Theoretical predictions of hydrolysis and complex formation of group-4 elements Zr, Hf and Rf in HF and HCl solutions

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1 Radiochim. Acta 90, (2002) by Oldenbourg Wissenschaftsverlag, München Theoretical predictions of hydrolysis and complex formation of group-4 elements Zr, Hf and Rf in HF and HCl solutions By V. Pershina 1,, D. Trubert 2, C. Le Naour 2 andj.v.kratz 3 1 Gesellschaft für Schwerionenforschung, D Darmstadt, Germany 2 Institut de Physique Nucléare, F Orsay, France 3 Institut für Kernchemie, Universität Mainz, D Mainz, Germany (Received March 13, 2002; accepted in revised form April 12, 2002) Element 104 / Hydrolysis / Complex formation / Relativistic molecular calculations Summary. Fully relativistic molecular density-functional calculations of the electronic structures of hydrated, hydrolyzed and fluoride/chloride complexes have been performed for group-4 elements Zr, Hf, and element 104, Rf. Using the electronic density distribution data, relative values of the free energy change for hydrolysis and complex formation reactions were defined. The results show the following trend for the first hydrolysis step of the cationic species: Zr > Hf > Rf in agreement with experiments. For the complex formation in HF solutions, the trend to a decrease from Zr to Hf is continued with Rf, provided no hydrolysis takes place. At ph > 0, further fluorination of hydrolyzed species or fluorocomplexes has an inversed trend in the group Rf Zr > Hf, with the difference between the elements being very small. For the complex formation in HCl solutions, the trend is continued with Rf, so that Zr > Hf > Rf independently of ph. A decisive energetic factor in hydrolysis or complex formation processes proved to be a predominant electrostatic metal-ligand interaction. Trends in the K d (distribution coefficient) values for the group-4 elements are expected to follow those of the complex formation. 1. Introduction One of the fundamental properties of transition elements is the complex formation. It is known to increase in the transition elements groups with increasing atomic number. The study of the complex formation of the heaviest elements aims at finding out weather this trend is continued with the transactinides. Experimentally, the complex formation is studied by solvent extraction or anion exchange separations. Extraction sequences, or sequences in the distribution coefficient (K d ) values, provide an information about stabilities of the extracted complexes or a degree of hydrolysis: the complex formation of the transition elements is known to be influenced by strong hydrolysis. Experiments on the anion exchange separations of elements 104, 105, and 106 and their lighter homologs in the *Author for correspondence ( V.Pershina@gsi.de). groups, Zr/Hf, Nb/Ta, and Mo/W, respectively, from aqueous acidic solutions have accumulated interesting data on the behaviour of the transactinides in comparison with those of the lighter elements (for reviews, see [1, 2]). In some cases, trends in the complex formation and extraction were found to be continued with the transactinides, like, e.g., in the cation exchange separations (CIX) of group-6 elements [3], while in other cases some trends were found to be inversed in the groups, like, e.g., in the amine extraction of group-5 elements from HCl solutions (Nb > Db > Ta) [4]. Notwithstanding the fact that group-4 elements, Zr, Hf and Rf, were the most extensively studied, results have revealed quite a number of surprises and also disagreements (see the review of Kratz [1]). Czerwinski et al. [5] performed a series of manual liquidliquid extractions with tributylphosphate, TBP, in benzene to study the effect of HCl, Cl and H + ion concentration between 8 M and 12 M on the extraction of Zr 4+,Hf 4+, Th 4+,Pu 4+,andRf 4+. It was found that Rf extracts efficiently as the neutral tetrachloride into TBP from 12 M HCl like Zr, Th, and Pu while the extraction of Hf was relatively low and increased from 20% to 60% between 8 M and 12 M HCl. Extraction of Rf increased from 60% to 100% between 8 M and 12 M HCl, thus defining an extraction sequence Zr > Rf > Hf for the group-4 chlorides. Surprising results were obtained when the chloride concentration was varied at a constant H + concentration of 8 M. Above 10 M Cl concentration, the extraction of Rf decreased and behaved differently from Zr, Hf, and Th, and resembled that of Pu 4+. As these experiments [5] suffered from differences in the details of the chemical procedures applied to the different elements, e.g., different contact times and volumes used, it is important to confirm these findings in experiments that establish identical conditions for all homologous elements including Rf. It should be noted that only the organic phase was assayed for α decays of 261 Rf rather than also counting the aqueous phase. This is a high risk, as absorption of activity onto the walls of the equipment would then remain undiscovered and would lead to erroneous distribution coefficients. Kacher et al. [6] performed some additional chloride extractions into TBP/benzene with Zr, Hf, and Ti. The

2 870 V. Pershina et al. reported low extraction yields of Hf [5] could not be reproduced by Kacher et al. who reported that significant amounts of Hf (more than 50% in some cases) had been lost by absorption in [5]. Surprisingly, a similar loss of Rf due to absorption in the Czerwinski et al. work [5] was not suspected by Kacher et al., and so the latter authors, based on their new Zr, Hf, and Ti- results and based on the old [5] Rf results, suggested a revised sequence of extraction into TBP/benzene from around 8 M HCl as Zr > Hf > Rf > Ti [6]. In view of somewhat unsatisfactory situation with the conflicting Hf results in [5] and [6], and with the intention to establish an independent set of data characterizing the extraction sequence of Zr, Hf, and Rf from 8 M HCl into TBP, Günther et al. [7] have determined distribution coefficients of these elements, and the sequence Zr > Rf > Hf was observed. The anion exchange behaviour of Zr, Hf, and Rf was investigated by Haba et al. [8], and the sequence Rf > Zr > Hf was reported. Fluoride complexation of the group-4 elements was studied by Szeglowski et al.[9]in0.2mhf using the multicolumn technique, and it was shown that Rf is sorbed on an anion-exchange column as anionic fluoride complex. More recently, Trubert et al. [10] using the multi-column technique, found distribution coefficients of 49 for Rf, 43 for Zr, and 32 for Hf on an anion-exchange (AIX) column and 0.02 M HF in 0.4 M HCl, thus establishing the sequence Rf > Zr > Hf. Strub et al. [11] studied the cation- and anion exchange behaviour of Zr, Hf, Th, and Rf from mixed 0.1 M HNO 3 /HF solutions where the HF concentration was varied systematically. On cation-exchange columns, in 0.1M HNO 3 /<10 3 M HF, the group-4 elements are present as cations and K d > 10 3 are observed. In the concentration range between 10 3 M and 10 2 MHF,theK d of Zr and Hf decrease. At these concentrations, neutral or anionic fluoride complexes are being formed and are beginning to be eluted from the cation-exchange columns. The behaviour of Zr and Hf is very similar, with Hf sticking to the resin even at slightly higher HF concentration than Zr. In contrast, the decrease of the Rf distribution coefficients occurs between 10 2 and 10 1 MHF,i.e., at on order of magnitude higher HF concentration than for Zr and Hf. The decrease of the sorption of Th occurs at still higher HF concentration. Thus, the desorption of the group-4 elements with increasing HF concentration occurs in the sequence Zr > Hf > Rf > Th [11]. On anion-exchange columns, the sorption of Zr, Hf, and Th in 0.1 M HNO 3 /HF is low below 10 3 MHF.Itincreases [11] between 10 3 and 10 2 M HF (followed by a plateau), i.e., in the same range of HF concentrations as sorption decreases on the cation-exchange resin. For Th, no anionic fluoride complexes are known. Hence, Th is eluted from anion-exchange columns at all HF concentrations. Surprisingly, Rf behaves differently from Zr and Hf: ThereisnoriseoftheK d in 0.1M HNO 3 even up to 1 M HF [11]. By varying the concentration of HNO 3, i.e. by varying the NO 3 concentration which acts as counter ion competing for the binding sites on the anion-exchange resin, Strub et al. [11] were able to demonstrate that, nevertheless, anionic fluoride complexes of Rf are being formed. However, these are much more susceptible to a replacement by the counter ion NO 3 than are the fluoride complexes of Zr and Hf [11]. The extreme sensitivity of the anion exchange of the Rf fluoride complexes to the presence of the counter ion NO 3 was also demonstrated by Kronenberg [12] using the multi-column technique. In 0.01 M HF without HNO 3,aK d > 300 ml/g was determined, while in 0.1 M HNO 3 /0.5MHFa K d < 3mL/g was found in good agreement with [11]. As to the question of hydrolysis of the group-4 elements, two sets of experiments have been devoted to this question [13, 14]. In [13], by liquid-liquid extraction into TTA (thenoyltrifluoracetone), the trend in hydrolysis was Zr > Hf > Rf. Due to reasons critically analysed in [1], the study of sorption of the group-4 elements on cobalt ferrocyanide surfaces [14] resulted seemingly in a hydrolysis sequence Rf > Zr > Hf. In our previous theoretical research, we have considered complex formation and extraction of group-5 and -6 complexes from HF, HCl, and HBr solutions [15 18]. To help understand contradictive results for group-4 elements separations and/or to predict an outcome of new experiments, our theoretical research has been extended to the consideration of hydrolysis and complex formation of Zr, Hf, and Rf in HF and HCl solutions. Thus, in the current publication, we present results of the calculations of the electronic structures of various, hydrolyzed species and fluoride/chloride complexes of Zr, Hf and Rf. On their basis, we predict the stability of these complexes and their extraction at particular experimental conditions. Some information on hydrolysis and complex formation of Zr and Hf is given in Sect. 2. The model to predict complex formation and the relativistic densityfunctional theory (DFT) method, as well as details of the calculations are described in Sects. 3 and 4. Results are discussed in Sect. 5. A summary is presented in Sect Hydrolysis and complex formation of group-4 elements in HF and HCl solutions Z(IV) and Hf(IV) ions undergo extensive hydrolysis (at ph > 0), and the predominant solution species are polynuclear. The mononuclear ions M 4+ are predominant solution species only at trace metal-ion concentrations (< 10 4 M). At ph = 0, there is obviously 50% of the non-hydrolyzed and 50% of the hydrolyzed Zr species [19] formed according to the reaction M(H 2 O) 8 4+ MOH(H 2 O) H +. (1) With increasing ph, the step-wise hydrolysis (deprotonation) process reaches the formation of M(OH) 5 at ph > 6. At all steps, Zr proved to be more hydrolyzed than Hf, as hydrolysis constants show (Table 1). In HF solutions, Zr and Hf form very stable complexes with fluoride ion, much stronger than the ones with other halides or nitrate. Complex formation in HF solutions fol-

3 Theoretical predictions of hydrolysis and complex formation of group-4 elements 871 Table 1. Cumulative tracer hydrolysis constants (log K 1n ) for Zr, Hf [19] and Rf at ionic strength I = 1. Reaction log K Zr Hf Rf c M(H 2 O) MOH(H 2 O) 7 K ± ± ± a b 4 d M(H 2 O) M(OH) 2 (H 2 O) 6 K ± 1.7 M(H 2 O) M(OH) 3 (H 2 O) 5 K ± 2.9 M(H 2 O) 4+ 8 M(OH) 4 (H 2 O) 4 K ± 4.1 a: Ref. [20]; b: [21]; c: [13]; d: this work. lows the same pattern as hydrolysis does M(H 2 O) HF MF(H 2 O) HF... MF 6 + 8H 2 O + 6H +, (2) with the ultimate product MF 6. Stability constants of the fluoride complexes are listed in Tables 2 and 3 (Th is shown there for comparison). Complex formation in HCl solutions is a more complicated case due the polymerisation: no monomeric species of Zr were established up to 6 M HCl. Thus, even at tracer concentrations, polymeric species like [Zr 3 (OH) 4 ] 8+ or [Zr 4 (OH) 8 ] 8+ are formed at low HCl concentrations, and [Zr 3 (OH) 6 Cl 3 ] 3+ at about 1 2 M HCl. (When HClO 4 concentration is increased up to 2 4 M, the polymeric hydroxospecies are converted to the mononuclear ion Zr(H 2 O) 8 4+ ). At about 5 M HCl polymeric neutral species are formed with the ratio Zr to Cl of 1 to 4. Only beyond 6 M HCl are the anionic monochloro species ZrCl 6 formed (depolymerization starts at 2 M HCl) [23, 24]. To study hydrolysis of Rf, we considered only the first hydrolysis step (Eq. (1)) and, hence, only the hydrated and first hydrolyzed complexes. For the complex formation in HF solutions, almost all the types of complexes were considered: MF(H 2 O) 7 3+,MF 2 (H 2 O) 6 2+,MF 3 (H 2 O) 5 +, MF 4 (H 2 O) 4,MF 6, since CIX and AIX separations [10, 11] were performed in a large range of the HF concentrations. For the chlorination, MCl 6 (M = Zr,HfandRf)wasthe only type of complexes extracted by the AIX [8] and, hence, considered theoretically. We have also assumed the same type of complexes for Rf as those for Zr and Hf. Table 2. Cumulative complex formation constants (β i ) of group-4 elements and Th in HF solutions (4 M HClO 4 ) [22]. Ion log β 1 log β 2 log β 3 log β 4 log β 5 log β 6 Zr Hf Th Table 3. Consecutive complex formation constants (log K i ) of group-4 elements in HF solutions (4 M HClO 4 ) derived on the basis of the data of Table 2. Ion log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Zr Hf Th Models to predict hydrolysis and complex formation 3.1 Simple hydrolysis model A simple hydrolysis model suggests that log K for reaction (1) changes linearly with the ratio of the cationic charge to the M OH distance, or the radius of the cation [19] log K 1 = A + B z + log 2n + log 55.5, (3) d M OH where d M OH is the bond distance of the metal ion to the OH ligand, z is the formal charge of the ion and n is the number of water molecules of the hydrated complex (n = 8inthe case of group-4 cations). The model does, however, not explain a stronger hydrolysis of Zr in comparison with that of Hf, and a stronger hydrolysis of Nb in comparison with that of Ta, though the ionicradius(ir)ofzr 4+ (0.84 Å for CN = 8) is slightly larger than the IR of Hf 4+ (0.83 Å) and IR of Nb 5+ is equal to IR of Ta 5+ (0.64 Å for CN = 6), respectively [25]. For heavier transition-metal compounds with more covalent character of bonding, Eq. (3) may not be reliable any more. The following proposed model is free of those drawbacks (see [15 18]). 3.2 The model used by us to predict hydrolysis/complex formation In a fashion analogous to that of Kassiakoff and Harker [26], the following expression for the free energy of formation of the M x O u (OH) v (H 2 O) w (z 2u v)+ species from the elements was adopted and G f (u,v,w)/3.2rt = a i + a ij + log P log(u!v!w!2 w ) + (2u + v + 1) log 55.5 (4) log K = G r /2.3RT, (5) where G r is the free energy change of a hydrolysis reaction. The first term on the right hand side of Eq. (4), a i,is the nonelectrostatic contribution from M, O, OH, and H 2 O. The next term, a ij, is a sum of each pairwise electrostatic (Coulomb) interaction: E C = a ij = B ij Q i Q j /d ij, (6) where d ij is the distance between moieties i and j; Q i and Q j are their effective charges, and B = 2.3RT e 2 /ε is an

4 872 V. Pershina et al. independent constant. P is the partition function representing the contribution of structural isomers if there are any. The last two terms in Eq. (4) are statistical: one is a correction for the indistinguishable configurations of the species, and the other is a conversion to the molar scale of concentration for the entropy. aij and a i for each compound are obtained from the electronic density distribution data calculated using the DFT method described below. The differences in those values for the left and right parts of the equilibrium reactions plus the differences in the other terms of Eq. (4) will define log K of a reaction. The source of uncertainty in calculating E C (Eq. (6)) is ε. Therefore, for clarity, the energy of the Coulomb interaction will be calculated for the case of vacuum (see further tables). The value of ε, orb, will be defined later by using a fitting procedure. The last contribution to G r would be a change in the hydration energy G hydr beyond the first coordination sphere. Since reactions (1) and (2) have the same metal atom in the left and right parts, the differences G hydr should be very similar for all the elements and will not change the relative values of G r in the row Zr-Hf-Rf. 4. The DFT method and details of the calculations 4.1 The DFT method Calculations of the electronic structure of the above mentioned compounds were performed using the fully relativistic Density Functional Theory method with the General Gradient Approximation (GGA) for the exchangecorrelation energy [27, 28]. The method is a fully relativistic all-electron (or frozen core, FC, if necessary) code with the spin-orbit coupling explicitly included. It uses four-component basis functions which are transformed into molecular symmetry orbitals using double point groups. Molecular integrals between these functions are calculated in a numerical, three-dimensional grid. The most recent version of the method includes minimization of an error in the total energy [28] and the integration scheme of Boerrigter et al. [29], which, altogether, gives much more accurate values of total energies. The present calculations were performed within the FC approximation. The number of integration points was The Mulliken population analysis [30] was applied for the electronic density distribution: effective charge, Q M,and overlap population, OP, which is a direct measure of the covalent binding energy. The OP was then used to calculate the first term on the right-hand side of Eq. (4) OP = A a i,so that for a chemical reaction 4.2 Geometrical configurations and bond distances Since experiments on the heaviest-elements are conducted in a one-atom-at-a-time regime, we will deal here with mono-species. Thus, except for Zr(H 2 O) 8 4+ (D 2d -symmetry) (Fig. 1) and ML 6 (L = FandCl,O h -symmetry) known experimentally [23], geometrical configurations of the other species could only be assumed. We have also assumed that CN = 8 (CN is the coordination number) was kept for the fluorinated species of group-4 elements up to MF 4 (H 2 O) 4 (Fig. 2) and considered all possible positions of F in the coordination sphere (F cis, or F trans). Though the DFT method allows for optimizing bond lengths (R e ), this did not seem to be feasible and/or efficient for numerous large and/or negatively charged complexes of the heaviest elements. In addition, we were mostly interested in trends rather than in exact values of the complex formation constants. Therefore, experimental bond lengths where available (if not, a sum of ionic radii) were taken for the Zr and Hf compounds. For the Rf complexes, R e were estimated on the basis of calculated (optimised) R e for simpler Rf gas-phase compounds [31, 32]. In addition, by varying the metal-ligand distance for Rf in a reasonable interval, we can give an error bar for the obtained values of the free energies of reactions, or complex formation constants, and can be sure of the predicted trends. (This approach proved to be sufficient to provide reliable sequences in the complex Fig. 1. Geometrical configuration of M(H 2 O) 8 4+ (D 2d symmetry). E OP = a ij = A OP. (7) The correct application of the Mulliken analysis requires the use of minimal consistent basis sets for all the members of the series. Therefore, the minimal basis sets including valence ns 1/2, np 1/2, np 3/2 and (n 1)d 3/2, (n 1)d 5/2 orbitals were used for Zr, Hf and Rf. Fig. 2. Geometrical configuration of MF 4 (H 2 O) 4. (The darkest cycles are the F atoms).

5 Theoretical predictions of hydrolysis and complex formation of group-4 elements 873 formation and extraction, as was shown for group-5 and -6 elements [15 18]). Thus, for Zr(H 2 O) 4+ 8, a D 2d -symmetry (Fig. 1) and experimental R e (Zr OH 2 ) = Å, as was measured for [Zr 4 (OH) 8 (H 2 O) 16 ]Cl 8 12H 2 O [23, 24], were taken. For the octa-coordinated tetrafluoride complexes of Zr (Fig. 2), the Zr F distance of 2.19 Å was chosen as a sum of IR for CN = 8. For ZrF 6 and ZrCl 6, experimental R e for crystals (discrete hexa-fluoro- and chlorozirconate ions) of 2.04 Å and 2.44 Å were used, respectively [23]. For ZrOH(H 2 O) 3+ 7,theZr OH distance was varied from Å (a probable distance for the terminal OH groups, though the shortest OH bridging value is 1.97 Å) to Å (an experimental value for bridging OH groups in [Zr 4 (OH) 8 (H 2 O) 16 ]Cl 8 12H 2 O [23, 24]) and to 2.23 Å, being a sum of the IR. R e for the Hf compounds were taken as those of Zr less 0.01 Å in accordance with the differences in their IR [25]. The Rf-ligand bond lengths were taken 0.05 ± 0.2 Å larger than those of Hf (in accordance with the difference in R e calculated for HfCl 4 and RfCl 4 [31]). We have also considered tetrahedral ZrF 4 and HfF 4,as well as ZrCl 4 and HfCl 4 which obviously do not exist in the aqueous phase in this form. (As will be shown later, results show interesting trends). For ZrF 4 and HfF 4,experimental bond lengths for the gas phase of Å and Å (which is also equal to the sum of IR) [33] were used (for HfF 4,thesumofIRof1.89 Å was also considered). For ZrCl 4 and HfCl 4, R e of 2.32 and Å [34], respectively, were taken. R e (RfF 4 ) = 1.94 Å was assumed as a sum of IR and R e (RfCl 4 ) = 2.36 Å distance was optimized in the DFT calculations [31] (2.381 Å according to the relativistic effective core potential, RECP, calculations with the coupled cluster single and double excitations, CCSD, for the exchange-correlation part [32]). The complexes were also treated without surrounding (embedding), since the latter would have introduced more uncertainty in the results due to lack of knowledge of the nature and geometry of the surrounding. When calculating hydrolysis or complex formation constants, the effects of surrounding can be taken into account by an empirical fitting procedure, as will be shown later. 5. Results of the calculations and discussion 5.1 Energy levels and electronic density distribution The present calculations for the aqueous species of Zr, Hf and Rf have shown trends in the electronic structure and bonding to be very similar to those obtained for their gasphase compounds [35]. In Table 4, energies of the highest occupied MO (HOMO), the lowest unoccupied MO (LUMO) and the energy gap, E, between them obtained as a result of the calculations are given for M(H 2 O) 8 4+ (M = Zr, Hf and Rf) as an example. The Coulomb part of the binding energy, E C, and overlap populations (covalence) are given in Table 5. The calculated values show an increase in covalence and a decrease in ionicity of the metalligand bond from Zr through Rf. This increase is typical of other transactinide compounds and is accounted for by a relativistic contraction and stabilization of the 7s and 7p 1/2 valence orbitals resulting in their enhanced participation in the chemical bonding [35]. 5.2 Predictions of hydrolysis of Rf E C and OP for the first hydrolysis step (Eq. (1)) calculated on the basis of the E C and OP values of Table 5 are Table 4. Energies of HOMO, LUMO and E for M(H 2 O) 8 4+,where M = Zr, Hf and Rf. Energy, ev Zr Hf Rf E (HOMO) E (LUMO) E Table 5. Coulomb Part of the Binding Energy, E C (in ev), for various complexes of Zr, Hf and Rf. OP are given for some species in the parentheses. Complex R, Å Zr Hf Rf 4+ M(H 2 O) (8.93) 52.09(9.03) 50.93(9.11) 3+ MOH(H 2 O) 7 1) a (8.10) 56.70(8.21) 55.04(8.29) 2) b 57.05(8.16) 55.72(8.28) 54.15(8.35) 3) 2.23 c MF(H 2 O) MF 2 (H 2 O) 6 F cis F trans MF 3 (H 2 O) MF 4 (H 2 O) MF 4 (T d ) MF 6 1) 2.07 d ) 2.08 e ) 2.09 f 0.29 MCl 4 (T d ) MCl a: Zr OH (terminal); b: Zr OH (bridging); c: R (Zr OH) = a sum of IR; d f: different Rf F distances.

6 874 V. Pershina et al. given in Table 6. (Negative values of OP mean a decrease in the covalent interaction energy, so that a positive energy change E OP occurs). The data of Table 6 indicate that, as for group-5 and -6 complexes, changes in the covalent part of the interaction energy are very similar for the homologs, so that the free energy change of a reaction is defined mainly by a change in the electrostatic term, E C. Thus, on the basis of E C, the trend in hydrolysis for the first step can be predicted as Zr > Hf > Rf, with that for Zr and Hf being in agreement with experiment [19]. The log K 11 for Rf can then be determined using Eq. (1) and coefficients A and B obtained for the same type of reaction for Zr and Hf, e.g., log K 11 (Zr) = 0.83A B = 0.295, (8) log K 11 (Hf) = 0.82A B = 0.146, (9) using the data of Table 6. Solution of Eqs. (8) and (9) gives A = and B = Using A and B, as well as E C and OP for Rf from Table 6, log K 11 (Rf) 4. Since we did not make a geometry optimisation for the considered complexes, the obtained log K is an approximate value. It, however, generally agrees with log K 11 for Rf obtained in [13] using a linear correlation between hydrolysis and complex formation constants for the M(TTA) x complexes (TTA stands for the thenoyltrifluoroacetone). The E C values (Table 6) clearly indicate that variations in the metal-ligand distances in a reasonable interval do not change the trend in hydrolysis, so that for the first step, the trend to a decrease is definitely continued with Rf: Zr > Hf > Rf. We did not consider further hydrolysis steps, but analogously, one can imagine the same trend as that for the first step. 5.3 Predictions of the complex formation of Rf in HF aqueous solutions The complex formation of group-4 elements in HF solutions has been considered by us in more detail, since cation and anion exchange separations [10, 11] were performed in quite a wide range of the acid concentrations, hence involving various equilibria. At the very low HF concentrations, the formation of the F-containing positively charged species occurs according to reaction (2). With increasing acid concentration (above 10 2 M HF) negative complexes start to be formed. In the absence of hydrolysis, i.e., at ph below 0 (e.g. at 4M HClO 4 ), the complex formation reactions would follow the pattern given in Table 7, and trends in the calculated E C can be compared with those of the complex formation constants given in Table 2. Thus, for the fist step, E C corresponds well to log β 1 forzrandhf,aswellastologβ 6 for the formation of MF 6. For the intermediate steps, E C for Zr and Hf are very similar, so that the method, as would be the case for any other method, is not sensitive to such small differences. The E C values for Rf are, however, much more positive, which means that the complex formation of Rf is weaker than that of Zr and Hf and it does not depend on the variations of the Rf F distance, as the data of the last row show. Thus, one can definitely predict the following trend Zr > Hf > Rf for the complex formation of these elements in HF solutions provided no hydrolysis takes place. At higher ph, e.g. at0.1m HNO 3 (ph = 1), the complex formation occurs probably preferentially from the hydrolyzed species and has the pattern shown in Table 8. For reactions shown there, E C for Zr and Hf become very close to each other, since the ligands OH and F are very similar. Here, one can, nevertheless, notice that with the formation of the positively charged fluoride complexes, the trend is still Zr Hf > Rf (Rf has obviously larger E C ), while for the negatively charged complexes, the trend becomes reversed, so that Rf Zr > Hf, with the differences between the elements being minimal. For the step-wise formation of the fluoride complexes (Tables 9 through 12), the E C data show the same trend as the data of Table 8: for the formation of the positively charged or neutral fluoride complexes, the trend is Zr Hf > Rf (Rf has larger E C, while Zr and Hf are very similar in agreement with the consecutive constants of Table 3), while for the formation of the negatively charged MF 6 complexes, the trend is reversed, so that Rf Zr > Hf. Thus, Table 6. E C and OP for the first hydrolysis reaction M(H 2 O) MOH(H 2 O) 7 (M = Zr, Hf, and Rf). R e (Zr OH), Å E C,eV OP Zr Hf Rf Zr Hf Rf Table 7. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions (no hydrolysis). Reaction R (Rf F), Å Zr Hf Rf M(H 2 O) MF(H 2 O) 7 M(H 2 O) MF 2 (H 2 O) F cis F trans M(H 2 O) MF 3 (H 2 O) M(H 2 O) 4+ 8 MF 4 (H 2 O) M(H 2 O) 4+ 8 MF 4 (T d ) M(H 2 O) 4+ 8 MF

7 Theoretical predictions of hydrolysis and complex formation of group-4 elements 875 Table 8. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions. Reaction R, Å a Zr Hf Rf MOH(H 2 O) MF(H 2 O) 7 1) ) ) MOH(H 2 O) MF 2 (H 2 O) 6 1) F cis 2) ) MOH(H 2 O) MF 2 (H 2 O) 6 1) F trans 2) ) MOH(H 2 O) 3+ 7 MF 3 (H 2 O) + 1) ) ) MOH(H 2 O) 3+ 7 MF 4 (H 2 O) 4 1) ) ) MOH(H 2 O) 3+ 7 MF 4 (T d ) 1) ) ) MOH(H 2 O) 3+ 7 MF 6 1) ) ) a: Cases 1, 2 and 3 correspond to the different M OH distances: R (Zr OH) are as those indicated in Table 6, with R (Hf OH) and R (Rf OH) accordingly changed. one can conclude that when negatively charged complexes are formed from hydrolyzed or fluoride species, the trend becomes reversed in the group, though the differences between the species are very small. The theoretically obtained trend for the formation of the positively charged complexes is in agreement with the CIX separations from 0.1 M HNO 3 / M HF solutions for Zr, Hf and Rf [11], showing the following trends in the descending K d values Zr < Hf < Rf. Since [36] K d = K DM[RB + L ] p orgk i [L ] i p N 0 K n[l ] n, (10) where K DM is the ion association constant and K i is the complex formation constant, it would be desirable to give theoretical values of the K d (Rf) in relation to those of Zr or Hf. This can, in principal, be done using the following relations log K d (Hf) log K d (Rf) log K i (Hf) log K i (Rf) = Gr (Hf) G r (Rf) EC (Hf) E C (Rf), (11) since K d is defined mostly by the complex formation, K i, and a ratio of G r between homologous elements can be replaced by a ratio of their E C [18]. To define K d for the CIX separations is, however, very difficult, since many types of complexes of various positive charges are extracted, so that the quantitative assignment is problematic. For the case of the AIX sorption from 0.1 M HNO 3 / M HF (equilibria listed in Tables 8 through 12), the theoretically predicted trend in the K d values would be Rf Zr > Hf, with the differences between the elements being very small. The results of the AIX separations have, however, shown a different sequence in the K d values, Zr > Hf > Rf [11, 12], since for Rf, there is obviously a stronger competition of the counter ion NO 3 for the binding sites [1, 11, 12]. The theoretically obtained trend, Rf Zr > Hf, is, however, in good agreement with the AIX separation of Zr, Hf and Rf from 0.02 M HF/0.4MHClsolution (ph = 0 1) [10], and Eq. (11) would give K d (Rf) K d (Zr). Table 9. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions. Reaction R (Rf F), Å Zr Hf Rf MF(H 2 O) MF 2 (H 2 O) 6 F cis F trans MF(H 2 O) MF 3 (H 2 O) MF(H 2 O) 3+ 7 MF 4 (H 2 O) MF(H 2 O) 3+ 7 MF 4 (T d ) MF(H 2 O) 3+ 7 MF a b c a c: different Rf F distances. Table 10. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions. Reaction Geom. Zr Hf Rf MF 2 (H 2 O) MF 3 (H 2 O) 5 F cis F trans MF 2 (H 2 O) 2+ 6 MF 4 (H 2 O) 4 F cis F trans MF 2 (H 2 O) 2+ 6 MF 4 (T d ) F cis F trans MF 2 (H 2 O) 6 3+ MF 6 F cis F trans

8 876 V. Pershina et al. Table 11. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions. Reaction R (Rf F), Å Zr Hf Rf MF 3 (H 2 O) + MF 4 (H 2 O) MF 3 (H 2 O) + MF 4 (T d ) MF 3 (H 2 O) + MF Table 12. Coulomb part of the free energy change, E C (in ev), for the fluorination reactions. Reaction R (Rf F), Å Zr Hf Rf MF 4 (H 2 O) 4 MF 6 MF 4 (T d ) MF Table 13. Coulomb part of the free energy change for the chlorination reactions, E C (in ev). Reaction R e,å a Zr Hf Rf M(H 2 O) 4+ 8 MCl 4 (T d ) M(H 2 O) 4+ 8 MCl MOH(H 2 O) 3+ 7 MCl 4 (T d ) 1) ) ) MOH(H 2 O) 3+ 7 MCl 6 1) ) ) a: Cases 1, 2 and 3 correspond to the different M OH distances: R (Zr OH) are as those indicated in Table 6, with R (Hf OH) and R(Rf OH) accordingly changed. 5.4 Predictions of the complex formation of Rf in HCl solutions The complex formation in HCl solutions follows the pattern shown in Table 13. For the formation of the MCl 6 complexes, the E C values show the same trend as the E C values show for the MF 6 complexes without hydrolysis (Table 7), e.g., Zr > Hf > Rf, independently whether the process starts from the non-hydrolyzed or hydrolyzed species. In the absence of hydrolysis, the difference between Zr/Hf and Rf is only larger (about 0.5 ev) than in the case of the hydrolyzed cations (about 0.2 ev). Thus, by analogy with the HF case, the complex formation of Rf from hydrolyzed species is more similar to that of Zr and Hf, than in the absence of hydrolysis. From this point of view, the results of the experiments [8] on the AIX separation of Zr, Hf and Rf from 4 8 M HCl solutions (i.e., in the full absence of hydrolysis, ph < 0) showing the trend Rf > Zr> Hf cannot find its theoretical explanation. 5.5 Tetrahedral complexes of group-4 elements In addition to the real fluoride and chloride species in aqueous solutions, we have also considered the tetrahedral MF 4 and MCl 4 complexes existing in this state only in the gas phase. The results have remarkably revealed a full inversion of the trend in the group whatever the reaction is: Rf > Hf > Zr (see Tables 5 13). This information can be valuable to interpret extraction sequences of MF 4 or MCl 4, e.g., by TBP. Results of the extraction of MCl 4 by TPB at 8 M HCl [7] have, indeed, shown a partially reversed trend in the group Zr > Rf > Hf. However, to predict or explain the trend in the extraction by TBP in detail, the electronic structure of the MCl 4 (TBP) 2 (M = Zr, Hf and Rf) complexes should be calculated. 6. Conclusions Relativistic density-functional calculations of the electronic structures of hydrated and hydrolyzed species of Zr, Hf and Rf in aqueous solutions and the derived first hydrolysis constant have shown the trend to a decrease in hydrolysis to be continued with Rf: Zr > Hf > Rf. Analogous trends in hydrolysis were predicted for group-5 and -6 cations including Db and Sg [15, 18]. The result for group-4 elements is in agreement with experiments on hydrolysis of Zr, Hf [15] and estimates for Rf [13]. For the complex formation in HF solutions, trends depend on whether the process starts from non-hydrolyzed or hydrolyzed cations. In the first case, the trend to a decrease in the complex formation is continued with Rf, so that Zr > Hf > Rf, while in the second case, or for the step-wise fluorination, the trend to a decrease is preserved in the group for the positively charged complexes, while it gets reversed fromhftorffortheformationofmf 6 (Rf Zr > Hf). For the formation of the anionic chloro-complexes, the trend to a decrease is continued with Rf independently of ph: Zr > Hf > Rf.

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