Solubility of MoO3 in Aqueous Acid Chloride-Bearing Solutions at 573

Apr 27, 2018 - The best stability was demonstrated by solutions of the 0.10 m NaCl + (0–0.20) m HClO4 series, presumably because the oxidative chara...
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Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Solubility of MoO3 in Aqueous Acid Chloride-Bearing Solutions at 573 K Tatiana P. Dadze, Galina A. Kashirtseva, Mikhail P. Novikov, and Andrey V. Plyasunov* Institute of Experimental Mineralogy, Russian Academy of Sciences, ul. Osipyana, 4, 142432 Chernogolovka, Moscow Region, Russia S Supporting Information *

ABSTRACT: This study reports the measurements of the solubility of crystalline molybdenum trioxide, MoO3, in aqueous solutions of the mixture HCl−HClO4−NaCl (up to 1 m of chloride ion) at 573.2 K and pressures close to the saturation water vapor pressure by the autoclave method with the chemical analysis of quenched solutions. In total, there are 75 data points. The data treatment shows that the most probable chloride species of Mo(VI) in the studied solutions is MoO2(OH)2Cl− (the coordinated water is omitted), which agrees with the literature extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES) spectra studies of speciation of Mo(VI) in chloride solutions at elevated temperatures. For the solubility reaction MoO3(cr) + H2O(l) + Cl− = MoO2(OH)2Cl−, log10 K° = −(0.87 ± 0.2). 6.2 m of chloride have been reported.3 According to this work, in basic to near neutral chloride-bearing solutions, tetrahedral complexes containing the molybdate anion (MoO42−) predominate, with no complexation between MoO42− and chloride detected. However, in highly acidic solutions, the Mo(VI) speciation is dominated by distorted octahedral oxo-chloro complexes of a proposed stoichiometry MoO2Cln(H2O)4−n−(n−2). As the temperature rises, the number of chloride ligands increases in 6.21 m HCl solution from ∼1.3 at room temperature to ∼5.1 at 613 K, i.e., with the substitution of oxygen in the inner shell of complexes and the formation of species of the type MoOCl5−. The formation of Mo(VI)−Cl complexes also was expected on the basis of high-temperature solubility studies: Kudrin1 observed a sharp increase in the solubility of MoO2 (the mineral tugarinovite) at 623−723 K and various pressures at HCl molalities above 0.01 m. Our results9 on the solubility of MoO3 (the mineral molybdite) in HCl and HClO4 solutions at 573 K and 10 MPa showed that the effect of both acids is identical up to molalities of ∼0.03 m; however, at higher acid concentrations, the solubility of MoO3 in HCl

1. INTRODUCTION The thermodynamic properties of molybdenum in hightemperature aqueous solutions are of interest for hydrometallurgy and geochemistry. It is known that, over wide ranges of redox conditions, the species of molybdenum in the +6 oxidation state are the most important in aqueous solutions at various temperatures.1−3 Speciation of Mo(VI) in high-temperature dense aqueous solutions was investigated in a number of studies.1−8 Recently, our group9−11 studied the solubility of MoO3 and CaMoO4 in solutions of salts (NaClO4, NaCl) and dilute acids (HCl, HClO4). A combined treatment of our own and literature experimental results allowed us to recommend11 the standard thermodynamic properties of the molybdate ion and products of its monomer hydrolysis (MoO42−, HMoO4−, H2MoO4(aq)) over the temperature range 273.15−623.15 K and at the saturated water vapor pressure. In addition to pH, several anions govern the speciation of metals in aqueous solutions, and among them, the chloride ion is important in both natural and industrial processes. At ambient temperatures, the complexation of Mo(VI) with chloride is studied only fragmentarily: the review article12 cites just a single source13 of corresponding stability constants. At elevated temperatures, up to 658 K, results of the EXAFS (extended Xray absorption fine structure) and XANES (X-ray absorption near-edge structure) study of Mo(VI) solutions containing up to © XXXX American Chemical Society

Received: February 22, 2018 Accepted: April 20, 2018

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water was used to prepare aqueous solutions. The synthesis and characterization of crystals of MoO3 (molybdenum trioxide, the mineral molybdite) were described in detail previously.10 Patterns of X-ray powder diffraction were measured both before and after the experiments using a Bruker D8 Discovery diffractometer with Cu Kα and Co Kα radiation. The powder diffraction patterns of molybdenum trioxide, both before and after experiments, exhibit no peaks other than those for orthorhombic MoO3; see the Supporting Information file for XRD data for the initial phase and samples after experiments. The INCAEnergy EDXS (energy dispersive X-ray spectroscopy) microanalysis system was used for a chemical amalysis of the precipitate; see below. Aqueous solutions of HCl and HClO4 were prepared from commercial reagents volumetrically, NaCl solutionsby adding a weighed amount of a solid salt. Several series of solutions have been prepared to vary the concentrations of H+ and Cl− in test solutions, for example, the HCl series, the

solutions started increasing, while in the HClO4 solutions it continued to decline. No thermodynamic properties have been reported for chloride species of Mo(VI) in either of the discussed publications. The current study attempted to use the solubility method to determine the stoichiometry and Gibbs energies of Mo(VI) chloride complexes at 573 K and 10 MPa.

2. EXPERIMENTAL METHODS 2.1. Materials. All reagents used in this study are listed in Table 1 and were used without further purification. Bidistillate Table 1. Sample Description chemical name

source

mass-fraction purity

purification method

sodium chloride perchloric acid hydrochloric acid

Reakhim Labtekh Khimtitry

>0.998 >0.99 0.998

none none none

Table 2. Experimental Results on the Solubility of MoO3 in HClO4 ± HCl ± NaCl Solutions at 573 K and Pressure ∼10 MPaa no. of run 265 293 294 303 304 305 306 231 232 233 263 264 257 258 259 260 261 262 325 326 704c 705c 706c 713 693 715c 716c 156 204 205 225 226 229 230 157 206 207 227 228 687

m HClO4 (mol kg−1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

m HCl (mol kg−1)

m NaCl (mol kg−1)

log10 m(Mo) (mol kg−1)

weight of a datumb

no. of run

m HClO4 (mol kg−1)

0 0 0 0 0 0.01 0.01 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.025 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.2

0.1 0.1 0.1 0.1 0.1 0 0 0 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 1.0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

−1.50 −1.43 −1.43 −1.46 −1.44 −2.04 −2.04 −2.00 −2.03 −2.03 −2.07 −1.98 −1.61 −1.65 −1.51 −1.51 −1.49 −1.34 −1.39 −1.30 −1.29 −1.34 −1.36 −1.18 −1.64 −1.12 −1.09 −2.05 −1.87 −1.85 −1.98 −2.16 −1.90 −1.89 −1.79 −1.68 −1.68 −1.7 −1.71 −1.89

400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 100 400 400 400 100 100 100 100 0 400 400 100 400 400 400 100 400 400 400 400 400 400 400 100

698 703 707 708 287 289 290 291 292 285 277 278 279 280 281 295 296 699 283 284 700c 694 709 710 701c 695 696 702c 711 297 298 299 300 301 302

0 0 0 0 0.005 0.01 0.01 0.02 0.02 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.05 0.05 0.1 0.1 0.2 0.2

m HCl (mol kg−1) 0.2 0.2 0.2 0.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

m NaCl (mol kg−1)

log10 m(Mo) (mol kg−1)

weight of a datumb

0 0 0 0 0.1 0.1 0.1 0.1 0.1 0 0.03 0.03 0.08 0.08 0.13 0.13 0.13 0.2 0.23 0.23 0.4 0.5 0.5 0.5 0.7 0.75 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1

−1.78 −2.01 −1.67 −2.01 −1.43 −1.47 −1.50 −1.57 −1.61 −1.97 −1.67 −1.68 −1.52 −1.53 −1.46 −1.41 −1.36 −1.54 −1.245 −1.253 −1.31 −1.57 −1.22 −1.17 −1.39 −1.55 −1.57 −1.32 −1.22 −1.60 −1.60 −1.75 −1.78 −1.90 −1.92

100 100 100 100 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 0 400 400 400 100 0 400 400 400 400 400 400 400 400

a

Standard uncertainties u are u(T) = 2 K, ur(P) = 0.10, ur(m) = 0.02, and u(log10(m)) = 0.10. bThe last column of Table 2 gives the weight of a data point used in the data treatment; see below. cNo crystals of solid phase were visible in the container after experiment.

B

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Table 4. Molar Gibbs Energy, G°m, at 573 K, Ps (in kJ mol−1), of Basic Species Used in Modeling

mixtures 0.022 m HClO4 + (0−1.0) m NaCl, 0.025 m HCl + (0− 1.0) m NaCl, and 0.10 m NaCl + (0−0.20) m HClO4. 2.2. Experimental Methods. Experiments were carried out in autoclaves made of the titanium alloy VT-8 and of the internal volume ∼20 cm3. The experimental methods were identical to those of earlier experiments.9,10 Again, the duration of experiments was 17−24 h, which, according to the kinetic series,9 is sufficiently long to reach the steady-state solubility values. At the same time, a rather short exposition period prevented decomposition of the aqueous solutions of perchloric acid. Indeed, according to the study of rates of decomposition of aqueous solutions of perchloric acid by Henderson et al.,14 “...it can be stated that the rate is negligibly small...for the 0.2 m acid at 300°, as far as experiments of about a day’s duration are concerned”. The pressure in autoclaves was determined by the degree of filling of the autoclave. To avoid the vapor phase in the experimental vessel, the degree of filling slightly exceeded the density of solution in equilibrium with the vapor phase at the temperature of experiments, 573 K. The saturated vapor pressure of water at this temperature is ∼8.6 MPa, and the calculated pressures of our experiments are ∼10 MPa. The densities of NaCl solutions at 573 K and 10 MPa are taken from the recommendations of Archer;15 the densities of dilute (0.2 m and less) solutions of HCl and HClO4 solutions have been assumed to be equal to those of NaCl solutions of the same molality. Prior to experiments, autoclaves were loaded with the 20 wt % nitric acid solution with a degree of filling of 0.72 and kept for a day at 573 K to form a protective TiO2 layer. This step was found to be absolutely necessary when working with the aggressive acid solutions used in this study. 2.3. Analytical Methods. The concentration of molybdenum in quenched solutions after the experiment was determined colorimetrically on the spectrophotometer Spekol-11 at λ = 453 nm with the thiocyanate method, based on the formation of yellow color thiocyanate complexes of Mo(V), as described in our earlier publications.9,10 Some quenched solutions were filtered using a paper filter with pore sizes of 3−5 μm; however, it was found that Mo concentrations were practically identical in both filtered and unfiltered samples.

Na

Cl

Mo

O

33.58 ± 1.09 33.25 ± 1.15

36.19 ± 0.73 36.23 ± 0.77

3.36 ± 0.32 3.47 ± 0.35

15.30 ± 1.62 15.37 ± 1.76

ref

species

G°m

ref

H2O(l) H+ OH− Na+ Cl− NaCl(aq)

−263.86 0 −138.70 −282.23 −131.83 −425.91

a a a a a b

HCl(aq) ClO4− MoO42− HMoO4− H2MoO4(aq) MoO3(cr)

−144.66 −57.02 −826.25 −915.02 −934.35 −696.64

c a d d d e

The database Unitherm, built into the HCh program17,18 based on the HKF-model.20,21 bCalculated using log10 K° = 1.0822 for the reaction Na+ + Cl− = NaCl(aq). cHKF-model parameters taken from http://geopig.asu.edu/sites/default/files/slop15_0.dat. dDadze et al.11 e The database Unitherm, built into the HCh program,17,18 based on the handbook by Glushko et al.23

Table 5. Goodness of Fit Corresponding to Each Form under Consideration, Given as MSE (See eq 1) species

MSE

MoO2Cl+ MoO2Cl2(aq) MoO2(OH)OCl2− MoO2(OH)Cl(aq) MoO2(OH)2Cl−

0.230 0.190 0.177 0.180 0.123

quenched solutions with the initial concentration of HCl above 0.2 m contained a black suspension of titanium dioxide, indicating the corrosion of walls of the autoclaves with dissolution of the protective layer of TiO2. Such solutions showed no yellow coloring when prepared for the photometric analysis, probably due to coprecipitation of dissolved molybdenum. In a few cases, the quenched solutions had intensely blue color, indicating the formation of “molybdenum blue”, i.e., polymer complexes containing both Mo(VI) and its reduced forms. An addition of hydrogen peroxide to test solutions before runs did not improve the stability of quenched solutions. Data of runs with the formation of visible precipitates or “molybdenum blue” were rejected and not given in Table 2. The best stability was demonstrated by solutions of the 0.10 m NaCl + (0−0.20) m HClO4 series, presumably because the oxidative character of perchloric acid solutions preserved the integrity of the protective TiO2 layer. Solubility data in 0.2 m HCl solutions, as well as in all solutions containing more than 0.5 m NaCl, whether with dilute HCl or HClO4 solutions, showed poor reproducibility, partly because of possible uncontrolled crystallization during the filtration of quenched solutions. Although these solutions have been transparent, one cannot exclude the formation of small amounts of precipitates. We expect that, for such solutions, results with the highest measured Mo concentrations are the most reliable. In a few cases, marked by the superscript “c” in Table 2 below, no solid phase’s crystals were visible in the container after experiment. They could be displaced at the quenching and opening procedures, or the runs (with a typical weight of a sample of 100 mg) could be undersaturated; in the latter case, the “true” solubilities could be higher. Results of experimental runs are given in Table 2. In a single case (run no. 715, solubility of MoO3 in 0.025 m HCl + 1 m NaCl), the transparent quenched solution was filtered, a sample of 1 mL was taken for photometrical analysis, and the rest was slowly evaporated. The precipitate was analyzed with the INCAEnergy EDXS (energy dispersive X-ray spectros-

Table 3. Chemical Analysis of the Precipitate of Run No. 715, in Atomic % no. 1 no. 2

G°m

a

3. RESULTS AND DISCUSSION 3.1. Solubility Results for MoO3. The experimental solubility data obtained in this study are presented in Table 2

element

species

in molality units, where molality m is defined as the number of moles of a substance per 1 kg of water. The uncertainty of most experimental results is estimated to be within 0.1 log10 units. A specific issue of the current series of experimental runs is the stability of quenched solutions. In our previous studies of the solubility of MoO3 at 573 K and 10 MPa in HCl (up to 0.10 m), HClO4 (up to 0.22 m),9 and NaClO4 solutions (up to 2.21 m),10 the maximum Mo concentration was 0.036 m. All of those solutions, despite high concentrations of molybdenum, remained completely transparent and stable for at least up to a week. This was not the case in the current study. It was found that all C

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Figure 1. Experimental (symbols: black circles, this work; open triangles, Dadze et al.9) and calculated (curves) values of MoO3 solubility in acid chloride solutions of the series 0.10 m NaCl + m HClO4 (a), 0.025 m HCl + m NaCl (b), 0.022 m HClO4 + m NaCl (c), and HCl (d), assuming the dominance of different Mo(VI) forms; see text.

= 1.218 (kg mol−1)0.5). Although all experiments were carried out at pressures of ∼10 MPa, slightly exceeding the vapor pressure of water at a given temperature (8.6 MPa) in order to avoid the vapor phase, the thermodynamic calculations were carried out at the saturated water vapor pressure. In turn, the calculation of the equilibrium composition of the solution in the OptimA routine is performed. The list of aqueous species, specifying the studied system, includes H+, Cl−, OH−, ClO4−, HCl(aq), Na+, NaCl(aq), MoO42−, HMoO4−, and H2MoO4(aq), and their thermodynamic properties (the molar Gibbs energy, G°m) are given in Table 4. The data treatment was carried out assuming that HClO4 is a fully dissociated acid. This assumption is based on the Raman spectroscopic study of its solutions at temperatures up to 573 K by Ratcliffe and Irish,19 who concluded that perchloric acid (unlike nitric and sulfuric acids) remains a “strong” acid even at temperatures up to 573 K. The possible association between Na+ and ClO4− was not considered due to lack of corresponding data; however, the contribution of this ionic pair is expected to be small based on a known example of sodium chloride solutions. In order to employ the OptimA program, it is necessary to specify the stoichiometry of a species, whose Gibbs energy this program will optimize. As was shown earlier,11 at this T and P, the neutral species, MoO2(OH)2, dominates in acid molybdenumbearing chloride-free solutions. We considered a number of likely stoichiometries of chloride forms of Mo(VI), which could be formed by an interaction of Cl− or H+ + Cl− with the neutral species MoO2(OH)2. The list of possible forms included MoO2Cl+, MoO2Cl2(aq), MoO2(OH)Cl(aq), MoO2(OH)2Cl−, and MoO2(OH)OCl2−. Additional forms with two or more chlorides were not considered, because the analysis of the precipitate in the 0.025 m HCl + 1 m NaCl solution (see above) showed that the Mo:Cl ratio in the formed molybdenum

copy) microanalysis system; see Table 3 with results of two parallel analyses. Assuming that the excess of Cl compared with Na (introduced as NaCl) is associated with dissolved Mo, we found that the Mo:Cl ratio for the Mo species in solution was 1.2, with the expected error up to 30% of the value. 3.2. Treatment of Solubility Results. The data analysis was carried out using the program OptimA,16 which processes experimental solubility data to derive the standard Gibbs energies of specified aqueous complexes by minimizing the mean squared error (MSE), defined as n

MSE =

∑ wi(Yexp,i − Ycalc,i)2 i=1

(1)

where n is the number of experimental data points, Y is the decimal logarithm of the solubility value, and w is the weight of an experimental data point. (If the expected error of the decimal logarithm of solubility value is 0.05 log10 units, then w = 1/0.052 = 400. The weight of 0, see Table 2, is assigned to outliers, strongly deviating from results of parallel experiments.) The calculated Y values are obtained by computing the equilibrium composition of the solution by minimizing the total Gibbs energy of the system with the HCh program.17,18 The values of the activity coefficients of the ions are evaluated with the Debye−Hückel equation in the form given by eq 2 log10 γi = −

Zi 2AI 0.5 1 + 1.5I 0.5

(2)

where Zi is the charge of an ion i, I stands for the ionic strength of a solution, and A is the Debye−Hückel parameter, defined by the theory and calculated using the dielectric and thermophysical properties of water in the OptimA program (at 573.2 K and Ps, A D

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aqueous species was around 1. In each case, we assumed that only one chloride form of Mo exists in the solution. Results of modeling are presented in Table 5 and in Figure 1. As follows from Table 5, the chemical model with MoO2(OH)2Cl− provides the minimal value of MSE, i.e., the best description of experimental data. Results in Figure 1 generally support this form, especially taking into account that for the case of unstable quenched solutions, characterized by a range of measured Mo concentrations in parallel runs, results with the highest measured Mo concentrations are expected to be the most reliable. For the solubility reaction MoO3(cr) + H 2O(l) + Cl− = MoO2 (OH)2 Cl−

AUTHOR INFORMATION

Corresponding Author

*Phone: +7(49652)44425. Fax: +7(49652)49687. E-mail: [email protected]. ORCID

Andrey V. Plyasunov: 0000-0001-6786-029X Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

(3)

This research was partially supported by the Russian Foundation for Basic Research (Grant No. 15-05-2255).

we calculated log10 K° = −0.87 with an estimated uncertainty of about 0.2 log10 units. 3.3. Comparison with X-ray Data.3 As was discussed in the Introduction, Borg et al.3 used the EXAFS (extended X-ray absorption fine structure) and XANES (X-ray absorption nearedge structure) methods to study Mo(VI)-bearing acid chloride solutions at elevated temperatures, up to 658 K. These authors have concluded that the Mo(VI) speciation is dominated by the distorted octahedral oxo−chloro complexes of a proposed stoichiometry MoO2Cln(H2O)4−n−(n−2). At high chloride concentrations, around 6 m, chloride may substitute both water and oxygen in the inner shell of complexes with the formation of species of the type MoOCl5−. Importantly, the authors indicated that “the protonation has no detectable effect on the EXAFS and XANES spectra”;3 i.e., the presence of a OH group instead of H2O molecules in the coordination sphere of Mo(VI) could not be detected. Our11 and literature8 data show that under acidic conditions at small chloride concentrations the neutral Mo form MoO2(OH)2 dominates (ignoring the coordinated water); therefore, we think that MoO2(OH)2Cln(H2O)2−n−n is a more likely stoichiometry of a complex species in molybdenumbearing acid solutions with small to moderate chloride concentrations. In this case, the results of our solubility study are in agreement with the conclusions of the X-ray investigation.3

Notes

The authors declare no competing financial interest.

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REFERENCES

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4. CONCLUSIONS We studied the solubility of molybdenum trioxide, MoO3, in the HCl−HClO4−NaCl solutions (up to 1 m of chloride ion) at 573.2 K and pressures close to the saturated water vapor pressure by the autoclave method with the chemical analysis of quenched solutions. In total, there are 75 data points. Solubility data in 0.2 m HCl solutions, as well as in all solutions containing more than 0.5 m NaCl, whether with dilute HCl or HClO4 solutions, showed poor reproducibility, likely due to instability of quenched solutions. The analysis of the solubility data showed that the most probable chloride species of Mo(VI) in the studied solutions is MoO2(OH)2Cl− (the coordinated water is omitted), which agrees with literature EXAFS and XANES spectra studies of speciation of Mo(VI) in chloride solutions at elevated temperatures.3 For the solubility reaction MoO3(cr) + H2O(l) + Cl− = MoO2(OH)2Cl−, log10 K° = −(0.87 ± 0.2).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.8b00151. X-ray powder diffraction data for the initial MoO3 phase and samples from containers after the experiments (PDF) E

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potentiometry and spectroscopy measurements. Appl. Geochem. 2015, 55, 17−27. (17) Shvarov, Yu. V. Algorithmization of the numeric equilibrium modeling of dynamic geochemical processes. Geochem. Int. 1999, 37, 571−576. (18) Shvarov, Yu. V. HCh: new potentialities for the thermodynamic simulation of geochemical systems offered by Windows. Geochem. Int. 2008, 46, 834−839. (19) Ratcliffe, C. I.; Irish, D. E. Vibrational spectral studies of solutions at elevated temperatures and pressures. VI. Raman studies of perchloric acid. Can. J. Chem. 1984, 62, 1134−1144. (20) Tanger, J. C., IV; Helgeson, H. Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: revised equations of state for the standard partial molal properties of ions and electrolytes. Am. J. Sci. 1988, 288, 19−98. (21) Shock, E. L.; Helgeson, H. C. Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: correlation algorithms for ionic species and equation of state predictions to 5 kb and 1000 °C. Geochim. Cosmochim. Acta 1988, 52, 2009−2036. (22) Zimmerman, G. H.; Arcis, H.; Tremaine, P. R. Limiting conductivities and ion association constants of aqueous NaCl under hydrothermal conditions: experimental data and correlations. J. Chem. Eng. Data 2012, 57, 2415−2429. (23) Glushko, V. P.; Gurvich, L. V.; Bergman, G. A.; Veits, I. V.; Medvedev, V. A.; Khachkuruzov, G. A.; Yungman, V. S. Thermodynamic properties of individual substances, Vol. 4; Nauka: Moscow, 1982 (in Russian).

F

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