Reactions of Hexa-aquo Transition Metal Ions with the Hydrated

Article pubs.acs.org/JPCA

Reactions of Hexa-aquo Transition Metal Ions with the Hydrated Electron up to 300 °C Kotchaphan Kanjana, Bruce Courtin, Ashley MacConnell, and David M. Bartels* Notre Dame Radiation Laboratory & Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556 United States ABSTRACT: Reactions of the hydrated electron with divalent aqueous transition-metal ions, Cd2+, Zn2+, Ni2+, Cu2+, Co2+, Fe2+, and Mn2+, were studied using a pulse radiolysis technique. The kinetics study was carried out at a constant pressure of 120 bar with temperatures up to 300 °C. The rate constants at room temperature agree with those reported in the literature. The reaction of Cd2+ is approximately diffusion-limited, but none of the first-row transitionmetal ion reactions are diffusion-controlled at any temperature studied. The activation energies obtained from the Arrhenius plots are in the range 14.5−40.6 kJ/mol. Pre-exponential factors are quite large, between 1 × 1013 and 7 × 1015 M−1 s−1. There appears to be a large degree of entropy−enthalpy compensation in the activation of Zn2+, Ni2+, Co2+, and Cu2+, as the larger pre-exponential factors strongly correlate with higher activation energy. Saturation of the ionic strength effect suggests that these reactions could be long-range nonadiabatic electron “jumps”, but Marcus theory is incompatible with direct formation of ground state (M+)aq ions. A self-consistent explanation is that electron transfer occurs to excited states derived from the metal 4s orbitals. The ionic strength effect in the Mn2+ and Fe2+ reactions suggests that these proceed by short-range adiabatic electron attachment involving breakdown of the water coordination shell. and Mn2+ were investigated up to 300 °C in pressurized water using a pulse radiolysis technique. The results are discussed in the context of transition state theory and the Marcus Theory of electron transfer.

I. INTRODUCTION Reactions of transition-metal ions and small colloidal particles with the primary species of water radiolysis are implicated as part of the complex process of corrosion and activity transport in the water cooling loops of nuclear reactors.1,2 The kinetics information on all possible reactions in high-temperature highpressure water is required to complete the radiolysis simulation of these systems.3,4 The present study represents a first installment in reporting the rate constants for hydrated electron with aqueous transition-metal cations that might be present in the cooling loops. As it turns out, the rate constants themselves are of considerable interest in terms of the theory of hydrated electron reaction rates and electron transfer in general. After the discovery of the hydrated electron by Hart and Boag in 1962,5,6 a question immediately asked was how the species behaves in an electron transfer process. It has been proposed that the hydrated electron reactions amount to outer-sphere electron transfer,7−10 the process in which an electron translates directly between two uncoupled reactants without penetration through any bridges between the two centers.11−14 Because many transition-metal complexes, including hexa-aquo transition-metal ions, are well-known to exhibit outer-sphere electron transfer,15−17 this group of compounds was used as reaction partners in the room temperature study of reactions of hydrated electron.7,18−20 Surprisingly, there has been no systematic study of activation energies for these compounds. In the present study the reactions of the hydrated electron with transition-metal ions Cd2+, Zn2+, Cu2+, Co2+, Ni2+, Fe2+, © 2015 American Chemical Society

II. EXPERIMENTAL SECTION Pulse radiolysis experiments were performed using nanosecond electron pulses from an 8 MeV Linear Accelerator (LINAC) at the Notre Dame Radiation Laboratory. Water radiolysis produces hydrated electrons, OH radicals, and other minor products as in eq 1.21 The yields of the several species are a weak function of temperature up to 300 °C.4 The transient absorption of the hydrated electron was followed by a UV− visible PMT/monochromator detection system with nanosecond time response. The probe light source was a 75 or 150 W xenon lamp pulsed to high current for several hundred microseconds. H 2O ⇝ (e−)aq + (H+)aq + OH + H + H 2 + H 2O2 (1)

Deionized water with 18 MΩ cm resistivity (total organic carbon (CO2) < 10 ppb) from a Serv-A-Pure Co. cartridge system was used throughout the experiment. The perchlorate salts of transition-metal ions Mn2+, Co2+, Ni2+, Cu2+, Fe2+, and Received: September 9, 2015 Revised: October 22, 2015 Published: November 4, 2015 11094

DOI: 10.1021/acs.jpca.5b08812 J. Phys. Chem. A 2015, 119, 11094−11104

Article

The Journal of Physical Chemistry A

Figure 1. Decay of the hydrated electron in Co2+ aqueous solutions under different experimental conditions: (a) 25 °C, 18 Gy; (b) 300 °C, 5 Gy. Solid lines represent a least-squares fit to a single exponential. Room temperature pH is 5.5.

Figure 2. Plots of single exponential (e−)aq decay rates vs Co2+ concentrations at different doses: (a) 25 °C and (b) 300 °C.

Zn2+ were purchased from American Elements (99.99%). Cadmium perchlorate (99.999%) was from Sigma-Aldrich. Perchloric acid was added to pH 5.5 to ensure the fully protonated hexahydrate ions were present. In the case of Mn2+, 0.01 M methanol (Sigma-Aldrich, 99%) was added to scavenge OH radicals. In the Fe2+ experiment, 0.01 M sodium formate (Sigma-Aldrich, 99.99%) was added to ensure reducing conditions and eliminate any contribution from Fe3+. Ionic strength was controlled by addition of sodium perchlorate. The high temperature kinetics study was carried out using two high pressure Isco 260D syringe pumps, one of which contained a concentrated solution of the metal perchlorate salts and the other was filled with water at the same pH as the salt solution. Both solutions were sparged with Ar to remove oxygen. The solutions from the pumps were mixed at a tee connection. By varying the flow rates of the two pumps, with constant total flow of 3 mL/min, the concentration of the metal ions could be controlled. In experiments with excess sodium perchlorate, both syringes contained identical background salt concentrations. After the tee connection, the mixed solution flowed through a preheater coil to set the temperature, and then into a high-temperature high-pressure Hastelloy 276C cell of 1.1 cm path length with sapphire optical windows. The cell is attached to the electron beamline where the transient absorption was measured. Experiments on Fe2+ were carried out only up to 200 °C using a very similar cell of 316 stainless steel and 2.5 cm path length with fused silica windows because the Hastelloy cell was unavailable.

On the basis of the absorption of the hydrated electron at 720 nm with extinction coefficient22 on the order of 20 000 M−1 cm−1, the concentration of electrons generated by the radiation pulses was on the order of 1 μM. The corresponding typical radiation dose was on the order of 1−5 Gy.

III. RESULTS It is well-established that the divalent transition-metal ions (M2+), Cd2+, Zn2+, Cu2+, Co2+, Ni2+, Fe2+ and Mn2+, can be reduced by the hydrated electron. The reduction process yields the corresponding hyper-reduced transient species (M+)6,23−27 as described in reaction 2. (e−)aq + M2 + → M+

(2)

The rate of the reaction is determined from the hydrated electron decay in the aqueous solution of the divalent transition-metal ion of interest. The decay of the hydrated electron absorption was recorded at temperatures up to 300 °C near its room temperature absorption maximum28 at 720 nm. At each temperature the divalent ion concentration was varied and signals were recorded for two or three doses to check for second-order kinetic effects. Example decays of the hydrated electron in aqueous cobalt perchlorate solutions are shown in Figure 1. (The blank traces in the absence of scavenger are dominated by second-order recombination and so are not well fit by the single exponential functions.) It is clear that the rate constant strongly depends on the temperature. Decay rates were plotted against the concentrations of Co2+, giving the 11095


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The reaction distance of ca. 3 Å required to correct the Mn2+ data for ionic strength is surprisingly small given the size of the hexa-aquo transition-metal complexes. The M−O distance is known from X-ray studies to be 2.1−2.2 Å in all of the first-row transition-metal complexes,31 whereas the radius of gyration for a hydrated electron is known to be 2.45 Å near room temperature.28 If we add 1.6 Å for the oxygen van der Waals radius of the complex, a reasonable minimum “contact distance” might be ca. 6.2 Å. The 3 Å reaction distance implies the electron is already inside the coordination sphere, and a similar inference can be drawn for the Fe2+ reaction. Given the picture of a hydrated electron with a tetrahedral arrangement of four inner-shell water molecules,32 at least one of the metal ion coordination sites has been “taken over” by the electron at the transition state, but (from the definition of transition state) the overall difficulty of reaction is such that the electron can still diffuse away even from this short contact distance. To check whether the other complexes have a similar ionic strength sensitivity, experiments were carried out on Ni2+, Co2+, Cu2+, and Zn2+ in the presence of excess sodium perchlorate salt. (The rate constants were determined as in the other experiments with five different M2+ concentrations, but constant NaClO4.) The result for Co2+ is plotted in Figure 4.

reaction rate constants from the slope as illustrated in Figure 2 for three different applied doses. In a reaction between charged species, the observed reaction rate is affected by the ionic strength as described by the Brønsted−Debye equation for spherical reactants shown below,29,30 log(k μ) = log(ko) + 2AZ1Z 2 μ /(1 + β r12 μ ) A=

β=

(3)

e 3{2NA }1/2 2.303{8π }{ε0εkBT }3/2

2e 2NA ε0εkBT

(4)

(5)

where kμ and k0 are the rate constants at the ionic strength equal to μ and 0, respectively, A is the Debye−Hückel constant calculated using eq 4, and β is defined in eq 5. Z1 and Z2 are charges of reactants 1 and 2, respectively, μ is the ionic strength (M), r12 is the reaction distance (assumed equal to the sum of the radii of the species 1 and 2), NA is the Avogadro number, ε0 is the vacuum permittivity, ε is the dielectric constant of the solvent, and kB is the Boltzmann constant. According to the Brønsted−Debye equation, the rate constant at the ionic strength μ (kμ) of the reaction between two opposite-sign charged species such as in our (e−)aq + M2+ system will decrease as the concentration of charged scavenger increases. In most of the M2+ scavenger experiments, the rate constant is high enough and the scavenger concentration is low enough to ignore this correction. For Zn2+ near room temperature, corrections were at the 10−20% level. In the case of Mn2+ and Fe2+, the rate constant is relatively low and the necessary ionic strength correction is quite large over the entire temperature range. In Figure 3, the measured scavenging

Figure 4. Co2+ reaction rate with (e−)aq vs ionic strength (from sodium perchlorate) at 25 °C. The fit to the data implies a reaction distance of 8.2 ± 0.3 Å.

These rate constants are less sensitive to ionic strength overall, and the effect saturates more quickly. The Brønsted−Debye equation fits the data, with a reaction distance near 11 Å for Zn2+ and Cu2+ and 8 Å for Co2+. An experiment with Co2+ at 200 °C is fit with a somewhat larger distance near 10 Å after appropriate correction for temperature and dielectric constant. A larger distance at high temperature might be rationalized on the basis of the greater “size” of the hydrated electron inferred from its optical spectrum28 at 200 °C. On the basis of these experiments, it is reasonable to infer that Ni2+, Co2+, Cu2+, and Zn2+ react with (e−)aq by a “long-range” nonadiabatic electron jump mechanism. A long-range reaction has been inferred for the reaction of solvated electron with Cu2+ in ethylene glycol solvent.8 In contrast, Fe2+ and Mn2+ may react by a short-range “adiabatic electron attachment” mechanism as described above. The ionic-strength-corrected rate constants at room temperature for all of the reactions studied are in good agreement with those reported previously.6,26,27,29,33−35 All rate constants are

Figure 3. Ionic strength effect on the rate of the reaction (e−)aq + Mn2+ at 100 °C. The arrow indicates the change in the reaction rate after the ionic strength correction.

rates at 100 °C for (e−)aq + Mn2+ at given concentration/ionic strength are also compared with the ones after ionic strength correction using eq 3 for each individual point. The correction successfully linearizes the scavenging plot, assuming a reaction distance r12 = 3.0 ± 0.5 Å. Similar corrections are required for Mn2+ and Fe2+ over the entire temperature range, with the optimum distance for Fe2+ at r12 = 4.5 ± 1.0 Å. 11096


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The Journal of Physical Chemistry A Table 1. Temperature-Dependent Rate Constants at Zero Ionic Strength (k0) and kdiff up to 300 °C 10−10k0 (M−1 s−1) T (°C)

Mn

2+

2+

Fe

0.0055 ± 0.0005 0.018 ± 0.002 0.057 ± 0.005 0.12 ± 0.03 0.34 ± 0.06 1.0 ± 0.2 3.7 ± 0.8

25 50 100 150 200 250 300

0.011 ± 0.001 0.029 ± 0.003 0.23 ± 0.02 1.2 ± 0.1

Co

2+

1.0 ± 0.1 3.3 ± 0.2 9.2 ± 0.7 27 ± 3 63 ± 6 120 ± 12 170 ± 20

Ni2+

Cu2+

Zn2+

Cd2+

10−10kdiff (M−1 s−1)

2.2 ± 0.1 5.0 ± 0.3 13 ± 0.7 28 ± 2 63 ± 4 110 ± 10 190 ± 20

3.4 ± 0.3 5.6 ± 0.3 13 ± 1 22 ± 2 31 ± 2 42 ± 5 74 ± 7

0.16 ± 0.01

5.35 ± 0.25 10.0 ± 0.5 25.8 ± 1.3 51.7 ± 2.5 103 ± 5 192 ± 10 253 ± 13

5.7 11.0 31.9 74 147 261 427

3.0 ± 0.2 14 ± 1 59 ± 5 140 ± 15 230 ± 20

Table 2. Summary of Ionic-Strength-Corrected Rate Parameters for Reactions of (e−)aq with Aqueous Transition-Metal Complexes M2+ valence orbital

reaction −

(e )aq (e−)aq (e−)aq (e−)aq (e−)aq (e−)aq (e−)aq

+ + + + + + +

2+

Mn Fe2+ Co2+ Ni2+ Cu2+ Zn2+ Cd2+

5

3d 3d6 3d7 3d8 3d9 3d10 4d10

10−10k0 (M−1 s−1) (25 °C)

r12 (Å)

0.0055 ± 0.0005 0.011 ± 0.001 1.0 ± 0.1 2.2 ± 0.1 3.4 ± 0.3 0.16 ± 0.01 5.35 ± 0.25

3.0 ± 0.5 4.5 ± 1.0 8.2 ± 0.3 18 ± 3 10.9 ± 0.4 11.0 ± 0.5

reported in Table 1. Errors are estimated as twice the standard deviation of the linear scavenging slopes, plus 5% to account for uncertainties in temperature, random impurities, and other unknown systematic errors. (Generally, this amounts to about 10% total.) Values for r12 inferred from the Brønsted−Debye equation are summarized in Table 2 along with the Arrhenius parameters. It is worth noting that our initial experiments with Mn2+ severely overestimated the rate constant for solvated electrons because there was no scavenger present for OH radical. The OH radical very quickly reacts27,34 with Mn2+, forming a product that can easily be reduced by (e−)aq. Even though this is a second-order reaction, the corresponding decay kinetics of the hydrated electron appear to be pseudo-first-order with respect to the Mn2+ concentration for the dose range used in our experiments. With addition of 0.01 M methanol, formation of this product is prevented, and electrons react only with Mn2+ as intended. All of the metal complex scavenging reactions have been found to approximate Arrhenius behavior up to 300 °C, as demonstrated in Figure 5. The activation energies (Ea) and Arrhenius pre-exponential factors (A) obtained from the plots are also summarized in Table 2. The activation energies for the reactions fall in the range of 14.5−40.6 kJ/mol. The preexponential factors in most of the reactions are very large, ranging from 4 × 1012 to nearly 1016 M−1 s−1. In general, the size of the pre-exponential factor correlates with the size of the activation energy. The exception is Mn2+, which has a more normal pre-exponential factor. Because the rate constant for an overall bimolecular charge transfer reaction is a combination of the diffusion and electron transfer steps, the diffusion-controlled rate constant kdiff needs to be estimated to understand the rate-determining step of the process. The estimation is based on the Smoluchowski−Debye equation shown below.24,36

kdiff

⎧ ⎫ Z1Z 2e 2 ⎪ ⎪ ⎪ 4πNAr12D ⎪ r12 4πεkT ⎨ ⎬ = ⎞⎪ 1000 ⎪ ⎛⎜ ⎡ Z1Z 2e2 ⎤ ⎪ ⎝exp⎣⎢ r12 4πεkT ⎦⎥ − 1⎟⎠ ⎪ ⎩ ⎭

Ea (kJ/mol) 27.5 40.6 24.8 21.6 14.5 37.0 19.9

± ± ± ± ± ± ±

0.7 1.4 0.7 0.6 0.5 1.0 1.4

prefactor (A) (M−1 s−1) (3.8 (1.1 (4.4 (1.9 (1.6 (7.9 (1.7

± ± ± ± ± ± ±

0.9) 0.5) 1.3) 0.6) 0.5) 2.4) 0.5)

× × × × × × ×

1012 1014 1014 1014 1013 1015 1014

Figure 5. Arrhenius plots of the rate constant (ko) and diffusioncontrolled rate constant (kdiff) at temperatures up to 300 °C.

Here D is the relative diffusion coefficient, which we set to the electron diffusion coefficient (4.9 × 10−9 m2 s−1 at room temperature37) because the electron diffusion is very much faster than the divalent metal ion. The reaction distance (r12) is the sum of the radii of the hydrated electron and the metal ion, assuming that the reaction takes place at the contact of the two reactants (the hydrated electron and the metal ions coordinated by six water molecules). We use the same minimum room temperature estimate of r12 = 6.2 Å made earlier in considering the ionic strength effect. The experimental gyration radius of

(6) 11097


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The Journal of Physical Chemistry A the hydrated electron derived from the optical spectrum (that varies from 2.4 Å at room temperature to 3.5 Å at 300 °C)38 was used for higher temperatures. The calculated kdiff and (ionic-strength-corrected) experimental rate constants at temperatures up to 300 °C are listed in Table 1 and plotted in Figure 5. We emphasize this is a minimum estimate of kdiff. According to this comparison, it is clear that none of the reactions investigated are diffusion-controlled at temperatures up to 300 °C, except perhaps for Cd2+. The Cu2+ activation energy is incorrect for diffusion control. Nickel seems to have about the correct activation energy but is overall 2.5 times slower than Cd2+. The large reaction distances for Ni2+, Co2+, Cu2+, and Zn2+ inferred from the ionic strength effect (Table 2) seem to be incompatible with lack of diffusion control. Generally, reaction distances larger than the “hard sphere” encounter distance give rise to fast diffusion-limited rate constants. A large reaction distance might be compatible with smaller diffusion-limited reaction rate if a spin effect limits the fraction of “effective” encounters. However, only Ni2+ reaction has the correct activation energy to be explained this way. We do not know any of the details of the potential of mean force in encounters of hydrated electrons with hexa-aquo metal ions, but any “hardsphere contact distance” larger than ca. 9 Å seems out of the question. It is far more likely that the Brønsted−Debye equation is not completely valid in characterizing these reactions in this range of ionic strength, so that the reaction distances inferred are not quantitatively accurate. Nevertheless, the difference between Mn2+ and Fe2+ and the other ions seems qualitatively robust. For activation-controlled reactions, transition state theory provides the rate constant formula k = ν exp( −ΔG⧧/RT )

Figure 6. Pre-exponential factors for hydrated electron reactions with transition-metal ions plotted vs their activation energies. The correlation between log(A) and Ea for Ni2+, Co2+, Zn2+, and Cu2+ indicates entropy−enthalpy compensation in the solvation changes at the transition state.

IV. DISCUSSION Although there are probably several thousand reaction rates for the hydrated electron reported in the literature,27 there are many fewer reports of activation energy. Shortly after discovery of the species in the early 1960s several workers made a quick survey of activation energies, mostly in reactions of (e−)aq with neutral species.20,42−44 The astonishing result found was that virtually all activation energies fall in the range 12−16 kJ/mol, regardless of the actual rate constant. Anbar et al.43 reported “The specific rates of these reactions range over f ive orders of magnitude, and no correlation whatsoever could be found between ΔE and k(e−)aq+x” (where ΔE is the Arrhenius activation energy). To our knowledge this “strange attractor” of activation energies has not yet been explained. Only two measurements were made on first-row aqueous transition-metal ions. The apparent quite high activation energy in reaction of hydrated electron with Co2+ and with Mn2+, based on only two temperature points,44 was dismissed in the book of Hart and Anbar20 as probably an artifact of metal ion clustering in the samples. In subsequent decades, Freeman and co-workers45−52 made systematic measurements of solvated electron reaction rates and activation energies in water, various alcohols, and water/ alcohol mixtures, in the temperature range from the freezing point up to 100 °C. This group made a point to investigate “efficient scavengers” like nitrobenzene, inefficient scavengers like phenol, and charged species like Cu2+, Ag+, NO3−, and CrO4−. Lai and Freeman50 reported similar Arrhenius parameters for Cu2+ scavenger as we find in the present study. The reaction of (e−)aq with Ag+ was found48 to have an activation energy 19 kJ/mol and a large pre-exponential of 1 × 1014 M−1 s−1, which fits perfectly on the line in Figure 6. However, on further consideration we realized that these parameters are quite compatible with the diffusion limit for the (e−)aq scavenging by the singly charged metal ion. (Scavenging by Cd2+ is faster because of the larger coulomb attraction in that case.) A high temperature study by Mostafavi et al. of the Ag+ reaction agrees with this assessment.53

(7)

where ΔG⧧ is the free energy of activation and ν is an appropriate frequency factor.12,30 We can rewrite the activation free energy in terms of the entropy and enthalpy components ΔG⧧ = ΔH⧧ − TΔS⧧. In comparison with the empirical Arrhenius equation, we can approximately identify ΔH⧧ = Ea, whereas the pre-exponential includes an additional constant factor exp(ΔS⧧/R). Generally, very large pre-exponential factors such as those found in Table 2 are obtained when activation entropy ΔS⧧ is large and positive. It was noted above that there is a correlation between the size of the preexponentials and the activation energies in Table 2. In Figure 6 we plot the logarithm of the pre-exponential vs the activation energy for the reactions (omitting Cd2+ under the assumption the reaction is diffusion-limited). Surprisingly, this reveals an almost-linear relationship for the four compounds that may react by long-range electron transfer on the basis of ionic strength effect. The linear relationship is evidence of very strong entropy−enthalpy compensation, which is a common characteristic of ionic solvation and reactions in water.39−41 Assuming that ΔS⧧/R.= bΔH⧧, the natural logarithm of the pre-exponential is given by ln(A) = ln(A′) + bΔH⧧, where the slope b = 0.115 kJ−1. The common intercept at ΔH⧧ = 0 corresponds to A′ = 5 × 1011 M−1 s−1. It is reasonable to infer a commonality of mechanism for the hydrated electron reactions with Zn2+, Co2+, Ni2+, and perhaps Cu2+ based on the correlation of Figure 6. Mn2+ and Fe2+ clearly do not fit the correlation, but we already discovered by way of the ionic strength effect that these compounds react by a short-range mechanism. 11098


Article

The Journal of Physical Chemistry A How can we rationalize these reaction rates? Fundamentally, most hydrated electron reactions are electron transfer processes, in which the electron moves from a potential well formed by the solvent molecules into another potential well localized on a scavenger molecule. The Marcus Theory provides a relation, given as eqs 8 and 9,11,12,14 for the rate constant of an electron transfer process in terms of the overall reaction free energy ΔF and a “reorganization energy” λ. The reorganization energy is a constrained free energy parameter directly related to the probability for reaching a transition state configuration where the electron energy is the same in either the donor or acceptor state.12 As a zero-order approximation we can take the hydrated electron to be a negatively charged sphere with a “Born radius” of ca. 4.6 Å calculated from its hydration free energy.28 (Note this is significantly larger than the 2.45 Å radius of gyration used in the diffusion limit estimate, where a lower limit was desired.) The activation energy (ΔG⧧) can be written as in eq 9 where W(r12) is the electrostatic work term for bringing the charged species together, λ is the reorganization energy and ΔF is the Gibbs free energy change for the reactants “in contact” after correcting the long-range electrostatic work. In eq 8, H12 is the matrix element for coupling the two postulated electron states at the transition state configuration where ΔE = 0. k=

2π |H12|2 ℏ

⧧ 1 e−[ΔG / kBT ] 4πλkBT

λ ΔF ΔF 2 ΔG = W (r12) + + + 4 2 4λ

ΔS ⧧ =

⎛ λ 2 − ΔF 2 ⎞ +⎜ ⎟T ΔSλ ⎝ 4λ 2 ⎠

⎛ ⎞ ΔF λ = 2⎜ΔG⧧ − − W ( r )⎟ ⎝ ⎠ 2 ±

(9)

ϕ λ + T T

(11)

2 ⎞⎞ 1 ⎛ ⎜⎛ ⧧ ΔF ⎜ −4 ΔG − − W (r )⎟⎟ − 4ΔF 2 ⎠⎠ 2 ⎝ ⎝ 2

(14)

W(r) and ΔF can be calculated using eqs 15 and 16, respectively, where Zd and Za are the charge of donor and acceptor.

There appears to be a large degree of entropy−enthalpy compensation in the Co2+, Ni2+, Cu2+, and Zn2+ reactions (cf. Figure 6), which is not detected in the vast majority of hydrated electron reactions. Entropy−enthalpy compensation is inherent in the water−water reorganization that accompanies the solvation of ions. 39,40 Therefore, a simple qualitative interpretation of the present data is that water molecules are partially “released” from the coordination sphere as a part of reaching the transition state for these reactions. Ghorai and Matyushov41 have shown that entropy−enthalpy compensation is also a general feature of the solvent reorganization (free) energy λ in electron-transfer reactions. They write the entropy and enthalpy components as

ΔSλ =

(13)

On the basis of eq 12, in the case where λ > |ΔF|, a positive activation entropy can result from positive ΔSr and/or positive ΔSλ. However, a linear relation between ΔS⧧ and ΔH⧧, independent of ΔF, such as suggested by Figure 6 can only be obtained from Marcus theory if λ ≫ |ΔF| . If we could assume that entropy is a minor fraction of the overall activation free energy, ΔG⧧ would be approximately equivalent to Ea obtained from the Arrhenius plots in the previous section. Figure 6 suggests that activation entropy is actually quite significant, so on the basis of that correlation we take ΔG⧧ = Ea × (1 − 0.115 kJ−1 × R × 298 K) for room temperature. Solving the quadratic eq 9 for λ gives eq 14.

⧧

(10)

(12)

⎛ (λ + ΔF )2 ⎞ ⎛ λ + ΔF ⎞ ⎟T ΔS ΔH ⧧ = ⎜ ⎟+⎜ r 4λ ⎠ ⎝ 2λ ⎠ ⎝

(8)

ΔHλ = 2λ + ϕ

⎛ λ 2 − ΔF 2 ⎞ ⎛ λ + ΔF ⎞ ⎜ ⎟ΔS + ⎜ ⎟ΔSλ r ⎝ 2λ ⎠ ⎝ 4λ 2 ⎠

W (r ) =

ZdZae 2 εrda

ΔF = ΔG° + (Za − Zd − 1)

(15)

e2 εrda

(16)

Here rda is the donor−acceptor reaction distance and ΔG0 is derived from the experimental redox potential of the metal ion/ hydrated electron couple. Choosing an appropriate reaction distance is critical for application of the Marcus formulas, because the results depend on the difference of large energy numbers. In our favor is the fact that solvation structure for the hydrated electron in water is settled by recent ab initio MD simulations.32,54 A tetrahedral motif, with OH bonds from four water molecules directed toward the center of a small cavity, seems to match most of the experimental properties.32 The central four water molecules are essential for defining (e−)aq, and likewise the six coordinating water molecules are essential for defining the (M2+)aq complex. The shortest possible distance of approach would place the second solvation shell of (e−)aq at the position of the inner (M2+)aq coordination shell. The reaction distance would be ca. 4.6 + 2.2 = 6.8 Å. Alternatively, one may consider that the second solvation shell of (e−)aq shares the second shell of (M2+)aq, in which case the distance is ca. 4.6 + 4.4 = 9 Å. We choose the latter number because it is in reasonable agreement with the ionic strength effect experiments. Given ΔG⧧ from the experiment together with W(r) and ΔF calculated from eqs 15 and 16, the reorganization energy (λ) for a nonadiabatic electron transfer reaction can be estimated. Assuming linear response,12 λ can be expressed as the sum of

where ϕ is an “energy of solvent restructuring”, which precisely subtracts out of the free energy of reorganization λ. λ/T reflects statistics of the solute−solvent interaction, whereas ϕ/T originates in restructuring of the solvent induced by the solute. In agreement with this, they find that the (positive) observed entropy ΔSλ in their simulations of a model electron transfer reaction is the result of mutual cancellation of these two large numbers. The Marcus theory expression for ΔG⧧ can be rewritten in terms of the entropy components of the overall reaction (ΔSr) and reorganization energy (ΔSλ), where we identify the temperature derivative of λ as the entropy ΔSλ: 11099


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The Journal of Physical Chemistry A Table 3. Reorganization Energies λ from the Marcus Theory by Assuming a Ground State Product reactions −

(e )aq (e−)aq (e−)aq (e−)aq

+ + + +

2+

Co Ni2+ Cu2+ Zn2+

M2+/+ E0 (V)

ΔF′ (eV)

W(r) (eV)

λ (eV)

λout (eV)

Ra (Å)

−0.44 > E0 > −0.95 −0.57 > E0 > −1.9 0.15 −1.9 > E0 > −2.8

−1.9 > ΔF > −2.4 −0.9 > ΔF > −2.3 −2.98 −0.03 > ΔF > −0.93

−0.041 −0.041 −0.042 −0.041

3.7−4.4 2.3−4.1 4.6 1.3−2.8

0.90 0.91 0.93 0.91

4.2 4.1 4.0 4.1

An alternative mechanism for electron transfer between metal ions has been suggested by Fletcher.11,62,63 Rather than fluctuations of electron energy induced by solvent dipoles, he proposes that the fluctuations are provided by the local fields induced by the ionic atmosphere. It seems very plausible that this mechanism could apply to the very polarizable solvated electron. However, Fletcher points out that his proposal would result in a negative activation entropy, whereas we have been trying to explain a large positive entropy. In the foregoing analysis we have started with the assumption that electron transfer occurs without any significant rearrangement of the metal complexes. That is, the initial (M+)aq products of the reductions by hydrated electron have been assumed to be octahedral hexa-aquo complexes derived from the (M2+)aq complexes, with only minor changes in bond lengths. (More generally, we have followed Marcus in assuming that linear response applies.12,64,65 Breaking bonds is usually highly nonlinear.) However, it is reported that (Cu+)aq is 2-fold coordinated,66,67 whereas the Cu2+ water complex is present in five- or six-coordination.55,67−69 A loss of water from the coordination sphere is required in this reaction, and it could be the rate-limiting step. If the other three reactions of Figure 6 have similar mechanisms, it suggests monovalent ions Zn+, Co+, and Ni+ do not exist in a hexa-aquo form. In fact, recent gas phase cluster infrared spectroscopy experiments have inferred that Co(H 2 O) n + 70 and Zn(H 2 O) n + 71 are only threecoordinate, whereas Ni(H2O)n+ is four-coordinate.72 Additional water molecules form hydrogen bonds to the first-shell water molecules in these small gas phase clusters, rather than coordinating to the metal. A computational estimate of the redox potentials for (Ni+)aq also found a four-coordinate geometry of this species.73 What if the first step of the (e−)aq reactions with Ni2+, Co2+, Cu2+, and Zn2+ is the loss of a water molecule from the coordination shell when the hydrated electron happens to be nearby? The nonadiabatic electron transfer might then occur quite readily and subsequent rearrangement would form the most stable (M+)aq complex. The initial loss of water from the hydration sphere could certainly occur with some degree of enthalpy−entropy compensation, which might explain Figure 6. If this two-step reaction idea were correct, these (e−)aq reaction rates should have some kinetic similarity to the exchange rate of solvent molecules with the coordination layer. Study of ligand exchange in transition-metal complexes has a very long history. The exchange of water in first row transition metals is naturally one of the most studied, primarily by NMR techniques. The subject is reviewed by Richens74 in 2005. Investigation of the exchange process by ab initio methods is also feasible, and this was reviewed by Rotzinger,75 also in 2005. Unfortunately, we can find no correlation of the solvent exchange rates with our hydrated electron reaction rates. In the case of Cu2+, average lifetime of H2O in the coordination sphere is measured on the time scale of several hundred picoseconds. For Co2+ the exchange time scale is hundreds of nanoseconds, and for Ni2+ the time scale is hundreds of

inner-sphere reorganization energy (λin) from change of bond lengths, and outer-sphere reorganization energy (λout) from solvent rearrangement: λ = λin + λout. Using the radii of the metal ions31,55 the outer sphere reorganization energy λout can be estimated from the formula derived by Marcus for electron transfer between two charged spheres:12 ⎛ 1 1 ⎞⎛ 1 1 1⎞ − ⎟⎜ + − λout = ⎜ ⎟ εs ⎠⎝ 2R d 2R a rda ⎠ ⎝ ε∞

(17)

Here we take the second shell M−O distance of the hexa-aquo complex to define Ra.31 (We should note that the (Cu2+)aq structure is not strictly octahedral, so the number used for Ra is an average of two distinct second-shell distances in the X-ray scattering.) Rd is the hydrated electron Born radius28 of 4.6 Å, which also coincides with its second coordination shell oxygen g(r) peak.54 Ultimately, λin is the unknown parameter estimated from this calculation, which can be used to interpret the nature of a specific electron transfer reaction. Applying eq 14 results in two solutions of λ. Only the value of λ that allows a positive λin is considered acceptable. The reduction potential E0 = −2.87 V is used for the hydrated electron.27 Unfortunately, standard redox potentials for the M2+/M+ couples are known only for copper. The possible range of redox potentials of the other transition-metal couples are based on a search of the literature and results from our unpublished work.56 The upper limit chosen for the Zn2+/+ couple is based on the demonstration that the monovalent ion Zn+ and can reduce Tl+ to Tl0 (Tl+/0 E0 = −1.9 V).57 A hydrated electron can clearly reduce Zn2+ irreversibly, providing the absolute lower limit for E0 of ca. −2.8 V. In the case of Co2+/+, the lower and upper limits for E0 come from the fact that Co2+ can oxidize the nickel complex NMD+− (E0 = −0.95 V)58 and reduce methyl viologen (E0 = −0.44 V). Ni2+ can be reduced by the CO2−• radical anion (E0 = −1.9 V)59 to establish a lower limit. The upper limit is based on our observation that Ni+ can reduce fluorescein (E0 = −0.57 V).60 The Cu2+/+ E0 = +0.15 V, is well-established from electrochemistry measurement.61 The results from this calculation can be seen in Table 3. In spite of the large uncertainty in most of the E0 values, we are able to establish that the inequality λ > |ΔF| should apply in these reactions so that positive activation entropy is possible within the Marcus framework. (A similar conclusion can be reached with the smaller estimates for the ionic reaction radii.) However, given the very large ΔGo values, reorganization energies must be similar in magnitude to |ΔF| to explain the observed activation energies. We cannot justify λ ≫ |ΔF|. It follows from the analysis presented earlier that standard Marcus theory cannot account for the linear entropy−enthalpy compensation behavior seen in Figure 6, if we assume a ground state product ion characterized by the equilibrium reduction potentials. 11100


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Table 4. Fit Parameters from Applying the Marcus Theory by Assuming Common Overlap Integrals and Entropy−Enthalpy Compensation reaction (e−)aq (e−)aq (e−)aq (e−)aq

+ + + +

Co2+ Ni2+ Cu2+ Zn2+

M2+/+ E0eff (V)

ΔGo (eV)

W(r) (eV)

λout (eV)

ΔSλin (meV K−1)

ΔHλin (eV)

−2.78 −2.81 −2.67 −2.74

−0.085 −0.056 −0.181 −0.126

−0.041 −0.041 −0.041 −0.041

0.91 0.91 0.91 0.91

0.538 0.114 0 1.468

0.315 0.067 0 0.859

microseconds.74 Not only are the time scales for water exchange impossible to explain by our hydrated electron reactions, but also the order of activation energies is incorrect.74 Up to this point we have focused only on the activation energies and entropies and have not remarked on the intercept of the line in Figure 6. A common pre-exponential factor of ca. 5 × 1011 M−1 s−1 (apart from entropy) suggests a common value for the overlap integral |H12|2 in eq 8. This is hardly to be expected if the electron directly transfers to the ground state of the product ions. The reduction of Zn2+, a 3d10 ion, must result in a product with HOMO derived from the metal 4s (or 4p) orbital.76 The ground state products for the other ions certainly place the electron into a 3d orbital. It seems plausible that a “4s” excited state orbital might be accessible to the electron in the Co2+, Ni2+, and Cu2+ complexes as well, to account simultaneously for the large reaction distances, large preexponential factors (requiring favorable overlap integral), and entropy-enthalpy compensation. To check the feasibility of this proposal, we fit the rate constant data for Co2+, Ni2+, Cu2+, and Zn2+ with a reduced model based on the Marcus equations given earlier. The outer sphere reorganization energy is taken as 0.91 eV (c.f. Table 4) independent of temperature, because the temperature dependence of eq 17 is small. As we assume temperature-independent reaction distances, the electrostatic work term W12 is −0.041 eV at room temperature, and scaled according to the temperaturedependent dielectric constant (eq 15). All of the entropy− enthalpy compensation is ascribed to the inner-sphere reorganization, using the constraint ΔHλin = αΔSλin. The value of the constant α is taken as common to all of the reactions, because we presume λin characterizes the partial release of water molecules from the metal coordination shell and reorganization in the hydrated electron structure. The pre-exponentials are fit as a common factor, scaled by (λT)−1/2 as called for in eq 8. For lack of knowledge of the reaction thermochemistry, we assume temperature-independent ΔGo. The global fit of the data based on this constrained model is shown in Figure 7, and the fit parameters are summarized in Table 4. The fit is performed up to 200 °C because the assumptions clearly break down above this temperature. (Nonphysical parameters are required to fit the higher temperatures.) Just as suggested by our earlier analysis, to obtain the entropy−enthalpy compensation, the reorganization energies λin + λout are significantly larger than the reaction free energy. The pre-exponential for the Cu2+ reaction was allowed to vary independently of the others as this resulted in a much improved fit. The best fit was actually obtained with λin = 0 for this reaction. We presume the (Cu2+)aq ion is different due to its nonoctahedral geometry.55,67 The reaction free energies fall into the range −0.056 to −0.181 eV. Subtracting these from the −2.87 eV reduction potential of the hydrated electron gives effective reduction potentials E0eff for the metal ion product states in the range −2.81 to −2.67 eV.

A298 (M−1 s−1) 3.4 3.6 4.3 3.0

× × × ×

1013 1013 1012 1013

Figure 7. Marcus theory fit of rate constants for hydrated electron reactions with Co2+, Ni2+, Cu2+, and Zn2+ up to 200 °C. Major assumptions include common values for the overlap integrals |H12|2, common enthalpy−entropy compensation of the inner sphere reorganization energy, and temperature-independent reaction distances and ΔGo.

Using the constrained Marcus theory calculation, we arrive at the result that the driving force for the initial electron transfer is small. If the initial products were stable, it would be possible to measure an equilibrium between the reduced ion and the (e−)aq. This is perfectly consistent with the proposal that the initial electron transfer occurs into a 4s-derived state of the hexa-aquo ions, which (we infer) all have reduction potentials slightly more positive than (e−)aq. The reduced hexa-aquo ions will then shed water molecules to form the much more thermodynamically stable (M+)aq forms. Marcus pre-exponential factors A298 are given in Table 4 for room temperature. The numbers are about 1013 s−1, consistent with a very favorable electron transfer overlap integral, which is only likely for a totally symmetric 4s metal atom state.11,12 So does the Marcus theory really work to describe the hydrated electron reaction rates with Co2+, Ni2+, Cu2+, and Zn2+? If we could make this claim, it would be a significant advance in understanding hydrated electron reaction rates. The quality of the fit seen in Figure 7 would suggest that Marcus theory can be made to work in this case, but we have to recall that temperature dependence of hydrated electron and metal ion structural and thermodynamic properties are completely omitted from the simple model. The electron is treated as a charged sphere. We simply find the ΔGo parameters that fit the data, subject to the enthalpy−entropy compensation constraint. So it is equally possible that Marcus theory will prove to be deficient, e.g., from a breakdown of the linear response assumption.64,65,77,78 If Marcus theory does work for this series 11101


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The Journal of Physical Chemistry A of (e−)aq reaction rates, we should ask why it does, given that activation energies for hydrated electron reactions fail in general to correlate with ΔGo.42,43 The answer may lie in the rather small free energy change deduced in the present series, which is certainly unusual for (e−)aq reactions.

(3) Bartels, D. M.; Henshaw, J.; Sims, H. E. Modeling the Critical Hydrogen Concentration in the AECL Test Reactor. Radiat. Phys. Chem. 2013, 82, 16−24. (4) Elliot, A. J.; Bartels, D. M. The Reaction Set, Rate Constants and G-Values for the Simulation of the Radiolysis of Light Water Over the Range 20° to 350°C Based on Information Available in 2008. AECL Report 153-127160-450-001; Atomic Energy of Canada Ltd.: Mississauga, Ontario, Canada, 2009. (5) Hart, E. J.; Boag, J. W. Absorption Spectrum of Hydrated Electron in Water and in Aqueous Solutions. J. Am. Chem. Soc. 1962, 84, 4090−4095. (6) Baxendale, J. H.; Fielden, E. M.; Keene, J. P. The Pulse Radiolysis of Aqueous Solutions of some Inorganic Compounds. Proc. R. Soc. London, Ser. A 1965, 286, 320−336. (7) Anbar, M.; Meyerstein, D. Effect of Ligands on Reactivity of Metal Cations Towards the Hydrated Electron: Part 1-the Effect of Ethylenediaminetetraacetic Acid. Trans. Faraday Soc. 1969, 65, 1812− 1817. (8) Schmidhammer, U.; Pernot, P.; De Waele, V.; Jeunesse, P.; Demarque, A.; Murata, S.; Mostafavi, M. Distance Dependence of the Reaction Rate for the Reduction of Metal Cations by Solvated Electrons: A Picosecond Pulse Radiolysis Study. J. Phys. Chem. A 2010, 114, 12042−12051. (9) Anbar, M.; Hart, E. J. On Reactivity of Hydrated Electrons Toward Inorganic Compounds. Adv. Chem. Ser. 1968, 81, 79. (10) Anbar, M.; Hart, E. J. The Reactivity of Metal Ions and Some Oxy Anions Toward Hydrated Electrons. J. Phys. Chem. 1965, 69, 973−977. (11) Fletcher, S. The Theory of Electron Transfer. J. Solid State Electrochem. 2010, 14, 705−739. (12) Nitzan, A. Chemical Dynamics in Condensed Phases; Oxford University Press: Oxford, U.K., 2012. (13) Wilkins, R. G. Kinetics and Mechanism of Reactions of Transition Metal Complexes, 2nd ed.; VCH: New York, 1991. (14) Bakac, A. Physical Inorganic Chemistry: Reactions, Processes and Applications; Wiley: Hoboken, NJ, 2010. (15) Endicott, J. F.; Brubaker, G. R.; Ramasami, T.; Kumar, K.; Dwarakanath, K.; Cassel, J.; Johnson, D. Electron-Transfer Reactivity in some Simple Cobalt(III)-Cobalt(II) Couples. Franck-Condon vs. Electronic Contributions. Inorg. Chem. 1983, 22, 3754−3762. (16) Ferraudi, G. J.; Endicott, J. F. Excited State Redox Chemistry of Polypyridyl Chromium(III) Complexes. A Determination of the Chromium(III)–(II) Self-Exchange Rate. Inorg. Chim. Acta 1979, 37, 219−223. (17) Endicott, J. F.; Kumar, K.; Ramasami, T.; Rotzinger, F. P. Structural and Photochemical Probes of Electron Transfer Reactivity. Prog. Inorg. Chem. 1983, 30, 141−187. (18) Navon, G.; Meyerstein, D. Reduction of Ruthenium(III) Hexaammine by Hydrogen Atoms and Monovalent Zinc, Cadmium, and Nickel Ions in Aqueous Solutions. J. Phys. Chem. 1970, 74, 4067− 4070. (19) Meyerstein, D.; Mulac, W. A. Effect of Ligands on Reactivity of Metal Cations Towards Hydrated Electron Part 2.-Effect of Glycine, Ethylenediamine and Nitrilotriacetic Acid. Trans. Faraday Soc. 1969, 65, 1818−1826. (20) Hart, E. J.; Anbar, M. The Hydrated Electron; Wiley-Interscience: New York, 1970. (21) Spinks, J. W. T.; Woods, R. J. An Introduction to Radiation Chemistry, 3rd ed.; Wiley Interscience: New York, 1990. (22) Hare, P. M.; Price, E. A.; Bartels, D. M. Hydrated Electron Extinction Coefficient Revisited. J. Phys. Chem. A 2008, 112, 6800− 6802. (23) Baxendale, J. H.; Dixon, R. S. Some Unusual Reductions by the Hydrated Electron. Proc. Chem. Soc. (London) 1963, 148−149. (24) Buxton, G. V.; Sellers, R. M. Pulse Radiolysis Study of Monovalent Cadmium, Cobalt, Nickel and Zinc in Aqueous Solution: Part 1-Formation and Decay of the Monovalent Ions. J. Chem. Soc., Faraday Trans. 1 1975, 71, 558−567.

V. SUMMARY The kinetic and thermodynamic characteristics of the hydrated electron reactions with divalent transition-metal ions Cd2+, Zn2+, Cu2+, Co2+, Ni2+, Fe2+, and Mn2+ have been investigated by a high-temperature high-pressure pulse radiolysis study. Except perhaps for Cd2+ none of the reactions investigated are diffusion-controlled at temperatures up to 300 °C. The ionic strength effect suggests that the relatively slow reactions of Mn2+ and Fe2+ are short-range adiabatic electron attachment processes, whereas Cu2+, Co2+, Ni2+, and Zn2+ could be longrange nonadiabatic electron transfers. The Arrhenius plots provide the activation energies in the range 14.5−40.5 kJ/mol, and pre-exponential factors from 1 × 1012 to 7 × 1015 M−1 s−1. The activation energies are unusual for hydrated electron reactions that otherwise tend to be in the 12−15 kJ/mol range.20,42 The pre-exponential factors are quite large and correlate very well with the activation energies for Cu2+, Co2+, Ni2+, and Zn2+, implying a large degree of entropy−enthalpy compensation. A plausible interpretation is that one or more water molecules must first dissociate from the hexa-aquo complex at the transition state for nonadiabatic electron transfer to occur, but the well-established (very slow) solvent exchange rates seem to rule out this two-step reaction idea. The Marcus Theory of electron transfer has been applied to estimate the reorganization energy of the long-range reactions. Except for Zn2+, it is clear that the nonadiabatic electron transfer cannot occur directly to the ground electronic state of a hyper-reduced (M+)aq ion. A self-consistent explanation for the series of “long-range” reaction rates, is that electron transfer occurs to the 4s-derived molecular orbitals, and the reduced ion then sheds two or more water molecules to form the final product. We presume these 4s orbitals are not thermally accessible in the case of Mn2+ and Fe2+, so that adiabatic electron attachment, involving a breakdown of the water coordination layers, becomes the preferred reaction pathway.

■

AUTHOR INFORMATION

Corresponding Author

*D. M. Bartels. E-mail: [email protected]. Phone: (574) 6315561. Fax: (574) 631-8068. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy through award DE-FC0204ER15533. This is manuscript number 5083 of the Notre Dame Radiation Laboratory.

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REFERENCES

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Reactions of Hexa-aquo Transition Metal Ions with the Hydrated

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