Cation Hydration and Ion Pairing in Aqueous Solutions of MgCl2 and

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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution 2

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Cation Hydration and Ion Pairing in Aqueous Solutions of MgCl and CaCl Sergej Friesen, Glenn Hefter, and Richard Buchner J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b11131 • Publication Date (Web): 02 Jan 2019 Downloaded from http://pubs.acs.org on January 3, 2019

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The Journal of Physical Chemistry

Cation Hydration and Ion Pairing in Aqueous Solutions of MgCl2 and CaCl2 Sergej Friesen,† Glenn Hefter,‡ and Richard Buchner∗,† †Institut für Physikalische und Theoretische Chemie, Universität Regensburg, Regensburg, Germany ‡Chemistry Department, Murdoch University, Murdoch, WA 6150, Australia E-mail: [email protected]

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Abstract Broadband dielectric relaxation spectroscopy (DRS) has been used to investigate aqueous solutions of MgCl2 and CaCl2 up to concentrations of about 1.8 mol L−1 at 25 °C over the frequency range 0.07 ≤ ν/GHz ≤ 89. Detailed analysis of the dominant solvent mode centered at ∼20 GHz showed that both Mg2+ and Ca2+ are strongly solvated, each immobilizing ∼20 water molecules on the DRS timescale. This is consistent with the formation of two well-defined hydration layers around both cations. The hydration shell of Ca2+ (aq) was found to be slightly more labile compared with Mg2+ (aq). Two or three low intensity solute-related modes were observed at frequencies ≲ 10 GHz for MgCl2 (aq) and CaCl2 (aq), respectively. Two of these modes were attributed to the formation of double-solvent-separated and solvent-shared 1:1 ion pairs. The third mode (observed at very low frequencies and only for CaCl2 solutions) was thought to be due to an ion-cloud relaxation. No evidence was found for “slow” water or, consistent with the strong cation hydration, for contact ion pairs. The overall association constants for MgCl+ (aq) and CaCl+ (aq) calculated from the ion-pairing constants were in good agreement with literature values obtained from other techniques.

Introduction The alkaline earth metal ions magnesium(II) and calcium(II) are ubiquitous in nature. 1 For example, Mg2+ is the second most common cation in seawater 2 (and most other natural water bodies), 3 whilst Ca2+ is the most abundant cation in the human body. 4 Both ions play especially important roles in biological systems. Thus, Mg2+ is essential for stabilization of proteins, polysaccharides, lipids and DNA/RNA molecules, while Ca2+ is critical for bone formation and a plays a key role in signal transduction. 4 As for all ions, the chemical, and hence biological, behaviour of Mg2+ and Ca2+ in aqueous solution is governed by a subtle interplay between their level of hydration and their interactions with other ions, especially those of opposite charge. 4,5 In more general 2


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terms, the chemical characteristics of Mg2+ (aq) and Ca2+ (aq) reflect a balance between solute-solvent and solute-solute interactions: relatively minor changes in either can result in dramatic differences in behaviour and function. 6 As would be expected from their importance and ready availability, the aqueous solutions of MgCl2 and CaCl2 have been studied extensively by traditional methods such as conductometry and potentiometry, 7,8 and using modern instrumental methods such as Xray 9–12 and neutron 13,14 diffraction, EXAFS, 15–17 and of course by computer simulations of varying degrees of sophistication. 13,14,18–24 Much has been learnt from these investigations regarding the structure and, to a lesser degree, the dynamics of these solutions. With regards to hydration there is a general consensus that Mg2+ (aq) is surrounded by two well-formed shells of water molecules whose dynamics differ considerably from those of bulk water. There is rather more uncertainty about the character of the hydration of Ca2+ (aq) although most studies conclude that it is somewhat less strongly solvated than Mg2+ (aq). Both MgCl2 (aq) and CaCl2 (aq) have been used extensively to test advances in the theory of electrolyte solutions, 8 as they have long been considered archetypal strong 2:1 electrolytes. Nevertheless there is persistent evidence, mostly from traditional techniques, of a small amount of ion association. 25 As even a very small formation (ion-pairing) constant can significantly alter chemical speciation in moderately concentrated solutions, it is important to establish the reality, or otherwise, of their existence. Dielectric relaxation spectroscopy (DRS), which measures the response of a sample to an oscillating, low-amplitude electric field, is a powerful technique for studying the structure and dynamics of aqueous electrolyte solutions on the to nano- to pico-second timescale. 26 Appropriate analysis of the frequency- and concentration-dependent responses of electrolyte solutions to the applied field yields detailed information about the hydration of dissolved ions and the effects of those ions on the solvent. Furthermore, DRS is a very sensitive, albeit not very accurate, probe of ion pairing, with a unique ability to detect

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solvent-separated ion pairs. 27 The present study reports a detailed investigation of the dielectric spectra in the frequency region 0.07 ≤ /GHz ≤ 89, of aqueous solutions of MgCl2 and CaCl2 at 25 °C and concentrations up to ∼ 1.8 mol L−1 . To the best of our knowledge, apart from two fairly old studies 28,29 made over more limited frequency ranges, this is the first broadband DRS study of the aqueous solutions of these two important salts.

Experimental The salts MgCl2 ⋅6H2 O (Merck, Germany, purity > 99 %) and CaCl2 ⋅2H2 O (Carl Roth, Germany, purity > 99 %) were employed without further purification. Solutions were prepared gravimetrically, without buoyancy corrections, from an appropriate stock solution using degassed Millipore Milli-Q water (electrical resistivity ≥ 18 MΩ cm). A vibrating-tube densimeter (DMA 5000 A, Anton Paar, Austria) was used to determine solution densities, ρ, at (25 ± 0.005) °C with an uncertainty of ±5 ⋅ 10−6 g cm−3 . Dynamic viscosities, η, were determined with a rolling-ball viscometer (AMVn, Anton Paar, Austria) at (25 ± 0.01) °C with a relative uncertainty of ≤ 2 %. Electrical conductivities, κ, were measured with an overall relative uncertainty of 0.015 using the equipment and procedures described previously. 30 The temperature was controlled to ±0.005 °C with a Huber Unistat 707 thermostat. The densities, viscosities and conductivities of the investigated solutions are given as a function of solute concentration, c, in Tables S1 and S2 of the Supporting Information. Dielectric spectra were recorded using two experimental setups. A waveguide interferometer with a variable-pathlength transmission cell held at (25 ± 0.02) °C (Lauda RKS thermostat) was used in the frequency region 60 ≤ ν/GHz ≤ 89, as described elsewhere. 31 At ν ≤ 50 GHz a frequency-domain reflectometer, consisting of an Agilent E8364B vector network analyzer (VNA) and Agilent 85070E-20 (0.05–20 GHz) and 85070-50 (5–50 GHz) dielectric probes, mounted into cells thermostated at (25 ± 0.01) °C (Huber CC-505 thermo-

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stat) was used. 32 A three-point calibration of the VNA was performed using air, purified mercury and water as open, short-circuit and load standards respectively. Additionally, due to the increasing deviations of the dielectric properties of the investigated samples at higher c from those of the reference liquid, a secondary calibration was used to improve the accuracy of the spectra. This was achieved via a complex Padé approximant using water, dimethylacetamide (DMA) and propylene carbonate (PC) as secondary standards. ˆ The experimentally accessible quantity in DRS is the generalized permittivity, η(ν), which can be expressed as ˆ η(ν) = η ′ (ν) − iη ′′ (ν) = εˆ(ν) − i

κ κ = ε′ (ν) − i [ε′′ (ν) + ] 2πνε 0 2πνε 0

(1)

where κ is the dc conductivity, arising from the steady-state migration of the ions in the electric field, and ε 0 is the permittivity of free space. The desired quantity to be obtained from DRS spectra is εˆ(ν), the complex permittivity, which includes all the frequencydependent contributions. The dc conductivity term in eq 1, scaling with ν−1 , determines the low frequency limit of the spectra, νmin , at which the uncertainty in the conductivity contribution to η ′′ (ν) exceeds the magnitude of the dielectric loss ε′′ (ν). Accordingly, νmin increases with increasing conductivity, i.e., with increasing solute concentration. For the present solutions, values of νmin in the range: 0.07 ≤ νmin /GHz ≤ 0.66 were found. It should be mentioned that the low-frequency limit for the 85070E-20 probe is, according to the manufacturer, 0.2 GHz. However, for low-conductivity (dilute in the present context) samples, it has been shown that νmin can be extended below this limit by careful calibration with established standards. 33 Note also that when evaluating eq 1, κ is treated as an adjustable parameter because fringing-field effects of the VNA probes cause it to differ from experimental values. Nevertheless, the latter were always used as starting estimates, and spectra for which the fitted and experimental κ values differed by >10 % were rejected. To extract molecular-level information from the dielectric spectra it is necessary to

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employ an appropriate relaxation model. In this work εˆ(ν) was described using a sum of n Havriliak–Negami (HN) band-shape functions 34 −1 1−α j β j

FHN,j (ν) = [(1 + (i2πντj )

) ]

(2)

thereby yielding n

εˆ(ν) = ε ∞ + ∑ S j FHN,j (ν)

(3)

j=1

Simultaneous fitting of the the real and imaginary parts of εˆ(ν), using a non-linear least squares procedure based on the Levenberg-Marquardt algorithm, 35 yields the amplitude S j , relaxation time τj , and shape parameters α j and β j for each relaxation mode j. For α = 0 and β = 1 the HN band-shape function reduces to a Debye (D) equation. The static permittivity of the solutions, ε = limν→0 ε′ (ν) = ε ∞ + ∑ j S j , with ε ∞ = limν→∞ ε′ (ν) being the infinite-frequency permittivity, which includes all intramolecular polarization contributions. To obtain an overview of the relaxation processes likely to be occurring in the present solutions, the computational procedure of Zasetzky 36 was employed. This approach yields the relaxation-time distribution function, P(τ), for each spectrum and thus gives an “unbiased” estimate of the number, location and intensity of the probable relaxation modes. The results of this analysis for the present solutions are shown in Figures S1 and S2 (Supporting Information). In the subsequent fitting process, up to five HN modes were tested for each spectrum. The quality of fit for each model was primarily assessed by the magnitude of the reduced error function, χ2r . 35 In addition, the selected model also had to be physically realistic, produce parameters that varied smoothly with c, and involve a minimal number of modes.

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90

a

80 70

50

'(

)

60

40 30 20 10 0

b

35 30

)

25

''(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 15 10 5 0 0.1

1

10

100

/ GHz

Figure 1: (a) Relative permittivity, ε′ (ν), and (b) dielectric loss, ε′′ (ν), spectra of CaCl2 (aq) in the concentration range 0.04865 ≤ c/M ≤ 1.786 at 25 °C. The solid lines represent the fit of the D+D+D+D+D model. For visual clarity the experimental data ( ) are shown only for selected samples. Arrows indicate increasing c.

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Results and discussion Assignment of modes Consistent with the Zasetsky plots (Figures S1 and S2), a sum of up to four Debye processes (D+D+D+D model) for MgCl2 (aq) and up to five (D+D+D+D+D model) for CaCl2 (aq) solutions provided the best fits of the observed spectra. The parameters obtained from these fits are shown in Tables S3 and S4. As usual, modes are numbered from lower to higher frequencies. For clarity in the following discussion j will be used to identify only MgCl2 (aq) modes; modes in CaCl2 (aq) solutions will be designated as k. Further, for reasons that will become obvious, the lowest frequency process observed for CaCl2 (aq) solutions will be defined as the k = 0 mode.

30

25

3

)

20

''(

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15

10

0

5

1

2

4

0 0.1

1

10

100

/ GHz

Figure 2: Dielectric loss spectrum, ε′′ (ν), of 0.4349 M CaCl2 (aq) at 25 °C. Experimental values are denoted by symbols, the solid line represents the overall fit. Shaded areas indicate the contributions to the spectrum from the various D modes. Higher-frequency modes. The dominant mode (j, k = 3) in the spectra, centered at ∼198


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26 GHz, can be unambiguously assigned, on the basis of its location and intensity, to the cooperative reorientation of bulk water molecules. 26,37 The amplitude of this mode, S3 , decreased systematically with increasing solute concentration for both systems (Figures 3 and S5). Also apparent in some, but not all, of the spectra (Figures S1 and S2) is a low intensity mode (j, k = 4) centered well above the present upper-frequency limit. Dielectric studies of neat water up to THz frequencies 37 locate this mode at ∼ 500 GHz; it is currently attributed to the fast switch of H-bond connections between adjacent H2 O molecules via a triangular transition state. 38 Since the present spectra were limited to ν ≤ 89 GHz it was not possible to resolve this process completely, as has been found for many aqueous electrolyte systems. 33,39,40 Accordingly, its relaxation time, τ4 , was fixed at the neat water value 41 of 0.278 ps (except for c(CaCl2 ) ≤ 0.085 M; Table S4). An alternative way of incorporating the effects of this “fast water” process is to include them in ε ∞ . 33,39 However, for the present spectra this approach yielded unreasonable fitting parameters for both salt systems. So, except for the solutions just noted, ε ∞ (c) was fixed at the neat water value, i.e., ε ∞ (0) = 3.52. 41 The assumption that ε ∞ (c) ≈ ε ∞ (0) is consistent with recent THz-spectroscopy measurements on aqueous salt solutions. 42 The ap

apparent bulk-like water amplitude, Sb , was therefore calculated as: ap

Sb = S3 + S4 + ε ∞ (c) − ε ∞ (0) ≈ S3 + S4

(4)

ap

The systematic decrease in S3 , the major component of Sb , with increasing concentration (Figures 3 and S5) reflects the replacement of polar water molecules in the solutions by non-polar ions, and the immobilization of water molecules in the hydration shells of the dissolved cations. 26 Lower-frequency modes. The lower-frequency modes in the present DR spectra (j, k ≤ 2) are solute-related, i.e., they are present only in the salt solutions. For MgCl2 (aq), two solute-modes (j = 1, 2), whose amplitudes varied with c were observed (Table S3, Figures

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80

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70 60 50

S

3

30

, S

j

40

S

12

2

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8

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6

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2

1 1

4 0.0

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0.2

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0.6

c / mol L

0.8

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-1

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/ ps

c 1

100

2

j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

0.0

3

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

c / mol L

-1

Figure 3: (a) Static permittivities (⋆) and relaxation intensities, S j (j = 1 ∎, j = 2 ▲, j = 3 ), and (c) relaxation times, τj (j = 1 ∎, j = 2 ▲, j = 3 ) as function of c(MgCl2 ). The hollow symbols indicate the parameter was fixed during fitting. Inset (b) shows S1 at low c. Dashed lines are visual guides only.

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3 and S1). All of the ions present in these solutions are symmetrical, consequently, they do not have permanent dipole moments and thus do not contribute (directly) to the DR spectra. It is therefore reasonable, given their locations and intensities, to assign both of these modes to (solvated) ion pairs, having lifetimes similar to their rotational correlation times. Analogous modes (k = 1, 2) were observed (Table S4, Figures S2 and S5) for CaCl2 (aq) and again can reasonably be attributed to ion-pairs. However, an additional slow process (for convenience labeled k = 0) variably centered at 15-270 MHz (∼1100-600 ps) was also detected, at least in some spectra. While this mode might also be related to ion-pairs, the long relaxation time, the rise and subsequent decline of its amplitude, S0 , at low c, followed by its disappearance altogether from the spectra at c ≳ 0.45 M, are more consistent with an ion-cloud (IC) relaxation (the Debye-Falkenhagen effect 43 ) rather than ion pairs. Such contributions are always present in DR spectra at low frequencies although they are usually small and become negligible at higher c. 40,44 Unfortunately, there is at present no experimental or theoretical way to separate IC contributions from the overall dielectric response. The absence of a similar effect for MgCl2 (aq) may reflect the more tightly bound hydration sheath of Mg2+ cf. Ca2+ (see below). Such stronger hydration would be expected to result in a larger effective radius and thus a lower surface charge density, which would lead to a weaker retarding force on its surrounding ion cloud. Combined with the lower conductivity of MgCl2 cf. CaCl2 (the IC relaxation time scales with 1/κ) 45 this might suffice to shift the IC relaxation to ν < νmin . Alternatively, the difference between the MgCl2 and CaCl2 spectra may simply reflect the accumulation of experimental uncertainties in the DR measurements at low ν in strongly conducting solutions (eq 1). A quantitative theory of ion-cloud relaxation is needed to decide this.

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Ion hydration Effective hydration numbers. The experimental amplitude of a re-orientating dipole, Si , is connected to its concentration, ci , through 41

ci =

ε + Ai (1 − ε) 3kB Tε 0 ⋅ ⋅ Si ε NA µ2eff,i

(5)

which can be used to calculate the apparent concentration of bulk water, cb , from the DR spectra. In eq 5, kB is the Boltzmann constant, NA is Avogadro’s constant, T the Kelvin temperature, and Ai and µeff,i are respectively the cavity field factor and the effective dipole moment of the species i. The cavity field factor accounts for the dipole shape √ and µeff,i is obtained as: µeff,i = gi µi /(1 − f i αi ), where µi is the permanent dipole moment, f i the reaction field factor and αi the polarizability. The factor gi accounts for possible orientational correlation among the dipoles i. 46 This quantity is usually unknown and not independently accessible from experiment. However, for the bulk-water relaxation in low-to-moderately concentrated solutions it can be side-stepped by normalizing eq 5 with respect to neat water. 47 This assumption breaks down at high c 47 and is not valid for H2 O molecules in the hydration shell of ions 48 but it should work reasonably well for ap

ap

evaluating the present Sb values. It should be noted that Sb (eq 4) cannot be inserted directly into eq 5 because it needs to be corrected for “kinetic depolarization”, which arises from the inhibition of rotation of solvent molecules caused by the movement of ions in the applied electric field. Details of the procedure used to make this correction 49 are given in the SI. The effective total hydration number, Zt , of the dissolved ions is determined as the difference between the total (analytical) water concentration, cw , and the “free” bulk water concentration, cb , per unit of solute concentration:

Zt =

cw − cb c 12


(6)

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As defined, Zt encompasses all the water dipoles whose dynamics are slowed down sufficiently by the presence of the solute so as to make them distinguishable from bulk water. Depending on the strength of the solute-solvent interactions, this can either produce a “slow-water” mode at lower (than bulk water) frequencies or cause some water molecules to “disappear” entirely from the DR spectra. 26 The latter are often referred to as irrotationally–bound (ib) water and are effectively frozen on the DR timescale. For both of the present systems, no slow water mode was discernible and therefore Zt = Zib . The Z values determined in this way refer to the whole salt. To estimate the more interesting ionic values it is necessary to assume that ionic additivity: Zib (MA2 ) = Zib (M2+ ) + 2 ⋅ Zib (A− )

(7)

is obeyed at all concentrations. Although eq 7 is strictly valid only at infinite dilution, it is often found to hold at least approximately at finite concentrations for many salts. 26 In addition to ionic additivity, it is necessary to make an appropriate “single ion” assumption by fixing the Z value of one ion. Dielectric measurements on salts such as Et4 NCl(aq) 47 indicate Zib (Cl− , aq) ≈ 0 at all concentrations, 26 which value was adopted here. The values of Zib (M2+ , aq) obtained for the two cations via eqs. 6 and 7 are shown as functions of concentration in Figure 4. 50 Their magnitudes and variation with c are remarkably similar. (Differences between Na+ (aq) and K+ (aq), for example, are much greater). 40,41 At c ≲ 1 M, the values of Zib (c) for both Mg2+ and Ca2+ were almost independent of concentration (Figure 4). Extrapolation of these values using a weighted least squares 0 (Mg2+ ) = 20 ± 1 and Z 0 (Ca2+ ) = linear fit yielded infinite dilution hydration numbers, Zib ib

19 ± 1. Given that true uncertainties are likely to be greater than the regression values, 51 these two results are in essence identical. At c ≈ 1 M, Zib (c) for both cations undergoes a small but discernible change (Figure 4), resulting in a linear decrease at higher c. Similar declines in Zib have been observed in other electrolyte solutions 33,40,41,52 and are usually

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ib

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, aq)

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40

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0 0.0

0.2

0.4

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0.8

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c / mol L

-1

Figure 4: Effective hydration numbers, Zib , of (a) Mg2+ and (b) Ca2+ at 25 °C. The dashed lines are linear fits. The arrows indicate the estimated overlap-onset concentration, c,* of the overlap of the second hydration shells of the cations. Unfilled symbols denote data omitted in the fitting procedure.

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attributed to hydration-sheath overlap. In this context it is instructive to look at a simple geometrical model of such overlaps. The minimum distance required to prevent overlap of ion hydration sheaths can be estimated as: dmin = [r+ + (n+ ⋅dw )] + [r− + (n− ⋅dw )]

(8)

where r± is the (crystallographic) radius of a bare ion ( = 72, 100 and 181 pm for Mg2+ , Ca2+ and Cl− , respectively), 53 dw = 285 pm is the diameter of a water molecule 54 and n± are the presumed number of hydration shells around the cation or anion. Overlap-onset concentrations, c* calculated via eq 8 for various plausible values of n+ , are shown in Table 1; a value of n− = 0 was assumed throughout for the weakly hydrated Cl− (aq). Table 1: Overlap-onset concentrations, c* in mol L−1 , for MgCl2 (aq) and CaCl2 (aq) as a function of the number of hydration shells around the cation, n+ . n+

c*(MgCl2 )

c*(CaCl2 )

1 2 3

3.75 1.05 0.43

3.22 0.95 0.40

The concentration at which the primary hydration layer of the cations (n+ = 1) begins to overlap is > 3 M for both salts, which is outside the present concentration range. In contrast, the values of c* corresponding to overlap-onset of second hydration shells (n+ = 2) coincide reasonably well with the break in the Zib (c) curves at c ≈ 1 M (Figure 4). Consistent with the weaker hydration of Ca2+ (see later), the slope at c > 1 M is more negative for Ca2+ (–5.7 M−1 ) than Mg2+ (–3.0 M−1 ). The apparent decrease in Zib (M2+ , aq) in the region 0.2 ≲ c/M ≲ 0.5 (Figure 4) might be indicative of 3rd shell overlap (Table 1) but, given the likely uncertainties, is probably an experimental artifact. 0 values for Mg2+ (aq) and Ca2+ (aq) Comparison with other data. Table 2 lists the present Zib

along with the corresponding DRS results for comparable cations. While the present values 15



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are slightly higher than those of the other divalent cations, it should be noted that the latter were mostly derived from measurements on more strongly associated (mostly sulfate) salts 0 (M2+ , and, as such, are probably under-estimates. As would be expected, the values of Zib 0 (M+ , aq) ≈ 0 to 6. 26 aq) are much larger than those of monovalent cations, for which Zib 0 (M3+ , aq) for trivalent cations are rather fewer and show more scatter but are Values of Zib

mostly considerably higher than those for M2+ (aq), lying within the range 18 to 44. 26,33 0 , deterTable 2: Ionic radii, r+ , and effective hydration numbers at infinite dilution, Zib mined by DRS for divalent cations.

Cation

r+ /pma

Ca2+ 100 2+ Cd 95 2+ Co 75 Cu2+ 73 2+ Mg 72 2+ Ni 69 a From Marcus 53

0 (M2+ ) Zib present work literature 19 – – 12 29 – 17 26 – 19 26,29 20 14; 52 15 29 – 17 26

salt Cl− Cl− SO2− 4 − Cl , SO2− 4 Cl− , SO2− 4 SO2− 4

It is well established, from X-ray, 11,55 neutron 13 and Raman scattering 21,56 investigations, and molecular dynamics (MD) simulations, 10,21,23 that Mg2+ has a primary coordination number, CN1 = 6, with strictly octahedral geometry. 55,56 Studies using X-ray diffraction and MD simulations have also detected a second hydration shell with CN2 = 12, 55 as would be anticipated on simple electronic grounds. Support for the presence of a second hydration shell comes also from a comparison of the experimental MgO6 skeleton mode with results from cluster calculations. 56 Consistent with these findings, the present value of 0 (Mg2+ , aq) ≈ 20 also indicates the formation of a well-defined second hydration shell Zib

and possibly even a small amount of ordering of water molecules beyond that shell. The hydration of Ca2+ is less clear-cut. Reported values of the primary coordination number vary over the range: 6 ≤ CN1 ≤ 10 55 with a recent Raman study favouring CN1 = 6

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with octahedral symmetry. 57 The larger values, cf. Mg2+ , presumably reflect the greater size (surface area) of the bare calcium ion enabling more “nearest neighbours” to be fitted into the first shell. 55 The existence of a second hydration shell is again supported by Raman spectroscopy. 57 Numerical estimates for CN2 are, surprisingly, in reasonable agreement. Thus, a combined EXAFS, X-ray scattering and MD study by Jalilehvand et al. 17 reported CN2 = 16 − 18. Badyal et al. 58 estimated CN2 = 19 − 20 from neutron diffraction measurements, while Hofer et al. 22 found CN2 = 19 using MD simulations. Given the differences observed for CN1 , at least some of this agreement must be fortuitous. Nevertheless, taking their average gives a total (= CN1 + CN2 ) of 24 to 28 water molecules collectively located within the first and second hydration shells of Ca2+ (aq). These numbers 0 (Ca2+ , aq) ≈ 19 but it must be remembered are somewhat larger than the present result of Zib

that Zib is a measure of water dynamics not geometry. It can be speculated that the larger value of CN1 for Ca2+ might result in a much less strongly bound second shell, thereby causing Zib to be less than the maximum expected from electronic considerations alone. That Ca2+ is less strongly hydrated than Mg2+ is evident from its considerably less favorable Gibbs energy of hydration: ∆hyd G○ = −1527 kJ mol−1 (Ca2+ ) vs. −1837 kJ mol−1 (Mg2+ ). 5 Further, the residence time of water molecules in the first hydration shell of Ca2+ (aq) is on the sub-nanosecond timescale 18–20,55,59–62 (with a slightly higher rate of exchange between the first and second shells) 24,61 whereas that for Mg2+ (aq) is in the microsecond region. 21,55,62,63 Solvent relaxation time. The relaxation time of the bulk water mode, τ3 , replotted on a more usable scale in Figure 5, decreases approximately linearly with solute concentration for both of the present salt systems. Such a decrease, albeit generally not linear, appears to be typical for inorganic salts. 33,39,64–66 First noting that water dynamics are decoupled from viscosity in aqueous electrolyte solutions, 26,67 the decrease in τ3 with increasing solute concentration (Figure 5) probably reflects a slight disruption of the bulk water structure by the presence of the (hydrated) ions. Note that the simulations of Rinne et al. 48 suggest that

17



the collective dynamics of hydration water is faster than bulk water. Thus, the observed shift of τ3 (c) might arise from a growing contribution of this collective hydration-shell mode although we could not resolve a separate relaxation for such a phenomenon. That the slope of τ3 (c) in Figure 5 is more negative for the calcium salt: –1.24 ps L mol−1 vs. –0.94 ps L mol−1 for MgCl2 (aq), probably reflects the greater size of Ca2+ (aq) cf. Mg2+ (aq).

18


Page 18 of 35

Page 19 of 35

8.5

3

2

(MgCl (aq)) / ps

a 8.0

7.5

7.0

8.5

2

(CaCl (aq)) / ps

b

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8.0

7.5

7.0

6.5

6.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

c / mol L

-1

Figure 5: Linear regressions (lines) of the dominant bulk water relaxation time, τ3 , as a function of the total salt concentration, c, in (a) MgCl2 (aq) and (b) CaCl2 (aq). The open symbol is for pure water.

19



Page 20 of 35

Ion-pairing As noted in the Introduction, MgCl2 and CaCl2 have long been considered as archetypal strong 2:1 electrolytes. Nevertheless, the appearance of solute-dependent modes at low frequencies in their DR spectra is prima facie evidence for at least some degree of ion pairing, with the following three provisos. First, all free ions present in the solutions must be fully symmetrical and thus not contribute to the spectra. Second, while ion-cloud relaxation undoubtedly contributes to the spectra at low ν its contribution must be small and become negligible at higher ν and higher c. Third, no distinct “slow water” or other plausible mode should contribute significantly to the spectra in this region. As all of these criteria have been met, it is safe to conclude that the finite amplitudes of modes j,k = 1 and 2 indicate the presence of small amounts of ion pairs. Ion pairing in electrolyte solutions is best regarded as occurring via the well-established Eigen-Tamm mechanism, 68,69 in which free solvated ions initially associate, with their hydration shells essentially intact, to form double-solvent-separated ion pairs (2SIPs). These may then lose their intervening water molecules, forming successively solventshared (SIPs) and contact ion pairs (CIPs). Assuming only 1:1 species are formed, this process can be written for the present systems as: M2+ (aq) + Cl− (aq) ⇌ [M2+ (OH2 )(OH2 )Cl− ] (aq) K2SIP

free ions

2SIP

⇌ [M2+ (OH2 )Cl− ] (aq) ⇌ [M2+ Cl− ] (aq)

KSIP

KCIP

SIP

(9)

CIP

It is readily shown that the overall ion-association equilibrium constant, KA , measured by traditional thermodynamic techniques and by conductometry, is related to the above as:

KA =

cIP = K2SIP + K2SIP KSIP + K2SIP KSIP KCIP c+ c−

(10)

where cIP (= c2SIP + cSIP + cCIP ) is the total concentration of all IPs formed. It follows that 20


Page 21 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the KA value obtained via eq 10 must be the same as that measured by traditional methods. This provides a stern test of the KIP values obtained by DRS. It should be noted that, at its present level of technological development, DRS is not a chemical speciation technique of choice. The great merit of DRS lies in its ability to detect and distinguish between the various types of IPs, and its unique sensitivity: 2SIP > SIP > CIP. 27 Concentrations of detected IPs were obtained from the spectra via eq 5, using the observed amplitudes S1 and S2 . The effective dipole moments of the different species, µeff,IP , and the corresponding field factors, AIP and f IP , were derived using the procedure of Barthel et al. 70 Polarizabilities of the IPs were estimated as: αIP = α+ + α− + mαH2 O , where m = 0, 1 or 2 for CIPs, SIPs and 2SIPs, respectively. Polarizabilities of the bare ions 53,71 were taken to be 7.13 ⋅ 10−32 Å3 for Mg2+ , 6.3 ⋅ 10−31 Å3 for Ca2+ , 3.42 ⋅ 10−30 Å3 for Cl− and 1.44 ⋅ 10−30 Å3 for H2 O. 72 The permanent dipole moments of the IPs, µIP , were calculated using the center of hydrodynamic stress (CHS) as the pivot. 73 This requires detailed knowledge of the size and shape of the IPs. These are unknown for the present MCl+ (aq) species but can be estimated approximately using the approach of Dote and Kivelson. 74 The calculated values of µeff, IP for all the 1:1 IPs are given in Table S5. The KA values calculated from eqs. 5 and 10 were fitted as a function of the stoichio○, metric ionic strength, I(= 3c), to obtain the standard state (infinite dilution) quantity, KA

using an extended Guggenheim-type equation 75 √ 2A ∣z z ∣ I + − DH ○ √ log KA = log KA − + B ⋅ I + C ⋅ I 3/2 1+ I

(11)

In eq 11, z± , are the ion charge numbers, ADH = 0.5115 (L mol−1 )1/2 is the Debye-Hückel constant for activity coefficients in water at 25 °C , and B and C are adjustable parameters.

21



4

a

3 2

logKi

1 0

-1 -2 -3 -4 -5 -6 4

b

3 2 1

logKi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 35

0

-1 -2 -3 -4 -5 -6 0

1

2

3

4

5

-1

I / mol L

Figure 6: Overall association constant, KA ( ), and step-wise association constants, K2SIP (∎) and KSIP (▲), for the formation of 1:1 IPs for: (a) MgCl2 (aq) and (b) CaCl2 (aq) at 25 °C as functions of the stoichiometric ionic strength, I. The solid lines indicate the fit to eq 11; the dashed line is an empirical linear fit.

The mode centered at ∼ 1.1 GHz could not be resolved at high c for either system due to its decreasing intensity (Figures 3 and S4; Tables S3 and S4). Values of S1 under such

22


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conditions were therefore estimated using an empirical exponential fit of the data at lower c. Likewise, the value of S2 at the lowest c(CaCl2 ) was obtained via a simple polynomial fit of S2 (c) at higher c. Note that the initial rise and subsequent decay of S1 with increasing c is fully consistent with the sequential formation of IP species (eq 9). 52 From their locations, modes 1 and 2 could be assigned to the presence of 2SIPs and SIPs, respectively. This assignment is consistent with the Zib values discussed above, which clearly indicate the existence of two hydration shells around each cation, and by the physical plausibility of the KA values derived. The formation of SIPs does not change the concentration of free ions since it only involves the loss of a water molecule (eq 9) and thus is favored by the decreasing analytical water concentration. The corresponding step-wise association constant could be described by an empirical linear regression: log KSIP (I) = ○ + D⋅I (Figure 6). The values of K log KSIP 2SIP were fitted with eq 11. Since no evidence for ○ significant CIP formation was found, K2SIP was fixed in this fit to the calculated value of ○ ○ /(1 − K ○ ), following from eq 10. K2SIP = KA SIP

The standard equilibrium constants obtained in these ways for the two systems are ○ . The empirical fit parameters presented in Table 3 together with the literature values of KA

B, C, D are shown in Table S6. As would be expected from the status of MgCl2 and CaCl2 ○ , for both systems is as classical strong electrolytes, the overall ion association constant, KA ○ values for MgCl+ (aq) and vanishingly small. The absence of a difference between the KA

CaCl+ (aq), which would be expected on Coulombic grounds, reflects the attenuation of such effects by the hydration shells. Given the present accuracy of DR measurements, and the general difficulties of quantifying weak complexation, 76 it is important not to over-interpret the present results. Nevertheless, it is gratifying that they agree broadly with the rather scattered values obtained from potentiometry 77–81 and MD simulations 62 (Table 3). That the present results are slightly higher than the literature constants might reflect the presence of unresolved (small) ion-cloud contributions to the IP modes. Accordingly, the constants reported here

23



Page 24 of 35

should be regarded as upper limits. ○ , K○ ○ Table 3: Standard formation constants, KA 2SIP and KSIP , for 1:1 ion pairs in MgCl2 (aq) and CaCl2 (aq) at 25 °C.a

MgCl+ (aq)

○ logKA ○ logK2SIP ○ logKSIP

pw 0.5 ± 0.2 0.4 ± 0.2 −0.6 ± 0.1

CaCl+ (aq)

lit. 62

77,b

-0.19; -0.13 ; 0.28 78,c ; 0.57 79

– –

pw 0.5 ± 0.1 0.3 ± 0.1 −0.2 ± 0.1

lit. 81

-0.292; -0.17 77,b ; ∼0 62,d ; 0.39 78,c ; 0.4; 79 0.43 79

– –

○ and K ○ ○ ○ −1 = present work; KA 2SIP in L mol ; K2SIP was calculated from KA and ○ and was fixed during fitting; b logK in 1 M NaNO (aq); c logK at I = 0.4 M; KSIP 3 A A d Values varied between –0.1 and 0.15. a pw

Other processes. In the view of the weakness of the 1:1 IPs (Table 3), significant formation of higher order species such as triple ions, MCl02 (aq), can be ruled out, at least over the present concentration range. At much higher salt concentrations, dehydration and packing effects become important and the notion of distinct individual species becomes less clearcut. Within the covered concentration range, c < 1.8 M, no evidence was found in the spectra for the existence of CIPs in either system (Figures 2 and S4). Thus, although the relatively short relaxation time of mode 2 (τ2 ≈ 19 ps) might be consistent with the formation of CIPs, their inclusion in the speciation model produced unrealistic KA values. With respect to MgCl2 (aq) solutions, the absence of CIPs is consistent with the strong hydration of Mg2+ (aq) and a variety of experimental 10,11,13,21 and computational studies. 82,83 For CaCl2 (aq) the situation is less straightforward: some investigations 12,19,20,57,83,84 claim the existence of CIPs, at least at higher concentrations (c ≳ 4 M), while many others 9,15–17,58 report negligible formation, consistent with present findings and the hydration of Ca2+ (aq). Another possibility that must be considered is the existence, or not, of a distinct 24


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“slow water” mode. Such contributions are usually located at about 15 – 30 ps in the DR spectrum 26 and therefore might be associated with mode 2. While some slow water is almost certainly present in MCl2 (aq) solutions, the current precision of DR spectra is insufficient to resolve it. All that can be said at this stage is that the intensity of mode 2 is mostly due to SIPs, with some contribution from slow water and, perhaps, very minor contributions from IC relaxation and CIPs.

Conclusion Broadband DRS measurements on the aqueous solutions of MgCl2 and CaCl2 indicate that both Mg2+ and Ca2+ are strongly hydrated, with each possessing two well-defined hydration shells. At infinite dilution both cations immobilized ∼ 20 water molecules on the DRS timescale, but no evidence was found for the existence of a “slow water” relaxation. Indications for slightly weaker hydration for Ca2+ compared to Mg2+ were found. The presence of two solute-dependent modes at low frequencies provides prima facie evidence for the formation of ion pairs in these classical strong electrolyte solutions. Detailed analysis of these modes indicated they could be assigned to double-solvent-separated and solvent-shared IPs. The overall ion association constants calculated from the DR spectra were in good agreement with literature data. No evidence for the significant formation of contact IPs was observed.

Acknowledgement S.F. gratefully acknowledges a PhD stipend of the Fonds der Chemischen Industrie.

25



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Cation Hydration and Ion Pairing in Aqueous Solutions of MgCl2 and

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