of rare earth cobaltites, RCoO3. Zhurnal Fizicheskoj Khimii. 1984;58(1):50–3. 18. Navrotsky A. Energetics of Phase Transition in AX, ABO3 and AB2O4 Compounds. New York: Academic Press; 1981. 71 p. (O’Keefe M., Navrotsky A., Editors. Structure and Bonding in Crystals). 19. Shannon RD. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr Sect A. 1976;32(5):751–67. DOI:10.1107/S0567739476001551 20. Ushakov SV, Cheng J, Navrotsky A, Wu JR, Haile SM. Formation Enthalpies of Tetravalent Lanthanide Perovskites by High Temperature Oxide Melt Solution Calorimetry. MRS Online Proceedings Library Archive. 2002;718. DOI:10.1557/PROC-718-D7.17 21. Morss LR, Mensi  N.  Enthalpy of  Formation of  Barium Lanthanide(IV) Oxides: BaCeO3, BaPrO3, and BaTbO3. Boston, MA: Springer; 1982. 279 p. (G. J. McCarthy, H. B. Silber, J. J. Rhyne, Editors. The Rare Earths in Modern Science and Technology). DOI:10.1007/978-1-4613-3406-4_56 50 22. NIST Standard Reference Database Number 69 [Internet]. Washington: National Institute of Standards and Technology; 2018 [modified October 2018; cited 2020 April 26]. Available from: https://doi.org/10.18434/T4D303 23. Todd SS, Lorenson RE. Heat Capacities at  Low Temperatures and Entro- pies at  298.16°K. of  Metatitanates of  Barium and Strontium. J Am Chem  Soc. 1952;74(8):2043–5. DOI:10.1021/ja01128a054 24. Navrotsky A. Thermochemistry of crystalline and amorphous phases related to ra- dioactive waste. Netherlands: Kluwer Academic Publishers; 1998. 267 p. (P. A. Sterne, A. Gonis, A. A. Borovoi, Editors. Actinides and the environment) 25. SpringerMaterials [Internet]. New York; 2020 [modified 2020 April 28; cited 2020 April 28]. Available from: https://materials.springer.com/ 26. Termicheskie konstanty veshestv [Internet]. Moscow: Moscow State University; 2020 [modified 2020 April 28; cited 2020 April 28]. Available from: http://www. chem.msu.ru/cgi-bin/tkv.pl 27. Goudiakas J, Haire RG, Fuger J. Thermodynamics of lanthanide and actinide per- ovskite-type oxides IV. Molar enthalpies of formation of MM′O3 (M = Ba or Sr, M′ = Ce, Tb, or Am) compounds. J Chem Thermodyn. 1990;22(6):577–87. DOI:10.1016/0021–9614(90)90150-O 28. Antunes I, Amador U, Alves A, Correia MR, Ritter C, Frade JR, Pérez-Coll D, Mather GC, Fagg DP. Structure and Electrical-Transport Relations in Ba(Zr,Pr)O3−δ Perovs- kites. Inorg Chem. 2017;56(15):9120–31. DOI:10.1021/acs.inorgchem.7b01128 29. Jacobson AJ, Tofield BC, Fender BEF. The structures of BaCeO3, BaPrO3 and BaTbO3 by neutron diffraction: lattice parameter relations and ionic radii in O-perovskites. Acta Cryst B. 1972;28(3):956–61. DOI:10.1107/S0567740872003462 30. Glasser L, Jenkins HDB. Predictive thermodynamics for condensed phases. Chem Soc Rev. 2005;34(10):866–74. DOI:10.1039/B501741F 31. Kurosaki K, Konings RJM, Wastin F, Yamanaka S. The low-temperature heat capacity and entropy of SrZrO3 and BaZrO3. J Alloys Compd. 2006;424(1):1–3. DOI:10.1016/j.jallcom.2005.09.096 32. Ahrens M, Maier J. Thermodynamic properties of BaCeO3 and BaZrO3 at low tem- peratures. Thermochim Acta. 2006;443(2):189–96. DOI:10.1016/j.tca.2006.01.020 33. Cordfunke EHP, van der Laan RR, van Miltenburg JC. Thermophysical and ther- mochemical properties of BaO and SrO from 5 to 1000 K. J Phys Chem of Solids. 1994;55(1):77–84. DOI:10.1016/0022–3697(94)90186–4 34. Konings RJM, Beneš O, Kovács A, Manara D, Sedmidubský D, Gorokhov L, et al. The Thermodynamic Properties of the f-Elements and their Compounds. Part 2. The Lanthanide and Actinide Oxides. J Phys Chem Ref Data. 2014;43(1):013101. DOI:10.1063/1.4825256