Heat Capacity and Thermodynamic Properties of Photocatalitic Bismuth Tungstate, Bi₂WO₆

Abstract

The photocatalytic activity of Bi₂WO₆ Aurivillius phase has been widely exploited for the degradation of a wide range of gaseous and aqueous molecules, as well as microorganisms, under the influence of visible irradiation. Strategies for the development of doped and co-doped bismuth tungstate materials require the thermodynamic data on this phase. The heat capacity of bismuth tungstate, Bi₂WO₆, was investigated using a DSC microcalorimeter on polycrystalline powder samples in the temperature range from 313 to 1103 K (40–830 °C) in two separate runs. The samples were synthesized by solid-state reaction from pure binary oxides at 1033 K (760 °C) in a platinum crucible with cover. The high temperature C_p(T) data were fitted by the Maier–Kelley equation and, from this relation, the standard molar heat capacity of γ-Bi₂WO₆ polymorph was estimated to be at 298.15 K 176.8 ± 3.9 J·K⁻¹·mol⁻¹. A reversible second-order transition of Bi₂WO₆ phase was observed in the experimental temperature range, with a peak close to 940 K (667 °C). Additionally, the extrapolation of C_p(T) to 0 K was proposed using a method based on the multiple Einstein model. Thermodynamic properties (heat capacity C_p(T), entropy S°(T), enthalpy H°(T), Gibbs energy G°(T)) of crystalline γ-Bi₂WO₆ were calculated in the temperature range of 298.15–1123 K (25–850 °C).

1. Introduction

Metal oxide semiconductors have been extensively explored as photoanodes over the past two decades. During this time, over one hundred metal oxide phases have been identified in the oxygen evolution reaction. Among them, a class of bismuth-based compounds, primarily BiVO₄, Bi₂WO₆ and BiFeO₃, has been identified as promising photoanode materials for the use in solar energy conversion [1,2] due to their chemical and electrochemical stability under oxidizing conditions [3]. These materials have also been found to be useful for producing pure hydrogen, a key step towards a sustainable hydrogen economy. The photocatalytic activity of the Bi₂WO₆ Aurivillius phase, since its discovery in 1999 [4], has been extensively used to degrade a wide range of gaseous and aqueous molecules and even microorganisms under the influence of visible irradiation [5,6].

Bi₂WO₆, as a photoanode material, can decompose water into hydrogen and oxygen during water splitting due to its special properties. These special features include suitable band edge positions for potential photoelectrochemical water splitting [7], chemical stability over a wide pH range (from 0.26 to 9) [8,9] and flat band potential below 0.25 V vs. RHE (Reversible Hydrogen Electrode), which is a key benchmark for the behavior of semiconductors in a photoelectron catalysis and solar energy applications [8,10].

Bismuth tungstate, Bi₂WO₆, the n-type semiconductor (~2.8 eV band gap), has a suitable crystal and electronic structure for photocatalysis under visible light. The layered structure consists of alternating bismuth–oxygen [Bi₂O₂]²⁺ ionic layers and perovskite-like [WO₄]²⁻ ionic layers [11,12] which promote the separation of photoexcited electron–hole pairs, increasing the photocatalytic efficiency.

The comprehensive review of Bi₂WO₆-based photocatalysts applied in clean energy production and environmental remedial action was published by Zhu et al. [13].

The ferroelectric Bi₂WO₆ phase is chemically and structurally complex; therefore, several crystal structure models have been proposed: I4/mmm [14], B2cb [15] and P21ab (referred to as Pca21) [16,17]. At low temperatures, Bi₂WO₆ is a material with an internally ordered structure that forms an electric dipole and it was assumed that this structure belongs to the polar P2₁ab space group [18,19,20]. With increasing temperatures, a reconstructive transformation from the polar state to the nonpolar state occurs, and it is not a typical ferroelectric transition. After the end of second-order phase transition ~933 K (660 °C), the orthorhombic phase belongs to the symmetry of the Aba2 space group, described in [18,21,22,23].

At 1233 K (960 °C), the reconstructive reversible transformation between the orthorhombic and monoclinic phase occurs. The monoclinic phase does not belong to Aurivillius structure and is unfavorable from a technical point of view due to its destructive effect on the phase structure. Information on the structural, photophysical and photocatalytic properties is summarized in the review by Elaouni et al. [24].

Although many studies have been carried out on the physical properties and crystal structure of compounds in the pseudobinary Bi₂O₃–WO₃ system, little attention has been paid to their thermodynamic analysis. And so, the phase diagrams reported so far on this system [25,26,27,28] contradict each other in terms of the existence and stability of ternary phases in the system. The Bi₂O₃–WO₃ phase diagram appears to be fairly complicated. Hoda and Chang [26] show the presence of four intermediate ternary oxide phases, namely, Bi₁₄WO₂₄, Bi₁₄W₂O₂₇, Bi₂WO₆ and Bi₂W₂O₉. Additionally, Bi₁₄W₂O₂₇ forms an extensive range of solid solutions in the system and Bi₁₄WO₂₄ represents the endmember of an isostructural solid solution extending from compositions around Bi₃₀WO₄₈ [29]. Ling et al. [27] observed two mixed oxide phases in the Bi-rich part of the pseudobinary system, except for the Bi₂WO₆ congruent melting stoichiometric compound: a stoichiometric phase Bi₁₄WO₂₄ and a solid solution between endmembers Bi₄₂W₈O₈₇ and Bi₃₀W₄O₅₇. The recent phase equilibria reported on this system [28] using differential thermal analysis (DTA) of bismuth tungsten oxides confirmed broad agreement with phase diagram data of Speranskaya [25] with five stable compounds. Finlayson et al. [28] concluded that other compounds in the Bi₂O₃–WO₃ phase system also show similar optical properties to Bi₂WO₆, so they also have the potential to exhibit photocatalysis. In particular, Bi₇WO_13.5 and Bi₂₄WO₃₉ are of interest because their broad tungsten solubility ranges open up the possibility of further optimization. For this purpose, the trusted thermodynamic data of the phases in the Bi₂O₃–WO₃ system are necessary. In 2006, a new Aurivillius bismuth tungstate, Bi₃W₂O_10.5, was synthesized and characterized by single-crystal X-ray diffraction by Muktha and Row [30].

In addition to inconsistent information on phase relations, many thermochemical properties of bismuth tungstates remain poorly recognized. The only available experimental thermochemical data on these compounds, standard molar enthalpy of formation, concern the compound Bi₂WO₆ [31]. Careful experimental study of thermochemical properties of bismuth tungstates is essential to predicting their stability and effectiveness in reactive environments. It can also enable the assessment of the phase relationships within the pseudobinary system. In general, for many potential applications of materials, knowledge of their thermal properties is of great importance. Thermodynamic data influence photocatalytic applications by determining material stability, reaction efficiency and energy conversion. It is worth noting that heat capacity is a critical factor, for example, in thermoelectric materials, where it is used to calculate the figure of merit (zT). The thermal conductivity of a material can be derived from heat capacity, density and diffusivity data, so heat capacity is an important parameter for designers and used in designing efficient heat management systems and reactor configurations for industrial-scale applications. In addition to the above mentioned thermal properties, the knowledge of one of the most fundamental property, i.e., Gibbs energy (G), is imperative to predict the formation and stability of the compounds in various physicochemical environments. In particular, Gibbs free energy explains the feasibility and direction of the reaction. This also helps in establishing phase diagrams of various systems.

In this context, in this work, for the first time, the heat capacity of Bi₂WO₆ was measured in a wide temperature range, from 313 to 1103 K (40–830 °C), using a differential scanning calorimeter (DSC). The high-temperature data were described by the well-known Maier–Kelley equation [32]. Additionally, an extrapolation of C_p(T) to 0 K was proposed using a method based on the modified Einstein model [33] to estimate the standard entropy of Bi₂WO₆, which was necessary to calculate the Gibbs energy function. Furthermore, the second-order phase transition was detected in the temperature range close to 940 K (667 °C). The further investigation of the thermal properties of the system is complicated due to discrepancies in the number of phases, their polymorphs and the characteristic transition temperatures.

2. Literature Data

2.1. Structure Data

Three phase transitions were found in pure bismuth tungstate, Bi₂WO₆. The first one was found between two polar and nonpolar orthorhombic phases (γ ⟶ γ‴), which occur in the temperature range 640–660 °C (913–933 K) with a space group change from P21ab to B2cb (nonstandard setting of Aba2). It corresponds to a second-order transition (the propagation of displacive modes along c axis is related to this difference). Next, ferroelectric transition between the polar and nonpolar orthorhombic phase (γ‴ ⟶ γ″) occurs at about 1203 K (930 °C) [24], and a first-order phase transition (γ″ ⟶ γ′) to monoclinic nonferroelectric phase with the space group A2/m at 1233 K (960 °C) was observed [34,35,36,37]. In the case of room temperature structure (γ form), the nonstandard setting of space group Pca2₁ was also used [17,20] to be consistent with the convention adopted in Aurivillius phase ferroelectrics, according to which the long axis is designated as c and the polar axis as a. It was confirmed by the use of high-resolution powder neutron diffraction [34] that a transition from space group P21ab (γ form) to B2cb (γ‴ form) in Bi₂WO₆ compound corresponds to the loss of one octahedral tilt mode within the perovskite-like WO₄ layer of the orthorhombic structure. The medium-temperature form γ‴ has a higher B2cb symmetry, but its structure is similar to the orthorhombic γ phase.

The second-order phase transition between polar and nonpolar phases, corresponding to the ferroelectric Curie point, occurs around 1203 K (930 °C). Lastly, first-order transition before melting with significant reorganization of the crystal structure, involving the breaking and reforming of primary chemical bonds, takes place at 1233 K (960 °C). This high-temperature monoclinic γ′ phase is very complex [34].

2.2. Thermodynamic Data

The DTA study of Newkirk et al. [38] showed a reversible first-order thermal effect at 1208 ± 5 K (935 °C) with a heat of transition of 4.184 kJ·mol⁻¹. The experimental error in measuring the transition enthalpy by the DTA method is usually significant.

Phapale et al. [31] measured the standard molar enthalpy of formation of solid Bi₂WO₆ compounds, ${\mathrm{\Delta }}_f{H}^{\mathrm{o}}(298.15),$ using a Calvet calorimeter (HT-1000, Setaram, Lyon, France) with the liquid Na₂O+MoO₃ (3:4 M) solvent. The purity of the solid compound was determined by the XRD method [30]. The obtained result was amended in the corrigendum [31]. Using the standard molar enthalpies of formation of pure Bi₂O₃ and WO₃ from the literature [39], the ${\mathrm{\Delta }}_f{H}^{\mathrm{o}}(298.15),$ value of Bi₂WO₆ was recalculated to be –1440.53 ± 8.38 kJ·mol⁻¹ [31].

3. Experiment

3.1. Material and Synthesis

Bismuth tungstate, Bi₂WO₆, was prepared by solid-state reaction using pure oxides, Bi₂O₃ (99.9995%, USA) and WO₃ (99.99%, USA), respectively, as starting materials. A stoichiometric oxide mixture was ground in the cyclic mortar (30 cycles/min) with 10 mm ceramic balls (zirconia) and pressed into 5 mm diameter pellets, then fired in air at 1033 K (760 °C) for 24 h in a platinum crucible. To drive the solid reaction to completion, the samples were reground, pressed and annealed twice. The phase composition of samples was determined by the XRD (X-Ray Diffraction) method using Rigaku diffractometer (Mini Flex II) (Rigaku, Tokyo, Japan) with Cu Kα radiation. XRD patterns were collected in angular range 2θ from 10° to 100°, with scan step 0.01.

The results are presented in Figure 1. The lattice parameters of Bi₂WO₆ were determined by Rietveld refinement using software GSAS II version 5800 (Argonne National Laboratory, Lemont, IL, USA) [40]. The pseudo-Voigt peak shape function was adopted. The short vertical markers represent the peak positions from [41] database. The parameters of oxygen atoms are difficult to refine because of their small scattering factors [18]. The lattice constants of Bi₂WO₆ structure are as follows: a = 5.43391(1) Å, b = 16.42596(2) Å, c = 5.45620(1) Å, (Z = 4, V = 487.005 ų), which proved that this product is orthorhombic. The lattice parameters of powder material were found to be similar with these reported in JCPDS card no. 01-079-2381. The obtained structure of the low-temperature polar (orthorhombic) phase of russellite agrees very well with the literature data of Knight [17] with powder neutron diffraction and Okudera et al. [20] with the Weissenberg method: 5.43726; 16.43018; 5.45842 (code ID 9011799) and 5.4345; 16.4324; 5.4558 (code ID 2108252), respectively [35]. No impurity lines from pure binary oxides or any other impurities were observed in the XRD spectrum of Bi₂WO₆.

The typical morphology of Bi₂WO₆ powder samples is shown in Figure 2. Microstructural studies were carried out on the Hitachi SU-70 scanning microscope (SEM) (Hitachi High-Technologies Corporation, Tokyo, Japan). No other impurity phases were detected.

3.2. Experimental Procedure

The molar heat capacity of lead tungstate Bi₂WO₆ has been measured using Netzsch 401 F1 Pegasus differential microcalorimeter DSC (Netzsch GmbH, Selb, Germany), calibrated by measuring the temperature of fusion of several pure (99.999%) metals (In, Sn, Zn, Al, Ag, Au) on heating. The measured melting point (onset) of these metals is compared with the known melting point temperature. By definition, the calibration point used is the extrapolated melting peak onset, as it is independent of the sample mass. Each metallic standard was measured three times. The temperature uncertainty obtained from calibration in this work is ±1 K, and the temperature detection sensitivity of DSC (the S-thermocouple resolution) is 0.01 K.

Temperature calibration is necessary because the thermocouple measuring the sample temperature is located on the probe holder below the contact surface of the calorimetric pan. Additionally, at the high-temperature range, the uncertainty increases due to radiation. To minimize other factors of influence, all single calibrations should be performed in the same conditions.

The high-temperature calibration on pure copper gave results even ~5 K lower that the nominal melting point. Therefore, only gold meets the high requirements of accurate temperature calibration.

To determine the high-temperature dependence of C_p(T), the method recommended by the American Society for Testing and Materials (ASTM) E1269 [42,43] was adopted. To evaluate the heat capacity, three successive scans on the DSC apparatus are required under the same experimental conditions. The platinum pans with Pt lids were applied as a sample and reference crucibles in C_p(T) measurements. The same pair of crucibles were used for the baseline, standard and sample measurements. The accuracy of the heat capacity measurement was increased by using the calorimetric pans with masses placed as close to each other as possible and placed in the same positions on the platinum DSC sensor for each scan. For accurate heat capacity experiments, the baseline scan should be repeated three times to ensure stability and reproducibility of DSC measurements. The first scan is an obligatory run with the empty platinum pans to establish a baseline of the apparatus. Following good practice, both sample and the reference Pt crucibles were covered with Pt lids.

As a heat capacity reference, the C_p(T) of synthetic sapphire standard disk of 5.2 mm diameter and 1 mm thick (certified by Netzsch GmbH, Selb, Germany) was established during the second run (Appendix A,

Table A1. Experimental data for the sapphire obtained at P = 0.1 MPa ¹.

T, K	Cp(T) 2, J·mol−1·K−1	T, K	Cp(T) 2, J·mol−1·K−1	T, K	Cp(T) 2, J·mol−1·K−1
324	84.3	604	112.9	884	122.6
334	86.1	614	113.3	894	122.3
344	87.8	624	113.9	904	122.7
354	89.4	634	114.2	914	123.5
364	90.9	644	114.7	924	123.1
374	92.5	654	115.2	934	123.3
384	94.0	664	115.3	944	123.4
394	95.2	674	116.0	954	124.0
404	96.5	684	116.4	964	124.3
414	97.7	694	116.9	974	124.2
424	98.9	704	117.0	984	124.4
434	100.1	714	117.3	994	124.7
444	101.1	724	117.8	1004	124.4
454	102.0	734	118.2	1014	125.0
464	103.0	744	118.6	1024	125.0
474	104.0	754	118.9	1034	125.4
484	104.8	764	119.0	1044	125.5
494	105.6	774	119.4	1054	125.3
504	106.4	784	120.1	1064	126.0
514	107.1	794	120.2	1074	125.8
524	107.9	804	120.2	1084	126.0
534	108.6	814	120.6	1094	126.4
544	109.3	824	120.8	1104	126.7
554	109.8	834	121.3	1114	126.8
564	110.4	844	121.3	1124	126.9
574	111.1	854	121.6	1134	126.8
584	111.7	864	121.8	1144	126.7
594	112.4	874	122.1	1154	127.0

¹ Standard uncertainties, are u(T) = 1 K and u(P) = 0.85 kPa. ² The extended uncertainty of u(C_p) is 0.02 C_p(T).

). In such cases, in the experimental temperature range, the heat flow was calibrated to the C_p(T) of sapphire comparing the obtained DSC results with tabled values [42,43].

To stabilize the instrument system and correct the baseline drift, ten-minute isothermal segments at the beginning and at the end of every calibration heat cycle were adopted. This stabilization period ensures the temperature is uniform before a measurement begins and allows any instrument drift to be characterized, which is crucial for accurate data analysis and heat capacity calculations.

In the third scan, the sample of the testing material was investigated.

As required, the measurements for the baseline, samples and reference material (sapphire) were performed in three separate heating cycles. Only curves with a difference between the second and third run of less than 1% were considered in further C_p(T) analysis. The heat capacity of synthetic sapphire calculated from the third curve compared to the values in the NBS (National Bureau of Standards) tables [42,43] can be considered as a test of the sensitivity and accuracy of the current sample measurement. Therefore, the relative error of the observed values against the corresponding values of NBS certificate was found to be less than ±1% and this value was adopted as uncertainty of measurements.

The measurements were carried out with heating and cooling rates of 10 K·min⁻¹ on the sample disks to maximize the thermal contact with the pan. To ensure good contact with the bottom of platinum pan and good heat flow, the samples were polished on one side by successive steps of finer and finer abrasives (polishing disk and the polishing cloth) to progressively create a smooth light-reflecting surface. No aqueous suspensions were used to ensure no contamination from polishing agents.

DSC measurement precision is limited not only by instrument calibration but also by experimental factors like heating rate and sample mass. The mass of the 5 mm diameter sample was comparable to the mass of the sapphire standard which was 84.5 mg. To determine heat capacity, the heating runs were assumed in the following cycles: thermal equalization at 313 K (40 °C) for 10 min, heating up to 1103 K (830 °C) with an isothermal stop for 10 min after heating. A cooling rate of 10 K·min⁻¹ was applied to temperature of 313 K (40 °C). The heating and cooling procedure was also repeated three times.

At the beginning of each DSC experiment, the apparatus was evacuated and filled twice with argon. The small orifices in the platinum lids guarantee the gas exchange in Pt crucibles during DSC tube evacuation and protective gas flow. The controlled 20 mL/min argon (Ar 6+, Air Products, Warsaw, Poland) flux flow was performed in all the experiments on the Netzsch DSC 404 instrument in the experimental temperature range. After the experiment, no traces of reaction were noticed between the samples and the bottom of the platinum calorimetric pan. Moreover, after the experiment, no traces of Bi₂WO₆ (neither Bi₂O₃ or WO₃ oxides) evaporation or deposition on the crucible or cover were observed. The weighted mass losses due to vaporization were less than 0.03%, showing insignificant mass change. The maximal difference between C_p values obtained in two separate temperature runs (run I and run II) was about 0.8%.

According to described procedure, the heat capacity function C_p(T) of the studied material was determined from three scans of continuous temperature change using the Netzsch Proteus ver. 5.2.1 (Netzsch, Germany) software with sensitivity related to sapphire.

4. Results and Discussion

The results of the heat capacity C_p(T) measurements obtained during two separate runs in the temperature range from 313 to 1103 K (40–830 °C) are shown in Figure 3 and are collected in

Table A2. Experimental data for the Bi₂WO₆ obtained by ASTM 1269 method at P = 0.1 MPa ¹.

First Run 2				Second Run 2
T, K	Cp(T), J·K−1·mol−1	T, K	Cp(T), J·K−1·mol−1	T, K	Cp(T), J·K−1·mol−1	T, K	Cp(T), J·K−1·mol−1
379	190.5	749	215.4	375	190	745	214.7
389	191.3	759	215.8	385	191	755	214.6
399	192.7	769	215.8	395	192	765	215.4
409	194.1	779	216.8	405	194	775	215.7
419	195.6	789	216.4	415	195	785	215.9
429	196.8	799	216.3	425	197	795	215.6
439	197.7	809	216.5	435	197	805	216.5
449	198.6	819	216.6	445	198	815	216.4
459	199.5	829	216.9	455	200	825	217.1
469	200.4	839	217.5	465	200	835	217.0
479	201.1	849	217.6	475	201	845	217.2
489	201.7	859	218.0	485	202	855	217.4
499	202.4	869	218.2	495	202	865	217.7
509	203.1	879	218.3	505	203	875	218.4
519	203.6	889	219.0	515	204	885	218.6
529	204.3	899	218.8	525	204	895	219.0
539	205.0	909	219.8	535	205	905	219.0
549	205.7	919	220.5	545	205	915	219.4
559	206.1	929	221.4	555	206	925	220.2
569	206.8	939	223.0	565	207	935	221.9
579	207.3	949	219.7	575	207	945	221.5
589	208.1	959	219.2	585	208	955	217.7
599	208.8	969	218.2	595	208	965	216.9
609	209.4	979	217.6	605	209	975	217.5
619	209.7	989	217.1	615	210	985	217.2
629	210.4	999	217.0	625	210	995	217.5
639	210.8	1009	217.2	635	211	1005	218.2
649	211.1	1019	217.3	645	211	1015	217.8
659	211.7	1029	217.0	655	211	1025	218.9
669	212.1	1039	217.1	665	212	1035	219.1
679	212.5	1049	217.2	675	212	1045	218.2
689	213.0	1059	217.3	685	213	1055	218.8
699	213.4	1069	217.4	695	213	1065	218.9
709	213.5	1079	217.9	705	213	1075	218.7
719	213.6	1089	217.5	715	214	1085	218.2
729	214.3	1099	218.0	725	214	1095	218.5
739	214.7			735	214

¹ Standard uncertainties are u(T) = 1 K and u(P) = 0.85 kPa. ² The extended uncertainty of u(C_p) is 0.02·C_p (T).

(in Appendix A) with consecutive ten degree intervals. In fact, every registered DSC signal consists of several hundred points. Molar mass of Bi₂WO₆ equal to 697.7972 g·mol^–1 was taken for the C_p(T) calculations.

The DSC signal shows weak endothermic heat effect up to the temperature 961 K (688 °C) (Figure 3). The C_p signal shows the peak at temperature T_max = 940 ± 1 K (667 °C). The continuous variation in the molar heat capacity in the vicinity of the extremum allows it to be considered as a second-order phase transition, and the λ-shaped heat capacity peak suggests that thermodynamic fluctuations affect the molar heat capacity of Bi₂WO₆ near the phase transition. The heat capacity jump in the phase transition region is ΔC_p(T_max) ≈ 6 J·mol^–1·K^–1. Yanovskii et al. [37] observed an inflection at 913 K (640 °C) and 933 K (660 °C) during the DSC heating scans of Bi₂WO₆ ceramic. A similar, very weak signal near 935 K (662 °C) was at the DTA curve detected by Watanabe [44]. From the earlier studies on the bismuth tungstate polymorphs [37,44], it has been proven that the phase transition is second-order in nature (polar–nonpolar [35]) and involves structural reversible order–disorder transition from P21ab to B2cb [24] (γ ⟶ γ‴).

4.1. High-Temperature Heat Capacity Function

The heat capacity of pure phases in the high-temperature range (T ≥ 298.15 K) listed in the reviews are the outcomes of various preliminary fitting and smoothing procedures for enthalpies, H(T), and the respective heat capacities, C_p(T) = (dH(T)/dT)_p. Instead of real experimental values, these C_p(T) values are based on relatively simple analytical expressions. For example, the well-known Maier–Kelley [32] equation is adopted:

$$C_p\left(T\right)=a+b\hspace{0.17em}T+c\hspace{0.17em}{T}^{-2}+d\hspace{0.17em}{T}^2$$

(1)

where a, b, c and d are the empirical parameters (as a rule a and b are positive and c is negative).

At high temperatures, the dominating effects such as lattice anharmonicity and expansion effects are described by the linear term in Equation (1). The quadratic term is required in many cases to describe the nonlinear behavior of the C_p(T) dependences close to the melting point. In the present analysis, Equation (1) was fitted with three and four terms, respectively, defined by a, b, c, d parameters (Figure 3) depending on the temperature range. The part of temperature dependence of heat capacity below the second-order transformation has been described by four terms Equation (2). In the case of detected phase transitions, the excess heat capacity Δ^exC_p related to this phase transition can be determined by subtracting the measured heat capacity values [45] and the heat capacity derived from the modeled line of the C_p = f(T) for both γ and γ‴ phases.

Taking into account the experimental heat capacity in the region of the γ phase stability (370–855 K), the fitted function of C_p(T) (Equation (1)) has the following form:

$$\begin{array}{l}{C}_p\left(T\right)=187.73\hspace{0.17em}(\pm 3.88)+6.4455\hspace{0.17em}(\pm 0.8681)·{10}^{-2}·T-2.4392537\\ \hspace{1em}\hspace{1em}\hspace{1em} (\pm 0.0211701)·\hspace{0.17em}{10}^6·T^{-2}+3.0123\hspace{0.17em}(\pm 0.0531)\cdot {10}^{-5}·T^2,\mathrm{J}·\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}·{\mathrm{K}}^{-1}\end{array}$$

(2)

The correlation coefficient for Equation (2) was determined to be 0.99946. The confidence intervals of parameters of fitting to high-temperature data are also attached. All values obtained in this extrapolation are characterized by Student’s t-distribution with 95% confidence intervals.

In Figure 3, the experimental temperature dependence of heat capacity of γ-Bi₂WO₆ is compared with calculated C_p values from Equation (2). The standard heat capacity at 298.15 K for one mole of Bi₂WO₆ was estimated to be 176.8 ± 3.9 J·mol⁻¹·K⁻¹ on the basis of Equation (2). The result calculated from Equation (2) is lower than the standard heat capacity calculated from the Neumann–Kopp rule (NKR) [46] $C_p^{\mathrm{o}}(298.15)$ = 184.9 J·K⁻¹·mol⁻¹. The values of standard molar heat capacity of pure oxides, $C_p^{\mathrm{o}}(\mathrm{B}{\mathrm{i}}_2{\mathrm{O}}_3)$ and $C_p^{\mathrm{o}}(W{O}_3)$, necessary for calculation of NKR, were taken from the review of Leitner et al. [47]. The difference between $C_p^{\mathrm{o}}(298.15)$ determined from NKR and from present data is ~4.5%.

Additionally, the values of Bi₂WO₆ heat capacity estimated by Neumann–Kopp’s rule (NKR) using the C_p data for pure oxides [48] were compared with the measured data in Figure 3. As shown in this figure, the heat capacity of NKR is greater than the experimentally obtained C_p values, and the percentage difference is ~4.3% at 400 K. The deviation increases with increasing temperature and is ~4.9% at 700 K.

These results can be compared, for example, with the differences in the experimental and calculated heat capacity of orthorhombic Bi₂MoO₆ using the Neumann–Kopp method [49]. The experimental heat capacity is lower than the estimated data, so that at the mid-range temperature, the percentage deviation is 3.3% at 568 K and increases with increasing temperature to 6.1% at 754 K. Also, Aiswarya et al. [50] compared the $C_p^{\mathrm{o}}(298.15)$ of triclinic CuV₂O₆ obtained by experimental measurements and the values calculated by NKR approximation and found that they were 157.6 and 174.9 J·K⁻¹·mol⁻¹, respectively. This means that the percentage difference is ~10% at 298.15 K. Negative deviation increases with increasing temperature.

Heat capacity data of BaTa₂O₆ ceramic were estimated by using NKR at the temperatures ranging from 298.15 to 1300 K and compared with experimental data of Ilhan et al. [51]. A good agreement was observed between the values fitted to experimental data and NKR heat capacities with an average positive deviation of less than 1%. The fitted and NKR calculated $C_p^{\mathrm{o}}(298.15)$ values of BaTa₂O₆ were estimated to be 182.3 J·K⁻¹·mol⁻¹ and 184.9 J·K⁻¹·mol⁻¹, respectively. Fitted standard heat capacity value showed good agreement with NKR $C_p^{\mathrm{o}}(298.15)$ with the mean error of 1.4%.

Aiswarya et al. [49] also show the variation in heat capacity of Bi₂Mo₃O₁₂ and Bi₆Mo₂O₁₅ compounds with temperature. For both compounds, the negative deviation from NKR was observed, (from 2.7% at 370 K to 7.0% at 800 K) and (from 1.4% at 380 K to 2.4% at 750 K), respectively.

Aiswarya et al. [50] also estimated the heat capacity of copper vanadates by the NKR method. For the case of orthorhombic Cu₂V₂O₇ phase_, good agreement with experimental data was found. However, $C_p^{\mathrm{o}}(298.15)$ Cu₂V₂O₇ computed from experimental data and obtained by the NKR method shows a negative deviation, and are 202.2 J·K⁻¹·mol⁻¹, and 217.2 J·K⁻¹·mol⁻¹, respectively (difference 6.9% at 298.15 K). On the other hand, significant negative deviation was also observed in the data reported by Denisova et al. [52] in the entire temperature range. The experimental C_p data of Cu₅V₂O₁₀ are also compared with estimated by NKR and those reported by Denisowa et al. [53]. A positive deviation from NKR was observed in almost the entire experimental temperature range [50]. Again, at 298.15 K, the C_p deviation is negative and differs by ~5.7%, and Denisova data [53] show lower values over almost the entire experimental temperature range. It should be noted here that C_p(T) data for complex oxides obtained by different scientific groups often differ by more than 15%.

The heat capacity values of Cu₁₁V₆O₂₆ compound at 298.15 K are 793.4 J·K⁻¹·mol⁻¹ based on the experimental expression and 862.7 J·K⁻¹·mol⁻¹ from the NKR estimation, respectively (difference 8.7% at 298.15 K). The experimental heat capacity values [50] are slightly lower than those predicted by NKR, and in the mid-range of the measured temperatures, there is good agreement between the experimental and estimated data. The above examples show that 4% is an acceptable difference between NKR and experimental data. It also emphasized that heat capacities of binary oxides are difficult to compare with C_p of the constituent elements, because one of them is oxygen.

Both positive and negative deviations from the Neumann–Kopp rule are possible, reflecting changes in the vibrational frequencies of atoms in the mixed oxide compounds compared to their component binary oxides. Predicting the heat capacity of compound oxides from the heat capacities of the constituent pure oxides works well in cases of the same crystal structures of the compound and the constituent components. At room temperature, both Bi₂O₃ and WO₃ polymorphs occur in a monoclinic structure compared to the orthorhombic structure of bismuth tungstate. Therefore, deviations are expected in case of significant difference between the molar volumes of the compound and constituent oxides.

The sources of the difference between experimental values of the heat capacity and those obtained from NKR estimation were discussed by Leitner et al. [54]. They proposed an additive expression for C_p(T) which takes into account several factors influencing the isobaric molar heat capacity of the solids, such as phonon contribution $C_{ph}$, lattice dilatations term $C_{dil}$ and $C_{other}$—the other contributions originating from electrons, vacancy formation, magnetic order, etc. In the case of Bi₂WO₆ ionic crystals, the influence of oxygen nonstoichiometry related to the formation of $V_O^{\dot{}\dot{}}$ vacancy cannot be excluded. However, the small contributions of electrons and the formation of $V_{Bi}^{‴}$ vacancies are expected. The phonon contribution C_ph was considered [54], as the sum of heat capacities modeled by the harmonic approximation and term describing the anharmonicity of vibration modes.

The dilatation term C_dil can be expressed as a function of temperature, molar volume V_m, the isothermal compressibility, β and isobaric volume expansion α. Negative deviation ~4% obtained in this work can be connected with the negative value of the dilatation term. If the formation of a ternary oxide results in a lower volume expansion coefficient or higher compressibility compared to the sum of its constituents, the dilatation contribution to the heat capacity will be negative. It should also be noted that in the Bi₂WO₆ phase, the additional effect on C_p will be related to electrical polarization, as it is a ferroelectric material with a noncentrosymmetric crystal structure up to Curie temperature 930 K (1203 °C). That causes spontaneous polarization, giving it piezoelectric properties and potential applications in memory devices. Leitner et al. [54] showed that for the combination Δα < 0 and Δβ > 0, the mixed oxide will exhibit the negative deviations from NKR, while for other combinations, this cannot be directly determined.

In analysis, the term $C_{other}$ related to the phase polarization can be identified with the excess heat capacity, Δ^exC_p. The entropy change calculated by the integration of the ratio Δ^exC_p/T in the phase transition region [45] is ΔS ≈ 0.6 J·K⁻¹·mol⁻¹.

The contribution of harmonic phonon vibrations, which dominate at low temperatures, approaches the Dulong–Petit limit of 3sR (s is the number of atoms per formula unit and R is the universal gas constant) at high temperatures. It is known from the literature [54] that at elevated temperatures, the dilatation and anharmonic correction terms are responsible for the deviation from the NKR predictions. Discussing the deviation of heat capacity of Bi₂MoO₆ from NKR according to the analysis of Leitner et al. [54], Aiswarya et al. [49] pointed out that the bond distances in binary oxides are relatively shorter than those in ternary oxides, because the binary oxides are more closely packed [49]. Therefore, for ternary oxides, when comparing the experimental C_p with heat capacity value obtained using NKR, the lower experimental heat capacity is expected.

In general, the difference observed between the measured and NKR estimated data of heat capacities could be attributed to the differences arising in the dilatation terms due to expansion or contraction in the molar volumes of the unit cells, as well as due to variation in the isothermal compressibility and isobaric volume expansion [54]. Especially at high temperatures, the relations between V_m, α and β as well as the internal anharmonic corrections are important. It should be noted that an accurate calculation of the dilatation contribution is only possible based on rarely available high-temperature data on the α and β parameters of binary and ternary oxides.

It should also be mentioned that the molar heat capacity of Bi₂WO₆ (Table A2) in the entire analyzed temperature range does not exceed the limit of 3sR (s = 9 for Bi₂WO₆), resulting from Dulong–Petit law, 224.5 J·K⁻¹·mol⁻¹. The heat capacity of mixed oxides can be lower than the high-temperature Dulong–Petit limit due to the difference in their dilatation contributions to heat capacity. At high temperatures, the heat capacity of many materials is dominated by lattice vibrations, but anharmonicity, thermal expansion and other contributions become significant. In mixed oxides, a negative deviation in the dilatation contribution compared to the sum of their constituent oxides can cause the total heat capacity to be lower than the expected Dulong–Petit value.

4.2. Extrapolation of Heat Capacity Function

To estimate the heat capacities of binary oxides in the solid state, empirical contribution methods as the sum of contributions of individual atoms or ions are used. For various groups of substances, such contributions are evaluated from the experimental data by an optimization procedure. The earliest attempts to physically predict the heat capacity data from the low (helium) region up to the 298.15 K were proposed by Einstein and Debye [34] who modeled the C_p as a function of T due to harmonic acoustic and optical phonons. To take into account the anharmonicity of the crystal lattice vibrations, a hybrid description combining the Debye and Einstein models was proposed [55,56,57,58,59,60,61].The use of the Debye function is quite difficult to implement in thermodynamic databases. Therefore, the method of assuming multiple Einstein temperatures ${\theta }_i^E$ proposed by Voronin and Kutsenok [62] is convenient for extrapolating C_p data in modeling multicomponent systems. After excluding any magnetic transitions, the molar heat capacity of the crystalline phase, C_p(T), can be expressed as follows:

$$C_p(T)=3\hspace{0.17em}s\hspace{0.17em}R{\displaystyle \mathbf{\sum }_{i=1}^k{f}_i\frac{{\left({\theta }_i^E/T\right)}^{2}exp\left({\theta }_i^E/T\right)}{{\left(exp\left({\theta }_i^E/T\right)-1\right)}^2}}+aT+b{T}^n$$

(3)

where s is a number of atoms per formula unit (for Bi₂WO₆, s = 9) and $f_i,{\mathit{\theta }}_i^E,a,b$ (i = 1, 2, 3…) are adjustable parameters obtained from C_p(T) experimental data fit. ${\theta }_i^E$ is the Einstein temperature for the ith mode of vibration and f_i is the corresponding pre-factor. The additional term for mutual low- and high-temperature fitting, aT, is an electronic part of heat capacity, which represents non-thermodynamic information of the electron density of states at the Fermi level. The term bTⁿ in Equation (3) accounts for high-order anharmonic lattice vibrations. In this case, the polynomial representation also accounts for the conversion from C_V to C_p. To be able to match a wider phase spectrum, Chen and Sundman [63] adopted the bT⁴ term instead of the initially recommended [64] bT² term. Onderka [65] adopted Equation (3) with k = 1 and n = 2 to estimate the heat capacity of bismuth silicates. It turned out that extrapolation for k = 1 is not sufficiently accurate because of Einstein function limitations. Therefore, in the present work, assuming no phase transitions in γ-Bi₂WO₃ phase below 300 K, the high-temperature experimental heat capacity data were extrapolated to 0 K using Equation (3) with k = 2 and n = 4. The parameters for extrapolating C_p(T) to 0 K based on the multiple Einstein model (3) are summarized in

Table 1. Parameters of experimental C_p data fit according Equation (3). The confidence intervals are attached.

i	${\mathsf{\theta }}_{\mathit{i}}^{\mathit{E}}$ K	fi	a, J·K−2·mol−1	b, J·K−5·mol−1
1	757.4 ± 0.7	0.3725 ± 0.0423	0.0199 ± 0.0070	−4.450·10−12 ± 3.295·10−12
2	172.0 ± 0.4	0.5570 ± 0.0208

All values obtained in this table are characterized by Student’s t-distribution with 95% confidence intervals. The correlation coefficient was determined to be 0.99998.

. The standard molar heat capacity, $C_p^{\mathrm{o}}(298.15)$ calculated from Equation (3) is 177.6 ± 5.7 J·K⁻¹·mol⁻¹.

It should be emphasized that the value of standard molar heat capacity calculated from this extrapolation is in good agreement with the value estimated by the Neumann-Kopp method, which is 184.9 J·K⁻¹·mol⁻¹ [46,47] (giving a difference of ~4%). Furthermore, the difference between $C_p^{\mathrm{o}}(298.15)$ values calculated from Equations (2) and (3) is only 0.4%.

It should be noted that such determined standard molar heat capacity of Bi₂WO₆ is close similar to that obtained by Aiswarya et al. [49] for monoclinic Bi₂MoO₆ compound given as 177.1 J·K⁻¹·mol⁻¹. The $C_p^{\mathrm{o}}(298.15)$ values of triclinic CuV₂O₆ [50] and tetragonal BaTa₂O₆ [51] compounds obtained by experimental measurements are 157.6 J·K⁻¹·mol⁻¹ and 182.3 J·K⁻¹·mol⁻¹, respectively.

Using Equation (3), the standard molar entropy of Bi₂WO₆, $S^{\mathrm{o}}(298.15),$ was estimated to be 226.3 ± 8.3 J·K⁻¹mol⁻¹. This value is comparable to the entropy approximated [66] by NKR from the sum of the entropies of the component binary oxides [67,68], which is found to be 225.4 J·K⁻¹mol⁻¹. The attempts to estimate $S^{\mathrm{o}}(298.15)$ using modified Einstein equation for k = 1 and modified Debye equation [65] gave the results ~38% lower. The difference between estimated and approximated values $S^{\mathrm{o}}(298.15)$ using Equations (2) and (3) is only 0.4%, which shows that the C_p(T) function (3) can be used to estimate the standard entropy of the Bi₂WO₆ phase. Taking into account the difference between the obtained result of the NKR and Einstein estimates (k = 2), in the absence of low-temperature C_p data [69], the data estimated by the Einstein equation were adopted for further thermodynamic calculations.

It should be emphasized again that calculating the exact standard molar entropy from the heat capacity function [69,70] requires experimental data on the C_p(T) of this compound at low temperatures.

4.3. Thermodynamic Functions of Bi₂WO₆

Using Equation (3) in conjunction with known thermodynamic relations, the thermodynamic functions of Bi₂WO₆ were calculated using standard molar entropy, $S^{\mathrm{o}}(298.15),$ determined from low-temperature approximation and standard molar enthalpy of formation, ${\mathrm{\Delta }}_f{H}^{\mathrm{o}}(298.15),$ of Phapale et al. [31]. The values of molar—heat capacity, entropy S^o(T), enthalpy H^o(T) and Gibbs energy G^o(T) of γ-Bi₂WO₆ phase—are summarized in

Table 2. Molar heat capacity data and calculated thermodynamic functions of γ-Bi₂WO₆.

T, K	Cp(T)	So(T) − So(298.15)	So(T)	Ho(T) − Ho(298.15)	Ho(T)	Go(T)
	J·K−1·mol−1	J·K−1·mol−1	J·K−1·mol−1	kJ·mol−1	kJ·mol−1	kJ·mol−1
298.15	176.8	0.0	226.3	0.0	−1440.5	−1508.0
300	177.3	1.1	227.4	0.3	−1440.2	−1508.4
350	186.7	29.2	255.5	9.4	−1431.1	−1520.5
400	193.4	54.6	280.9	19.0	−1421.6	−1533.9
450	198.6	77.7	304.0	28.8	−1411.8	−1548.6
500	202.7	98.8	325.1	38.8	−1401.7	−1564.3
550	206.0	118.3	344.6	49.0	−1391.5	−1581.0
600	208.8	136.3	362.6	59.4	−1381.2	−1598.7
650	211.1	153.1	379.4	69.9	−1370.7	−1617.3
700	213.1	168.9	395.2	80.5	−1360.0	−1636.7
750	214.8	183.6	409.9	91.2	−1349.3	−1656.8
800	216.2	197.5	423.8	102.0	−1338.6	−1677.6
850	217.4	210.7	437.0	112.8	−1327.7	−1699.1

.

5. Conclusions

The heat capacity of bismuth tungstate Bi₂WO₆ was determined for the first time using a DSC on a powder sample synthesized by solid-state reaction in the temperature range from 313 to 1103 K (40–830 °C). The C_p of Bi₂WO₆ at 298.15 K (25 °C), estimated in this study from the Maier–Kelley model (Equation (2)) is 176.8 ± 3.9 J·K⁻¹·mol⁻¹.

The modified Einstein model [62,63] (k = 2) was proposed to extrapolate the $C_p^{\mathrm{o}}(T)$ of bismuth tungstate on the basis of high-temperature data set. The standard molar heat capacity, $C_p^{\mathrm{o}}(298.15)$ calculated from this extrapolation (Equation (3)) is 177.6 ± 5.7 J·K^–1·mol^–1. The difference between the standard molar heat capacities calculated from Equations (2) and (3) is about 0.4%. The standard molar entropy of compound Bi₂WO₆, $S^{\mathrm{o}}(298.15),$ approximated from Equation (3) is 226.3 ± 8.3 J·K⁻¹mol⁻¹.

Thermodynamic functions of Bi₂WO₆ were calculated using standard molar entropy determined in this work and standard molar enthalpy of formation of Phapale et al. [31]. The values of calculated molar heat capacity and thermodynamic molar functions of entropy S^o(T), enthalpy H^o(T) and Gibbs energy G^o(T) over the temperature range of 298.15–1123 K (25–850 °C) are presented Table 2. More accurate thermodynamic data can be obtained after analysis of calorimetric low-temperature data (4–300 K).

A reversible second-order phase transition (polar–nonpolar [45]) was observed in the experimental temperature range, with a maximum at 940 ± 1 K (667 °C). The end of the weak endothermic thermal effect was observed near 961 K (688 °C).

Recent works are ongoing to develop new heat capacity estimation techniques using machine learning algorithms. For example, Kauwe et al. [71] described the development of a high-throughput supervised machine learning-based tool that allows the prediction of temperature-dependent heat capacity by correlating with a range of elemental and atomic properties. The conventionally accepted heat capacity prediction methods, such as DFT-based (Density Functional Theory) calculations [72], cation/anion contribution methods [73] and NKR estimates are limited by accuracy or speed. Therefore, the most reliable thermodynamic data, which is the experimental heat capacity, should be used to increase the accuracy of computer throughput calculations.

Machine learning methods enable extremely fast prediction of the heat capacity of thousands of compounds, which allows for easy implementation in integrated computational materials engineering (ICME) [74,75,76]. In this context, it is being considered as a replacement for Neumann–Kopp predictions as a high-throughput screening tool to help identify new materials suitable for engineering processes.