Calorimetric Determination of Heat Capacity, Entropy and Enthalpy of Mixed Oxides in the System CaO–SrO–Bi2O3–Nb2O5–Ta2O5

1. Introduction

Mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅ possess many extraordinary electric, magnetic and optical properties for which they are used in fabrication of various electronic components. For example Sr₂(Nb,Ta)₂O₇ and (Sr,Ca)Bi₂(Nb,Ta)₂O₉ are used for ferroelectric memory devices, CaNb₂O₆, Sr₅(Nb_1–xTa_x)₄O₁₅ and Bi(Nb,Ta)O₄ for microwave dielectric resonators and Ca₂Nb₂O₇ as non-linear optical materials and hosts for rare-earth ions in solid-state lasers. Ternary strontium bismuth oxides SrBi₂O₄, Sr₂Bi₂O₅, and Sr₆Bi₂O₉ are of considerable interest due to a visible light driven fotocatalytic activity.

To assess the thermodynamic stability and reactivity of these oxides under various conditions during their preparation, processing and operation, a complete set of consistent thermodynamic data, including heat capacity, entropy and enthalpy of formation, is necessary. Some of these data are available in literature. Akishige et al. [1] have been measured the heat capacities of Sr₂Nb₂O₇ and Sr₂Ta₂O₇ single crystals in the temperature range 2-600 K. The results have been only plotted and the values of S°_m(298) have not been calculated. A commensurate transformation of Sr₂Nb₂O₇ at T_INC = 495 K has been observed accompanied by changes in enthalpy and entropy of ΔH = 291 J mol^-1 and ΔS = 0.587 J K^-1 mol^-1. The heat capacity of Sr₂Nb₂O₇ has been also measured by Shabbir at al. [2] in the temperature range 375-575 K. They have observed a phase transition at T_INC = 487 ± 2 K connected with ΔH = 147 ± 14 J mol^-1 and ΔS = 0.71 ± 0.10 J K^-1 mol^-1. The heat capacities of polycrystalline and monocrystalline SrBi₂Ta₂O₉ and Sr_0,85Bi_2,1Ta₂O₉ have been measured by Onodera at al. [3–5] at 80-800 K. Morimoto at al. [6] have reported the results of the heat capacity measurements of SrBi₂(Nb_xTa_1-x)₂O₉ (x = 0, 1/3, 2/3 a 1). The temperature dependences of heat capacities show lambda-transitions with maxima at the Currie temperature TC = 570 ± 1 K, 585 ± 2 K, 625 ± 3 K a 690 ± 2 K for x = 0, 1/3, 2/3 and 1, respectively. Using EMF (electromotive force) measurements, Raghavan has obtained the values of the Gibbs energy of formation from binary oxides, Δ_oxG, for some niobates [7,8] and tantalates [9,10] of calcium. His results are summarized in Table 1. The same technique has been employed by Dneprova et al. [11] for Δ_oxG measurement for CaNb₂O₆ and Ca₂Nb₂O₇. Their results presented in Table 1 are not significantly different from the results of Raghavan. Using the CALPHAD approach [13], Yang et al. [14] have assessed thermodynamic data for various mixed oxides in the SrO–Nb₂O₅ system. The same approach has been used by Hallstedt et al. for the assessment of thermodynamic data of mixed oxides in the systems CaO–Bi₂O₃ [14] and SrO–Bi₂O₃ [15]. Besides equilibrium data, values of the enthalpy of formation [16] of mixed oxide have been considered. Later on, these systems have been studied by EMF method by Jacob and Jayadevan [17,18] and temperature dependences of Δ_oxG for various mixed oxides have been derived. These data have been included into the thermodynamic re-assessment of the CaO-SrO-Bi₂O₃ system [19].

Oxide	ΔoxG; (kJ mol^–1)	T; (K)	ΔoxH; (kJ mol^–1)	ΔoxS; (J K^-1 mol^–1)	Ref.
CaNb2O6	–75.82 – 0.03345T	1245-1300	–75.82	33.45	[7]
Ca2Nb2O7	–178.44	1256			[8]
Ca3Nb2O8	–209.94	1256			[8]
CaTa4O11	–36.982 – 0.029T	1250-1300	–36.98	29.0	[9]
CaTa2O6	–65.14	1250			[10]
Ca2Ta2O7	–102.82	1250			[10]
Ca4Ta2O9	–165.05	1250			[10]
CaNb2O6	–175.73 + 0.02259T	1100-1276	–175.73	–22.59	[11]
Ca2Nb2O7	–212.54 – 0.02218T	1100-1350	–212.54	22.18	[11]
Sr2Nb10O27	–1125.69 + 0.35069T	298-5000	–1125.69	–350.69	[12]
SrNb2O6	–325.04 + 0.05865T	298-5000	–325.04	–58.65	[12]
Sr2Nb2O7	–367.43 + 0.03993T	298-5000	–367.43	–39.93	[12]
Sr5Nb4O14	–746.72 + 0.05101T	298-5000	–746.72	–51.01	[12]
Ca5Bi14O26	–125.90 – 0.055T	298-1300	–125.9	55.0	[19]
CaBi2O4	–27.60 – 0.003T	298-1300	–27.6	3.0	[19]
Ca4Bi6O13	–97.60 – 0.008T	298-1300	–97.6	8.0	[19]
Ca2Bi2O5	–42.20 – 0.003T	298-1300	–42.2	3.0	[19]
SrBi2O4	–63.86 – 0.0018T	298-1300	–63.86	1.8	[19]
Sr2Bi2O5	–118.75 + 0.024T	298-1300	–118.75	–24.0	[19]
Sr3Bi2O6	–109.60 + 0.0024T	298-1300	–109.60	–2.4	[19]

Table 1.

Published values of Δ_oxG, Δ_oxH a Δ_oxS for some mixed oxides in the system CaO-SrO-Bi₂O₃-Nb₂O₅-Ta₂O₅

This review brings a summary of our results [20–30] focused on calorimetric determination of heat capacity, entropy end enthalpy of mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅. Temperature dependences of molar heat capacity in a broad temperature range were evaluated from the experimental heat capacity and relative enthalpy data. Molar entropies at T = 298.15 K were calculated from low temperature heat capacity measurements. Furthermore, the results of calorimetric measurements of the enthalpies of drop-solution in a sodium oxide-molybdenum oxide melt for several stoichiometric mixed oxides in the above mentioned system are reported from which the values of enthalpy of formation from constituent binary oxides were derived. Finally, some empirical estimation and correlation methods (the Neumann-Kopp’s rule, entropy-volume correlation and electronegativity-differences method) for evaluation of thermodynamic data of mixed oxides are tested and assessed.

2. Experimental

Nineteen mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅ with stoichiometry CaBi₂O₄, Ca₄Bi₆O₁₃, Ca₂Bi₂O₅, SrBi₂O₄, Sr₂Bi₂O₅, CaNb₂O₆, Ca₂Nb₂O₇, SrNb₂O₆, Sr₂Nb₂O₇, Sr₂Nb₁₀O₂₇, Sr₅Nb₄O₁₅, BiNbO₄, BiNb₅O₁₄, BiTaO₄, Bi₄Ta₂O₁₁, Bi₇Ta₃O₁₈, Bi₃TaO₇, SrBi₂Nb₂O₉, and SrBi₂Ta₂O₉ were prepared, characterized and examined. The samples were prepared by conventional solid state reactions from high purity precursors (CaCO₃, SrCO₃ Bi₂O₃, Nb₂O₅ and Ta₂O₅). A three step procedure was used consisting of an initial calcination run of mixed powder precursors and subsequent double firing of prereacted mixtures pressed into pellets. The phase composition of the prepared samples was checked by X-ray powder diffraction (XRD). XRD data were collected at room temperature with an X’Pert PRO (PANalytical, the Netherlands) θ-θ powder diffractometer with parafocusing Bragg-Brentano geometry using CuK_α radiation (λ = 1.5418 nm). Data were scanned over the angular range 5–60° (2θ) with an increment of 0.02° (2θ) and a counting time of 0.3 s step^–1. Data evaluation was performed by means of the HighScore Plus software.

The PPMS equipment 14 T-type (Quantum Design, USA) was used for the heat capacity measurements in the low temperature region [31-35]. The measurements were performed by the relaxation method [36] with fully automatic procedure under high vacuum (pressure ~10^–2 Pa) to avoid heat loss through the exchange gas. The samples were compressed powder pellets. The densities of the samples were about 65 % of the theoretical ones.

The samples were mounted to the calorimeter platform with cryogenic grease Apiezon N (supplied by Quantum Design). The procedure was as follows: First, a blank sample holder with the Apiezon only was measured in the temperature range approx. 2–280 K to obtain background data, then the sample plate was attached to the calorimeter platform and the measurement was repeated in the same temperature range with the same temperature steps. The sample heat capacity was then obtained as a difference between the two data sets. This procedure was applied, because the heat capacity of Apiezon is not negligible in comparison with the sample heat capacity (~8 % at room temperature) and exhibits a peak-shaped transition below room temperature [37]. The manufacturer claims the precision of this measurement better then 2 % [38]; the control measurement of the copper sample (99.999 % purity) confirmed this precision in the temperature range 50–250 K. However, the precision of the measurement strongly depends on the thermal coupling between the sample and the calorimeter platform. Due to unavoidable porosity of the sample plate this coupling is rapidly getting worse as the temperature raises above 270 K and Apiezon diffuses into the porous sample. Consequently, the uncertainty of the obtained data tends to be larger.

A Micro DSC III calorimeter (Setaram, France) was used for the heat capacity determination in the temperature range of 253–352 K. First, the samples were preheated in a continuous mode from room temperature up to 352 K (heating rate 0.5 K min^–1). Then the heat capacity was measured in the incremental temperature scanning mode consisting of a number of 5–10 K steps (heating rate 0.2 K min^–1) followed by isothermal delays of 9000 s. Two subsequent step-by-step heating were recorded for each sample. Synthetic sapphire, NIST Standard reference material No. 720, was used as the reference. The uncertainty of heat capacity measurements is estimated to be better than ±1 %.

Enthalpy increment determinations were carried out by drop method using high-temperature calorimeter, Multi HTC 96 (Setaram, France). All measurements were performed in air by alternating dropping of the reference material (small pieces of synthetic sapphire, NIST Standard reference material No. 720) and of the sample (pressed pellets 5 mm in diameter) being initially held at room temperature, through a lock into the working cell of the preheated calorimeter. Endothermic effects are detected and the relevant peak area is proportional to the heat content of the dropped specimen. The delays between two subsequent drops were 25–30 min. To check the accuracy of measurement, the enthalpy increments of platinum in the temperature range 770–1370 K were measured first and compared with published reference values [39]. The standard deviation of 22 runs was 0.47 kJ mol^–1, the average relative error was 2.0 %. Estimated overall accuracy of the drop measurements is ±3 %.

The heats of drop-solution were determined using a Multi HTC 96 high-temperature calorimeter (Setaram, France). A sodium oxide-molybdenum oxide melt of the stoichiometry 3Na₂O + 4MoO₃ was used as the solvent. The ratio of solute/solvent varied from 1/250 up to 1/500. The measurements were performed at temperatures of 973 and 1073 K in argon or air atmosphere. The method consists in alternating dropping of the reference material (small spherules of pure platinum) and of the sample (small pieces of pressed tablets 10–40 mg), being initially held near room temperature (T₀), through a lock into the working cell (a platinum crucible with the solvent) of the preheated calorimeter at temperature T. Two or three samples were examined during one experimental run. The delays between two subsequent drops were 30–60 min. The total heat effect (Δ_dsH) includes the heat of solution (Δ_solH), the heat content of the sample (Δ_TH), and, for the carbonates, the heat of decomposition (Δ_decompH) to form solid CaO or SrO and gaseous CO₂. Using appropriate thermochemical cycles, the values of the enthalpy of formation of mixed oxides from the binary oxides and from the elements at 298 K were evaluated. The temperature dependence of the heat capacity of platinum [39] was used for the calculation of the sensitivity of the calorimeters.

2.1. Characterization of prepared samples

The XRD analysis revealed that the prepared samples were without any observable diffraction lines from unreacted precursors or other phases. The lattice parameters of the oxides were evaluated by Rietveld refinement [40] and are summarized in Table 2 together with the values of theoretical density calculated from the lattice parameters.

2.2. Evaluation of temperature dependence of heat capacity at low temperatures

The fit of the low-temperature heat capacity data (LT fit) consists of two steps. Assuming the validity of the phenomenological formula C_pm = βT ³ + γ_elT, at T → 0 where β is proportional to the inverse cube root of the Debye temperature Θ_D and γ_elT is the Sommerfeld term, we plotted the C_pm/T vs. T ² dependence for T < 8 K to estimate the Θ_D and γ_el values. Since all compounds under study are semiconductors with a sufficiently large band gap, the non-zero γ_el values are supposed to be either due to some metallic impurities or to a series of Schottky-like transitions resulting from structure defects. Nevertheless, they are negligible in most cases (typically < 0.5 mJ K^–2 mol^–1) and can be ignored in further analysis. As an example, the results of heat capacity measurements on CaNb₂O₆ and LT fit for T < 10 K is shown in Fig. 1.

Oxide	a (nm)	b (nm)	c (nm)	α (°)	β (°)	γ (°)	d (g cm^–3)	Ref.
CaBi2O4	1.66143	1.15781	1.39915	90	134.03	90	6.631	[20]
Ca4Bi6O13	0.59308	1.73512	0.72192	90	90	90	6.540	[20]
Ca2Bi2O5	1.01074	1.01249	1.04618	116.88	107.16	92.98	6.468	[20]
SrBi2O4	1.92635	0.43437	0.61444	90	95.50	90	7.392	[29]
Sr2Bi2O5	1.42935	0.61715	0.76478	90	90	90	6.628	[29]
CaNb2O6	1.49698	0.57472	0.52202	90	90	90	4.760	[26]
Ca2Nb2O7	0.76853	1.33587	0.54959	90	90	98.29	4.496	[26]
SrNb2O6	0.77209	0.55930	1.09821	90	90.37	90	5.174	[24]
Sr2Nb2O7	0.39544	2.67735	0.57004	90	90	90	5.206	[26]
Sr2Nb10O27	3.715	3.697	0.3943	90	90	90	5.653	^a)
Sr5Nb4O15	0.56576	0.56576	1.14536	90	90	120	5.490	[27]
BiNbO4	0.56893	1.1728	0.49915	90	90	90	7.297	[21]
BiNb5O14	1.76762	1.72072	0.39610	90	90	90	4.948	^b)
BiTaO4	0.56394	1.1776	0.49626	90	90	90	9.149	[21]
Bi4Ta2O11	0.66159	0.76528	0.98781	101.39	90.10	89.99	9.306	[28]
Bi7Ta3O18	3.40162	0.76054	0.66354	90	109.16	90	9.395	[28]
Bi3TaO7	0.54711	0.54711	0.54711	90	90	90	9.327	[28]
SrBi2Nb2O9	0.55160	0.55087	2.51020	90	90	90	7.275	[22]
SrBi2Ta2O9	0.55224	0.55266	2.50124	90	90	90	8.801	[22]

Table 2.

Structural characterization of prepared samples

In the second step of the LT fit, both sets of the C_pm data (relaxation time + DSC) were considered. Analysis of the phonon heat capacity was performed as an additive combination of Debye and Einstein models. Both models include corrections for anharmonicity, which is responsible for a small, but not negligible, additive term at higher temperatures and which accounts for the difference between isobaric and isochoric heat capacity. According to literature [41], the term 1/(1 – αT ) is considered as a correction factor.

The acoustic part of the phonon heat capacity is described using the Debye model

$$C_{\text{phD}}=\frac{9R}{1-{\alpha }_{\text{D}}T}{\left(\frac{T}{{\Theta }_{\text{D}}}\right)}^3{\displaystyle {\int }_0^{x_{\text{D}}}\frac{x^4\text{exp(}x\text{)}}{{\left[\text{exp(}x\text{)}-1\right]}^2}\text{d}x}$$E1

where R is the gas constant, Θ_D is the Debye characteristic temperature, α_D is the coefficient of anharmonicity of acoustic branches and x_D = Θ_D/T. Here the three acoustic branches are taken as one triply degenerate branch. Similarly, the individual optical branches are described by the Einstein model

Figure 1.

Temperature dependence of C_pm/T function for CaNb₂O₇ at low temperatures

$$C_{\text{phE}i}=\frac{R}{1-{\alpha }_{\text{E}i}T}\frac{x_{\text{E}i}^2\exp (x_{\text{E}i})}{{\left[\exp (x_{\text{E}i})-1\right]}^2}$$E2

where α_Ei and x_Ei = Θ_Ei/T have analogous meanings as in the previous case. Several optical branches are again grouped into one degenerate multiple branch with the same Einstein characteristic temperature and anharmonicity coefficient. The phonon heat capacity then reads

$$C_{\text{ph}}=C_{\text{phD}}+{\displaystyle \sum _{i=1}^{3n-3}{C}_{\text{phE}i}}$$E3

All the estimated values were further treated by a simplex routine and a full non-linear fit was performed on all adjustable parameters.

The values of relative enthalpies at 298.15 K, H_m(298.15) – H_m(0), were evaluated from the low-temperature C_pm data (LT fit) by numerical integration of the C_pm(T) dependences from zero to 298.15 K. Standard deviations (2σ) were calculated using the error propagation law. The values of standard molar entropies at 298.15 K, S_m(298.15), were derived from the low-temperature C_pm data (LT fit) by numerical integration of the C_pm(T)/T dependences from zero to 298.15 K. A numerical integration was used with the boundary conditions S_m = 0 and C_pm = 0 at T = 0 K. Standard deviations (2σ) were calculated using the error propagation law. All calculated values are summarized in Table 3.

Oxide	Cpm(298); (J K^–1 mol^–1)	Hm(298)–Hm(0); (J mol^–1)	Sm(298); (J K^–1 mol^–1)	ΔoxS; (J K^-1 mol^–1)	Ref.
CaBi2O4	151.3	26470 ± 158	188.5 ± 3.3	1.9	[20]
Ca4Bi6O13	504.1	85079 ± 507	574.1 ± 8.8	–23.8	[20]
Ca2Bi2O5	197.4	33735 ± 201	231.3 ± 2.9	6.6	[20]
SrBi2O4	155.6	29601 ± 169	206.1 ± 1.1	4.0	[29]
Sr2Bi2O5	201.9	38199 ± 219	261.2 ± 1.4	5.5	[29]
CaNb2O6	171.8	28159 ± 170	167.3 ± 0.9	–8.1	[26]
Ca2Nb2O7	218.1	35631 ± 215	212.4 ± 1.2	–1.1	[26]
SrNb2O6	170.2	28722 ± 174	173.9 ± 0.9	–17.0	[24]
Sr2Nb2O7	216.6	37977 ± 266	238.5 ± 1.3	–5.9	[26]
Sr2Nb10O27	746.8	124150 ± 740	759.7 ± 4.1	–33.9	[27]
Sr5Nb4O15	477.2	83340 ± 490	524.5 ± 2.8	–18.4	[27]
BiNbO4	121.3	22120 ± 134	147.9 ± 0.8	5.0	[21]
BiNb5O14	386.8	62639 ± 362	397.2 ± 2.1	–25.8	[23]
BiTaO4	119.3	22021 ± 132	149.1 ± 0.8	3.3	[21]
Bi4Ta2O11	363.2	66566 ± 384	449.6 ± 2.3	9.5	[28]
Bi7Ta3O18	602.7	109760 ± 634	743.0 ± 3.8	8.6	[28]
Bi3TaO7	235.2	44265 ± 254	304.3 ± 1.6	10.0	[28]
SrBi2Nb2O9	286.4	49230 ± 292	327.2 ± 1.7	–12.2	[22]
SrBi2Ta2O9	286.6	49060 ± 289	339.2 ± 1.8	–5.9	[22]

Table 3.

Heat capacity, relative enthalpy, entropy and entropy of formation from binary oxides at temperature 298.15 K of various mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅

A comparison is given in Table 4 of the values of entropy of formation from binary oxides Δ_oxS at 298 K calculated from our results and those from literature. The values of Δ_oxS are calculated using the relation

$${\Delta }_{\text{ox}}S=S_{\text{m}}\text{(MO)}-{\displaystyle {\sum }_i{b}_i{S}_{\text{m}}(\text{BO,}i\text{)}}$$E4

where S_m(MO) and S_m(BO,i) stand for the molar entropies of a mixed oxide and a binary oxide i, respectively, and b_i is a constitution coefficient representing the number of formula units of a binary oxide i per formula unit of the mixed oxide. The following values were used for calculation: S_m(CaO,298.15 K) = 38.1 J K^–1 mol^–1 [42], S_m(SrO,298.15 K) = 53.58 J K^–1 mol^–1 [43], S_m(Bi₂O₃,298.15 K) = 148.5 J K^–1 mol^–1 [44], S_m(Nb₂O₅, 298.15) = 137.30 J K^–1 mol^–1 [45] S_m(Ta₂O₅, 298.15) = 143.09 J K^–1 mol^–1 [45]. Furthermore, S_m(Sr₂Nb₂O₇, 298.15) = 238.5 J K^–1 mol^–1 from this work can be directly compared with the value 232.37 J K^–1 mol^–1 obtained by numeric integration of the C_pm(T)/T dependences from zero to 298.15 K given in Ref. [1]. It should be noted that the values of entropy assessed by thermodynamic optimization of phase equilibrium data are generally considered as less reliable as the values derived from low temperature heat capacity measurements. It is due to possible strong correlation between the enthalpy and entropy contributions to the Gibbs energy. So the obvious discrepancies between our values and data from assessments [12,19] could be explain in this way.

Oxide	ΔoxS^a); (J K^-1 mol^–1)	ΔoxS; (J K^-1 mol^–1)	Ref.
CaNb2O6	–8.1	33.45	[7]
–22.59	[11]
Ca2Nb2O7	–1.1	22.18	[11]
Sr2Nb10O27	–34.0	–350.69	[12]
SrNb2O6	–17.0	–58.65	[12]
Sr2Nb2O7	–6.0	–39.93	[12]
Sr5Nb4O14	–18.0	–51.01	[12]
CaBi2O4	1.9	3.0	[19]
Ca4Bi6O13	–23.8	8.0	[19]
Ca2Bi2O5	6.6	3.0	[19]
SrBi2O4	4.0	1.8	[19]
Sr2Bi2O5	5.5	–24.0	[19]

Table 4.

The values of entropy of formation from binary oxides at 298.15 K: a comparison of our results and data from literature

It should be noted that the thorough analysis of the Debye and Einstein contributions to the heat capacities reveals that the different vibrational modes contribute to the total values of Δ_oxS to a different extent and partial compensation is possible in some cases.

2.3. Evaluation of heat capacity at temperatures above 298 K

For the assessment of temperature dependences of C_pm above room temperature, the heat capacity data from DSC and the enthalpy increment data from drop calorimetry were treated simultaneously by the linear least-squares method (HT fit). The temperature dependence of C_pm was considered in the form

$$C_{p\text{m}}=A+BT+C/T^2$$E5

thus the related temperature dependence of ΔH_m(T) = H_m(T) – H_m(T₀) is given by equation

$$\Delta H_{\text{m}}(T)=H_{\text{m}}(T)-H_{\text{m}}(T_0)={\displaystyle {\int }_{T_0}^T{C}_{p\text{m}}\text{d}T}=A\left(T-T_0\right)+B\left(T^{\text{2}}-T_0^2\right)/2-C\left(1/T-1/T_0\right)$$E6

The sum of squares which is minimized has the following form

$$\begin{array}{c}F={\displaystyle {\sum }_{i=1}^{N(C_p)}{w}_i^2{\left[C_{p\text{m,}i}-A-B{T}_i-C/T_i^2\right]}^2}+\\ +{\displaystyle {\sum }_{j=1}^{N(\Delta H)}{w}_j^2{\left[\Delta H_{\text{m,}j}-A\left(T_j-T_{0,j}\right)-B\left(T_j^2-T_{0,j}^2\right)/2+C\left(1/T_j-1/T_{0,j}\right)\right]}^2}\to \min \end{array}$$E7

where the first sum runs over the C_pm experimental points while the second sum runs over the ΔH_m experimental points. Different weights w_i (w_j) were assigned to individual points calculated as w_i = 1/δ_i (w_j = 1/δ_j) where δ_i (δ_j) is the absolute deviation of the measurement estimated from overall accuracies of measurements (1 % for DSC and 3 % for drop calorimetry). Both types of experimental data thus gain comparable significance during the regression analysis. To smoothly connect the LT fit and HT fit data the values of C_pm(298.15) from LT fit were used as constraints and so Eq. (7) is modified

$$F_{\text{constr}}=F-\lambda \left[C_{p\text{m}}(298.15)-A-298.15B-C/{298.15}^2\right]\to \min$$E8

The numerical values of parameters A, B and C are now obtained by solving a set of equations deduced as derivatives of F_constr with respect of these parameters and a multiplier λ which are equal to zero at the minimum of F_constr. Assessed values of parameters A, B and C of Eq. (4) for mixed oxides are presented in Table 5.

As an example, the results of heat capacity measurements and relative enthalpy measurements on Bi₇Ta₃O₁₈ [28] are shown in Fig. 2. Empirical estimation according to the Neumann-Kopp’s rule (NKR) is also plotted for comparison.

The empirical Neumann-Kopp’s rule (NKR) is frequently used for estimation of unknown values of the heat capacity of mixed oxides [46–48]. According to NKR, heat capacity of a mixed oxide is calculated as a sum of heat capacities of the constituent binary ones

$$C_{p\text{m}}(\text{MO)}={\displaystyle {\sum }_i{b}_i{C}_{p\text{m}}(\text{BO,}i\text{)}}$$E9

It was concluded [47,48] that NKR predicts the heat capacities of mixed oxides remarkably well around room temperature but the deviations (mostly positive) from NKR become substantial at higher temperatures. Mean relative error of the estimated values of C_pm(298.15 K) is 1.4 %. Calculated temperature dependences of Δ_oxC_pm = C_pm(MO) – b_iC_pm(BO,i) for various mixed oxides in the systems CaO–Nb₂O₅, SrO–Nb₂O₅ and Bi₂O₃–Ta₂O₅ are shown in Fig. 3.

Oxide	Cpm = A + B·T + C/T^2 (J K^–1 mol^–1)			Temperature range (K)	Ref.
A	10^3 B	10^–6 C
CaBi2O4	157.161	38.750	–1.546	298-1000	[20]
Ca4Bi6O13	550.808	114.890	–7.201	298-1200	[20]
Ca2Bi2O5	226.096	33.374	–3.432	298-1100	[20]
SrBi2O4	161.97	45.936	–1.7832	298–1100	[29]
Sr2Bi2O5	197.48	87.463	–1.9282	298–1200	[29]
CaNb2O6	200.40	34.32	–3.45	298-1500	[26]
Ca2Nb2O7	257.20	36.21	–4.435	298-1400	[26]
SrNb2O6	200.47	29.37	–3.473	298-1500	[24]
Sr2Nb2O7	248.00	43.50	–3.948	298-1400	[26]
Sr2Nb10O27	835.351	227.648	–13.904	298-1400	[27]
Sr5Nb4O15	504.796	147.981	–6.376	298-1400	[27]
BiNbO4^a)	128.628	33.400	–1.991	150-1200	[21]
BiNb5O14	455.840	60.160	–7.734	298-1400	[23]
BiTaO4^b)	133.594	25.390	–2.734	150-1200	[21]
Bi4Ta2O11	445.8	5.451	–7.489	298–1400	[28]
Bi7Ta3O18	699.0	52.762	–9.956	298–1400	[28]
Bi3TaO7	251.6	67.05	–3.237	298–1400	[28]
SrBi2Nb2O9	324.470	63.710	–5.076	298-1400	[22]
SrBi2Ta2O9	320.220	64.510	–4.700	298-1400	[22]

Table 5.

Parameters of temperature dependence of molar heat capacities of various mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅

Figure 2.

Temperature dependence of heat capacity (a) and relative enthalpy (b) of Bi₇Ta₃O₁₈ (3NR means the Dulong-Petit limit).

2.4. Evaluation of enthalpy of formation

The heats of drop-solution for the calcium and strontium carbonates and for the bismuth and niobium oxides were measured first. These data are necessary for the evaluation of the Δ_oxH values for the mixed oxides, and furthermore, these data could be compared with the literature data [49–52]. For the AECO₃ carbonates, the measured heat effect consists of three contributions:

$${\Delta }_{\text{ds}}H\left({\text{AECO}}_{\text{3}},T\right)={\Delta }_{T}H\left({\text{AECO}}_{\text{3}},T_0\to T\right)+{\Delta }_{\text{decomp}}H\left({\text{AECO}}_{\text{3}},T\right)+{\Delta }_{\text{sol}}H\left(\text{AEO},T\right)$$E10

The measurements were performed at 973 K. The values of Δ_dsH(AECO₃, 973 K) are given in Table 6 along with the values of Δ_dsH(AEO, 973 K), which were derived based on the following thermochemical cycle (T₀ ≈ 298 K):

$${\text{AECO}}_{\text{3}}\text{(s,}{T}_{\text{0}}\text{)}\to \text{AEO(melt,}T{\text{) + CO}}_{\text{2}}\text{(g,}T\text{),}{\Delta }_{\text{ds}}H{\text{(AECO}}_{\text{3}}\text{)}$$E11

$${\text{AECO}}_{\text{3}}\text{(s,}{T}_{\text{0}}\text{)}\to \text{AEO(s,}{T}_{\text{0}}{\text{) + CO}}_{\text{2}}\text{(g,}{T}_0\text{),}{\Delta }_{\text{decomp}}H{\text{(AECO}}_{\text{3}}\text{)}$$E12

$${\text{CO}}_{\text{2}}\text{(g,}{T}_0\text{)}\to {\text{CO}}_{\text{2}}\text{(g,}T\text{),}{\Delta }_{T}H{\text{(CO}}_{\text{2}}\text{)}$$E13

$$\text{AEO(s,}{T}_{\text{0}}\text{)}\to \text{AEO(melt,}T\text{),}{\Delta }_{\text{ds}}H\text{(AEO)}$$E14

$${\Delta }_{\text{ds}}H\text{(AEO)}={\Delta }_{\text{ds}}H{\text{(AECO}}_{\text{3}}\text{) }-{\Delta }_{\text{decomp}}H{\text{(AECO}}_{\text{3}}\text{) }-{\Delta }_{T}H{\text{(CO}}_{\text{2}}\text{)}$$E15

The values Δ_decompH(CaCO₃, 298 K) = 178.8 kJ mol^–1, Δ_decompH(SrCO₃, 298 K) = 233.9 kJ mol^–1 and Δ_TH(CO₂, 298 → 973 K) = 32.0 kJ mol^–1 [53] were used for the calculations.

Next, the Δ_dsH values of the binary oxides Bi₂O₃ and Nb₂O₅ were measured. Because the dissolution of Nb₂O₅ and of the mixed oxides at 973 K proceeds rather slowly, the higher temperature of 1073 K was used. The measured values Δ_dsH are also given in Table 6.

The experimental values of Δ_dsH for SrCO₃ and CaCO₃ are in quite good agreement with the literature data [49–51]. On the other hand, our results and the published [52] values of Δ_dsH(Nb₂O₅) are quite different. It should be noted that a more endothermic value Δ_decompH(SrCO₃, 298 K) = 249.4 kJ mol^–1 is presented in the literature [45], which results in more exothermic value for Δ_dsH(SrO) by 15.5 kJ mol^–1.

Δ_dsH for the mixed oxides was measured at 1073 K. The following thermochemical cycle was used for the calculation of Δ_oxH for calcium and strontium niobates (T₀ ≈ 298 K):

$${\text{AE}}_x{\text{Nb}}_{\text{2}}{\text{O}}_{5+x}\text{(s,}{T}_{\text{0}}\text{)}\to x\text{AEO(melt,}T{\text{) + Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{(melt,}T\text{),}{\Delta }_{\text{ds}}H\text{(AEO)}$$E16

$$\text{AEO(s,}{T}_{\text{0}}\text{)}\to \text{AEO(melt,}T\text{),}{\Delta }_{\text{ds}}H\text{(AEO)}$$E17

$${\text{Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{(s,}{T}_{\text{0}}\text{)}\to {\text{Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{(melt,}T\text{),}{\Delta }_{\text{ds}}H{\text{(Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{)}$$E18

$$x\text{AEO(s,}{T}_{\text{0}}{\text{) + Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{(s,}{T}_{\text{0}}\text{)}\to {\text{AE}}_x{\text{Nb}}_{\text{2}}{\text{O}}_{5+x}\text{(s,}{T}_{\text{0}}\text{),}{\Delta }_{\text{ox}}H{\text{(AE}}_x{\text{Nb}}_{\text{2}}{\text{O}}_{5+x}\text{)}$$E19

Figure 3.

Temperature dependences of Δ_oxC_pm for various mixed oxides in the systems CaO–Nb₂O₅, and SrO–Nb₂O₅ (a) and Bi₂O₃–Ta₂O₅ (b)

$${\Delta }_{\text{ox}}H{\text{(AE}}_x{\text{Nb}}_{\text{2}}{\text{O}}_{5+x}\text{)}=x{\Delta }_{\text{ds}}H\text{(AEO) + }{\Delta }_{\text{ds}}H{\text{(Nb}}_{\text{2}}{\text{O}}_{\text{5}}\text{) }-{\Delta }_{\text{ds}}H{\text{(AE}}_x{\text{Nb}}_{\text{2}}{\text{O}}_{5+x}\text{)}$$E20

An analogous scheme was applied to calculate Δ_oxH(BiNbO₄). All of the experimental and calculated values are summarized in Table 7. The Δ_oxH(298 K) values derived from high-temperature EMN measurements [7,8,11] for the CaO-Nb₂O₅ oxides and the assessed values from the phase diagram for the SrO-Nb₂O₅ oxides [12] are also presented in Table 7.

Substance	T (K)	ΔdsH (kJ mol^–1)^a)	ΔdsH (kJ mol^–1)
CaCO3	973	128.4 ± 10.1 (10)	119.70 ± 1.02^b)
CaO	973	–82.39	–90.70 ± 1.69^b)
CaO	1073	–77.04^c)
SrCO3	973	131.4 ± 9.1 (7)	130.16 ± 1.66 d); 134.48 ± 1.89 e)
SrO	973	–134.47	–135.82 ± 2.48 d); –131.42 ± 1.89 e)
SrO	1073	–129.25^f)
Bi2O3	973	26.0 ± 2.9 (12)
Bi2O3	1073	39.6^g)
Nb2O5	1073	141.8 ± 6.0 (11)	91.97 ± 0.78^h)

Table 6.

Enthalpy of drop-solution in 3Na₂O + 4MoO₃ melts [30]

Substance	T (K)	ΔdsH (kJ mol^–1)^a)	ΔoxH(298 K); (kJ mol^–1)^b)	ΔoxH (298 K); (kJ mol^–1)
CaNb2O6	1073	196.8 ± 20.7 (8)	–132.0 ± 23.8	–159.8 c); –130.1 d)
Ca2Nb2O7	1073	195.7 ± 27.8 (8)	–208.0 ± 31.9	–147.3 c); –177.5 e)
SrNb2O6	1073	180.50 ± 15.7 (4)	–167.9 ± 19.1	–325.0^f)
Sr2Nb2O7	1073	167.54 ± 34.7 (4)	–289.2 ± 37.5	–367.4^f)
BiNbO4	1073	132.61 ± 8.9 (7)	–41.9 ± 11.1

Table 7.

Enthalpies of drop-solution in 3Na₂O + 4MoO₃ melt (Δ_dsH) and enthalpy of formation from constituent binary oxides (Δ_oxH) [30]

Our values for the calcium niobates are in good agreement with Raghavan’s data [7,8], while the data from Dneprova et al. [11] are quite different. Moreover, a relation, Δ_oxH(CaNb₂O₆) > Δ_oxH(Ca₂Nb₂O₇), that holds for the values from the work of Dneprova et al. is rather unexpected. The Δ_oxH values for strontium niobates obtained based on the binary SrO-Nb₂O₅ phase diagram evaluation [12] are substantially more exothermic than our calorimetric data. These large differences in the Δ_oxH values are not surprising in view of simultaneous differences in the Δ_oxS values from the assessment [12] and those derived from low temperature dependences of the molar heat capacity of SrNb₂O₆ and Sr₂Nb₂O₇ [24,26].

3. Empirical correlation S–V

A linear correlation between the standard molar entropy at 298.15 K and the formula unit volume V_f.u has been proposed by Jenkins and Glaser [54–56]. This approach was used in this work for mixed oxides in the CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅ system. The linear relation is obvious (see Fig. 4) and the straight line almost naturally passes through the origin:

$$S_{\text{m}}(\text{J}{\text{K}}^{-1}{\text{mol}}^{-1})=1680.5V_{\text{f}\text{.u}\text{.}}({\text{nm}}^3\text{f}\text{.u}{\text{.}}^{-1})$$E21

The average relative error in entropy is 8.2 %, the binary oxides CaO and Nb₂O₅ show the deviations around 20 %. It should be noted that, in this set of values, the simple analogy of NKR (Eq.(9)) provides a better prediction with an average relative error in entropy of 4.2 %.

Eq. (21) can be used for estimation of missing data. So, the estimated value S_m(Sr₂Ta₂O₇) = 256.06 J K^–1 mol^–1 can be compared with the value 245.41 J K^–1 mol^–1 obtained by numeric integration of the C_pm(T)/T dependences from zero to 298.15 K given in Ref. [1] (relative deviation of –4.3 %). Simple calculation S_m(Sr₂Ta₂O₅) = 2S_m(SrO) + S_m(Ta₂O₅) = 250.25 J K^–1 mol^–1 gives more reliable value (relative deviation 2.0 %).

4. Empirical estimation of enthalpy of formation

There are other mixed oxides in the system CaO–SrO–Bi₂O₃–Nb₂O₅–Ta₂O₅ for which the values of enthalpy of formation Δ_fH or enthalpy of formation from binary oxides Δ_oxH have not yet been determined. As a rough estimate, the values of Δ_oxH calculated according to an empirical method proposed by the authors [56] can be used. In the case of Ca, Sr and Bi niobates the following relation holds for Δ_oxH:

$\frac{{\Delta }_{\text{ox}}H}{n_{\text{Nb}}+n_{\text{Me}}}=-2\cdot 96.5\alpha y{x}_{\text{Nb}}{x}_{\text{Me}}^{\delta }{\left(X_{\text{Nb}}-X_{\text{Me}}\right)}^2$E22

where X_Nb and X_Me (Me = Ca, Sr or Bi) are Pauling’s electronegativities of the relevant elements, x_Nb and x_Me are the molar fractions of the oxide-forming elements (x_Nb = n_Nb/(n_Nb + n_Me) etc.), y is the number of oxygen atoms per one atom of oxide-forming elements and α and δ are the model parameters. Using Pauling’s electronegativities, X_Nb = 1.60, X_Ca = 1.00, X_Sr = 0.95, and X_Bi = 2.02, and the calorimetric values of Δ_oxH obtained in this work, the values of α = 2.576 and δ = 1.50 were derived from the least-squares fit. The estimated Δ_oxH values for calcium and strontium niobates are shown in Fig. 5. The values of Δ_oxH that were calculated according to an empirical method proposed by Zhuang et al. [57] are displayed for comparison.

Figure 4.

Correlation between the standard molar entropy at 298.15 K and the formula unit volume V_f.u. for various mixed oxides in the system CaO–SrO– Bi₂O₃–Nb₂O₅–Ta₂O₅ (data from table Table 3)

5. Conclusion

The above presented data derived from calorimetric measurements became the basis for thermodynamic database FS-FEROX [58] compatible with the FactSage software [59,60]. Missing data for other stoichiometric mixed oxides were estimated by the empirical methods described before: the Neumann-Kopp’s rule for heat capacities, the entropy-volume correlation for molar entropies and electronegativity-differences method for enthalpies of formation. At the same time, thermodynamic description of a multicomponent oxide melt was obtained analyzing relevant binary phase diagrams published in literature. The database and the FactSage software were subsequently used for various equilibrium calculations including binary T-x phase diagrams and ternary phase diagrams in subsolidus region. Thermodynamic modeling of SrBi₂Ta₂O₉ and SrBi₂Nb₂O₉ thin layers deposition from the gaseous phase were also performed to optimize the deposition conditions.

Figure 5.

Values of enthalpy of formation of the mixed oxides from constituent binary oxides in the CaO–Nb₂O₅ (a) and SrO–Nb₂O₅ (b) systems (lines serve only as a guide for the eyes)

Acknowledgment

This work was supported by the Ministry of Education of the Czech Republic (research projects N° MSM6046137302 and N° MSM6046137307). Part of this work was also supported from the Grant Agency of the Czech Republic, grant No P108/10/1006. Low temperature experiments were performed in MLTL (http://mltl.eu/), which is supported within the program of Czech Research Infrastructures (project no. LM2011025).