**1. Introduction**

24 Nitroxides – Theory, Experiment and Applications

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[22] Fawzi NL, Fleissner MR, Anthis NJ, Ka´lai T, Hideg K, Hubbell WL, Clore GM (2011) A Rigid Disulfide-Linked Nitroxide Side Chain Simplifies the Quantitative Analysis of

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Stable nitroxide radicals have proved to be helpful in solving many problems in chemistry, biochemistry, biophysics, material science etc. as model compounds. To understand in detail the great variety of different chemical reactions, a good knowledge of the underlying "simple" exchange reactions related to nitroxide radicals is necessary. This chapter focuses on electron-self and spin exchange reactions of various nitroxide radicals in different solvents. The corresponding rate constants, the activation parameters, like activation enthalpies and volumes are obtained from temperature- and pressure dependent ESR-line broadening effects. A short introduction to dynamic ESR-spectroscopy is given.

### **1.1. From ESR spectrum to exchange rate constant**

There are several chemical reactions giving rise to dynamic line shape effects, such as spinexchange, also known as Heisenberg exchange, electron self-exchange and proton or counter ion transfer. A theoretical illustration of how an ESR spectrum may be affected by chemical reactions is presented in figure 1. Here, the interconvertion between two states, a and b, can influence the spectrum dramatically as the exchange rate increases.

Apart from the two extreme limits of infinitely slow or fast exchange, usually three regions are considered: slow, intermediate and fast. In the slow region the two lines broaden as if the life, or relaxation, times of the spins had been reduced due to quenching. When the exchange enters the intermediate region, the interconvertion rate leads to an averaging of the lines and correspondingly to line shifts. At one point, the exchange so fast that the spectrum appears as a single line due to overlap of the two original ones. This marks the beginning of the fast region, where a further increase in the exchange rate leads to narrowing of the single line.

© 2012 Grampp and Rasmussen, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Grampp and Rasmussen, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Figure 1.** Theoretical illustration of the first derivative ESR spectra arising from the interconversion between forms *a* and *b*. (a) Slow limit. (b) Moderately slow region. (c) Intermediate region showing line shifts. (d) Moderately fast region. (e) Fast limit.

A chemical reaction corresponding to this illustration could, for example, be an exchange reaction like the one shown in equation (1), where the forms *a* and *b* are interpreted as being molecules having different nuclear spin configurations.

$$\mathbf{A\_{(a)}} + \mathbf{A\_{(b)}^{\bullet \cdot \cdot}} \xrightarrow{k\_{\rm ab}} \mathbf{A\_{(a)}^{\bullet \cdot \cdot} + \mathbf{A\_{(b)}}} + \mathbf{A\_{(b)}} \tag{1}$$

Transferring the unpaired electron from one molecule to the other thus corresponds to the interchange between the two forms. Note, that reactions between molecules of the same form, e.g. *a*, also take place. If this happens no ESR line broadening is expected, but nevertheless the electron transfer takes place and these 'uneventful' reactions must be taken into account. The resulting line width in the case of slow exchange then becomes:

$$
\Delta B = \Delta B^o + \frac{1 - p\_j}{\left| \mathcal{V}\_e \mathcal{T} \right|} \tag{2}
$$

with pj being the probability that the reactants have the same nuclear spin configuration, here *j*.

In general, an ESR spectrum may show more than just a few lines (states), of which some are originating from transitions involving degenerate energy levels. This can give rise to substantially different values of pj and it is important to recognize that when a self-exchange reaction occurs, one expects the ESR lines to have different widths. Strictly speaking, equation (2) is therefore only valid for a single state and its corresponding ESR line.

Looking at equation (1) from the point of view of chemical kinetics it is seen that although the self-exchange reaction is bimolecular, one may express it using a first order rate law since the concentration of diamagnetic species is constant. Recalling the relationship between life times and first order rate constants one obtains

$$
\tau = \frac{1}{k} = \frac{1}{k\_{obs}[A]} \tag{3}
$$

which in combination with equation (2) and taking the ESR signal in its first derivative Lorentzian form results in the following relationship between line broadening and rate constant:

$$k\_{obs} = \frac{\sqrt{3}\pi \left| \mathcal{V}\_e \right| \left( \Delta B\_{pp} - \Delta B\_{pp}^o \right)}{\left( 1 - p\_j \right) \left[ \! \left[ A \right] \! \right]} \tag{4}$$

where *<sup>o</sup> pp B* and *pp B* are the peak-to-peak widths of the first derivative line in the absence and presence of the self-exchange reaction.

Similarly, an expression of kobs may be derived for the fast exchange region, where the broadening of the singe ESR line obeys

$$
\pi \left( \Delta B\_{pp} - \Delta B\_{pp}^o \right) = \frac{4\pi \left| \mathcal{V}\_\varepsilon \right| \nabla\_2}{3k\_{obs} \left[ A \right]} \tag{5}
$$

Here <sup>2</sup> is the second moment of the ESR spectrum, that is

26 Nitroxides – Theory, Experiment and Applications

shifts. (d) Moderately fast region. (e) Fast limit.

here *j*.

molecules having different nuclear spin configurations.

**Figure 1.** Theoretical illustration of the first derivative ESR spectra arising from the interconversion between forms *a* and *b*. (a) Slow limit. (b) Moderately slow region. (c) Intermediate region showing line

A chemical reaction corresponding to this illustration could, for example, be an exchange reaction like the one shown in equation (1), where the forms *a* and *b* are interpreted as being

Transferring the unpaired electron from one molecule to the other thus corresponds to the interchange between the two forms. Note, that reactions between molecules of the same form, e.g. *a*, also take place. If this happens no ESR line broadening is expected, but nevertheless the electron transfer takes place and these 'uneventful' reactions must be taken

> 1 *<sup>o</sup> <sup>j</sup> e*

 

*<sup>p</sup> B B*

with pj being the probability that the reactants have the same nuclear spin configuration,

In general, an ESR spectrum may show more than just a few lines (states), of which some are originating from transitions involving degenerate energy levels. This can give rise to

into account. The resulting line width in the case of slow exchange then becomes:


(2)

$$\nabla\_2 = \sum\_j p\_j \left(\overline{B} - B\_j\right)^2 \tag{6}$$

with *B* being the center field of the spectrum and Bj the resonant magnetic field strength of the j'th ESR line.

Note that the fact that an ESR spectrum belonging to an exchange reaction appears as a single line does not always imply that the fast limit is reached. Such systems may still be in the intermediate region and equation (5) does not hold. In an effort to decide if a given system is indeed exhibiting fast exchange, the following relation may be used [1].

$$Z = \frac{\sqrt{3}}{2} \cdot \frac{\left(\Delta B\_{pp} - \Delta B\_{pp}^{o}\right)}{\sqrt{\nabla\_{2}}} \le 0, 2\tag{7}$$

That is, if Z≤0.2 the system may safely be considered to be in the fast exchange region.

In the intermediate exchange region, the pseudo-first order rate constant, *k* = *τ* <sup>−</sup>1 = *k*obs[A], can been extracted from the experimental spectra by means of density matrix simulations of the dynamically broadened line shapes [2]. Assuming a weakly coupled spin system undergoing degenerate electron exchange in the high temperature limit and a relaxation superoperator diagonal in the subspace of induced transitions, *i.e.* neglecting cross relaxations, the evolution of the relevant terms of the spin density operator in the rotating frame, *σ*, is governed by:

$$\begin{split} \frac{\partial \sigma\_{\beta\{\boldsymbol{\nu}\},\boldsymbol{\alpha}\{\boldsymbol{\nu}\}}}{\partial t} &= \left[ i \left( \alpha\_{\boldsymbol{\alpha}\{\boldsymbol{\nu}\},\beta\{\boldsymbol{\nu}\}} - \alpha \right) - R\_{\beta\{\boldsymbol{\nu}\},\boldsymbol{\alpha}\{\boldsymbol{\nu}\}} - \frac{1}{\tau} \right] \sigma\_{\beta\{\boldsymbol{\nu}\},\boldsymbol{\alpha}\{\boldsymbol{\nu}\}} \\ &+ \frac{2}{N\tau} \sum\_{\boldsymbol{\nu}' \neq \boldsymbol{\nu}} \sigma\_{\beta\{\boldsymbol{\nu}'\},\boldsymbol{\alpha}\{\boldsymbol{\nu}'\}} + \frac{i\alpha\_{1,\boldsymbol{\epsilon}} \alpha\_{\boldsymbol{\varepsilon}}}{2k\_{\boldsymbol{B}}T\boldsymbol{\text{N}}}, \end{split} \tag{8}$$

where {*ν*} denotes the set of quantum numbers describing the nuclear spin states and *N* the dimensionality of the Hilbert space [3]. *R<sup>β</sup>*{*ν*}*α*{*<sup>ν</sup>*} is the element of the relaxation matrix corresponding to the induced transition and *ωα*{*ν*}*,β*{*<sup>ν</sup>*} = *ωα*{*ν*}−*ωβ*{*<sup>ν</sup>*} = *(E<sup>α</sup>*{*ν*}− *E<sup>β</sup>*{*ν*}*)/ħ*. The absorption signal is obtained from the expectation value of the out of phase magnetization assuming steady state conditions, *i.e.* 

$$Y = Tr\left(\hat{\mathcal{S}}\_y \hat{\sigma}\right) \propto \operatorname{Im} \frac{-i \sum\_k \alpha\_k \mathbf{u}\_k F\_k}{1 + \frac{2}{N\tau} \sum\_k \mathbf{u}\_k F\_k} \tag{9}$$

where the sum now runs over all completely equivalent spin packets of degeneracy *nk* and resonance frequency *ωk*. *N* equals the sum over all *nk* and *τ* denotes the average lifetime of the nuclear spin configurations, *i.e. τ* <sup>−</sup>1 = *k* = *k*obs [A]. Furthermore, *Fk* is defined by *Fk* <sup>−</sup><sup>1</sup> = i*(ω<sup>k</sup>* − *ω)* − *T*2*,k*−<sup>1</sup> − *τ*−1, with *T*2*,k* denoting the transversal relaxation time for spin packet *k* and *ω* the microwave irradiation frequency, respectively.

Equation (9) has been implemented in Matlab and Fortran 95. Local (trusted region Newton, Simplex) as well as global (Lipschitz, adaptive simulated annealing) optimization algorithms have been employed to fit the experimental spectra. Field modulation effects (100 kHz) have been accounted for by using pseudo-modulation as introduced by Hyde [4].

#### **1.2. Experimental considerations**

To obtain the best possible results from ESR line broadening investigations, a certain amount of experimental diligence is required. It may seem superfluous to mention this here, but as it is said: 'God is in the detail'. Albeit information on this subject is ample in literature [5, 6], some key points shall be reiterated.

Any spectrometer setting that might influence the line shape, e.g. modulation amplitude or microwave power, should remain unchanged throughout a series of measurements. Temperature control can be very advantageous as a means of reducing the experimental error. The reason why can be illustrated in the following example: Assuming an Arrheniuslike behaviour with an activation energy of 15 kJ mol-1, a deviation of the temperature by 1 K around 298K corresponds to a change in rate constant of approximately 2%.

28 Nitroxides – Theory, Experiment and Applications

,

*t*

the microwave irradiation frequency, respectively.

 

assuming steady state conditions, *i.e.* 

**1.2. Experimental considerations** 

[5, 6], some key points shall be reiterated.

frame, *σ*, is governed by:

In the intermediate exchange region, the pseudo-first order rate constant, *k* = *τ* <sup>−</sup>1 = *k*obs[A], can been extracted from the experimental spectra by means of density matrix simulations of the dynamically broadened line shapes [2]. Assuming a weakly coupled spin system undergoing degenerate electron exchange in the high temperature limit and a relaxation superoperator diagonal in the subspace of induced transitions, *i.e.* neglecting cross relaxations, the evolution of the relevant terms of the spin density operator in the rotating

, ,,

 

  1

 

(8)

<sup>−</sup><sup>1</sup> = i*(ω<sup>k</sup>*

 

1,

*i*

*e e B*

*kkk*

*n F*

(9)

*N k k k*

*i nF*

*N k TN*

*i R*

 

'

 

ˆ 'ˆ Im

*Y Tr S*

*y*

the nuclear spin configurations, *i.e. τ* <sup>−</sup>1 = *k* = *k*obs [A]. Furthermore, *Fk* is defined by *Fk*

 

', '

<sup>2</sup> , <sup>2</sup>

where {*ν*} denotes the set of quantum numbers describing the nuclear spin states and *N* the dimensionality of the Hilbert space [3]. *R<sup>β</sup>*{*ν*}*α*{*<sup>ν</sup>*} is the element of the relaxation matrix corresponding to the induced transition and *ωα*{*ν*}*,β*{*<sup>ν</sup>*} = *ωα*{*ν*}−*ωβ*{*<sup>ν</sup>*} = *(E<sup>α</sup>*{*ν*}− *E<sup>β</sup>*{*ν*}*)/ħ*. The absorption signal is obtained from the expectation value of the out of phase magnetization

<sup>2</sup>

where the sum now runs over all completely equivalent spin packets of degeneracy *nk* and resonance frequency *ωk*. *N* equals the sum over all *nk* and *τ* denotes the average lifetime of

− *ω)* − *T*2*,k*−<sup>1</sup> − *τ*−1, with *T*2*,k* denoting the transversal relaxation time for spin packet *k* and *ω*

Equation (9) has been implemented in Matlab and Fortran 95. Local (trusted region Newton, Simplex) as well as global (Lipschitz, adaptive simulated annealing) optimization algorithms have been employed to fit the experimental spectra. Field modulation effects (100 kHz) have been accounted for by using pseudo-modulation as introduced by Hyde [4].

To obtain the best possible results from ESR line broadening investigations, a certain amount of experimental diligence is required. It may seem superfluous to mention this here, but as it is said: 'God is in the detail'. Albeit information on this subject is ample in literature

Any spectrometer setting that might influence the line shape, e.g. modulation amplitude or microwave power, should remain unchanged throughout a series of measurements. Temperature control can be very advantageous as a means of reducing the experimental error. The reason why can be illustrated in the following example: Assuming an Arrhenius-

1

*k*

Other things that are often overlooked are the position of the sample in the ESR resonator as well as the so-called filling factor. These are particularly important when working with lossy solvents, as the effective microwave power felt by the paramagnetic species, often expressed by the Q factor, is strongly dependent on the amount of solvent used [7].

Finally, a few remarks on the design of the experiment: In the case of self-exchange, the concentration of the radical is kept constant while varying that of the diamagnetic compound. It is advisable to keep the concentration of the radical low in order to avoid effects from spin exchange, especially when interested in temperature or pressure effects. Often radical concentrations of around 10-4 mol l-1 are used, whereas the concentration of the diamagnetic compound preferably spans several orders of magnitude. Paramagnetic exchange is treated analogously.

### **1.3. Experimental techniques in pressure dependent ESR-measurement**

Several commercially available temperature-control units exist for magnetic resonance spectroscopic measurements, ranging from liquid helium to very high (1100oC) temperatures.

In contrast units for pressure dependent experiments must be constructed by the experimentalist themselves.<sup>1</sup> Fortunately, several excellent books are published on highpressure techniques covering a great variety of experimental measurements and different spectroscopies [8, 9]. To our knowledge no special review concerning ESR-spectroscopy under high pressure exists, but several articles describing high-pressure cells for NMR- and ESR-spectroscopy. Many experimental details developed for high-pressure NMR-cells can be adapted for ESR-spectroscopy often in simpler versions [10-13]. Several publications deal with high-pressure magnetic resonance cells for solid state investigations [14-21]. Even highpressure cells for ENDOR-studies [22], for K-band (23 GHz) ESR-cavities [23] and for helix ESR-resonators [24, 25] are reported.

Since this article focuses on liquid samples, only details of pressure-dependent papers will be considered here. A short overview on different investigations is given.

A detailed description for a high-pressure ESR-cell up to 900 MPa using quartz glass capillaries is reported by Yamamoto et al. [26]. An apparatus for studying ESR of fluids in a flow cell under high pressure and high temperature is described by Livingston and Zeldes [27]. Measurements up to 12.4 MPa and 500oC are used to study polymerization and catalysis [28]. Even a combination of high-pressure and rapid mixing stopped-flow techniques is realized [29]. A simple system for measurements in water up to 60 MPa is described by Cannistraro [30].

<sup>1</sup> The only commercial high-pressure ESR-cell was offered by: High Research Center, Unipress Equipment Division, Warszawa, Poland.

Important Russian contributions to high pressure ESR-cells for solid state investigations appeared in the early 1970s [31, 32].

An early contribution published by Maki et al. describes the temperature- and pressure dependence on the spin exchange kinetics and the change in ESR-line widths of the di-tertbutyl nitroxide (DTBN) radical in various solvents [33, 34]. Also the nitrogen ESR-hyperfine splitting, aN, of DTBN was measured in solution as a function of the pressure [35]. Freed et al. looked at the pressure dependence of ordering and spin relaxation in liquid crystals [36].

Solvated electrons and their reaction behavior under high pressure is published by Schindewolf et al. together with experimental details of the high-pressure cell [37]. The line width of vanadyl acetylacetonate was investigated in a number of non-hydrogen bonded solvent together with its anisotropic interactions [38]. Biochemical spin label studies in solution up to 300 MPa are reported together with a detailed cell construction [39]. Well studied kinetic phenomena like the cation migration within p-quinone radical anions have been studied up to 63.7 MPa. Activation volumes for different cations are reported [40]. Even CIDEP-experiments were performed to get information on the pressure dependence of the spin-lattice relaxation time [41].

Several papers deal with investigations on various nitroxides. The pressure of the spin exchange rate constants of 2,2,6,6-tetramethyl-4-oxo-1-piperidinoxyl (4-oxo-TEMPO) was measured in different solvents, both unpoar and polar. Up to 58.8 MPa the rate constants vary between 109 – 1010 M-1s-1 [42]. Experimental and calculated activation volumes are compared for these reactions [43]. These investigations were extended to DTBN and 2,2,6,6 tetramethyl-1-piperidinoxyl radical (TEMPO) [44] and finally to 4-Hydroxy-TEMPO, 4- Amino-TEMPO, Carboxy-PROXYL, Carbamoyl-PROXYL, 5-DOXYL and 10-DOXYL. From the rate constants and the activation volumes obtained the authors were able to calculate the corresponding exchange integral J for each reaction [45]. The pressure dependence of the inclusion equilibrium of diphenylmethyl t-butyl nitroxide and DTBN with β- and γcyclodextrins were studied in detail [46,47]. The effect of pressure and the solvent dependence of the intramolecular spin-exchange of biradicals with two nitroxide fragments linked by a long flexible chain were obtained from ESR.-lifetimes studies of the radical fragments inside and outside the cage. The nearly cyclic conformation in the cage is reported as the favorable one in solution [48]. A short review in Japanese appeared on the spin exchange kinetics of nitroxides [49]. Applications of the spin-label technique at high pressures is reported up to the high pressure of 700 MPa [50]. Recently Hubbell et al. [51] published the design of a high-pressure cell using polytetrafluoroethylene (PTFE) coating fused silica capillary tubes up to 400 MPa. Bundles of these capillaries are placed inside an ESR-cavity. This cell is adapted from an NMR-cell designed by Yonker et al. [52].

Another recent high pressure setup was used to investigate electron self-exchange reactions [53]. The experimental arrangement used by the authors is illustrated in figure 2. A handdriven high-pressure pump is connected with the medium exchanger, separating sample solution and pressure medium, using ethylene glycol as pressure medium. The sample-side of the system can evacuated for sample changing and cleaning.

ESR Spectroscopy of Nitroxides: Kinetics and Dynamics of Exchange Reactions 31

**Figure 2.** Diagram of a typical arrangement for experiments at elevated pressures [53].

## **2. Electron self-exchange reactions**

The ESR line broadening method was first used by Ward and Weissman [54] in 1957 when they investigated the self-exchange reaction between naphthalene and its radical anion. In the following decade, Weissman continued his work, adding information on further systems as well as refining the theoretical treatment of the experiment [55]. Other important contributors were Miller and Adams [56], who investigated several systems using electrochemically generated radicals. These authors were among the first to use this technique to make a comparison between homogeneous and heterogeneous electron transfer according to the Marcus Theory [57]. Since then, numerous other authors have made their contributions, covering many aspects of the self-exchange reaction, including temperature dependencies and effects from counter ions and solvent.

### **2.1. Theory**

30 Nitroxides – Theory, Experiment and Applications

appeared in the early 1970s [31, 32].

the spin-lattice relaxation time [41].

Important Russian contributions to high pressure ESR-cells for solid state investigations

An early contribution published by Maki et al. describes the temperature- and pressure dependence on the spin exchange kinetics and the change in ESR-line widths of the di-tertbutyl nitroxide (DTBN) radical in various solvents [33, 34]. Also the nitrogen ESR-hyperfine splitting, aN, of DTBN was measured in solution as a function of the pressure [35]. Freed et al. looked at the pressure dependence of ordering and spin relaxation in liquid crystals [36]. Solvated electrons and their reaction behavior under high pressure is published by Schindewolf et al. together with experimental details of the high-pressure cell [37]. The line width of vanadyl acetylacetonate was investigated in a number of non-hydrogen bonded solvent together with its anisotropic interactions [38]. Biochemical spin label studies in solution up to 300 MPa are reported together with a detailed cell construction [39]. Well studied kinetic phenomena like the cation migration within p-quinone radical anions have been studied up to 63.7 MPa. Activation volumes for different cations are reported [40]. Even CIDEP-experiments were performed to get information on the pressure dependence of

Several papers deal with investigations on various nitroxides. The pressure of the spin exchange rate constants of 2,2,6,6-tetramethyl-4-oxo-1-piperidinoxyl (4-oxo-TEMPO) was measured in different solvents, both unpoar and polar. Up to 58.8 MPa the rate constants vary between 109 – 1010 M-1s-1 [42]. Experimental and calculated activation volumes are compared for these reactions [43]. These investigations were extended to DTBN and 2,2,6,6 tetramethyl-1-piperidinoxyl radical (TEMPO) [44] and finally to 4-Hydroxy-TEMPO, 4- Amino-TEMPO, Carboxy-PROXYL, Carbamoyl-PROXYL, 5-DOXYL and 10-DOXYL. From the rate constants and the activation volumes obtained the authors were able to calculate the corresponding exchange integral J for each reaction [45]. The pressure dependence of the inclusion equilibrium of diphenylmethyl t-butyl nitroxide and DTBN with β- and γcyclodextrins were studied in detail [46,47]. The effect of pressure and the solvent dependence of the intramolecular spin-exchange of biradicals with two nitroxide fragments linked by a long flexible chain were obtained from ESR.-lifetimes studies of the radical fragments inside and outside the cage. The nearly cyclic conformation in the cage is reported as the favorable one in solution [48]. A short review in Japanese appeared on the spin exchange kinetics of nitroxides [49]. Applications of the spin-label technique at high pressures is reported up to the high pressure of 700 MPa [50]. Recently Hubbell et al. [51] published the design of a high-pressure cell using polytetrafluoroethylene (PTFE) coating fused silica capillary tubes up to 400 MPa. Bundles of these capillaries are placed inside an

ESR-cavity. This cell is adapted from an NMR-cell designed by Yonker et al. [52].

of the system can evacuated for sample changing and cleaning.

Another recent high pressure setup was used to investigate electron self-exchange reactions [53]. The experimental arrangement used by the authors is illustrated in figure 2. A handdriven high-pressure pump is connected with the medium exchanger, separating sample solution and pressure medium, using ethylene glycol as pressure medium. The sample-side The theoretical aspects of electron transfer in solution originate in the 1950s with the pioneering work of Marcus [58-60], who offered a description of the energetics of both the self-exchange and the general reaction. Since then the theory has greatly evolved [61-63], as shall be discussed in the theoretical part of this chapter, where especially solvent (dynamic) effects are of interest.

In general, self-exchange denotes the one-electron transfer between the partners of a redox couple, which results in no net chemical reaction. The mechanism is shown in figure 3, using a radical anion, A- , and its neutral parent compound, A.

**Figure 3.** The reaction mechanism of an electron transfer reaction.

The bimolecular electron transfer rate constant, ket = KAkex, is expressed as

$$k\_{et} = K\_A \kappa\_{el} \nu\_n \exp\left(\frac{-\Delta G^\*}{RT}\right) \tag{10}$$

Here, KA = 4πNAσ<sup>2</sup> δσ is the formation constant of the precursor complex, which in turn is expressed by Avogadro's number, NA, the reaction distance σ and the reaction zone thickness δσ. Further in equation (10), κel is the electronic transmission coefficient, υn the nuclear collision frequency and ΔG\* energy of activation according to Marcus.

Figure 4 depicts the energy surfaces of precursor and successor complexes, simplified in a two-dimensional representation of energy as a parabolic function of reaction coordinate. From this it is possible to express the activation energy as G\*=λ/4-VPS, using the reorganization energy, λ, and the resonance splitting energy, VPS.

Strictly speaking, the Marcus equation only applies to the precursor and successor complexes. In most cases, however, no information is available about these and therefore it is desirable to express the Marcus equation in terms of the reactants, A and A- . In the case where both reactants have nonzero charges, zA and zD, a work term, W, accounting for the Coulombic work needed to bring the reactants together must be included,

$$\mathcal{W} = \frac{\mathbb{Z}\_A \mathbb{Z}\_D \boldsymbol{\varepsilon}\_0^2}{4\pi \varepsilon\_0 \varepsilon\_S \sigma} \tag{11}$$

with e0 representing the electronic charge, εS the dielectric constant of the medium, σ the reaction distance.

**Figure 4.** Energy diagram of the self-exchange reaction

32 Nitroxides – Theory, Experiment and Applications

**Figure 3.** The reaction mechanism of an electron transfer reaction.

The bimolecular electron transfer rate constant, ket = KAkex, is expressed as

exp *et A el n <sup>G</sup> k K*

Here, KA = 4πNAσ<sup>2</sup> δσ is the formation constant of the precursor complex, which in turn is expressed by Avogadro's number, NA, the reaction distance σ and the reaction zone thickness δσ. Further in equation (10), κel is the electronic transmission coefficient, υn the

Figure 4 depicts the energy surfaces of precursor and successor complexes, simplified in a two-dimensional representation of energy as a parabolic function of reaction coordinate. From this it is possible to express the activation energy as G\*=λ/4-VPS, using the

Strictly speaking, the Marcus equation only applies to the precursor and successor complexes. In most cases, however, no information is available about these and therefore it

where both reactants have nonzero charges, zA and zD, a work term, W, accounting for the

<sup>0</sup> 4 *A D S*

with e0 representing the electronic charge, εS the dielectric constant of the medium, σ the

*zze <sup>W</sup>* 

2 0

(11)

 

nuclear collision frequency and ΔG\* energy of activation according to Marcus.

is desirable to express the Marcus equation in terms of the reactants, A and A-

Coulombic work needed to bring the reactants together must be included,

reorganization energy, λ, and the resonance splitting energy, VPS.

reaction distance.

\*

(10)

. In the case

*RT*

  Resulting, the expression of the Marcus equation takes the following form:

$$
\Delta \mathbf{G}^\* = \mathbf{W} + \frac{\lambda}{4} - V\_{\text{PS}} \tag{12}
$$

The reorganization energy, λ, is an energetic measure of the changes the reactants and their surroundings undergo as the reaction proceeds.

Commonly, the reorganization energy is split into two contributions, the inner and outer reorganization energies, λi and λo , respectively.

$$
\mathcal{X} = \mathcal{X}\_{\sharp} + \mathcal{X}\_{o} \tag{13}
$$

λi, takes into account the changes in bond lengths and/or angles in the reactants undergoing electron transfer. In the classic high temperature limit, where all vibrational frequencies fulfil νn << kBT/h (not confusing νn with the collision frequency), λi is given by:

$$\lambda\_{\rm i}^{\rm co} = \sum\_{\rm j} \frac{\mathbf{f}\_{\rm k}^{\rm j} \cdot \mathbf{f}\_{\rm l}^{\rm j}}{\mathbf{f}\_{\rm k}^{\rm j} + \mathbf{f}\_{\rm l}^{\rm j}} \Big(\Delta \mathbf{q}\_{\rm j}\big)^2 \tag{14}$$

Here, fj R,P is the force constant of bond j, either in the reactants, R, or the products P, and Δqj expresses the change in length of bond j. Unfortunately this approach has certain drawbacks when organic redox couples are considered, as it becomes increasingly difficult to find all important vibrations contributing to the reaction coordinate. The situation worsens further as the high temperature approximation is not always valid, meaning that corrections for quantum-mechanical tunnelling should be included. This was pointed out by Holstein [64], and used by several other authors like Bixon and Jortner [61, 65], and Sutin [66] yielding an equation of the following type:

$$\mathcal{A}\_i(T) = \mathcal{X}\_i^\circ \frac{4k\_B T}{h\overline{\nu}} \tanh\left(\frac{h\overline{\nu}}{4k\_B T}\right) \tag{15}$$

where =5∙1013 s-1 is used as the mean vibration frequency for organic redox couples.

A method of calculating *<sup>i</sup>* , without explicitly knowing the changes in bond-lengths and force constants, has been published by Nelsen *et al.* [67,68], needing the energies of four distinct points on the energy surfaces of products and reactants. By calculating the energies of the reactants in their most stable geometrical configuration, as well as in the most stable configuration of the products, together with analogous calculations for the products, *<sup>i</sup>* may be obtained.

The outer reorganization energy, λo, which is sometimes also referred to as the solvent reorganization energy, describes the change in polarization of the solvent molecule surrounding the activated complex. Marcus showed that the outer reorganization energy may be expressed as in equation (16), using a simple dielectric continuum model and assuming that the reactants are spherical.

$$
\lambda\_o = \frac{\mathbf{e}\_0^2 \mathbf{N}\_A}{4\pi\varepsilon\_0} g(r\_\prime\sigma) \cdot \chi \tag{16}
$$

Here, apart from the permittivity of vacuum, ε0, new parameters are the function g(r,σ) which depends on the geometry of the transition state, and γ, the so-called Pekar factor, which depends on the dielectric properties of the solvent.

In its simplest form, assuming rA=rA-=r, using spherical models of the reactants, g(r, σ) is given by:

$$\log\left(r,\sigma\right) = \frac{1}{\mathbf{r}} - \frac{1}{\sigma} \tag{17}$$

Often the reaction distance, σ, is assumed to be that of close contact, here being twice the radius of the reactants: σ=2r. For organic molecules, this approach is often too simple, as the geometry can no longer be assumed to be spherical. If, in stead, an ellipsoidal model is used, r is replaced by the mean elliptic radius, : *r*

$$\overline{r} \approx \frac{\left(a^2 - c^2\right)^{\chi\_2^{\chi\_1}}}{F\left(\varphi, a\right)}\tag{18}$$

with the elliptic semi axes fulfilling a > b > c and F(φ,α) being elliptic integrals of the first kind, using <sup>1</sup> 2 2 2 arcsin *ac a* / and <sup>1</sup> <sup>2</sup> 22 22 arcsin / *ab ac* [69]. A good approximation has been found to be: <sup>1</sup> <sup>3</sup> *r abc* ( ) .

Another approach has been suggested by Compton and co-workers [70, 71] who, instead of calculated radii, use experimentally determined hydrodynamic radii.

As intermolecular orientation becomes more difficult to describe, the reaction distance may also be modified using the ellipsoid model [72, 73], replacing σ, with the apparent reaction distance, σ':

$$\frac{1}{\sigma'} = \frac{1}{\sigma} \left[ 1 + \frac{2c^2 - a^2 - b^2}{3\sigma^2} + \frac{4bc}{\sigma^3} + \frac{4\left(8c^4 + 3\left(a^4 + b^4\right) - 8c^2\left(a^2 + b^2\right) + 2a^2b^2\right)}{15\sigma^4} \right] \tag{19}$$

The second contribution to the reorganization energy, the solvent dependent Pekar factor, γ, in turn, is given by

$$\gamma = \frac{1}{\varepsilon\_{\oplus}} - \frac{1}{\varepsilon\_{S}} \tag{20}$$

where εS and ε∞ are the static and optical dielectric constants, respectively. Unfortunately, reliable experimental values of ε∞, can be very hard to come by, so it has become common practice to use the refractive index, nD, in an approximation: <sup>2</sup> *D n* .

Apart from the obvious solvent dependence of the reorganization energy, the rate constant of an electron transfer reaction may also be influenced by the solvent in terms of the nuclear frequency factor, νn.

Originally, Marcus expressed νn using a simple gas-phase collision model [74], yielding equation (21),

$$\nu\_n = \sigma^2 N\_A \left(\frac{16\pi RT}{M}\right)^{\frac{1}{2}} \tag{21}$$

with the only new parameter being the molar mass, M.

34 Nitroxides – Theory, Experiment and Applications

assuming that the reactants are spherical.

r is replaced by the mean elliptic radius, :

kind, using <sup>1</sup>

approximation has been found to be: <sup>1</sup>

A method of calculating *<sup>i</sup>*

may be obtained.

given by:

=5∙1013 s-1 is used as the mean vibration frequency for organic redox couples.

force constants, has been published by Nelsen *et al.* [67,68], needing the energies of four distinct points on the energy surfaces of products and reactants. By calculating the energies of the reactants in their most stable geometrical configuration, as well as in the most stable configuration of the products, together with analogous calculations for the products, *<sup>i</sup>*

The outer reorganization energy, λo, which is sometimes also referred to as the solvent reorganization energy, describes the change in polarization of the solvent molecule surrounding the activated complex. Marcus showed that the outer reorganization energy may be expressed as in equation (16), using a simple dielectric continuum model and

> 2 0 A

0 e N <sup>λ</sup> , <sup>γ</sup> <sup>4</sup>πε *g r*

Here, apart from the permittivity of vacuum, ε0, new parameters are the function g(r,σ) which depends on the geometry of the transition state, and γ, the so-called Pekar factor,

In its simplest form, assuming rA=rA-=r, using spherical models of the reactants, g(r, σ) is

1 1 , <sup>r</sup>

Often the reaction distance, σ, is assumed to be that of close contact, here being twice the radius of the reactants: σ=2r. For organic molecules, this approach is often too simple, as the geometry can no longer be assumed to be spherical. If, in stead, an ellipsoidal model is used,

> 

*F* 

with the elliptic semi axes fulfilling a > b > c and F(φ,α) being elliptic integrals of the first

Another approach has been suggested by Compton and co-workers [70, 71] who, instead of

<sup>3</sup> *r abc* ( ) .

, *a c* 1 2 2 2

and <sup>1</sup>

*g r* 

*r*

*r*

o

which depends on the dielectric properties of the solvent.

2 2 2

calculated radii, use experimentally determined hydrodynamic radii.

arcsin *ac a* /

(16)

(17)

(18)

<sup>2</sup> 22 22

arcsin / *ab ac* [69]. A good

, without explicitly knowing the changes in bond-lengths and

where 

> The solvent dependence is introduced by taking the formation of the precursor complex into account, expressing the total frequency factor as combination of those of the solvent, νo and the reactants, νi, weighted by the respective reorganization energies.

$$\nu\_n = \left(\frac{\nu\_i^2 \mathcal{X}\_i + \nu\_o^2 \mathcal{X}\_o}{\mathcal{X}}\right)^{\gtrless} \tag{22}$$

Equation (22) is valid only for rapid dielectric relaxation in the solvent. For slow dielectric relaxation, the longitudinal relaxation time of the solvent, τL, must be included as the solvent relaxation now happens on the same time scale as the reaction. This observation was first made by Kramers [75], who spoke of a solvent dynamic effect or solvent friction, and the theory was later expanded by authors like Zusman [76, 77], Jortner and Rips [61, 78, 79], Marcus and Sumi [63] as well as several others.

The longitudinal relaxation time may be determined from the usual dielectric parameters of the solvent [80],

$$
\tau\_{\rm L} = \frac{\mathcal{E}\_{\rm co}}{\mathcal{E}\_{\rm S}} \tau\_{\rm D} \tag{23}
$$

with, τD being the Debye relaxation time, which itself may be expressed using the molar volume, VM, and the viscosity, η, of the solvent:

$$
\pi\_D = \frac{3V\_M \eta}{RT} \tag{24}
$$

Assuming that λi << λo, the nuclear frequency factor then becomes:

$$\nu\_{\pi} = \frac{1}{\tau\_L} \left(\frac{\mathcal{A}\_o}{4\pi RT}\right)^{\frac{1}{2}} \tag{25}$$

So far it has been supposed that the electron transfer reaction was adiabatic, assuming the transmission factor, κel, to be unity. This applies to all reactions studied in this work, but for completeness, the expression valid in the case of diabatic electron transfer [69, 76, 81, 82] is given here:

$$
\kappa\_{el} \nu\_n = \frac{2\pi V^2}{\hbar \mathcal{N}\_A \left(4\pi \mathcal{k}\_o \mathcal{R} T\right)^{\frac{1}{2}}} \tag{26}
$$

It is now possible to discern the nature of the electron transfer by the use of simple mathematical treatment, calculating the experimental value, (κelνn)obs as suggested by Weaver [83, 84],

$$\left(\kappa\_{el}\nu\_{n}\right)\_{\text{obs}} = k\_{ct,\text{obs}}K\_A^{-1}\exp\left(\frac{\Delta G\_{\text{calc}}^{\bullet}}{RT}\right) \tag{27}$$

where ket,obs is the experimentally determined rate constant and *Gcalc* the activation energy calculated using equation (12). Double logarithmic plots determine the type of electron transfer observed. In the general Marcus case, a constant value of ln (κelνn)obs is expected, whereas the diabatic and adiabatic cases both produce straight lines with slopes of unity when plotted against ln (γ-1/2) or ln (γ1/2τL-1), respectively.

It is well-known that, kex depends on temperature and pressure. The temperature dependence is seen directly in equation (28), but also in the dependencies of the activation energy, via γ, and that of the nuclear frequency factor.

$$k\_{ex} \left( p\_\prime T \right) = \kappa\_{el} \nu\_n \left( p\_\prime T \right) \exp \left( \frac{-\Delta G^\* \left( p\_\prime T \right)}{RT} \right) \tag{28}$$

The temperature dependence of νn is often neglected in the general Marcus case and that exhibiting diabatic solvent dynamics. However, when dealing with adiabatic solvent dynamics, the temperature dependence must be included, due to the fact that τL may vary strongly with temperature, according on the solvent in question. In some cases even, this variation may be of more than an order of magnitude, as in the case of propylene carbonate, where τL is 42.6 ps at 228 K and 2.6 ps at 308 K [85].

The pressure dependence is usually expressed using the activation volume, Δ‡V:

36 Nitroxides – Theory, Experiment and Applications

given here:

Weaver [83, 84],

volume, VM, and the viscosity, η, of the solvent:

L D *S* 

with, τD being the Debye relaxation time, which itself may be expressed using the molar

3 *<sup>M</sup>*

<sup>2</sup> 1 4 *o*

*<sup>L</sup> RT* 

 

So far it has been supposed that the electron transfer reaction was adiabatic, assuming the transmission factor, κel, to be unity. This applies to all reactions studied in this work, but for completeness, the expression valid in the case of diabatic electron transfer [69, 76, 81, 82] is

1

<sup>1</sup>

<sup>2</sup> 2 4

*V N RT* 

*<sup>G</sup> k K*

*A o*

It is now possible to discern the nature of the electron transfer by the use of simple mathematical treatment, calculating the experimental value, (κelνn)obs as suggested by

, exp *obs*

calculated using equation (12). Double logarithmic plots determine the type of electron transfer observed. In the general Marcus case, a constant value of ln (κelνn)obs is expected, whereas the diabatic and adiabatic cases both produce straight lines with slopes of unity

It is well-known that, kex depends on temperature and pressure. The temperature dependence is seen directly in equation (28), but also in the dependencies of the activation

> \* , , , exp *ex el n G pT k pT pT RT*

The temperature dependence of νn is often neglected in the general Marcus case and that exhibiting diabatic solvent dynamics. However, when dealing with adiabatic solvent

 

  2

*calc*

*RT*

 

*V RT* 

(23)

(24)

(25)

(26)

(27)

the activation energy

(28)

*D*

 

*n*

*el n*

 

<sup>1</sup>

 

when plotted against ln (γ-1/2) or ln (γ1/2τL-1), respectively.

energy, via γ, and that of the nuclear frequency factor.

where ket,obs is the experimentally determined rate constant and *Gcalc*

*el n et obs A*

 

Assuming that λi << λo, the nuclear frequency factor then becomes:

$$
\Delta^{\ddagger}V = -RT\left(\frac{\partial \ln k\_{ex}}{\partial P}\right)\_T \tag{29}
$$

In terms of transition state theory, the activation volume may be expressed using the volume of the transition state, ‡V and the molar volumes of the reactants, VA and VA-, cf. equation (30),

$$
\Delta^{\ddagger} V =^{\ddagger} V - \left( V\_A + V\_{A^{\bullet-}} \right) \tag{30}
$$

and thus represents the changes in volume involved in the electron transfer process. This is also seen in figure 5, where the volume profile of a reaction with a positive volume of activation is shown.

**Figure 5.** Volume profile of an electron transfer reaction

Generally, one may also define a reaction volume, ΔrV, as the difference in the volumes of products and reactants, but for self-exchange reactions, this becomes zero.

As in the case of ΔG\*, one may seek to describe the activation volume by separating the various contributions. This has been discussed in great detail by Swaddle [10, 86-88], who

derived expressions of the contributions from inner and outer reorganization, Δ‡Vi and Δ‡Vo, Coulombic work, Δ‡VCoulomb, Debye-Hückel effects, Δ‡VDH, solvent dynamics, Δ‡VSD, and finally a contribution from the non-Coulombic part of the precursor complex formation constant, Δ‡VPREC. Summing up the different parts, the volume of activation of the selfexchange reaction is calculated as expressed in equation (31).

$$
\Delta^{\ddagger} V\_{\text{calc}} = \Delta^{\ddagger} V\_i + \Delta^{\ddagger} V\_o + \Delta^{\ddagger} V\_{\text{Coulomb}} + \Delta^{\ddagger} V\_{DH} + \Delta^{\ddagger} V\_{SD} + \Delta^{\ddagger} V\_{PREC} \tag{31}
$$

Each of these contributions is treated separately by differentiation of the respective energetic expressions.

The contribution to the activation volume, arising from inner reorganization is usually neglected, as the increase in volume of the acceptor almost cancels the decrease in volume of the donor. Stranks [89] showed that in most cases, a value of 0.6 cm3 mol-1 may be taken for Δ‡Vi, which is usually close to the experimental uncertainty of Δ‡V. The contribution from Δ‡VPREC is also expected to be small since both σ and δσ are practically pressure independent [88].

Taking the Marcus expression for the outer reorganization energy, λo, and assuming that only the Pekar factor, γ, is pressure dependent, and additionally taking into account compression of the solvent, Δ‡Vo, must be given as shown in equation (32), remembering that λo only contributes with a quarter of its value to the activation energy.

$$
\Delta^{\ddagger} \mathbf{V}\_{o} = \frac{1}{4} \left( \frac{\partial \mathcal{X}\_{o}}{\partial \mathcal{P}} \right)\_{T} = \frac{\mathbf{e}\_{0}^{2} \mathbf{N}\_{A}}{16 \pi \varepsilon\_{0}} \left[ \left( \frac{1}{r} - \frac{1}{\sigma} \right) \left( \frac{\partial \mathcal{Y}}{\partial \mathcal{P}} \right)\_{T} - \gamma \frac{\mathcal{P}}{3\sigma} \right] \tag{32}
$$

The Pekar factor is approximated as 2 1 *S n* using the refractive index, n, and static dielectric constant, εS, of the solvent and the isothermal compressibility of the solvent, β, is defined via its density, ρ, as <sup>1</sup> *<sup>P</sup> <sup>T</sup>* .

By simple differentiation of the work term given in equation (11), the Coulombic contribution to the activation volume is found.

$$
\Delta\Lambda^{\ddagger}V\_{Coulomb} = \frac{z\_A z\_D e\_0^2 N\_A}{4\pi\varepsilon\_0\sigma} \left(\frac{\partial\varepsilon\_s^{-1}}{\partial P}\right)\_T \tag{33}
$$

The expression for Δ‡VDH is derived from the extended Debye-Hückel theory, and takes the following, rather cumbersome, form:

$$
\Delta^{\ddagger} V\_{DH} = \frac{RT z\_A z\_D C I^{\not\not\simeq}}{\left(1 + BaI^{\not\simeq}\right)^2} \left[ \left(\frac{\partial \ln \varepsilon}{\partial P}\right)\_T \left(3 + 2BaI^{\not\simeq}\right) - \beta \right] \tag{34}
$$

with a, B; and C being the Debye-Hückel constants and I the ionic strength.

These two contributions are of opposite sign and usually, for values of I between 0.1 and 0.5, tend to cancel each other quite conveniently [86, 87]. In the case where neutral molecules are involved, as it is in the experimental part of this work, both Δ‡VCoulomb and Δ‡VDH are zero.

Recalling that the solvent dynamic effect is manifested in the pre-exponential term of the Arrhenius equation, the following definition offers itself readily:

$$
\Delta^{\ddagger} V\_{SD} = RT \left( \frac{\partial \ln \left( \kappa\_{el} \nu\_n \right)}{\partial P} \right)\_T \tag{35}
$$

This may then be further elaborated in the cases of diabatic and adiabatic solvent dynamics, as shown shortly. Note that even in the general Marcus case, with κelνn depending on σ, a slight pressure dependence is expected, but as argued earlier, this may be neglected. However, due to the logarithmic form of equation (35), it is now considerably simpler to describe the influence of σ, as all constants in equation (21) are conveniently eliminated. Assuming that σ compresses approximately as the solvent, i.e. <sup>1</sup> <sup>3</sup> , the contribution to the activation volume should be obtainable with relative ease.

In the diabatic case given in equation (36), Δ‡VSD becomes:

38 Nitroxides – Theory, Experiment and Applications

expressions.

[88].

exchange reaction is calculated as expressed in equation (31).

derived expressions of the contributions from inner and outer reorganization, Δ‡Vi and Δ‡Vo, Coulombic work, Δ‡VCoulomb, Debye-Hückel effects, Δ‡VDH, solvent dynamics, Δ‡VSD, and finally a contribution from the non-Coulombic part of the precursor complex formation constant, Δ‡VPREC. Summing up the different parts, the volume of activation of the self-

‡ ‡‡ ‡ ‡ ‡ ‡

Each of these contributions is treated separately by differentiation of the respective energetic

The contribution to the activation volume, arising from inner reorganization is usually neglected, as the increase in volume of the acceptor almost cancels the decrease in volume of the donor. Stranks [89] showed that in most cases, a value of 0.6 cm3 mol-1 may be taken for Δ‡Vi, which is usually close to the experimental uncertainty of Δ‡V. The contribution from Δ‡VPREC is also expected to be small since both σ and δσ are practically pressure independent

Taking the Marcus expression for the outer reorganization energy, λo, and assuming that only the Pekar factor, γ, is pressure dependent, and additionally taking into account compression of the solvent, Δ‡Vo, must be given as shown in equation (32), remembering

0

dielectric constant, εS, of the solvent and the isothermal compressibility of the solvent, β, is

By simple differentiation of the work term given in equation (11), the Coulombic

<sup>0</sup> 4 *AD A S*

The expression for Δ‡VDH is derived from the extended Debye-Hückel theory, and takes the

<sup>1</sup>

<sup>2</sup> <sup>1</sup>

*T*

ln 3 2

 

4 16πε 3

*S*

1 2

*T*

2

*P* 

(32)

 

using the refractive index, n, and static

(33)

(34)

*<sup>T</sup> <sup>T</sup> P rP*

that λo only contributes with a quarter of its value to the activation energy.

<sup>1</sup> e N 1 1 <sup>V</sup>

*n*

*<sup>P</sup> <sup>T</sup>* 

‡ 0

1 2

*A D*

with a, B; and C being the Debye-Hückel constants and I the ionic strength.

1

2

*RTz z CI <sup>V</sup> BaI <sup>P</sup> BaI*

*z z eN <sup>V</sup>*

*Coulomb*

‡ 0 A

*o*

o

The Pekar factor is approximated as 2 1

contribution to the activation volume is found.

‡

*DH*

 .

defined via its density, ρ, as <sup>1</sup>

following, rather cumbersome, form:

2

*V VVV V V V calc i o Coulomb DH SD PREC* (31)

$$
\Delta^{\ddagger}V\_{SD} = RT \left( \frac{\partial}{\partial P} \left( \ln \frac{2\pi V^2}{\hbar \mathcal{N}\_A \left( 4\pi \mathcal{k}\_o \mathcal{R} T \right)^{\ddagger}} \right) \right)\_T = -\frac{RT}{2} \left( \frac{\partial \ln \mathcal{Y}}{\partial P} \right)\_T \tag{36}
$$

with all pressure independent quantities eliminated as above. In cases of strong diabaticity, where a noticeable dependence of kex on σ is expected, one must include the pressure dependence of VPS, which follows the principle discussed in the Marcus case, and shall not be further elaborated here.

For electron transfer exhibiting adiabatic solvent dynamic behaviour, the situation becomes slightly more complicated, as now also the longitudinal relaxation time contributes to Δ‡VSD.

$$
\Delta^\ddagger V\_{SD} = RT \left[ \frac{\partial}{\partial P} \left( \ln \frac{1}{\tau\_L} \left( \frac{\lambda\_o}{4 \pi R T} \right)^\circ \right) \right]\_T = RT \left[ \frac{1}{2} \left( \frac{\partial \ln \gamma}{\partial P} \right)\_T - \left( \frac{\partial \ln \tau\_L}{\partial P} \right)\_T \right] \tag{37}
$$

Returning to electron self-exchange reactions, where the activation volume consists of the five parts, which have just been introduced, by far the greatest contribution is expected from Δ‡Vo. As already mentioned, Δ‡VCoulomb and Δ‡VDH tend to cancel each other, or even vanish when an uncharged molecule is reacting, and Δ‡Vi is generally negligible. Consequently, one may write:

$$
\Delta^\ddagger V = \Delta^\ddagger V\_o + \Delta^\ddagger V\_{SD} \tag{38}
$$

In other words, the activation volume may be interpreted as owing to a term described by the energy barrier crossed during the reaction, Δ‡Vo, and a term linked to the actual crossing of the barrier, Δ‡VSD. Even here, the contribution from the pre-exponential factor is unimportant in the general Marcus case and in that of diabatic solvent dynamics, leaving only the adiabatic case with an appreciable contribution.

Typical values of Δ‡V are found to be in the range of -30 to +30 cm3 mol-1 [10], corresponding to a retardation or acceleration of the reaction of about a factor of 4 at 100 MPa as compared to ambient pressure (0,1 MPa).

### **2.2. Experimental**

As mentioned above, much work has been done using ESR line broadening experiments to study self-exchange reactions; however, only in surprisingly few cases nitroxides have been involved. Nitroxides are readily oxidized [90-92] to form the diamagnetic compound needed in the self exchange with the neutral radical. In most cases, the ESR spectra remain within the slow exchange region due to the relatively large nitrogen hyperfine splitting constant. The reported results have dealt with the self-exchange of various nitroxide couples in acetonitrile [93] as well as of that of TEMPO with its oxidized counterpart in several different solvents [94]. For the latter couple, the temperature dependence of the exchange reaction has also been investigated [95].

Some typical spectra are shown in figure 6 together with the accompanying plot to determine kobs. Subsequently, the electron transfer rate constant, ket, can be obtained applying a correction for diffusion as shown in equation (39).

$$\frac{1}{k\_{\rm obs}} = \frac{1}{k\_{et}} + \frac{2}{k\_d} \tag{39}$$

The factor of two seen on the last term of equation (39), as opposed to that of one which is seen in more general reactions, is due to the nature of the self-exchange reaction.

**Figure 6.** Left: Central lines of the ESR spectra of TEMPO in acetonitrile. Concentrations of TEMPO+ Clˉ were 0 mM (bottom) 8 mM (middle) and 18 mM (top). Right: Concentration dependence of the change in peak-to-peak line width in water (squares) and propylene carbonate (circles).

The experimental results of references [94] and [95] serve as a good example of how ESR line broadening experiments can be utilized to study solvent effects on electron transfer.


Table 1 presents the electron transfer rate determined as given above, together with relevant solvent properties and the quantities needed to investigate the presence of a solvent effect.

Numbers in brackets correspond to the relevant reference.

40 Nitroxides – Theory, Experiment and Applications

to ambient pressure (0,1 MPa).

reaction has also been investigated [95].

applying a correction for diffusion as shown in equation (39).

**2.2. Experimental** 

only the adiabatic case with an appreciable contribution.

of the barrier, Δ‡VSD. Even here, the contribution from the pre-exponential factor is unimportant in the general Marcus case and in that of diabatic solvent dynamics, leaving

Typical values of Δ‡V are found to be in the range of -30 to +30 cm3 mol-1 [10], corresponding to a retardation or acceleration of the reaction of about a factor of 4 at 100 MPa as compared

As mentioned above, much work has been done using ESR line broadening experiments to study self-exchange reactions; however, only in surprisingly few cases nitroxides have been involved. Nitroxides are readily oxidized [90-92] to form the diamagnetic compound needed in the self exchange with the neutral radical. In most cases, the ESR spectra remain within the slow exchange region due to the relatively large nitrogen hyperfine splitting constant. The reported results have dealt with the self-exchange of various nitroxide couples in acetonitrile [93] as well as of that of TEMPO with its oxidized counterpart in several different solvents [94]. For the latter couple, the temperature dependence of the exchange

Some typical spectra are shown in figure 6 together with the accompanying plot to determine kobs. Subsequently, the electron transfer rate constant, ket, can be obtained

> 1 12 *obs et d k kk*

The factor of two seen on the last term of equation (39), as opposed to that of one which is

**Figure 6.** Left: Central lines of the ESR spectra of TEMPO in acetonitrile. Concentrations of TEMPO+ Clˉ were 0 mM (bottom) 8 mM (middle) and 18 mM (top). Right: Concentration dependence of the change

in peak-to-peak line width in water (squares) and propylene carbonate (circles).

seen in more general reactions, is due to the nature of the self-exchange reaction.

(39)

**Table 1.** Self-exchange rate constants TEMPO/TEMPO+ in pure solvents at 293K

The values of \* *Gcalc* listed in table 1 were determined using a λi of 33 kJ mol-1 and λo of 191 ×γ kJ mol-1. Both values were calculated [95] as shown above, and have been confirmed experimentally by other authors [100].

Values of (κelνn)obs were obtained using equation (27), and as it is clearly not solvent independent, the general Marcus type of electron transfer is immediately ruled out and the existence of a solvent dynamic effect is successfully demonstrated. The nature of this is shown in figure 7 where the appropriate double logarithmic plots are given.

From figure 7 it can clearly be seen that the presence of a diabatic solvent effect can be excluded, since a line with a slope of unity would be needed to confirm the corresponding assumption. The line shown in the right hand plot of the figure has been fixed to this slope and thereby shows that an adiabatic solvent effect is likely in the case of self-exchange in the TEMPO/TEMPO+ system.

Since the literature values of the dielectric properties of the solvents are often subject to large uncertainties error bars have been introduced in each diagram to illustrate the range of

values corresponding to errors of 10% in each γ and τL. In the case of the former an error of such magnitude is highly unexpected, but for the latter, 10% might even be considered conservative in some cases.

**Figure 7.** Diagrams assuming diabatic (left) and adiabatic (right) solvent dynamic effects. X-axis error bars correspond to a 10% error on each γ and τL. The line has a fixed slope of one.

In fact, regarding the longitudinal relaxation time needed in the calculations concerning the adiabatic case a few remarks must be made. In many solvents and electrolyte solutions, high precision experiments have indicated the existence of several relaxation pathways in the solvent. Usually it is assumed that only the slowest of the relaxation processes will influence the electron transfer reaction [76, 101] and the τL to be used in investigations of solvent dynamic effects, is thus the largest of the relaxation times:

$$
\tau\_{\rm iL} = \frac{\mathcal{E}\_{i\rm io}}{\mathcal{E}\_{i\rm iS}} \tau\_{\rm iD} \quad i = 1, 2, \dots \tag{40}
$$

As an example, take methanol [98] which has two relaxations with parameters at 293K: εS = ε1S = 32.5, ε1∞ = ε2S = 5.91, ε2∞ = ε∞ = 4.9 and τD1 = 51.49 ps, τD2 = 7.09 ps, leading to τL1 = 9.36 ps and τL2 = 5.88 ps. For comparison, both older and newer works [102-105] report only a single relaxation process for methanol, resulting in a τL which ranges from 4.58 ps to 10.6 ps. The reason that these latter experiments have only detected one relaxation time is, that they have been conducted using a relatively narrow frequency range and thus a long extrapolation for determination of ε∞.

Having established that the solvent dynamic effect is adiabatic, a plot of ln (ketτLγ-1/2) vs. γ allow the calculation of g(r,σ') since,

$$\ln\left(k\_{ct}\tau\_L\gamma^{-\not p\not\!}\right) = const-\frac{\lambda\_o}{4RT} = const-\frac{e\_0^2\,\mathrm{g}\left(r,\sigma'\right)}{16\pi\varepsilon\_0 k\_B T}\gamma\tag{41}$$

From the slope of this plot, while taking r = *r* = 3.11 Å, the reaction distance becomes σ' = 5.0 Å, which is in reasonably good agreement with the value of σ' = 5.4 Å found in the theoretical calculations [94]. The experimental value of σ' corresponds to KA = 0.21 l mol-1.

conservative in some cases.

determination of ε∞.

allow the calculation of g(r,σ') since,

values corresponding to errors of 10% in each γ and τL. In the case of the former an error of such magnitude is highly unexpected, but for the latter, 10% might even be considered

**Figure 7.** Diagrams assuming diabatic (left) and adiabatic (right) solvent dynamic effects. X-axis error

In fact, regarding the longitudinal relaxation time needed in the calculations concerning the adiabatic case a few remarks must be made. In many solvents and electrolyte solutions, high precision experiments have indicated the existence of several relaxation pathways in the solvent. Usually it is assumed that only the slowest of the relaxation processes will influence the electron transfer reaction [76, 101] and the τL to be used in investigations of solvent

> iL iD 1, 2, *<sup>i</sup> iS*

As an example, take methanol [98] which has two relaxations with parameters at 293K: εS = ε1S = 32.5, ε1∞ = ε2S = 5.91, ε2∞ = ε∞ = 4.9 and τD1 = 51.49 ps, τD2 = 7.09 ps, leading to τL1 = 9.36 ps and τL2 = 5.88 ps. For comparison, both older and newer works [102-105] report only a single relaxation process for methanol, resulting in a τL which ranges from 4.58 ps to 10.6 ps. The reason that these latter experiments have only detected one relaxation time is, that they have been conducted using a relatively narrow frequency range and thus a long extrapolation for

Having established that the solvent dynamic effect is adiabatic, a plot of ln (ketτLγ-1/2) vs. γ

<sup>1</sup> <sup>2</sup>

From the slope of this plot, while taking r = *r* = 3.11 Å, the reaction distance becomes σ' = 5.0 Å, which is in reasonably good agreement with the value of σ' = 5.4 Å found in the theoretical calculations [94]. The experimental value of σ' corresponds to KA = 0.21 l mol-1.

*o*

4 16

*RT k T*

, ln

*egr k const const*

 

*i*

(40)

2 0

0

(41)

*B*

 

bars correspond to a 10% error on each γ and τL. The line has a fixed slope of one.

dynamic effects, is thus the largest of the relaxation times:

*et L*

 

**Figure 8.** 'Traditional' diagram for illustration of the adiabatic solvent dynamic effect. The slope of the dotted line corresponds to a reaction distance of 5.0 Å.

The listed rate constants were used to determine the activation energy of the reaction in each of the solvents, which, due to the adiabatic solvent dynamics, was done by plotting ln (ketτLγ-1/2T1/2) vs. T-1 as can be seen in table 2. For comparison, the values obtained from common Arrhenius plots (ln ket vs T-1) are also shown. The table also contains the longitudinal relaxation energies of the solvent as well as the quantity \* *Hcalc* , which represents a theoretical prediction of the activation enthalpy, taking into account the temperature dependency of the reorganization energy [95].

The concerns regarding the physical properties of solvents become even more pronounced when dealing with temperature dependencies. Once again, the longitudinal relaxation time provides the greatest problem as the relaxation energy, HL, varies tremendously in literature. Using the example of methanol again, HL could only be obtained from data using a single relaxation time, whereas for THF no less than four values are reported, ranging from -8.3 kJ mol-1 [106] to 11.1 kJ mol-1 [107].

Returning to the experimental activation energies and enthalpies, it is seen that the relation Ea = H\* + HL is fulfilled within the experimental error, as expected due to the Arrheniuslike temperature dependence of the involved quantities, with any deviations being mainly due to the slight temperature dependence of γ. The results illustrate the importance of knowing whether or not solvent dynamics are present in an electron transfer system before determining the activation parameters. The activation energies from the traditional Arrhenius plots do not correspond to Marcus' G\* due to the involvement of τL. To make matters worse, neither does the energy obtained from the plots of ln (ketτLγ-1/2T1/2) vs. T-1,

since both λi and λo are temperature dependent. For this reason the values in the table have been labeled as enthalpies in order to signify that they correspond to the temperature independent part of G\*. Taking the temperature dependence of G\* into account, it is possible to reach a theoretical estimate of this enthalpy, \* *Hcalc* [95]. As can be seen, for acetonitrile, benzonitrile, tetrahydrofuran and water, good agreement is found between experiment and theory. This is not the case for the three remaining solvents, where large discrepancies are found. Except for propylene carbonate, the main reason for the poor agreement is likely to be the quality of the dielectric data as already discussed.


a) Activation energy determined from a traditional Arrhenius plot (ln ket vs. T-1)

b) Activation energy determined assuming adiabatic solvent dynamics (ln ketτLγ-1/2T1/2 vs. T-1)

c) Longitudinal relaxation energy obtained from the empirical relationship ln ln *<sup>L</sup> L <sup>H</sup> <sup>b</sup> RT* .

d) From ref [95].

Numbers in brackets correspond to the relevant reference.

**Table 2.** Characteristic energies of the of the TEMPO/TEMPO+ self-exchange in pure solvents at 293K

## **3. Spin exchange reactions**

Spin exchange and its effect on the acquired ESR spectra is well understood, experimentally as well as theoretically [108-115]. A large amount work has been done using nitroxide radicals, e.g. by Plachy and Kivelson who studied the spin exchange between di-tertiarybutyl nitroxide radicals in n-pentane and propane [116]. Another interesting work is one of Bales, where the spin exchange between various nitroxide radicals was used to study the structure of micelles as well as chemical reactions taking place within them [117].

Other authors have studied intramolecular spin exchange in nitroxide biradicals [118-122] or the so-called paramagnetic exchange between a nitroxide radical and a paramagnetic line broadening agent [123-126]. Several reports have been given on temperature and pressure dependent spin exchange reactions [33, 42-45].

Due to the wealth of information available on spin exchange of nitroxide radicals, here, focus shall be on the pressure dependence of the spin exchange reaction, as this arguably has received the least attention in the past.

#### **3.1. Theory of pressure dependent spin exchange**

With respect to the pressure dependence of the spin exchange reaction, an approach similar to the one shown for self-exchange can be made. Thus, the activation volume takes the form shown below, containing contributions from diffusion, Coulombic work and Debye-Hückel effects.

$$
\Delta^{\ddagger} V\_{\text{calc}} = \Delta^{\ddagger} V\_{\text{diff}} + \Delta^{\ddagger} V\_{\text{Coulomb}} + \Delta^{\ddagger} V\_{\text{DH}} - RT\beta \tag{42}
$$

Δ‡VColumb and Δ‡VDH can be calculated as shown above and Δ‡Vdiff is obtained analogously to equation (29):

$$
\Delta^{\ddagger} V\_{diff} = -RT \left( \frac{\partial \ln k\_{diff}}{\partial P} \right)\_T \tag{43}
$$

The expression of kdiff varies with the used theoretical description of diffusion, but often a simple Smoluchowski approach, as above, is sufficient. The last term in eq. 5 takes into account the compression of the solvent due to elevated pressures and typically has a value of 1 – 3 cm³/mol.

#### **3.2. Experimental results**

44 Nitroxides – Theory, Experiment and Applications

Solvent

d) From ref [95].

since both λi and λo are temperature dependent. For this reason the values in the table have been labeled as enthalpies in order to signify that they correspond to the temperature independent part of G\*. Taking the temperature dependence of G\* into account, it is possible to reach a theoretical estimate of this enthalpy, \* *Hcalc* [95]. As can be seen, for acetonitrile, benzonitrile, tetrahydrofuran and water, good agreement is found between experiment and theory. This is not the case for the three remaining solvents, where large discrepancies are found. Except for propylene carbonate, the main reason for the poor

Energy / kJ mol-1

Acetonitrile [94] 18.2±0.7 18.1±0.7 1.0 [96] 19 Benzonitrile [94] 23.9±0.7 14.1±0.4 11.0 [85] 15 Deuterium Oxide [94] 15±2 -4.3±0.5 19.0 [97] 25 Methanol [95] 21±2 8.8±0.5 12.8 [102] 31 Propylene Carbonate [94] 13±1 -7±1 20.9 [85] 18 Tetrahydrofuran [94] 6.7±0.8 16.6±0.4 -8.3 [106] 19 Water [94] 6.5±0.2 22.7±0.5 -14.9 [99] 24 Water [95] 6.8±0.1 22.9±0.4 -14.9 [99] 24

**Table 2.** Characteristic energies of the of the TEMPO/TEMPO+ self-exchange in pure solvents at 293K

Spin exchange and its effect on the acquired ESR spectra is well understood, experimentally as well as theoretically [108-115]. A large amount work has been done using nitroxide radicals, e.g. by Plachy and Kivelson who studied the spin exchange between di-tertiarybutyl nitroxide radicals in n-pentane and propane [116]. Another interesting work is one of Bales, where the spin exchange between various nitroxide radicals was used to study the

Other authors have studied intramolecular spin exchange in nitroxide biradicals [118-122] or the so-called paramagnetic exchange between a nitroxide radical and a paramagnetic line broadening agent [123-126]. Several reports have been given on temperature and pressure

structure of micelles as well as chemical reactions taking place within them [117].

Ea a) <sup>Δ</sup>H\* b) HL c) \* *Hcalc* d)

*L*

*<sup>H</sup> <sup>b</sup> RT*

.

agreement is likely to be the quality of the dielectric data as already discussed.

a) Activation energy determined from a traditional Arrhenius plot (ln ket vs. T-1)

Numbers in brackets correspond to the relevant reference.

dependent spin exchange reactions [33, 42-45].

**3. Spin exchange reactions** 

b) Activation energy determined assuming adiabatic solvent dynamics (ln ketτLγ-1/2T1/2 vs. T-1) c) Longitudinal relaxation energy obtained from the empirical relationship ln ln *<sup>L</sup>*

Spin exchange rate constants, kobs = kse, obtained in some pressure dependent studies are shown in table 3 together with the corresponding activation volumes for spin exchange and diffusion.

From the pressure dependent rate constants, volumes of activation were determined for each system according to equation (29).

The differences between the values of Δ‡Vexp and Δ‡Vdiff vary tremendously with the solvent and radical used. Several explanations for this have been offered. For example, it was noted in [43] that for non-polar solvents, the agreement between the volumes is very good, whereas it fails for protic polar solvents. The explanation for this is suggested to lie in the formation of the encounter complex prior to the spin exchange. If the solvent assists in this, e.g. through hydrogen bonds, a noticeable reduction in the experimental activation volume is to be expected.

The theoretical description of kse has been thoroughly dealt with in literature [108, 109]. relating it to the diffusional rate constant, kdiff, by

$$k\_{se} = p f\_S k\_{diff} \tag{44}$$

where p is the probability of spin exchange upon collision of the radicals and fS is a steric factor that takes into account the anisotropy of the spin exchange, i.e., its dependence on the orientation of the colliding particles.


Numbers in brackets correspond to the relevant reference.

**Table 3.** Rate constants and activation volumes of selected spin exchange reactions.

The probability factor can be determined as shown in equation (45), using the exchange integral, J, and the collision time τ1 as well as a factor pmax which depends on the value of the spin and the spin-lattice relaxation time of the radical [108].

$$p = p\_{\max} \frac{J^2 \tau\_1^2}{1 + J^2 \tau\_1^2} \tag{45}$$

The pressure dependence of the probability factor has been shown to differ from one solvent to another [44]. For many solvents, p is practically pressure independent while for others, such as acetone or hexane, it varies greatly with pressure. This is believed to be connected to the strength of the exchange. Systems with a pressure independent p, are believed to exhibit strong exchange, i.e. p1, whereas those showing a pressure dependent p are in the weak exchange region. This illustrates the large discrepancies between Δ‡Vexp and Δ ‡Vdiff in those solvents.


For reactions where the spin exchange is realised by the use of a paramagnetic broadening agent, the situation is similar. Some data obtained from this kind of experiments are shown in table 4.

Numbers in brackets correspond to the relevant reference.

46 Nitroxides – Theory, Experiment and Applications

orientation of the colliding particles.

4-hydroxy-

solvents.

Compound Solvent kse

Numbers in brackets correspond to the relevant reference.

where p is the probability of spin exchange upon collision of the radicals and fS is a steric factor that takes into account the anisotropy of the spin exchange, i.e., its dependence on the

/ 109 l mol-1 s-1

Acetonitrile [95] 3.9±0.3 9.4±0.7 8,1 Methanol [95] 2.8±0.2 8.5±0.9 8.8

n-Hexane [45] 10.7 13.3 23.2

n-Hexane [44] 11.5 12.1 23.2 Methanol [42] 5.5±0.2 6.7±1.1 8.8 Water [42] 1.9±0.1 -7.6±0.4 -2.5 Benzene [43] 5.3±0.3 18.4±1.0 19.1 Toluene [42] 5.27±0.09 13.7±0.9 14.2 o-Xylene [43] 4.2±0.2 15.3±1.2 15.5

n-Hexane [45] 8.94 11.5 23.2

TEMPO Acetone [44] 7.8 10.7 15.8

TEMPO Acetone [45] 6.42 8.1 15.8

4-oxo-TEMPO Acetone [44] 8.1 8.7 15.8

4-amino-TEMPO Acetone [45] 5.82 12.6 15.8

DTBN Acetone [44] 9.4 7.4 15.8

The probability factor can be determined as shown in equation (45), using the exchange integral, J, and the collision time τ1 as well as a factor pmax which depends on the value of the

*p p <sup>J</sup>*

The pressure dependence of the probability factor has been shown to differ from one solvent to another [44]. For many solvents, p is practically pressure independent while for others, such as acetone or hexane, it varies greatly with pressure. This is believed to be connected to the strength of the exchange. Systems with a pressure independent p, are believed to exhibit strong exchange, i.e. p1, whereas those showing a pressure dependent p are in the weak exchange region. This illustrates the large discrepancies between Δ‡Vexp and Δ ‡Vdiff in those

2 2 1 2 2 <sup>1</sup> <sup>1</sup> *max J*

(45)

**Table 3.** Rate constants and activation volumes of selected spin exchange reactions.

spin and the spin-lattice relaxation time of the radical [108].

Δ‡Vexp / cm3 mol-1

Δ‡Vdiff / cm3 mol-1

**Table 4.** Rate constants and activation volumes of selected paramagnetic exchange reactions.

When the paramagnetic exchange takes place between a nitroxide and a tetragonal transition metal complex, only slight differences between Δ‡Vexp and Δ‡Vdiff are observed, thus indicating strong exchange. An exception is the reaction between 10-DOXYL and [CoCl2(C3H7OH)2], where the steric hindrance around the >N-O group leads to weak exchange.

However, changing the structure of the complex to orthogonal, a change in the effectiveness of the paramagnetic exchange is observed. The steric hindrance is now increased and many complexes show weak exchange.
