Villiaumite, NaF, solubility

Villiaumite dissolution is a simple dissociation reaction:

(1)$${\rm Na}{\rm F}_{( {{\rm cr}} ) }{\rm \;}\to {\rm Na}_{( {{\rm aq}} ) }^ + { + } {\rm \;F}_{( {{\rm aq}} ) }^- $$

Twenty-two solubility measurements of NaF from 18 reports have been compiled by Reynolds and Belsher (Reference Reynolds and Belsher2017) for 25°C. They screened outliers using statistical methods and concluded the best value was 0.987 m. Prior to their work the solubility was usually based on Ivett and Vries (Reference Ivett and Vries1941) who reported saturation at 0.983 m and was earlier reported by Payne (Reference Payne1937) with the same value. The recommended value from Reynolds and Belsher is S = 0.987 m and their standard error is ±0.004. With reliable mean activity coefficient data, the logK_sp and Δ_rG° can be calculated (Nordstrom and Munoz, Reference Nordstrom and Munoz1994). A summary of the solubility studies is tabulated in Table 2 along with mean activity coefficient data.

Table 2. Villiaumite solubilities and mean activity coefficients reported at 25°C.

Seven mean activity coefficients of NaF solutions at saturation and 25°C were found in the literature of which four were primary measurements. The values vary between 0.569 (Hernández-Luis et al., Reference Hernández-Luis, Galleguillos and Vázquez2004) and 0.6211 (Aghie and Samaie, Reference Aghie and Samaie2006). Hamer and Wu (Reference Hamer and Wu1972) regressed the data of Ivett and Vries (Reference Ivett and Vries1941) and Robinson (Reference Robinson1941) to obtain their value of 0.574. If we eliminate the data of Ivett and Vries (Reference Ivett and Vries1941) and Aghamie and Samaie (Reference Aghie and Samaie2006) because they are outliers and take the mean value among the remaining four (Robinson, Reference Robinson1941; Taghikhani et al., Reference Taghikhani, Modarress and Vera2000; Hernández-Luis et al., Reference Hernández-Luis, Galleguillos and Vázquez2004; Faridi and Guendouzzi, Reference Faridi and Guendouzzi2015), the result is 0.571 ± 0.002. The activity is then 0.5636 ± 0.004 and the logK_sp = –0.4981 ± 0.006. For dissolution, Δ_rG° = 2.843 ± 0.034 kJ/mol from equation 2. These results are often more precise and accurate than what can be achieved from calorimetry.

(2)$$\log K_{{\rm sp}} = -\displaystyle{{{\rm \Delta }_{\rm r}G^o} \over {2.303RT}}$$

In the CODATA book on key thermodynamic values, Cox et al. (Reference Cox, Wagman and Medvedev1989) chose the solubility value from Payne (Reference Payne1937) and the mean activity coefficient from Hamer and Wu (Reference Hamer and Wu1972). These were probably the best values available at that time and nearly identical to the values recommended here. Their value for Δ_rG° = 2.837 ± 0.050 transforms to logK _sp = –0.4969 ± 0.0088.

Villiaumite solubility from calorimetric data

Villiaumite solubility can also be calculated from the Gibbs energies of formation for the species designated in reaction 1 using calorimetric data and equation 2. There is an important caveat attached to using equation 2. The Gibbs energies of reactants and products are always much larger in absolute magnitude than the Gibbs energy of the reaction. Consequently, the Gibbs energy of the reaction is usually the difference between two large numbers, and it gets overwhelmed by the cumulative errors. For accurate and precise data, any logK _sp value based on calorimetric data should always be supported by solubility data and solubility data can be used to better constrain calorimetric data for the same system.

Using the data compiled in Table 3, we find that the logK _sp is either –0.3645 or +0.0701 depending on whether one chooses the Δ_fG°(NaF) value of −545.081 kJ/mol from Chase (Reference Chase1998) or –543.494 kJ/mol from Wagman et al. (Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982), respectively. The same value from Wagman et al. (Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982) was also used in Robie and Hemingway (Reference Robie and Hemingway1995). For these values, the errors listed are reflecting something of the precision of the measurements, not the accuracy because these Gibbs energies do not overlap when errors are considered.

Table 3. Thermodynamic properties for villiaumite dissolution at 25°C.

If one uses all the Δ_fG° values (ions and solid) from Wagman et al. (Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982) for calculating Δ_rG° for the dissolution reaction, the results are very similar to those obtained from solubility and isopiestic data (Table 3). Both the Gibbs energies of the ion and the solid phase are offset by the same amount from the more recent Gibbs energies from CODATA and Chase (Reference Chase1998). These differences appear to be related to the older references for the Wagman NBS tables (Wagman, Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982) and the NBS Technical Note series that preceded it (prepared 1968). It only takes a small error in the calorimetric data to change substantially the logK _sp value. The combined uncertainty in the Gibbs energy values for the ions, Na⁺ and F^–, is quite small, only about ±0.7, mostly from the F^– ion. These uncertainties in ion values should not be the source of the discrepancy because it would call into question several heat measurements on fluoride (and sodium) solutions that are in very close agreement (Cox et al., Reference Cox, Wagman and Medvedev1989). For the solid phase, a consistency check was performed to confirm that the Δ_fG°(NaF) could be calculated from

$${\rm \Delta }_{\rm r}G^\circ ( {{\rm NaF}} ) = {\rm \Delta }_{\rm r}H^\circ ( {{\rm NaF}} ) -T{\rm \Delta }_{\rm r}S^\circ ( {{\rm NaF}} ) $$

for the data of Chase (Reference Chase1998). The result was indeed confirmed. It should be because the data in Chase (Reference Chase1998) had undergone Second Law and Third Law methods of evaluation.

The next step is to consider the sources of data used by Chase (Reference Chase1998) and by Wagman et al. (Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982). Unfortunately, the NBS tables do not have references, but we do know that the references are older than 1968. The NIST–JANAF tables (Chase, Reference Chase1998) have shown nine references from which heat measurements can be used to obtain the Δ_fH°(NaF). The measurements were reported between 1884 and 1965. These were in remarkably precise agreement giving the reported value of 575.384 ± 0.8 kJ/mol which is used in this study as more reliable than other sources.

The value of 2.843 ± 0.034 kJ/mol for the Δ_rG° of the solubility equilibrium is considered more accurate and that leaves the matter of clearing up the discrepancy between the derivation of the Δ_rG° from the solubility determination and from the derivation based on calorimetry. Because the Δ_fH°(NaF) is the largest contributor to the value of Δ_fG°(NaF) and seems highly reliable from the NIST–JANAF analysis, the thermodynamic properties of the ions need to be examined.

The Δ_fG°(Na⁺_(aq)) in Table 3 is shown from several sources, but the most direct and careful measurement is reported from Lewis and Kraus (Reference Lewis and Kraus1910) and reproduced in Lewis and Randall (Reference Lewis and Randall1923) based on measurements of the sodium electrode for the reaction:

$${\rm N}{\rm a}_{( {\rm s} ) }{\rm \;}\to {\rm Na}_{( {{\rm aq}} ) }^ + { + } \;{\rm e}^-$$

which were accomplished with a sodium and a sodium amalgam electrode and measurements of the half-cell potential of the amalgam electrode. The uncertainty on that potential is considered to be 1 millivolt in the potential measurement (Lewis and Kraus, Reference Lewis and Kraus1910; Latimer Reference Latimer1952), which leads to an uncertainty of only ±0.15 kJ/mol for the Δ_fG°(Na⁺_aq) and cannot be a major source of uncertainty for the villiaumite dissolution reaction. Hence, we are left with the Δ_fG°(F^–_(aq)) as the source of the greater uncertainty. There is a 2.7 kJ/mol difference between the value reported by CODATA and the value reported by the NBS tables. Because the two sources of data agree on the entropy and the entropy has little effect on the Gibbs energy value, the Δ_fG°(F^–_(aq)) must be examined.

Using the Δ_rG° from solubility and mean activity coefficient data along with the Δ_fG°(NaF) from Chase (Reference Chase1998) and the Δ_fG°(Na⁺_(aq)) from Lewis and Randall (Reference Lewis and Randall1923), the result is

$$\Delta _{\rm f}G^\circ ( {{\rm F}_{( {{\rm aq}} ) }^- } ) = \Delta _{\rm r}G^\circ{ + } \Delta _{\rm f}G^\circ ( {{\rm NaF}} ) -\Delta _{\rm f}G^\circ ( {{\rm Na}_{( {\rm aq} ) }^ + } ) = -280.518\;{\rm kJ}/{\rm mol}$$

The Δ_fH°(F^–_(aq)) can be calculated from

$$\Delta _{\rm f}H^\circ ( {{\rm F}_{( {{\rm aq}} ) }^- } ) = {\rm \;}\Delta _{\rm f}G^\circ ( {{\rm F}_{( {{\rm aq}} ) }^- } ) -{\rm T}\Delta _{\rm f}S^\circ ( {{\rm F}_{( {{\rm aq}} ) }^- } ) = {-}334.346\,\,{\rm kJ}/{\rm mol}$$

which agrees quite closely with the experimental value of –334.05 + 0.34/–0.69 kJ/mol reported by Nuttall et al. (Reference Nuttall, Churney and Kilday1978). A consistency check shows that the data from this study in Table 3 results in a Δ_rH° = 0.891 ± 0.034 kJ/mol for NaF dissolution which compares well with data compiled by Cox et al. (Reference Cox, Wagman and Medvedev1989) who report experimental values of 0.8 to 0.99 kJ/mol (after correction to infinite dilution). It also demonstrates that it is not possible to maintain consistency between the data evaluation of Chase (Reference Chase1998) and that of Cox et al. (Reference Cox, Wagman and Medvedev1989) without the modifications shown in this study, even though internal consistency is maintained within each evaluation.

Cryolite, Na₃AlF₆, solubility

Only six reports on cryolite solubility have been found in the published literature. The first was that of Frere (Reference Frere1936) in which cryolite solubility was measured with variable concentrations of Al and Fe salts at 25°C. One measurement in pure water was reported but the concentration was an order of magnitude too high and a correction was published later by Mockrin (Reference Mockrin1950) to be 0.390 g/kg_solution. The next study (Buchwald, Reference Buchwald1939) reported a concentration of 0.391 g/L_solution at 25°C. The study by Tananaev and Talipov (Reference Tananaev and Talipov1939) reported 0.4175 g/kg_solution at 25°C and was reported again by Tananaev and Vinogradova (Reference Tananaev and Vinogradova1957). A dissertation by A. F. Köhl (Reference Köhl1926) reported a solubility similar to that of Buchwald (Reference Buchwald1939) and Tananaev and Talipov (Reference Tananaev and Talipov1939). Solubilities at other than 25°C were reported by Buchwald (Reference Buchwald1939) and Tananaev and Talipov (Reference Tananaev and Talipov1939) but are not listed in Table 4.

Table 4. Reported solubilities of cryolite, Na₃AlF₆, at 25°C with references.

Using the PHREEQC code (v.3.4.0) with molar volume data to calculate density, the solubility results from Frere (Reference Frere1936), Buchwald (Reference Buchwald1939) and Tananaev and Talipov (Reference Tananaev and Talipov1939) gave pH values close to 7 and logK _sp values very close to –34.0 (Table 5) which are remarkably consistent with the more carefully done studies by Roberson and Hem (Reference Hem1968 and Reference Roberson and Hem1973).

Table 5. Comparison of logK _sp values. * Indicate values chosen for averaging. Roman numerals refer to the experimental run number listed in Roberson and Hem (Reference Roberson and Hem1968). Arabic numerals refer to Method 1 or 2 and are explained in the text for PHREEQC recalculations of their data.

Using sodium and fluoride ion-selective electrodes, Roberson and Hem (Reference Roberson and Hem1968) measured sodium and fluoride ion activities, dissolved fluoride was also measured colorimetrically, together with dissolved sodium and aluminium by atomic absorption spectrometry. They made five determinations of cryolite solubility with variable pH, aluminium and fluoride concentrations. They reported that one of the five determinations was an outlier and averaged the other four to find logK _sp = –33.84 ± 0.1. In agreement with the authors, the outlier is excluded here as well in averaging values, not just because it was an outlier but because it was a solution with the highest pH (8.46) and may have had interferences from hydroxidofluorido complexes in solution, or on surfaces, or Al(OH)₃ as a surface phase. This solution (number III) was much closer to amorphous gibbsite solubility equilibrium than any of the other samples.

Because they mixed reagents in different proportions, added perchlorate as an inert electrolyte, and covered a range of pH, they did not measure the simple solubility in pure water. Their solutions were made to be supersaturated and allowed to age for 8 to 15 months. Calculation of activities were obtained without the benefit of a geochemical code so that their data was rerun for speciation in the present study using PHREEQC to check charge balances and to corroborate their results with slightly different complex stability constants and activity coefficients. The ionic strengths of the solutions were all very close to 0.11 m so that activity coefficient variations were not a concern. There are two ways of doing this test. One way is to take their reported concentration values and recalculate the speciation and the logK _sp, (Method 1). The other way is to accept their measured sodium and fluoride activities, which were the original measurements, and change the sodium and fluoride concentrations until the PHREEQC calculated activities matched those reported (Method 2). Both methods were employed, and the results are compiled in the Supplementary Appendix. When optimising for two ion activities, an exact match was usually not possible, in which case the fluoride free-ion activity took precedent over the sodium free-ion activity because it has proven to be a more accurate and precise method.

In a later study, Roberson and Hem (Reference Roberson and Hem1973) used natural cryolite from Greenland to determine the solubility from undersaturation. The same analytical and calculational techniques were used as in their earlier study with similar ageing periods. For this study, the solubility in pure water was measured for six runs of ageing times from 8 to 17 months. Their results were also run through PHREEQC in the present study to check charge balances and to confirm their interpretation. Only the concentration data for Na, F and Al were used because ion-selective electrode activities were not available for all samples. After recalculating all experimental values from six reports, minus outliers, the grand mean for the logK _sp from undersaturation and oversaturation is –33.90 ± 0.19, which converts to a Δ_rG° of 193.524 ± 1 kJ/mol.

Fluorite, CaF₂, solubility

Numerous solubility measurements of fluorite solubility are found in the literature and those measured at or close to 25°C are listed in Table 6 along with the logK _sp values, both reported and calculated from PHREEQC when the solubility value is given.

Table 6. Reported values for fluorite solubility in pure water at or near 25°C. References marked with an ‘x’ were identified as outliers and excluded from averaging. ‘n’ refers to the number of measurements.

¹ Average between two methods of measuring the fluoride concentration.

Excluding the same outliers (‘x’ in Table 6) that were excluded in the CODATA review, but updating with some of the new measurements, we have averaged the solubility determinations and, separately, the solubility product constants because not all of the solubility data is available to derive solubility products. Where there is a difference between the solubility products calculated from PHREEQC and those reported, we have chosen the PHREEQC results for consistency. Particular weight has been given to the data of Gardner and Nancollas (Reference Gardner and Nancollas1976) because they obtained both dissolution and precipitation rate data, they determined both dissolved Ca and dissolved F in congruent concentrations, they determined the heat of precipitation, and they obtained a large data set of precise values. The data points were not tabulated, but they could be read from the graph as an average between the dissolution curve and the precipitation curve as they nearly plateaued to a constant value. Additional weight was also afforded the results of Knowles-Van Cappellen et al. (Reference Knowles-Van Cappellen, Van Cappellen and Tiller1997) because that was also a kinetic growth study of fluorite in which they monitored the reaction with Ca and F ion-selective electrodes and obtained identical results as Gardner and Nancollas (Reference Gardner and Nancollas1976) for the equilibrium value.

From 12 reports of fluorite solubility, the mean is 16.0 ± 0.9 mg/L, which converts to a logK of –10.54 ± 0.08. From 18 reports of logK, the mean is –10.57 ± 0.08. Taking into consideration the dissolution and precipitation rate studies of Gardner and Nancollas (Reference Gardner and Nancollas1976) and Knowles-Van Cappellen et al. (Reference Knowles-Van Cappellen, Van Cappellen and Tiller1997), and the large number of measurements by Henry (Reference Henry2018), the mean from these three sources is –10.57. Consequently, this study recommends a logK = –10.57 ± 0.08 and a Δ_rG° = 60.34 ± 0.45 kJ/mol based on solubility data.

Fluorite solubility from calorimetric data

For the solubility reaction,

$${\rm Ca}{\rm F}_{{\rm 2( cr) }}{\rm \;}\to {\rm \;Ca}_{{\rm ( aq) }}^{{\rm 2\ + \ }} {\rm} + {\rm 2F}_{{\rm ( aq) }}^ \hbox{-} $$

we need the Gibbs energies of the three species, solid fluorite and the Ca²⁺ and F^– aqueous ions. The largest source of uncertainty is the Gibbs energy of the solid phase. Thermodynamic evaluations of fluorite properties are listed in Table 7.

Table 7. Enthalpy of formation, entropy, heat capacity and Gibbs energy of formation for fluorite from major compilations.

The fluorite enthalpies and Gibbs energies of formation from CODATA (Garvin et al., Reference Garvin, Parker and White1987) and Robie and Hemingway (Reference Robie and Hemingway1995) are nearly identical because Robie and Hemingway (Reference Robie and Hemingway1995) took their values from the NBS tables (Wagman et al., Reference Wagman, Evans, Parker, Schuman, Halow, Bailey, Churney and Nuttall1982) which was the same source as those from CODATA. Also, the CODATA evaluation considered more than 15 solubility measurements and averaged them after rejecting several other measurements that were outside a reasonable range of values, some that were too high and others that were too low. By comparison, Chase (Reference Chase1998) based their Gibbs energy for fluorite on only two averaged solubility measurements, those from Smyshlyaev and Edeleva (Reference Smyshlyaev and Edeleva1962) and Kohlrausch (Reference Kohlrausch1908). Those are among the more reliable solubility measurements and in close agreement, however, they then combined those data with Gibbs energies for the ions from an unpublished evaluation by V. Parker for Δ_fG°(Ca²⁺) and a value for Δ_fG°(F^–) consistent with Δ_fG°(HF) dissolution data. For some reason, the CODATA key values for the ions (Cox et al., Reference Cox, Wagman and Medvedev1989) were not used. The consistency is rather good with only a 2 kJ/mol difference between the two sets of values, but 2 kJ/mol can propagate to a large uncertainty in the logK _sp. The same 2 kJ/mol difference is seen in the enthalpy values. Because the solubility reaction has a lower uncertainty than that for solid fluorite, it would be better to derive the Δ_fG°(CaF₂) from solubility and aqueous ion data.

The standard state thermodynamic values for F^– have already been addressed. The most direct and potentially the most reliable value for Δ_fG°(Ca²⁺) would be from measurements of the calcium metal (amalgam) electrode. Unfortunately, there have been challenging problems with the electrode, mostly from preventing oxidation and showing that it is reversible. These concerns have been described by Butler (Reference Butler1968) who was able to show reversibility for the Ca amalgam electrode and evaluating his results along with research reports concluded that the standard potential, E° = 1.996 ± 0.002 V. He was not able to obtain a quantitative measurement to deduce Δ_fG°(Ca²⁺), but he did indicate a semi-quantitative value that was in agreement with the estimate of –2.87 V from Latimer (Reference Latimer1952) and he cited the work of Jakusewski and Taniewska-Osinska (Reference Jakusewski and Taniewska-Osinska1962) whose measurements on the Pb–PbCO₃–CaCO₃–CaCl₂ cell with a calomel reference gave a value of −2.868 V. This latter value converts to –553.43 kJ/mol for Δ_fG°(Ca²⁺) and almost exactly coincides with estimates solely based on solubility, enthalpies of reaction, and entropies from CODATA (Δ_fG°(Ca²⁺) = –552.8 ± 1.1 kJ/mol). In the present evaluation, a median value of –553.0 ± 0.5 kJ/mol will be used and recommended. Finally, using the recommended value for the Gibbs energy of the dissolution reaction and solving for the Gibbs energy of formation of fluorite, Δ_fG°(CaF₂) = –1174.38 kJ/mol. This value falls conveniently midway between the values recommended by CODATA and those recommended in the NIST–JANAF tables, 4^th edition (Chase, Reference Chase1998). Using this Gibbs energy and the entropies of fluorite and the elements from CODATA, the Δ_fH°(CaF₂) becomes –1227.097 kJ/mol which is also midway between the enthalpy of formation values recommended by CODATA and by NIST–JANAF. This result gives further confidence that this network of minerals and aqueous species are internally consistent.