Calcium isotope fractionation in liquid chromatography with crown ether resins

Abstract

Breakthrough mode liquid chromatography was employed to study calcium isotope fractionation. Highly porous silica beads, the inner pores of which were embedded with a benzo-18-crown-6 ether resin or a benzo-15-crown-5 ether resin, were used as column packing material. For both the resins, enrichment of heavier isotopes of calcium was observed in the frontal part of their respective calcium chromatograms. The values of the isotope fractionation coefficient were on the order of 10⁻³ and 10⁻⁴ for the benzo-18-crown-6 ether and benzo-15-crown-5 ether resins, respectively. The observed isotope fractionation coefficient was dependent on the concentration of hydrochloric acid in the calcium feed solution; a higher hydrochloric acid concentration resulted in a smaller fractionation coefficient value. The present calcium isotope effects were most probably mass dependent, indicating they came from isotope effects based on molecular vibration. Molecular orbital calculations supported the present experimental results in a qualitative fashion.

Keywords:

• calcium
• calcium isotopes
• calcium isotope fractionation
• benzo-18-crown-6 ether
• benzo-15-crown-5 ether
• isotope effects
• separation factor
• reduced partition function ratio
• molecular orbital calculations

1. Introduction

Naturally occurring calcium (Ca) consists of six isotopes, ⁴⁰Ca, ⁴²Ca, ⁴³Ca, ⁴⁴Ca, ⁴⁶Ca, and ⁴⁸Ca, among which ⁴⁰Ca with double magic numbers is the most abundant. Ca isotopes have applications in various research fields ranging from medicine to astrophysics. For instance, ⁴⁸Ca is an important nuclide to synthesize super-heavy elements and is to be used in research on neutrino-less double beta decay [1]. Due to its low natural abundance (0.19%), however, it must be concentrated prior to applications. ⁴⁷Ca, a radioisotope of Ca, which is produced from ⁴⁸Ca by neutron bombardment, has certain medical and biological research applications because of its short half-life (four days) and convenient β and γ spectra [2].

Several methods for Ca isotope separation and enrichment have been reported. The electromagnetic method (Calutron) is too costly and impractical for large-scale production of enriched Ca isotopes. Another method is a redox system method which is based on reported Ca isotope effects between two different oxidation states of Ca (0 and + 2) for the redox systems with host materials of Ca insertion being mercury (Hg) [3] or graphite [4]. The former redox system with aneffective isotope fractionation coefficient, ε, of 8 × 10⁻³ for the ⁴⁸Ca/⁴⁰Ca isotopic pair is attractive for small pre-enrichment of heavier Ca isotopes [3], but the use of toxic Hg entails biological and environmental problems and this system would be difficult to apply to large-scale isotope enrichment. The redox system with graphite [4] to obtain enriched Ca isotopes is even less technically feasible than the Hg system.

In general, chromatographic processes based on chemical exchanges are desirable for large-scale production of enriched isotopes due to operational simplicity and easy accumulation of small single-stage isotope fractions. As chromatographic column packing materials, those having crown ethers and cryptands as functional groups and ion exchange resins including chelating resins, have been reported. Ion exchange chromatographic processes using strongly acidic cation exchange resins showed only small ε values and are likely impractical [5,6]. Processes using chromatographic columns packed with polymer-bound cryptands, although showing ε values on the order of 10⁻³, are also impractical as a consequence of slow exchange rates [7–9].

A chromatographic Ca isotope separation process with a crown ether resin was shown to be an improvement over those with cryptands [9,10]. Recently, accumulation of Ca isotope fractions was developed for liquid chromatography using benzo-18-crown-6 ether resin as column packing material, and ε values on the order of 10⁻³ were reported [11,12]. Liquid chromatography with crown ether resins may be a promising process for large-scale production of enriched Ca isotopes.

In the present paper, which constitutes a part of our study on isotope effects of the group 2 metals in crown ether systems [13], we investigated Ca isotope fractionation in chromatographic systems with crown ether resins in detail. In addition, we conducted molecular orbital (MO) calculations for elucidation of the Ca isotope effects observed in those experiments. The chromatographic experiments and MO calculations and their results are described here.

2. Experimental

2.1. Crown ether resins and other materials

Two kinds of crown ether resins, benzo-18-crown-6 ether (B18C6) resin, and benzo-15-crown-5 ether (B15C5) resin, both synthesized at the Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, were employed. The B18C6 resin, embedded in the inner pores of highly porous silica beads (diameter: 40 to 60 μm), is a copolymer ofphenol and monobenzo-18-crown-6 ether that provide functional groups [14]. The B15C5 resin is similar to the B18C6 resin except that the functional groups are monobenzo-15-crown-5 ether instead of monobenzo-18-crown-6 ether [11]. All reagents were of analytical-reagent grade and used without further purification.

2.2. Measurements of distribution coefficients (K_d) ofCa

Distribution coefficients of Ca are defined as

where c _resin and c _solution are Ca²⁺ ion concentrations in the resin phase (mmol g⁻¹) and in the solution phase (mmol cm⁻³), respectively, and they were determined chromatographically. The B18C6 resin was packed in a Pyrex glass column (100 cm length × 0.8 cm diameter) so that the resin column height was about 30 cm. The packed resin was washed with aqueous hydrochloric acid (HCl) solution with the same HCl concentration as that of Ca feed solution. The aqueous HCl solution containing calcium chloride (CaCl₂) (feed solution) was introduced continuously from the top of the column ata constant flow rate which was controlled by ahigh-pressure pump (Nihon Seimitsu Kagaku NP-KX-110U high-pressure pump). The effluent from the bottom of the column was collected into small fractions and analyzed for Ca contents. The experimental temperature was kept constant by circulating thermostatted water through a jacket surrounding the column.

For the B15C5 resin, a similar experimental procedure was followed except that the resin column height was 100 cm instead of 30 cm for the B18C6 resin.

The distribution coefficient was calculated using the equation

where ΔV = V _1/2 − V ₀ and m is the weight of the resin packed in the column. Here, V _1/2 is the effluent volume at which the Ca²⁺ ion concentration equals half of its concentration in the feed solution after the breakthrough point, V_p and V ₀ is the corresponding volume when no Ca²⁺ ion adsorption occurs.

2.3. Ca isotope fractionation experiments

Chromatography was done in the breakthrough mode using two or five Pyrex glass columns (100 cm length and 0.8 cm diameter) packed with the resin andconnected in series with polytetrafluoroethylene (PTFE) tubes (1 mm diameter). The resin column was first washed with HCl solution with the same HCl concentration as that of Ca feed solution. The aqueous HCl solution containing CaCl₂ (feed solution) of the natural abundances of the Ca isotopes was introduced continuously from the top of the first column at a constant flow rate controlled with a high-pressure pump (Nihon Seimitsu Kagaku NP-KX-105U high-pressure pump) and it flowed through all the columns. The effluent from the bottom of the final (second or fifth) column was collected into small fractions and analyzed for Ca contents and Ca isotopic compositions. The experimental temperature was kept constant at 10, 25, 40, or 50°C by circulating thermostatted water through jackets surrounding the columns.

Experimental conditions are summarized in Table1.

Table 1. Experimental conditions and results other than isotopic data.

Run No.	Resin	Temp./°C	HCl concn. mol dm^−3	Ca^2 + ion concn. of the feed/mol dm^−3	Resin column height/cm	Flow rate/cm^3 h^−1	V 1/2/cm^3	Q/mmol	Q/Vr /mmol cm^−3
Ca18-1	B18C6	50	0.0	0.0513	193.0	6.2	83.8	0	0
Ca18-2		50	9.0	0.0480	193.0	6.0	131.0	2.27	0.172
Ca18-3		50	10.5	0.0476	193.0	5.7	161.4	3.70	0.280
Ca18-4		50	12.0	0.0466	193.0	6.0	217.8	6.24	0.473
Ca18-5		40	9.0	0.0468	193.0	6.0	140.1	2.59	0.196
Ca18-6		25	9.0	0.0478	193.0	6.0	150.4	3.18	0.242
Ca18-7		10	9.0	0.0481	193.0	5.9	160.2	3.67	0.279
Ca18-8		50	9.0	0.0474	488.5	5.9	331.2	5.65	0.169
Ca18-9		40	9.0	0.0470	488.5	5.9	348.1	6.39	0.192
Ca18-10		25	9.0	0.0476	488.5	6.0	379.0	7.94	0.238
Ca15-1	B15C5	50	4.5	0.0475	492.8	6.2	196.6	0	0
Ca15-2		50	10.5	0.0484	491.0	5.9	218.8	1.73	0.033
Ca15-3		50	12.0	0.0490	495.8	5.8	237.6	2.13	0.039

2.4. Analyses

The Ca²⁺ ion concentrations in effluent fractions and feed solutions were measured after appropriate dilution by inductively coupled plasma atomic emission spectrometry (ICP-AES; Seiko Instruments SPS 7700 ICP-atomic emission spectrometer). For the Ca isotope analysis, the chemical form of Ca in effluent fractions and feed solutions was converted from CaCl₂ to calcium iodide (CaI₂) through anion exchange (Muromac, 1 × 8, 200–400 mesh, OH form) followed by addition of hydroiodic acid (HI) [4,12]. Ca isotopic ratios (⁴²Ca/⁴⁰Ca, ⁴³Ca/⁴⁰Ca, ⁴⁴Ca/⁴⁰Ca and ⁴⁸Ca/⁴⁰Ca) of the selected fractions of the effluents in the frontal part of the Ca adsorption zones were determined by thermal ionization mass spectrometry (Finnigan MAT261 mass spectrometer) [4,12]. The standard deviations were typically 0.08 to 0.21%, 0.13 to 0.32%, 0.09 to 0.18% and 0.13 to 0.32% for the ⁴²Ca/⁴⁰Ca, ⁴³Ca/⁴⁰Ca, ⁴⁴Ca/⁴⁰Ca and ⁴⁸Ca/⁴⁰Ca isotopic pairs, respectively.

2.5. MO calculations

To elucidate the Ca isotope effects observed in the present chromatographic Ca isotope fractionation experiments in a qualitative fashion, MO calculations of the reduced partition function ratios (rpfrs) of Ca species expected to be involved in the experiments were carried out.

The Ca isotope exchange reaction that gives rise tothe present Ca isotope effects may be expressed, for the ⁴²Ca/⁴⁰Ca, ⁴³Ca/⁴⁰Ca, ⁴⁴Ca/⁴⁰Ca, and ⁴⁸Ca/⁴⁰Ca isotopic pairs, as

where Ca²⁺(aq) and Ca²⁺(crown) denote the Ca²⁺ ions in the aqueous solution and crown ether phases, respectively. The equilibrium constant K_{x /40} (x = 42, 43, 44 or 48) of Equation (3) is given as the ratio of two rpfrs for the x/40 isotopic pair [15]:

where (s/s′)f_{x /40}(aq) is the rpfr of the ^x Ca/⁴⁰Ca isotopic pair in the aqueous solution phase and (s/s′)f_{x /40}(crown) is that in the crown ether phase.

The general expression for the rpfr is, under Born–Oppenheimer and harmonic oscillator approximations, given as [15],

where u_i  = hcω_i /(kT) and u′_i  = hcω′_i /(kT); f, the degree offreedom of molecular vibration; h, the Planck constant; c, the velocity of light; ω_i and ω′_i , the wave numbers of the ith molecular vibration of the heavier and the lighter isotopic species, respectively; k, the Boltzmann constant; and T, the absolute temperature [15]. Thus, the rpfr for the given species and for the given isotopic pair can be theoretically estimated by knowing all the isotopic vibrational frequencies of the species.

As model Ca species in aqueous solution, we considered a Ca²⁺ ion with the solvation number of six or seven, surrounded by water molecules and additionally a chloride (Cl⁻) ion, [Ca²⁺(H₂O)₆], [Ca²⁺(H₂O)₆](H₂O), [Ca²⁺(H₂O)₇], [Ca²⁺(H₂O)₅Cl⁻], and [Ca²⁺(H₂O)₆Cl⁻], in which water (H₂O) molecules and a Cl⁻ ion in the brackets denote that they are in the primary solvation sphere of the Ca²⁺ ion, directly interacting with the Ca²⁺ ion. As model Ca species inthe crown ether phase of the B18C6 resin system, weconsidered clusters consisting of a B18C6 molecule and a CaCl₂ without or with up to two HCl molecules ([B18C6 + CaCl₂], [B18C6 + CaCl₂ + HCl], and [B18C6 + CaCl₂ + 2HCl]). Similarly, [B15C5 + CaCl₂], [B15C5 + CaCl₂ + HCl], and [B15C5 + CaCl₂ + 2HCl] were considered as model Ca species in the crown ether phase of the B15C5 resin system. Inclusion of HCl molecules in the crown ether phases was inferred from the results of strontium (Sr) isotope fractionation in a solvent extraction system reported by Shibahara et al. [16]. For both the B18C6 and B15C5 resins, the existence of organic networks of the resins and H₂O molecules was ignored.

Calculations were made at the B3LYP/6-31G(d) level of theory using the Gaussian 03 and 98 program packages (Gaussian Inc.) [17]. In our previous paper onboron (B) isotope fractionations between the boric acid (B(OH)₃) molecule and the monoborate anion (B(OH)₄ ⁻) [18], we showed that a more advanced MO theory with a higher basis level basis set did not necessarily yield better rpfr results. The B3LYP theory is probably the most commonly used one in computational chemistry nowadays, and the 6-31G(d) basis set, although not a large one, may be one of the most standard basis sets. The Gauss View (Gaussian Inc.) and the Free Wheel programs (Butch Software Studio) were used for the graphics. No frequency correction was made since we were only concerned with relative magnitudes of the rpfrs of Ca species and of equilibrium constants of Ca isotope exchange reactions. For each species considered, the vibrational analysis was performed after the structure optimization, and using the calculated frequencies, we estimated the rpfrs and equilibrium constants for the Ca isotope exchange reactions.

3. Results and discussion

3.1. Distribution coefficients of Ca

The measured K _d value of Ca²⁺ ion was plotted against the HCl concentration in the feed solution in Figure 1. As is seen in the figure, the B18C6 resin works as a Ca²⁺ ion adsorbent at pH values of about 4 and higher, and the B15C5 resin does so at pH values of about 6 and higher. For both resins, the measured K _d value is an increasing function of the HCl concentration at both 25 and 40°C, which means the resins are better Ca ion adsorbents at a higher HCl concentration. This observation is reasonable since Ca²⁺ ions are considered to be adsorbed on the crown ethers as the neutral species, CaCl₂. At a given pH value, the K _d value is much larger for the B18C6 resin than for the B15C5 resin, meaning that the Ca adsorption capacity of the former is much larger than the latter and consequently, the former is a much better Ca²⁺ ion adsorbent than the latter. At a given pH value, the K _d value is slightly larger at a lower temperature for the two resins, which indicates that the adsorption of Ca²⁺ ion from the aqueous feed solution onto the resins is slightly exothermic.

Figure 1. Plots of K _d as a function of HCl concentration in the feed solution at Ca²⁺ ion concentration of 0.05 mol dm⁻³. □, B18C6 resin at 25°C; ▪, B18C6 resin at 40°C; ○, B15C5 resin at 25°C; •, B15C5 resin at 40°C.

The K _d value was plotted against the concentration of Ca²⁺ ion in the feed solution in Figure 2. It is a decreasing function of the concentration of Ca²⁺ ion for both resins at the HCl concentrations of 9 and 12 mol dm⁻³. At a given Ca²⁺ ion concentration, the K _d value is much larger for the B18C6 resin than for the B15C5 resin.

Figure 2. Plots of K _d as a function of Ca²⁺ ion concentration in the feed solution at a given HCl concentration at 25°C. □, B18C6 resin at HCl concentration of 9 mol dm⁻³; ▪, B18C6 resin at HCl concentration of 12 mol dm⁻³; ○, B15C5 resin at HCl concentration of 9 mol dm⁻³; •, B15C5 resin at HCl concentration of 12 mol dm⁻³.

As a summary of the K _d measurements, the B18C6 resin is a much better Ca²⁺ ion adsorbent than the B15C5 resin. This is most likely attributable to the difference in the hole size between the two crown ethers. While a B18C6 molecule with the hole (ring) size of 260 to 320 pm can accommodate the Ca²⁺ ion with the diameter of 228 pm in its hole, the hole size (170 to 220 pm) of a B15C5 molecule is slightly smaller than the diameter of the Ca²⁺ ion [19]. A higher HCl concentration is preferable for a larger adsorption capacity for the Ca²⁺ ion, but temperature has only a slight effect.

3.2. Chromatograms and isotope accumulation curves

The results of the Ca isotope fractionation experiments, excluding the isotopic data, are summarized in Table 1. The total Ca²⁺ ion adsorption capacity, Q, of the resin column in each experiment was calculated by the equation

where c ₀ is the Ca²⁺ ion concentration of the feed solution. The parameter Q/V_r is the adsorption capacity per unit volume of the resin column, where V_r is the volume of the resin column.

Breakthrough mode chromatograms for the experiments with the B18C6 resin are depicted in Figures 3 and 4. Figure 3 compares the chromatograms with varying HCl concentrations in the feed solution while the other experimental conditions were nearly kept constant (Runs Ca18-1, Ca18-2, Ca18-3, and Ca18-4). As expected from the results of K _d measurements, V_p and V _1/2, and consequently, Q and Q/V_r , too, increase with increasing HCl concentration in the feed solution for a constant Ca²⁺ ion concentration in the feed solution and for a constant resin column height at a constant temperature. Figure 4 shows the temperature dependence of the chromatograms for a constant Ca²⁺ ion concentration in the feed solution and for a constant resin column height at a constant flow rate. The shapes of the chromatograms are seemingly nearly independent of temperature, but V_p and V _1/2, and consequently, Q and Q/V_r , too, slightly decrease with increasing temperature.

Figure 3. Chromatograms with the benzo-18-crown-6 ether (B18C6) resin with varying HCl concentration. —□—, Run Ca18-1; —•—, Run Ca18-2; —○—, Run Ca18-3; —▴—, Run Ca18-4.

Figure 4. Chromatograms with the benzo-18-crown-6 ether (B18C6) resin at different temperatures and at different resin bed heights. —▵—, Run Ca18-2; —•—, Run Ca18-5; —□—, Run Ca18-6; —▾—, Run Ca18-7; —▴ — Run Ca18-8; —○—; Run Ca18-9; —▪—, Run Ca18-10.

Breakthrough mode chromatograms for the experiments with the B15C5 resin are depicted in Figure 5. The V_p and V _1/2 values are both larger at higher HCl concentration in the feed solution for a constant Ca²⁺ ion concentration in the feed and for a constant resin column height. The V_p and the V _1/2 values of Run Ca15-1 are equivalent to those of a chemical species like the Cl⁻ ion that is not adsorbed onto the B15C5 resin at all, which means that no Ca²⁺ ions are adsorbed onto the B15C5 resin when HCl concentration is 4.5 mol dm⁻³ and lower.

Figure 5. Chromatograms with the benzo-15-crown-5 ether(B15C5) resin. —□—, Run Ca15-1; —○—, Run Ca15-2; —▴—, Run Ca15-3.

The results shown in Figures 3 to 5 are all consistent with the K _d data shown in Figures 1 and 2.

Except for Runs Ca15-1 and Ca18-1 that showed practically no adsorption capacity for Ca²⁺ ions, the others were analyzed for their Ca isotopes. The results of the selected Runs in which five columns were used are shown in Figures 6 –9. In each of those figures, the chromatogram is shown in the lower half of the figure and the isotopic data are given in the upper half. The isotopic data are plotted as local enrichment factor (LEF) of a fraction of the effluent defined as the ^x Ca/⁴⁰Ca (x = 42, 43, 44 or 48) isotopic ratio of that fraction divided by the corresponding ratio of the feed solution:

Figure 6. Chromatogram and LEF profiles of Run Ca18-8. ▪, ⁴²Ca/⁴⁰Ca isotopic pair; ○, ⁴³Ca/⁴⁰Ca isotopic pair; □, ⁴⁴Ca/⁴⁰Ca isotopic pair; •, ⁴⁸Ca/⁴⁰Ca isotopic pair.

In Figures 6 to 8, the results with the B18C6 resin, Runs Ca18-8, Ca18-9, and Ca18-10, are drawn. The LEF value of an isotopic pair in general decreases monotonously with increasing effluent volume; it is generally the largest in the breakthrough-point fraction and asymptotically and promptly approaches unity with increasing effluent volume. The LEF values larger than unity mean the heavier isotopes are preferentially fractionated into the solution phase while the lighter ones remain in the crown ether phase. The degree of deviation of LEF from unity is larger for a larger value of x. That is, a larger mass difference between the isotopic pair results in larger Ca isotope effects.

The results of Run Ca15-2 using the B15C5 resin are depicted in Figure 9. The shapes of the isotope accumulation profiles are roughly similar to those with the B18C6 resin. A close look at the plotted points, however, reveals that the LEF profiles, especially thoseof the ⁴³Ca/⁴⁰Ca and ⁴⁸Ca/⁴⁰Ca isotopic pairs, are significantly shifted from the expected smooth andmonotonously decreasing function of the effluent volume. We also note that the maximum degrees of isotope fractionation reached are much lower in Figure9 than in Figures 6 to 8, which indicates that the B18C6 resin is a better Ca isotope separator than the B15C5 resin.

Figure 9. Chromatogram and LEF profiles of Run Ca15-2. ▪, ⁴²Ca/⁴⁰Ca isotopic pair; ○, ⁴³Ca/⁴⁰Ca isotopic pair; □, ⁴⁴Ca/⁴⁰Ca isotopic pair; •, ⁴⁸Ca/⁴⁰Ca isotopic pair.

Figure 8. Chromatogram and LEF profiles of Run Ca18-10. ▪, ⁴²Ca/⁴⁰Ca isotopic pair; ○, ⁴³Ca/⁴⁰Ca isotopic pair; □, ⁴⁴Ca/⁴⁰Ca isotopic pair; •, ⁴⁸Ca/⁴⁰Ca isotopic pair.

Figure 7. Chromatogram and LEF profiles of Run Ca18-9. ▪, ⁴²Ca/⁴⁰Ca isotopic pair; ○, ⁴³Ca/⁴⁰Ca isotopic pair; □, ⁴⁴Ca/⁴⁰Ca isotopic pair; •, ⁴⁸Ca/⁴⁰Ca isotopic pair.

3.3. Evaluation of the magnitude of Ca isotope effects

To evaluate the magnitudes of the Ca isotope effects observed in the present study, the fractionation coefficient, ε = S − 1, was calculated. The single-stage separation factor, S, is defined as

where ( ^x Ca/⁴⁰Ca)_solution and ( ^x Ca/⁴⁰Ca)_crown denote the ^x Ca/⁴⁰Ca (x = 42, 43, 44 or 48) isotopic ratios in theaqueous solution phase and in the crown ether phase, respectively, and ε can be calculated using thechromatographic experimental data by the equation [20],

In Equation (9), R_i and q_i are the atomic fraction of the isotope of interest and the amount of Ca in the ith fraction of the effluent, respectively, R ₀ is the atomic fraction of the isotope of interest in the feed solution, and the summation is taken over all the fractions where isotope fractionation is observed. The values of ε and ε per unit mass difference between the two isotopes, ε/ΔM, are listed in Table 2 together with some other ε data obtained in chromatographic studies found in the literature [5–8,10–12]. The errors on the ε values are often large. The uncertainty in the value of ε originates mostly from R_i  − R ₀ in Equation (9) since this difference is usually very close to zero. We could not determine the ε values for the ⁴³Ca/⁴⁰Ca and ⁴⁸Ca/⁴⁰Ca isotopic pairs of Run Ca15-2 due to the too large experimental uncertainties.

Table 2. Fractionation coefficients obtained in this work and in some previous chromatographic works.

Adsorbent	Resin type	Run No.	Temp./°C	HCl concn./mol dm^−3	ε/10^−3				ε/ΔM/10^−3				References
					^42Ca/^40Ca	^43Ca/^40Ca	^44Ca/^40Ca	^48Ca/^40Ca	^42Ca/^40Ca	^43Ca/^40Ca	^44Ca/^40Ca	^48Ca/^40Ca
Crown ether	B18C6	Ca18-2	50	9	0.9 ± 0.2	3.2 ± 1.5	2.7 ± 0.3	3.6 ± 1.2	0.43	1.04	0.68	0.45	This work
		Ca18-3	50	10.5	1.2 ± 0.6	2.4 ± 0.1	1.9 ± 0.5	4.3 ± 0.3	0.61	0.79	0.47	0.54
		Ca18-4	50	12	0.9 ± 0.2	1.1 ± 0.2	1.6 ± 0.2	2.8 ± 0.2	0.47	0.38	0.40	0.35
		Ca18-5	40	9	1.3 ± 0.5	3.0 ± 0.9	2.4 ± 0.3	6.0 ± 1.0	0.68	0.99	0.61	0.76
		Ca18-6	25	9	1.1 ± 0.2	2.4 ± 0.7	2.4 ± 0.2	4.3 ± 0.5	0.57	0.80	0.60	0.54
		Ca18-7	10	9	1.3 ± 0.1	1.9 ± 0.4	2.6 ± 0.2	4.4 ± 0.5	0.67	0.64	0.66	0.55
		Ca18-8	50	9	1.4 ± 0.5	1.3 ± 0.1	2.6 ± 0.3	4.0 ± 0.1	0.69	0.45	0.64	0.50
		Ca18-9	40	9	1.3 ± 0.1	2.2 ± 0.4	2.1 ± 0.6	4.0 ± 1.0	0.63	0.75	0.54	0.50
		Ca18-10	25	9	1.4 ± 0.1	2.1 ± 0.4	2.2 ± 0.2	4.2 ± 0.9	0.68	0.69	0.56	0.53
	B15C5	Ca15-2	50	10.5	0.26 ± 0.23	–	0.81 ± 0.38	–	0.13	–	0.20	–
		Ca15-3	50	12	0.18 ± 0.38	0.20 ± 0.21	0.40 ± 0.22	0.72 ± 0.26	0.088	0.066	0.099	0.089
	B18C6		35	9				3.1					[12]
	B18C6		27	9				1.9					[11]
	18-crown-6		20				1.7–2.5						[10]
Cryptand	[2B.2.2]		20				5.7–7.5	10.4–12.9					[7,8]
Ion exchanger	Strongly acidic		25		0.041	0.047	0.08	0.18					[5]
	(−SO3 ^−H^+)		25		0.11	0.16	0.21	0.37
			25		0.05	0.067	0.096	0.16
	(−SO3 ^−H^+)						1.1	1.8					[6]

The findings in Table 2 as well as in Figures 6 to 9 may be summarized as follows:

1.	In the present study, the heavier isotopes are preferentially fractionated into the solution phase, while the lighter ones are fractionated into the crown ether phase in both the B18C6 and B15C5 resin systems. The direction of this Ca isotope fractionation is common to all the systems listed.
2.	The values of ε on the order of 10−3 and those of ε/Δ M on the order of 10−4 were obtained with the B18C6 resin; they are on the same order as those of previous studies with the same resin [11,12] and for 18-crown-6 resin [10], but larger than those of studies with ion exchange resins [5,6]. The cryptand systems [7,8] yielded larger ε values than crown ether systems. However, we note that, while the cryptand systems were basically operated in organic solvent media, our experiments were all carried out in aqueous solutions, which is a very large advantage obtained using the present silica-based resins.
3.	The ε and ε/ΔM values with the B15C5 resin, which have very large errors, are one order of magnitude smaller than the corresponding values obtained with the B18C6 resin.
4.	Admittedly roughly, we can say the ε value seems nearly proportional to the isotopic mass difference, ΔM, i.e. the ε/Δ M value is nearly constant in a given Run. This indicates that the present Ca isotope effects are solely mass dependent ones and are little affected by nuclear size and shape effects (nuclear field shift effects) [21]. The data supporting the present mass dependent Ca isotope effects can be provided by three-isotope plots. An example is shown in Figure 10 where the LEF value minus one for the x Ca/40Ca isotopic pair (x = 43, 44 or 48) of Run Ca18-8 is plotted against that for the 42Ca/40Ca isotopic pair. The slopes of the curves were calculated to be 1.44, 1.90, and 3.44 for the x Ca/40Ca isotopic pair (x = 43, 44, or 48), respectively. The corresponding theoretical values, estimated on the basis that the observed isotope effects arise solely from molecular vibration, are 1.47, 1.91, and 3.50, respectively. The smallness of the deviation of the experimental slope from the theoretical one supports our conclusion that the isotope effects in the present systems are normal mass dependent ones.
5.	In general, a higher HCl concentration in the feed solution yields a smaller ε value both in the B18C6 resin system (Runs Ca18-2, Ca18-3, and Ca18-4) and in the B15C5 resin system (Runs Ca15-2 and Ca15-3) at a given temperature. The degree of HCl concentration dependence ofthe ε value is more substantial with the B15C5 resin than with the B18C6 resin. We previously observed a similar HCl concentration dependence of the ε value in chromatographic Sr isotope fractionation experiments with the B15C5 resin [13].
6.	Contrary to our expectation, the temperature dependence of the ε value is not clear (Runs Ca18-2, Ca18-5, Ca18-6, Ca18-7 and Runs Ca18-8, Ca18-9, Ca18-10). The temperature range may be too narrow for the temperature dependence of the ε value to show up clearly.

3.4. MO calculations

Our major purpose for the present MO calculations was to elucidate the findings (1), (3), and (5) above. For all the Ca species considered, no imaginary frequency was obtained, which means that the optimized structures are all at the global or local minimum of the potential energy surface. Electronic energies corrected for the zero point energies are summarized in Table 3 for some Ca species considered.

Table 3. Comparison of energies of Ca species.

Entry	Ca species	Energy/kJ mol^−1	Difference
			Entry	Energy/kJ mol^−1
1	[Ca^2+(H2O)6](H2O)	−3182205.5	1–2	−17.9
2	[Ca^2+(H2O)7]	−3182187.5
3	[Ca^2+(H2O)6Cl^−]	−4190773.2	3–4	−12.2
4	[Ca^2+(H2O)5Cl^−](H2O)	−4190761.0
5	[B18C6 + CaCl2] (I)	−7018529.1	5–6	54.8
6	[B18C6 + CaCl2] (II)	−7018583.9
7	[B15C5 + CaCl2] (I)	−6614801.7	7–8	−8.6
8	[B15C5 + CaCl2] (II)	−6614793.1

To determine the hydration number of the aqueous Ca²⁺ ion, we first optimized the structures of [Ca²⁺(H₂O)₆](H₂O) (i.e. six H₂O molecules in the primary hydration sphere and one in the secondary hydration sphere) and [Ca²⁺(H₂O)₇] (i.e. seven H₂O molecules in the primary hydration sphere), and compared the energies at the optimized structures. Asa result, [Ca²⁺(H₂O)₆](H₂O) was found to be more stable by 17.9 kJ mol⁻¹ than [Ca²⁺(H₂O)₇], as shown in Table 3. We thus regarded [Ca²⁺(H₂O)₆] as the simply hydrated Ca species in aqueous HCl solution. The hydration number of 6 is within the range of 5 to 13 reported as hydration number of aqueous Ca²⁺ ion [22]. The optimized structure of [Ca²⁺(H₂O)₆] is shown in Figure 11a.

Figure 10. Three-isotope plot of Run Ca18-8. ○, x = 43; □, x = 44; •, x = 48.

Figure 11. The optimized structures of (a) [Ca²⁺(H₂O)₆] and (b) [Ca²⁺(H₂O)₆Cl⁻].

In Ca²⁺ ion-bearing aqueous HCl solution, part of the Ca²⁺ ions form complex species with Cl⁻ ions,

where (aq) indicates these species are hydrated. As the structure of the species, calcium monochloride cation (CaCl⁺(aq)), we considered [Ca²⁺(H₂O)₅Cl⁻] and [Ca²⁺(H₂O)₆Cl⁻]. Since the hydration number of the aqueous Ca²⁺ ion in the primary hydration sphere wassix in our calculations above, we first assumed thecoordination structures of [Ca²⁺(H₂O)₅Cl⁻] and [Ca²⁺(H₂O)₆]Cl⁻ in which the Cl⁻ ion was in the secondary salvation sphere. The converged structure ofthe latter was, however, [Ca²⁺(H₂O)₆Cl⁻] with thecoordination number of seven and not [Ca²⁺(H₂O)₆]Cl⁻ with the coordination number of six. The optimized [Ca²⁺(H₂O)₆Cl⁻] was more stable by 12.2 kJ mol⁻¹ than [Ca²⁺(H₂O)₅Cl⁻](H₂O) as shown in Table 3. Thus, we decided to regard [Ca²⁺(H₂O)₆Cl⁻], an inner-sphere complex in which the Cl⁻ ion directly interacts with the Ca²⁺ ion, as thecomplex species of the Ca²⁺ ion with the Cl⁻ ion in aqueous HCl solution. The optimized structure of [Ca²⁺(H₂O)₆Cl⁻] is depicted in Figure 11b .

The stability constant of Equation (10) is 1.2 at theionic strength of 0.7 mol dm⁻³ and 25°C [23]. Applying this value for the present systems with HCl concentrations of 9 mol dm⁻³ and Ca²⁺ ion concentration of 0.05 mol dm⁻³, the percentage of the complex species is calculated to be 91.6%. Since the ionic strength in our chromatographic experiments was 9 to 12 mol dm⁻³, quite different from 0.7 mol dm⁻³, this value is not directly applicable to the present systems. It must be true, however, that the majority of Ca²⁺ ions in our chromatographic experiments are in the complex form with Cl⁻ ions, CaCl⁺(aq), and the proportion of the complex form is higher at a higher HCl concentration.

The optimized structures of [B18C6 + CaCl₂], [B18C6 + CaCl₂ + HCl] and [B18C6 + CaCl₂ + 2HCl], the model species expected for the crown ether phase ofthe B18C6 system, are shown in Figures 12a–c, respectively. In each of these structures, the six oxygen atoms of the B18C6 molecule and the Ca²⁺ ion are nearly coplanar, since the size of the Ca²⁺ ion fits that of the hole (ring) formed by the six oxygen atoms [19]. For [B18C6 + CaCl₂], we considered two conformers, Conformer I in which the two Cl⁻ ions of CaCl₂ are bonded to the Ca²⁺ ion from one side of the plane, and the other (Conformer II) in which one of the two Cl⁻ ions is bonded to the Ca²⁺ ion from one side of the plane and another Cl⁻ ion from the other side of the plane. Calculations showed that Conformer II is more stable by 54.8 kJ mol⁻¹ than Conformer I, and so, we regarded Conformer II as [B18C6 + CaCl₂], which is depicted in Figure 12a. In [B18C6 + CaCl₂ + HCl], the HCl molecule interacts with a Cl⁻ ion of CaCl₂ through its H atom. In [B18C6 + CaCl₂ + 2HCl], each of the two HCl molecules interacts with each of the two Cl⁻ ions of CaCl₂. Although no experimental data exist, we expect the proportion of [B18C6 + CaCl₂ + HCl] and [B18C6 + CaCl₂ + 2HCl] increases with increasing HCl concentration.

Figure 12. The optimized structures of (a) [B18C6 + CaCl₂], (b) [B18C6 + CaCl₂ + HCl] and (c) [B18C6 + CaCl₂ + 2HCl].

The optimized structures of [B15C5 + CaCl₂], [B15C5 + CaCl₂ + HCl], and [B15C5 + CaCl₂ + 2HCl], the model species expected for the crown ether phase ofthe B15C5 system, are shown in Figures 13 a–c, respectively. In each of these structures, the five oxygen atoms of the B15C5 molecule are nearly coplanar and the Ca²⁺ ion is located slightly above the plane. This is because the size of a Ca²⁺ ion is larger than that of the hole (ring) formed by the five oxygen atoms of the B15C5 molecule [19]. For [B15C5 + CaCl₂], we considered two conformers, Conformer I in which the two Cl⁻ ions of CaCl₂ are bonded to the Ca²⁺ ion from the same side of the plane as the Ca²⁺ ion is located and Conformer II in which one of the two Cl⁻ ions is bonded to the Ca²⁺ ion from one side of the plane and another Cl⁻ ion from the other side of the plane. Contrastively to the case of [B18C6 + CaCl₂], Conformer I is more stable by 8.6 kJ mol⁻¹ than Conformer II, and so, we decided to regard Conformer I as [B15C5 + CaCl₂], which is depicted in Figure 13a. As in the case of [B18C6 + CaCl₂ + HCl] and [B18C6 + CaCl₂ + 2HCl], the HCl molecule interacts with a Cl⁻ ion of CaCl₂ through its H atom in [B15C5 + CaCl₂ + HCl], and each of the two HCl molecules interacts with each of the two Cl⁻ ions of CaCl₂ in [B15C5 + CaCl₂ + 2HCl]. Although no experimental data exist, we expectthe proportion of [B15C5 + CaCl₂ + HCl] and [B15C5 + CaCl₂ + 2HCl] increases with increasing HCl concentration.

Figure 13. The optimized structures of (a) [B15C5 + CaCl₂], (b) [B15C5 + CaCl₂ + HCl] and (c) [B15C5 + CaCl₂ + 2HCl].

The rpfrs for the ⁴⁸Ca/⁴⁰Ca isotopic pair of the considered Ca species at 50°C are summarized in Table4. First of all, we notice that the rpfrs for the species in the solution phase ([Ca²⁺(H₂O)₆] and [Ca²⁺(H₂O)₆Cl⁻]) are larger than those for the species in the crown ether phases ([B18C6 + CaCl₂], [B18C6 + CaCl₂ + HCl] and [B18C6 + CaCl₂ + 2HCl] in theB18C6 system and [B15C5 + CaCl₂], [B15C5 + CaCl₂ + HCl], and [B15C5 + CaCl₂ + 2HCl] in the B15C5 system). This means that the sum of the forces acting on the Ca²⁺ ion in the solution phase is larger than the sum in the crown ether phases, and as a consequence, the heavier Ca isotopes are preferentially fractionated into the aqueous solution phase and the lighter ones into the crown ether phases in both the B18C6 and the B15C5 systems if the aqueous and crown ether phases are in equilibrium with each other concerning Ca isotopes [15]. The MO results are thus consistent with the results of the chromatographic Ca isotope fractionation experiments (finding (1) in Section 3.3). InTable 4, we also listed the rpfr value of Ca²⁺(H₂O)₇, a higher energy model of the aqueous Ca²⁺ ion; its value is larger than those of the species inthe crown ether phases. Thus, even if we regard Ca²⁺(H₂O)₇ as the aqueous Ca²⁺ ion in the solution phase, the qualitative discussion above does not change at all.

Table 4. The values of the reduced partition function ratios for the ⁴⁸Ca/⁴⁰Ca isotopic pair at 50°C.

Phase	Species	rpfr for ^48Ca/^40Ca
Solution	Ca^2+(H 2 O)6	1.02522
	Ca^2+(H 2 O)7	1.02417
	Ca^2+(H 2 O)6Cl^−	1.02325
Crown ether (B18C6)	B18C6 + CaCl2	1.01751
	B18C6 + CaCl2 + H2O	1.01706
	B18C6 + CaCl2 + HCl	1.01713
	B18C6 + CaCl2 + 2HCl	1.01691
Crown ether (B15C5)	B15C5 + CaCl2	1.01816
	B15C5 + CaCl2 + HCl	1.01845
	B15C5 + CaCl2 + 2HCl	1.01925
	B15C5 + CaCl2 + 2HCl + H2O	1.01888

We also notice in Table 4 that the rpfrs of the Ca species in the crown ether phase of the B15C5 system are larger than those of the B18C6 system, from which we can expect that, under similar experimental conditions, the Ca isotopic fractionation coefficient is larger for the B18C6 system than for the B15C5 system. This also qualitatively agrees with the present experimental results (finding (3) in Section 3.3.).

The rpfr of [Ca²⁺(H₂O)₆] is larger than that of [Ca²⁺(H₂O)₆Cl⁻]; the former minus the latter is 1.02522 − 1.02325 = 0.00197. This indicates that the rpfr value of the solution phase decreases with increasing HCl concentration since the proportion of [Ca²⁺(H₂O)₆Cl⁻] is expected to be higher at a higher HCl concentration. The decreasing order of the rpfrs of the species in the crown ether phase of the C18B6 system is: [B18C6 + CaCl₂] > [B18C6 + CaCl₂ + HCl] > [B18C6 + CaCl₂ + 2HCl]; the differences between the neighboring values are 0.00038 and 0.00022, respectively. This indicates that, like the rpfr of the resin phase of the B18C6 system, the rpfr of the solution phase also decreases with increasing HCl concentration. However, comparing the value 0.00197 with 0.00038 and 0.00022, the dependence of the rpfr value on the HCl concentration is expected to be much more significant in the solution phase than in the crown ether phase. Thus, the present MO calculations indicate that the ε value decreases with increasing HCl concentration, which is in qualitative agreement with the experiments (finding (5) in Section3.3.).

The decreasing order of the rpfrs of the species in the crown ether phase of the C15B5 system is: [B15C5 + CaCl₂ + 2HCl] > [B15C5 + CaCl₂ + HCl] > B15C5 + CaCl₂]. This means that, contrastively to the B18C6 system, the existence of HCl in the crown ether phase enhances the rpfr of that phase in the B15C5 system; the rpfr of the crown ether phase is an increasing function of HCl concentration. Coupled with its effect on the rpfr of the solution phase, the increase in HCl concentration decreases the ε value in the B15C5 system, probably in a more significant way than in the B18C6 system. This is again consistent with the experiments (finding (5) in Section 3.3.).

As a summary of the MO calculations above, if weassume the existence of such Ca species as [Ca²⁺(H₂O)₆], [Ca²⁺(H₂O)₆Cl⁻], [B18C6 + CaCl₂], [B18C6 + CaCl₂ + HCl], [B18C6 + CaCl₂ + 2HCl], [B15C5 + CaCl₂], [B15C5 + CaCl₂ + HCl] and [B15C5 + CaCl₂ + 2HCl], most of the trends in the Ca isotope effects observed in the present chromatographic experiments are, albeit in a qualitative fashion, explainable.

In additional MO calculations, we examined effects of the existence of H₂O molecules in the crown ether phases. The results are also included in Table 4. Twoclusters, [B18C6 + CaCl₂ + H₂O] and [B15C5 + CaCl₂ + 2HCl + H₂O] were considered as examples. In the optimized structure of [B18C6 + CaCl₂ + H₂O], the H₂O molecule was H-bonded to a Cl⁻ ion of CaCl₂, and it was H-bonded to the Cl⁻ ion of an HCl molecule in [B15C5 + CaCl₂ + 2HCl + H₂O]. In both the cases, the rpfr value of the cluster with a H₂O molecule is slightly smaller than that of corresponding cluster without it, which indicates that the presence of H₂O molecules slightly decreases the rpfr values in the crown ether phases.

4. Conclusion

The following statements summarize our present study.

1.	Heavier isotopes of calcium were enriched in the frontal part of the chromatograms obtained in the chromatographic experiments operated in the breakthrough mode in which the benzo-18-crown-6 ether resin synthesized in the inner pores of highly porous silica beads functioned as the absorbent of calcium ions. A similar tendency in calcium isotope fractionation was observed in the chromatographic experiments with the benzo-15-crown-5 ether resin. The fractionation coefficient for the given isotopic pair, x Ca/40Ca (x = 42, 43, 44, or 48), was larger with the benzo-18-crown-6 ether resin than with benzo-15-crown-5 ether resin, which was consistent with the molecular orbital calculations on the calcium isotopic reduced partition function ratios of the calcium species assumed to be present in the solution and crown ether phases of the chromatographic experiments. The calcium species assumed for the calculations were a simply hydrated calcium ion and an aqueous calcium ion directly interacting with a chloride ion in the solution phase, and those in the crown ether phase were calcium chloride interacting with a crown ether molecule (benzo-18-crown-6 ether or benzo-15-crown-5 ether) accompanied by zero, one or two hydrochloric acid molecules.
2.	The observed calcium isotope effects were most probably mass dependent; the values of the fractionation coefficients per unit mass difference between two calcium isotopes were nearly constant, albeit sometimes with large experimental uncertainties, and the slopes of the three-isotope plot curves were very close to the theoretical values that were inferred from the isotope effectsbased solely on the molecular vibration.
3.	A higher hydrochloric acid concentration in the feed solution resulted in a smaller value of the fractionation coefficient in both the benzo-18-crown-6 ether and the benzo-15-crown-5 ether systems. Support for this was provided by the results of the molecular orbital calculations in aqualitative fashion.