Electron Transfer Reactions - ACS Publications - American Chemical

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10 Nuclear Factors in Main-Group Electron Transfer Reactions David M. Stanbury Department of Chemistry, Auburn University, Auburn, AL 36849

The generalfieldof aqueous electron transfer reactions involving main -groupmolecules is reviewed from the perspective of correlating self -exchange rate constants with calculated internal reorganization ener -gies. Oxidation of SCN via SCN is used as an example of high reactivity, in which the rates are limited by diffusive product separation. The cross relationship of Marcus theory leads to a lower limit of about 5 x10 M s for k for the SCN/SCN system. Oxidation of NH OH via NH OH occurs quite differently, with a record-low self-exchange rate constant (5 x 10 M s ). A comprehensive review of the self -exchange rates of main-group redox couples is presented, and the degree of correlation between log k andλ is examined. -

4 -1 -1

2

-

11

2

+

-13

-1

-1

11

i

R E A C T I V I T Y IS T H E H E A R T O F C H E M I S T R Y , as Henry Taube has often said.

Much of his early research was oriented toward the reactivity of aqueous maingroup species such as O and B r , with an important component being an interest i n distinguishing between single- and multielectron redox reactions. This chapter provides a current overview of half of the picture (single-electron reactions of main-group species), with an emphasis on the role of nuclear fac tors in determining the rate constants. It is now evident that two important fea tures have emerged. First, these nuclear factors can have an overwhelming effect. Second, simple methods now exist that can be used to estimate their magnitudes. The present state of affairs is grounded i n Taube's early distinction between inner-sphere and outer-sphere electron transfer reactions (J). One crucial aspect of this distinction is that reactions can be virtually assured of having single-electron steps if the mechanism is outer sphere. The only known s

© 1997 American Chemical Society

2

165

166

E L E C T R O N TRANSFER REACTIONS

exceptions to this rule are the reactions of B r with a Ni(II) complex and of S 0|" with [ R u ( N H ) p z ] (2, 3). (In both cases the proposed mechanisms entail successive single-electron steps with the second step occurring before cage escape of the free radical.) As a consequence, by studying redox reactions between main-group species and "outer-sphere" metal complexes, one is investigating single-electron reactions of the main-group species, that is, free radical reactions. Marcus theory provides the framework for analyzing the nuclear factors in such reactions, but making use of this theory generally requires accurate stan dard potentials for the species participating in the rate-limiting electron trans fer step. Accurate standard potentials for the metal complexes are widely avail able, and formal potentials are readily determined by, for example, cyclic voltammetry. For inorganic free radicals, however, the situation is not as sim ple. The difficulties arise because most main-group one-electron redox processes are electrochemically irreversible. As an example, one-electron oxi dation of SOf" leads via S O j to either SOf" or S Of-, and simple cyclic voltam metry shows an oxidation wave for SOf~ but no return wave. The C 1 0 / C 1 0 system is one of the very rare cases displaying reversible cyclic voltammetry. Various methods have been developed to estimate the requisite reduction potentials, for example, by use of thermochemical cycles based on gas-phase heats of formation, but these methods are not generally of sufficient accuracy to assess the nuclear factors via Marcus theory. The concurrent developments of pulse radiolysis and flash photolysis have been essential here. These two complementary techniques have permitted the generation and characteriza tion of most of the important free-radical intermediates, as well as the determi nation of the kinetics of their decay processes. These two methods have also made it possible to determine the standard potentials of the free radicals with considerable accuracy, as has been summarized in a review (4). In its form as the widely used cross relationship, Marcus theory also makes extensive use of self-exchange rate constants. A major effort since the 1960s has been the development of a large library of "outer-sphere" coordina tion complex reagents and the measurement of their self-exchange rate con stants. It is now possible to select a group of electron transfer reagents for a given main-group substrate that will have a high likelihood of generating use ful kinetic data. With this information in hand the stage is set for investigating the subject matter of this chapter. We begin with a description of two recent studies: (1) the oxidation of S C N " , which is an example of a system with very little nuclear reor ganization (5), and (2) the oxidation of N H O H , which suffers from such a large nuclear reorganization energy that major challenges beset its experimental study (6). We finish with a survey of the reported self-exchange rate constants for main-group redox couples, their estimated internal reorganization energies, and the degree of correlation between these two measures of reactivity. 2


2

n

2

3

10

4+

2

2

2

2

10.

STANBURY

Nuclear Factors in Main-Group Electron Transfer

167

Oxidation of SCN~ The general features of outer-sphere oxidation of SCN~ are well established and have been reviewed by Nord (7). The reactions have the stoichiometry


6M

0 X

+ S C N " + 4 H 0 -> 6 M 2

+ SOf" + H C N + 7 H

r e d

+

(1)

and they obey the general two-term rate law - d [ M ] / d * = (2^[SCN-] + 2fc [SCN"] )[M ] 0X

2

2

(2)

0X

The k term is seen most commonly, but there are a few examples where the k term appears. Most workers adopt the following mechanism to explain these results: 2

l

M M M

o x

o x

o x

+ SCN" £ M

+ 2SCN" ? M + (SCN) -> M 2

+ SCN

r e d

r e d

r e d

+ (SCN)

l9

2

+ (SCN)

S C N + S C N " £ (SCN)

x

fc ,

(4)

k_

2

2

(5)

h

2

k , k^ K 4

2

3(SCN) + 4 H 0 -> 5 S C N " + H C N + S O f + 7 H 2

(3)

k k_

2

r a d

fast

+

(6) (7)

The observed rate law is obtained when the fc step is fast and the reverse steps k_ and k_ can be ignored, as is usually the case. Of greatest relevance to the present chapter is the k rate constant, which corresponds to a simple outer-sphere electron transfer process. Prior to our recent study (5) there were only three reports of such rate constants for reac tions that could reasonably be assigned outer-sphere mechanisms, and these were the reactions of [ I r C y - , [Fe(bpy) ] , and [ C o W O ] - {8-10). We noticed as an interesting feature of these reactions that they all appeared to have values of k_ that were approximately diffusion-controlled. These values of were estimated by using the published values of k an estimate of the electron transfer equilibrium constant Κ derived from the reduction poten tials of the oxidants and the S C N radical, and the relationship Κ = k^k^. This observation raised the question of whether, by appropriate selection of an oxi dant, a reaction could be found in which the back electron transfer process would be slower than the diffusion limit. This line of reasoning led to the choice of [Ni(tacn) ] as an oxidant (tacn = 1,4,7-triazacyclononane). This complex is well-known as a potent outersphere oxidant with a standard reduction potential (E°) of 0.94 V vs. normal 3

x

2

1

2

3

3+

12

5

40

x

v

λ

λ

2

3+

168


hydrogen electrode ( N H E ) (II). More importantly, it has a low self-exchange rate constant (k = 6 χ 10 M " s ) (II, 12), which led to the hope that its oxi dation of S C N " would proceed with activation-controlled kinetics (i.e., that would be less than diffusion-controlled). We found (5) that the oxidation of S C N " by [Ni(tacn) ] was conveniendy rapid and that its products corresponded to the usual stoichiometry as in eq 1. However, the kinetics proved to be relatively complex. Under conditions of a large excess of S C N " the normal rate law, eq 2, leads to pseudo-first-order kinetics. But the reaction with [Ni(tacn) ] was far from pseudo-first-order and gave a relatively good fit to a pseudo-second-order rate law. This effect was quickly traced to the powerful inhibitory effect of [Ni(tacn) ] , which is a product of the reaction. This mechanism, under conditions in which the k_ and k_ steps cannot be neglected, leads to the rate law 3

22

1

-1

3+


2

2

3+

2

2+

x

2

d[M ] 0X

^[SCN-l+fcalSCN-j^fcaKradtMoxftSCN-]

=

M M

d t

r

e

d

] + ^ [ s c N - l N g i t

MC^tMjISCN-]

When the denominator is dominated by the terms first order i n [ M ] , eq 8 predicts pseudo-second-order kinetics, as observed. We deduced that for some reason the and k_ processes were quite important, leading to the product inhibition. A method that we have employed before in such situations is to use a spin trap as a free-radical scavenger (13-15); if the spin trap is sufficiently reactive it will intervene in the kinetics and prevent product inhibition. In the present case we found that P B N (A7-feri-butyl-a-phenylnitrone) was not suffi ciently reactive, but that excellent pseudo-first-order kinetics could be obtained with small concentrations (5 mM) of D M P O (5,5-dimethyl-l-pyrroline N-oxide). red

2

In the presence of D M P O the reaction showed only mild inhibition by [Ni(tacn) ] , and the effect could be safely ignored under conditions in which no additional [Ni(tacn) ] was present. Under these conditions the pseudofirst-order rate constants obeyed a simple two-term dependence on [SCN~] as in eq 2, with fc = 0.046 ± 0.003 M " s" and k = 2.04 ± 0.12 M " s" at 25 °C and 0.1 M ionic strength. At this stage we had met one important objective: to find another reaction in which the fc term of the rate law could be measured. The next objective was to determine whether the fc_ process was at the diffusion limit. As for the other examples discussed previously, this rate con stant was estimated by use of the known reduction potentials, along with the principle of detailed balancing and the measured value of Jt . The outcome was a calculated value of 9.7 χ 10 M " s for k_ which is essentially the diflusioncontrolled result. Although this result showed no evidence for an activation-controlled com ponent to the value of k it was still informative from the point of view of the cross relationship of Marcus theory. This cross relationship, as we use it, is given by (16) 2

2+

2

2+

1

x

1

2

2

x

x

1

9

1

1

-1

l9

1

10.

STANBURY

Nuclear Factors in Main-Group Electron Transfer *12 = ( * l l « 2 / l 2 ) \\nK +


1 2

0)

2

{w -w )/RT]

l2

l2

2

2l

(10)

4[ln(fc fc /Z ) + (u +u? )/Rr]

1 2

W

Wi

1 / 2

169

11

2

22

= exp[{-w - w l2

?11

+ w

2l

n

22

+

w )/2RT]

(11)

22

(12)

Wij = (4.23Ζ .Ζ .)/{ΰ[1 + 0.328β(μ )]} ΐ

1/2

;

In eqs 9-12, k is the electron transfer rate constant for the cross reaction, while k and fc are the component self-exchange rate constants, and K is the electron transfer equilibrium constant. This cross relationship yields rate constants that do not take into account the possibility of diffusion control. If one calculates a value of k by use of eqs 9-12, the result obtained will be a lower limit i f the value of k (or k_ ) is limited by diffusion control. When applied to the oxidations of S C N " by [Ni(tacn) ] , [IrClg] ", and [Fe(bpy) ] , the values of k obtained are 5 χ 10 , 3 χ 10 , and 2 χ 10 M s" , respectively. Because these are lower limits, the highest lower limit (5 χ 10 M s ) is the most meaningful. In a prior publication Nord et al. (17) reported that a k value of 1 χ 10 M s gave a satisfactory account of the rates for [IrClg] " and [Fe(bpy) ] , but this result did not take the work terms into account i n apply ing the cross relationship. 12

n

l2

22

n

l2

l2

2

4

n

3+

2

3

2

3

- 1

3+

1

4

_ 1

_1

n

7

3

_ 1

_1

2

3+

Taken in the context of other main-group self-exchange rate constants, our inferred value of k (> 5 χ 10 M " s ) for the S C N / S C N " system is among the highest, and is comparable to the value of 4 χ 10 M " s derived for the N / N system from similar reactions {14). In both cases there is virtually no structural difference between the two oxidation states of these linear species because the transferring electron resides in a nonbonding orbital (15,18). As noted already, a value for k was also determined experimentally. This value corresponds to an electron transfer process that differs drastically from the k process because it involves concurrent S-S bond formation i n generat ing the ( S C N ) intermediate. A n interpretation of this process is outside the scope of the present chapter, but a full analysis in terms of Marcus theory has been presented (5). By analysis of the kinetic inhibition by [Ni(tacn) ] it was also possible to extract the value of fc , another electron transfer reaction, but this discussion too is outside the scope of the present chapter. 4

n

1

_1

4

1

_1

3

3

2

1

2

2

2+

3

Oxidation of NH OH 2

Unlike for S C N " , reports of outer-sphere oxidation of N H O H are rather rare, and prior to our study (6) the only examples were the oxidations by [Fe(CN) ] , [IrClg] -, and [W(CN) ] . The original report {19) of simple one-electron oxida tion by [ F e ^ N ^ ] was subsequently shown to be i n error because of unrecog2

6

2

8

3 -

3_

3_

170


nized catalysis by trace amounts of copper ions (20). With this correction, the intrinsic rate of oxidation was immeasurably slow. Two studies had been pub lished on the oxidation by [IrClg] , but they differed in their rate laws (21, 22); neither of these papers reported the trace copper catalysis noted elsewhere (4). Thus only the oxidation by [W(CN) ] appeared relevant, although even in this study there was no mention of whether a test for copper catalysis was per formed (23). In our studies of the oxidation of S Of~ by various outer-sphere reagents we found that copper catalysis could be suppressed by the addition of oxalate (24, 25), and so we decided to investigate whether this method could be applied to the direct oxidation of N H O H by [IrClg] - (6). Prehminary studies revealed that both C u and F e were highly effec tive catalysts i n the oxidation of N H O H by [IrClg] " but that their effects could be thoroughly inhibited by addition of C O f " . With this precaution taken, a full study of the reaction showed that the stoichiometry and rate law between p H 4.2 and 6.8 are given by 2-


8

3_

2

2

2

2 +

2 +

2

2

2

[ I r C l ] " + N H O H -> [IrClg] " + ± N + 2 H + H 0 2

6

+

3

d[IrCl|-]

3

+

2

(13)

2

fciK [N(-l)][IrCl|-]

=

a

*

[H ]+K +

a

In this rate law, N ( - l ) refers to hydroxylamine, regardless of its state of proto nation, and K refers to the acid dissociation of N H O H . A value of 24 ± 5 M s was evaluated for k The proposed mechanism was a

- 1

+

3

-1

v

NH OH ^NH OH + H +

3

2

Λ

[IrCl ] - + N H O H ? [IrCl ] " + N H O H 6

2

2

NH OH 2

6

+

?

3

2

NH 0+H 2

NH O^NHOH 2NHOH-^N + 2H 0 2

+

k

2

(16)

Κ

v

λ

(17)

+

K

2

(15)

Κ

+

i s o

(18)

fast

(19)

According to this mechanism, the fc rate constant corresponds to an outersphere electron transfer process. A n estimate of the equilibrium constant for this process was obtained by combining the known standard reduction poten tial for [IrClg] " with an estimate of 0.42 V for E° for the N H O H / N H O H system. This estimate was obtained by a thermochemical cycle that was based on an ab initio calculation of the standard enthalpy of formation (Afl°) for N H 0 (g), a semiempirical calculation of its entropy, a rough guess of its hydration free energy, a literature value for Κ^ , and the National Bureau of Standards (NBS) value of the standard free energy of formation (Afi°) for N H O H (aq). x

2

2

+

2

2

Τ2ίά

2

10.

STAN BURY

Nuclear Factors in Main-Group Electron Transfer

171

The outcome was a value of 9 χ 10 for K which explains why the reaction is not subject to product inhibition. Application of the cross relationship of Marcus theory with eqs 9-12 to the measured value of k led to a self-exchange rate constant of 5 χ 10~ M s for the N H O H / N H O H couple. This result stands in stark contrast to that found for the S C N / S C N " system, and in fact it is the lowest such rate constant ever reported (excluding reactions involving concerted bond cleavage). We have just discussed self-exchange rate constants as a measure of the intrinsic reactivity of an outer-sphere redox couple. A n alternative and com monly used method is to calculate the reorganizational barrier, λ, which is taken as 4AG*. Moreover, it is generally considered that this barrier is given by the sum of the barriers arising from solvent reorganization, λ , and internal reorganization, λ such that λ = λ + λ It is to be expected that λ is princi pally determined by the size of the molecules involved; by analogy with other systems of comparable size (26) an estimate of 120 k j m o l can be made for the N H O H / N H O H system. The self-exchange rate constant calculated from the Marcus cross relationship leads to a value of 530 k j m o H for λ, which implies a value of 410 k j m o l for We used ab initio calculations as an independent check of The method taken was first to optimize the structures of N H O H and N H O H and calcu late their energies. Then, the energy of the transition state was determined by optimizing its structure under the double constraint that the two molecules have identical structures and that they be separated far enough from each other that there would be negligible electronic interaction between the two. The first set of calculations showed that there are major structural differences between the two molecules: N H O H is pyramidal at Ν and has a 1.45 Â N - O bond length, whereas N H O H is planar and has a 1.30 Â N - 0 bond length. These qualitative features would lead one to expect a large value of λ The quantitative results at the QCISD(T)/6-311+G(d,p) level of theory gave a value of 414 k j m o l for λ , which is in remarkably close agreement with that deduced from the kinetics of the oxidation of N H O H by [IrClg] . 7

l9

13

1

+


2

_ 1

_1

2

0

{9

0

Γ

0

-1

2

+

2

-1

2

+

2

2

2

+

Γ

- 1

{

2

2-

Overview of Self Exchange Rates and Nuclear Factors The examples just reviewed describe approaches to evaluating the selfexchange rates for main-group electron transfer reactions. The two specific examples are notable in that one of them (SCN/SCN") appears to be near the upper limit of reactivity when expressed as k , whereas the other ( N H O H / N H O H ) serves as an example of extremely low reactivity. These results, as well as all other such known data, are summarized in Table I, and alongside of these rate constants are presented the corresponding values of λ insofar as they are available. It should be recognized that the rate constants presented have all been determined by applying the Marcus cross relationship to reac tions with coordination complexes. In the two cases for which the selfn

2

2

{

+

172


Table I. Self-Exchange Rate Constants vs. Calculated ^ System

+


2

2

oNCscyf-*co /co 2

2

H0 /H0 N0 /N0 2

2

2

2

cycii

COj/COfSOj/SOf03/03

NO+/NO 2

2

Br /Br 2

2

cio /cio 2

2

HC0 /HC0 2

2

OH/OH-

so /so 2

2

N3/N5

SCN/SCN-

s Oâ/s o|2

1

n

H+/H NH OH /NH OH

o /o

X^fkjmol- )

bgk

2

NOTE: X

i

4.7 5.4 8.3

414 447 70 129 305 147 44 102 104 94 154 94 67 0 0 0

measures internal reorganizational barrier.

exchange rate constants have been measured directly ( N 0 / N 0 and 0 / 0 ) , substantially greater k values have been recorded, the significance of which will be discussed. Another hmitation is that the values of k tabulated have not generally been corrected for the effects of solvent-barrier nonadditivity. This effect arises when the cross-reaction takes place between two reagents of widely differing size, and we have shown that this effect can be a significant factor i n reactions of the type considered here (IS). Yet another limitation is that many of the self-exchange rate constants have not been corrected for work terms; as the 0 / 0 system shows, this omission can lead to errors as large as a factor of 100. However, the point of Table I is to discern rough trends, and for this purpose the existing data merit some consideration. In what follows we discuss the individual entries in Table I. 2

2

2

2

n

n

2

2

H / H . As recendy reported by Kelly et al. (27), Schwarz has estimated the self-exchange rate constant for this redox couple (

Electron Transfer Reactions - ACS Publications - American Chemical

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