A Raman spectroscopic investigation of speciation in La₂(SO₄)₃(aq)

Abstract

Raman spectroscopic measurements have been made of aqueous solutions of La(ClO₄)₃, La₂(SO₄)₃, and Na₂SO₄ in water and heavy water, in the terahertz frequency region (40–1400 cm⁻¹) and down to low concentrations (0.000263 mol L⁻¹). Temperature dependent measurements of a 0.0098 mol L⁻¹ La₂(SO₄)₃ solution have been carried out from 23–98 °C. In solutions of La(ClO₄)₃ with water and heavy water, the [La(OH₂)₉]³⁺ and [La(OD₂)₉]³⁺ have been characterized and a weak, strongly polarized band observed at 343 cm⁻¹ and 326 cm⁻¹ respectively assigned to the ν₁ LaO₉ mode, the breathing mode of the clusters. In La₂(SO₄)₃(aq), in addition to the ν₁-SO₄²⁻ mode at 980 cm⁻¹, a pronounced band component at 991 cm⁻¹ has been assigned to an inner-sphere complex (ISC) and a similar ν₁-SO₄²⁻ band contour has been observed in La₂(SO₄)₃ solutions in D₂O. Sulfate may act as a monodentate ligand. Conformation of this assignment is provided by the component at 312 cm⁻¹ of the [La(OH₂)₈OSO₃]⁺ species in addition to the band at 343 cm⁻¹ for the fully hydrated cluster, [La(OH₂)₉]³⁺. After subtraction of the component of the ISC at 991 cm⁻¹, the ν₁-SO₄²⁻ band in La₂(SO₄)₃(aq) showed systematic differences from that in Na₂SO₄(aq). This is consistent with a ν₁-SO₄²⁻ band at 983.3 cm⁻¹ that can be assigned to the existence of an outer-sphere complex (OSCs). The observed change of the degree of sulfato-complex formation with dilution reflects the stepwise sulfato-complex formation. A K₃-value has been determined at 0.9 of the equilibrium between OSC and ISC. Temperature dependent measurements on a dilute La₂(SO₄)₃ solution has shown that the concentration of the La³⁺ sulfato-complex rises with increasing temperature while at the same time the concentration of the “free” sulfate diminished. The sulfato-complex formation is an endothermic process absorbing heat with increasing temperature. The following thermodynamic parameters for the rate determining equilibrium, [La(OH₂)SO₄]⁺ ↔ [LaOSO₃]⁺ has been determined: ΔH⁰ = 18.6 kJ mol⁻¹ and ΔS⁰ = 62.1 J mol⁻¹ K⁻¹.

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1. Introduction

Lanthanum sulfate enneahydrate, La₂(SO₄)₃·9H₂O, is the most common hydrate of lanthanum sulfate, La₂(SO₄)₃·nH₂O (n = 1–9)^1–3 and is the only stable hydrate in contact with its solution between 0° and 100°. La₂(SO₄)₃ is sparingly soluble at room temperature and the solubility diminishes with temperature. Lanthanum exists in aqueous solution exclusively in the trivalent state and the La³⁺ ion is strongly hydrated due to its high charge to radius ratio. The coordination number (CN) in the primary hydration shell for La³⁺ in aqueous solution has been determined at nine (data in Table 1, ref. 4). The water molecules of the nonahydrate results in a hydration structure of a tricapped trigonal prism (TTP) with D₃ symmetry.⁵ The O-atoms of the three waters in the equatorial plane (capping position) are separated from the cation by a bond distance of 2.64 Å, while six water molecules at the vertices of the trigonal prism have a La–O bond distance of 2.515 Å.^5,6 A recent Raman spectroscopic investigation revealed that in concentrated La(ClO₄)₃(aq) ion pairs are formed while in dilute solutions the undisturbed [La(H₂O)₉]³⁺ species exist.

This study has been undertaken to characterize the species formed in aqueous La³⁺–sulfate solutions over a very broad concentration range and at elevated temperatures. Sulfato-complex formation has been characterized in solutions on a variety of divalent and trivalent metal sulfates^7–12 and the question arises whether these complexes also occur with La³⁺(aq). The investigation of La₂(SO₄)₃ solutions proved to be especially challenging because lanthanum sulfate is only sparingly soluble in water and the salt shows a retrograde solubility with temperature increase. The study on the aqueous La₂(SO₄)₃ is the first Raman experimental investigation one this dilute system. The following aqueous systems have been measured using Raman spectroscopy at 23 °C: La(ClO₄)₃, La₂(SO₄)₃ and Na₂SO₄. Specifically, we have been interested in the vibrational characterization of the La³⁺–aqua stretching band at low concentration and the possible formation of ion pairs/complexes between La³⁺ and sulfate. Dilute Na₂SO₄ solutions have been measured to characterize the sulfate modes free of sulfato-complex species. A La₂(SO₄)₃ solution at 0.0098 mol L⁻¹ has been measured from 23 °C to 98 °C in order to characterize the sulfato-complex formation as a function of temperature. Thermodynamic parameters such as enthalpy and entropy of the sulfato-complex formation have been determined from the measured K₃ values as a function of temperature.

2. Experimental details

2.1. Preparation of solutions

Lanthanum perchlorate solutions were prepared from La₂O₃ (Sigma-Aldrich, 99.9%) and HClO₄ in a beaker until all oxide dissolved. The lanthanum ion content was analysed by complexometric titration.¹³ The solution density was determined with a pycnometer at 23 °C and the molar ratios water per salt were calculated (R_w-values). A La(ClO₄)₃ stock solution was prepared at 2.488 mol L⁻¹ (R_w = 15.72)⁴ and the solution was slightly acidic with a pH value at ∼3.5. From the stock solution, the following dilution series was prepared: 0.622 mol L⁻¹ (R_w = 81.54), 0.498 mol L⁻¹ (R_w = 103.43) and 0.249 mol L⁻¹ (R_w = 213.75). The solutions were analyzed for dissolved chloride with a 5% AgNO₃ solution. The absence of a white AgCl precipitate proved that the stock solution was free of Cl⁻. For Raman spectroscopic measurements the solutions were filtered through a fine sintered glass frit (1–1.6 μm pore size). The solutions showed no Tyndall effect and were “optically empty” (see e.g.ref. 14).

Lanthanum sulfate is only sparingly soluble in water. A 0.0376 mol L⁻¹ La₂(SO₄)₃ stock solution (0.0378₄ mol kg⁻¹) was prepared from La₂(SO₄)₃·9H₂O and was used to further prepare dilute solutions. The solution density of the stock solution and its La³⁺ were determined as described above. The pH of the stock solution was determined with a glass electrode equal to 5.52. Nine solutions were prepared from this stock solution: 0.0302 mol L⁻¹, 0.0189 mol L⁻¹, 0.0093 mol L⁻¹, 0.00754 mol L⁻¹, 0.00606 mol L⁻¹, 0.00377 mol L⁻¹, 0.00189 mol L⁻¹, 0.00121 mol L⁻¹, 0.00094 mol L⁻¹ and 0.00026 mol L⁻¹. The lanthanum ion content was analysed by complexometric titration and the solution density was determined with a 5.000 mL pycnometer at 23 °C.

2.2. Spectroscopic measurements

Raman spectra were measured in the macro chamber of the T 64000 Raman spectrometer from Jobin Yvon in a 90° scattering geometry at 23 °C. These measurements have been described elsewhere in detail.^15,16 Briefly, the spectra were excited with the 514.5 nm line of an Ar⁺ laser at a power level of 1100 mW at the sample. After passing the spectrometer in subtractive mode, with gratings of 1800 grooves per mm, the scattered light was detected with a cooled CCD detector. I_VV and I_VH spectra were obtained with fixed polarisation of the laser beam by rotating the polarisator at 90° between the sample and the entrance slit to give the scattering geometries:

I_VV = I(Y[ZZ]X) = 45α′² + 4γ′² (1)

I_VH = I(Y[ZY]X) = 3γ′². (2)

The isotropic spectrum, I_iso is then constructed:

I_iso = I_VV − 4/3I_VH. (3)

The depolarization ratio, ρ, of the modes were determined according to eqn (4):

ρ = I_VH/I_VV = 3γ′²/(45α′² + 4γ′²). (4)

The polarization analyser was calibrated with CCl₄ before each measuring cycle and adjusted if necessary. The depolarisation ratio of the ν₁ mode of CCl₄ at 459 cm⁻¹ was determined at 0.0036 ± 0.0005 (15 independent measurements). The depolarization ratios of the depolarized CCl₄ bands at 217 cm⁻¹ and 315 cm⁻¹ have been determined at 0.75 ± 0.02.

In order to obtain spectra defined as I([small nu, Greek, macron]) which are independent of the excitation wavenumber ν_L, the measured Stokes intensity should be corrected for the scattering factor (ν_L − ([small nu, Greek, macron]))³. When counting methods are used, the measured count rates have to be corrected with the factor (ν_L − ([small nu, Greek, macron]))³. The spectra were further corrected for the Bose–Einstein temperature factor, B = [1 − exp(−h[small nu, Greek, macron]c/kT)] and the frequency factor, [small nu, Greek, macron], to give the so called reduced or R([small nu, Greek, macron]) spectrum. It is also possible to calculate the isotropic spectrum in R-format from the corrected R_VV and R_VH spectra according to eqn (5):

R([small nu, Greek, macron])_iso = R([small nu, Greek, macron])_VV − 4/3R([small nu, Greek, macron])_VH. (5)

In the low wavenumber region the I([small nu, Greek, macron]) and R_Q([small nu, Greek, macron]) spectra are significantly different and only the spectra in R-format are presented. It should be noted that one of the advantages of using isotropic R-spectra is that the baseline is almost flat in the 50–700 cm⁻¹ wavenumber region allowing relatively unperturbed observation of the presence of any weak modes.^15,16

The spectra were fitted to a sum of Lorentzian–Gaussian product functions and the fitting procedure has been described elsewhere in detail.¹⁷

Temperature dependent Raman measurements were carried out in sealable fused quartz cuvettes, 10 × 10 mm, with 3.5 mL volume from Hellma (Müllheim, Germany) using a home-built oven. The oven and the setup have been described in detail in ref. 17.

2.3. Quantitative Raman measurements of the sulfato complex formation

In order to be able to quantify species in solution without using an internal standard, the relative molar scattering factors (RMSF) for those bands must be determined.¹⁸ The RMSF for ν₁-SO₄²⁻ at 980 cm⁻¹ for “free” SO₄²⁻ in (NH₄)₂SO₄ solutions were measured in solutions in which a known amount of NH₄ClO₄ had been dissolved as an internal standard (ClO₄⁻ band ν₁ at 931 cm⁻¹). From the integrated band intensities, A_i, and the concentration of the standard, the relative integrated band intensities, I_R, with I_R = (A₉₈₀/A₉₃₁)C_NH4ClO4 were calculated. The quantity I_R is directly proportional to the concentration of the species of interest and thus: I_R = JC. From the slope of the curve, I_R = JC, the so called relative molar scattering coefficient, J was obtained. For ν₁ SO₄²⁻ at 980 cm⁻¹, it follows J₉₈₀ = 0.788 and for bound sulfate at 991 cm⁻¹: J₉₉₁ = 0.760.

The equilibrium concentrations of [SO₄²⁻]_free and [SO₄²⁻]_bound were obtained from the ratios of the areas of “free” SO₄²⁻ mode at 980 cm⁻¹ (this mode includes contributions from the fully hydrated sulfate, in the outer-outer-sphere and outer-sphere^30–33) and the bound SO₄²⁻ mode at 991 cm⁻¹ and the relative molar scattering factors. The sum of the equilibrium concentrations [SO₄²⁻]_free plus [SO₄²⁻]_bound is equal to C₀, the stoichiometric concentration of MnSO₄ (in mol kg⁻¹). The integrated band intensities are A₉₈₀ = J₉₈₀[SO₄²⁻]_free and A₉₉₁ = J₉₉₁[SO₄²⁻]_bound = J₉₉₁(C₀ − [SO₄²⁻]_free), where J₉₈₀ and J₉₉₁ are the relative molar scattering factors for the bands at ∼980 and 991 cm⁻¹. For the ratio of the integrated intensities, A₉₈₀/A₉₉₁ was calculated as follows:

and after rearranging for [SO₄²⁻]_free it follows:

With the equilibrium concentration of [SO₄²⁻]_free determined and C₀, the stoichiometric concentration of La₂(SO₄)₃ known, the degree of bound sulfate, α, may be calculated by eqn (8):

α = (C₀ − [SO₄²⁻]_free)/C₀. (8)

3. Results and discussion

3.1. The unassociated [La(OH₂)₉]³⁺(aq)

The La³⁺(aq) ion is strongly hydrated in aqueous solution due to its high charge to radius ratio. A recent DFT study on a [Ln(H₂O)₉]³⁺ cluster imbedded in a polarizable dielectric continuum, taking into account the bulk water resulted in a tricapped trigonal prism (TTP) polyhedron of symmetry D₃.⁴ The oxygen atoms of three water molecules in the equatorial plane are separated from La³⁺ by a bond distance of 2.64 Å, while six water molecules at the vertices of the trigonal prism are found at an average La–O bond distance at 2.515 Å.^5,6 The hydration sphere of La³⁺(aq) is labile and a water-exchange rate constant k_ex at 25 °C was given at ≥2 × 10⁸ s⁻¹‡ (estimated from H₂O–SO₄²⁻ interchange rates) and a water residence time τ = ∼5 ns follows.^19,20 Because of the labile coordination sphere for La³⁺ there is the possibility of a counterion (anion) coordination replacing water from the coordination polyhedron.²¹

Although perchlorate is considered a non-complex forming anion, in La(ClO₄)₃ solutions at higher salt concentrations (1.5 mol L⁻¹), it was shown that perchlorate forms outer sphere-ion pairs, [La(OH₂)₉]³⁺·ClO₄⁻ and at yet higher concentrations perchlorate penetrates into the first hydration sphere of the La³⁺.⁴ In La(ClO₄)₃(aq) at 2.488 mol L⁻¹ the mole ratio solute to water is 15.7 and this amount of water is barely enough to completely hydrate the La³⁺ ion while the remaining 6.7 water molecules hydrate the three ClO₄⁻ ions. In such a concentrated solution contact ion pair formation is simply forced on La³⁺. Therefore, it is necessary to measure dilute La(ClO₄)₃ solutions but this measurement is hampered by the fact that the La–O breathing mode is very weak and its Raman scattering coefficient, 0.030, is low.

A vibrational analysis of the LaO₉ skeleton modes of the [La(OH₂)₉]³⁺ species (D_3h symmetry) has been carried out recently.⁴ Twenty four normal modes (n.m.'s) are expected for the LaO₉ skeleton modes of the nonahydrate [La(OH₂)₉]³⁺. The irreducible representation of the vibrational modes is as follows: Γ_vib(D_3h) = 3a′₁(Ra) + a′₂ + 5e′(Ra, i.r.) + a′′₁ + 3a′′₂(i.r.) + 3e′′(Ra). Two stretching modes and a bending mode are expected with character a′₁ and these modes are Raman active only. Two stretching modes and three bending modes occur with character e′ and these modes are infrared and Raman active. A stretching mode and two bending modes for character a′′₂ are expected and these n.m.'s are infrared active only while three modes (one stretch and two bending modes) with character e′′ are only Raman active. The modes with character a′₂ and a′′₁ are torsional modes and are not active. However, it was evident that only the symmetric La–O stretching mode appears in the Raman effect. The remaining LaO₉ modes are too weak to be identified in Raman and this has also been observed for other metal ion–oxygen modes in aqueous solution.⁴ The La(ClO₄)₃ solution spectra reveals a Raman mode for the [La(OH₂)₉³⁺] species, which is very weak, strongly polarized and broad at 343 cm⁻¹. A 0.249 mol L⁻¹ La(ClO₄)₃ solution is presented in Fig. 1. In neither NaClO₄(aq) nor HClO₄(aq) was this mode observed and must therefore stem from a symmetrical vibration connected to a La–O stretching band.

Fig. 1 Raman scattering profiles of a 0.249 mol L⁻¹ (R_w = 214) La(ClO₄)₃(aq). Upper panel: R_pol and R_depol scattering. Note the very weak band at 343 cm⁻¹ due to the symmetric breathing mode of [La(OH₂)₉]³⁺. In addition, the deformation modes of ClO₄⁻(aq) which dominate the spectrum at 460 cm⁻¹ and 628 cm⁻¹ are shown. The band at 178 cm⁻¹ (R_VV) is due to the restricted O–H⋯O bonds of the water H-bond network. Lower panel: Isotropic scattering in R-format. Note the very weak band at 343 cm⁻¹ due to the symmetric breathing L–O mode. The isotropic mode at 182 cm⁻¹ is due to the restricted O–H⋯O bonds of the water H-bond network.

The perchlorate modes are well characterized so only a brief description is given.⁴ The ClO₄⁻ ion possesses T_d symmetry and has nine modes of internal vibrations spanning the representation Γ_vib(T_d) = a₁(Ra) + e(Ra) + 2f₂(Ra, i.r.). All n.m. are Raman active, but in i.r. only the f₂ modes are active. The spectra of La(ClO₄)₃(aq) show the predicted four Raman-active bands for the tetrahedral ClO₄⁻(aq). The ν₁(a₁) ClO₄⁻ band centred at 931.5 cm⁻¹ is totally polarized (ρ = 0.005) whereas ν₃(f₂) ClO₄⁻ centred at 1105 cm⁻¹ is depolarized as are the deformation modes ν₄(f₂) ClO₄⁻ at 631 cm⁻¹ and ν₂(e) ClO₄⁻ at 463 cm⁻¹ (Fig. 1).

In addition to the isotropic mode, ν₁ LaO₉ at 343 cm⁻¹ (fwhh = 49 ± 2 cm⁻¹) a very weak, broad mode centered at 170 ± 10 cm⁻¹ can be observed in aqueous La(ClO₄)₃ solution (isotropic Raman scattering). The mode can also be seen in pure water at ∼175 cm⁻¹ and is moderately intense but slightly polarized. This mode has been assigned to a restricted translational mode of the H-bonded water molecules and is strongly anion and concentration dependent.^4,22 A La(ClO₄)₃ solution in heavy water shows, in addition to the perchlorate modes, a strongly polarized band at 326 cm⁻¹ which is due to the ν₁ LaO₉ at 326 cm⁻¹ (fwhh = 47 ± 2 cm⁻¹) of the deuterated analog of the nonahydrate, [La(OD₂)₉]³⁺(D₂O).

Relative intensity measurements confirm that the scattering intensity of ν₁ La–O mode is very weak and this may be the reason why the mode has been observed as an obscured shoulder in aqueous La³⁺ salt solutions in previous publications.²³ The scattering coefficient, S_h for the ν₁ La–O mode at 0.030 is small.

3.2. The unassociated SO₄²⁻(aq)

Unassociated SO₄²⁻(aq), in (NH₄)₂SO₄(aq) for instance, has been described in detail.²⁴ Although in Na₂SO₄ solutions at higher concentrations, ion pairs such as outer-outer sphere and outer-sphere ion-pairs are formed, in dilute solutions <0.1 mol L⁻¹ the equilibrium concentrations are small relative to “free” sulfate. In a 0.101 mol L⁻¹ Na₂SO₄ solution ∼13% of the total sulfate exists as ion pairs, mainly as outer-outer-sphere ion pairs while 87% are considered “free” or unassociated, SO₄²⁻(aq).²⁵ With further dilution, the fraction of ion pairs diminishes significantly and reaches zero at infinite dilution. Therefore, dilute Na₂SO₄ solutions shall be briefly discussed and compared with La₂(SO₄)₃ solutions. An overview Raman spectrum of a 0.101 mol L⁻¹ Na₂SO₄(aq) from 100–1900 cm⁻¹ is presented in Fig. 2. The spectral characteristics of the non-associated SO₄²⁻(aq) ion have been conducted.²⁶ Here, however, a brief discussion of the results relevant to the ligated sulfate is presented.

Fig. 2 Overview Raman profiles of a 0.101 mol L⁻¹ Na₂SO₄ solution. The I_VV and I_VH scattering orientations (upper panel) are given and in the lower panel the isotropic scattering.

The non-associated SO₄²⁻(aq) ion possesses T_d symmetry and thus nine modes of internal vibration, having the representation: Γ_vib(T_d) = a₁(R) + e(R) + 2f₂(R, IR). All vibrational modes are Raman active, one mode with character a₁ is totally polarized, and three modes with character e and f₂ are depolarized but only the f₂ modes are IR allowed.

In the IR spectrum only two sulfate bands are active, namely ν₄(f₂) and ν₃(f₂) located at 618 cm⁻¹ and 1104 cm⁻¹ respectively²⁴ but all four modes are Raman active. Detailed Raman spectroscopic characterizations of the SO₄²⁻(aq) bands are given in ref. 24. The ν₁(a₁)-SO₄²⁻ mode is the strongest band in the sulfate spectrum. The band for a 0.101 mol L⁻¹ Na₂SO₄(aq) centred on 980.69 cm⁻¹ (fwhh = 6.05 ± 0.05 cm⁻¹) is almost totally polarized (ρ = 0.0035) whereas the ν₃(f₂)-SO₄²⁻ band centred on 1109 cm⁻¹ (fwhh = 65.0 cm⁻¹) is depolarized. The deformation modes ν₄(f₂)-SO₄²⁻ at 614 cm⁻¹ (fwhh = 31 cm⁻¹) and ν₂(e)-SO₄²⁻ at 448 cm⁻¹ (fwhh = 35 cm⁻¹) are depolarized as well.

At low frequencies, a broad and weak mode at 180 cm⁻¹ (fwhh = 168 cm⁻¹) was observed in the isotropic R-spectrum.²⁴ This mode is also observable in pure water at 175 cm⁻¹ and has been attributed to H-bonded water molecules.^26,27 In Na₂SO₄(aq) it may be assigned to the restricted translational modes of both OH⋯O and OH⋯(OSO₃)²⁻ hydrogen-bonds thereby the contribution of the OH⋯(OSO₃)²⁻ hydrogen-bonds increases with increasing Na₂SO₄ concentration. In a concentrated, saturated Na₂SO₄ solution, 1.617 mol L⁻¹, the restricted translational mode, ν OH⋯O was observed at 185 cm⁻¹ with a fwhh at 130 cm⁻¹ while in dilute solutions the restricted translational mode is determined by the OH⋯O hydrogen bonds in water and only slightly influenced by OH⋯(OSO₃)²⁻ hydrogen-bonds. The corresponding bending mode of this unit appears at ∼60 cm⁻¹.^26,27

The ν₁-SO₄²⁻ band shape in dilute Na₂SO₄(aq) is symmetrical in both Raman polarization arrangements but other band parameters vary slightly with concentration (Fig. 3, panel A and B, data with open squares). For example, at 23 °C for a 1.617 mol L⁻¹ Na₂SO₄ solution (R_w = 32.75), ν_max for this isotropic band appears at 981.04 cm⁻¹ with a fwhh = 6.60 cm⁻¹, for a 0.101 mol L⁻¹ solution (R_w = 552.2) ν_max appears at 980.69 cm⁻¹ with a fwhh = 6.05 cm⁻¹. For an infinitely dilute solution (extrapolation to zero concentration) the ν₁-SO₄²⁻ band appears at 980.45 cm⁻¹ and a fwhh = (5.55 ± 0.05) cm⁻¹. For present purposes, it is relevant that the symmetric profile of the ν₁-SO₄²⁻ mode was independent of the salt concentration over the entire range studied (≤1.617 mol L⁻¹). The ν₁-SO₄²⁻ band is the strongest band in the SO₄²⁻(aq) – Raman spectrum and the intensity ratio of ν₁-SO₄²⁻ band area to ν₃-SO₄²⁻ band area is 9 : 1.

Fig. 3 Concentration effects at 23 °C on the ν₁-SO₄²⁻ mode at ca. 980 cm⁻¹ in Na₂SO₄ solutions (open circles) and in La₂(SO₄)₃ solutions (after subtraction of the 991 cm⁻¹ component, filled circles); (A and C): peak position (ν_max); (B and D): fwhh.

The band contour of the antisymmetric S–O stretching mode, ν₃-SO₄²⁻ is also symmetrical in both polarization arrangements but band parameters vary slightly with concentration. For example, at 23 °C for a 1.617 mol L⁻¹ Na₂SO₄ solution, ν_max is found at (1112 ± 2) cm⁻¹ and shifted to slightly lower wavenumbers as the concentration was decreased, reaching (1106 ± 2) cm⁻¹ at infinite dilution. The fwhh decreased from (76 ± 2) cm⁻¹ for a 1.617 mol L⁻¹ salt solution to (67 ± 2) cm⁻¹ for a 0.100 mol L⁻¹ solution reaching (64 ± 2) cm⁻¹ at infinite dilution. It is important to state that the symmetric profile of the ν₃-SO₄²⁻ band which is completely depolarized is independent of the Na₂SO₄ concentration over the entire range studied (Fig. 2).

3.3. Raman spectra of La₂(SO₄)₃(aq)

The Raman spectra of La₂(SO₄)₃ in aqueous solution show additional features to those in the unassociated sulfate spectra and the [La(OH₂)₉]³⁺(aq) mode which have been described above. La₂(SO₄)₃ is only sparingly soluble in water and the Raman signals are therefore of low intensity and the spectra are dominated by the solvent, water. Nevertheless these additional spectroscopic features could be observed and are now discussed in detail.

An overview Raman spectrum of the 0.0376 mol L⁻¹ La₂(SO₄)₃ solution is presented in Fig. 4. Because of the low solute concentration, the sulfate bands appear weak and the scattering spectrum is dominated by the bands of the solute, water. The ν₁(a₁) SO₄²⁻ stretching profile shows two band components at 981 cm⁻¹ and at 991 cm⁻¹. The weak anti-symmetric stretching mode, ν₃ SO₄²⁻ at 1112 cm⁻¹ appears broader compared to a comparable Na₂SO₄ solution. Furthermore, in the terahertz frequency range, the La–O breathing mode, normally at 343 cm⁻¹, appears in the difference spectrum of the 0.0376 mol L⁻¹ La₂(SO₄)₃ solution as a broad feature at ∼328 cm⁻¹ with two separate component bands, one at 343 cm⁻¹ assigned to the La–O breathing mode of [La(OH₂)₉]³⁺(aq) and one at 312 cm⁻¹ assigned to hydrated La³⁺ where sulfate occurs in the first hydration shell, [La(OH₂)₈OSO₃]⁺(aq) (Fig. 5). With further dilution of La₂(SO₄)₃ the very small scattering coefficient of this La–O breathing mode prevents further observation of this band.

Fig. 4 Raman scattering profiles (polarized and depolarized) of a 0.0376 mol L⁻¹ La₂(SO₄)₃ solution in the low wavenumber region from 200–1900 cm⁻¹. In addition to the sulfate modes at 448, 614, 981 and 1112 cm⁻¹ the spectrum is dominated by the solvent with its two very broad librational bands at ∼455 and ∼720 cm⁻¹ of water and the deformation mode at 1637 cm⁻¹ (I_pol scattering). Note that the ν₁ SO₄²⁻ stretching band, the strongest sulfate band is split into two component bands at 982 and 991 cm⁻¹. Both SO₄²⁻ deformation modes at 451 and 614 cm⁻¹ appear on the broad librational band profile as weak bands. The ν₃ SO₄²⁻ antisymmetric stretching band at 1112 cm⁻¹ is also of weak intensity.

Fig. 5 Band fit result on a difference Raman scattering profile (polarized scattering) of a 0.0376 mol L⁻¹ La₂(SO₄)₃ solution from which the water background was subtracted. In addition to the difference Raman spectrum of the broad band feature, the sum curve of the fit and the two band components at 312 cm⁻¹ and 343 cm⁻¹ are shown.

The split of the ν₁(a₁) SO₄²⁻ stretching band into a band at 981 cm⁻¹ and a pronounced shoulder at 991 cm⁻¹ becomes immediately obvious in Fig. 6B. In Na₂SO₄(aq) the strongly polarized mode at 991 cm⁻¹ is absent while in a variety of divalent and trivalent metal sulfate solutions sulfato-complex species could be observed.^8,11,12,24 In Fig. 7, a concentration plot of ν₁(a₁) SO₄²⁻ band profiles are given, showing the concentration dependence of four representative Raman spectra from 0.0303–0.000263 mol L⁻¹. With dilution, the equilibrium concentration of the band at 991 cm⁻¹ diminishes while the one of the “free” sulfate rises. The band parameters for the 980 cm⁻¹ band and the band at 991 cm⁻¹ is given in Table S1.† It is interesting to note that both sulfate stretching modes are strongly polarized and in the anisotropic scattering both component bands appear at 981 and 991 cm⁻¹ (Fig. 6B) and the polarization degrees are 0.004 and 0.006 respectively. This spectroscopic observation contradicts the observation made in a similar system, MgSO₄(aq), for which a dynamic exchange model was proposed to explain the occurrence of the band at 995 cm⁻¹. (The 995 cm⁻¹ band in MgSO₄(aq), however, is the band of a sulfato-complex species.^24,28) The deformation bands of sulfate, ν₂ SO₄²⁻ at 448 cm⁻¹ and the band ν₄ at 614 cm⁻¹ are broadened compared to the ones in Na₂SO₄(aq) of comparable sulfate concentrations (Fig. 2, 3 and 6A). These two deformation bands are depolarized in the Raman scattering as shown in Fig. 2 and 6A. The anti-symmetric stretch ν₃ SO₄²⁻ is much broader than Na₂SO₄(aq) of comparable sulfate concentration and shows a band contour with peak positions at 1088 cm⁻¹, 1127 cm⁻¹ and 1158 cm⁻¹ and in the broad band at 1109 cm⁻¹ for “free” sulfate (compare band contour of Na₂SO₄(aq) spectra). The measured anisotropic band profile of ν₃ SO₄²⁻ is shown in Fig. S1† together with the sum curve of the band fit. All these spectroscopic features of the sulfate in La₂(SO₄)₃ solutions lead to the conclusion that sulfate must have penetrated the first hydration shell of La³⁺(aq) forming a well-defined La³⁺–sulfato complex. In the terahertz frequency range of La₂(SO₄)₃(aq) of a 0.376 mol L⁻¹ solution, as mentioned above, the La–O band, of the LaO₉ skeleton appears as a broad feature at ∼328 cm⁻¹ while the fully hydrated, [La(OH₂)₉]³⁺ occurs at 343 cm⁻¹. The downshift of the La³⁺–sulfato complex, [La(OH₂)₈OSO₃]⁺ compared with the fully hydrated cation was detected in other trivalent-metal sulfate solutions. It should be pointed out that the scattering intensity of the broad La–O features is very small and the solutions are quite dilute. Only in Al₂(SO₄)₃, Ga₂(SO₄)₃, and In₂(SO₄)₃ solutions which are considerably more soluble than La₂(SO₄)₃ and therefore much easier to detect, is such a split into the aqua mode of the fully hydrated species observable without the sulfate and partially hydrated metal–sulfato complex species.^8,11,12

Fig. 6 (A) Raman scattering profiles (polarized, depolarized and isotropic) of a 0.0376 mol L⁻¹ La₂(SO₄)₃ solution in the low wavenumber region from 100–900 cm⁻¹. Shown are the restricted translation band O–H⋯O at 181 cm⁻¹ and the very broad and dominating bands at ca. 455 and 720 cm⁻¹ due to the librational bands of H₂O. The two weak deformation modes of SO₄²⁻ at 448 and 614 cm⁻¹ are “sitting” on the broad librational band structure. (B) Raman scattering profiles of a 0.0376 mol L⁻¹ La₂(SO₄)₃ solution. Upper panel: Wavenumber rage from 850–1250 cm⁻¹ showing the ν₁ SO₄²⁻ and ν₃ SO₄²⁻ stretching region of sulfate. Lower panel: Enlarged wavenumber region from 920–1040 cm⁻¹ depicting the S–O stretching band region. Note the split of the S–O stretching band into a band at 981.5 cm⁻¹ of “free” SO₄²⁻ and coordinated SO₄²⁻ at 991.5 cm⁻¹ appearing as a well pronounced shoulder. Underneath is the depolarization degree as a function of wavenumbers.

Fig. 7 Dilution series of polarized Raman scattering profiles of selected La₂(SO₄)₃ solutions in the wavenumber range from 920–1060 cm⁻¹ (ν₁ SO₄²⁻ stretching region). From bottom to top: (A) 0.000263 mol L⁻¹, (B) 0.00121 mol L⁻¹, (C) 0.00606 mol L⁻¹ and (D) 0.0302 mol L⁻¹ La₂(SO₄)₃ solution. Note the split of the S–O stretching band into a band at 981.3 cm⁻¹ of “free” SO₄²⁻ and coordinated SO₄²⁻ at 991 cm⁻¹ as a separate band component.

An overview Raman spectrum of a 0.035 mol L⁻¹ La₂(SO₄)₃ solution in heavy water is presented in Fig. S2 (ESI†). The spectrum is dominated by the solvent D₂O(l) and the deformation mode at 1204 cm⁻¹, a broad combination band at 1554 cm⁻¹ (ν₂ + ν_L) and the O–D stretching profile with a band doublet at 2382 and 2477 cm⁻¹ as well as a shoulder at 2680 cm⁻¹. A detailed spectrum is given in Fig. S3† showing the ν₁(a₁) SO₄²⁻ band complex with two band components at 981 cm⁻¹ and 991.4 cm⁻¹. This ν₁(a₁) SO₄²⁻ band complex observed in La₂(SO₄)₃(D₂O) compares with the spectra found in solutions of light water (compare Fig. 4). This finding reinforces the assignment of sulfate band components as bands of “free” sulfate and “bound” sulfate at 981 cm⁻¹ and 991 cm⁻¹ respectively.

3.4. Quantitative analysis of sulfato-complex formation in La₂(SO₄)₃(aq)

The log β₁ value for the hydrolysis of [La(OH₂)₉]³⁺ according to: [La(H₂O)₉]³⁺ + H₂O ↔ [La(H₂O)₈OH]²⁺ + H₃O⁺ is at −8.5 for 25 °C.^29,30 Therefore, La³⁺(aq), in contrast to Fe³⁺(aq) which acts as a strong acid in solution (log β₁ = −2.19 (ref. 31)) is not markedly hydrolyzed. A 0.01 mol L⁻¹ La₂(SO₄)₃ solution shows a pH value of 5.53 and therefore the formation of hydrogen sulfate, HSO₄⁻ may be neglected.

The association between La³⁺(aq) and SO₄²⁻(aq) ions is generally thought to occur via a three step process, referred to as the Eigen mechanism,^32,33 in which the strongly hydrated ions combine initially to form an outer-outer-sphere complex. The two interposing water molecules are then lost successively, forming outer-sphere and ultimately inner-sphere complexes. This process is summarized in the following reaction scheme:

where K_i (i = 1 to 3) are the ‘stepwise’ constants for the corresponding equilibria. The first equilibrium step is the diffusion controlled step and defined as:the second step, the intermediate step, resulting in outer-sphere complex is defined as:and the third step, the rate determining step, is defined as:

(The rate constants of the forward and backward reaction, the rate determining step via La³⁺(OH₂)SO₄²⁻ ↔ La³⁺OSO₃²⁻ + H₂O, may be formulated via K₃ = k_f/k_b. The rate constants for forward (k_f) and backward (k_b) reactions were published from earlier ultrasound absorption measurements.^19,20,34,35)

The overall association constant, K_A, as measured by traditional techniques such as potentiometry or conductivity, corresponds to the equilibrium:

La³⁺(aq) + SO₄²⁻(aq) ↔ LaSO₄⁺(aq) (13)

where LaSO₄⁺(aq) refers to all forms of associated {La³⁺, SO₄²⁻} species of 1 : 1 stoichiometry in aqueous solution (thermodynamics makes no distinction between dissolved species of identical stoichiometry but with differing levels of hydration^24,28). The K_A value, the overall association constant for the three step mechanism is related to the individual K_i values viaeqn (14):

K_A = K₁ + K₁K₂ + K₁K₂K₃ (14)

Three K_i values were obtained from ultrasound absorption (UA) experiments^19,20 namely K₁ = 435, K₂ = 1.96 and K₃ = 3.7. The reported K_A value, 4200, derived from UA is too large compared with the recommended K_A value at 3620 at 25 °C.^36–38

The ν₁-SO₄²⁻ band can be fitted with three band components, a component of the ‘free’ SO₄²⁻(aq) band at ∼980 cm⁻¹, a component at 983 cm⁻¹ representing the SO₄²⁻ in the outer-sphere-complex and the mode of the sulfato-complex at 991 cm⁻¹ (Fig. 6B). The sulfato-complex band is quite distinct, while the band of the outer sphere complex is severely overlapped by the SO₄²⁻(aq) band at 980 cm⁻¹. The SO₄²⁻ band of the outer-outer-sphere complex is not distinguishable from the band of the free sulfate.

The degree of the sulfato-complex formation α, represents the spectroscopically-determined fraction of sulfate present as sulfato complex according to eqn (8). Fig. 8 shows the present quantitative Raman results as a function of the stoichiometric solute concentration C₀. The proportion of the inner-sphere complex formed appears almost insensitive to the concentration > 0.003 mol L⁻¹ but below this concentration vanishes more severely. This paradox can be resolved if it is assumed that the inner-sphere complex, [LaOSO₃]⁺ is in equilibrium with the outer-sphere complex, [La(OH₂)SO₄]⁺ and not with the “free” sulfate. The Raman spectroscopically determined K₃ value was determined according to eqn (12). In Table S1 (ESI†) the data of the analytical curve fitting on the ν₁-SO₄²⁻ band profile of the two band components at 981 cm⁻¹ and 991.1 cm⁻¹ are given such as integrated band intensities, A₉₈₁ and A₉₉₁, as well as the degree of sulfato-complex formation of the equilibrium between the outer-sphere complex and the inner-sphere-sulfato complex. From these data a K₃-value was determined at 0.90 ± 0.1 at 23 °C and it appears that the Raman derived K₃ value is comparable to the UA-values considering the uncertainty of the data from the UA measurements.^19,20,34,35

Fig. 8 Degree of sulfato complex-formation, α, as a function of the stoichiometric La₂(SO₄)₃ concentration at 23 °C.

Temperature dependent measurements. A 0.0094 mol L⁻¹ La₂(SO₄)₃ solution was measured from 23 to 98 °C and the temperature dependent isotropic scattering profiles are given in Fig. 9. Generally, the peak position of the ν₁-SO₄²⁻ band complex shifts to slightly lower wavenumbers with temperature increase. The band of “free” sulfate at 23 °C peaks at 981.0 cm⁻¹ and shifts to 976.5 cm⁻¹ at 98 °C. A similar peak shift was observed for the band of the coordinated sulfate which peaked at 991 cm⁻¹ at 23 °C and shifts to 987 cm⁻¹ at 98 °C. Furthermore, the band complex (and the band components) broadens with temperature. In addition to these two already resolved band components a third band component assigned to sulfate of the outer-sphere ion pair, La³⁺(H₂O)SO₄²⁻ was detected at 983 cm⁻¹ at 23 °C which shifted to 978.6 cm⁻¹ at 98 °C. The results of a three component band fit on the ν₁-SO₄²⁻ band complex are given in Table S2 (ESI†). The most striking fact, however, is the rise of the band component of the coordinated sulfate band, the so called bound sulfate, and the degree of sulfato-complex formation rises from 0.25 at 23 °C to 0.42 at 98 °C.

Fig. 9 Temperature dependence of a series of isotropic Raman scattering profiles of a 0.0093 mol L⁻¹ La₂(SO₄)₃ solution in the wavenumber range from 900–1080 cm⁻¹ (ν₁ SO₄²⁻ stretching region). From bottom to top: 24.0 °C, 50.0 °C, 75.0 °C and 98.0 °C. Note that with temperature increase the bands at 980.3 cm⁻¹ of “free” SO₄²⁻(aq) and coordinated SO₄²⁻ at 991 cm⁻¹ shift slightly to lower wavenumbers. The intensity of the band at 980.5 cm⁻¹ of “free” SO₄²⁻(aq), however, decreases markedly with the rise in temperature while the intensity of band of the complex bound sulfate at 991 cm⁻¹ increases considerably. At 24 °C ∼25% of the sulfate occurs in the La³⁺–sulfato complex species while at 98.0 °C ∼42% of the sulfate is bound as a sulfato-complex.

The increase of the La³⁺–sulfato complex with enhancing temperature shows that the rate determining third step of reaction (10) is an endothermic process by which heat is absorbed, making the enthalpy change positive. The change of the equilibrium constant as a function of temperature follows the equation:

Plotting ln K₃ as a function of 1/T and assuming that the change of the heat capacity in this small temperature range is zero leads to the enthalpy and the entropy of the last step of the sulfato-complex formation, the outer-sphere, inner-sphere sulfato complex formation according to La³⁺(OH₂)SO₄²⁻ ↔ La³⁺OSO₃²⁻. The following quantitative results were derived from the temperature-dependence of ln(K₃) as a function of the reciprocal temperature (in Kelvin): ΔH⁰ = 18.6 kJ mol⁻¹ and ΔS⁰ = 62.1 J mol⁻¹ K⁻¹. The enthalpy for the rate determining step of eqn (9) compares favorably with the extrapolated value, 15 kJ mol⁻¹, for ΔH⁰ taken from ref. 39.

Finally, it should be mentioned that Malatesta and co-workers⁴⁰ showed, by comparing activity coefficient data on La₂(SO₄)₃(aq) and La(ClO₄)₃(aq), that in the sulfate system of La³⁺ “there is evidence of short-range interactions that add to long-range interactions for lanthanum and bivalent metal sulfates, as if sulfate ions were displacing water from the hydration shells of the cations”. This is exactly what the Raman spectroscopic data show.

4. Conclusions

Raman spectroscopic measurements have been made of aqueous solutions of La(ClO₄)₃, La₂(SO₄)₃, and Na₂SO₄ in water and heavy water, in the terahertz frequency region (40–1400 cm⁻¹) and down to low concentrations (0.000263 mol L⁻¹). Temperature dependent measurements have been carried out on a 0.0098 mol L⁻¹ La₂(SO₄)₃ solution from 23–98 °C. In solutions of La(ClO₄)₃ with water and heavy water, the [La(OH₂)₉]³⁺ and [La(OD₂)₉]³⁺ have been characterized and a weak, strongly polarized band observed at 343 cm⁻¹ and 326 cm⁻¹ respectively assigned to the ν₁ LaO₉ mode, the breathing mode of the clusters. The Raman spectroscopic data suggest that the [La(OH₂)₉]³⁺ ion is thermodynamically stable in dilute perchlorate. No inner-sphere complexes in these dilute solutions could be detected spectroscopically.

In La₂(SO₄)₃(aq), in addition to the ν₁-SO₄²⁻ mode at 980 cm⁻¹, a pronounced band component at 991 cm⁻¹ has been assigned to an inner-sphere complex (ISC). The sulfate is most likely bound as a monodentate ligand ([LaOSO₃]⁺). Conformation of this assignment is provided by the component at 312 cm⁻¹ of the [La(OH₂)₈OSO₃]⁺ unit in addition to the band at 343 cm⁻¹ for the fully hydrated cluster, [La(OH₂)₉]³⁺. A similar ν₁-SO₄²⁻ band contour has been observed in La₂(SO₄)₃ solutions in heavy water. After subtraction of the component of the ISC at 991 cm⁻¹, the ν₁-SO₄²⁻ band in La₂(SO₄)₃(aq) showed systematic differences from that in (NH₄)₂SO₄(aq) and dilute Na₂SO₄(aq). This is consistent with a ν₁-SO₄²⁻ band at 983.3 cm⁻¹ that can be assigned to the existence of an outer-sphere complex ion (OSCs). The band profile of the weaker antisymmetric S–O stretching mode, ν₃-SO₄²⁻ shows asymmetry and has been fitted with four band components including the component of the “free” ν₃-SO₄²⁻ band reinforcing the existence of the ligated sulfate. In aqueous La₂(SO₄)₃ solutions thermodynamically stable ISC have been detected down to very low concentrations at 0.00026 mol L⁻¹ and in addition an OSC, [La(OH₂)SO₄]⁺ and the free non-ligated sulfate. The observed change of the equilibrium concentration of the ISC, [(LaOSO₃)]⁺ with dilution reflects the stepwise sulfato-complex formation. A K₃-value has been determined at ∼0.9 of the equilibrium between OSC and ISC.

Temperature dependent measurements (23–98 °C) on a 0.0098 mol L⁻¹ La₂(SO₄)₃ solution has shown that the concentration of the La³⁺ sulfato-complex rises with increasing temperature while at the same time the concentration of the “free” sulfate diminished. The sulfato-complex formation is an endothermic process absorbing heat with increasing temperature. The following thermodynamic parameters for the rate determining equilibrium, [La(OH₂)SO₄]⁺ ↔ [LaOSO₃]⁺ + H₂O, has been determined: ΔH⁰ = 18.6 kJ mol⁻¹ and ΔS⁰ = 62.1 J mol⁻¹ K⁻¹.

References

1. R. A. Vanderpool, M. A. Khan and R. Frech, J. Mol. Struct., 1991, 245, 255–273.
2. M. H. Choi, M. K. Kim, V. Jo, D. W. Lee, I. W. Shim and K. M. Ok, Bull. Korean Chem. Soc., 2010, 31, 1077–1080.
3. M. S. Wickleder, Chem. Rev., 2002, 102, 2011–2087.
4. W. W. Rudolph and G. Irmer, Dalton Trans., 2015, 44, 295–305,  10.1039/c4dt03003f.
5. J. Näslund, P. Lindqvist-Reis, I. Persson and M. Sandström, Inorg. Chem., 2000, 39, 4006–4011.
6. I. Persson, P. D'Angelo, S. de Panfilis, M. Sandström and L. Eriksson, Chem.–Eur. J., 2008, 14, 3056–3066.
7. W. W. Rudolph, D. Fischer, G. Irmer and C. C. Pye, Proceedings of the XXI International Conference on Raman Spectroscopy, ed. R. Whithnall and B. Z. Chowdhry, IM Publications, Charlton, Chichester, 2008, pp. 672–673.
8. W. W. Rudolph, R. Mason and C. C. Pye, Phys. Chem. Chem. Phys., 2000, 2, 5030–5040.
9. W. W. Rudolph and G. Irmer, Dalton Trans., 2013, 3919–3935.
10. W. W. Rudolph and G. Irmer, Dalton Trans., 2013, 14460–14472.
11. W. W. Rudolph and C. C. Pye, Phys. Chem. Chem. Phys., 2002, 4, 4319–4327.
12. W. W. Rudolph, D. Fischer, M. R. Tomney and C. C. Pye, Phys. Chem. Chem. Phys., 2004, 6, 5145–5155.
13. A. I. Vogel, A Text-Book of Quantitative Inorganic Analysis, Longman, London, 3rd edn, 1961.
14. F. H. Spedding, M. J. Pikal and B. O. Ayers, J. Phys. Chem., 1966, 70, 2440–2449 ; see p. 2441.
15. W. W. Rudolph and G. Irmer, Appl. Spectrosc., 2007, 61, 1312–1327.
16. W. W. Rudolph, D. Fischer and G. Irmer, Appl. Spectrosc., 2006, 60, 130.
17. W. W. Rudolph, J. Chem. Soc., Faraday Trans., 1998, 94, 489–499.
18. W. W. Rudolph, Z. Phys. Chem., 1996, 194, 73–95.
19. D. P. Fay, D. Litchinsky and N. Purdie, J. Phys. Chem., 1969, 73, 544–552.
20. N. Purdie and C. A. Vincent, Trans. Faraday Soc., 1967, 63, 2745–2757.
21. L. Helm and A. E. Merbach, Chem. Rev., 2005, 105, 1923–1959.
22. G. E. Walrafen, J. Chem. Phys., 1962, 36, 1035–1042.
23. H. Kanno, J. Alloys Compd., 1993, 192, 271–273.
24. W. W. Rudolph, G. Irmer and G. Hefter, Phys. Chem. Chem. Phys., 2003, 5, 5253–5261.
25. R. Buchner, S. G. Capewell, G. Hefter and P. M. May, J. Phys. Chem. B, 1999, 103, 1185–1192.
26. W. W. Rudolph and G. Irmer, Appl. Spectrosc., 2007, 61, 1312–1327.
27. M. H. Brooker, G. Hancock, B. C. Rice and J. Shapter, J. Raman Spectrosc., 1989, 20, 683–694.
28. R. Buchner, W. W. Rudolph and G. Hefter, J. Chem. Phys., 2005, 123, 034508 ( J. Chem. Phys. , 2006 , 124 , 247101 ).
29. C. F. Baes and R. E. Mesmer, The Hydrolysis of Cations, Wiley-Interscience, New York, 1976.
30. G. D. Klungness and R. H. Byrne, Polyhedron, 2000, 19, 99–107.
31. A. Stefasson, Environ. Sci. Technol., 2007, 41, 6117–6123.
32. M. Eigen and K. Tamm, Z. Elektrochem., 1962, 66, 93–107.
33. M. Eigen and K. Tamm, Z. Elektrochem., 1962, 66, 107–121.
34. J. Reidler and H. B. Silber, J. Phys. Chem., 1973, 77, 1275–1280.
35. B. Voleišienė, G. Miglinienė and A. Voleišis, J. Acoust. Soc. Am., 1999, 105, 962.
36. S. A. Wood, Chem. Geol., 1990, 82, 159–186.
37. S. A. Wood, Chem. Geol., 1990, 88, 99–125.
38. J. Schijf and R. H. Byrne, Geochim. Cosmochim. Acta, 2004, 68, 2825–2837.
39. D. P. Fay and N. Purdie, J. Phys. Chem., 1970, 74, 1160–1166.
40. F. Malatesta, F. Bruni and N. Fanelli, Phys. Chem. Chem. Phys., 2002, 4, 121–126.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra16900c

‡ The datum for k_ex at 25 °C was given incorrectly in ref. 4 page 298.

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