Unveiling the Cerium(III)/(IV) Structures and Charge-Transfer Mechanism in Sulfuric Acid

Abstract

The Ce³⁺/Ce⁴⁺ redox couple has a charge transfer (CT) with extreme asymmetry and a large shift in redox potential depending on electrolyte composition. The redox potential shift and CT behavior are difficult to understand because neither the cerium structures nor the CT mechanism are well understood, limiting efforts to improve the Ce³⁺/Ce⁴⁺ redox kinetics in applications such as energy storage. Herein, we identify the Ce³⁺ and Ce⁴⁺ structures and CT mechanism in sulfuric acid via extended X-ray absorption fine structure spectroscopy (EXAFS), kinetic measurements, and density functional theory (DFT) calculations. We show EXAFS evidence that confirms that Ce³⁺ is coordinated by nine water molecules and suggests that Ce⁴⁺ is complexed by water and three bisulfates in sulfuric acid. Despite the change in complexation within the first coordination shell between Ce³⁺ and Ce⁴⁺, we show that the kinetics are independent of the electrode, suggesting outer-sphere electron-transfer behavior. We identify a two-step mechanism where Ce⁴⁺ exchanges the bisulfate anions with water in a chemical step followed by a rate-determining electron transfer step that follows Marcus theory (MT). This mechanism is consistent with all experimentally observed structural and kinetic data. The asymmetry of the Ce³⁺/Ce⁴⁺ CT and the observed shift in the redox potential with acid is explained by the addition of the chemical step in the CT mechanism. The fitted parameters from this rate law qualitatively agree with DFT-predicted free energies and the reorganization energy. The combination of a two-step mechanism with MT should be considered for other metal ion CT reactions whose kinetics have not been appropriately described.

1. Introduction

Charge transfer (CT) is integral to many processes such as energy storage, chemical conversion, and biological reactions.¹ Theories of CT are crucial to interpret experiments and to predict rates and trends. The Marcus theory (MT) of electron transfer (E) was developed to describe homogeneous self-exchange E reactions.¹⁻³ MT and its extensions rationalize E across liquid–liquid interfaces,^1,4 for biological systems,^1,5,6 and for reactions at electrode interfaces.⁷⁻¹⁰ In this work, we use MT to understand a heterogeneous E reaction that initially seems to have discrepancies between the observed kinetics and aqueous ionic structure. Specifically, we study the Ce³⁺/Ce⁴⁺ redox couple in sulfuric acid on different electrodes, which shows notable differences in electrolyte complexation¹¹ between the Ce³⁺ and Ce⁴⁺ oxidation states and extreme asymmetry in the observed redox kinetics. A system is considered asymmetric when the CT coefficient (α) is far from 0.5,¹² and Ce³⁺/Ce⁴⁺ is reported to have α < 0.3.¹³⁻²⁰ The high voltage and tunable redox potentials achievable by Ce³⁺/Ce⁴⁺ lead to its myriad uses including volumetric analysis and ceric oxidimetry,^21,22 chemical oxidation, and energy storage;²³⁻²⁸ thus, understanding the Ce³⁺/Ce⁴⁺ reaction mechanism is important to improve redox kinetics. The work herein highlights the necessity of understanding the cerium ion structure and ligand exchange with the electrolyte to explain the Ce³⁺/Ce⁴⁺ redox kinetics and mechanism. Furthermore, the methodology established to study the cerium CT mechanism is applicable to other redox couples that so far have not been adequately described, such as the V⁴⁺/V⁵⁺ redox reaction.^29,30

The Ce³⁺/Ce⁴⁺ CT mechanism has not been satisfactorily determined, leading to unresolved questions about structure–kinetic relationships and the origin of the highly asymmetric α of Ce³⁺/Ce⁴⁺ in common electrolytes. We have shown that a structural change in the first coordination shell occurs between the Ce³⁺ and Ce⁴⁺ oxidation states,¹¹ in agreement with the literature showing that Ce³⁺ is preferentially coordinated by water in most acids³¹⁻³⁴ and Ce⁴⁺ is complexed by anions in acids.³⁵⁻⁴¹ The density functional theory (DFT)-calculated energy of this complexation explains the shift in the Ce³⁺/Ce⁴⁺ redox potential with acid.^11,23 However, we identified small differences in the extended X-ray absorption fine structure spectroscopy (EXAFS) of Ce³⁺ in sulfuric acid compared to other acids, and we were unable to identify the exact structure of Ce⁴⁺ in sulfuric acid through EXAFS, which motivated additional EXAFS studies.

The Ce³⁺/Ce⁴⁺ CT has typically been studied as a single outer-sphere E step despite evidence of uneven complexation. A recent study of Ce³⁺/Ce⁴⁺ kinetics on a gold electrode in sulfuric acid corroborated the reports of asymmetry.¹³ However, this asymmetric behavior could not be accounted for even with an asymmetric model of MT. Electrostatic effects in the region beyond the outer Helmholtz plane were proposed to control the observed kinetics through a rate-determining step (RDS) involving ligand dissociation. Although this finding highlights the importance of considering chemical steps in the overall mechanism, the derived rate law is inconsistent with the experimentally observed exchange current densities as a function of Ce⁴⁺ concentration that we report here.

We hypothesize that the Ce³⁺/Ce⁴⁺ CT can be explained by considering ligand-exchange and electron-transfer steps in series. This type of mechanism is an example of a mechanism with chemical (C) and E steps (e.g., CE, EC, or CEC mechanisms).^42,43 The CE reaction mechanism is shown in Scheme 1a, where M is a metal ion undergoing an electron transfer and X^– is a complexing ligand. From Scheme 1a, the first step is a ligand exchange with free energy ΔG₁. We define ligand exchange as the replacement of a water molecule with a complexing ligand in the first solvation shell of M.⁴⁴ Then, an electron transfer occurs with free energy ΔG₂, where the species before and after the E are complexed by the same molecules. When the electrode potential U is equal to the standard redox potential of the overall reaction, U°, ΔG₁ is equal to the negative of ΔG₂.

Scheme 1. CE Reaction Mechanism.

(a) Steps for a CE mechanism and (b) free energy curves for the species involved in the mechanism. The metal ion, M, undergoes a CE mechanism between the oxidation states of M^z+ and M^(z–1)+, with M^z+ undergoing a ligand exchange (C step) before the E step. System free energy is depicted at the electrode potential U = U°. The redox potential of the overall reaction is U°, and the redox potential of the electron transfer step is U_E^°. The free energy of the E step is . This difference is related to the C step energy, ΔG₁, through the Nernst equation. The reorganization energy, λ₂, is defined as the energy required to change the reactant and solvent nuclear configuration to the configurations of the product. To inform the Ce³⁺/Ce⁴⁺ CT mechanism, we identify the (c) Ce³⁺ and Ce⁴⁺ structures using EXAFS, molecular dynamics EXAFS (MD-EXAFS), and DFT calculations and the (d) Ce³⁺/Ce⁴⁺ kinetics (exchange current density, i₀, and activation barrier, E_a) in sulfuric acid using Tafel plots and electrochemical impedance spectroscopy (EIS) to extract CT resistances (Ω) at different temperatures.

Multiple studies have developed models for the kinetics and current–potential responses of coupled electron-transfer chemical reactions,^{7,42,43,45−50} as well as tested the validity of these mechanisms by fitting them to the experimental data of organic compounds^42,46,51 and heterogeneous metal ion complexes.^7,49 MT has been used to understand the E step in coupled E and C systems such as organic compounds,^46,52 transition metal ions,^7,53 transition metal oxides,⁵⁰ and coupled redox-inactive metal ion–organic electron acceptor pairs.⁵⁴Scheme 1b shows how the free energy parabolas derived from standard MT can be used to describe the free energies of the species involved in a CE mechanism. The difference in values of the free energy curves at the reaction coordinate value of 1 gives information about the reorganization energy of the E step, λ₂. Importantly, several of these studies note that an ongoing challenge in identifying CT mechanisms is obtaining accurate structural information of reactants and products⁵³ as well as the intermediate species⁴⁶ undergoing CT.

Herein, we confirm the Ce³⁺ and Ce⁴⁺ structures and present a CT mechanism in sulfuric acid. We use the EXAFS and MD-EXAFS spectra of the Ce L₃-edge and K-edge to address unresolved uncertainties about the structure of Ce³⁺ and Ce⁴⁺ in sulfuric acid (Scheme 1c) and confirm the favorability of these structures through DFT modeling. Ce³⁺ coordinates with nine water molecules as [Ce^III(H₂O)₉]³⁺ and Ce⁴⁺ likely complexes with six water molecules and three bisulfates as [Ce^IV(H₂O)₆(HSO₄)₃]⁺. We show that the Ce³⁺/Ce⁴⁺ CT kinetics are similar as a function of Ce⁴⁺ concentration and temperature on platinum (Pt) and glassy carbon (GC). We use two independent methods to obtain the standard rate constants and CT coefficients for the Pt and GC rotating disk electrodes (RDEs), namely, the Tafel method and the CT resistance method (Scheme 1d). The similar rate constants, CT coefficients, and activation energies on two different electrode surfaces implies an outer-sphere CT mechanism. We measure a low cathodic CT coefficient of α = 0.23, in qualitative agreement with prior reports of an asymmetric CT reaction (Table S1). We explain this behavior through a mechanism where the anion-complexed Ce⁴⁺ species rapidly undergoes a ligand exchange to form [Ce^IV(H₂O)₉]⁴⁺, which is then followed by a rate-determining outer-sphere E between [Ce^IV(H₂O)₉]⁴⁺ and [Ce^III(H₂O)₉]³⁺. We derive a rate law based on this CE mechanism that results in a better fit to the experimentally observed kinetic data than a Butler–Volmer (BV) rate law. Using this CE rate law, we extract parameters such as reorganization energy and Gibbs free energy of the ligand exchange step. We find agreement between the experimentally fitted and DFT-predicted reorganization energies and ligand-exchange free energies and show that the fitted ligand-exchange free energy agrees with the shift in redox potential observed for the Ce³⁺/Ce⁴⁺ redox couple between HClO₄ (a non-complexing acid) and H₂SO₄.²³ These findings demonstrate the necessity of considering ligand-exchange energetics to rationalize CT kinetics and shed light on the Ce³⁺/Ce⁴⁺ CT reaction, which will aid various electrochemical applications.

2. Results and Discussion

2.1. Structures of Ce³⁺ and Ce⁴⁺ in Sulfuric Acid from EXAFS

We use Ce K-edge and L₃-edge EXAFS to resolve uncertainties in the Ce³⁺ and Ce⁴⁺ structures in sulfuric acid from our prior work solely at the L₃-edge.¹¹ The first uncertainty was the Ce³⁺–O coordination number (CN) of 9, which was previously obtained without using a known solid crystalline standard for calibration.¹¹ The second uncertainty was whether Ce³⁺ in sulfuric acid had Ce–sulfate or bisulfate complexation. The third uncertainty was in the accuracy of the fit of the EXAFS data of Ce⁴⁺ in sulfuric acid at the Ce L₃-edge, which did not prove the presence of sulfate or bisulfate in the first coordination shell of Ce⁴⁺.¹¹ Herein, we use CeCl₃·7H₂O and CeO₂ standards to confirm our previously reported Ce–O CNs and use the Ce K-edge and additional L₃-edge EXAFS data to show evidence that Ce³⁺ coordinates solely with water and Ce⁴⁺ complexes with three bisulfate anions. The Ce K-edge allows for quality data at higher k values than the L₃-edge, where interference from the L₂-edge occurs for lanthanides between 9 and 15 Å^–1. Thus, the K-edge gives more accurate structural information for the CNs and scattering distances between Ce⁴⁺ and its surrounding atoms.³³ Measuring at an additional edge allows for co-fitting both sets of EXAFS data, which will improve the fit statistics. We also collect EXAFS data at the Ce L₃-edge to elucidate the influence of Ce ion concentration and acid concentration on the cerium structure. Our MD-EXAFS predictions of different possible cerium complexes help to interpret the experimental spectra.

We show that the Ce³⁺–O CN in sulfuric acid is 9 using co-fits of EXAFS data at the Ce L₃- and K-edges. To obtain accurate CNs for Ce³⁺ solutions, we first record the EXAFS data of a Ce³⁺ standard, CeCl₃·7H₂O, at both Ce L₃- and K-edges. We determine an amplitude reduction factor, S₀², value of 1.5 for Ce³⁺ at the Ce K-edge by fitting the CeCl₃·7H₂O standard using ARTEMIS,⁵⁵ as shown in Figure 1a. The fit of the magnitude component in the R space and the fit in the k space of the CeCl₃·7H₂O standard and fitting parameters are included in the Supporting Information (Figure S1 and Table S2). The fit of the CeCl₃·7H₂O standard at the Ce L₃-edge is included in the Supporting Information (Figure S2 and Table S2), and the S₀² value for the Ce L₃-edge was 1.1. Using these S₀² values for the Ce K- and L₃-edges, we co-fit the EXAFS spectra of a 0.05 M Ce³⁺ + 2 M H₂SO₄ solution at the Ce K-edge, as can be seen in Figure 1b, and a 0.1 M Ce³⁺ + 2 M H₂SO₄ solution at the Ce L₃-edge, and we obtain a Ce³⁺–O CN of 8.7 ± 0.6. Different concentrations were necessary at the two Ce edges to optimize the signal while avoiding energy attenuation through the sample. The CN is consistent with the value of 9 that we previously reported for Ce³⁺ in acidic solutions including sulfuric acid.¹¹ From the fit, we also obtain a Ce³⁺–O scattering distance of 2.541 ± 0.004 Å, which agrees within 0.01% of our previously reported Ce³⁺–O distance in sulfuric acid.¹¹ The Ce K-edge fits in the R- and k-space (Figure S3), and the Ce L₃-edge fits (Figure S4 and Table S3) are shown in the Supporting Information. By using both K- and L₃-edges and a standard, the fit here gives additional confidence to our previous report on the Ce³⁺–O CN and distance in sulfuric acid.

Figure 1.

Ce K-edge k²·χ(R) EXAFS spectra and fits for different Ce³⁺ species. (a) CeCl₃·7H₂O standard (orange solid line) with fit (red dashed line) and Ce–O and Ce–Cl path contributions (shifted in the y-axis). (b) 0.025 M Ce₂(CO₃)₃ in 2 M H₂SO₄ (blue solid line) with fit using the Ce–O path (red dashed line). The inset structure shows [Ce^III(H₂O)₉]³⁺ from a MD snapshot. (c) 0.025 M Ce₂(CO₃)₃ in 2 M H₂SO₄ (blue solid line), 2 M MSA (black solid line), and 2 M TFSA (light green solid line). The insets are zoomed-in portions of the EXAFS spectra. (d) Simulated MD-EXAFS spectra of [Ce^III(H₂O)₉]³⁺ (blue solid line) and [Ce^III(H₂O)₈(SO₄)]⁺ (brown solid line) for comparison to the experimental data in (a–c). The insets are zoomed-in portions of the EXAFS spectra. During the preparation of Ce solutions with Ce₂(CO₃)₃, the carbonate reacts with H⁺ to form CO₂, which we then remove by sparging with inert nitrogen before collecting the measurements.

We also confirm that the dominant Ce³⁺ structure in H₂SO₄ is [Ce^III(H₂O)₉]³⁺ by comparing the EXAFS spectra of Ce³⁺ to MD-EXAFS spectra. The co-fit of the EXAFS spectra of Ce³⁺ in 2 M H₂SO₄ did not improve with the addition of a Ce³⁺–S scattering pathway (Figure S5). We compare our experimental Ce K-edge EXAFS data of Ce³⁺ in 2 M acids in Figure 1c. Ce³⁺ coordinates only with water in triflic acid (TFSA)³³ and methanesulfonic acid (MSA),¹¹ so the spectra of Ce³⁺ in TFSA and MSA can serve as Ce³⁺–water coordinated references. We observe no shift in the peak centered at ∼1.85 Å and a slight shift to the right in the peak at 2.1 Å in H₂SO₄ compared to MSA or TFSA. From the MD-EXAFS spectra of [Ce^III(H₂O)₉]³⁺ and [Ce^III(H₂O)₈(SO₄)]⁺ in Figure 1d, we can see a shift to shorter distances in both peaks when Ce³⁺ is complexed by sulfate because the average Ce³⁺–O scattering distance is shortened. Because we do not see this shift to the left in our experimental EXAFS spectra of Ce³⁺ in H₂SO₄, as shown qualitatively in Figure 1c, or quantitatively from the fits of Ce³⁺ in TFSA, MSA, and H₂SO₄ at the Ce K-edge (Figure S6 and Table S3), Ce³⁺ does not complex with sulfate or bisulfate. Additionally, the EXAFS spectra do not change with sulfuric acid concentration (Figure S7). Based on this evidence that Ce³⁺ does not complex with sulfate or bisulfate, and the evidence that the Ce³⁺–O CN is 9, we conclude that Ce³⁺ exists as [Ce^III(H₂O)₉]³⁺, which is consistent with our previous DFT-predictions.¹¹

Comparisons between the experimental Ce⁴⁺ EXAFS and MD-EXAFS data of possible Ce⁴⁺ species in H₂SO₄ imply that Ce⁴⁺ complexes with either sulfate (SO₄^2–) or bisulfate (HSO₄^–) anions. From Figure 2a, we observe a shift in the Ce–O scattering peak as we go from the EXAFS spectra of Ce³⁺ to Ce⁴⁺ in H₂SO₄ at the Ce K-edge (confirmed at the Ce L₃-edge, Figure S8). The shift in Ce–O distance is 0.16 Å based on the co-fits of Ce³⁺ and Ce⁴⁺ at both edges. We show the MD-EXAFS spectra of [Ce^III(H₂O)₉]³⁺ and two possible Ce⁴⁺ complexes [Ce^IV(H₂O)₈SO₄]²⁺ and [Ce^IV(H₂O)₉]⁴⁺ in Figure 2b. Those Ce⁴⁺ MD-EXAFS complexes were selected to illustrate the expected effect of sulfate complexation in the first coordination shell of Ce⁴⁺ relative to solely water coordination since the exact Ce⁴⁺ anion complex structure was not known. The Ce–O shift of 0.16 Å that we observe experimentally between Ce³⁺ and Ce⁴⁺ is expected because the Ce⁴⁺ ionic radius is 0.14–0.20 Å smaller than that of Ce³⁺.⁵⁶ The experimental value of 0.16 Å is closer to the 0.13 Å shift between MD-EXAFS [Ce^IV(H₂O)₈SO₄]²⁺ and [Ce^III(H₂O)₉]³⁺ compared to the 0.11 Å shift between [Ce^IV(H₂O)₉]⁴⁺ and [Ce^III(H₂O)₉]³⁺. The DFT-predicted shifts in Ce–O from [Ce^III(H₂O)₉]³⁺ to bisulfate complexes ([Ce^IV(H₂O)₈HSO₄]⁺ and [Ce^IV(H₂O)₆(HSO₄)₃]⁺) are 0.14 Å, also close to the experimental shift (Table S4). More importantly, scattering peaks at ∼2.6 Å appear for Ce⁴⁺ in H₂SO₄ that are not present for Ce³⁺, as shown in the inset in Figure 2a. From the inset in Figure 2b, similar peaks appear in the MD-EXAFS spectra of [Ce^IV(H₂O)₈(SO₄)]²⁺ that are not observed for [Ce^IV(H₂O)₉]⁴⁺ or [Ce^III(H₂O)₉]³⁺. We confirm that the peaks in this region of the MD-EXAFS spectra are due to sulfate scattering by comparing the MD-EXAFS-generated spectrum for [Ce^IV(H₂O)₈SO₄]²⁺ with and without sulfate scattering pathways included from Figure 2c. The S atom in either of the bisulfate complexes also is at the same distance as the S from sulfate. The peaks at ∼2.6 Å we see for Ce⁴⁺ in 2 M H₂SO₄ in the K-edge spectra are also present in the Ce⁴⁺ EXAFS at the Ce L₃-edge (Figure S8), and the peak heights increase with increasing H₂SO₄ concentration (Figure S9), further suggesting that these peaks are associated with an anion, which would be either SO₄^2– or HSO₄^–. To further probe the anion complexation of Ce⁴⁺ in H₂SO₄, including the type of anion and CN, we next fit the Ce⁴⁺ EXAFS data.

Figure 2.

Ce K- and L₃-edge k²·χ(R) EXAFS and fits for different Ce³⁺ and Ce⁴⁺ species. (a) 0.025 M Ce₂(CO₃)₃ (blue solid line) and 0.025 M Ce₂(CO₃)₃ oxidized to Ce⁴⁺ in 2 M H₂SO₄ (dark green solid line), with the inset showing the zoomed-in region of spectra. (b) Simulated MD-EXAFS spectra of [Ce^III(H₂O)₉]³⁺ (blue solid line), [Ce^IV(H₂O)₉]⁴⁺ (gray solid line), and [Ce^IV(H₂O)₈(SO₄)]²⁺ (dark green solid line), with the inset showing the zoomed-in region of spectra and green-colored ΔCe–O distance representing the shift in Ce–O distance from MD-EXAFS [Ce^III(H₂O)₉]³⁺ to [Ce^IV(H₂O)₈(SO₄)]²⁺ and gray-colored ΔCe–O distance representing the shift in Ce–O distance from MD-EXAFS [Ce^III(H₂O)₉]³⁺ to [Ce^IV(H₂O)₉]⁴⁺. (c) Simulated MD-EXAFS spectra of [Ce^IV(H₂O)₈(SO₄)]²⁺ (dark green solid line) and [Ce^IV(H₂O)₈(SO₄)]²⁺ with paths associated with sulfate scattering removed (light green solid line), with the inset showing the zoomed-in region of spectra. (d) 0.025 M Ce₂(CO₃)₃ oxidized to Ce⁴⁺ in 2 M H₂SO₄ (dark green solid line) with the best fit (red dashed line) and Ce–O and Ce–S path contributions (shifted in the y-axis). The inset is the proposed [Ce^IV(H₂O)₆(HSO₄)₃]⁺ structure. (e) 0.025 M Ce₂(CO₃)₃ oxidized to Ce⁴⁺ (orange solid line) and 0.05 M Ce₂(CO₃)₃ oxidized to Ce⁴⁺ (dark green solid line), both in 2 M H₂SO₄, with the inset showing the zoomed-in region of spectra. (f) Simulated MD-EXAFS spectra of [Ce^IV(H₂O)₈(SO₄)]²⁺ (dark green solid line) and [(H₂O)₈Ce^IV–Ce^IV(H₂O)₈]⁸⁺ (purple solid line), with the inset showing the zoomed-in region of spectra.

Ce⁴⁺ likely complexes with three bisulfates in the first coordination shell based on EXAFS fits and DFT-predicted complexation free energies. To fit our experimental Ce⁴⁺ K- and L₃-edge EXAFS, we measure a CeO₂ standard to determine a value for S₀² (Figure S10, with fitting results included in Table S5), which gives us confidence in the total CN of Ce⁴⁺. By co-fitting 0.05 M Ce⁴⁺ in 2 M H₂SO₄ EXAFS data at the Ce K-edge and 0.1 M Ce⁴⁺ in 2 M H₂SO₄ EXAFS data at the Ce L₃-edge with Ce–O and Ce–S scattering pathways, as shown in Figure 2d, we confirm evidence of sulfate or bisulfate in the first coordination shell because the Ce–S scattering pathway fits the peaks centered at 2.6 Å. The result of the co-fit at the Ce L₃-edge is shown in Figure S8. The co-fit with just a Ce–O scattering shell for both edges results in a worse fit (Figure S11). All parameters for Ce⁴⁺ EXAFS fitting are given in the Supporting Information (Table S6). From the co-fit with a Ce–S scattering pathway included, the Ce⁴⁺–O distance is 2.382 ± 0.006 Å and the Ce⁴⁺–S distance is 3.671 ± 0.016 Å. We identify a Ce⁴⁺–O CN of 8.6 ± 0.5 and a Ce⁴⁺–S CN of 3.0 ± 0.7.

It is not possible to distinguish from EXAFS whether the Ce–S path is due to a sulfate or a bisulfate anion in the first coordination shell because the Ce–S scattering distance is similar for Ce⁴⁺–sulfate and Ce⁴⁺–bisulfate complexes, so we turn to DFT modeling to determine whether sulfate or bisulfate complexation is more favorable. Previously, we compared the DFT-predicted ligand-exchange free energies for Ce⁴⁺ complexed with one or two sulfates or one or two bisulfates and found that of these four options, the [Ce^IV(H₂O)₈SO₄]²⁺ species was the most energetically favorable.¹¹ Here, we extend this analysis to compare the free energies of Ce⁴⁺ complexed with three sulfates and Ce⁴⁺ complexed with three bisulfates. We find that the free energy of the [Ce^IV(H₂O)₆(HSO₄)₃]⁺ species is the most energetically favorable Ce⁴⁺–bisulfate-complexed species considered, and its energy is comparable to that of the [Ce^IV(H₂O)₈SO₄]²⁺ species (Figure S12). Also, considering the relative acid dissociation constants of HSO₄^– and SO₄^2–,⁵⁷ HSO₄^– will be present at 99 times greater concentration than SO₄^2–. Although studies have proposed a Ce⁴⁺ structure with three sulfates,³⁶ this structure is unfavorable based on our DFT calculations. The DFT-predicted ligand-exchange free energy for [Ce^IV(H₂O)₆(HSO₄)₃]⁺ is also in good agreement with the experimentally observed shift in redox potential for the Ce³⁺/Ce⁴⁺ redox couple from 1 M HClO₄ (a non-complexing acid⁵⁸) to 1 M H₂SO₄ of −28.9 kJ/mol. Because this structure is the most energetically favorable based on DFT and matches our experimental CNs of nine oxygens and three sulfurs for each Ce atom, we conclude that the dominant Ce⁴⁺ species is [Ce^IV(H₂O)₆(HSO₄)₃]⁺.

We do not detect cerium dimers for 0.05–1.0 M Ce⁴⁺ in 2 M H₂SO₄, unlike what has been proposed for Ce⁴⁺ in HClO₄⁵⁹⁻⁶² and HNO₃.^39,40 In Figure 2e, we see that increasing Ce⁴⁺ concentration from 0.05 to 0.1 M does not result in an increase in any features around 4.0 Å, the distance at which the Ce–Ce scattering pathway from dimers is expected to appear based on the MD-EXAFS spectra of a dimer species, as shown in the inset in Figure 2f. The lack of dimers in H₂SO₄ may be due to the stronger anion complexation of Ce⁴⁺ single ions in H₂SO₄ compared to in HNO₃ and HClO₄.

Our findings from the Ce K-edge and additional Ce L₃-edge EXAFS in sulfuric acid support that Ce³⁺ coordinates with nine water molecules, while Ce⁴⁺ complexes with at least one sulfate or bisulfate.¹¹ Our EXAFS measurements indicate that a structural change occurs within the first coordination shell of Ce between Ce³⁺ and Ce⁴⁺. Thus, a CT mechanism for the cerium redox reaction must include an inner-sphere ligand exchange. In the next section, we present kinetic results that show that despite this inner-sphere structural change, the Ce³⁺/Ce⁴⁺ redox reaction behaves as if outer sphere.

2.2. Kinetic Measurements of the Ce³⁺/Ce⁴⁺ CT and Modeling with BV

We measure the Ce³⁺/Ce⁴⁺ CT coefficient, standard rate constant, and activation energy on two electrode surfaces, Pt and GC, under identical reaction conditions and show the results are consistent with an outer-sphere electron-transfer RDS. Although Ce³⁺/Ce⁴⁺ redox kinetics have been measured on multiple electrodes (Table S1), the experimental conditions varied, making it difficult to draw conclusions about the relative rates on different electrodes. We emphasize the importance of controlled mass transport conditions and consideration of multiple possible reactions when extracting kinetic parameters¹² as well as the utility of comparing kinetic parameters obtained through different measurement techniques. Here, we use two independent methods (Tafel method and the CT resistance method) to ensure comparable and accurate steady-state kinetic measurements. We study the Ce⁴⁺ reduction rates to avoid convolution with oxygen evolution or electrode oxidation during Ce³⁺ oxidation and control the mass transport to allow extraction of kinetically limited rates. See the Experimental Methods section for more details.

To obtain the Ce³⁺/Ce⁴⁺ kinetic parameters and a rate law on a Pt electrode in sulfuric acid, we use the Tafel method to measure exchange current densities, i₀, as a function of Ce⁴⁺ concentration and temperature, as shown in Figure 3a,b, respectively. The data in Figure 3c shows the cathodic Tafel slope as a function of Ce⁴⁺ concentration. The Ce³⁺/Ce⁴⁺ exchange current densities for Pt extracted from the CT resistance method (Figure S13) agree within 31% of the Tafel method. The exchange current densities increase with increasing Ce⁴⁺ concentration until [Ce⁴⁺] = 0.04 M and then decrease. The exchange current densities increase exponentially with temperature.

Figure 3.

Tafel analysis of Ce³⁺/Ce⁴⁺ kinetic measurements on Pt and GC electrode surfaces. Kinetic measurements and fit (solid line) using the BV equation of the Ce³⁺/Ce⁴⁺ redox couple. Exchange current densities, i₀, extracted from Tafel plots as a function of Ce⁴⁺ concentration for (a) Pt RDE and (d) GC RDE. i₀ extracted from Tafel plots as a function of temperature for (b) Pt RDE and (e) GC RDE. Cathodic Tafel slopes as a function of Ce⁴⁺ concentration for (c) Pt RDE and (f) GC RDE. Ce⁴⁺ concentrations were determined by titration. Reported values are averaged from three runs, with error bars representing 1 standard deviation from the average value in both horizontal and vertical directions. Data in (a,c,d,f) collected at room temperature in 2 M H₂SO₄ at a total cerium concentration of 0.05 M. Data in (b,e) collected at a total cerium concentration of 0.05 M with a Ce⁴⁺ concentration of 0.026 and 0.025 M, respectively. An Ag/AgCl reference electrode and a graphite rod counter electrode were used for all measurements. Measurements were done at 2000 rpm of the RDE. The rate constant k₀, the cathodic CT coefficient α, and the activation energy E_a were obtained through minimizing the NMSE of the data.

Differences in the i₀ values from the Tafel and CT resistance methods are comparable to our previous studies for the V²⁺/V³⁺ system (∼40%)⁶³ as well as others for Ce³⁺/Ce⁴⁺ on Pt in 4.5 M MSA (35% difference in i₀ between Tafel and CT methods).⁶⁴ We attribute these differences to uncertainty in fitting of the EIS data, which depends on the selection of an appropriate equivalent circuit and the use of the correct OCV value without influence from competitive side reactions. Because the Tafel method is used more frequently in the literature and is less prone to error, we use the Tafel exchange current density values to study the Ce³⁺/Ce⁴⁺ CT mechanism. We note that despite the 30% difference in i₀, the activation barriers extracted from each method are within 10% of each other on Pt.

The exchange current densities for Pt reveal that the Ce³⁺/Ce⁴⁺ redox reaction is asymmetric, with a low α that is unusual for metal ion CTs and a standard rate constant and activation energy that agree with the reported values (Table S1). We attempt to fit our kinetic data through the BV formulism to describe the kinetic current density, i_K,BV (eq 1a), which assumes a one-step E and relates exchange current density, i_0,BV (eq 1b), to the Ce³⁺ and Ce⁴⁺ concentrations, where k₀ is the standard rate constant and α is the cathodic CT coefficient.

1a

1b

where U is the electrode potential, U° is the standard equilibrium potential (1.44 V vs SHE), and U_eq is the equilibrium potential corresponding to the [Ce³⁺] and [Ce⁴⁺] conditions at which the measurement was taken. The normalized mean-square error (NMSE) fit of eq 1a to the exchange current densities using k₀ and α as fitting parameters is shown in Figure 3a. The k₀ value obtained from the fit is 1.43 × 10^–4 ± 7 × 10^–6 cm/s at 298 K, which aligns with many of the standard rate constants reported in the literature for Ce³⁺/Ce⁴⁺ in sulfuric acid (Table S1). The activation energy E_a on Pt from fitting the data in Figure 3b is 48.3 ± 21 kJ/mol, which, while having a large degree of uncertainty, agrees within 35% of an activation energy calculated from cerium redox standard rate constants on Pt in sulfuric acid.⁶⁵ The value of α is 0.23 ± 0.005, agreeing with reports for Pt (Table S1) and indicating the asymmetric nature of the cerium redox reaction. The CT behavior of cerium with a Pt electrode is the same between 0.01 and 0.05 M cerium (Figures S14–S16), with the magnitude of exchange current densities dependent on the total concentration of cerium.

The large cathodic Tafel slopes on Pt corroborate the low value of α and therefore the asymmetry of the cerium redox reaction, but the BV equation does not capture the Tafel slopes’ dependence on Ce⁴⁺ concentration. The cathodic Tafel slopes shown in Figure 3c are between 220 and 320 mV/decade as a function of [Ce⁴⁺], which correspond to α values between 0.18 and 0.27. These α values are similar to those obtained from fitting eqs 1a and 1b to the data in Figure 3a, giving further evidence that Ce³⁺/Ce⁴⁺ is highly asymmetric. However, unlike the BV equation where α is constant, here we see a trend in cathodic Tafel slope with concentration. We also measure cathodic Tafel slopes as a function of temperature (Figure S17) and find that the Tafel slope increases with temperature. The BV fit does not entirely capture the dependence of the Tafel slope on temperature. The inability of the BV equation to capture the cathodic Tafel slopes’ dependence on Ce ion concentration suggests that a more accurate rate law than that described through BV kinetics is necessary to describe the Ce³⁺/Ce⁴⁺ redox kinetics on Pt.

We repeat the same kinetic analysis as Pt for a GC electrode and find that the redox reaction is asymmetric on GC as well. Fitting eqs 1a and 1b to the Tafel exchange current densities as a function of [Ce⁴⁺] shown in Figure 3d, we determine a k₀ value of 2.99 × 10^–5 ± 1.30×10^–6 cm/s and an α value of 0.23 ± 0.008. The exchange current densities derived from CT resistances are given in the Supporting Information (Figure S18). The α value of GC aligns well with the reported values, but the standard rate constant is lower than most of the previously reported values in sulfuric acid (Table S1). Based on our Tafel exchange current density versus temperature data shown in Figure 3e, the activation energy on GC is 41.2 ± 14.0 kJ/mol, which is 7.1 kJ/mol smaller than that of Pt. To our knowledge, this is the first time the activation energy of Ce³⁺/Ce⁴⁺ has been compared between two electrodes under the same conditions. The magnitude of the cathodic Tafel slopes as a function of Ce⁴⁺ concentration ranges from 170 to 250 mV/decade as shown in Figure 3f, which corresponds to α values between 0.24 and 0.35. Although the fitted value of α qualitatively agrees with the values predicted from the cathodic Tafel slopes, the BV fit is unable to capture the decrease in Tafel slope with [Ce⁴⁺]. Also, the BV fit only partially captures the dependence of the cathodic Tafel slopes with temperature (Figure S19). To further probe the CT mechanism of Ce³⁺/Ce⁴⁺, it is beneficial to compare the kinetic behavior on the Pt and GC electrodes.

The similar exchange current densities, CT coefficients, activation energies, and cathodic Tafel slopes for the Pt and GC RDEs in Figure 3 imply that the cerium redox kinetics behave as an outer-sphere reaction. The exchange current densities and rate constants of Pt and GC are within a factor of 5 of one another, comparable to the differences for various outer-sphere redox couples on different electrode surfaces (Table S7). Even for well-known outer-sphere redox reactions such as [Fe(CN)₆]^3–/[Fe(CN)₆]^4– in 1.0 M KCl,⁶⁶ the electrode materials affect the standard rate constants of various outer-sphere redox couples by a factor of 2–9.⁶⁶⁻⁶⁸ These electrode effects are ascribed to a variety of factors, including the Frumkin effect,⁶⁹ a metal’s electronic properties, for example, electronic spillover distance, work function, intrinsic electric field,⁷⁰ and the interaction of adsorbed water on the electrode surface with electroactive species.⁷¹⁻⁷³ The activation barriers on the Pt and GC RDEs in this work are similar, suggesting a small electrode influence on the kinetics, further underscoring the outer-sphere behavior of the Ce³⁺/Ce⁴⁺ electron transfer. Additionally, the Pt and GC RDE cathodic Tafel slopes fall within the same range of 150–320 mV/decade and the α values are both 0.23, highlighting not only a similar kinetic behavior but also the extreme asymmetry in the CT on both electrodes.

From our structural and kinetic results, the Ce³⁺/Ce⁴⁺ reaction is unlikely to be purely a one-step electron-transfer reaction. Although obeying an outer-sphere kinetic behavior, the value of α falls far below the expected value of 0.5, suggesting that a rate law based on a one-step electron transfer is inappropriate to model the Ce³⁺/Ce⁴⁺ redox kinetics. Additionally, the BV rate law predicts constant cathodic Tafel slopes with [Ce⁴⁺], which we do not observe on either Pt or GC. Our structural data from Figures 1 and 2 also suggests that solely a one E step mechanism cannot adequately describe the cerium redox kinetics because of the structural change that must occur from [Ce^III(H₂O)₉]³⁺ and [Ce^IV(H₂O)₆(HSO₄)₃]⁺.

2.3. Proposed Mechanism for the Ce³⁺/Ce⁴⁺ Redox Reaction in Sulfuric Acid

We propose a mechanism that aligns with both our structural and kinetic data and discuss the implications of the resulting fit of our experimental data to a rate law derived from the mechanism. The appropriate CT mechanism and rate law for the Ce³⁺/Ce⁴⁺ redox reaction must satisfy the following nine criteria observed from our structural and kinetic data:

• 1.A structural change occurs in addition to an electron transfer.
• 2.Ce³⁺ preferentially coordinates with water as [Ce^III(H₂O)₉]³⁺.
• 3.Ce⁴⁺ favorably complexes with three bisulfates as [Ce^IV(H₂O)₆(HSO₄)₃]⁺.
• 4.There is minimal influence of the electrode on kinetic activity.
• 5.There is a maximum in the exchange current density as a function of Ce⁴⁺ concentration (with the total cerium concentration fixed) occurring between 60 and 80% [Ce⁴⁺] relative to the total cerium concentration.
• 6.The exchange current density increases with increasing temperature.
• 7.The cathodic Tafel slope is large, with values between 170 and 320 mV/decade.
• 8.The cathodic Tafel slope decreases with an increasing ratio of Ce⁴⁺ concentration to the total cerium concentration.
• 9.The cathodic Tafel slope increases with increasing temperature.

In Table 1, we summarize six different possible mechanisms and whether each mechanism and its corresponding rate law matches the nine criteria. We begin with the simplest mechanism and expand complexity only when needed to describe the data.⁷⁴ We show that a CE mechanism where electron transfer is the RDS is the simplest mechanism that satisfies criteria 1–9. The rate law uses MT to describe the electron-transfer step. Derivations for all rate laws considered are in the Supporting Information (see Tables S8 and S9).

Table 1. Different Possible Mechanisms and Agreement or Disagreement of the Mechanism and the Corresponding Rate Law to the Structural and Kinetic Data Observed Experimentally for the Ce³⁺/Ce⁴⁺ CT^a.

		criteria: observedCe3+/Ce4+ structural and kinetic behavior
mechanism and rate law		1	2	3	4	5	6	7	8	9
one-step E	BV	yes	yes	yes	yes	yes	yes	yes	no	yes
	MT	no	—	—	yes	no	yes	yes	yes	yes
two-step	CE Mech, E RDS	yes	yes	yes	yes	yes	yes	yes	yes	yes
	EC Mech, E RDS	yes	no	yes	yes	yes	yes	yes	yes	yes
	EC Mech, C RDS	yes	yes	yes	yes	no	no	no	no	no
	CE Mech, C RDS	yes	yes	yes	yes	no	no	no	no	no

^aMechanisms are separated into one and two step. E RDS refers to the electron-transfer step being the RDS, whereas C RDS refers to ligand exchange being the RDS. The numbers in the criteria row correspond with the list of criteria at the beginning of this section. The dashes included for the MT rate law indicate that these criteria are not considered since they are dependent on criterion 1 being met.

The BV and the MT rate laws for a one-step electron-transfer mechanism do not meet all the criteria established from our experimentally observed structural and kinetic data (Table 1). The BV rate law does not adequately describe the behavior of the cathodic Tafel slope as a function of [Ce⁴⁺] concentration (criterion 8) because the Tafel slope from BV is constant with potential. Although the BV rate law describes the Ce³⁺/Ce⁴⁺ CT asymmetry through the low value of α, the BV rate law is only an empirical model, and as a result, the fitting parameters are unphysical. The MT rate law better captures the behavior of the cathodic Tafel slopes; however, it does not allow for a structural change to occur between the redox species (criterion 1), and it does not satisfy criterion 5 (Figure S20). For a symmetric MT rate law, the exchange current density maximum occurs at 50% Ce⁴⁺. As has been previously shown,¹³ although asymmetric MT can mathematically describe the observed asymmetry in the kinetics, the reorganization energies from fitting the data are unreasonably large (see the Supporting Information for further discussion of an asymmetric MT rate law). Additionally, the NMSE of the MT fit is higher for both Pt and GC data than the BV fit (Table S9). Since the rate laws for a one-step E mechanism do not satisfy all the observed structural and kinetic data, it is necessary to consider mechanisms that include more than a one-step electron transfer.

We next consider different two-step mechanisms, which we refer to as CE and EC mechanisms. In the CE mechanism, the Ce⁴⁺ species, [Ce^IV(H₂O)₆(HSO₄)₃]⁺, undergoes ligand exchanges with water to form [Ce^IV(H₂O)₉]⁴⁺ (eq 2a). [Ce^IV(H₂O)₉]⁴⁺ is subsequently reduced to form [Ce^III(H₂O)₉]³⁺ (eq 2b). Note that the C step would not in reality occur in a single elementary step, as it involves the exchange of three bisulfates with three water molecules. If one of the C steps were rate limiting, then we would not be able to aggregate the bisulfate exchange into one step as we have in eq 2a. However, we will show below that the bisulfate-exchange C steps are likely quasi-equilibrated; thus, we can condense the ligand exchange into a single step.

2a

2b

In the EC mechanism, the E step occurs first between the Ce⁴⁺ and Ce³⁺ complexes, followed by ligand exchange of Ce³⁺ (see the Supporting Information for details on the EC mechanism). For both the CE and EC mechanisms, we consider two possible rate laws, based on assuming the RDS is either the E or C. We show the rate law for the kinetic current density (i_K,CE,E) for the CE mechanism when E (eq 2b) is the RDS in eq 3a. represents the concentration of the water-coordinated Ce⁴⁺ (eq 3b), and A, B, and C are defined in eqs 3c–3e.

3a

3b

3c

3d

3e

Here, K₁ is the equilibrium constant of the C step, n is the number of electrons transferred, assumed to be 1, F is Faraday’s constant, Z₂ is the preexponential factor for the E step, refers to the concentration of the bulk Ce⁴⁺ species, that is, [Ce^IV(H₂O)₆(HSO₄)₃]⁺, λ₂ is the reorganization energy of the E step, R and T have their usual meanings, η is the overpotential applied (U – U_eq), and is the concentration of the bulk Ce³⁺ species, that is, [Ce^III(H₂O)₉]³⁺. Figure 4a shows the free energies of the species involved in the CE mechanism, as well as the physical meaning of some of the fitting parameters used in the rate law (λ₂ and K₁). As shown by the free energy parabolas, standard MT expressions are used to describe the E step.

Figure 4.

Proposed CE mechanism and fit of rate law to kinetic data on Pt assuming that the electron-transfer step (E) is rate determining. (a) Free energy profiles at the equilibrium Ce³⁺/Ce⁴⁺ potential (U = U°) in H₂SO₄ for the three species involved in the proposed CE mechanism for Ce³⁺/Ce⁴⁺ CT: [Ce^IV(H₂O)₆(HSO₄)₃]⁺ (green solid line), [Ce^IV(H₂O)₉]⁴⁺ (gray solid line), and [Ce^III(H₂O)₉]³⁺ (light blue solid line). Parabolas represent the free energies of the species as a function of reaction coordinate, as defined through MT. The Ce³⁺/Ce⁴⁺ electron transfer is described by the reorganization energy λ₂ and the transition-state free energy, ΔG₂^⧧, and the ligand exchange is described by the equilibrium constant K₁. We propose that electron transfer between [Ce^IV(H₂O)₉]⁴⁺ and [Ce^III(H₂O)₉]³⁺ is the RDS; that is, [Ce^IV(H₂O)₆(HSO₄)₃]⁺ and [Ce^IV(H₂O)₉]⁴⁺ are quasi-equilibrated. (b,c) Fit (dark blue solid line) of the Ce³⁺/Ce⁴⁺ exchange current densities (blue circles) in the 0.05 M Ce/2 M H₂SO₄ solution on the Pt RDE using the rate law in eqs 3a–3e derived for the CE mechanism assuming that E is RDS as a function of (b) Ce⁴⁺ concentration and (c) temperature. (d) Fit (blue solid line) of the Ce³⁺/Ce⁴⁺ cathodic Tafel slopes (blue circles) on the Pt RDE using the rate law in eqs 3a–3e derived for the CE mechanism assuming that the E is RDS as a function of Ce⁴⁺ concentration. All fits of the data in (b–d) were obtained through minimizing the NMSE of the data.

Of the rate laws derived from the two-step mechanisms, the only one that meets all criteria is a CE-type mechanism with the E as the RDS (CE, E RDS), as summarized in Table 1. Because the C step involves ligand exchange, and the only Ce³⁺ species included in the CE mechanism is the [Ce^III(H₂O)₉]³⁺ species, criteria 1 and 2 are satisfied. The rate law does not incorporate any electrode-dependent properties; thus, criterion 4 is also met. By fitting the exchange current densities and cathodic Tafel slopes for both Pt and GC RDEs to expressions for i₀ and Tafel slopes derived from eq 3a, we see that the rate law satisfies all the observed kinetic criteria. The rate law equation for i_K,CE,E indicates that i₀ will be asymmetric with Ce⁴⁺ concentration and increase with temperature (criteria 5 and 6), as shown in Figure 4b,c for Pt. From Figure 4d, the rate law modeled Pt cathodic Tafel slopes agree with the observed Tafel slopes (criteria 7 and 8), and the Tafel slopes also increase with increasing temperature (criterion 9, Figure S21). The parameters used in the fit of the Pt experimental data are shown in Figure 4b–d. The NMSE of the CE, E RDS rate law fit to Pt was the lowest for all rate laws at 0.049. The E step reorganization energy is reasonable at 100.2 kJ/mol, and the equilibrium constant, K₁, is small at 1.1 × 10^–7, indicating that the ligand exchange from [Ce^IV(H₂O)₆(HSO₄)₃]⁺ to [Ce^IV(H₂O)₉]⁴⁺ is unfavorable, as expected from our EXAFS structural data. Additionally, the observed shift in redox potential for the Ce³⁺/Ce⁴⁺ redox couple from 1 M HClO₄ to 1 M H₂SO₄ is 0.30 V,²³ which corresponds to an equilibrium constant of 8.4 × 10^–6, assuming that the shift is due only to Ce⁴⁺-anion complexation in sulfuric acid. Thus, criterion 3 is also met. The pre-exponential factor Z₂ is equal to the product of the electronic transmission coefficient, precursor equilibrium constant, and nuclear frequency factor (see the Supporting Information),¹² and values for the apparent preexponential factor for metal ions at metal–aqueous interfaces are reported to be on the order of 10³ to 10⁵ cm/s.⁷⁵ Our fitted Z₂ value of 4.22 × 10⁴ cm/s for Pt is within this range. The fit of the CE, E RDS rate law to the GC data is included in the Supporting Information (Figure S22), with the lowest NMSE of all rate laws fit to the GC data. The fitted Z₂ and λ₂ values for GC are smaller than those for Pt, and the value of K₁ for GC is 2 orders of magnitude larger than the Pt equilibrium constant (Table S9). This K₁ value does still correctly predict that Ce⁴⁺ is complexed by an anion in sulfuric acid and the CE, E RDS rate law is still the best fitting rate law to the GC data.

We were unable to experimentally confirm the existence of the [Ce^IV(H₂O)₉]⁴⁺ species. Experimental determination of intermediates in catalysis and electrochemistry through spectroscopy is notoriously difficult because of their small populations and/or short lifetimes.⁴⁶ We assert that the fact that the CE mechanism is the simplest mechanism that is consistent with our experimental kinetics is evidence of the plausible existence of [Ce^IV(H₂O)₉]⁴⁺, in accordance with the common use of Occam’s razor in mechanistic studies.⁷⁶ Additionally, in our previous work,¹¹ we demonstrated that the shift in redox potential observed for Ce³⁺/Ce⁴⁺ from HClO₄ to H₂SO₄ agrees with our calculated change in Gibbs free energy, assuming the [Ce^IV(H₂O)₉]⁴⁺ species in HClO₄ and an anion-complexed species in H₂SO₄. We note, however, that previous studies suggest a Ce⁴⁺-hydrolyzed dimer forms in HClO₄.^59,61,62 Additional spectroscopic studies of the proposed intermediate and DFT-based calculations of other possible structures such as monomeric and dimeric hydroxides would help test our hypothesized mechanism.

The rate laws derived from the CE and EC mechanisms in which the C step is the RDS and the rate law for the EC mechanism with the E step as RDS do not satisfy the observed structural and kinetic behavior (Table 1) and thus are eliminated as possible mechanisms for the Ce³⁺/Ce⁴⁺ CT. Details of the fits of both the Pt and GC data for these rate laws are included in Figures S23–S25. The rate laws with the rate-determining C step result in an incorrect linear dependence of i₀ on [Ce⁴⁺]. If a series of C steps were assumed to occur instead of a single C step to go from [Ce^IV(H₂O)₆(HSO₄)₃]⁺ to [Ce^IV(H₂O)₉]⁴⁺, and any of these steps were the RDS, then the rate law would still display a linear dependence of i₀ on [Ce⁴⁺]. Since our observed i₀ does not have a linear dependence with [Ce⁴⁺], we conclude that none of the C steps are rate determining and can therefore be condensed into a single quasi-equilibrated reaction as shown in eq 2a. In recent work that studied the Ce³⁺/Ce⁴⁺ kinetics in sulfuric acid on a gold electrode,¹³ it was concluded that the E step was preceded by a rate-determining ligand dissociation step. From their proposed rate law, exchange current density is linear with Ce⁴⁺ concentration, as we see for our C RDS rate laws. Therefore, we can rule out the rate-determining ligand dissociation step proposed in this prior work based on its failure to meet criterion 5. The C RDS rate laws also do not capture the values of the cathodic Tafel slopes, the decrease in Tafel slopes with an increase in Ce⁴⁺ concentration, or the increase in Tafel slopes with temperature (criteria 7–9). Considering the EC mechanism with the E step assumed to be the RDS (EC, E RDS), the fits to the experimental Pt and GC data (Figure S25) indicate that Ce³⁺ will favorably complex with an anion, contradicting our experimental EXAFS and thus failing criterion 2. Also, the fitted value of the reorganization energy is unreasonably high at 663 kJ/mol (Table S9). With all this evidence taken together, we reject that these three rate laws and conclude that only the CE, E RDS rate law is consistent with the experimental data. On the principle of Occam’s razor,⁷⁷ which asserts that the simplest explanation that satisfies all observed data is preferable, we do not consider more complex mechanisms with additional C or E steps.

The parameters K₁ and λ₂ extracted from the fit to the experimental data of the CE, E RDS rate law compare favorably to DFT-predicted values, allowing us to use DFT to analyze the contributions to the total reorganization energy and possibly make predictions for different systems. Our DFT-predicted K₁ for the ligand exchange from [Ce^IV(H₂O)₆(HSO₄)₃]⁺ to [Ce^IV(H₂O)₉]⁴⁺ is 9.8 × 10^–9 as compared to the fitted value of 1.1 × 10^–7 for Pt. These equilibrium constants agree qualitatively with the equilibrium constant derived from the Ce³⁺/Ce⁴⁺ redox potential in H₂SO₄ relative to HClO₄. We also calculate the reorganization energy using DFT for the water-coordinated Ce³⁺/Ce⁴⁺ CT to be 87.2 kJ/mol (Table S9). This reorganization energy value falls between the experimentally fitted λ₂ values for GC (71.8 kJ/mol) and Pt (100.2 kJ/mol). From DFT, the inner-sphere reorganization energy from the change in the Ce water-coordinated species (e.g., metal–ligand bond length), λ_i, is 40.4 kJ/mol, and the outer-sphere reorganization energy related to the reorganization of the solvent,^12,78 λ_o, is 46.8 kJ/mol (Table S10). See the Supporting Information (Figure S26) for further discussion on the reorganization energy calculations. Our ability to computationally determine the ligand exchange and reorganization energies for the Ce³⁺/Ce⁴⁺ CT has implications for future efforts to enhance Ce kinetics. By coupling calculated reorganization energy information with known ligand-exchange equilibrium constants, we could use the CE, E RDS rate law to screen acids to determine which would yield optimal kinetic activities. Assuming that the CE, E RDS rate law holds in other acids, we would expect acids with stronger complexing anions (smaller K₁) to exhibit slower Ce³⁺/Ce⁴⁺ kinetics than acids with weaker complexing anions. If the reorganization energies were calculated for Ce-anion complexed species, we could also calculate the expected rates in acids if the dominant mechanism switches to be EC (see the Supporting Information).

Although the CE, E RDS model fits the Pt experimental data and DFT values, we observe discrepancies between the Pt and GC fitted parameters, which could be because we ignore the effect of temperature on Z₂ in all considered rate laws. The λ₂ and Z₂ values for GC are smaller than those of Pt (Table S9), whereas the K₁ value for GC is larger. Although we expect a difference in Z₂ between Pt and GC due to factors such as the Frumkin effect, the metal’s electronic properties, and the effect of water adsorption on the electrode, it is unexpected that there would be different λ₂ and K₁ values. In our model, we captured the temperature dependence of the kinetic activity through the λ₂ and K₁ parameters and assumed the pre-exponential value Z₂ to be independent of temperature. However, it has been noted in the case of weak electronic coupling that the nuclear frequency factor has a T^1/2 dependence on temperature,^2,79 and Z₂ is proportional to the nuclear frequency factor. Thus, it is possible that one reason for the difference in λ₂ and K₁ between Pt and GC is because we ignore the temperature dependence of Z₂. Better treatment of the preexponential factor’s dependence on the temperature through more sophisticated treatments⁷⁵ as well as kinetic measurements on additional electrodes would be needed in future mechanistic studies of the cerium redox couple.

3. Conclusions

We demonstrate how determining the cerium ions’ structures and characterizing the kinetic behavior as a function of experimental conditions can be used to identify a CT mechanism that is consistent with both experimentally observed and computationally predicted behavior. We show that a rate law where a chemical step (exchange of bisulfate ligands with water) is followed by a rate-determining electron-transfer step, described using MT, successfully captures the extreme asymmetry of the Ce³⁺/Ce⁴⁺ kinetics. Additionally, this mechanism is consistent with the differences in complexation observed for the Ce³⁺ and Ce⁴⁺ oxidation states. The agreement between the rate law-modeled behavior and observed kinetic behavior demonstrates the importance of understanding ion structures, considering possible mechanisms, and the utility of MT in mechanistic studies. Given the agreement between experimental and computational work, we propose that structural information from either DFT calculations or experiment can be combined with our proposed rate law to predict the activity of the Ce³⁺/Ce⁴⁺ redox couple in other electrolytes. The use of MT coupled with a CE- or EC-type mechanism, which has been considered previously for other systems, and informed by detailed structural data could also be applied to understand other redox couples that have seemingly anomalous empirical kinetic parameters such as V⁴⁺/V⁵⁺.

4. Experimental and Computational Methods

4.1. Experimental Methods

4.1.1. Solution Preparation

The Ce³⁺ solution preparation methods and chemical sources are described in detail elsewhere.¹¹ To prepare solutions of mixed oxidation state, that is, some Ce³⁺ and some Ce⁴⁺ present, or entirely Ce⁴⁺, cerium(III) carbonate hydrate (same source as used for Ce³⁺ solutions) was added to 2 M H₂SO₄ and stirred until all cerium had dissolved and a clear solution had formed. While it has been noted that cerium complexes with carbonate in solution,⁸⁰ those studies were conducted in basic solutions, whereas our studies were conducted in a strongly acidic solution and with a step to remove carbonate before spectroscopic or kinetic studies. As discussed in our prior work,¹¹ when using cerium(III) carbonate in an acid solution, the carbonate will react with the available protons to form CO₂ which is vented out of the solution with sufficient N₂ sparging. The redox potentials of cerium carbonate complexes range from 0.198 to 0.2595 V versus SHE,⁸⁰ whereas the redox potentials we measured during kinetic measurements ranged between 1.39 and 1.49 V versus SHE, which is consistent with cerium complexation in H₂SO₄. We previously demonstrated that the Ce³⁺ UV–vis peaks did not change significantly whether Ce³⁺ were prepared from Ce₂(CO₃)₃ or Ce(CF₃SO₃)₃, suggesting that Ce³⁺ is predominantly coordinated by the same species in each of these, that is, water.¹¹ Here, we also show that the EXAFS of a 0.05 M Ce³⁺ solution in 2 M TFSA does not change whether Ce₂(CO₃)₃ or Ce(CF₃SO₃)₃ is used to prepare the solution (Figure S27). Then, using a two-compartment glass electrochemical cell discussed in more detail elsewhere,¹¹ Ce³⁺ ions were electrochemically oxidized to Ce⁴⁺ using a titanium-based anode from De Nora (coating type DN-240 or DN-300) as the working electrode, a graphite rod (Alfa Aesar, 99.9995% metals basis) as the counter electrode, and a double-junction Ag/AgCl electrode (saturated KCl, Pine Research) as the reference electrode until the desired ratio of Ce⁴⁺ to Ce³⁺ was achieved. The concentrations of Ce³⁺ and Ce⁴⁺ were measured using titration, as described in detail elsewhere.¹¹

4.1.2. Electrochemistry Cell Setup

For room-temperature kinetic measurements on either the Pt or GC working electrodes, the same electrochemical cell was used that was used for solution preparation, with the same reference and counter electrodes. For activation barrier measurements, a jacketed two-compartment electrochemical cell (Adams & Chittenden Scientific Glass) was used. A refrigerated/heated bath circulator (Fisher Scientific) controlled the temperature of the water. The temperature of the solution was measured before kinetic measurements. A VSP potentiostat (Biologic Science Instruments USA) was used to supply voltage and measure kinetics. Before kinetic measurements, the working electrode compartment solution was sparged for at least 15 min with nitrogen (Metro Welding Supply Corp, pure compressed nitrogen) to minimize oxygen contamination, and the solution was continuously blanketed with nitrogen. A Modulated Speed Rotator (model AFMSRCE, Pine Research) was used to control the rotation rate of the RDEs, which were inserted into the E5TQ ChangeDisk Tip (Pine Research). We compared Ce³⁺/Ce⁴⁺ kinetic measurements in 2 M H₂SO₄ with 0.05 M total cerium concentration prepared from Ce₂(CO₃)₃ and Ce(SO₄)₂ and found no significant difference in activity (see Figure S28), suggesting that Ce³⁺ and Ce⁴⁺ are not complexing with carbonate species in our measurements. To eliminate the effect of cerium crossover, no kinetic measurements were taken beyond 48 h from when the solution had been added to the cell (Figure S29).

4.1.3. Experimental XAFS Data Collection, Normalization, and Fitting

XAFS measurements consisting of both X-ray near-edge spectroscopy (XANES) and EXAFS were collected at the Advanced Photon Source at Argonne National Laboratory (L₃-edge collected at 20 BM and K-edge collected at 20 ID-B,C). The data analyzed in this paper included EXAFS spectra of a CeO₂ standard and 0.1 M Ce⁴⁺ in 2 M H₂SO₄ at the Ce L₃-edge from our previous work¹¹ (newly normalized) and additional EXAFS spectra of cerium species at the Ce L₃- and K-edges. In the Supporting Information, the additionally collected EXAFS spectra at the Ce L₃-edge of 0.1 M Ce⁴⁺ in 2 M H₂SO₄ are shown in Figures S8b,f and S9 and labeled “Ce⁴⁺ in varied [H₂SO₄].”

The Ce³⁺ and Ce⁴⁺ solutions used in the additional XAFS measurements were shipped to the beamline in either glass vials or capillary tubes. To ensure that the solutions of Ce⁴⁺ maintained their oxidation state, they were shipped in dry ice and stored in a freezer with temperatures less than 0 °C. The frozen solutions were thawed and syringed into solution holders immediately before EXAFS measurements began. The XANES of the Ce⁴⁺ solutions were compared to CeO₂ XANES (Figure S30) to confirm 100% Ce⁴⁺ oxidation state. The CeCl₃·7H₂O and CeO₂ standards were prepared by mixing CeCl₃ (Strem Chemicals, Inc., 99.9% pure) and CeO₂ (Alfa Aesar, 99.99%), respectively, with boron nitride (BN, Sigma Aldrich) in air, grinding using a mortar and pestle, and then forming a pellet using a pellet press with a pressure of 10,000 psi. The ratios of either CeCl₃ or CeO₂ to BN for each edge are reported in Table S11.

For the Ce L₃-edge XAFS measurements, the same monochromator was used as previously described,¹¹ as were the processes for harmonic contamination suppression and incident beam intensity measurement. For all Ce L₃-edge XAFS spectra except the 0.05 M Ce⁴⁺/2 M H₂SO₄ solution XAFS spectra in Figure 2f, data collection was performed in the transmission mode. A chromium (Cr) foil was used as a reference to verify energy reproducibility because its K-edge energy (5989 eV) is close to the Ce L₃-edge energy (5723 eV). The Ce L₃-edge XAFS of the 0.05 M Ce⁴⁺/2 M H₂SO₄ solution in Figure 2f was collected in the fluorescence mode due to low transmission signal, using a 13-element germanium fluorescence detector. The Ce K-edge XAFS studies were carried out at 295 K. The incident beam was monochromatized using a pair of Si(311) crystals. Higher order harmonics were suppressed by detuning the monochromator to reduce the incident X-ray intensity by approximately 15%. Argon-filled ion chambers were used for the I₀, I_t, and I_ref detectors. All Ce K-edge XAFS spectra were collected in the transmission mode, and the XANES spectra of a CeO₂ standard was used to confirm energy reproducibility several times throughout the data collection. At least two scans were collected for each sample. For the Pt L₃-edge XAFS data discussed in the Supporting Information, the XAFS measurements were collected in a polyacrylate electrochemical cell designed for beamline measurements described in more detail elsewhere.⁸¹

The XAFS data were normalized using the software ATHENA,⁵⁵ and all normalization parameters used are listed in Table S12. The repeating scans for each sample were merged after energy alignment and normalization. The Fourier transformed EXAFS data used an R_bkg parameter of 1.4 Å for all K-edge samples and 1.2 Å for all L₃-edge samples. All plotted EXAFS data have unadjusted R space values. No attempt was made to correct for multi-electron excitation (MEE) effects in the L₃-edge data based on our previous findings that correcting for MEE did not result in improved fitting results.¹¹

The normalized and k²-weighted EXAFS spectra were analyzed using the software ARTEMIS,⁵⁵ with R and k ranges specified in Table S13. FEFF9 was used to generate paths from the known structures of CeCl₃·7H₂O (sourced from the Cambridge Structural Database^82,83) and CeO₂ (source included in our previous work¹¹) to fit the standards for Ce³⁺ and Ce⁴⁺. For the CeCl₃·7H₂O standard, Ce–O and Ce–Cl were used in the fit. The amplitude reduction factor (S₀²) was first fixed at a value of 1.0, while the value of shift in threshold energy (ΔE₀) and path specific values (the shift in scattering distance (ΔR), the Debye–Waller factor (σ²), and the CN) were obtained by fitting. The CN were fixed to sum to 9, and these fitted results were then set as fixed parameters to obtain S₀². Fits of the CeCl₃·7H₂O data for the K- and L₃-edges are shown in Figures S1 and S2, respectively, and fitted parameter values are included in Table S2. For CeO₂, two Ce–O shells (Ce–O₁ and Ce–O₂) and one Ce–Ce shell were included in the fit. The S₀², ΔE₀, and path-specific ΔR and σ² were obtained by fitting, but the CN of Ce–O₁ was set to 8, the CN of Ce–Ce was set to 12, and the CN of Ce–O₂ was set to 24. Fits of the CeO₂ data are shown in Figure S10, and the fitted parameter values are included in Table S5.

To fit the k² weighted EXAFS data of cerium solutions, FEFF9 was used to generate paths with self-consistency in JFEFF from DFT-predicted cerium structures. The final fit of Ce³⁺ in H₂SO₄ was a Ce–O scattering shell co-fit to the Ce L₃-edge EXAFS spectra of 0.1 M Ce³⁺/2 M H₂SO₄ and Ce K-edge EXAFS spectra of 0.05 M Ce³⁺/2 M H₂SO₄. A Ce–S scattering path was also considered in the Ce³⁺ fit in Figure S5. The values of S₀² and ΔE₀ were set as fixed global parameters and were obtained from the fits of the CeCl₃·7H₂O standard. The CN, ΔR, and σ² were all obtained by co-fitting. Ce K-edge data of 0.05 M Ce³⁺ in 2 M MSA, 2 M TFSA, and 2 M H₂SO₄ were fit with a Ce–O scattering shell with the same R range and k range as that used for the K-edge data in the co-fit (Table S13). The final fit of Ce⁴⁺ in H₂SO₄ was a co-fit using the Ce L₃-edge EXAFS spectra of 0.1 M Ce⁴⁺/2 M H₂SO₄ and the Ce K-edge EXAFS spectra of 0.05 M Ce⁴⁺/2 M H₂SO₄, with Ce–O and Ce–S shells. The co-fit with only a Ce–O scattering shell is shown in Figure S11. Four path-specific parameters (CN, ΔR, σ², and the third cumulant, σ³) were obtained by co-fitting, and values for S₀² and ΔE₀, which were set as fixed global parameters, were obtained from the fit of the CeO₂ standard. See Table S13 for R and k ranges.

4.1.4. Electrode Pretreatment

To prepare the RDEs for kinetic measurements, a polishing, sonication, and electrochemical cleaning procedure was followed for each electrode material. For GC, the RDE was polished for 3 min using a 0.3 μm alumina slurry (Allied High Tech Products, Inc., DeAgglomerated) and then sonicated (Fisher Scientific, 2.8 L Ultrasonic Bath) in water purified with the MilliporeSigma Synergy UltrapureWater Purification System (18.2 MΩ cm resistivity) for 45 min. EIS was used to measure the solution resistance, and then a series of cyclic voltammograms (CVs) were conducted in the range 0.36 V versus RHE to 1.16 V versus RHE, that is, the non-faradaic region, as a function of scan rate in 2 M H₂SO₄. If no impurities were detected from the CVs, the double-layer capacitance could be calculated and the GC was ready for kinetic measurements of the cerium redox couple. For Pt, which was more sensitive to oxidation and thus required a more rigorous cleaning procedure, the RDE was first polished for 3 min using a 0.3 μm alumina slurry and then sonicated in Millipore water for 45 min. Then, the RDE was dried and polished for 3 min using a 0.05 μm alumina slurry (Allied High Tech Products, Inc., DeAgglomerated) before being sonicated again in Millipore water for 45 min. The Pt RDE was subjected to 50 electrochemical cleaning cycles in 2 M H₂SO₄ (CV with a potential range of −0.35 V vs RHE to 1.56 V vs RHE and a scan rate of 100 mV/s) to remove any contaminants on the Pt surface. Following the cleaning cycles, the Pt RDE hydrogen underpotential deposition (HUPD) peaks were characterized, and the electrochemically active surface area (ECSA) was calculated by dividing the amount of charge associated with the desorption of a monolayer of adsorbed hydrogen on the Pt surface by the specific charge of Pt (210 μC cm^–2). If the HUPD peaks demonstrated any oxygen contamination or contamination on the Pt surface, for example, extremely low or misshapen hydrogen desorption peaks, then the cleaning cycles were repeated with additional N₂ sparging and blanketing at higher flow rates until HUPD peaks were stable. The Pt kinetic data was only used if the ECSA calculated from the HUPD peaks was greater than or equal to the geometric surface area of the Pt RDE (0.196 cm²) and had a corresponding roughness factor that was less than 2.5.

4.1.5. Kinetic Measurements

To measure the kinetic activity of the Pt and GC electrodes, the exchange current density of the reaction, i₀, the Tafel slope, b, and the activation energy, E_a, were extracted. Two independent methods of obtaining the exchange current density were used: the Tafel method and the CT resistance method. In the Tafel method, a series of fixed potentials were applied to the working electrode, and the resulting steady-state currents were measured to achieve a polarization curve. The applied potentials were compensated for solution resistance using EIS measurements. The steady-state currents were then normalized by surface area to obtain steady-state current densities. For Pt measurements, the ECSA from HUPD measurements were used to normalize the currents, whereas the GC kinetic activity was normalized by its geometric surface area of 0.196 cm². From the steady-state current densities, a Tafel plot was constructed from the Tafel equation (eq 4, where a is a constant, b is the Tafel slope, and η is the overpotential). The Tafel slope was extracted from the Tafel plot by fitting a linear trendline in the overpotential range of −0.250 to −0.118 V¹² and extrapolating to η = 0 to obtain i₀.

4

In the CT resistance method, the CT resistance, R_ct, was extracted from a Nyquist plot using EIS. The solution resistance, R_s, and CT resistance, R_ct, were found by fitting an electrical circuit, assumed here to be the Randles circuit, and finding the intercepts of the x-axis of the semicircle, which is the real portion of the impedance. R_s is the left-most x-intercept (i.e., high frequencies), and R_ct is the x-intercept at the right-most side of the semicircle (i.e., low frequencies). Then, eq 5 was used to calculate i₀, where R is the ideal gas constant, T is the temperature, A is the surface area (ECSA from HUPD for Pt, geometric for GC), and n refers to the number of electrons that are transferred during CT.

5

Obtaining similar exchange current density values from the Tafel method and the CT method lends confidence in the kinetic values reported, and so the exchange current densities were measured using both methods for all values reported in this study. All exchange current densities obtained from the CT resistance method for the Pt and GC electrodes can be found in Figures S13 and S18.

A competitive redox reaction that will occur at similar potentials as the Ce³⁺/Ce⁴⁺ electron transfer is the oxygen evolution reaction (OER).⁸⁴ To avoid measuring the OER activity, only the reduction currents of Ce⁴⁺ to Ce³⁺, which occur at potentials at which OER is negligible, were used to determine i₀ from the Tafel method. As noted, oxygen reduction contributions are mitigated by purging the electrolyte with nitrogen.

To ensure that the data reported herein were kinetically controlled, all activity measurements were collected using RDEs. The observed activity for the GC and Pt RDEs was no longer dependent on rotation rate at 2000 rpm (Figure S31), so all reported kinetic activity was collected at a rotation speed of 2000 rpm. Additionally, a Koutecký–Levich analysis was conducted for the Pt RDE (Figure S32) at a 0.02 M Ce⁴⁺ concentration (total Ce concentration of 0.05 M), and it was found that using the kinetically limited current from the Koutecký–Levich analysis resulted in an exchange current density that was less than 1% different from the exchange current density extracted from the Tafel method. Last, the kinetic activities for Pt were found to be independent of the electrochemical reactor design by confirming that the exchange current densities achieved for a Pt RDE were similar to those achieved for a Pt wire with rapid stirring (Figure S33).

Several precautions were taken to ensure that the kinetic data were accurate and reproducible and the concentrations of Ce³⁺ and Ce⁴⁺ were accurate. As mentioned, only electrode surfaces with no indication of impurities from the CV were used for analysis. To ensure that the kinetic measurements were reproducible, the reported i₀ and Tafel slope values were determined from the average of three measurement runs at a specific experimental condition (either Ce⁴⁺ concentration or temperature), with uncertainty represented by the standard deviation away from the average. The accuracy of the concentration of Ce⁴⁺ relative to the total amount of cerium was determined from three methods for each data point: (1) titration, (2) using the open-circuit voltage (OCV) and obtaining concentrations from the Nernst equation, and (3) the UV–vis spectrum of the solution in the working electrode compartment of the electrochemical cell compared to other UV–vis spectra at similar Ce⁴⁺ concentrations. Only data points that had Ce⁴⁺ concentration results that were less than 10% different between titration and OCV, and with the UV–vis spectrum in qualitative agreement with the expected peak intensity from Ce⁴⁺, were used for analysis. The UV–vis spectra of Ce³⁺ and Ce⁴⁺ are discussed in more detail in our previous work.¹¹

4.1.6. Kinetic Data Fitting

To fit the kinetic rate laws to the experimentally collected kinetic data, the NMSE was calculated for each data set (i₀[Ce⁴⁺], i₀[T], b[Ce⁴⁺], b[T]) using eq 6, where y_obs,i is the experimentally observed data point i, y_mod,i is the modeled data point i, , is the mean of the experimentally observed data points, is the mean of the modeled data points, and N is the number of i data points.

6

The modeled data points were calculated using the rate laws discussed in the Supporting Information by using the Tafel method to extract exchange current densities and cathodic Tafel slopes from modeled kinetic currents as a function of overvoltage. The NMSE for each set of data was then weighted equally and summed together, and a solver function was used to minimize the total summed error by varying relevant fitting parameters. For the BV rate law, the fitting parameters were standard rate constant, k₀, CT coefficient, α, and activation energy, E_a. For the two-step EC and CE mechanisms, the fitting parameters were the reorganization energy, a preexponential factor, and the equilibrium constant of the C step. The uncertainty in the fitted parameters was the standard error (eq 7) of the value, estimated through the jackknife method,⁸⁵ in which one observed data point is removed and the fitted value (T_j) is determined for that subset j. The process is repeated for all M subsets, and the average of the fitted value for all subsets (T_j,avg) is used in eq 7 to calculate the standard error.

7

4.2. Computational Methods

4.2.1. MD-EXAFS Spectra Calculations

The MD-EXAFS spectra of [Ce^III(H₂O)₉]³⁺, [Ce^IV(H₂O)₉]⁴⁺, [Ce^III(H₂O)₈SO₄]⁺, and [Ce^IV(H₂O)₈SO₄]²⁺ complexes were generated by averaging the EXAFS signals of geometry snapshots of the complex in solution over a molecular dynamics trajectory. Car–Parrinello molecular dynamics (CPMD) were performed in the NWChem software⁸⁶ to generate structures of the complexes in solution (Figure S34), which were then used as inputs to the FEFF9 code⁸⁷ to calculate EXAFS signals due to scattering paths. The CPMD simulations were performed in an 8 × 8 × 8 ų periodic box using the PBE functional in the canonical ensemble (Figure S35). Simulations of each cerium complex were run for 50 ps post-equilibration, and one snapshot was taken each ps to generate 50 geometry snapshots. The cerium complexes were explicitly solvated with 15 water molecules (to give a solution density of ≈1.0 g/cm³). All hydrogen atoms were given a fictitious mass of 2 amu to decrease the frequency of O–H bond vibrations, allowing a larger time step of 5.0 au to be used for computational tractability. The Nosé–Hoover thermostat⁸⁸ was used to maintain the temperature of the system at 300 K throughout the simulation, and the periods of the ionic and electronic thermostats were set to 1200 a.u. All non-cerium atoms were treated with the Hamann pseudopotential,⁸⁹ and the cerium ion was treated with the Troullier–Martins pseuodopotential.⁹⁰

Each sampled geometry snapshot was fed into the FEFF9 code to calculate an EXAFS spectrum. Each spectrum was calculated using a cluster radius of 6 Å centered on the cerium ion. The 50 geometries were averaged to produce a final EXAFS spectrum of the complex in solution. To generate spectra without sulfate scattering pathways, those paths were omitted during FEFF9 runs.

4.2.2. Reorganization Energy and Anion Complexation Free Energy Calculations

Reorganization energies were calculated for the [Ce^III(H₂O)₉]³⁺/[Ce^IV(H₂O)₉]⁴⁺ CT (Table S10). All calculations were performed in the NWChem software unless otherwise noted. DFT-optimized structures of the cerium ion and its first coordination shell (nine water molecules) were used at each step in the process. For all geometry optimizations and calculations, a mixed basis set was used with the Stuttgart RSC 1997 ECP basis set⁹¹ for the cerium ion and the 6-31+G* basis set⁹² for all non-cerium atoms. This basis set was used to maintain computational tractability and accuracy. The calculated reorganization energy was shown to converge with this basis set (see converged outer-sphere reorganization energies, Figure S36). Implicit solvation was included using conductor-like screening model (COSMO) with default parameters.⁹³ The B3LYP functional with Grimme’s D3 dispersion correction⁹⁴ was employed for all electron-transfer calculations.⁹⁴⁻⁹⁶

The reorganization energy λ includes the inner- (λ_i) and outer-sphere (λ_o) contributions (eq 8), which were calculated separately. λ_i was approximated by Nelsen’s four-point method (eq 9),⁹⁷ in which single-point energy calculations of the oxidized and reduced species geometries (with COSMO) at both the oxidized and reduced charge states were used to estimate the energetics of inner-sphere reorganization.

8

9

where R^ox and R^red are the optimized geometries of the oxidized and reduced species, respectively, and E_ox and E_red are the energies of the oxidized and reduced states, respectively, evaluated at the given geometry. Energies of each species were evaluated at each state using the computational settings described above. See the Supporting Information for further discussion of reorganization energy calculations (Table S10).

The outer-sphere reorganization energy, λ_o, was obtained within the PCM framework⁹⁸ in the GAMESS⁹⁹ software. The PCM framework was employed to calculate solvation free energy at different polarization potentials (eq 10).

10

where G_ox^eq and G_ox are the free energy of the oxidized geometry evaluated at the oxidized and reduced polarization potentials, respectively. G_red^eq and G_red are the free energy of the reduced geometry evaluated at the reduced and oxidized polarization potentials, respectively.

In this framework, the electrode surface was modeled as a perfect conductor, and water was implicitly treated as the solvent, meaning that the dielectric constant assumed in the λ_o calculation was the dielectric constant of water. Cavity surfaces that contain the cerium complex were defined using the Gauss–Bonet tessellation procedure, and cavity sizes for each atom were generated using their van der Waal radius using the default GAMESS radii, except cerium, which was estimated as twice the covalent radius, 4.08 Å.¹⁰⁰ The outer-sphere reorganization energies were calculated in the Born–Oppenheimer limit using inertial polarization;¹⁰¹ however, the self-consistent limit was shown to yield results less than 1 kJ/mol different for the systems studied. The solvated radius of the water molecules used to define the distance of the cavity from the electrode surface was obtained from the experiment¹⁰² to be 3.00 Å. Last, the anion complexation free energies for the [Ce^IV(H₂O)₆(SO₄)₃]^2– and [Ce^IV(H₂O)₆(HSO₄)₃]⁺ species (Figure S12) were calculated in the same manner as our previous paper.¹¹

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.2c00484.

• Previous cerium kinetic literature, Ce³⁺ and Ce⁴⁺ EXAFS data, Ce⁴⁺ DFT predictions, Ce³⁺/Ce⁴⁺ kinetic data, rate law derivations, DFT-predicted reorganization energies, and experimental and computational procedures (PDF)

Author Present Address

^∥ Oak Ridge National Laboratory, Electrification and Energy Infrastructures Division, 1 Bethel Valley Rd, Oak Ridge, TN, 37830, United States

Author Contributions

N.S. conceived the project. C.A.B. carried out all kinetic experimental measurements and characterization. M.B. suggested and performed the K-edge EXAFS measurements, and C.A.B. completed the data reduction and fitting with input from M.B. D.H. performed all the first-principles modeling calculations under the guidance of B.R.G. All authors were involved in the analysis and writing of the manuscript. CRediT: Cailin Anne Buchanan data curation, formal analysis, investigation, methodology, validation, visualization, writing-original draft; Dylan Herrera data curation, formal analysis, investigation, visualization, writing-review & editing; Mahalingam Balasubramanian data curation, methodology, writing-review & editing; Bryan R. Goldsmith funding acquisition, methodology, project administration, resources, software, supervision, validation, writing-review & editing; Nirala Singh conceptualization, funding acquisition, methodology, project administration, resources, supervision, validation, visualization, writing-review & editing.

This work was supported by University of Michigan (UofM) start-up funds of B.R.G. and N.S. This work was in part supported by the Dow Sustainability Fellows Program at the UofM. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Sector 20 operations are supported by the US Department of Energy and the Canadian Light Source.

The authors declare no competing financial interest.