Trends in Ground-State Entropies for Transition Metal Based Hydrogen Atom Transfer Reactions

Abstract

Reported herein are thermochemical studies of hydrogen atom transfer (HAT) reactions involving transition metal H-atom donors M^IILH and oxyl radicals. [Fe^II(H₂bip)₃]²⁺, [Fe^II(H₂bim)₃]²⁺, [Co^II(H₂bim)₃]²⁺ and Ru^II(acac)₂(py-imH) [H₂bip = 2,2’-bi-1,4,5,6-tetrahydropyrimidine, H₂bim = 2,2’-bi-imidazoline, acac = 2,4-pentandionato, py-imH = 2-(2’-pyridyl)-imidazole)] each react with TEMPO (2,2,6,6-tetramethyl-1-piperidinoxyl) or ^tBu₃PhO^• (2,4,6-tri-tert-butylphenoxyl) to give the deprotonated, oxidized metal complex M^IIIL, and TEMPOH or ^tBu₃PhOH. Solution equilibrium measurements for the reaction of [Co^II(H₂bim)₃]²⁺ with TEMPO show a large, negative ground-state entropy for hydrogen atom transfer, −41 ± 2 cal mol⁻¹ K⁻¹. This is even more negative than the ΔS^o_HAT = −30 ± 2 cal mol⁻¹ K⁻¹ for the two iron complexes and the ΔS^o_HAT for Ru^II(acac)₂(py-imH) + TEMPO, 4.9 ± 1.1 cal mol⁻¹ K⁻¹, as reported earlier. Calorimetric measurements quantitatively confirm the enthalpy of reaction for [Fe^II(H₂bip)₃]²⁺ + TEMPO, thus also confirming ΔS^o_HAT. Calorimetry on TEMPOH + ^tBu₃PhO^• gives ΔH^o_HAT = −11.2 ± 0.5 kcal mol⁻¹ which matches the enthalpy predicted from the difference in literature solution BDEs. A brief evaluation of the literature thermochemistry of TEMPOH and ^tBu₃PhOH supports the common assumption that ΔS^o_HAT ≈ 0 for HAT reactions of organic and small gas-phase molecules. However, this assumption does not hold for transition metal based HAT reactions. The trend in magnitude of |ΔS^o_HAT| for reactions with TEMPO, Ru^II(acac)₂(py-imH) << [Fe^II(H₂bip)₃]²⁺ = [Fe^II(H₂bim)₃]²⁺ < [Co^II(H₂bim)₃]²⁺, is surprisingly well predicted by the trends for electron transfer half-reaction entropies, ΔS^o_ET, in aprotic solvents. This is because both ΔS^o_ET and ΔS^o_HAT have substantial contributions from vibrational entropy, which varies significantly with the metal center involved. The close connection between ΔS^o_HAT and ΔS^o_ET provides an important link between these two fields and provides a starting point from which to predict which HAT systems will have important ground-state entropy effects.

Introduction

The transfer of a hydrogen atom, reaction 1, is one of the most fundamental chemical transformations. It is a cornerstone of organic free-radical chemistry, from combustion to the in

$$\mathrm{A}-\mathrm{H}+{\mathrm{B}}^{\ast }\to {\mathrm{A}}^{\ast }+\mathrm{H}-\mathrm{B}$$ (1)

vitro and in vivo action of antioxidants.¹ In recent years, it has become clear that this reaction is also involved in a variety of metal-mediated oxidations, including coordination complexes, metalloenzyme active sites, and metal-oxide surfaces.^2-5 For example, both plants and animals employ lipoxygenases to catalyze the selective hydroperoxidation of 1,4 diene units in fatty acids by using hydrogen transfer to an iron(III) hydroxide species.⁶ Reaction 1 has also been implicated in catalysis by other metalloenzymes such as cytochrome P450,⁷ methane monooxygenases⁸ and class I ribonucleotide reductases.^3a,9 Cobaloximes, cobalt-porphyrins, and chromium cyclopentadienyl compounds effect chain transfer in living radical polymerizations using reaction 1.¹⁰ Developing a fundamental understanding of hydrogen transfer is thus broadly important.

Reaction 1, which we will call hydrogen atom transfer (HAT),¹¹ is part of a broad class of processes involving proton and electron transfer, often called proton-coupled electron transfer (PCET).^12-14 Theoretical treatments of PCET, like their antecedent theories of electron transfer (ET)¹⁵ and proton transfer (PT),¹⁶ use free energies (ΔG) as measures of reaction driving force.⁵ In contrast, analyses of HAT reactions have typically used the enthalpic driving force, as in the Bell-Evans-Polanyi equation (BEP) that relates the activation barrier to the ΔH.^1a,17 ΔH^o for an HAT reaction is the difference in bond dissociation enthalpies (BDEs) between the reactant A–H and the product B–H. In our view, the BEP correlation is a primary historical reason why chemists have focused on BDEs, rather than bond dissociation free energies (BDFEs).

The focus on enthalpies to understand HAT reactions is surprising since reactivity typically is correlated with free energies, as in linear free energy relationships (LFERs).¹⁸ These treatments are equivalent when the entropies of reaction are close to zero (when ΔS^o = 0, ΔG^o = ΔH^o), as has been assumed in most treatments of hydrogen atom transfer (with a few exceptions¹⁹). The assumption that ΔS^o ≈ 0 is also part of the foundation for the increasingly common determination of BDEs from solution pK_a and E_1/2 values, as popularized by Bordwell and co-workers.^20-25

The assumption that ΔS^o is ≈ 0 appears to hold for HAT reactions of small molecules in the gas phase²⁶ and of larger organics in solution,²⁷ but HAT reactions of two iron complexes have recently been shown to have very large |ΔS^o_HAT|.²⁸ For instance, H-atom transfer from [Fe^II(H₂bip)₃]²⁺ to TEMPO (H₂bip = 2,2’-bi-1,4,5,6-tetrahydropyrimidine) has ΔS^o₄ = −30 ± 2 cal mol⁻¹ K⁻¹.²⁸ In this case, TΔS^o₄ = −8.9 kcal mol⁻¹ at 298 K, a change in K_eq of 4 × 10⁶. This large |ΔS^o_HAT| originates primarily from a change in vibrational entropy upon redox change at the iron center,²⁸ which suggests that HAT reactions of other metal systems may also have large values of |ΔS^o|.

Herein, we report calorimetric and equilibrium measurements of ground state enthalpies and entropies for a series of HAT reactions.²⁹ These reactions involve iron, cobalt, and ruthenium complexes with unsaturated nitrogen ligands, of the general type shown in eq 2. These thermodynamic measurements are used to elucidate trends in the magnitude of ground-state entropies for hydrogen atom transfer reactions, ΔS^o_HAT.

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Results

I. Equilibrium studies

A. Co^II(H₂bim) + TEMPO

[Co^II(H₂bim)₃]²⁺ [Co^II(H₂bim); H₂bim = 2,2’-bi-2-imidazoline; 10 mM] reacts with the stable nitroxyl radical TEMPO (3−15 equiv) in CD₃CN to give an equilibrium mixture with [Co^III(Hbim)(H₂bim)₂]²⁺ [Co^III(Hbim)] and TEMPOH (eq 3; N-N = H₂bim). Equilibrium is reached within approximately 48 hours at 298 K under these conditions. In the reverse direction,

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Co^III(Hbim) plus excess TEMPOH gives complete formation of Co^II(H₂bim) and TEMPO over the course of 4 hours. This reaction has been very briefly described in a previous report.³⁰ The equilibrium constant K₃ has been determined by integrating ¹H NMR spectra of reaction mixtures. All four species have easily observable ¹H NMR spectra over the temperature range studied, even though Co^II(H₂bim) and TEMPO are both paramagnetic. The average for 3 experiments gives K₃ = (5.9 ± 0.8) × 10⁻³ at 298 K, ΔG^o₃ = 3.0 ± 0.4 kcal mol⁻¹ (Table 1). This value for ΔG^o₃ is within the error of the previous estimate (+0.3 ± 3 kcal mol⁻¹) derived from the relevant pK_a and E^o values.³⁰ The large error in the previous estimate is due to the poor electrochemical response of Co^III(H₂bim).³¹

Table 1.

Equilibrium constants and ground state thermodynamics for hydrogen atom transfer reactions in MeCN.

AH + B	Methoda	Keq (298 K)	ΔGoHAT (kcal mol−1)	ΔHoHAT (kcal mol−1)	ΔSoHAT (cal mol−1 K−1)
FeII(H2bip) + TEMPO	VHb	1.7 ± 0.3	−0.3 ± 0.2	−9.4 ± 0.6	−30 ± 2
	Cal	--	--	−8.9 ± 0.6	--
	BDFE	∼2	−0.5 ± 1	--	--
FeII(H2bim) + TEMPO	VHb	(2.0 ± 0.3) × 10−4	5.0 ± 0.2	−4.1 ± 0.3	−30 ± 2
	BDFE	∼0.9 × 10−4	5.5 ± 1.0	--	--
RuII(py-imH) + TEMPO	VHc	(1.8 ± 0.2) × 103	−4.4 ± 0.1	−3.0 ± 0.3	4.9 ± 1.1
	BDFE	(2.0 ± 1.5) × 103	−4.5 ± 0.4	--	--
CoII(H2bim) + TEMPO	VH	(5.9 ± 0.8) × 10−3	3.0 ± 0.4	−9.3 ± 0.4	−41 ± 2
	BDFE	∼0.6	0.3 ± 3d	--	--
RuII(py-imH) + tBu3PhO•	BDE	--	--	−15 ± 1	--
RuII(hfac)2(py-imH) + tBu3PhO•	VH	0.062 ± 0.013	1.6 ± 0.1	--	--
CoII(H2bim) + tBu3PhO•	BDE	--	--	-24 ± 4	--
TEMPOH + tBu3PhO•	Cal	--	--	−11.2 ± 0.5	--
	BD(F)E	--	−10 ± 1	−11.5 ± 1.4	−2 ± 3

^aMethod: VH (van't Hoff) = Temperature dependence of K_eq from van't Hoff plots; Cal = calorimetric; BDFE/BDE: ΔG^o_HAT = BDFE[AH] - BDFE[BH] and/or ΔH^o_HAT = BDE[AH] - BDE[BH], using values from Table 2.

^bData from reference ²⁸.

^cData from reference ³².

^dEstimated from BDFEs derived from pK_a and E^o values, see Table 2.

K₃ was measured from 274 − 313 K and found to vary by an order of magnitude over this 40 °C range (Figure 1A). Van't Hoff analysis yields ΔH^o₃ = −9.3 ± 0.4 kcal mol⁻¹ and ΔS^o₃ = −41 ± 2 cal mol⁻¹ K⁻¹ (Table 1). For each sample, after measurements were complete at the high and low temperatures, the NMR tubes were allowed to re-equilibrate at room temperature (294 K). In each case, the ratio of species re-adjusted to values consistent with the predicted K₃ at 294 K (Figure 1A), indicating that this is a true equilibrium. Over a week at these concentrations, there is no observable decrease in the cobalt mass balance relative to the NMR integration standard, though there is slight decomposition (ca. 5%) of the excess TEMPOH.

Figure 1.

Van't Hoff plots (A) for Co^II(H₂bim) + TEMPO (eq 3) (●), with (■) indicating reactions that were initially run at high and low temperatures and then re-equilibrated back to 294 K; and (B) for the much less temperature dependent Ru^II(py-imH) + TEMPO ⇄ Ru^III(py-im) + TEMPOH (eq 6). Figure B adapted, with permission, from reference ^32b.

B. [Fe^II(H₂bip)₃]²⁺, [Fe^II(H₂bim)₃]²⁺, and Ru^II(acac)₂(py-imH) + TEMPO

Equilibrium constants have been reported for the reactions of TEMPO with [Fe^II(H₂bip)₃]²⁺ [Fe^II(H₂bip)], [Fe^II(H₂bim)₃]²⁺ [Fe^II(H₂bim)], and Ru^II(acac)₂(py-imH) [Ru^II(py-imH))] (eqs 4 - 6; H₂bip = 2,2’-bi-1,4,5,6-tetrahydropyrimidine; acac = 2,4-pentanedionato, py-imH = 2-(2’-pyridyl)imidazole).^28,32 In the iron systems, K₄ and K₅ in MeCN were determined both by static methods (as above) and from the ratio of the opposing second-order rate constants; the ruthenium K₆ was measured by UV-vis spectroscopic titration. The temperature dependence of these equilibrium constants yield the ΔH^o and ΔS^o values given in Table 1. K₆ is much less temperature dependent than either the iron or cobalt systems discussed above, varying by barely a factor of 1.5 over the 41°C temperature range examined 269 − 310 K (Figure 1B). For the ruthenium hexafluoro-acac derivative [Ru^II(hfac)₂(py-imH)], HAT to the stable and isolable³³ free radical ^tBu₃PhO^• (eq 7): K₇ = 0.062 ± 0.013 at 298 K (hfac = 1,1,1,5,5,5-hexafluoro-2,4-pentanedionato, CF₃C(O)CHC(O)CF₃).³²

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$$\left[{\mathrm{Ru}}^{\mathrm{II}}{\left(\mathrm{hfac}\right)}_2\left(\mathrm{py}\text{-}\mathrm{imH}\right)\right]+{}^t\mathrm{Bu}_3\mathrm{PhO}\stackrel{K_7}{\rightleftarrows }\left[{\mathrm{Ru}}^{\mathrm{III}}{\left(\mathrm{hfac}\right)}_2\left(\mathrm{py}\text{-}\mathrm{im}\right)\right]+{}^t\mathrm{Bu}_3\mathrm{PhOH}$$ (7)

II. Calorimetry

Solution calorimetry experiments have been done to independently confirm the ground-state enthalpy values determined from the van't Hoff analyses above. A Setaram C-80 Calvet calorimeter was outfitted with a pair of Hastelloy dual chamber reversal cells and run under isothermal conditions. In a typical experiment, separate solutions of Fe^II(H₂bip) and TEMPO were thermally equilibrated under nitrogen in separate chambers of one cell. Excess TEMPO (8−34 equiv) were used to ensure that the reaction went to completion. The second cell contained an identical volume of MeCN and acted as a reference. Inversion of the calorimeter mixed the solutions and initiated the reaction. The heat flux signal changes rapidly after the reaction is initiated and then gradually returns to its equilibrium value (Figure 2). Integration of this signal over the course of several hours gives the total heat released, which can be converted to ΔH^o using the reagent concentrations. Both reagents started as solutions in order to avoid the contributions from the heat of solution for the solid reagent. Instead, the heats of dilution were measured, which are typically much smaller contributions to the overall heat flux.³⁴

Figure 2.

Heat flow curve () and sample temperature () for the reaction between 3.2 mM Fe^II(H₂bip) + 0.10 M TEMPO in MeCN. The “x” marks indicate the integration limits used to extract the enthalpy of reaction.

The average of three measurements of the heat of reaction of Fe^II(H₂bip) and TEMPO (eq 4) gave ΔH^o₄ = −8.9 ± 0.6 kcal mol⁻¹. The observed heat of reaction was found to be independent of [TEMPO], indicating that heat of dilution for TEMPO, ΔH^o_dil[TEMPO], is small. ΔH^o_dil[Fe^II(H₂bip)] was measured independently and also found to be negligible. The value of ΔH^o₄ from calorimetry is in excellent agreement with that determined previously from van't Hoff analysis of K₄, −9.4 ± 0.6 kcal mol⁻¹. Thus, calorimetry provides a direct and independent confirmation of the heat of reaction and also, because ΔG^o₄ is well known, the entropy of reaction, ΔS^o = −30 ± 2 kcal mol⁻¹.

Attempts to measure other heats of metal HAT reactions unfortunately all proved problematic. For the reaction of Fe^III(Hbim) and TEMPOH, the exoergic direction for reaction 5, ¹H NMR and UV-Vis spectra of product mixtures after calorimetric measurements showed decomposition of the iron product. This decomposition was found to be strongly exothermic, and overwhelmed the small endothermic signal expected. The cobalt/TEMPO reaction (eq 3) is too slow to be reliably measured directly by the Calvet calorimeter apparatus, so the reaction of Co^II(H₂bim) with ^tBu₃PhO^• was investigated instead. This reaction cleanly forms the HAT products Co^III(Hbim) and ^tBu₃PhOH over the few minutes required for kinetic measurements. On the multiple-hour timescale of the calorimetry experiment, however; the UV-Vis spectra showed further reaction of the excess ^tBu₃PhO^•. This further reactivity results in calorimetric molar reaction enthalpy values that vary linearly with the amount of excess ^tBu₃PhO^• present, and this heat signal masks the enthalpy of the simple HAT reaction. Conditions with stoichiometric reagents or with excess Co^II(H₂bim) were also unsuccessful. Calorimetric measurements of the reactions of Ru^II(py-imH) with TEMPO (eq 6) or ^tBu₃PhO^• gave irreproducible heat flux signals, with large shifts in the baseline heat flux before and after mixing. Similar baseline shifts were also observed in the heat of dilution of Ru^II(py-imH) experiments. These experiments unfortunately have to be run at the edge of the sensitivity of the calorimeter due to the low solubility of Ru^II(py-imH) in MeCN.

As part of these studies, the heat of H-atom transfer from TEMPOH to ^tBu₃PhO^• to form TEMPO + ^tBu₃PhOH was measured (eq 8). These products are quantitatively formed and the

$$\mathrm{TEMPOH}+{}^t\mathrm{Bu}_3{\mathrm{PhO}}^{\cdot }\stackrel{K_8}{\rightleftarrows }\mathrm{TEMPO}+{}^t\mathrm{Bu}_3\mathrm{PhOH}$$ (8)

product mixture is stable overnight at 30 °C, by ¹H NMR and UV-Vis spectra. The directly measured heats of dilution in MeCN (ΔH^o_dil) for both TEMPOH and ^tBu₃PhO^• are small, on the order of 2% of the total heat for the HAT reaction. The average of three experiments gave ΔH^o₈[calorimetry] = −11.2 ± 0.5 kcal mol⁻¹.

Discussion

I. Overview of Hydrogen Atom Transfer Thermochemistry

Hydrogen atom transfer (HAT^11,29) reactions in solution (eq 9) can be written as the sum of two pseudo-half-reactions, eqs 10 and 11. This is analogous to describing an electron transfer (ET) process as two ET half-reactions, except that in HAT these are complete reactions since they are not relative to a reference electrode. For HAT, each half-reaction is the definition of the bond dissociation enthalpy (BDE) or bond dissociation free energy (BDFE). Strictly speaking, BDEs and BDFEs are gas-phase quantities but it is convenient to consider analogs in solution, BDE_s and BDFE_s in solvent ‘s’. The driving force for eq 9 is therefore the difference between the BDFEs (or BDEs if working with enthalpy) of the two half reactions, eq 12.

$$\mathrm{A}-\mathrm{H}+{\mathrm{B}}^{\cdot }\to {\mathrm{A}}^{\cdot }+\mathrm{H}-\mathrm{B}\Delta {G^{\mathrm{o}}}_{\mathrm{HAT}},\Delta {H^{\mathrm{o}}}_{\mathrm{HAT}},\Delta {S^{\mathrm{o}}}_{\mathrm{HAT}}$$ (9)

$$\mathrm{A}-\mathrm{H}\to {\mathrm{A}}^{\cdot }+{\mathrm{H}}^{\cdot }\mathrm{BDFE}{\left[\mathrm{AH}\right]}_{\mathrm{s}},\mathrm{BDE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}$$ (10)

$$\mathrm{B}-\mathrm{H}\to {\mathrm{B}}^{\cdot }+{\mathrm{H}}^{\cdot }\mathrm{BDFE}{\left[\mathrm{BH}\right]}_{\mathrm{s}},\mathrm{BDE}{\left[\mathrm{BH}\right]}_{\mathrm{s}}$$ (11)

$$\Delta {G^{\mathrm{o}}}_{\mathrm{HAT},\mathrm{s}}=\mathrm{BDFE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}-\mathrm{BDFE}{\left[\mathrm{BH}\right]}_{\mathrm{s}}$$ (12)

$$\mathrm{BDFE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}=\mathrm{BDE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}-\mathrm{T}\Delta {S^{\mathrm{o}}}_{\mathrm{s}}$$ (13)

$$\Delta {S^{\mathrm{o}}}_{\mathrm{s}}=S^{\mathrm{o}}{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}}+S^{\mathrm{o}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}}-S^{\mathrm{o}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}$$ (14)

$$S^{\mathrm{o}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}=S^{\mathrm{o}}{\left[\mathrm{AH}\right]}_{\mathrm{g}}+\Delta {S^{\mathrm{o}}}_{\text{solv}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}$$ (15)

The BDFE_s and BDE_s are related by the entropy of reaction ΔS^o_s (eq 13), which is the difference between absolute entropies of the component species in solution (eq 14). S^o[AH]_s is the sum of the gas-phase entropy of AH (S^o[AH]_g) and the entropy of solvation of AH_g (ΔS^o_solv[AH]_s), eq 15. Analogous definitions apply to S^o[H^•]_s and S^o[A^•]_s. S^o[AH]_g has contributions from the entropy of translations (depending on mass), rotations (depending on moment of inertia), vibrations (depending on frequencies), and electron spins.³⁷ ΔS^o_solv[AH]_s will vary given the polarity and hydrogen-bonding ability of the solvent used.³⁸

Organic HAT reactions, as noted above, have for many decades been analyzed using BDEs and the Bell-Evans-Polanyi (BEP) correlation of activation energies with ΔH^o_HAT.^1a,17,39 The differences between the BEP enthalpy treatment and more typical linear free-energy relationships (LFERs)^18,19 have not previously been of serious concern because HAT reactions have typically been assumed to have ΔS^o_s close to zero.²⁶ This assumption derives from the parallel assumption that (for most species) AH and A^• have similar absolute entropies (eq 16) because they have similar mass, size, and charge. As discussed elsewhere, equation 16 appears

$$S^{\mathrm{o}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}\approx S^{\mathrm{o}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}}$$ (16)

to hold for small gas phase molecules and for solution phase organic compounds^26,27 but does not hold for reactions of Fe^II(H₂bip), Fe^II(H₂bim),²⁸ and (as reported here) Co^II(H₂bim). The magnitude of ΔS^o_HAT has not been extensively explored for transition metal complexes.⁴⁰ The four different transition metal hydrogen atom donors studied here allow us to study the metal-based trends associated with ground-state entropies, and to determine if large ΔS^o_HAT is a general phenomenon for transition metal complexes.

II. Bond Dissociation Enthalpies and Free Energies

TEMPOH/TEMPO and ^tBu₃PhOH/^tBu₃PhO^• are convenient and common hydrogen atom transfer reagents, and they are reference points for much of the thermochemistry described here. It is therefore important to assess the ‘best’ current values for their O-H BDFEs and BDEs, as is done in the following sections and summarized in Table 2. In addition, the excellent agreement among the different approaches to these values supports the validity of each method.

Table 2.

Bond dissociation free energies (BDFEs) and enthalpies (BDEs) in MeCN.

AH	BDEMeCN (kcal mol−1)	BDFEMeCN (kcal mol−1)	Reference
tBu3PhOH	83 ± 1	77 ± 1	see text, 34
TEMPOH	71.5 ± 0.5	66.5 ± 0.5	see text, 23, 35
FeII(H2bip)	62.0 ± 1.7a	66.0 ± 1.7b	28, 30
FeII(H2bim)	67 ± 2a	72 ± 2b	36
RuII(py-imH)	68 ± 1a	62 ± 1b	32
RuII(hfac)2(py-imH)	--	79.6 ± 1b	32
CoII(H2bim)	62 ± 1a,c	69.5 ± 0.9c	see text

^aBDE_MeCN = ΔH^o_{(AH + TEMPO)} - BDE[TEMPOH]_MeCN.

^bFrom eq 21 using pK_a and E^o values.

^cFrom K_eq and ΔH^o values in Table 1; these are in agreement with values calculated from eq 21 (using pK_a and E^o), which are less precise for Co^II(H₂bim) because of the large uncertainty in E^o.³⁰

IIa. Solution BDEs of ^tBu₃PhOH from Calorimetry

The literature on gas phase and solution phase BDEs of phenols is extensive.⁴¹ Data are available in a variety of solvents, and it is important in comparisons and thermochemical cycles to use values in the same solvent.⁴² In 1969, calorimetry determined the heat of transfer of two H atoms from diphenylhydrazine to 2 ^tBu₃PhO^• in both benzene and CCl₄.³⁴ When coupled with the known solution heats of formation for PhNHNHPh and PhNNPh (the latter having been re-evaluated since 1969),⁴³ this gives ΔH^o_f[^tBu₃PhO^•]_s - ΔH^o_f[^tBu₃PhOH]_s, (28.09 ± 0.08 and 28.01 ± 0.12 kcal mol⁻¹ for C₆H₆ and CCl₄ respectively) which can be converted to a BDE_s using eq 17.

$${\mathrm{BDE}}_{\mathrm{s}}=\Delta {H^{\mathrm{o}}}_f{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}}\text{-}\Delta {H^{\mathrm{o}}}_f{\left[\mathrm{AH}\right]}_{\mathrm{s}}+\Delta {H^{\mathrm{o}}}_f{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}}$$ (17)

The last term in eq 17, ΔH^o_f[H^•]_s, is the sum of the gas-phase heat of formation of H^• (ΔH^o_f[H^•]_g = 52.103 ± 0.001 kcal mol⁻¹)⁴⁴ and its enthalpy of solvation (ΔH^o_solv[H^•]_s). The solvation of H₂ is considered to be a good model for solvation of H^•45 so ΔH^o_solv[H^•]_s can be approximated by ΔH^o_solv[H₂]_s. ΔH^o_solv[H₂]_s in benzene and CCl₄ are not known, but should be similar to those in toluene and 1,2-dichloroethane (1.38 and 1.81 kcal mol⁻¹, respectively).^46,47 Using these values in eq 17 gives solution BDEs for ^tBu₃PhOH of 81.6 ± 0.4 kcal mol⁻¹ in benzene and 82.0 ± 0.5 kcal mol⁻¹ in CCl₄.

The BDE of ^tBu₃PhOH in MeCN differs from the values in benzene and CCl₄ by the differences in solvation (eq 18). For H^•, the difference in solvation between benzene and MeCN

$$\begin{array}{cc}\hfill {\mathrm{BDE}}_{\mathrm{s}1}\text{-}{\mathrm{BDE}}_{\mathrm{s}2}=& \Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}1}\text{-}\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[\mathrm{AH}\right]}_{\mathrm{s}1}+\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}1}\hfill \\ \hfill & \text{-}(\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}2}\text{-}\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[\mathrm{AH}\right]}_{\mathrm{s}2}+\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}2})\hfill \end{array}$$ (18)

is very small, 0.18 kcal mol⁻¹.⁴⁶ Ingold and coworkers have proposed that the difference in solvation between a phenol and its radical is due primarily to differences in hydrogen bonding to the solvent^41b which they estimate using Abraham's empirical hydrogen-bond strength model.^48,49 For AH = ^tBu₃PhOH, the Ingold/Abraham H-bonding model estimates {ΔH^o_solv[A^•]_MeCN - ΔH^o_solv[AH]_MeCN} - {ΔH^o_solv[A^•]_C6H6 - ΔH^o_solv[AH]_C6H6} ≈ 1.3 kcal mol⁻¹, ⁵⁰ and therefore BDE[^tBu₃PhOH]_MeCN = 83 ± 1 kcal mol⁻¹.

IIb. BDE and BDFE of ^tBu₃PhOH in MeCN from Gas Phase Data

The BDE_MeCN and BDFE_MeCN for ^tBu₃PhOH can also be determined from gas phase values and estimates for the appropriate solvation terms (eqs 18, 19). The gas phase BDE of ^tBu₃PhOH has been critically reviewed^41c and found to be 8.8 ± 0.95 kcal mol⁻¹ weaker than BDE[PhOH]_g, which is 88.7 ± 0.5 kcal mol⁻¹.^41a Therefore, the BDE[^tBu₃PhOH]_g = 79.9 ± 1.1 kcal mol⁻¹.

$$\mathrm{BDE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}=\mathrm{BDE}{\left[\mathrm{AH}\right]}_{\mathrm{g}}+\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}}\text{-}\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}+\Delta {H^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}}$$ (19)

$$\mathrm{BDFE}{\left[\mathrm{AH}\right]}_{\mathrm{s}}=\mathrm{BDFE}{\left[\mathrm{AH}\right]}_{\mathrm{g}}+\Delta {G^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{A}}^{\cdot }\right]}_{\mathrm{s}}\text{-}\Delta {G^{\mathrm{o}}}_{\text{solv}}{\left[\mathrm{AH}\right]}_{\mathrm{s}}+\Delta {G^{\mathrm{o}}}_{\text{solv}}{\left[{\mathrm{H}}^{\cdot }\right]}_{\mathrm{s}}$$ (20)

ΔH^o_solv[^tBu₃PhO^•]_MeCN - ΔH^o_solv[^tBu₃PhOH]_MeCN has been estimated by two methods. The Ingold/Abraham H-bond model gives 1.43 kcal mol⁻¹. ⁴⁹ Alternatively, ΔH^o_solv[^tBu₃PhO^•]_MeCN - ΔH^o_solv[^tBu₃PhOH]_MeCN can be evaluated computationally. Bakalbassis, et al. found that chemically accurate values of BDEs for several phenols could be obtained with the (RO)B3LYP level of theory using a non-standard basis set (see Experimental for details).⁵¹ In our laboratory, this method yields BDE[^tBu₃PhOH]_g,DFT = 79.3, in good agreement with the experimental value above. Application of a polarizable continuum solvation model (PCM) yields the slightly negative value for ΔH^o_solv[^tBu₃PhO^•]_MeCN - ΔH^o_solv[^tBu₃PhOH]_MeCN of −0.4 kcal mol⁻¹. This is a consequence of the larger dipole moment of ^tBu₃PhO^•. Together, eq 19, the BDE[^tBu₃PhOH]_g of 79.9 kcal mol⁻¹ and the average of the two solvation models above (0.5 ± 1.3 kcal mol⁻¹) give BDE[^tBu₃PhOH]_MeCN = 82 ± 2 kcal mol⁻¹, consistent with the independently-determined calorimetric value above.

The bond dissociation free energy for ^tBu₃PhOH can also be determined using eq 20, starting with BDFE[^tBu₃PhOH]_g = 71.8 ± 1 kcal mol⁻¹.⁵² As above, PCM calculations were used to compute ΔG^o_solv[^tBu₃PhO^•]_MeCN - ΔG^o_solv[^tBu₃PhOH]_MeCN = −0.4 kcal mol⁻¹ (see Experimental for details). Ingold's model gives similarly small values. When used in eq 20, with ΔG^o_solv[H^•]_MeCN ≅ ΔG^o_solv[H₂]_MeCN = 5.12 kcal mol⁻¹,⁴⁶ the BDFE[^tBu₃PhOH]_MeCN is 77 ± 1 kcal mol⁻¹.

IIc. BDFE of ^tBu₃PhOH in MeCN from pK_a and E^o Measurements

An alternative measure of BDFE_MeCN is obtained from a thermochemical cycle using a pK_a and an E^o (eq 21). Bordwell

$${\mathrm{BDFE}}_{\mathrm{MeCN}}\left({\text{kcal mol}}^{-1}\right)=\mathrm{nF}{E}^{\mathrm{o}}(\mathrm{vs}.{\mathrm{Fc}}^{+∕0})+2.303\mathrm{RTp}{K}_{\mathrm{a}}+54.9$$ (21)

and co-workers popularized the method in DMSO,²² and Tilset worked out the cycle for MeCN.²⁰ The two reported values for E^o[^tBu₃PhO^•] in MeCN vs. Cp₂Fe^+/0, −707 mV⁵³ and −689 mV,⁵⁴ average to −0.70 V. The pK_a in DMSO for ^tBu₃PhOH (17.8)⁵⁵ can be converted into a value of 27.5 in MeCN using the linear relationship found by Kutt and coworkers.⁵⁶ Putting these values into eq 21 gives BDFE[^tBu₃PhOH]_MeCN = 76 ± 1.3 kcal mol⁻¹. The measured equilibrium constant for reaction of ^tBu₃PhO^• with Ru^II(hfac)₂(py-imH), equation 7 above, provides another measure of BDFE[^tBu₃PhOH]_MeCN, since the BDFE of the ruthenium complex is known from its pK_a and an E^o values.³² Application of eq 12 to this data gives a BDFE[^tBu₃PhOH]_MeCN of 78 ± 1 kcal mol⁻¹. The close agreement among these three BDFE values (77 ± 1, 76 ± 1, and 78 ± 1 kcal mol⁻¹) supports the validity of all three approaches and indicates a consensus BDFE[^tBu₃PhOH]_MeCN of 77 ± 1 kcal mol⁻¹ (Table 2).

IId. BDFE and BDE of TEMPOH in MeCN

A calorimetric study of the reaction of TEMPO with diphenylhydrazine in benzene,³³ following the analysis above for ^tBu₃PhOH, gives⁵⁷ BDE[TEMPOH]_C6H6 = 70.0 ± 0.8 kcal mol⁻¹ and BDE[TEMPOH]_MeCN = 71.5 ± 0.9 kcal mol⁻¹. This BDE can be converted to a BDFE of 66.4 ± 0.5 kcal mol⁻¹ via eqs 13 and 14 assuming that S^o[TEMPO]_MeCN ≈ S^o[TEMPOH]_MeCN (an example of eq 16) and ΔG^o_solv[H^•]_MeCN = 5.12 kcal mol⁻¹. This BDFE[TEMPOH]_MeCN can also be calculated directly using the known pK_a and E^o values (eq 21), which gives 66.9 ± 0.5 kcal mol⁻¹ (see Supporting Information). BDFE[TEMPOH]_MeCN is also determined by the equilibrium constants for reactions of Fe^II(H₂bip),²⁸ Fe^II(H₂bim),²⁸ or Ru^II(py-imH)³² with TEMPO (the BDFEs for each metal complex independently determined from pK_a and E^o values). The average value from these equilibration experiments, 66.5 ± 0.5 kcal mol⁻¹, is essentially the same as the values from the TEMPOH pK_a and E^o and from calorimetry. This agreement supports the consensus BDFE value (Table 2) and the assumption that S^o[TEMPO]_MeCN ≈ S^o[TEMPOH]_MeCN.

IIe. Solution BDFE and BDE of Metal Complexes in MeCN

For each of the four metal complexes examined here, the BDFE_MeCN was determined using eq 21 and the relevant pK_a and E^o values (which have been previously reported^32,30). As noted above, these values are consistent with the measured equilibrium constants and the BDFEs of TEMPOH and ^tBu₃PhOH. The BDE_MeCN (enthalpies) shown in Table 2 were calculated from the BDFE_MeCNs and the experimentally determined ΔS^o_MeCN for reaction with TEMPO, attributing all of this ΔS^o to the metal complex ‘half reaction’ (eq 11).

III. Comparison between Calorimetry and van't Hoff methodologies

The calorimetric experiments confirm the enthalpy of reaction of Fe^II(H₂bip) + TEMPO (eq 4) measured by solution equilibrium and kinetic data. The ΔH^o₄[calorimetry] = −8.9 ± 0.6 kcal mol⁻¹, is within error of ΔH^o₄[van't Hoff] = −9.4 ± 0.6 kcal mol⁻¹.²⁸ This quantitative agreement confirms the large negative ΔS^o = −30 ± 3 cal mol⁻¹ K⁻¹.²⁸ Similarly, the calorimetric heat of H-atom transfer from TEMPOH to ^tBu₃PhO^• (eq 8), ΔH^o₈[calorimetry] = −11.2 ± 0.5 kcal mol⁻¹, agrees with the value calculated from the literature solution BDEs in MeCN (Table 2), −11.5 ± 1.4 kcal mol⁻¹. Calorimetric studies of Ru^II(py-imH), Fe^III(Hbim), and Co^II(H₂bim) plus TEMPO were unsuccessful because the reactions did not meet the experimental requirements of high solubilities and good long-term stability of the reaction mixtures. For these systems, the van't Hoff methodology is easier and more reliable for determining the ground state thermodynamics, as long as equilibrium constants are measurable.

IV. Origins of the ΔS^o for H-atom transfer

The four reactions of transition metal H-atom donors with the same atom acceptor, TEMPO, have widely varying values of ΔS^o (Table 1). The reaction of Ru^II(py-imH) shows only a small positive entropy, ΔS^o₆ = 4.9 cal mol⁻¹ K⁻¹. The reactions with Fe^II(H₂bip), Fe^II₂bim), Co^II(H₂bim) show much larger negative values: ΔS^o_HAT = −30, −30, and −41 cal mol⁻¹ K⁻¹ respectively. These are very substantial values of |ΔS^o_HAT|, in contradiction with the common assumption that the entopic contribution to HAT is not significant. Thinking of these reactions as the sum of two quasi-half reactions (eqs 9-11), ΔS^o_HAT ≠ 0 requires that S^o[AH]_s ≠ S^o[A^•]_s (in contradiction to eq 16) for either the metal or organic redox couple.

The independent measurements of the BDE_MeCN and the BDFE_MeCN in Section II give (after subtracting TS^o[H^•]_MeCN), {S^o[TEMPO]_MeCN - S^o[TEMPOH]_MeCN} = 1 ± 4 cal mol⁻¹ K⁻¹ and {S^o[^tBu₃PhO^•]_MeCN - S^o[^tBu₃PhOH]_MeCN} = 5 ± 7 cal mol⁻¹ K⁻¹. In addition, HAT from TEMPOH to ^tBu₃PhO^• (eq 8), has ΔH^o₈ = −11.2 ± 0.5 kcal mol⁻¹ and ΔG^o = −10.5 ± 1.3 kcal mol⁻¹ so ΔS^o₈ = −2 ± 3 cal mol⁻¹ K⁻¹. The difference between ΔS^o₈ and {S^o[TEMPO]_MeCN - S^o[TEMPOH]_MeCN} further limits the entropy for {S^o[^tBu₃PhO^•]_MeCN - S^o[^tBu₃PhOH]_MeCN} to 3 ± 5 cal mol⁻¹ K⁻¹. These data demonstrate that S^o[AH]_s – S^o[A^•]_s ≈ 0 for both TEMPOH and ^tBu₃PhOH. Therefore, the unusual entropy contributions in our HAT reactions come from the metal redox couple.

The metal redox couples for reactions 3 − 6 show the following trend for |ΔS^o|: Ru^II(py-imH) << Fe^II(H₂bip) ≈ Fe^II(H₂bim) < Co^II(H₂bim). The Fe and Co reactions have negative entropies, indicating that the M^III(HL) complex is more ordered than M^II(H₂L). For the iron systems, previous experimental and computational studies²⁸ showed that the large |ΔS^o| originates primarily from changes in the vibrational entropy (ΔS^o_vib) upon oxidation of the iron. The calculations showed that the primary contributors are ca. 30 low-frequency (ν ≤ kT = 207 cm⁻¹ at 298 K) torsions and bends that change frequency between the Fe^II and Fe^III compounds.²⁸ Alternative origins of the large |ΔS^o| such as ion pairing or solvent effects were ruled out.

For Co^II(H₂bim) + TEMPO, the ΔS^o_HAT (−41 ± 2 cal mol⁻¹ K⁻¹) is even more negative than for the two iron reactions. In solution, Co^II(H₂bim) is entirely high-spin, while Co^III(Hbim) is entirely low-spin.³¹ In the idealized octahedral case, this is a change in multiplicity from a 12-fold degenerate ⁴T_1g Co^II electronic state to a non-degenerate ¹A_1g Co^III state, an electronic entropy of Rln(12) or ΔS^o_elec = −4.9 cal mol⁻¹ K⁻¹. This maximum value of ΔS^o_elec (spin-orbit coupling and the D₃ symmetry of Co^II(H₂bim) will lower the degeneracy) is still a minor contribution to the observed ΔS^o_HAT. The Co and Fe complexes are very similar: in structure (in both systems the M-N bond lengths are ∼0.1 Å shorter in the M^III derivative);^31,36 in acidity (similar pK_a values in MeCN); and in hydrogen bonding (the ΔG^o for formation of a hydrogen-bonded adduct between TEMPOH and either Co^III(Hbim) or Fe^III(Hbim) differs by less than 0.1 kcal mol⁻¹).^28,58 This similarity suggests a common vibrational origin for the entropy in both systems. The entropy would be expected to be larger for Co, since the high-spin to low-spin conversion should cause even larger frequency changes.^67,60a An increase of ca. 10 cal mol⁻¹ K⁻¹ for the addition of a spin-change is not unreasonable based on the electron transfer entropies discussed below. The ruthenium complexes, with a 4d transition metal, are all low-spin and have stronger bonds and therefore fewer low-frequency vibrational modes. With fewer modes ≤ kT, the vibrational entropy will be much reduced, as observed: Ru^II(py-imH) + TEMPOH has ΔS^o_HAT = 4.9 ± 1.1 cal mol⁻¹ K⁻¹. Thus the trend for |ΔS^o_HAT|, Ru^II(py-imH) << Fe^II(H₂bip) ≈ Fe^II(H₂bim) < Co^II(H₂bim), is consistent with a dominant role for vibrational entropy in these reactions.

V. Trends in Electron Transfer Entropies

The entropies of electron transfer half reactions (ΔS^o_ET, eq 22) have been determined for a wide range of complexes and found to depend on the nature of the metal center, the associated redox change, the coordinating ligands, and the surrounding solvent and counterions.^59-65 We note that, by convention, the ET half-reaction in eq 22 is written as a reduction, opposite to the way the HAT reactions are written here (eqs 3-8, 11, 12).

$${\mathrm{L}}_{\mathrm{n}}{\mathrm{M}}^{\mathrm{III}}+e^{-}\to {\mathrm{L}}_{\mathrm{n}}{\mathrm{M}}^{\mathrm{II}}\Delta {S^{\mathrm{o}}}_{\mathrm{ET}}$$ (22)

Vibrational entropy has been shown to be important in both electron transfer^66,67 and spin-equilibrium processes,^68,69 when there are changes in metal-ligand bonding upon redox or spin change. Richardson and Sharpe estimated that in the gas phase the vibrational contribution to the total electron transfer entropy ranges from as low as 9% in RuO₄ (where there are few vibrations and many are at high frequency) up to 42% in [Fe(CN)₆]^4-.^67a Vibrational entropy is a substantial part of the measured ΔS^o_ET = 29 ± 3 cal mol⁻¹ K⁻¹ for the Fe^II(H₂bim)/Fe^III(H₂bim) redox couple,²⁸ and of the ΔS^o_HAT observed for the HAT reactions of Fe^II(H₂bim) and Fe^II(H₂bip). These entropies are close in magnitude because they both arise from the Fe^III complexes being more rigid and having fewer low-frequency vibrational modes than Fe^II.²⁸ In general, when vibrational entropy is a significant contributor, there should be a strong parallel between the entropies for ET and HAT reactions in a given system.

Differences in electron transfer entropies between redox two couples, when measured in the same solvent for reagents of the same charge, are primarily due to vibrational and electronic entropies.^60a For example, {ΔS^o_ET[Co(tacn)₃]^3+/2+_solv - ΔS^o_ET [Ru(tacn)₃]^3+/2+_solv} is 16 ± 3 cal mol⁻¹ K⁻¹ for solv = DMSO, acetone, water and four other solvents (tacn = 1,4,7-triazacyclononane).^60a,70,71 This difference is independent of solvent because the electronic and vibrational entropies do not depend on the solvent. {ΔS^o_vib[Co(tacn)₃]^3+/2+ - ΔS^o_vib[Ru(tacn)₃]^3+/2+} has been estimated as 12.7 cal mol⁻¹ K⁻¹, and the remainder of the difference can be accounted for by ΔS^o_elec.^60a

The results described in this report indicate that, when comparing one transition metal system to another, the same trends are observed in the half-reaction entropies for both ΔS^o_ET and |ΔS^o_HAT|. The Co^III/Co^II couples have the largest ΔS^o_ET, with the exception of a few lanthanide and actinide complexes,^59d,64,72 because they involve a spin-state change (low-spin Co^III to high-spin Co^II) in addition to an oxidation state change. ΔS^o_ET for Co^III/Co^II couples ranges from 30 − 50 cal mol⁻¹ K⁻¹ in organic solvents.^{59f,60,61,67,72,73} For similar complexes, ΔS^o_ET values for Co derivatives are typically 10 − 20 cal mol⁻¹ K⁻¹ larger than those for the Fe analogs. In our measurements of HAT entropies, the Co reaction has the largest |ΔS^o_HAT| (ca. −41 cal mol⁻¹ K⁻¹), 11 cal mol⁻¹ K⁻¹ more negative than the iron analogs in the same reaction with TEMPO. Fe^III/II couples have ΔS^o_ET of typically 15 − 30 cal mol⁻¹ K⁻¹ in organic solvents (except for those with an accompanying spin change).^59,60,67 Ru^III/Ru^II couples have ΔS^o_ET in a similar range as Fe^III/Fe^II couples (∼10−30 cal mol⁻¹ K^{−1 59,60,66a,67,72}). Comparing complexes in the same solvent, low-spin Fe^III/II couples are quite similar to the Ru^III/II analogs, while iron couples that exhibit spin-equilibrium or are high-spin only have larger ΔS^o_ET than the related Ru^III/II couples (probably due to larger vibrational entropies).⁷⁴

Thus the trend for ΔS^o_ET is Co > high-spin Fe > low-spin Fe ≈ Ru.^59b This is the same as the trend in –ΔS^o_HAT described above for HAT (recall that the half reactions are by convention written in opposite directions for ET (eq 22) and HAT (eq 10), so the signs are opposite). This trend is a result of the major contribution of ΔS^o_vib, for both HAT and ET reactions of transition metal complexes. The close connection between ΔS^o_ET and ΔS^o_HAT provides valuable insight in cases where only one or the other has been measured. In particular, it suggests that ground-state entropy effects will be important for HAT reactions of high-spin first-row transition metals.

VI. Relevance to Biological Systems and HAT Analyses

The observation of significant ground-state entropies for HAT reactions appears to be general for first-row transition metal coordination complexes. This leads to a significant temperature dependence of ΔG. For instance, in a reaction with ΔS^o = −30 cal mol⁻¹ K⁻¹ such as observed for the Fe systems above, ΔG shifts by more than 1 kcal mol⁻¹ between 5 and 45°C and K_eq shifts by almost an order of magnitude. This effect does not appear to have been incorporated into most applications of modern PCET theories to either small molecule or enzymatic systems.¹⁹ In particular, variation in ΔG indicates changes in the shape of the free energy surface which should affect processes involving hydrogen tunneling. Does the variation in ΔG play a role, for instance, in the unusual temperature dependence of the kinetic isotope effect for HAT from a fatty acid to the non-heme iron center in lipoxygenase enzymes? Since large values of |ΔS^o_ET| have been observed in biological systems,^75,76 the close connection between ΔS^o_ET and ΔS^o_HAT described here suggests that there are significant entropic contributions in HAT and PCET reactions of metalloproteins.

Conclusions

Ground state entropy changes for hydrogen atom transfer reactions, ΔS^o_HAT, vary substantially depending on the reaction. Values reported vary from −41 ± 2 cal mol⁻¹ K⁻¹ for HAT from a cobalt(II) 2,2’-bi-2-imidazoline complex to the nitroxyl radical TEMPO (eq 3), to – 2 ± 3 cal mol⁻¹ K⁻¹ for HAT from TEMPOH to the stable aryloxyl radical ^tBu₃PhO^• (eq 8). These values have been determined by Van't Hoff analysis of equilibrium data, by calorimetric measurements, and using thermochemical cycles. These data and those from previous reports show that the magnitude of |ΔS^o_HAT| for reactions with TEMPO have the following trend: Co^II(H₂bim) > Fe^II(H₂bip) = Fe^II(H₂bim) > Ru^II(py-imH) ≳ ^t_Bu3PhOH ≈ 0. The analysis presented here supports the long-standing assumptions that ΔS^o_HAT ≈ 0 and that S^o[AH] ≈ S^o[A^•] for HAT reactions of organic and small gas phase molecules, but not for transition metal complexes. Analyses of transition metal HAT reactions need to take into account the frequently large reaction entropies and should not be based just on bond dissociation enthalpies. The trend in ΔS^o_HAT for the metal complexes is the same as that observed for electron transfer half-reaction entropies in aprotic solvents ΔS^o_ET, and the magnitudes of these values are often similar as well. This striking analogy is a result of both the HAT and ET values being significantly influenced by vibrational entropy contributions. The more extensive database of electron transfer entropies therefore provides guidelines for initial predictions of hydrogen transfer entropies.

Experimental Section

General Considerations

All manipulations were carried out under anaerobic conditions in MeCN using standard high-vacuum line and nitrogen-filled glovebox techniques unless otherwise noted. NMR spectra were acquired on Bruker Avance-500, DRX-499, Avance-300, or Avance-301 spectrometers. Static UV-Visible spectra were obtained using either a Hewlett-Packard 5483 spectrophotometer equipped with an eight-cell holder thermostatted with a Thermo-Neslab RTE-740 waterbath, or using a Shimadzu UV-2401 PC dual beam instrument. Spectra are reported as λ_max, nm [ε, M⁻¹ cm⁻¹] and were blanked relative to pure MeCN. Air-sensitive samples were prepared in the glovebox and their spectra taken using either quartz cuvettes attached to Teflon-stoppered valves (Kontes) or injectable screw-capped cuvettes with silicone/PFTE septa (Spectrocell). Septa were replaced after each experiment. Rapid kinetic measurements were taken using an OLIS USA stopped-flow instrument equipped with the OLIS-rapid scanning monochromator and UV-Vis detector, and thermostatted by a Neslab RTE-111 waterbath. All errors are reported as ±2σ̵ based on fits weighted with the errors propagated from experimental measurements.

Materials

Low water content CH₃CN (< 10 ppm H₂O; Allied Signal / Burdick and Jackson brand) was taken from a steel keg sparged with Ar and dispensed through the glovebox. CD₃CN (Cambridge Isotopes Laboratories) was dried by stirring overnight with CaH₂, vacuum transferring and stirring briefly (< 1 hr) over P₂O₅, vacuum transferring back over CaH₂ for ca. 30 min and then storing in the glove box free of drying agent. Other solvents were dried using a ”Grubbs-type” Seca Solvent System installed by GlassContour.⁷⁷ 2,2,6,6-Tetramethyl-1-piperidinyloxy (TEMPO; Acros Organic and Aldrich) was sublimed at room temperature under static vacuum before use. [Fe^II(H₂bip)₃][ClO₄]₂, (Fe^II(H₂bip));³¹ [Fe^III(H₂bip)₂(Hbip)][ClO₄]₂ (Fe^III(Hbip));³¹ [Co^II(H₂bim)₃][ClO₄]₂, (Co^II(H₂bim));³¹ [Co^III(Hbim)(H₂bim)₂][ClO₄]₂ (Co^III(Hbim));³¹ [Ru^II(acac)₂(py-imH)], (Ru^II(py-imH));³² [Ru^III(acac)₂(py-im)], (Ru^III(pyim));³² 2,2,6,6-tetramethyl-1-piperidinylhydroxide (TEMPOH);^28,78 and 2,4,6-tri-tert-butyl phenoxyl (^tBu₃PhO^•)³³ were prepared and characterized following literature procedures. All other reagents were purchased from Aldrich and used as received. Caution: The perchlorate salts used herein are potentially explosive and should be handled with care in small quantities only. They should not be heated when dry or subjected to friction or shock, such as scratching with a non-Teflon-coated spatula.

Solution K_eq measurements

Equilibrium experiments for Fe^II(H₂bip) and Fe^II(H₂bim) + TEMPO have been previously reported.²⁸ K₃ for Co^II(H₂bim) + TEMPO ⇄ Co^III(Hbim) + TEMPOH were directly measured by ¹H NMR spectroscopy. A set of J-Young capped NMR tubes were charged with varying amounts of TEMPO (5 − 10 mg, 0.03 − 0.06 mmol), leaving one tube empty to act as a ‘time zero’ spectrum. To each of these tubes was added a 0.4 mL aliquot of a stock solution of Co^II(H₂bim) (11 mM in CD₃CN containing 3−10 μL CH₂Cl₂ as an integration standard). After mixing, the tubes were quickly removed from the glovebox and placed in a thermal bath. Changes in the NMR spectrum were monitored over the course of 2−7 days depending on the temperature. Data acquisition was stopped when the integrations for each species remained constant for a minimum of 12 hours. The tubes were then placed at room temperature (21 °C) and allowed to re-equilibrate for 2−7 days, again monitored by ¹H NMR spectroscopy. Reliable integrations were obtained using Mestre-C by manually phasing and baselining the spectrum from +60 to −3 ppm, as well as adjusting the phase and bias for each integral. Integrations were found to be reproducible to ± 3−5% based on the standard deviation of three spectra taken in rapid succession on a tube at equilibrium. K₄ values were obtained from the ratio of the following integration regions: Co^II(H₂bim) 24.7 − 21.2 ppm, Co^III(Hbim) 4.85 − 2.68 ppm, TEMPO 21.2 − 10.8 ppm, TEMPOH 1.73 − 1.34 ppm, from at least two spectra.

The equilibrium constant for [Ru^II(hfac)₂(py-imH)] + ^tBu₃PhO^• ⇄ [Ru^III(hfac)₂(py-im)] + ^tBu₃PhOH (eq 7) was measured by UV–vis titration using the method used in reference ³² to determine K₇, using a solution of Ru^II(hfac)₂(py-imH) (0.027 mM, 2.5 mL) titrating with ^tBu₃PhO^• (6.7 mM) until 10 equiv (10 μL = 1 equiv). The UV–vis data were analyzed using the absorbance at 481 nm to determine an equilibrium constant K₇ = 0.062 ± 0.013 (ΔG^o₇ = 1.6 ± 0.1 kcal mol⁻¹) from the average of two runs. Since ΔG^o₇ = BDFE[Ru^II(hfac)₂(py-imH)]_MeCN – BDFE(^tBu₃PhOH)_MeCN and BDFE[Ru^II(hfac)₂(py-imH)] = 79.6 ± 1.0 (from E^o and pK_a values³²), this yields BDFE(^tBu₃PhOH)_MeCN = 78 ± 1.

Calorimetry

Experiments were done on a Setaram C-80 Calvet Calorimeter outfitted with a pair of Hastelloy C276 Reversal Mixing cells (utilizing the larger 2.5 mL reaction cup and graphited Teflon seals) under isothermal conditions. In the calorimeter, each cell is surrounded by an electrical heater coil. The difference in current required to maintain a constant temperature between the two cells is related to the heat evolved from the reaction. The methodology used is a modification of literature procedures.^34,35 The calorimeter was set at 30.0 °C and had an actual sample temperature of 29.6 ± 0.1 °C. The calorimeter had been previously calibrated with a Joule-effect vessel, and the calibration was checked intermittently using the standardized aqueous heat of solution for KCl.

In a glovebox, the inner cup of one cell charged with 5−15 mg of limiting reagent in 2.0 mL of MeCN and covered with a Hastelloy cap. The outer chamber was charged with 1.0 mL of the reagent in slight excess (1.1 − 2.0 equiv) and the remainder of the cell assembled. See Table 3 for typical concentrations. The reference cell was assembled under identical conditions using a total of 3.0 mL of MeCN. The sealed cells were removed from the glovebox and thermally equilibrated in the calorimeter until both the flux and the temperature had reached steady-state (ca. 1−2 hours). This was deemed the experimental baseline, and the reaction was initiated by rotation of the calorimeter body, which inverted the cells and allowed the two solutions to mix. The heat flux was then recorded until the baseline was once-again achieved. The resulting fluxogram was integrated using SetSoft-2000 to give the heat evolved (Table 3). After the reaction was complete, the sample cell was removed from the calorimeter and returned to the glovebox. The final reaction mixture was diluted in a Kontes-valve cuvette (0.1 mL sample in ca. 2.0 mL MeCN) and an optical spectrum was obtained, to determine that the reaction had gone to completion and remained uncontaminated by air. Reactions were repeated a minimum of three times to achieve the desired level of reproducibility. Control reactions for the heats of dilution were measured for each reagent under similar concentration regimes to the actual reaction conditions. In most cases these were found to be small effects.

Table 3.

Standard conditions and integration times for calorimetry reactions.

Reaction	Inner Chamber	Outer Chamber	Integration time
FeII(H2bip) + TEMPO a	2.8 mM FeII(H2bip)	23 − 109 mM TEMPO a	∼ 10000 s
RuII(py-imH) + tBu3PhO•	5 − 10 mM RuII(py-imH)	11 mM tBu3PhO•	∼ 2000 s
RuII(py-imH) + TEMPO	10 mM RuII(py-imH)	11 mM TEMPO	∼ 4000 s
CoII(H2bim) + tBu3PhO•	6.2 mM CoII(H2bim)	4.4 mM tBu3PhO•	∼ 8000 s
TEMPOH + tBu3PhO•	11 mM TEMPOH	14.7 mM tBu3PhO•	∼ 4000 s

^aAt 29.6 °C, Fe^II(H₂bip) + TEMPO has K_eq = 1.7 ± 0.3. In order to ensure that the reaction went to completion, 8 − 34 equiv of TEMPO were used.

Calculations

All calculations were performed using Gaussian03.^{79 t}Bu₃PhOH and ^tBu₃PhO^• were optimized in C₁ symmetry and geometries were confirmed to be local minima by vibrational analysis. Following literature precedent,⁵¹ the B3LYP functional was used with the 6−31+G basis set with additional (p) polarization functions on hydrogen atoms only. Radical species were computed using the restricted-open shell (RO) formalism. Bakalbassis, et al. found this (RO)B3LYP/6−31+G(,p) method (the non-standard basis set nomenclature indicates that polarization functions are included for hydrogen atoms only and not heavy atoms) produced chemically accurate values of BDE for several phenolic compounds.⁵¹ Solution enthalpies and free energies were obtained from geometry optimizations and frequency analyses including a polarizable continuum model (PCM)⁸⁰ of acetonitrile, as implemented in Gaussian03.⁸¹ Free energies of solution, ΔG_solv, were also obtained from PCM single point calculations on the gas phase optimized geometries, with the inclusion of the SCFVAC keyword; for these calculations atomic radii from the United Atom Topological Model (UAHF) were used. The values of ΔG^o_solv[^tBu₃PhO^•]_MeCN - ΔG^o_solv[^tBu₃PhOH]_MeCN computed via these two methods are essentially identical (within ca. 0.01 kcal mol⁻¹).