Ion Association and Hydration in Aqueous Solutions of Copper(II

J. Phys. Chem. B 2006, 110, 14961-14970

14961

Ion Association and Hydration in Aqueous Solutions of Copper(II) Sulfate from 5 to 65 °C by Dielectric Spectroscopy Chandrika Akilan and Glenn Hefter* Chemistry - DSE, Murdoch UniVersity, Murdoch, WA 6150, Australia

Nashiour Rohman and Richard Buchner* Institut fu¨r Physikalische und Theoretische Chemie, UniVersita¨t Regensburg, D-93040 Regensburg, Germany ReceiVed: April 4, 2006; In Final Form: May 23, 2006

Aqueous solutions of copper(II) sulfate have been studied by dielectric relaxation spectroscopy (DRS) over a wide range of frequencies (0.2 j ν/GHz e 89), concentrations (0.02 e m/mol kg-1 j 1.4), and temperatures (5 e t/°C e 65). The spectra show clear evidence for the simultaneous existence of double-solvent-separated, solvent-shared, and contact ion pairs at all temperatures, with increasing formation especially of contact ion pairs with increasing temperature. The overall ion association constant K°A corresponding to the equilibrium: Cu2+(aq) + SO42-(aq) h CuSO40(aq) was found to be in excellent agreement with literature data over the investigated temperature range. However, the precision of the spectra and other difficulties did not allow a thermodynamic analysis of the formation of the individual ion-pair types. Effective hydration numbers derived from the DRS spectra were high but consistent with simulation and diffraction data from the literature. They indicate that both ions influence solvent water molecules beyond the first hydration sphere. The implications of the present findings for previous observations on copper sulfate solutions are briefly discussed.

1. Introduction Aqueous solutions of copper(II) sulfate are of major importance in agriculture where they are widely used as antifungal and antibacterial agents1,2 and for the correction of copper deficiency in soils, plants, and animals.3,4 Such solutions also find application as a mordant for dyeing,5 as an activator for the froth flotation of various ores,6 as a molluscicide, and for algae control.3 The major use of hot copper sulfate solutions is in the recovery of copper from its ores and, especially, for copper refining and electroplating.6 Indeed, it could be said that there are few technological activities that do not make at least some use of copper sulfate solutions.5 The physicochemical properties of copper(II) sulfate solutions have, not surprisingly, been well investigated. Large amounts of data exist on activity coefficients, densities, solubilities, heat capacities, and many other thermodynamic quantities, as well as other properties such as conductivities and viscosities.7,8 Copper sulfate belongs to a group of electrolytes that in aqueous solution can be described as modestly associated.8 Thus, an important factor in understanding the properties of such solutions is the extent of ion association and how it varies with concentration and temperature. It has generally been accepted that CuSO4(aq) solutions, like those of the other divalent sulfates,9 are appreciably associated at moderate concentrations, with only one significant equilibrium (Scheme I) KA

Cu2+(aq) + SO42-(aq) y\z CuSO40(aq)

(I)

This equilibrium has been extensively studied, with more than 70 papers over the last 60 or so years reporting quantitative * To whom correspondence should be addressed. E-mail: G.Hefter@ murdoch.edu.au; [email protected].

thermodynamic information using a wide variety of techniques.10 Indeed, the Cu2+/SO42- system has been discussed at length as a paradigm for the study of ion association in aqueous solution in many of the classical texts on electrolytes.8,9,11 Nevertheless, the value of KA, the overall association constant for Scheme I, has proven to be remarkably difficult to determine with the sort of precision that would be expected for such an apparently simple system.10 There have even been suggestions from time to time12,13 that no complexation occurs in copper sulfate solutions. The reasons for this situation are manifold and subtle but include the effects of activity coefficient variation and the use of inappropriate measurement techniques.10,14 Both of these problems are related, at least in part, to the possible existence of solvent-separated ion pairs. As discussed in detail elsewhere for the closely related Mg2+/SO42- system,15,16 the (often unsuspected) presence of such species can undermine some of the usual assumptions that are made in the interpretation of solution equilibrium data. According to Eigen and Tamm,17,18 the association of strongly solvated ions in aqueous solution occurs via a three-step process in which the free hydrated ions initially combine with their (inner) hydration sheaths essentially intact to form a doublesolvent-separated ion pair (2SIP). This is followed by successive losses of water molecules to form a solvent-shared (SIP) and then a contact (CIP) ion pair, as shown in Scheme II K1

Mm+(aq) + Ln-(aq) y\z [M free ions

m+

K2

(OH2)(OH2)Ln-](aq) y\z 2SIP K3

[Mm+(OH2)Ln-](aq) y\z [Mm+Ln-](aq) (II) CIP SIP

10.1021/jp0620769 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/07/2006

14962 J. Phys. Chem. B, Vol. 110, No. 30, 2006 Unfortunately, there are very few techniques available that can routinely detect solvent-separated ion pairs. Of these, the most widely applicable are undoubtedly the ultrasonic19 and dielectric20,21 relaxation methods. Quite a few ultrasonic investigations have been made of CuSO4(aq)22-24 but none recently and all were at 25 °C. The more comprehensive studies of Bechteler et al.22 and Fritsch et al.23 are in good agreement although (due to equipment limitations) the latter could only detect two equilibria, corresponding to steps 1 and 2 in Scheme II, whereas Bechteler et al.22 detected all three. In contrast, only one dielectric relaxation study, at various temperatures from 5 to 55 °C but at only one concentration (∼1 mol/L), has been made.25 Despite the technological limitations of the time, with respect to both the available frequency range and measurement accuracy, and the absence of data at other concentrations, Pottel25 was able to identify the presence of 2SIPs and SIPs but could derive only limited thermodynamic information for them. In view of the importance of copper(II) sulfate solutions and the likely influence of ion association on their physicochemical properties, a detailed investigation of CuSO4(aq) using modern broadband dielectric relaxation spectroscopy (DRS) over wide ranges of concentration and temperature has been undertaken. 2. Experimental Section Dielectric spectra consisting of 101 points at equally spaced increments of log ν over the frequency range νmin e ν/GHz e 20 were measured at Murdoch University with a HewlettPackard model HP 85070M dielectric probe system, consisting of a HP 8720D vector network analyzer (VNA) and a HP 85070M dielectric probe kit, as described previously.26,27 The HP 85070M dielectric software was used to control the instrument and to calculate the dielectric permittivity, ′(ν), and the total loss η′′(ν), from the experimentally determined relative complex reflection coefficient. The accuracy of these measurements is estimated to be around 3% for ′(ν) and 5% for η′′(ν), with a reproducibility 2-4 times better.26 The minimum frequency of investigation, νmin, was governed by the conductivity contribution to the loss spectrum. As such, it varied with solute concentration and temperature but typically was in the range 0.2-0.5 GHz. All VNA spectra were recorded at least twice using independent calibrations with air, mercury (as a short circuit), and water as the standards. The calibration parameters for water at 5, 25, 45, and 65 °C were taken from Buchner et al.26 The temperature of the solutions was controlled with a Hetofrig (Denmark) circular-thermostat with precision of (0.02 °C and an accuracy of about (0.05 °C (NIST-traceable). Higher frequency data were determined at Regensburg University using two waveguide interferometers (IFMs): A-band (27 e ν/GHz e 39) and E-band (60 e ν/GHz e 89) as described in detail elsewhere.28,29 Temperatures were controlled, with similar precision and accuracy to those at Murdoch, using a Lauda (Germany) model RKS 20 circulator-thermostat. Solutions were prepared by weight without buoyancy corrections. Millipore (Milli-Q) water was used throughout. CuSO4‚ 5H2O (AR grade) was used as received (Ajax Chemicals, Australia, and Merck, Germany). Exact metal ion concentrations were determined to (0.2% by complexometric titration against EDTA (BDH, U.K., concentrated volumetric standard).30 Density data required for the conversion from molality (m, mol/kg solvent) to molarity (c, mol/L solution) were taken from the literature.7

Akilan et al.

Figure 1. Conductivity, κ, of CuSO4(aq) at t/°C ) 5 (1), 25 (2), 45 (3), and 65 (4): as determined from the present η′′(ν) data (b) and representative data from conventional conductance measurements32 (4).

3. Data Analysis The complex permittivity spectra ˆ (ν) ) ′(ν) - i′′(ν) consisted of the VNA data and (where recorded) IFM measurements as a function of ν. The loss component ′′(ν) is not directly observable and must be obtained from the measurable total loss η′′(ν) ) ′′(ν) + κ/2πν0, where κ is the solution conductivity and 0 is the permittivity of free space. As previously described,26 conductivity corrections to η′′(ν) to obtain ′′(ν) were made by treating κ as an adjustable parameter. Nevertheless, the values of κ so obtained, which are listed in Supporting Information Tables 1-4, are in reasonable agreement (Figure 1) with the somewhat limited and scattered bulk solution conductivity data available in the literature.31-33 Typical dielectric spectra of CuSO4(aq) for representative concentrations at 5 and 45 °C are given in Figure 2. The effects of temperature are more clearly seen in Figure 3, which presents DR spectra at two concentrations over the full temperature range studied (5 e t/°C e 65). These spectra were analyzed by simultaneously fitting the in-phase (relative permittivity, ′(ν), parts a and c of Figure 3 and parts a and c of Figure 4) and out-of-phase (loss, ′′(ν), parts b and d of Figure 3 and parts b and d of Figure 4) components to plausible relaxation models based on the sum of n distinguishable dispersion steps, as described in detail previously.16 Debye equations were found to be the most appropriate of the various models tried for all of the relaxations. Thus, the fitting equation used for all of the present spectra was n

ˆ (ν) )

j - j+1

+ ∞ ∑ j)1 1 + i2πτ

(1)

j

The very fast water relaxation, centered on 400 GHz at 25 °C34 and barely observable in the present spectra even at lower temperatures, moves to higher frequencies with increasing temperature. Thus, the present spectra were fitted with a 5-Debye model (n ) 5) for the lower temperatures (5 and 25 °C) and a 4-Debye model (n ) 4) for the higher temperatures (45 and

Aqueous Solutions of Copper(II) Sulfate

Figure 2. Dielectric permittivity (a and c) and loss (b and d) spectra for CuSO4(aq) at 5 °C (a and b) and 45 °C (c and d), at concentrations m/mol kg-1 ) 0.05 (1), 0.10 (2), 0.20 (3), 0.40 (4), 0.65 (5), and 1.01 (6).

65 °C). As previously described,16 the infinite-frequency permittivity, ∞, was used as an adjustable parameter, together with the limiting permittivities for each process, j, and the corresponding relaxation times, τj. The static permittivity of the sample is then ) ∞ + Σ Sj (with ∞ ) n+1) and Sj ) j j+1 is the amplitude (relaxation strength) of process j. The values of j, ∞, and τj obtained from the fits of all the present spectra for CuSO4(aq) are summarized in Supporting Information Tables 1-4 together with the corresponding values of the reduced error function χr2 as a measure of the quality of fit.35 Note that, at 5 and 25 °C, τ5 was fixed at the value of the fast water relaxation process in pure water.34 At 45 and 65 °C, as already mentioned, the value of τ5 is so small that this process was not observable in the present spectra. As previously described,16,36,37 values of the bulk water relaxation time, τ4(c), obtained by fitting the combined VNA + IFM spectra were interpolated, using appropriate functions, to the concentrations of the spectra for which only VNA data were recorded. At

J. Phys. Chem. B, Vol. 110, No. 30, 2006 14963

Figure 3. Dielectric permittivity (a and c) and loss (b and d) spectra for CuSO4(aq) at m/mol kg-1 ) 0.05 (a and b) and 0.40 (c and d), at temperatures t/°C ) 5 (1), 25 (2), 45 (3), and 65 (4).

5 °C, the CIP relaxation time, τ3(c), had to be fixed at 30 ps to resolve that process because the dominant and strongly temperature-dependent bulk water relaxation (τ4 ≈ 14 ps) was in general too close to allow separation in a free-running fit. On the other hand, the temperature dependence of τ4 facilitated separation of the ion-pair processes at 45 and 65 °C, which partly compensated the higher uncertainties arising from the increased solution conductivity. Typical examples of the decomposition of the observed loss spectra into their component processes are given in Figure 4 for representative temperatures. 4. Results and Discussion 4.1. Ion Association: General Features. As mentioned in the Introduction, the Cu2+/SO42- system has been widely seen as a paradigm for the study of ion association in solu-

14964 J. Phys. Chem. B, Vol. 110, No. 30, 2006

Akilan et al.

kg-1

Figure 4. Dielectric loss spectra, ′′(ν), of 0.20 mol CuSO4(aq) at (a) 5 °C and (b) 65 °C. Experimental data (points) are fitted with a 5D model (eq 1) at 5 °C and a 4D model at 65 °C (see text). Designations 2SIP, SIP, and CIP indicate contributions of doublesolvent separated, solvent-shared, and contact ion pairs, respectively; s1 and s2 designate the cooperative and the fast relaxations of water, respectively.

tion.8,9,11,38,39 The extensive data available for Scheme I have been critically reviewed recently for IUPAC and show a widerthan-expected variation which is at least partly related to the presence of solvent-separated ion pairs (2SIPs and SIPs).10 As far as thermodynamic methods are concerned, the formation of solvent-separated ion pairs is unimportant because there is no thermodynamic distinction between dissolved species with the same stoichiometry but differing levels of solvation.40 This means that thermodynamic methods measure only the overall association constant, KA, corresponding to Scheme I, with (assuming activity coefficients to be constant)

KA ) mIP/m+m-

(2)

where mIP ) m2SIP + mSIP + mCIP is the total ion-pair concentration and m+ and m- are, respectively, the free cation and anion concentrations. It is readily shown that, again ignoring activity coefficients

KA ) K1 + K1K2 + K1K2K3

(3)

with

K1 )

m2SIP m+m-

(4a)

K2 )

mSIP m2SIP

(4b)

K3 )

mCIP mSIP

(4c)

and

Similar considerations apply to conductance data, at least for symmetrical electrolytes where all ion-pair types are considered

Figure 5. Solute contribution to the dielectric loss spectra, ′′IP(ν), of CuSO4(aq) at 25 °C and at concentrations: m/mol kg-1 ) 0.020 (1), 0.050 (2), 0.10 (3), 0.25 (4), 0.35 (5), 1.01 (6), and 1.42 (7). The frequencies νj ) 1/(2πτj) define the location of the loss peaks for the various ion-pair types.

to be (equally) nonconducting.41 Conductance measurements therefore produce KA values that are directly comparable with thermodynamic results. On the other hand, the presence of the various ion-pair types does create some difficulties for both the activity coefficient corrections and the conductivity expressions that are typically used in the analysis of conductance data.41 These problems have long been recognized and discussed at length on several occasions.39,42,43 Their major effect is to increase the uncertainty in the calculated values of KA even for the highest quality data. For spectroscopic measurements, however, the situation is quite different. As discussed in detail elsewhere,14 the major spectroscopic methods (NMR, Raman, and UV-vis) typically detect only CIPs. The association constant determined from such measurements is thus typically smaller, sometimes (but not always) dramatically so, than the values obtained from thermodynamic or conductance data.14 For example, conventional analysis of high-quality Raman spectra yields K°A(CuSO40 (aq)) ≈ 10,44 which is more than an order of magnitude lower than the thermodynamic/conductivity values.10 Furthermore, although the CIP concentrations determined from Raman spectroscopy are identical with those obtained by other techniques (such as DRS) the equilibrium constant is not directly comparable with either KA or Ki.15,16 The present dielectric spectra (see, for example, Figures 2 and 3) indicate the presence of all three ion-pair types at appropriate conditions of solute concentration and temperature. Typical decompositions of representative loss spectra are given in Figure 4. The existence of the various ion-pair types is confirmed, as for other spectroscopic techniques, by the growth and decay of spectral features as functions of concentration. This is seen more clearly in Figure 5, which plots the total solute contribution to the loss spectra, ′′IP(ν), as a function of concentration. Also included in Figure 5 are the average peak frequencies of the loss spectrum associated with each ion-pair species, corresponding to the relaxation times of τj/ps ≈ 400 (2SIP), 100 (SIP), and 27 (CIP) for j ) 1, 2, and 3, respectively.



4.2. Ion Association: Quantitative Analysis. The amplitudes S1, S2, and S3, arising from the presence of 2SIPs, SIPs, and CIPs, respectively, were analyzed as described previously16,37 using the modified Cavell equation45

3( + (1 - )AIP) kBT0 (1 - RIPfIP)2 cIP ) SIP NA µ 2

(5)

IP

where kB, NA, and T are, respectively, the Boltzmann’s constant, Avogadro’s number, and thermodynamic (Kelvin) temperature; RIP, fIP, and µIP are the polarizabilities, field factors, and dipole moments of the individual ion-pair species; and AIP is a geometric factor that can be calculated from the shape and size of each of the ion-pair types.27,45 Note that the analysis via the Cavell equation gives concentrations cj in mol/L (M). Ion pairs were assumed to be ellipsoids with semiprincipal axes a ) r+ + r- + qr(H2O) where q ) 0 (CIP), 1 (SIP), or 2 (2SIP); b ) c ) r- ) 230 pm, with r+ ) 73 pm and r(H2O) ) 142.5 pm. The ion-pair polarizabilities were likewise estimated as RIP ) R+ + R- + qR(H2O), where R+ ) 0.515 Å3, R- ) 5.47 Å3, and R(H2O) ) 1.444 Å3. All radii and polarizabilities were taken from Marcus.46 The dipole moments of the ion pairs, µIP, were calculated from presumed linear geometries as described previously,16,45 yielding µ2SIP ) 79.4 D, µSIP ) 53.9 D, and µCIP ) 22.3 D (1 D ) 3.33564 × 10-30 Cm). All other values required for the evaluation of eq 5 are readily calculated using the appropriate equations described elsewhere.27,45 The concentrations of the various ion-pair types obtained via eq 5 over the investigated solute concentration and temperature ranges are summarized in Figures 6, 7, and 8 as smooth curves (for representational clarity). The parameters of these empirical curves are collected in Supporting Information Table 5. To illustrate the quality of the fits, error bars representing 2σfit are given for each curve, along with data points at one or two temperatures. Because of the relative weakness of the overall complexation equilibrium, the amplitudes of the various ion-pair types were never very large. Coupled with the inherent difficulties in separating strongly overlapping processes (Figure 4), this means that the uncertainties in the ion-pair concentrations were considerable and care must be taken not to over-interpret them. For example, the apparently odd sequence of c2SIP with respect to temperature in Figure 6 (25 °C > 5 °C > 45 °C > 65 °C) may well be a reflection of the accumulated uncertainties in the data processing (see the relevant error bars on the curves in Figure 6) rather than a real effect. Similar reservations can be expressed about the “abnormal” shape of cCIP(c) at 5 °C (Figure 8, curve 1) compared with those at the other temperatures (Figure 8, curves 2, 3, and 4). Despite these problems, the general features of the data are fully consistent with the proposed model. As required by the existence of the sequential equilibria in Scheme II, both the 2SIP (Figure 6) and SIP (Figure 7) concentrations go through a maximum as a function of the total solute concentration, c, with that for the 2SIPs occurring at lower c. The concentrations of the CIPs on the other hand (Figure 8) simply increase with c as would be expected. The variations at 5 and 65 °C in the concentration of each ion-pair type relative to the total solute concentration are given in Figure 9. These data and those in Figures 6-8 clearly show that the overall association constant increases with temperature. This is consistent with the endothermic (positive) value of ∆rH°.10 The use of the ion-pair concentrations calculated from

Figure 6. Concentrations of double-solvent separated ion pairs, c2SIP, for CuSO4(aq) as a function of the total solute concentration, c, at temperatures t/°C ) 5 (1, 2), 25 (2), 45 (3), and 65 (4). Error bars represent 2σfit for each of the smoothed curves. Data points are omitted for curves 2, 3, and 4 for representational clarity.

Figure 7. Concentrations of solvent-shared ion pairs, cSIP, for CuSO4(aq) as a function of the total solute concentration, c, at temperatures t/°C ) 5 (1, b), 25 (2), 45 (3, 2), and 65 (4). Error bars represent 2σfit for each of the smoothed curves. Data points are omitted for curves 2 and 4 for representational clarity.

eq 5, converted to molalities, enables calculation of Ki, the equilibrium constants for each of the steps of Scheme II and hence the overall association constant KA (eqs 3 and 4) for Scheme I at each temperature and solute concentration. The KA values obtained in this way were fitted to an extended Guggenheim-type equation8

log KA ) log K°A -

2ADH|z+z-|xI 1 + AKxI

+ BKI + CKI3/2 (6)


Akilan et al.

Figure 8. Concentrations of contact ion pairs, cCIP, for CuSO4(aq) as a function of the total solute concentration, c, at temperatures t/°C ) 5 (1), 25 (2, b), 45 (3), and 65 (4, 2). Error bars represent 2σfit for each of the smoothed curves. Data points are omitted for curves 1 and 3 for representational clarity.

where K°A is the standard state (infinite dilution) association constant, ADH is the Debye-Hu¨ckel constant for activity coefficients, YK (Y ) A, B, or C) are adjustable parameters, and I ) (1/2) Σ mizi2 ) 4(m - mIP) is the actual solution ionic strength (with the subscript i here representing any charged species in solution, of charge number zi). The results are plotted in Figure 10, and the fitting parameters are listed in Table 1. Values of ADH at the required temperatures were taken from Archer and Wang.47 Note that use of the Guggenheim equation is for convenience only and does not imply that it has any special validity. Note also that any variation in the activity coefficients of the neutral ion pairs (which almost certainly occurs) is subsumed by the fitting parameters, YK. The sensitivity of K°A to the exact form of the activity coefficient expression (eq 6) is well-known for the Cu2+/SO42system and has been discussed at length on a number occasions.10,38,39,42 Furthermore, there are particular problems attendant on the determination of the rather small ion-pair amplitudes at low concentrations (especially at 5 and 65 °C: the first because of the weaker association at lower T, the second because of the increasing conductivity correction) and the modest precision of even the best DR data. These factors introduce considerable uncertainty into the derived association constants. Nevertheless, the KA values obtained are smooth and systematic functions of I at all temperatures (Figure 10). As found for other systems,16,27 the present values, valid for the “self-medium” of the pure electrolyte solutions, are lower at higher I values than those measured in the presence of a large excess of a supposedly indifferent electrolyte.10 This difference is usually attributed to activity coefficient effects.16,27 The values (on the mol kg-1 concentration scale) of the standard Gibbs energy, enthalpy, entropy, and heat capacity changes for the overall association reaction (cf. Scheme I) were obtained from a conventional thermodynamic analysis of the

Figure 9. Concentrations of ion pairs (ci) relative to the total solute concentration (c) at (a) 5 °C and (b) 65 °C. Note that by definition ci/c ) 0 at c ) 0 for all curves.

K°A(T) data based on the equations

∆rG° ) -RT ln K°A

(7)

∆rG°(T) ) ∆rH°298 + ∆rC°p (T - T*) - T[∆rS°298 + ∆rC°p ln(T/T*)] (8) where T* ) 298.15 K and ∆rC°p is assumed to be independent of temperature.10 The values so derived (Table 2, Figure 11) are in excellent agreement with those recommended by IUPAC10 and with the most recent determination by Besˇter-Rogacˇ et al.43 (reanalyzed in mol/kg concentration units) employing highprecision conductivity measurements. Given the large uncertainties in the DRS data, the agreement between the present and literature values for the thermodynamic quantities, however gratifying, must be partly fortuitous. This becomes apparent from a consideration of the stepwise equilibrium constants (Scheme II). Values of Ki were calculated from



Figure 10. Overall association constants, KA (mol kg-1 scale), for Scheme I as a function of ionic strength, I, at 5 °C (1, 1, ×0.2 for representational clarity), 25 °C (2, b, ×1), 45 °C (3, 2, ×2.5), and 65 °C (4, 9, ×5).

TABLE 1: Standard Overall Association Constant, K°A, and Parameters AK, BK, and CK of eq 6 for CuSO4(aq)a T

K°A

AK

BK

CK

278.15 298.15 318.15 338.15

195 ( 140 245 ( 20 299 ( 41 386 ( 142

2.19 1.00b 1.35 1.61

-1.08 -0.163 -0.48 -0.63

0.33 0.025 0.12 0.18

a

mol

Figure 11. Gibbs energy change, ∆rG°, for the ion association reaction (cf. Scheme I) for CuSO4(aq) as a function of temperature, T. The solid line gives the fit of the present DRS data (b) with eq 8. The triangles (2, 1) show ∆rG° from two data treatments of ref 43. The broken line and the 298 K point (]) indicate the values of ∆rG° calculated from the recommended thermodynamic parameters of ref 10.

Units: T in K; K°A, BK in kg mol-1; AK in (kg mol -1)1/2; CK in (kg -1 3/2 b ) . Fixed.

TABLE 2: Standard Thermodynamic Parameters (mol kg-1 concentration scale) for the Association of Cu2+(aq) and SO42-(aq) at 25 °Ca log K°A -∆rG° ∆rH° ∆rS° ∆rCp°

ref 10

ref 43

this work

2.34 ( 0.05 13.4 ( 0.3 7.3 ( 1.0 68.4 ( 0.7 272b

2.39 ( 0.17 13.7 ( 1.0 6.5 ( 0.5 68 ( 2 250 ( 70

2.4 ( 0.2 13.6 ( 1.1 8.2 ( 0.5 73 ( 2 (79 ( 40)c

a Units: K°A in kg mol-1; ∆rG°, ∆rH° in kJ mol-1; ∆rS°, ∆rC°p in J K-1 mol-1. b Essentially constant over a wide temperature range.10 c Value not considered reliable due to overall uncertainties.

the smoothed species molalities using the appropriate expressions (eq 4). The dependence of the Ki values on I at 25 °C is shown in Figure 12. Note that at m g 0.4 mol/kg the 2SIP contribution is too small to provide a reliable estimate of m2SIP and hence of K1 and K2. The values of K1 and K2 at such molalities, shown as broken lines in Figure 12, were extrapolated from the fits at m < 0.4 mol/kg and are included only as an indication of the trend. Similarly, the downturn in K3 at low I is almost certainly an artifact since such a constant (eq 4) would not be expected to show a significant dependence on ionic strength, particularly at low I. Also included in Figure 12 are the K°i values reported from the two most comprehensive ultrasonic absorption studies.22,23 Given the uncertainties in both the ultrasonic and DR methods, this level of agreement in such a complicated system provides strong support for the validity of the present interpretation.

Figure 12. Stepwise stability constants Ki for the formation of the ion-pair types (cf. scheme II and eq 4) for CuSO4(aq) at 25 °C: present DRS results shown as solid lines; ultrasonic relaxation data of ref 22 (b) and ref 23 (2). The broken lines are extrapolated and are included only as a visual aid (see text).

On the other hand, it becomes apparent from the corresponding analysis of the DR data at other temperatures (not shown) that the Ki values do not vary smoothly with temperature. This is almost certainly a reflection of the real uncertainties in the data. It does not appear to be possible with the currently available accuracy of DRS to derive sensible thermodynamic parameters for the stepwise processes of the present system. An interesting feature of the present data is the absence of any evidence for the formation of triple ions. This contrasts


Akilan et al.

Figure 13. Solute relaxation times τ1, τ2, and τ3 for CuSO4(aq) at temperatures t/°C ) 5 (O, b), 25 (0, 9), 45 (4, 2), and 65 (3, 1).

with the behavior of other MSO4(aq) solutions previously investigated by broadband DRS (M ) Mg16, Ni37, and Co37) which all show significant formation of M2SO42+(aq) triple ions even at modest salt concentrations.48 4.3. Ion-pair Relaxation Times. The relaxation times for the three solute-related processes, τj (j ) 1 to 3), are summarized in Supporting Information Tables 1-4 and plotted in Figure 13. Within the limits of experimental error all three are independent of solute concentration. The slight increase/decrease in τ2 as c f 0 at 5/65 °C is not significant and probably arises from correlations among the fitting parameters. Even so, all three ion-pair relaxation times decrease with increasing temperature consistent with decreasing solution viscosities. The uncertainties in the data, coupled to the difficulties of separating these overlapping processes, do not permit the determination of meaningful activation energies. 4.4. Solvent Relaxation and Ion Hydration. As shown in detail elsewhere,26,27 DRS can be used to derive effective (“irrotationally bound”) hydration numbers, Zib

Zib ) (c°s - capp s )/c

(9)

where capp is the apparent water () solvent, subscript “s”) s concentration as obtained from the DRS data and c°s is the analytical (total) concentration of water in the solution. Values of capp were calculated via the solvent-normalized Cavell s equation

capp s )

2 2(c) + 1 (0) (1 - Rsfs(c)) c°s(0) S (c) 2(0) + 1 (c) (1 - Rsfs(0))2 Ss(0) s

(10)

as described in detail for MgSO4(aq),16 with correction for kinetic depolarization of the ions assuming slip boundary conditions.26,27 As justified previously,37 the dispersion amplitude of unbound water, Ss(c) ) S4 + S5 ) 4 - ∞ was derived by assuming ∞ ) ∞(0), the pure water value at the appropriate temperature. The ∞(0) values were taken from Ho¨lzl et al.34 but are included for convenience in Supporting Information Tables 1-4.

Figure 14. Effective hydration numbers, Zib(CuSO4(aq)) (symbols) at temperatures t/°C ) 5 (1), 25 (2), 45 (3), and 65 (4). Lines are weighted fits; symbols have been omitted from curves 2 and 3 for representational clarity.

The values of Zib so obtained are plotted in Figure 14 and show that the total effective solvation decreases smoothly with increasing concentration. The error bars correspond to the standard deviation of the solvent amplitude obtained from fits of 4 - ∞ with appropriate polynomials forced through the value for pure water, (0) - ∞(0). A striking feature of the data in Figure 14 is the strong increase of the infinite dilution value, Zib(0), with decreasing temperature. The trend from Zib(0) ≈ 41 at 5 °C to ≈29 at 25 °C, ≈22 at 45 °C, and ≈21 at 65 °C is consistent with the DRS solvation numbers reflecting a balance between ion-solvent interactions and thermal motion. The Zib(0) values substantially exceed the sum of the first-shell coordination numbers of Cu2+ (CN+ ≈ 649,50) and SO42- (CN≈ 849) dictated by geometric constraints. This indicates that water molecules beyond the first and possibly even the second hydration shell are also restricted in their response to the imposed electric field. At higher solute concentrations, Zib drops to values close to (5 and 25 °C) or even below (45 and 65 °C) the sum of the first-shell coordination numbers (CN+ + CN-). This is consistent with the breakdown of well-defined second hydration shells associated with increased ion association, and in particular with the increased formation of CIPs, with rising temperature. Splitting Zib into its ionic contributions (Figure 15) was only possible at 25 °C, where Zib(SO42-) values are available.27 On the basis of these ionic values, which are ultimately based on the assumption that Zib(Cl-) ) 0,26,51 it appears that, at c f 0, Cu2+ “immobilizes” approximately 19 water molecules at 25 °C. This is in good agreement with the sum of first- and second-shell hydration numbers from computer simulations (6 + 11.7)50 and from scattering experiments (6 + 11).49 As the solute concentration increases, Zib(Cu2+) rapidly drops to ∼10 and remains constant at c J 0.4 M. Similar, albeit less pronounced, behavior of Zib(M2+) has also been observed for Mg2+ 16 and Ni2+ and Co2+.37 The present results do not, unfortunately, shed any light on the ongoing controversy52 about the coordination (inner sphere) geometry of Cu2+(aq) since the present DRS data are broadly


J. Phys. Chem. B, Vol. 110, No. 30, 2006 14969 investigator to suggest that “it may become desirable to adopt reasonable but arbitrary conVentions (present authors’ italics) which would make K values unambiguous”.42 As noted previously,16 activity coefficient equations that take no account of the actual species present cannot be much more than exercises in numerology, with little physical significance. The existence of solvent-separated ion pairs also explains why the KA and K°A values that have been obtained from extensive high-quality measurements of the Cu2+/SO42- system by UVvis spectrophotometry are lower than those obtained from reliable conductivity and thermodynamic data.10,14 Finally, although not the focus of the present study, the presence of the various ion-pair types is of considerable significance in kinetic studies. Not only would such species be expected to affect homogeneous reaction kinetics involving CuSO4(aq) but also, potentially much more importantly, the heterogeneous kinetics involved, for example, in the electrodissolution and electrodeposition steps of the copper refining process.

Figure 15. Effective hydration numbers at 25 °C, Zib(CuSO4), symbols and corresponding weighted fit, (1), Zib(Cu2+) (2), and Zib(SO42-)27 (3).

consistent with any of the proposed geometries. However, as noted above, Zib(CuSO4) increases considerably at lower temperatures, especially at c f 0, although the error limits are large (Figure 14). Such behavior which is attributable, at least in part, to an increase in Zib(Cu2+) is quite different from that of MgSO4(aq) where a change in temperature from 5 to 65 °C has almost no effect on Zib.53 This might reflect increased ordering of the first hydration shell of Cu2+(aq) and thus in the subsequent shells but proof of this would require detailed structural studies over a range of temperatures and none is available at present.52 5. Concluding Remarks: Implications of the Present Work The present DRS data clearly demonstrate that, as for MgSO4(aq),16 NiSO4(aq),37 and CoSO4(aq),37 all three ion-pair types, 2SIPs, SIPs, and CIPs, exist simultaneously (in varying amounts) in aqueous solutions of copper(II) sulfate at all practicable concentrations. The relative amounts of these species vary but all of them persist over the studied temperature range (Figure 4). Where comparison is possible, the present findings are quantitatively corroborated by earlier thermodynamic (Figure 11) and ultrasonic absorption (Figure 12) studies. The existence of such species explains many observations. For example, the failure of early Raman spectroscopy studies12 to detect ion association in CuSO4(aq) would appear to be not only due to the relatively wide slit-widths then available (making observation of overlapping peaks difficult) but also because the species formed were mostly solvent-separated ion pairs (2SIPs and SIPs) that are not detected by Raman measurements of the strong ν1(a1) mode of the SO42- ion.15 Modern high-quality Raman spectra14,44 detect CIP concentrations that are in broad agreement with the present DRS results. The difficulties experienced in obtaining a unique theoretical description of even the most high-quality conductivity data39,43 are also closely related to the presence of solvent-separated ion pairs. This is because such species make difficult the meaningful definition of the integration limits (distance parameters) that are required.41 Similar considerations apply to the estimation of activity coefficients: a problem that has been recognized for at least 50 years and which has led at least one authoritative

Acknowledgment. The authors thank the Humboldt Foundation for a Fellowship (for N.R.), Dr. W. W. Rudolph for making available some of his unpublished Raman data, Prof. W. Kunz for the provision of laboratory facilities at Regensburg, and Murdoch University for partial financial support. Supporting Information Available: Tables of conductivities, limiting permittivities, relaxation times, and the reduced error function of CuSO4(aq) as a function of solute molality at 5, 25, 45, and 65 °C are presented. Empirical fitting equations and a table of the parameters a1-a4 of the equations are also given. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Hayes, W. J. Pesticides Studied in Man; William and Wilkins: Baltimore, MD, 1982. (2) Worthing, C. R., Ed. In The Pesticide Manual; British Crop Protection Council: Croydon, U. K., 1983. (3) Nriagu, J. O., Ed. In Copper in the EnVironment; Wiley-Interscience: New York, 1979. (4) Hartley, D., Kidd, H., Eds. In The Agrochemicals Handbook; Royal Society of Chemistry: Nottingham, U. K., 1983. (5) Kroschwitz, J., Ed. Kirk-Othmer Encyclopedia of Chemical Technology, 5th ed.; Wiley-Interscience: Hoboken, NY, 2004; Vol. 7, pp 767783. (6) Biswas, A. K.; Davenport, W. G. ExtractiVe Metallurgy of Copper, 3rd ed.; Pergamon/Elsevier: Oxford, U. K., 1994. (7) Zaytsev, I. D.; Aseyev, G. G. Properties of Aqueous Solutions of Electrolytes; CRC Press: Boca Raton, FL, 1992. (8) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth: London, 1970. (9) Nancollas, G. H. Interactions in Electrolyte Solutions; Elsevier: New York, 1966. (10) Powell, K. J.; Brown, P. L.; Byrne, R. H.; Gajda, T.; Hefter, G. T.; Sjo¨berg, S.; Wanner, H. Pure Appl. Chem., in press. (11) Davies, C. W. Ion Association; Butterworth: London, 1962. (12) Hester, R. E.; Plane, R. A. Inorg. Chem. 1964, 3, 769. (13) Malatesta, F.; Zamboni, R. J. Solution Chem. 1997, 26, 791. (14) Hefter, G. T. Pure Appl. Chem. 2006, 78, 1571. (15) Rudolph, W. W.; Irmer, G.; Hefter, G. T. Phys. Chem. Chem. Phys. 2003, 5, 5253. (16) Buchner, R.; Chen, T.; Hefter, G. J. Phys. Chem. B 2004, 108, 2365. (17) Eigen, M.; Tamm, K. Z. Elektrochem. 1962, 66, 93. (18) Eigen, M.; Tamm, K. Z. Elektrochem. 1962, 66, 107. (19) Kaatze, U.; Hushcha, T. O.; Eggers, F. J. Solution Chem. 2000, 29, 299. (20) Buchner, R.; Barthel, J. Annu. Rep. Prog. Chem. C 2001, 97, 349. (21) Buchner, R. In NoVel Approaches to the Structure and Dynamics of Liquids; Samios, J., Durov, V. A., Eds.; NATO Science Series II; Kluwer: Dordrecht, The Netherlands, 2004; Vol. 133.

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Ion Association and Hydration in Aqueous Solutions of Copper(II

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