Critical Evaluation of the Thermodynamic Properties of Na₂Cr₂O₇, K₂Cr₂O₇, Na₂Mo₂O₇, K₂Mo₂O₇, Na₂W₂O₇, and K₂W₂O₇ Open Access

Combustion power plants and recovery boilers in the pulp and paper industry^1–4 are mainly made of steels and stainless steels⁵ with Ni, Cr, Mo, W and V as alloying elements.^6–8 Under operating conditions, these steel parts interact with oxygen, hydrogen, sulfur, carbon, chlorine, and alkaline salts (NaCl, KCl, Na₂SO₄, K₂SO₄, Na₂CO₃, K₂CO₃, etc.). These various species are particularly aggressive and may lead to the formation of salt deposits. If the local temperature is high enough, a molten salt solution can form and be extremely corrosive. This limits the durability of such installations and reduces their conversion efficiency. Moreover, conditions can lead to the solid oxide layers of the alloying elements (Cr, Mo, W, and V) complexing with the ions in solution to form dichromates, dimolybdates or ditungstates in the solid or liquid state, thus reducing the passivating layers and promoting further corrosion. As an example, at temperatures suitable for the combustion process, Na₂Cr₂O₇(L) [or K₂Cr₂O₇(L)] can be formed when Cr₂O₃(sol) interacts with NaCl(L) [or KCl(L)],⁹ resulting in the decrease of the protective properties of the oxide layer. In addition, K₂Cr₂O₇ may decrease the melting temperature through the formation of binary salt mixtures.¹⁰ Similarly, in environments with a favorable oxygen potential, Na₂W₂O₇(sol) may be formed due to the interaction of tungsten with Na.¹¹ 

In order to help our understanding of the corrosion phenomena, a thermodynamic model, as Gibbs energy functions of the different solid and liquid phases, for the Na⁺, K⁺//Cl⁻, SO₄²⁻, CO₃²⁻, CrO₄²⁻, Cr₂O₇²⁻, MoO₄²⁻, Mo₂O₇²⁻, WO₄²⁻, W₂O₇²⁻, O²⁻ system is being developed. This requires as a first step the assessment of the thermodynamic properties [standard enthalpy of formation Δ_fH°₂₉₈ from the elements in their stable standard state at 298.15 K and 1 atm, absolute (third law) entropy S°₂₉₈ referenced at 298.15 K and 1 atm, and heat capacity C_p as a function of temperature] of all relevant pure salts. In a previous paper,¹² the thermodynamic properties and phase transitions for sodium and potassium chromates, molybdates, and tungstates were assessed using the available data from the literature and newly obtained differential scanning calorimetry (DSC) measurements. The present article is devoted to a similar assessment for the sodium and potassium dichromates, dimolybdates and ditungstates, which may be formed through reactions of the type 2 A₂MO₄ ⇋ A₂M₂O₇ + A₂O (where A = Na, K and M = Cr, Mo, W). The crystal structures of those compounds were also taken into account in our literature search since this information is useful to identify possible solid solutions forming with other considered salts.

A critical review of the literature related to the crystal structures and space groups of the six pure compounds is provided in this section.

Crystal structure information for phases of Na₂Cr₂O₇ is summarized in Table 1.

XRD analyses of Panagiotopoulos and Brown¹³ revealed the presence of two phases for Na₂Cr₂O₇. The low-temperature form Na₂Cr₂O₇(I) has a triclinic crystal structure with $P1̄$ space group while the high-temperature phase Na₂Cr₂O₇(II) has a triclinic crystal structure corresponding to $A1̄$ space group. Na₂Cr₂O₇(II) is formed by the displacement of two distinct layers (010) in Na₂Cr₂O₇(I); these layers become equivalent in Na₂Cr₂O₇(II) and the resulting high-temperature layer lies almost halfway between the two low-temperature layers.¹³ 

Investigations trying to identify phase transformations of Na₂Cr₂O₇ are very contradictory. Indeed, thermal analyses between 70 and 300 °C by Robinson et al.¹⁴ suggested no phase transformations for Na₂Cr₂O₇. Afonskii¹⁵ and Samuseva et al.¹⁶ reported a single solid-solid phase transition at 240 °C. Subsequent differential thermal analysis (DTA) and high-temperature refractive index measurements conducted by Vesnin and Khripin¹⁷ indicated the presence of four solid phases. Two of them were identified for the first time and their crystal structures were not specified.¹⁷ 

In the present work, in agreement with the study of Panagiotopoulos and Brown,¹³ two different phases were introduced for Na₂Cr₂O₇. These phases have very similar crystal structures, and the corresponding transition may be a second-order transition. In the present work, this solid-solid transition was assumed to be a first-order transition, and the latent heat of transformation was estimated as 1.42 kJ mol⁻¹ as will be discussed later. This is a reasonably low value.

Crystal structure information for phases of K₂Cr₂O₇ is summarized in Table 2.

In 1833, Mitscherlich²⁴ reported for the first time a phase transition for K₂Cr₂O₇ at about 250 °C. Thereafter, several publications were devoted to the identification of the phase transitions and corresponding crystal structures. Jaffray and Labary¹⁸ characterized three crystal forms using DTA, dilatometry and X-ray diffraction: a triclinic phase stable at room temperature, a monoclinic phase formed upon heating at about 269 ± 2 °C, and a metastable monoclinic phase formed from the high-temperature phase upon cooling below 240 °C.

According to the XRD analysis of Jaffray and Labary, the three phases have different crystal structures. No supplementary material was provided and no investigations on single crystals were performed.¹⁸ The metastable monoclinic form is known to be metastable at room temperature.¹⁸ 

The refractometric and thermographic measurements of Vesnin and Khripin¹⁷ led to solid-solid transition temperatures identical to those of Jaffray and Labary.¹⁸ According to Vesnin and Khripin,¹⁷ the metastable form must be a low-temperature form of the high-temperature (stable) monoclinic modification, and is thus identical to the monoclinic crystals precipitated from aqueous solutions.²⁰ 

According to Klement and Schwab,²⁰ the phase transition from the low-temperature form to the high-temperature form is reversible, and the stable and metastable monoclinic forms reported by Jaffray and Labary¹⁸ are identical since they have the same solubility in water. However, in the presence of moisture, the metastable form can quickly transform into the stable form, and it can always be associated with a small amount of the stable form although the solubility of these two forms can be very similar.¹⁷ Moreover, other reversibility checks showed that, in both transformations (namely from the low-temperature form to the high-temperature form and vice-versa), the stratification lines of the crystals and therefore the orientation of the [100] axis, around which the crystal has rotated, have been preserved. Consequently, Klement and Schwab²⁰ claimed that the metastable form reported by Jaffray and Labary¹⁸ does not exist.

The space group of the triclinic form at room temperature was determined by Rao using the statistical method of X-ray reflection intensity.¹⁹ The intensity distribution curves showed a center of symmetry corresponding to the space group classification $P1̄$⁠. Further XRD analysis by Kuz’min et al.,²¹ and by Brandon and Brown²² have confirmed the same space group.

The high-temperature phase is monoclinic with the $C2h5−P21/n$ space group, based on the XRD analysis of Klement and Schwab²⁰ and the electron diffraction analysis of Zhukova and Pinsker.²³ 

The exact crystal structure of the form obtained upon cooling of molten K₂Cr₂O₇ was not identified by Vesnin and Khripin,¹⁷ owing to the lack of data on the interplanar distances of the monoclinic modification at low temperatures.¹⁷ 

In the present work, two different phases were finally introduced for K₂Cr₂O₇. Owing to contradictory information, the so-called metastable monoclinic form was not taken into account (Table 2).

Crystal structure information for phases of Na₂Mo₂O₇ is summarized in Table 3.

XRD analyses conducted by Lindqvist et al.,²⁵ Seleborg et al.,²⁸ Singh Mudher et al.,²⁶ and Saraiva et al.²⁷ have shown that Na₂Mo₂O₇ is orthorhombic at room temperature with the Cmca space group.^25,27,28

At high temperature, XRD analyses performed by Singh Mudher et al.²⁶ and Saraiva et al.²⁷ revealed a monoclinic structure. However, no information was provided for the space group, and Singh Mudher et al.²⁶ reported that the symmetry of this phase has not yet been established.

Crystal structure information for phases of K₂Mo₂O₇ is summarized in Table 4.

The crystal structure of K₂Mo₂O₇ at room temperature was determined by XRD measurements performed by Magarill and Klevtsova.²⁹ A triclinic structure was obtained, with center of symmetry defined as P$1̄$⁠. Further refinement was performed using symmetry reduction in the P1 space group, but this had no significant effect on the results.

To this date, no information is available for the crystal structure and space group of K₂Mo₂O₇ at high temperatures. In the binary phase diagram K₂Mo₂O₇–K₂W₂O₇, Amadori³⁰ reported at high temperatures a complete solid solution, thus indicating that the high-temperature phases II of K₂Mo₂O₇ and K₂W₂O₇ have very similar crystal structures. However, the crystal structure of K₂W₂O₇ is unknown. According to Amadori,³⁰ the binary phase diagram K₂Cr₂O₇–K₂W₂O₇ displays at high temperatures a complete solid solution. This suggests that both K₂Mo₂O₇(II) and K₂W₂O₇(II) have the same crystal structure as K₂Cr₂O₇(II) (that is, monoclinic with space group P2₁/c or $C2h5−P21/n$⁠).

Crystal structure information for phases of Na₂W₂O₇ is summarized in Table 5.

XRD analysis revealed that the crystal structure of Na₂W₂O₇ is isostructural to that of Na₂Mo₂O₇.²⁵ According to Okada et al.,³¹ the coordinates of atoms in Na₂W₂O₇ are very similar to those of Na₂Mo₂O₇, which were determined by Seleborg et al.²⁸ In addition, XRD analyses by Okada et al.³¹ and Range and Haase³² showed that Na₂W₂O₇ exhibits at room temperature an orthorhombic structure corresponding to the space group Cmca³¹ or Cmc2₁.³² 

Using DTA, Nolte and Kordes³³ established that there are no solid-state phase transformations for this salt. On the other hand, according to the experimental binary phase diagrams Na₂Mo₂O₇–Na₂W₂O₇ of Fedorov and Mokhosoev^34,35 using the differential thermographic method (Kurnakov pyrometer), there is a continuous solid solution with a minimum at 90 mol. % Na₂Mo₂O₇ and 590 °C. This suggests the presence of a second phase for Na₂W₂O₇. Therefore, in the present work, two different phases were introduced for Na₂W₂O₇; Na₂W₂O₇(II) has the same crystal structure as monoclinic Na₂Mo₂O₇(II), whose space group remains unknown. The temperature for the solid-solid transition of Na₂W₂O₇ was estimated as 670 °C, as will be discussed later.

Crystal structure information for phases of K₂W₂O₇ is summarized in Table 6.

To our knowledge, there is no information on the number of phases and crystal structure of solid K₂W₂O₇.

As discussed previously, K₂Cr₂O₇ and K₂Mo₂O₇ both have two phases, and the two high-temperature forms have the same crystal structure as is already the case for the two low-temperature forms. According to the experimental binary phase diagram K₂W₂O₇–K₂Cr₂O₇ of Amadori,³⁰ there is a complete solid solution at high temperatures, which suggests that high-temperature K₂W₂O₇ has the same crystal structure as K₂Cr₂O₇(II) (that is, a monoclinic structure with P2₁/c or $C2h5−P21/n$ space group). Finally, two phases were considered for K₂W₂O₇ in the present work, and the low-temperature form I was assumed to be triclinic with $P1̄$ space group, by analogy with the low-temperature phases of K₂Cr₂O₇ and K₂Mo₂O₇. The temperature for the solid-solid transition of K₂W₂O₇ was assessed as 459 °C, as will be explained later.

Table 7 lists all data from the literature corresponding to the temperature and enthalpy change for the solid-solid transition of Na₂Cr₂O₇.

The measured values of the temperature and enthalpy of fusion of Na₂Cr₂O₇ collected from the literature are presented in Table 8.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for sodium dichromate Na₂Cr₂O₇ are provided in Table 9. For the pure liquid, below the temperature of fusion, the heat capacity (as a function of temperature) over a given temperature range was taken identical to that of the solid phase stable over this specific temperature range (i.e., C_p of the low-temperature phase I between 298.15 and 522.8 K, and C_p of the high-temperature phase II between 522.8 and 629.1 K). Similarly, for solid II, below the temperature of the solid-solid transition, the heat capacity was taken identical to that of solid I. For all three phases, the heat capacity above the maximum temperature T_max of the highest temperature range in Table 9 is assumed to be constant and equal to the value of C_p at T_max.

The same approach was used for all other compounds considered in this work. More details on the thermodynamic properties selected for Na₂Cr₂O₇ are given below.

The available measurements for the temperature and enthalpy change of the (I → II) and (II → L) transitions for Na₂Cr₂O₇ are shown in Tables 7 and 8, respectively. For each property, the measured values were plotted as a function of the experiment number in order to identify visually outliers (see Figs. S1–S3 in the supplementary material). A weighted average was then calculated from the retained measurements, with a weighting based on the expected accuracy of each experimental technique. The weighting factors used for the temperature and enthalpy change are given in Tables S3 and S4, respectively, in the supplementary material. The standard deviation σ was then calculated.

The outliers are identified by the letter a in Tables 7 and 8 and are shown in red color in Tables S1 and S2 in the supplementary material, and as empty circles in Figs. S1 and S3 in the supplementary material. In these scatter plots, the weighted average is shown as a solid line while the two dashed lines correspond to (weighted average ±2σ). It was checked that all outliers lie outside this range. The reported uncertainties in Tables 7 and 8 correspond to ±2σ. Owing to the lack of data, some temperatures and enthalpies of transition were sometimes estimated by us (the enthalpy change for the I → II transition, in the case of Na₂Cr₂O₇). No uncertainties were reported in those cases.

A similar approach was used for all other compounds investigated in this work.

Measurements of the enthalpy change for the solid-solid transition of Na₂Cr₂O₇ are not available in the literature. As a first approximation, the entropy change was assumed to be identical to that for K₂Cr₂O₇ (that is, 2.72 J mol⁻¹ K⁻¹ based on the weighted average of the experimental measurements from Refs. 17 and 33 as will be discussed later). This is the only compound of the type A₂M₂O₇ (with A = Na, K and M = Cr, Mo, W) for which the enthalpy change of the solid-solid transition was measured. The transition temperature has been evaluated as 522.8 K, which thus corresponds to an enthalpy change of about 1.42 kJ mol⁻¹. Note that this is a rough estimate since the high-temperature phases of Na₂Cr₂O₇ (triclinic) and K₂Cr₂O₇ (monoclinic) do not have the same crystal structure.

The value of the standard enthalpy at 298.15 K (Δ_fH°₂₉₈) for Na₂Cr₂O₇(I) was estimated as follows. The enthalpy of dissolution of Na₂Cr₂O₇(I) (⁠$Na2Cr2O7(I)+2OHaq−→2Naaq++2CrO4(aq)2−+H2O(l)$⁠) was measured by calorimetry⁴⁰ in the presence of a small excess of NaOH_(aq) as −89.41 kJ mol⁻¹.

Values of $ΔfH298◦(Naaq+)$⁠, $ΔfH298◦(CrO4(aq)2−)$⁠, $ΔfH298◦OHaq−$⁠, and $ΔfH298◦H2O(l)$ were taken from the FactPS database available in FactSage⁴¹ as −239.73,⁴² −881.15,⁴² −229.99,^42,43 and −285.83 kJ mol⁻¹,⁴⁴ respectively.

The standard enthalpy at 298.15 K of Na₂Cr₂O₇(I) was estimated by us using the method proposed by Hisham and Benson.⁴⁵ These authors derived equations of the type $1aΔfH298◦MaXb$ = $mΔfH298◦MCl+1−m2ΔfH298◦M2O+C$⁠, where m and C are two constants, and Δ_fH°₂₉₈ refers to the standard enthalpy at 298.15 K of the corresponding solid compound. They selected the corresponding chloride and oxide of a metal M as reference compounds since, for the majority of metals, the chloride and oxide values are known with good accuracy. For alkali dichromates, Hisham and Benson⁴⁵ obtained m = 1.3 and C = −516.72 kJ mol⁻¹. Using the Δ_fH°₂₉₈ values for NaCl and Na₂O from the FTsalt and FToxid databases available in FactSage⁴¹ (both taken from Barin’s compilation⁴⁴), Δ_fH°₂₉₈[Na₂Cr₂O₇(I)] was estimated as −1976.97 kJ mol⁻¹, which is very close to the value of −1978.21 kJ mol⁻¹ selected in the present work and derived from the calorimetric measurements of Nelson et al.⁴⁰ 

Owing to the lack of data, as a first approximation, the standard entropy at 298.15 K (S°₂₉₈) of Na₂Cr₂O₇(I) was assessed as 251.32 J mol⁻¹ K⁻¹ from the exchange reaction $Na2Mo2O7I+K2Cr2O7I→$ $Na2Cr2O7I$ + $K2Mo2O7I$⁠, for which ∆S = 0 was assumed at 298.15 K. The S°₂₉₈ values for Na₂Mo₂O₇(I), K₂Cr₂O₇(I) and K₂Mo₂O₇(I) selected in the present work will be discussed later.

Dellien et al.⁴⁶ assessed a S°₂₉₈ value of 267.78 J mol⁻¹ K⁻¹ for Na₂Cr₂O₇(I). This estimate was not considered in this work since the methodology was not described.

The heat capacity of Na₂Cr₂O₇(II) was derived from the heat content measurements (H_T − H₂₉₈) of Nguyen-Duy and Dancy³⁸ obtained by drop calorimetry over a limited temperature range (from 521 to 604 K), and was assumed to be valid from our selected temperature of 522.8 K for the I → II transition to our selected temperature of fusion of 629.1 K. The heat content measurements of the same authors for Na₂Cr₂O₇(L) between 629 and 654 K (i.e., a very limited temperature range) were discarded since they most likely correspond to much too high heat capacity values. Indeed, in the same study, Nguyen-Duy and Dancy³⁸ derived a C_p of 213.38 J mol⁻¹ K⁻¹ for NaNO₃(L) between 579 and 620 K. This is much higher than the value of 138.20 J mol⁻¹ K⁻¹ recommended above 579 K by Robelin et al.,⁴⁷ based on a critical evaluation of all available data from the literature (see Fig. 1). Finally, in the present work, the heat capacity of Na₂Cr₂O₇(L) above the melting temperature was taken as the average of the heat capacities of Na₂S₂O₇(L),⁴⁸ Na₂Mo₂O₇(L) and Na₂W₂O₇(L) (The heat capacities of the latter compounds will be discussed later.) This is a reasonable assumption since the heat capacities of Na₂SO₄(L),⁴⁸ Na₂CrO₄(L),¹² Na₂MoO₄(L)¹² and Na₂WO₄(L)¹² are very similar.

The heat capacity of Na₂Cr₂O₇(I) was assumed to be constant by analogy with Na₂Cr₂O₇(II), and was set to 267.00 J mol⁻¹ K⁻¹ to reproduce satisfactorily the heat content data of Nguyen-Duy and Dancy³⁸ for Na₂Cr₂O₇(II) (see Fig. 2). (The enthalpy change for the I → II transition was evaluated previously.) As seen in Table 9, the heat capacity selected for Na₂Cr₂O₇(II) is significantly higher than that of Na₂Cr₂O₇(I). It may be somewhat too large since it was derived from the heat content measurements of Nguyen-Duy and Dancy,³⁸ which were very doubtful for Na₂Cr₂O₇(L). However, based on the comparison between the data of the same authors for NaNO₃(II) and the calculations from Ref. 47 (see Fig. 1), it was not possible to derive a correction that could have been applied to the original heat content measurements for Na₂Cr₂O₇(II).

Calculated values of H_T − H₂₉₈ for Na₂Cr₂O₇ are shown along with the available measurements in Fig. 2.

Table 10 presents all data from the literature corresponding to the temperature and enthalpy change for the solid-solid transition of K₂Cr₂O₇.

The properties of fusion (temperature and enthalpy change) were measured by Refs. 14, 17, 18, 33, 36, 38, 39, and 49–56. The heat of fusion was measured by drop calorimetry,³⁸ DTA,³³ and DSC.^36,39 All experimental data from the literature are gathered in Table 11.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for potassium dichromate K₂Cr₂O₇ are given in Table 12.

Note that the unreasonable enthalpy change of 2033 kJ mol⁻¹ reported by Lax⁵⁰ for the solid-solid transition was discarded; this may very well be a typo. Also, the experimental technique used by this author was not described.

Owing to the lack of data, the standard enthalpy at 298.15 K (Δ_fH°₂₉₈) of K₂Cr₂O₇(I) was estimated as −2060.38 kJ mol⁻¹ using the equation of Hisham and Benson⁴⁵ with an approach similar to that described previously for Na₂Cr₂O₇. This estimate is expected to be reliable since very good agreement was observed for Na₂Cr₂O₇(I) between the estimated Δ_fH°₂₉₈ value and our selected value based on experimental data from the literature.

The standard entropy at 298.15 K (S°₂₉₈) of K₂Cr₂O₇(I) was derived by Popov and Kolesov⁵⁷ as 291.21 J mol⁻¹ K⁻¹ by integration of their experimental C_p values from 60 to 300 K obtained by calorimetry. Lax⁵⁰ reported a value of 291.20 J mol⁻¹ K⁻¹, which might be directly taken from Ref. 57. The value of Popov and Kolesov⁵⁷ was selected in the present work.

The heat capacity of K₂Cr₂O₇(II) was derived from the heat content measurements (H_T − H₂₉₈) of Nguyen-Duy and Dancy³⁸ obtained by drop calorimetry over a limited temperature range (from 536 to 639 K), and was assumed to be valid from our selected temperature of 540.3 K for the I → II transition to our selected temperature of fusion of 670.6 K. The heat content measurements of the same authors for K₂Cr₂O₇(L) between 681 and 706 K (i.e., a very limited temperature range) were not considered since they most likely correspond to much too high heat capacity values. Indeed, in the same article, Nguyen-Duy and Dancy³⁸ derived a C_p of 234.30 J mol⁻¹ K⁻¹ for KNO₃(L) between 518 and 616 K. This is substantially higher than the value of 142.70 J mol⁻¹ K⁻¹ recommended above 606 K by Robelin et al.,⁴⁷ based again on a critical evaluation of all available data from the literature (see Fig. 3). Finally, in this work, the heat capacity of K₂Cr₂O₇(L) above the melting temperature was taken as the average of the heat capacities of K₂S₂O₇(L),⁴⁸ K₂Mo₂O₇(L) and K₂W₂O₇(L). (The heat capacities of the latter compounds will be discussed later.) This is a reasonable assumption since the heat capacities of K₂SO₄(L),⁴⁸ K₂CrO₄(L),¹² K₂MoO₄(L)¹² and K₂WO₄(L)¹² are close to each other.

The heat capacity of K₂Cr₂O₇(I) was assumed to be constant by analogy with K₂Cr₂O₇(II), and was set to 219.66 J mol⁻¹ K⁻¹ in order to reproduce satisfactorily the room temperature C_p value of Popov and Kolesov⁵⁷ (see Fig. 5) and the heat content data of Nguyen-Duy and Dancy³⁸ for K₂Cr₂O₇(II) (see Fig. 4). (The enthalpy change for the I → II transition was evaluated previously.) As seen in Table 12, the heat capacity selected for K₂Cr₂O₇(II) is substantially higher than that of K₂Cr₂O₇(I). It may very well be too large since it was derived from the heat content measurements of Nguyen-Duy and Dancy,³⁸ which were very questionable for K₂Cr₂O₇(L). Nevertheless, based on the comparison between the data of the same authors for KNO₃(II) and the calculations from Ref. 47 (see Fig. 3), it was not possible to derive a correction that could have been applied to the heat content measurements for K₂Cr₂O₇(II).

Calculated values of H_T − H₂₉₈ are compared to the available measurements in Fig. 4. The calculated heat capacity at low temperatures for K₂Cr₂O₇(I) is shown in Fig. 5 along with the data of Popov and Kolesov⁵⁷ and the two series of data obtained by Nan et al.⁵⁸ using adiabatic calorimetry over the temperature range of 80–390 K.

Table 13 presents all data from the literature corresponding to the temperature and enthalpy change for the solid-solid transition of Na₂Mo₂O₇. Owing to the lack of relevant data, the enthalpy change was estimated as will be discussed later.

The properties of fusion (temperature and enthalpy change) of Na₂Mo₂O₇ collected from the literature are listed in Table 14. Owing to the lack of relevant data, the enthalpy of fusion was assessed as will be explained later.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for sodium dimolybdate Na₂Mo₂O₇ are presented in Table 15.

To our knowledge, measurements of the enthalpy change for the solid-solid transition and fusion of Na₂Mo₂O₇ are not available in the literature. As a first approximation, the entropy change for the solid-solid transition was assumed to be identical to that for K₂Cr₂O₇ (that is, 2.72 J mol⁻¹ K⁻¹ based on the weighted average of the experimental measurements from Refs. 17 and 33). The transition temperature for Na₂Mo₂O₇ has been evaluated as 829 K, which thus corresponds to an enthalpy change of about 2.25 kJ mol⁻¹. Note that this is a rough estimate since the low-temperature phases of Na₂Mo₂O₇ (orthorhombic) and K₂Cr₂O₇ (triclinic) do not have the same crystal structure.

The entropy of fusion of Na₂Mo₂O₇ was assumed to be identical to that of Na₂W₂O₇ (that is, 94.68 J mol⁻¹ K⁻¹ as will be discussed later). This is a reasonable assumption since the two phases Na₂Mo₂O₇(II) and Na₂W₂O₇(II) have the same crystal structure (with an undefined space group), the same alkali cation (Na⁺) and very similar radii for the other cations (Mo⁶⁺ = 0.59 Å and W⁶⁺ = 0.60 Å in an octahedral geometry).⁶⁵ The temperature of fusion of Na₂Mo₂O₇ has been estimated as 885.6 K, which thus corresponds to an enthalpy of fusion of about 83.86 kJ mol⁻¹.

Using a cycle of calorimetric reactions, Koehler et al.⁶⁶ evaluated the standard enthalpy at 298.15 K of Na₂Mo₂O₇(I) as −2244.30 kJ mol⁻¹. The molar enthalpy of dissolution of Na₂Mo₂O₇ in a NaOH solution was measured at 298.15 K by Tangri et al.⁶⁷ using an isoperibol calorimeter. The molar enthalpy of solution at infinite dilution was measured as −55.89 ± 0.69 kJ mol⁻¹,⁶⁷ and a standard enthalpy at 298.15 K of −2245.02 ± 0.81 kJ mol⁻¹ was then derived for Na₂Mo₂O₇(I). Using the same experimental technique, Suponitskii et al.⁶⁸ obtained Δ_fH°₂₉₈ = −2242.62 kJ mol⁻¹. Mathews et al.⁶⁹ performed electromotive force (EMF) measurements for the high temperature galvanic cell Pt, Ar + CO₂ + O₂, Na₂CO₃|NASICON|Na₂Mo₂O₇ + Na₂MoO₄, O₂ + CO₂ + Ar, Pt over the temperature range 650–800 K (where NASICON is a material for electrodes used in electrochemical cells). The cell reaction is Na₂Mo₂O₇(sol) + Na₂CO₃(sol) = 2Na₂MoO₄(sol) + CO₂(g), and the authors reported the Gibbs energy change ΔG^◦ = 53.78 − 0.16T kJ mol⁻¹ (±1 kJ) for this reaction. The calculated Gibbs energy change of the cell reaction is compared to the data points of Mathews et al.⁶⁹ in Fig. 6. In this figure, the two vertical dashed lines correspond to the solid-solid transitions Na₂MoO₄(I) = Na₂MoO₄(II) and Na₂CO₃(II) = Na₂CO₃(III), respectively.

The calculated Gibbs energy change of the cell reaction investigated by Mathews et al.⁶⁹ overestimates the measurements of these authors by about 8.74 kJ mol⁻¹, which is somewhat outside the reported experimental error limit. The thermodynamic properties (Δ_fH°₂₉₈, S°₂₉₈, and C_p) selected for Na₂Mo₂O₇(I) in the present study were all based on experimental data from the literature. Also, the thermodynamic properties of Na₂CO₃ were critically reviewed by Lindberg et al.⁷⁰ while those of both phases of Na₂MoO₄ were based on our careful evaluation of all experimental data from the literature.¹² For all these reasons, it is believed that the calculations shown in Fig. 6 have a reasonable accuracy.

Since the Δ_fH°₂₉₈ values given by Koehler et al.,⁶⁶ Tangri et al.⁶⁷ and Suponitskii et al.⁶⁸ are close to each other, their average was selected in the present work. The obtained value (−2243.98 kJ mol⁻¹) is very close to the value of −2244.25 kJ mol⁻¹ estimated using the equation of Hisham and Benson⁴⁵ (for alkali dimolybdates, m = 0.98 and C = −715.05 kJ mol⁻¹).

For the standard entropy at 298.15 K (S°₂₉₈) of Na₂Mo₂O₇(I), Weller and Kelley⁶³ reported a value of 250.62 ± 2.09 J mol⁻¹ K⁻¹, assessed by integration of their low-temperature heat capacity measurements obtained by calorimetry. The Einstein and Debye functions were used over the extrapolated temperature range of 0–51 K.

To our knowledge, no direct high-temperature C_p measurements are available in the literature. Therefore, the expression of the heat capacity as a function of temperature was directly taken from the SGPS database available in FactSage⁴¹ (validity from 298.15 to 888 K) and was assumed to be valid for all phases (I, II and liquid).

Calculated values of H_T − H₂₉₈ are shown along with the data of Iyer et al.⁵⁹ in Fig. 7. These data were obtained between 335 and 760 K using drop calorimetry in a high temperature Calvet calorimeter. The DTA study of the same authors from 298.15 K to about 881 K revealed no solid-solid transition.

The calculated heat capacity at low temperatures for Na₂Mo₂O₇(I) is shown in Fig. 8 along with the measurements of Weller and Kelley.⁶³ 

Table 16 lists all data from the literature corresponding to the temperature and enthalpy change for the solid-solid transition of K₂Mo₂O₇. Since these data are very limited, the transition temperature and enthalpy change were both estimated as will be described later.

The properties of fusion (temperature and enthalpy change) collected from the literature are listed in Table 17. Owing to the lack of relevant data, the enthalpy of fusion was assessed as will be explained later.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for potassium dimolybdate K₂Mo₂O₇ are presented in Table 18.

Based on all experimental data available in the literature, the melting temperature of K₂Mo₂O₇ was estimated as 498.0 ± 23.7 °C. This value is lower than the only solid-solid transition temperature value of 516.9–521.9 °C.⁷¹ Therefore, the latter measurement was discarded, and the solid-solid transition temperature T_trs was roughly estimated by assuming that the ratio $TfusTtrs$ for K₂Mo₂O₇ is identical to that for K₂Cr₂O₇. Strictly speaking, this assumption has no physical basis, but these two compounds have strong similarities. Indeed, both the low-temperature and high-temperature phases of K₂Mo₂O₇ and K₂Cr₂O₇ have the same crystal structure (with the same space group), the same alkali cation (K⁺) and very similar anionic radii (Mo₂O₇²⁻, Cr₂O₇²⁻). A T_trs value of about 348.1 °C was obtained.

To our knowledge, measurements of the enthalpy change for the solid-solid transition and fusion of K₂Mo₂O₇ are not available in the literature. By analogy with Na₂Mo₂O₇, as a first approximation, the entropy change for the solid-solid transition was assumed to be identical to that for K₂Cr₂O₇ (2.72 J mol⁻¹ K⁻¹). The transition temperature for K₂Mo₂O₇ has been evaluated as 621.3 K, which corresponds to an enthalpy change of about 1.69 kJ mol⁻¹. This is expected to be a reasonable estimate owing to the close similarity of the low-temperature and high-temperature phases of K₂Mo₂O₇ and K₂Cr₂O₇ mentioned above. Similarly, the entropy of fusion of K₂Mo₂O₇ was assumed to be identical to that of K₂Cr₂O₇ (58.22 J mol⁻¹ K⁻¹). This permitted us to assess an enthalpy of fusion of about 44.89 kJ mol⁻¹ from the previously evaluated temperature of fusion of 771.1 K.

To our knowledge, no experimental value of the standard enthalpy of formation at 298.15 K (Δ_fH°₂₉₈) of K₂Mo₂O₇(I) has been reported in the literature. This quantity was assessed by us as −2293.22 kJ mol⁻¹ from the equation of Hisham and Benson⁴⁵ with an approach similar to that described previously for Na₂Mo₂O₇.

The standard entropy at 298.15 K (S°₂₉₈) of K₂Mo₂O₇(I) was roughly estimated as 290.51 J mol⁻¹ K⁻¹ from the exchange reaction $Na2Mo2O7I+K2MoO4I→K2Mo2O7I+Na2MoO4I$⁠, assuming that ∆S = 0 at 298.15 K. The standard entropies at 298.15 K of Na₂MoO₄(I) and K₂MoO₄(I) were assessed by us previously,¹² and that of Na₂Mo₂O₇(I) was estimated by us in this work.

Owing to the lack of relevant data, the heat capacities as a function of temperature for the various phases of K₂Mo₂O₇ were assessed from the exchange reactions $Na2Mo2O7II+K2MoO4III→K2Mo2O7sol+Na2MoO4IV$ (for the two solid phases) and $Na2Mo2O7L+K2MoO4L→K2Mo2O7L+Na2MoO4L$ (for the liquid phase), assuming that ∆C_P = 0 at all temperatures.

To our knowledge, no data are available in the literature for the properties of solid-solid transition of Na₂W₂O₇. As will be explained later, the temperature and enthalpy change were estimated very roughly as 670 °C and 2.56 kJ mol⁻¹, respectively.

The properties of fusion (temperature and enthalpy change) collected from the literature are listed in Table 19. The enthalpy of fusion has only been measured by Nolte and Kordes,³³ using DTA.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for sodium ditungstate Na₂W₂O₇ are presented in Table 20.

The temperature for the solid-solid transition of Na₂W₂O₇ is unknown and was finally assessed as about 942.7 K by assuming that the ratio $TfusTtrs$ for Na₂W₂O₇ is identical to that for Na₂Mo₂O₇ (1.07). Again, this assumption has no physical basis, but these two compounds are very similar: both the low-temperature and high-temperature phases have the same crystal structure (with the same space group), the same alkali cation (Na⁺), and very similar radii for the other cations (Mo⁶⁺ = 0.59 Å and W⁶⁺ = 0.60 Å in an octahedral geometry)⁶⁵ which are also in the same column of the periodic table.

Again, as a first approximation, the entropy change for the solid-solid transition of Na₂W₂O₇ was assumed to be identical to that for K₂Cr₂O₇ (2.72 J mol⁻¹ K⁻¹). This permitted us to assess an enthalpy change of about 2.56 kJ mol⁻¹ from our estimated transition temperature of about 942.7 K for Na₂W₂O₇.

Using a cycle of calorimetric reactions, Koehler et al.⁶⁶ evaluated Δ_fH°₂₉₈ = −2485.30 kJ mol⁻¹ for Na₂W₂O₇(I). This is to our knowledge the only experimental value available in the literature.

Weller and Kelley⁶³ reported a S°₂₉₈ value of 254.39 ± 2.09 J mol⁻¹ K⁻¹ estimated by integration of their low-temperature heat capacity measurements obtained by calorimetry, with the use of the Einstein and Debye functions over the extrapolated temperature range of 0–51 K.

Using calorimetry, Liu et al.¹¹ performed heat content measurements (H_T − H₂₇₃) for Na₂W₂O₇ from 273.15 to 979 K. This permitted us to derive the heat capacity of both phases as a function of temperature (see Table 20), which was assumed to be valid up to our selected temperature of fusion of 1007.1 K.

Owing to the lack of data, the heat capacity of the liquid was estimated from the exchange reaction $Na2Mo2O7L+2Na2WO4L→$ $2Na2MoO4L$ + $Na2W2O7L$⁠, assuming that ∆C_P = 0 at all temperatures.

Calculated values of H_T − H₂₇₃ are compared to the measurements of Liu et al.¹¹ in Fig. 9. The calculated heat capacity at low temperatures for Na₂W₂O₇(I) is shown in Fig. 10 along with the measurements of Weller and Kelley.⁶³ 

No data were found in the literature for the properties of solid-solid transition of K₂W₂O₇. As will be discussed later, the temperature and enthalpy change were assessed very roughly as 459 °C and 1.99 kJ mol⁻¹, respectively.

Experimental properties of fusion for K₂W₂O₇ are presented in Table 21. The enthalpy of fusion was assessed by us as 52.87 kJ mol⁻¹, as will be explained later.

Our selected thermodynamic data (Δ_fH°₂₉₈, S°₂₉₈, and C_p) for potassium ditungstate K₂W₂O₇ are presented in Table 22.

The temperature for the solid-solid transition of K₂W₂O₇ has not been measured. It was finally estimated as about 731.7 K by assuming that the ratio $TfusTtrs$ for K₂W₂O₇ is identical to that for K₂Cr₂O₇ (1.24). As mentioned previously, such an assumption has no physical basis but these two compounds have similarities. The high-temperature phases have the same crystal structure (with the same space group) and this is expected to apply to the low-temperature phases too. Also, the two compounds have the same alkali cation (K⁺) and very similar anionic radii (W₂O₇²⁻, Cr₂O₇²⁻).

Again, as a first approximation, the entropy change for the solid-solid transition of K₂W₂O₇ was assumed to be identical to that for K₂Cr₂O₇ (2.72 J mol⁻¹ K⁻¹). This permitted us to estimate an enthalpy change of about 1.99 kJ mol⁻¹ from our assessed transition temperature of about 731.7 K for K₂W₂O₇. Similarly, the entropy of fusion of K₂W₂O₇ was assumed to be identical to that of K₂Cr₂O₇ (58.22 J mol⁻¹ K⁻¹). This corresponds to an enthalpy of fusion of about 52.87 kJ mol⁻¹ based on our previously evaluated temperature of fusion of 908.1 K.

The standard enthalpy at 298.15 K (Δ_fH°₂₉₈) of K₂W₂O₇(I) was estimated as −2554.00 kJ mol⁻¹ by Kaganyuk and Trachevskii⁷³ using the effective charges on atoms in molecules and ions, calculated for the condition of equalization of potentials. To our knowledge, this is the only estimate available in the literature. Unfortunately, Hisham and Benson⁴⁵ have not considered alkali ditungstates in their work. Our selected Δ_fH°₂₉₈ value for K₂Cr₂O₇(I) is about −82.17 kJ mol⁻¹ more negative than that for Na₂Cr₂O₇(I), and our selected Δ_fH°₂₉₈ value for K₂Mo₂O₇(I) is about −49.24 kJ mol⁻¹ more negative than that for Na₂Mo₂O₇(I). Applying the corresponding average shift of −65.71 kJ mol⁻¹ to our selected Δ_fH°₂₉₈ value for Na₂W₂O₇(I) led to a very rough estimate of −2551.00 kJ mol⁻¹ for the standard enthalpy at 298.15 K of K₂W₂O₇(I). The latter value compares very well to the estimate of Kaganyuk and Trachevskii,⁷³ which was finally selected in the present work. This estimate may have limited accuracy since the same authors assessed a Δ_fH°₂₉₈ value of −2329.00 kJ mol⁻¹ for Na₂Mo₂O₇(I), which is substantially more negative than our selected value of −2243.98 kJ mol⁻¹ based on experimental data from the literature.

Owing to the lack of data, S°₂₉₈ for K₂W₂O₇(I) was roughly estimated as 242.83 J mol⁻¹ K⁻¹ from the exchange reaction $K2Cr2O7I+2K2WO4I→2K2CrO4I+K2W2O7I$⁠, assuming that ∆S = 0 at 298.15 K. The standard entropies at 298.15 K of K₂WO₄(I) and K₂CrO₄(I) were assessed by us previously,¹² and that of K₂Cr₂O₇(I) was estimated by us in this work.

Using drop calorimetry, Chen et al.⁷² performed heat content measurements (H_T − H₂₇₃) for K₂W₂O₇ from 273.15 to 928.1 K. This permitted us to derive the heat capacity of both solid phases as a function of temperature (see Table 22), which was assumed to be valid up to our selected temperature of fusion of 908.1 K. Due to the lack of data, the same heat capacity expression was assumed to be valid for the pure liquid. Calculated values of H_T − H₂₇₃ are shown along with the measurements of Chen et al.⁷² in Fig. 11. As seen in this figure, the single measurement of these authors for the pure liquid is well reproduced.

Table 23 provides a summary of all transition temperatures and enthalpies of transition estimated in the present work, along with their assessed uncertainties.

Using our selected thermodynamic properties for the A₂MO₄¹² and A₂M₂O₇ (present work) compounds (with A = Na or K, and M = Cr, Mo or W), the Gibbs energy changes were calculated as a function of temperature for all reactions of the type 2A₂MO₄ ⇋ A₂M₂O₇ + A₂O. It permitted us to conclude that these reactions are very limited in the temperature range of interest for the targeted industrial applications (that is, up to above the liquidus temperatures). The inversion temperatures at which the Gibbs energy changes become zero and then negative are always higher than about 3976 °C (value for Na₂Mo₂O₇). At temperatures near or higher than these calculated inversion temperatures, the above reactions may be strongly shifted to the right with the formation of substantial oxide content in the liquid phase, and our thermodynamic model, which is being developed for the Na⁺, K⁺//Cl⁻, SO₄²⁻, CO₃²⁻, CrO₄²⁻, Cr₂O₇²⁻, MoO₄²⁻, Mo₂O₇²⁻, WO₄²⁻, W₂O₇²⁻, O²⁻ system, will no longer be valid since only dilute amounts of oxides are considered in the liquid phase. Note that, at such high temperatures (which have very limited interest), the gas phase would need to be included in the calculations.

Enthalpies of formation at 0 K for the compounds A₂M₂O₇ (where A = Na or K, and M = Cr, Mo or W) are available in the Open Quantum Materials Database (OQMD)^74,75 and in Materials Project.⁷⁶ These values were derived from Density Functional Theory (DFT) calculations. Sometimes, the calculated shifts between them and our selected standard enthalpies of formation at 298.15 K were large. To facilitate comparison, the reactions 2A₂MO_4(I) ⇋ A₂M₂O_7(I) + A₂O_(I) and A₂O_(I) + 2MO_3(I) ⇋ A₂M₂O_7(I) were both considered. Tables 24 and 25 provide calculations of the enthalpy changes at 298.15 and 0 K for these two reactions, respectively (with A = Na or K, and M = Cr, Mo or W). Our calculations at 298.15 K are based on the standard enthalpies of formation for the A₂MO₄ pure compounds, evaluated in our previous work,¹² and on those for the A₂M₂O₇ pure compounds obtained in the present study. The standard enthalpies of formation at 298.15 K for Na₂O and K₂O were taken from Barin's compilation,⁴⁴ while those for CrO₃ and MoO₃ were taken from the SGPS database⁴¹ in FactSage, and that for WO₃ was taken from JANAF.⁴³ The enthalpy changes at 0 K for the two reactions mentioned above were estimated using the enthalpies of formation at 0 K of the various compounds from OQMD^74,75 and from Materials Project.⁷⁶ 

Table 24 is related to the reactions 2A₂MO_4(I) ⇋ A₂M₂O_7(I) + A₂O_(I). With the exception of Na₂Cr₂O₇, the absolute values of the calculated shifts (Δ_rH°₂₉₈ − Δ_rH°₀) are always substantially smaller for Materials Project⁷⁶ than for OQMD.^74,75 In the former case, the highest shift (about −64.6 kJ mol⁻¹) corresponds to the compound Na₂W₂O₇.