This study outlines an extensive investigation of crystallographic data and thermodynamic properties (including solid-solid transition and fusion) for sodium dichromate, potassium dichromate, sodium dimolybdate, potassium dimolybdate, sodium ditungstate, and potassium ditungstate. A thorough literature review was conducted to obtain a good understanding of the data available in the literature, and a critical and complete evaluation has been performed from room temperature to above the melting temperatures. This work is one of the key steps towards the development of a thermodynamic model for the Na+, K+//Cl, SO42−, CO32−, CrO42−, Cr2O72−, MoO42−, Mo2O72−, WO42−, W2O72−, O2− system, relevant for high temperature corrosion in atmospheres containing O–H–S–C–Cl and alkali salts.

Combustion power plants and recovery boilers in the pulp and paper industry1–4 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 ΔfH°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 Cp 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.

TABLE 1.

Crystallographic data for phases of Na2Cr2O7

PhaseCrystal structureSpace groupT (°C)Reference
Na2Cr2O7(I) Triclinic P1̄ <250 13  
Na2Cr2O7(II) Triclinic A1̄ 250–356 13  
PhaseCrystal structureSpace groupT (°C)Reference
Na2Cr2O7(I) Triclinic P1̄ <250 13  
Na2Cr2O7(II) Triclinic A1̄ 250–356 13  

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.

TABLE 2.

Crystallographic data for phases of K2Cr2O7

PhaseCrystal structureSpace groupT (°C)Reference
K2Cr2O7(I) Triclinic ⋯ <267 18  
P1̄ 19  
Ci1P1̄ 20  
P1̄ 21  
P1̄ 22  
K2Cr2O7(II) Monoclinic ⋯ 267–398 18  
C2h5P21/n 20  
P21/c 23  
PhaseCrystal structureSpace groupT (°C)Reference
K2Cr2O7(I) Triclinic ⋯ <267 18  
P1̄ 19  
Ci1P1̄ 20  
P1̄ 21  
P1̄ 22  
K2Cr2O7(II) Monoclinic ⋯ 267–398 18  
C2h5P21/n 20  
P21/c 23  

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 C2h5P21/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.

TABLE 3.

Crystallographic data for phases of Na2Mo2O7

PhaseCrystal structureSpace groupT (°C)Reference
Na2Mo2O7(I) Orthorhombic Cmca D2h18a <556 25  
⋯ 26  
Cmca D2h18a 27  
Na2Mo2O7(II) Monoclinic ⋯ 556–613 26  
⋯ 27  
PhaseCrystal structureSpace groupT (°C)Reference
Na2Mo2O7(I) Orthorhombic Cmca D2h18a <556 25  
⋯ 26  
Cmca D2h18a 27  
Na2Mo2O7(II) Monoclinic ⋯ 556–613 26  
⋯ 27  
a

The space group reported by the authors represents the abbreviated notation of the international tables of the 1935 edition.78 

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.

TABLE 4.

Crystallographic data for phases of K2Mo2O7

PhaseCrystal structureSpace groupT (°C)Reference
K2Mo2O7(I) Triclinic P1̄ or P1 <348 29  
K2Mo2O7(II) Monoclinic (assumption) P21/c or C2h5P21/n (assumption) 348–498 This work 
PhaseCrystal structureSpace groupT (°C)Reference
K2Mo2O7(I) Triclinic P1̄ or P1 <348 29  
K2Mo2O7(II) Monoclinic (assumption) P21/c or C2h5P21/n (assumption) 348–498 This work 

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 P1̄. 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 C2h5P21/n).

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

TABLE 5.

Crystallographic data for phases of Na2W2O7

PhaseCrystal structureSpace groupT (°C)Reference
Na2W2O7(I) Orthorhombic Cmca <670 31  
Cmc21 32  
Na2W2O7(II) Monoclinic ⋯ 670–734 This work 
 (assumption)    
PhaseCrystal structureSpace groupT (°C)Reference
Na2W2O7(I) Orthorhombic Cmca <670 31  
Cmc21 32  
Na2W2O7(II) Monoclinic ⋯ 670–734 This work 
 (assumption)    

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 Mokhosoev34,35 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.

TABLE 6.

Crystallographic data for phases of K2W2O7

PhaseCrystal structureSpace groupT (°C)Reference
K2W2O7(I) Triclinic (assumption) P1̄ (assumption) <459 This work 
K2W2O7(II) Monoclinic (assumption) P21/c or C2h5P21/n (assumption) 459–635 This work 
PhaseCrystal structureSpace groupT (°C)Reference
K2W2O7(I) Triclinic (assumption) P1̄ (assumption) <459 This work 
K2W2O7(II) Monoclinic (assumption) P21/c or C2h5P21/n (assumption) 459–635 This work 

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 C2h5P21/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.

3.1.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for Na2Cr2O7

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

TABLE 7.

Transition temperature and enthalpy change of solid-solid transition for Na2Cr2O7

Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
248 ⋯ Thermal analysis (heating curves) at 5 °C/min 16  
240a ⋯ DTA 17 b 
251 ⋯ DSC at 10 °C/min 36  
249.6 ± 3.0 1.4 (see text) Weighted average This work 
Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
248 ⋯ Thermal analysis (heating curves) at 5 °C/min 16  
240a ⋯ DTA 17 b 
251 ⋯ DSC at 10 °C/min 36  
249.6 ± 3.0 1.4 (see text) Weighted average This work 
a

Outlier.

b

These authors reported four solid phases between room temperature and the melting temperature, with two other solid-solid transitions at 291 and 330 °C. The latter were not considered in this work.

3.1.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for Na2Cr2O7

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

TABLE 8.

Transition temperature and enthalpy of fusion for Na2Cr2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
356.7 ± 1.0 ⋯ Thermal analysis 37  
360 ⋯ Thermal analysis (heating curves) at 5 °C/min 16  
355.4 35.364 DSC at 10 °C/min 36  
351.9 25.104a Calorimetry (value derived from heat contents) 38  
354 35.7 ± 0.75 DSC (from 2 to 10 °C/min) 39  
356.0 ± 5.5 35.5 ± 0.3 Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
356.7 ± 1.0 ⋯ Thermal analysis 37  
360 ⋯ Thermal analysis (heating curves) at 5 °C/min 16  
355.4 35.364 DSC at 10 °C/min 36  
351.9 25.104a Calorimetry (value derived from heat contents) 38  
354 35.7 ± 0.75 DSC (from 2 to 10 °C/min) 39  
356.0 ± 5.5 35.5 ± 0.3 Weighted average This work 
a

Outlier.

3.1.3. Recommended thermodynamic data for sodium dichromate Na2Cr2O7

Our selected thermodynamic data (ΔfH°298, S°298, and Cp) 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., Cp of the low-temperature phase I between 298.15 and 522.8 K, and Cp 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 Tmax of the highest temperature range in Table 9 is assumed to be constant and equal to the value of Cp at Tmax.

TABLE 9.

Selected thermodynamic properties of Na2Cr2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)a298 (J mol−1 K−1)bCp (J mol−1 K−1)Reference
Na2Cr2O7(I) 298.15 −1978.21 251.32  This work 
298.15–522.8   267.00 
Na2Cr2O7(II) 298.15 −1976.79 254.03  This work 
298.15–522.8   267.00 
522.8–629.1   343.09 38  
Na2Cr2O7(L) 298.15 −1941.25 310.51  This work 
298.15–522.8   267.00 
522.8–629.1   343.09 38  
629.1–2500   242.20 + 0.048 12 (T/K) This work 
PhaseT range (K)ΔfH°298 (kJ mol−1)a298 (J mol−1 K−1)bCp (J mol−1 K−1)Reference
Na2Cr2O7(I) 298.15 −1978.21 251.32  This work 
298.15–522.8   267.00 
Na2Cr2O7(II) 298.15 −1976.79 254.03  This work 
298.15–522.8   267.00 
522.8–629.1   343.09 38  
Na2Cr2O7(L) 298.15 −1941.25 310.51  This work 
298.15–522.8   267.00 
522.8–629.1   343.09 38  
629.1–2500   242.20 + 0.048 12 (T/K) This work 
a

Enthalpy relative to the enthalpy of the elements in their stable standard states at 298.15 K.

b

Absolute (third law) entropy.

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 (ΔfH°298) for Na2Cr2O7(I) was estimated as follows. The enthalpy of dissolution of Na2Cr2O7(I) (Na2Cr2O7(I)+2OHaq2Naaq++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), ΔfH298OHaq, and ΔfH298H2O(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 1aΔfH298MaXb = mΔfH298MCl+1m2ΔfH298M2O+C, where m and C are two constants, and ΔfH°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 ΔfH°298 values for NaCl and Na2O from the FTsalt and FToxid databases available in FactSage41 (both taken from Barin’s compilation44), ΔfH°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 (HTH298) 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 Cp 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.

FIG. 1.

Calculated heat content HTH298 for NaNO3. Experimental data from Ref. 38 (○) and full lines calculated from Ref. 47.

FIG. 1.

Calculated heat content HTH298 for NaNO3. Experimental data from Ref. 38 (○) and full lines calculated from Ref. 47.

Close modal

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).

FIG. 2.

Calculated heat content HTH298 for Na2Cr2O7. Experimental data from Ref. 38 (○).

FIG. 2.

Calculated heat content HTH298 for Na2Cr2O7. Experimental data from Ref. 38 (○).

Close modal

Calculated values of HTH298 for Na2Cr2O7 are shown along with the available measurements in Fig. 2.

3.2.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for K2Cr2O7

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

TABLE 10.

Transition temperature and enthalpy change of solid-solid transition for K2Cr2O7

Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
236a ⋯ Cooling curves, automatic recording; microscopy (solidified melt sections) 49  
236.8 ± 0.5a ⋯ Pt resistance thermometer, galvanometer 14  
241.6 ± 1a ⋯ Thermal analysis 37  
269 ± 2 ⋯ DTA 18  
255 ± 2a,b 
264 ± 5 ⋯ Thermal analysis, HSM 20  
269 ± 2 ⋯ Electron diffraction 23  
270 1.723 Heating curves 17c 
ΔHtrs from Ref. 4  
241.6a 2033 (discarded: see text) ⋯ 50  
264 1.213 DTA at 1 °C/min 33  
269 ⋯ Raman spectroscopy 51  
268 ⋯ DSC at 10 °C/min 36  
267.2 ± 4.7 1.5 ± 0.5 Weighted average This work 
Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
236a ⋯ Cooling curves, automatic recording; microscopy (solidified melt sections) 49  
236.8 ± 0.5a ⋯ Pt resistance thermometer, galvanometer 14  
241.6 ± 1a ⋯ Thermal analysis 37  
269 ± 2 ⋯ DTA 18  
255 ± 2a,b 
264 ± 5 ⋯ Thermal analysis, HSM 20  
269 ± 2 ⋯ Electron diffraction 23  
270 1.723 Heating curves 17c 
ΔHtrs from Ref. 4  
241.6a 2033 (discarded: see text) ⋯ 50  
264 1.213 DTA at 1 °C/min 33  
269 ⋯ Raman spectroscopy 51  
268 ⋯ DSC at 10 °C/min 36  
267.2 ± 4.7 1.5 ± 0.5 Weighted average This work 
a

Outlier.

b

This temperature was measured when the same product was heated anew.

c

These authors reported four solid phases between room temperature and the melting temperature, with two other solid-solid transitions at 345 and 380 °C (enthalpy changes of 0.03 and 0.18 cal/g, respectively). Those two transitions were not considered in this work.

3.2.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for K2Cr2O7

The properties of fusion (temperature and enthalpy change) were measured by Refs. 14, 17, 18, 33, 36, 38, 39, and 4956. 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.

TABLE 11.

Transition temperature and enthalpy of fusion for K2Cr2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
400 ⋯ Heating curves; visual 52  
395 ⋯ Cooling curves, automatic recording; microscopy (solidified melt sections) 49  
397.5 ± 0.5 ⋯ Melting curves 53  
398.4 ⋯ Pt resistance thermometer, galvanometer 14  
398 ⋯ DTA 18  
398 36.694 Heating curves 17  
398 36.7 Recommended value in the compilation 50  
400 37.279 DTA at 1 °C/min 33  
400 ⋯ DTA 54  
398 ⋯ Raman spectroscopy 51  
394 ⋯ Cooling curves at 5 °C/min 55  
400 ⋯ DTA 56  
392.9 39.330 Calorimetry (value derived from heat contents) 38  
395.2 40.389 DSC at 10 °C/min 36  
394 40.07 ± 0.4 DSC (from 2 to 10 °C/min) 39  
397.5 ± 4.8 39.0 ± 3.4 Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
400 ⋯ Heating curves; visual 52  
395 ⋯ Cooling curves, automatic recording; microscopy (solidified melt sections) 49  
397.5 ± 0.5 ⋯ Melting curves 53  
398.4 ⋯ Pt resistance thermometer, galvanometer 14  
398 ⋯ DTA 18  
398 36.694 Heating curves 17  
398 36.7 Recommended value in the compilation 50  
400 37.279 DTA at 1 °C/min 33  
400 ⋯ DTA 54  
398 ⋯ Raman spectroscopy 51  
394 ⋯ Cooling curves at 5 °C/min 55  
400 ⋯ DTA 56  
392.9 39.330 Calorimetry (value derived from heat contents) 38  
395.2 40.389 DSC at 10 °C/min 36  
394 40.07 ± 0.4 DSC (from 2 to 10 °C/min) 39  
397.5 ± 4.8 39.0 ± 3.4 Weighted average This work 

3.2.3. Recommended thermodynamic data for potassium dichromate K2Cr2O7

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

TABLE 12.

Selected thermodynamic properties of K2Cr2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2Cr2O7(I) 298.15 −2060.38   45  
  291.21  57  
 298.15–540.3   219.66 This work 
K2Cr2O7(II) 298.15 −2058.91 293.92  This work 
 298.15–540.3   219.66  
 540.3–670.6   401.66 38  
 670.6–2000a   207.04 + 0.103 30 (T/K) − 436 000 (T/K)−2 This work 
K2Cr2O7(L) 298.15 −2019.87 352.15  This work 
 298.15–540.3   219.66  
 540.3–670.6   401.66 38  
 670.6–2000   207.04 + 0.103 30 (T/K) − 436 000 (T/K)−2 This work 
PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2Cr2O7(I) 298.15 −2060.38   45  
  291.21  57  
 298.15–540.3   219.66 This work 
K2Cr2O7(II) 298.15 −2058.91 293.92  This work 
 298.15–540.3   219.66  
 540.3–670.6   401.66 38  
 670.6–2000a   207.04 + 0.103 30 (T/K) − 436 000 (T/K)−2 This work 
K2Cr2O7(L) 298.15 −2019.87 352.15  This work 
 298.15–540.3   219.66  
 540.3–670.6   401.66 38  
 670.6–2000   207.04 + 0.103 30 (T/K) − 436 000 (T/K)−2 This work 
a

Above the temperature of fusion, the heat capacity of K2Cr2O7(II) was assumed to be equal to that of K2Cr2O7(L) to ensure a reasonable extrapolation of the Gibbs energy at high temperatures. In absence of this additional Cp range, the high-temperature phase would be calculated to be more stable than the pure liquid above 2096 K.

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 (ΔfH°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 ΔfH°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 Cp 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 (HTH298) 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 Cp 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.

FIG. 3.

Calculated heat content HTH298 for KNO3. Experimental data from Ref. 38 (○) and full lines calculated from Ref. 47.

FIG. 3.

Calculated heat content HTH298 for KNO3. Experimental data from Ref. 38 (○) and full lines calculated from Ref. 47.

Close modal

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 Cp 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).

FIG. 4.

Calculated heat content HTH298 for K2Cr2O7. Experimental data from Ref. 38 (○).

FIG. 4.

Calculated heat content HTH298 for K2Cr2O7. Experimental data from Ref. 38 (○).

Close modal

Calculated values of HTH298 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.

FIG. 5.

Calculated heat capacity at low temperatures for K2Cr2O7(I). Experimental data from Ref. 57 (□), and Ref. 58 (○ and ●).

FIG. 5.

Calculated heat capacity at low temperatures for K2Cr2O7(I). Experimental data from Ref. 57 (□), and Ref. 58 (○ and ●).

Close modal

3.3.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for Na2Mo2O7

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.

TABLE 13.

Transition temperature and enthalpy change of solid-solid transition for Na2Mo2O7

Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
556.9 ⋯ DSC at 10 °C/min 26  
554.9 ⋯ DTA at 10 °C/min 
555.9 ± 2.0 2.3 (see text) Weighted average This work 
Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
556.9 ⋯ DSC at 10 °C/min 26  
554.9 ⋯ DTA at 10 °C/min 
555.9 ± 2.0 2.3 (see text) Weighted average This work 

3.3.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for Na2Mo2O7

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.

TABLE 14.

Transition temperature and enthalpy of fusion for Na2Mo2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
612 ⋯ Differential thermographic (Kurnakov pyrometer) at 10 °C/min 34  
602.9 ± 5a ⋯ Calorimetry, drop method (using Calvet high-T) 59  
599.9a ⋯ DSC at 10 °C/min 26  
612 ⋯ Thermal analysis 60  
612 ⋯ Thermal analysis 61  
613.8 ⋯ DTA 62  
612.5 ± 1.6 83.9 (see text) Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
612 ⋯ Differential thermographic (Kurnakov pyrometer) at 10 °C/min 34  
602.9 ± 5a ⋯ Calorimetry, drop method (using Calvet high-T) 59  
599.9a ⋯ DSC at 10 °C/min 26  
612 ⋯ Thermal analysis 60  
612 ⋯ Thermal analysis 61  
613.8 ⋯ DTA 62  
612.5 ± 1.6 83.9 (see text) Weighted average This work 
a

Outlier.

3.3.3. Recommended thermodynamic data for sodium dimolybdate Na2Mo2O7

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

TABLE 15.

Selected thermodynamic properties of Na2Mo2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
Na2Mo2O7(I) 298.15 −2243.98   This work 
 250.62  63  
298.15–829   173.64 + 0.144 35 (T/K) 64  
Na2Mo2O7(II) 298.15 −2241.73 253.34  This work 
298.15–885.6   173.64 + 0.144 35 (T/K) 64  
Na2Mo2O7(L) 298.15 −2157.87 348.03  This work 
298.15–2000   173.64 + 0.144 35 (T/K) 64  
PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
Na2Mo2O7(I) 298.15 −2243.98   This work 
 250.62  63  
298.15–829   173.64 + 0.144 35 (T/K) 64  
Na2Mo2O7(II) 298.15 −2241.73 253.34  This work 
298.15–885.6   173.64 + 0.144 35 (T/K) 64  
Na2Mo2O7(L) 298.15 −2157.87 348.03  This work 
298.15–2000   173.64 + 0.144 35 (T/K) 64  

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 ΔfH°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.

FIG. 6.

Calculated Gibbs energy change ΔG for the cell reaction investigated by Mathews et al.69 Experimental data from Ref. 69 (○). A: Na2Mo2O7(I) + Na2CO3(II) = 2Na2MoO4(I) + CO2(g); B: Na2Mo2O7(I) + Na2CO3(II) = 2Na2MoO4(II) + CO2(g); C: Na2Mo2O7(I) + Na2CO3(III) = 2Na2MoO4(II) + CO2(g).

FIG. 6.

Calculated Gibbs energy change ΔG for the cell reaction investigated by Mathews et al.69 Experimental data from Ref. 69 (○). A: Na2Mo2O7(I) + Na2CO3(II) = 2Na2MoO4(I) + CO2(g); B: Na2Mo2O7(I) + Na2CO3(II) = 2Na2MoO4(II) + CO2(g); C: Na2Mo2O7(I) + Na2CO3(III) = 2Na2MoO4(II) + CO2(g).

Close modal

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 (ΔfH°298, S°298, and Cp) 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 ΔfH°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 Cp 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 HTH298 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.

FIG. 7.

Calculated heat content HTH298 for Na2Mo2O7(I). Experimental data from Ref. 59 (○).

FIG. 7.

Calculated heat content HTH298 for Na2Mo2O7(I). Experimental data from Ref. 59 (○).

Close modal

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

FIG. 8.

Calculated heat capacity at low temperatures for Na2Mo2O7(I). Experimental data from Ref. 63 (●).

FIG. 8.

Calculated heat capacity at low temperatures for Na2Mo2O7(I). Experimental data from Ref. 63 (●).

Close modal

3.4.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for K2Mo2O7

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.

TABLE 16.

Transition temperature and enthalpy change of solid-solid transition for K2Mo2O7

Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
516.9–521.9 ⋯ Calorimetry 71  
348.1 (see text) 1.7 (see text)  This work 
Ttrs (I → II) (°C)ΔHtrs (I → II) (kJ mol−1)Experimental methodReference
516.9–521.9 ⋯ Calorimetry 71  
348.1 (see text) 1.7 (see text)  This work 

3.4.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for K2Mo2O7

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.

TABLE 17.

Transition temperature and enthalpy of fusion for K2Mo2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
484 ⋯ Cooling curves; visual 30  
491.9 ± 5 ⋯ ⋯ Reference 1 of Ref. 71  
541.9–556.9a ⋯ Calorimetry 71  
516 ⋯ Thermal analysis 61  
495.5 ⋯ DTA 62  
498.0 ± 23.7 44.9 (see text) Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
484 ⋯ Cooling curves; visual 30  
491.9 ± 5 ⋯ ⋯ Reference 1 of Ref. 71  
541.9–556.9a ⋯ Calorimetry 71  
516 ⋯ Thermal analysis 61  
495.5 ⋯ DTA 62  
498.0 ± 23.7 44.9 (see text) Weighted average This work 
a

Outlier.

3.4.3. Recommended thermodynamic data for potassium dimolybdate K2Mo2O7

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

TABLE 18.

Selected thermodynamic properties of K2Mo2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2Mo2O7(I) 298.15 −2293.22   45  
 290.51  This work 
298.15–621.3   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
K2Mo2O7(II) 298.15 −2291.53 293.23  This work 
298.15–771.1   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
K2Mo2O7(L) 298.15 −2246.64 351.45  This work 
298.15–771.1   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
771.1–1500   165.30 + 0.159 41 (T/K) 
1500–2000   187.90 + 0.144 35 (T/K) 
PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2Mo2O7(I) 298.15 −2293.22   45  
 290.51  This work 
298.15–621.3   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
K2Mo2O7(II) 298.15 −2291.53 293.23  This work 
298.15–771.1   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
K2Mo2O7(L) 298.15 −2246.64 351.45  This work 
298.15–771.1   161.79 + 0.162 65 (T/K) + 729 690 (T/K)−2 
771.1–1500   165.30 + 0.159 41 (T/K) 
1500–2000   187.90 + 0.144 35 (T/K) 

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 Ttrs 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 Ttrs 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 (ΔfH°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+K2MoO4IK2Mo2O7I+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+K2MoO4IIIK2Mo2O7sol+Na2MoO4IV (for the two solid phases) and Na2Mo2O7L+K2MoO4LK2Mo2O7L+Na2MoO4L (for the liquid phase), assuming that ∆CP = 0 at all temperatures.

3.5.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for Na2W2O7

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.

3.5.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for Na2W2O7

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.

TABLE 19.

Transition temperature and enthalpy of fusion for Na2W2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
738 ⋯ Differential thermographic (Kurnakov pyrometer) at 10 °C/min 34  
730.9 95.353 DTA at 1 °C/min 33  
730.9 ⋯ Calorimetry 11  
734.0 ± 6.9 95.4 Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
738 ⋯ Differential thermographic (Kurnakov pyrometer) at 10 °C/min 34  
730.9 95.353 DTA at 1 °C/min 33  
730.9 ⋯ Calorimetry 11  
734.0 ± 6.9 95.4 Weighted average This work 

3.5.3. Recommended thermodynamic data for sodium ditungstate Na2W2O7

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

TABLE 20.

Selected thermodynamic properties of Na2W2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
Na2W2O7(I) 298.15 −2485.30   66  
 254.39  63  
298.15–942.7   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
Na2W2O7(II) 298.15 −2482.73 257.10  This work 
298.15–1007.1   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
Na2W2O7(L) 298.15 −2387.38 351.79  This work 
298.15–1007.1   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
1007.1–2000   308.18 This work 
PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
Na2W2O7(I) 298.15 −2485.30   66  
 254.39  63  
298.15–942.7   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
Na2W2O7(II) 298.15 −2482.73 257.10  This work 
298.15–1007.1   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
Na2W2O7(L) 298.15 −2387.38 351.79  This work 
298.15–1007.1   211.12 + 0.1072 (T/K) − 2 566 000 (T/K)−2 11  
1007.1–2000   308.18 This work 

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 ΔfH°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 (HTH273) 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 ∆CP = 0 at all temperatures.

Calculated values of HTH273 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 

FIG. 9.

Calculated heat content HTH273 for Na2W2O7. Experimental data from Ref. 11 (●).

FIG. 9.

Calculated heat content HTH273 for Na2W2O7. Experimental data from Ref. 11 (●).

Close modal
FIG. 10.

Calculated heat capacity at low temperatures for Na2W2O7(I). Experimental data from Ref. 63 (●).

FIG. 10.

Calculated heat capacity at low temperatures for Na2W2O7(I). Experimental data from Ref. 63 (●).

Close modal

3.6.1. Transition temperature (Ttrs) and enthalpy change (ΔHtrs) of solid-solid transition (I → II) for K2W2O7

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.

3.6.2. Temperature (Tfus) and enthalpy of fusion (ΔHfus) for K2W2O7

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.

TABLE 21.

Transition temperature and enthalpy of fusion for K2W2O7

Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
555a ⋯ Cooling curves; visual 30  
629.8 ⋯ Calorimetry, drop method (the corresponding value is the average of 604.5 and 655 °C) 72  
637 ⋯ Thermal analysis 61  
635.0 ± 7.8 52.9 (see text) Weighted average This work 
Tfus (II → L) (°C)ΔHfus (II → L) (kJ mol−1)Experimental methodReference
555a ⋯ Cooling curves; visual 30  
629.8 ⋯ Calorimetry, drop method (the corresponding value is the average of 604.5 and 655 °C) 72  
637 ⋯ Thermal analysis 61  
635.0 ± 7.8 52.9 (see text) Weighted average This work 
a

Outlier.

3.6.3. Recommended thermodynamic data for potassium ditungstate K2W2O7

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

TABLE 22.

Selected thermodynamic properties of K2W2O7

PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2W2O7(I) 298.15 −2554.00   73  
 242.83  This work 
298.15–731.7   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  
K2W2O7(II) 298.15 −2552.01 245.55  This work 
298.15–908.1   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  
K2W2O7(L) 298.15 −2499.14 303.77  This work 
298.15–2000   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  
PhaseT range (K)ΔfH°298 (kJ mol−1)298 (J mol−1 K−1)Cp (J mol−1 K−1)Reference
K2W2O7(I) 298.15 −2554.00   73  
 242.83  This work 
298.15–731.7   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  
K2W2O7(II) 298.15 −2552.01 245.55  This work 
298.15–908.1   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  
K2W2O7(L) 298.15 −2499.14 303.77  This work 
298.15–2000   207.97 + 0.109 46 (T/K) − 1 744 000 (T/K)−2 72  

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 (ΔfH°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 ΔfH°298 value for K2Cr2O7(I) is about −82.17 kJ mol−1 more negative than that for Na2Cr2O7(I), and our selected ΔfH°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 ΔfH°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 ΔfH°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+2K2WO4I2K2CrO4I+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 (HTH273) 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 HTH273 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.

FIG. 11.

Calculated heat content HTH273 for K2W2O7. Experimental data from Ref. 72 (○).

FIG. 11.

Calculated heat content HTH273 for K2W2O7. Experimental data from Ref. 72 (○).

Close modal

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

TABLE 23.

Summary table of recommended values

Transition Ttrs (°C)U(Ttrs) (°C)ΔHtrs (kJ mol−1)UHtrs) (kJ mol−1)
Na2Cr2O7(I → II) 249.6 3.0 1.4 ⋯ 
Na2Cr2O7(II → L) 356.0 5.5 35.5 0.3 
K2Cr2O7(I → II) 267.2 4.7 1.5 0.5 
K2Cr2O7(II → L) 397.5 4.8 39.0 3.4 
Na2Mo2O7(I → II) 555.9 2.0 2.3 ⋯ 
Na2Mo2O7(II → L) 612.5 1.6 83.9 ⋯ 
K2Mo2O7(I → II) 348.1 ⋯ 1.7 ⋯ 
K2Mo2O7(II → L) 498.0 23.7 44.9 ⋯ 
Na2W2O7(I → II) 669.5 ⋯ 2.6 ⋯ 
Na2W2O7(II → L) 734.0 6.9 95.4 ⋯ 
K2W2O7(I → II) 458.8 ⋯ 2.0 ⋯ 
K2W2O7(II → L) 635.0 7.8 52.9 ⋯ 
Transition Ttrs (°C)U(Ttrs) (°C)ΔHtrs (kJ mol−1)UHtrs) (kJ mol−1)
Na2Cr2O7(I → II) 249.6 3.0 1.4 ⋯ 
Na2Cr2O7(II → L) 356.0 5.5 35.5 0.3 
K2Cr2O7(I → II) 267.2 4.7 1.5 0.5 
K2Cr2O7(II → L) 397.5 4.8 39.0 3.4 
Na2Mo2O7(I → II) 555.9 2.0 2.3 ⋯ 
Na2Mo2O7(II → L) 612.5 1.6 83.9 ⋯ 
K2Mo2O7(I → II) 348.1 ⋯ 1.7 ⋯ 
K2Mo2O7(II → L) 498.0 23.7 44.9 ⋯ 
Na2W2O7(I → II) 669.5 ⋯ 2.6 ⋯ 
Na2W2O7(II → L) 734.0 6.9 95.4 ⋯ 
K2W2O7(I → II) 458.8 ⋯ 2.0 ⋯ 
K2W2O7(II → L) 635.0 7.8 52.9 ⋯ 

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 2A2MO4(I) ⇋ A2M2O7(I) + A2O(I) and A2O(I) + 2MO3(I) ⇋ A2M2O7(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 OQMD74,75 and from Materials Project.76 

TABLE 24.

Enthalpy changes at 298.15 and 0 K for the reactions 2A2MO4 ⇋ A2M2O7 + A2O (with A = Na or K, and M = Cr, Mo or W)

Reaction ΔrH°298 (kJ mol−1)ΔrH°074,75 (kJ mol−1)ΔrH°076 (kJ mol−1)rH°298 − ΔrH°076) (kJ mol−1)rH°298 − ΔrH°074,75) (kJ mol−1)
A = Na 286.79 304.42 312.04 −17.62 −25.25 
M = Cr 
A = K 385.52 383.04 383.35 2.48 2.18 
M = Cr 
A = Na 269.78 313.39 270.17 −43.61 −0.39 
M = Mo 
A = K 340.97 367.03 353.42 −26.05 −12.45 
M = Mo 
A = Na 186.19 298.91 250.78 −112.73 −64.58 
M = W 
A = K 247.70 348.03 270.83 −100.33 −23.12 
M = W 
Reaction ΔrH°298 (kJ mol−1)ΔrH°074,75 (kJ mol−1)ΔrH°076 (kJ mol−1)rH°298 − ΔrH°076) (kJ mol−1)rH°298 − ΔrH°074,75) (kJ mol−1)
A = Na 286.79 304.42 312.04 −17.62 −25.25 
M = Cr 
A = K 385.52 383.04 383.35 2.48 2.18 
M = Cr 
A = Na 269.78 313.39 270.17 −43.61 −0.39 
M = Mo 
A = K 340.97 367.03 353.42 −26.05 −12.45 
M = Mo 
A = Na 186.19 298.91 250.78 −112.73 −64.58 
M = W 
A = K 247.70 348.03 270.83 −100.33 −23.12 
M = W 
TABLE 25.

Enthalpy changes at 298.15 and 0 K for the reactions A2O + 2MO3 ⇋ A2M2O7 (with A = Na or K, and M = Cr, Mo or W)

Reaction ΔrH°298 (kJ mol−1)ΔrH°074,75 (kJ mol−1)ΔrH°076 (kJ mol−1)rH°298 − ΔrH°074,75) (kJ mol−1)rH°298 − ΔrH°076) (kJ mol−1)
A = Na −397.91 −401.48 −403.12 3.57 5.22 
M = Cr 
A = K −536.56 −539.74 −528.84 3.18 −7.73 
M = Cr 
A = Na −336.80 −310.10 −351.79 −26.70 14.99 
M = Mo 
A = K −442.52 −440.94 −437.17 −1.59 −5.35 
M = Mo 
A = Na −381.50 −261.48 −252.89 −120.02 −128.61 
M = W 
A = K −506.68 −380.63 −383.91 −126.05 −122.77 
M = W 
Reaction ΔrH°298 (kJ mol−1)ΔrH°074,75 (kJ mol−1)ΔrH°076 (kJ mol−1)rH°298 − ΔrH°074,75) (kJ mol−1)rH°298 − ΔrH°076) (kJ mol−1)
A = Na −397.91 −401.48 −403.12 3.57 5.22 
M = Cr 
A = K −536.56 −539.74 −528.84 3.18 −7.73 
M = Cr 
A = Na −336.80 −310.10 −351.79 −26.70 14.99 
M = Mo 
A = K −442.52 −440.94 −437.17 −1.59 −5.35 
M = Mo 
A = Na −381.50 −261.48 −252.89 −120.02 −128.61 
M = W 
A = K −506.68 −380.63 −383.91 −126.05 −122.77 
M = W 

Table 24 is related to the reactions 2A2MO4(I) ⇋ A2M2O7(I) + A2O(I). With the exception of Na2Cr2O7, the absolute values of the calculated shifts (ΔrH°298 − ΔrH°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.

Table 25 is related to the reactions A2O(I) + 2MO3(I) ⇋ A2M2O7(I). Overall, the absolute values of the calculated shifts (ΔrH°298 − ΔrH°0) are similar for Materials Project76 and OQMD.74,75 For Na2Cr2O7, K2Cr2O7, Na2Mo2O7 and K2Mo2O7, they always remain lower than 15 kJ mol−1 for Materials Project.76 However, there are substantial shifts of the order of −120 kJ mol−1 for Na2W2O7 and K2W2O7, independently of the origin of the values of the enthalpies of formation at 0 K. In Materials Project,76 the ΔfH°0 values for Na2W2O7 and K2W2O7 are substantially higher (by about 139.7 and 124.6 kJ mol−1, respectively) than our values of ΔfH°298 selected in the present work, whereas the ΔfH°0 value for WO3 is almost identical to the ΔfH°298 value taken from JANAF.43 In OQMD,74,75 the ΔfH°0 values for Na2W2O7 and K2W2O7 are lower (by about 77.8 and 51.6 kJ mol−1, respectively) than our values of ΔfH°298 selected in the present work, whereas the ΔfH°0 value for WO3 is significantly lower (by about 101.1 kJ mol−1) than the ΔfH°298 value taken from JANAF.43 

DFT calculations are usually well suited for metallic compounds, and inorganic salt compounds (in which bonds are very localized). To overcome the self-interaction error in transition-metal oxides such as A2M2O7 (where A = Na or K, and M = Cr, Mo or W), a correction is introduced to the DFT energy using parameters U (on-site Coulomb) and J (on-site exchange), or sometimes just their difference (U − J), known as the DFT+U.77 These parameters, U and J, are typically derived through semi-empirical methods. This approach was used in Materials Project76 for all six A2M2O7 compounds. For Na2Cr2O7, K2Cr2O7, Na2Mo2O7 and K2Mo2O7, the calculated shifts (ΔfH°298 − ΔfH°076) are relatively small (between −5.6 and −16.7 kJ mol−1). However, as reported previously, for Na2W2O7 and K2W2O7, the corresponding calculated shifts are substantially higher (about −139.7 and −124.6 kJ mol−1, respectively). As explained in the present work, limited data were available in the literature for the ΔfH°298 values of these two compounds, and thus our selected values might not be very accurate. However, the same trend was observed for the calculated shifts (ΔfH°298 − ΔfH°076) for the A2MO4 compounds: for Na2CrO4, K2CrO4, Na2MoO4 and K2MoO4, they are small (between −9.7 and 1.9 kJ mol−1), whereas for Na2WO4 and K2WO4 they are significantly higher (−44.7 and −53.2 kJ mol−1, respectively). As explained in our previous work,12 our ΔfH°298 values selected for these two compounds were based on a critical evaluation of several data from the literature, and are expected to be accurate. Therefore, in Materials Project,76 the ΔfH°0 values for Na2WO4, K2WO4, Na2W2O7 and K2W2O7 may not be very reliable, unless some transitions (unaccounted for) occur over the temperature range 0–298.15 K. For the reactions 2A2WO4 ⇋ A2W2O7 + A2O (with A = Na or K) considered in Table 24, the calculated shifts (ΔrH°298 − ΔrH°076) are not that large (about −64.6 and −23.1 kJ mol−1, respectively) since the corresponding calculated shifts (ΔfH°298 − ΔfH°076) mentioned above for A2W2O7 and A2WO4 partly cancel each other out.

In this work, thermodynamic properties (standard enthalpy at 298.15 K, standard entropy at 298.15 K, and heat capacity as a function of temperature) were selected for all condensed phases of the compounds Na2Cr2O7, K2Cr2O7, Na2Mo2O7, K2Mo2O7, Na2W2O7 and K2W2O7, based on a critical analysis of all available experimental data from the literature. Crystal structure data were also collected for all solid phases. Sometimes, some thermodynamic properties had to be estimated due to the lack of relevant data. A critical evaluation has been conducted previously for the thermodynamic properties of the compounds Na2CrO4, K2CrO4, Na2MoO4, K2MoO4, Na2WO4 and K2WO4.12 This permitted us to conclude that reactions of the type 2A2MO4 ⇋ A2M2O7 + A2O (where A = Na, K and M = Cr, Mo, W) are very limited at least up to the temperature of fusion of A2MO4. As mentioned previously, our selected heat capacities for Na2Cr2O7(I), Na2Cr2O7(II) and K2Cr2O7(II) are directly based on the heat content data of Nguyen-Duy and Dancy,38 which were very doubtful for both Na2Cr2O7(L) and K2Cr2O7(L). It is recommended to perform direct heat capacity measurements or heat content measurements for these three solid phases over relevant temperature ranges.

Evaluating the thermodynamic properties of the A2M2O7 (present work) and A2MO412 compounds is the first step towards the development of a thermodynamic model for the Na+, K+//Cl, SO42−, CO32−, CrO42−, Cr2O72−, MoO42−, Mo2O72−, WO42−, W2O72−, O2− system, which is relevant for high temperature corrosion of equipment such as heat-transfer tubes. This model combined with Gibbs energy minimization software will permit us to perform phase equilibria calculations and investigate high temperature corrosion phenomena (typical temperature range of 600–950 °C).

See the supplementary material for scatter plots for Na2Cr2O7, K2Cr2O7, Na2Mo2O7, K2Mo2O7, Na2W2O7, and K2W2O7.

Ms. Sara Benalia would like to thank the Canada Research Chair in Computational Thermodynamics for High Temperature Sustainable Processes held by Professor Patrice Chartrand. The authors would like to thank Dr. Paul Lafaye and Dr. Aïmen Gheribi for fruitful discussions.

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material.

1.
L.
Antoni
and
A.
Galerie
, “
Corrosion sèche des métaux—Cas industriels: Dépôts, milieux fondus
,” in
Matériaux Métalliques
(Éditions T.I.,
Techniques de L’ingénieur
,
Saint-Denis Cedex
,
2003
).
2.
A.-M.
Huntz
and
B.
Pieraggi
,
Oxydation des Matériaux Métalliques: Comportement à Haute Température
(
Hermes-Lavoisier
,
Paris
,
2003
).
3.
D.
Lindberg
, “
Thermochemistry and melting properties of alkali salt mixtures in black liquor conversion processes
,” in
Åbo Akademi
(
Åbo Akademi University
,
Åbo
,
2007
).
4.
M.
Hazi
, “
Processus d’interaction corrosion/érosion/dépôt dans les enceintes de traitement thermique des déchets
,”
Report No. 03-0223/1A
,
2006
.
5.
J.
Stringer
,
Annu. Rev. Mater. Sci.
7
,
477
509
(
1977
).
6.
J.
Goebel
,
F.
Pettit
, and
G.
Goward
,
Metall. Trans.
4
,
261
278
(
1973
).
7.
J.
Küpper
and
R. A.
Rapp
,
Mater. Corros.
38
,
674
682
(
1987
).
8.
A. K.
Misra
and
C. A.
Stearns
,
Report No. 0499-9320
,
NASA Lewis Research Center
,
Cleveland, OH
,
1983
.
9.
Y.
Niu
,
H.
Tan
, and
S.
Hui
,
Prog. Energy Combust. Sci.
52
,
1
61
(
2016
).
10.
J.
Lehmusto
,
D.
Lindberg
,
P.
Yrjas
,
B.-J.
Skrifvars
, and
M.
Hupa
,
Oxid. Met.
77
,
129
148
(
2012
).
11.
S.
Liu
,
Q.
Chen
, and
P.
Zhang
,
Thermochim. Acta
371
,
7
11
(
2001
).
12.
S.
Benalia
,
F.
Tesfaye
,
D.
Lindberg
,
D.
Sibarani
,
L.
Hupa
,
P.
Chartrand
, and
C.
Robelin
, “
Critical evaluation and calorimetric study of the thermodynamic properties of Na2CrO4, K2CrO4, Na2MoO4, K2MoO4, Na2WO4, and K2WO4
,”
J. Phys. Chem. Ref. Data
52
, 043102 (2023).
13.
N. C.
Panagiotopoulos
and
I. D.
Brown
,
Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem.
29
(
4
),
890
894
(
1973
).
14.
P. L.
Robinson
,
G. E.
Stephenson
, and
H. V. A.
Briscoe
,
J. Chem. Soc. Trans.
127
,
547
549
(
1925
).
15.
N. S.
Afonskii
,
Zh. Neorg. Khim.
7
,
2640
2641
(
1962
).
16.
R. G.
Samuseva
,
I. F.
Poletaev
, and
V. E.
Plyushchev
,
Zh. Neorg. Khim.
7
,
1146
1149
(
1962
).
17.
Y. I.
Vesnin
and
L. A.
Khripin
,
Zh. Neorg. Khim.
11
,
2216
2221
(
1966
).
18.
J.
Jaffray
and
A.
Labary
,
Compt. Rend.
242
,
1421
1422
(
1956
).
19.
G. S.
Rao
,
J. Indian Inst. Sci.
41
,
47
51
(
1959
).
20.
U.
Klement
and
G. M.
Schwab
,
Z. Kristallogr.
114
,
170
199
(
1960
).
21.
E. A.
Kuz’min
,
V. V.
Ilyukhin
, and
N. V.
Belov
,
Dokl. Akad. Nauk SSSR
173
,
1068
1071
(
1967
).
22.
J.
Brandon
and
I.
Brown
,
Can. J. Chem.
46
,
933
941
(
1968
).
23.
L. A.
Zhukova
and
Z. G.
Pinsker
,
Kristallografiya
9
,
44
49
(
1964
).
24.
E.
Mitscherlich
,
Poggend. Ann.
28
,
116
(
1833
).
25.
I.
Lindqvist
et al,
Acta Chem. Scand.
4
,
1066
1074
(
1950
).
26.
K. D.
Singh Mudher
,
M.
Keskar
,
K.
Krishnan
, and
V.
Venugopal
,
J. Alloys Compd.
396
,
275
279
(
2005
).
27.
G.
Saraiva
,
W.
Paraguassu
,
M.
Maczka
,
P.
Freire
,
F.
De Sousa
, and
J.
Mendes Filho
,
J. Raman Spectrosc.
42
,
1114
1119
(
2011
).
28.
M.
Seleborg
et al,
Acta Chem. Scand.
21
,
499
504
(
1967
).
29.
S. A.
Magarill
and
R. F.
Klevtsova
,
Kristallografiya
16
,
742
745
(
1971
).
30.
M.
Amadori
,
Atti Accad. Lincei Rend. Sci. Fis. Mat. Nat.
22
,
609
616
(
1913
).
31.
K.
Okada
,
H.
Morikawa
,
F.
Marumo
, and
S.
Iwai
,
Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem.
31
,
1200
1201
(
1975
).
32.
K.-J.
Range
and
H.
Haase
,
Acta Crystallogr., Sect. C: Cryst. Struct. Commun.
46
,
317
318
(
1990
).
33.
G.
Nolte
and
E.
Kordes
,
Z. Anorg. Allg. Chem.
371
,
149
155
(
1969
).
34.
P. I.
Fedorov
and
M. V.
Mokhosoev
,
Zh. Neorg. Khim.
6
,
123
124
(
1961
).
35.
P. I.
Fedorov
and
M. V.
Mokhosoev
,
Razdelenie Blizkikh po Svoistvam Redkikh Metal
(
State Scientific and Technical Publishing House of Literature on Ferrous and Non-Ferrous Metallurgy, Moscow,
1962
), pp.
224
232
.
36.
J. L.
Holm
, in
Proceedings of the 6th International Conference on Thermal Analysis
(Birkhäuser, Basel,
1980
), Vol.
2
, pp.
105
110
.
37.
A.
Lehrman
,
H.
Selditch
, and
P.
Skell
,
J. Am. Chem. Soc.
58
,
1612
1615
(
1936
).
38.
P.
Nguyen-Duy
and
E. A.
Dancy
,
Thermochim. Acta
39
,
95
102
(
1980
).
39.
K. H.
Breuer
and
W.
Eysel
,
Thermochim. Acta
57
,
317
329
(
1982
).
40.
T.
Nelson
,
C.
Moss
, and
L. G.
Hepler
,
J. Phys. Chem.
64
,
376
377
(
1960
).
41.
C. W.
Bale
,
E.
Bélisle
,
P.
Chartrand
,
S. A.
Decterov
,
G.
Eriksson
,
A. E.
Gheribi
,
K.
Hack
,
I. H.
Jung
,
Y. B.
Kang
,
J.
Melançon
,
A. D.
Pelton
,
S.
Petersen
,
C.
Robelin
,
J.
Sangster
,
P.
Spencer
, and
M. A.
Van Ende
,
Calphad
54
,
35
53
(
2016
).
42.
H. E.
Barner
and
R. V.
Scheuerman
,
Handbook of Thermochemical Data for Compounds and Aqueous Species
(
Wiley
,
New York
,
1978
).
43.
W.
Chase
, Jr.
,
J. Phys. Chem. Ref. Data Monogr.
9
,
1
1951
(
1998
).
44.
I.
Barin
,
Thermochemical Data of Pure Substances
(
VCH
,
Weinheim
,
1989
).
45.
M. W.
Hisham
and
S. W.
Benson
,
J. Phys. Chem.
92
(
21
),
6107
6112
(
1988
).
46.
I.
Dellien
,
F.
Hall
, and
L. G.
Hepler
,
Chem. Rev.
76
,
283
310
(
1976
).
47.
C.
Robelin
,
P.
Chartrand
, and
A. D.
Pelton
,
J. Chem. Thermodyn.
83
,
12
26
(
2015
).
48.
D.
Lindberg
,
R.
Backman
, and
P.
Chartrand
,
J. Chem. Thermodyn.
38
,
1568
1583
(
2006
).
49.
S.
Żemcżużny
,
Z. Anorg. Allg. Chem.
57
,
267
277
(
1908
).
50.
E.
Lax
,
Taschenbuch fuer Chemiker und Physiker-Band 1: Makroskopische Physikalischchemische Eigenschaften
(
Springer
,
Berlin
,
1967
).
51.
J. B.
Bates
,
L. M.
Toth
,
A. S.
Quist
, and
G. E.
Boyd
,
Spectrochim. Acta, Part A
29
,
1585
1600
(
1973
).
52.
E.
Groschuff
,
Z. Anorg. Chem.
58
,
102
112
(
1908
).
53.
H. S.
Roberts
,
Phys. Rev.
23
,
386
(
1924
).
54.
M.
Hassanein
and
E.
Kordes
,
Z. Anorg. Allg. Chem.
381
,
241
246
(
1971
).
55.
D. P.
Nguyen
and
E. A.
Dancy
,
J. Am. Ceram. Soc.
60
,
94
(
1977
).
56.
T.
Sekiya
and
H.
Okuda
,
J. Cryst. Growth
46
,
410
414
(
1979
).
57.
M. M.
Popov
and
V. P.
Kolesov
,
Zh. Obshch. Khim.
26
,
2385
2393
(
1956
).
58.
Z.
Nan
,
X.-Z.
Lan
,
L.-X.
Sun
, and
Z.-C.
Tan
,
Int. J. Therm. Sci.
42
,
657
664
(
2003
).
59.
V. S.
Iyer
,
R.
Agarwal
,
K. N.
Roy
,
S.
Venkateswaran
,
V.
Venugopal
, and
D. D.
Sood
,
Int. J. Therm. Sci.
22
,
439
448
(
1990
).
60.
E.
Groschuff
,
Z. Anorg. Chem.
58
,
113
119
(
1908
).
61.
F.
Hoermann
,
Z. Anorg. Allg. Chem.
177
,
145
186
(
1929
).
62.
P.
Caillet
,
Bull. Soc. Chim. Fr.
12
,
4750
4755
(
1967
).
63.
W. W.
Weller
and
K. K.
Kelley
,
Bureau of Mines Report of Investigations Report No. 6191
,
WA
,
1963
, p.
5
.
64.
O.
Knacke
,
O.
Kubaschewski
, and
K.
Hesselmann
,
Thermochemical Properties of Inorganic Substances
(
Springer-Verlag; Verlag Stahleisen
,
New York; Berlin, NY
,
1991
).
65.
R. D.
Shannon
,
Acta Crystallogr., Sect. A
32
,
751
767
(
1976
).
66.
M. F.
Koehler
,
L. B.
Pankratz
, and
R.
Barany
,
Bureau of Mines Report of Investigations Report No. 5973
,
WA
,
1962
, Vol.
13
.
67.
R. P.
Tangri
,
V.
Venugopal
, and
D. K.
Bose
,
Thermochim. Acta
198
,
259
265
(
1992
).
68.
Y. L.
Suponitskii
,
O. P.
Proshina
, and
M. K.
Karapet’yants
,
Zh. Fiz. Khim.
52
,
2956
2958
(
1978
).
69.
T.
Mathews
,
D.
Krishnamurthy
, and
T.
Gnanasekaran
,
J. Nucl. Mater.
247
,
280
284
(
1997
).
70.
D.
Lindberg
,
R.
Backman
, and
P.
Chartrand
,
J. Chem. Thermodyn.
39
,
1001
1021
(
2007
).
71.
Y. L.
Suponitskiy
,
E. S.
Zolotova
,
A. G.
Dyunin
, and
S. E.
Liashenko
,
Russ. J. Phys. Chem. A
92
,
397
400
(
2018
).
72.
Q.
Chen
,
S.
Liu
, and
P.
Zhang
,
J. Chem. Thermodyn.
31
,
513
519
(
1999
).
73.
D. S.
Kaganyuk
and
V. V.
Trachevskii
,
Ukr. Chem. J.
52
,
1135
1138
(
1986
).
74.
J. E.
Saal
,
S.
Kirklin
,
M.
Aykol
,
B.
Meredig
, and
C.
Wolverton
,
JOM
65
,
1501
1509
(
2013
).
75.
S.
Kirklin
,
J. E.
Saal
,
B.
Meredig
,
A.
Thompson
,
J. W.
Doak
,
M.
Aykol
,
S.
Rühl
, and
C.
Wolverton
,
npj Comput. Mater.
1
,
15010
(
2015
).
76.
A.
Jain
,
S. P.
Ong
,
G.
Hautier
,
W.
Chen
,
W. D.
Richards
,
S.
Dacek
,
S.
Cholia
,
D.
Gunter
,
D.
Skinner
,
G.
Ceder
, and
K. A.
Persson
,
APL Mater.
1
,
011002
(
2013
).
77.
L.
Wang
,
T.
Maxisch
, and
G.
Ceder
,
Phys. Rev. B
73
,
195107
(
2006
).
78.
T.
Hahn
,
U.
Shmueli
, and
J. W.
Arthur
,
International Tables for Crystallography
(
Reidel
,
Dordrecht
,
1983
), Vol.
1
.

Supplementary Material