Thermodynamics of formation of solid solutions between BaZrO3 and BaPrO3

A linear relationship between the standard enthalpy of formation from binary 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 dependence 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 ·K. 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, and those of S°298 and ΔfS°ox — less than 2.8 J · mol ·K and 3.5 J · mol·K, respectively, for BaZr1–xPrxO3 (x = 0.0–1.0).


Introduction
Partially substituted barium zirconates, BaZr 1-x M x O 3-δ (M = rare-earth or alkaline-earth element), are the state-ofthe-art proton-conducting electrolyte materials for intermediate-temperature solid oxide fuel cells [1][2][3]. These complex oxides possess high proton conductivity upon hydration, good chemical and mechanical stability. Among their known drawbacks are high grain boundary resistance, slow grain growth and, as a consequence, very high sintering temperatures (1900-2000 K) required for obtaining dense ceramics [4][5][6][7][8]. Praseodymium doping was suggested as a possible way not only to overcome these drawbacks [9] but also, due to potentially mixed-valent state of Pr, to obtain triple-conducting (electron-proton-oxide ion) and catalytically active electrode materials for highly efficient proton-conducting solid oxide fuel cells (PC SOFCs) [10,11]. In spite of the promising electrochemical properties of the BaZr 1-x Pr x O 3 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 thermodynamic properties of BaZr 1-x Pr x O 3 oxides such as enthalpy increments and constant-pressure 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 zirconates and was aimed to estimate the standard thermodynamic functions (enthalpy, entropy and Gibbs free energy) of formation of BaZr 1-x Pr x O 3 oxides.

Results and discussion
Typically, when it is necessary to experimentally determine the standard formation enthalpy of a compound, the solution calorimetry is the most straightforward method of choice. However, the dissolution of zirconates is quite a hard task, as our preliminary experiments showed. It requires using either highly corrosive mixtures of acids such as, for example, HF and HNO 3 employed by Huntelaar et al. [13], or high-temperature melts [14]. Importantly, in the latter case the solvent stirring is necessary since the dissolution kinetics is slow. Unfortunately, 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, resulted in irreproducible solution enthalpies of BaZr 1-x Pr x O 3 . Besides, the hydrofluoric acid resistant measurement cell for the solution calorimeter has to be custom-made and was not readily available. Because of these reasons, the standard formation enthalpies of BaZr 1-x Pr x O 3 zirconates were estimated using the well-known strong correlation between the formation enthalpy and Gold-schmidt's tolerance factor [15][16][17][18]. This correlation was shown to allow predicting reasonably good, i.e. very close to the experimental values, estimates of the formation enthalpies for many perovskite oxides.
The standard enthalpy of formation at 298.15 K, ∆ f H°o x , corresponding to the reaction AO+ BO 2 = ABO 3 (1) calculated for a number of A II B IV O 3 perovskite-type oxides, is shown in Fig. 1 as a function of Goldschmidt's tolerance factor, The values of the tolerance factor were calculated using the crystal radii reported by Shannon [19] with the following coordination numbers: 12 -A 2+ cation, 6 -for both B 4+ cation and O 2anion. The necessary thermodynamic data were taken from [20][21][22][23][24][25][26][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 BO 2 and ABO 3 oxides which possess different crystal structure depending on the nature of the A and B cations.
However, the differences in the crystal structure of both BO 2 and ABO 3 with different cations were not taken into account. The enthalpies of slight distortions of the perovskite structure in ABO 3 are generally small and were thought to be much less than the standard deviation of the estimated values. In turn, even though the crystal structure of BO 2 varies more than that of ABO 3 , judging by the good linearity of the ∆ f H°o x ,(t) dependence in Fig. 1, its influence should also be rather small. The linear dependence observed in Fig. 1 was least squares fitted. The resulting equation is the following: ( ) with the coefficient of determination R 2 = 0.98. The standard formation enthalpies of BaZr 1-x Pr x O 3 oxides calculated according 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 uncertainty (95% confidence level) of the ∆ f H°o x values reported in Table 1 is equal to 14 kJ·mol -1 . However, since the ex-perimental points corresponding to both BaZrO 3 and BaPrO 3 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 ∆ f H°o x values is likely to be somewhat better than this rather conservative estimate of 14 kJ·mol -1 .
As follows from Fig. 1 and Table 1, the standard formation enthalpy of zirconates BaZr 1-x Pr x O 3 increases with doping 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 tolerance factor, t, from the value of 1, characteristic of undoped BaZrO 3 possessing ideal cubic perovskite structure, to 0.946 for BaPrO 3 with orthorhombic distortions of the lattice. Similar, but significantly more pronounced trend -the decrease in ∆ f H°o x with the increase in x -was also reported for BaZr 1-x [14]. In contrast with BaZr 1-x Pr x O 3 , the structure of BaZr 1-x Y x O 3-δ is destabilized not only by the difference in crystal radii of Zr and Y, but also by the formation of the oxygen vacancies. Moreover, Ba-loss during synthesis procedure and associated Y redistribution between A-and B-sublattice, not to mention of ordering of oxygen vacancies, are also influencing the stability of BaZr 1-x Y x O 3-δ . These additional factors should be responsible for more abruptly It is also of interest that the mixing enthalpy of BaZr 1-x Pr x O 3 solid solution, calculated as  Table 1, most probably, as a result of both the abovementioned difference in the crystal structure of the end members and the size mismatch between Zr 4+ and Pr 4+ cations. However, the absolute value of Δ mix H° is well within the estimated level of uncertainty, indicating the behavior close to that of the ideal or regular (the maximum of Δ mix H° corresponds to x = 0.5) solution. This is consistent with a very small positive change of the molar volume upon mixing BaZrO 3 and BaPrO 3 [28]. The ideal (or regular)  (1 ) (1 ) , where R is the universal gas constant, the first term in the right hand side is the entropy of ideal mixing and   [14], (-117.44 ± 3.7) kJ·mol -1 [13] ** Experimental formation enthalpy f ox H ∆  = ( -70 ± 10) kJ·mol -1 [20], (-147 ± 8) kJ·mol -1 [21] The only unknown parameter in the Eq. (4) is the standard entropy of BaPrO 3 , 3 (BaPrO ) , S  which has to be estimated since no experimental value has been reported so far. To do this, we, first, tried to correlate the standard entropies available for some of the A II B IV O 3 oxides with their molar volumes in line with the so-called volumebased approach introduced by Glasser and Jenkins [30]. However, it was found that much better correlation can be established using relative changes of entropy and molar volume instead of their absolute values. These relative changes correspond to the formation from binary oxides (reaction (1)) and can be calculated as follows: where ω S and ω V are the relative changes of entropy and molar volume; f ox S ∆  and -are standard entropies and molar volumes of constituting binary oxides, respectively. ω S as a function of ω V is shown in Fig. 2 for the A II B IV O 3 oxides for which we have managed to find the literature values of the absolute entropies. Surprisingly good linear correlation can be observed between ω S and ω V . The two outliers are CaHfO 3 and BaTiO 3 . The reason for these deviations is unclear, but, taking into account the good linear trend for the rest of the A II B IV O 3 oxides, it seems that one can suggest some errors in the reference data reported for BaTiO 3 and CaHfO 3 .
The observed ω S (ω V ) linear dependence (see Fig. 2) was least squares fitted. The resulting equation is the following: 2 7.99 10 0.51 .
The coefficient of determination is R 2 = 0.97. The standard deviation of the fitted line from the points in Fig. 2 is 0.006 (note that BaTiO 3 and CaHfO 3 were not taken into account). The absolute entropy of the perovskite oxide A II B IV O 3 can be calculated using the Eq.   The entropies of formation from oxides, f ox , S ∆  listed in Table 1, obviously, also depend on the concentration of praseodymium: ( ) and so does their uncertainty, which increases with x from 1.
using estimated enthalpies and entropies, is also given in Table 1. The combined expanded uncertainty of f ox , G ∆  is determined 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 BaZr 1-x Pr x O 3 decreases with the amount of Pr.

Conclusions
The dependence of the standard enthalpy 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 A II B IV O 3 (A = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce, Pr, Tb, U, Pu, Am) oxides. This dependence was used to predict the f ox H ∆  values for praseodymiumsubstituted barium zirconates BaZr 1-x Pr x O 3 . The increase in x results in the distortions of the crystal lattice, decreasing the tolerance factor and making f ox H ∆  more positive. The values of the enthalpies of mixing, calculated regarding BaZr 1-x Pr x O 3 as a solid solution of BaPrO 3 in BaZrO 3 , were found to be indicative of the regular or ideal solu-tion behavior. Thus, to estimate the absolute entropy of BaZr 1-x Pr x O 3 using the expression for the entropy of ideal mixing, the absolute entropy of BaPrO 3 , not yet reported in the literature, had to be estimated first. We found that for some of the A II B IV O 3 (A = Ca, Sr, Ba; B = Ti, Zr, Hf, Ce) perovskites, for which the entropy values are known, an almost perfectly linear relationship exists between the relative changes of entropy and molar volume in the reaction of formation of A II B IV O 3 from AO and BO 2 . This relationship allowed predicting the entropy of BaPrO 3 with relative uncertainty of less than 2% of its value, the uncertainty being virtually determined by the uncertainties of the reference S°2 98 data for the corresponding binary oxides. With the knowledge of S°2 98 (BaPrO 3 ), not only the absolute entropy values, but also the standard entropies and Gibbs energies of formation of BaZr 1-x Pr x O 3 from binary oxides were calculated. The latter, though increasing with x in BaZr 1-x Pr x O 3 , are negative for all x from 0.0 to 1.0, so BaZr 1-x Pr x O 3 should be stable with respect to BaO, ZrO 2 and PrO 2 .
The methodology employed in predicting the enthalpy, f ox , H ∆  and, especially, the absolute entropy of BaZr 1-x Pr x O 3 can be applied to other similar oxides. We believe that, especially in the absence of experimental data, our work would be of interest 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-ZrO 2 -PrO 2 and related oxide systems.