Radiation Chemistry-II

25

Rate Constants and

Equilibrium

Constants

for E l e c t r o n T r a n s f e r R e a c t i o n s o f

Downloaded by CORNELL UNIV on October 21, 2016 | http://pubs.acs.org Publication Date: January 1, 1968 | doi: 10.1021/ba-1968-0082.ch025

A r o m a t i c M o l e c u l e s in S o l u t i o n SHIGEYOSHI ARAI and LEON M. DORFMAN The Ohio State University, Columbus, Ohio 43210

Absolute rate constants for electron transfer reactions o aromatic molecules in solution have been determined by the pulse radiolysis method for three additional pairs of aromatic compounds. In two of these cases in which an electron transfer equilibrium is established, the rate constant for the back reaction has also been determined. Th equilibrium constant has been estimated from the kinetic data. A correlation of the experimental rate constants with the theory for homogeneous electron transfer rates is considered. bsolute rate constants for electron transfer reactions of some aromatic molecules in solution have been reported in our earlier work (2) using the pulse radiolysis method. The transfer of an electron from various radical anions to a second aromatic compound in solution was observed directly. Of the rate constants for nine donor-acceptor pairs investigated, two were found to be lower than the diffusion controlled values, and a correlation with such parameters as the reduction potential difference of the pair was considered. These measurements have been extended to additional transfer pairs for which the reduction potential difference is small. The objective of this work, in addition to furnish ing new data for electron transfer rates, is to provide an adequate test of theories of the rate of homogeneous electron transfer in polar liquids (10, 11,12,13,14, 15,16,17). Experimental The details of our experimental method, using a Varian V-7715A linear accelerator as the pulsed electron source, have been adequately 378 Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.


25.

ARAI A N D D O R F M A N

379

Aromatic Molecules

described (2, 7 ) . The electron pulse used ranged from 100 nsec. to 1 /xsec. The fast optical detection method and the reaction cell arrange ment were the same as before ( 2 ) . The reactions were carried out i n isopropyl alcohol. This compound was selected as solvent because the natural lifetime of the aromatic radical anion with respect to protonation (1, 3) is longer than i n methyl alcohol or ethyl alcohol. The isopropyl alcohol was obtained from Matheson, Coleman, and Bell and was freshly distilled over sodium metal through a glass-packed column for each set of runs. Anthracene from Matheson, Coleman and Bell, pyrene from Aldrich Chemical Co., m terphenyl and p-terphenyl from City Chemical Corp. were purified as described ( 2 ) . 9,10-Dimethylanthracene obtained from Κ and Κ Lab oratories, Plainview, Ν. Y., was recrystallized from isopropyl alcohol solution. Results and Discussion The donor-acceptor pairs investigated were pyrene-anthrace, pyrene9,10-dimethylanthracene and m-terphenyl-p-terphenyl. The radical anions of these compounds may be observed at the appropriate optical absorp tion bands, the maxima of which are as follows: m-terphenyl, 7400 Α., a broad weak band; p-terphenyl, 8770 Α.; pyrene, 4900 Α.; anthracene, 7200 Α.; and 9,10-dimethylanthracene, 7300 A . For most of the systems we have studied in this and earlier work (1,2,3) these absorption bands have been known (4, 6, 8) from work on solutions i n which the radical anions are stable. In the systems involving pyrenide anion as donor the decay of the donor anion and the formation of the acceptor could be observed simultaneously since their absorption bands do not overlap extensively. Rate Constants. The sequence of reactions taking place in a twocomponent solute system, as discussed previously ( 2 ) , are the formation by attachment of the solvated electron, the decay by proton transfer and by counter-ion combination, and the electron transfer. e i~ + arene

= arene"

(1)

arene" + i-PrOH

= arene H - + t-PrO"

(2)

= arene H - + i-PrOH

(3)

= arene + arene,,"

(4)

8Q

arene" + i - P r O H arene " + arene a

b

2

+

a

4

At sufficiently high concentrations of arene so that Reaction 1 is com plete at the end of the pulse, and at sufficiently low pulse intensity so that the rate of Reaction 3 is negligible, the differential rate expression for the decay of the radical anion is adequately represented (2) b y : ~

d

{

£ ^ = *4.[A„-] [A„] + k [V] 2a

[i-PrOH]

Hart; Radiation Chemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

(5)

380

RADIATION CHEMISTRY

II

If the composition of the system is such that & ι [ Α ] > > fci&[A ], the radical anion A ~ w i l l have been formed preferentially at the end of the pulse. This condition obtained for most of the runs. These are, of course, equilibrium systems with respect to the electron transfer, and in those cases in which the equilibrium is not overwhelmingly on the side of the acceptor anion the back reaction may occur. If there is a significant rate of the back reaction, 4b, Equation 5 becomes: α

α

6

a

[A."] [ A ] - *

dt

b

4 b

[ A ] [A J b

+ k

[A."] [i-PrOH]

2&

If the formation curve for A " is observed, the analysis may be carried out by considering the analogous equation for d [ A ~] /dt. This was done for only the m-terphenyl-p-terphenyl system, i n which case the m-terphenylide band is obscured by a strong absorption of the p-terphenylide. b


(6)

b

30

Ο χ

-I

1

1

I

I

I

I

I

I

I

5

I

10

ANTHRACENE CONCENTRATION (MxlQ ) 4

Figure 1. Plot of { ^ [ A ] + ^ b [ A ] } , from decay curves of pyrenide anion, against concentration of an thracene. The concentration of pyrene is constant and is equal to 1.0 X 10~*M. The slope gives k, = 1.8 X 10 M> seer at 25°C. a

b

a

i(l

9

1



25.

ARAi

381

Aromatic Molecules

AND D O R F M A N

DIMETHYL ANTHRACENE (MxK) ) 4

Figure 2. Plot of {fc, [A ] 4- ^ [ A J } , from decay curves of pyrenide anion, against concentration of 9,10dimethylanthracene. The concentration of pyrene is constant and is equal to 1.1 X I0~ M. The slope gives k = 1.3 X WW seer at 25°C. a

6

2

1

ia

If the concentration of the acceptor, A , is sufficiently high, the rate of Reaction 2a is negligible compared with the electron transfer rate and the integrated form of Equation 6 is: b

[ A " L - [ A . - ] = C exp a

e

(* [A ] + * [A ])*

1

4 a

b

4 b

(7)

a

where [ A ~ ] is the concentration of A " at equilibrium and Ci is a constant. In terms of the optical density, D, this becomes: a

e

a

2.303 l o g

10

(D - D ) = - ( f c [ A ] + fc [A ])t + C t

4 a

e

b

4b

a

(8)

2

A plot of the left-hand side as a function of time gives a straight line with slope — ( f c [ A ] + ^ 4 b [ A ] ) . In this way, values of ( f c [ A ] + f c t b [ A ] ) are obtained for different [ A ] holding [ A ] constant. A plot of this term against [ A ] gives a straight line from the slope of which fc may be determined. The rate constant for the back reaction, fc , may be 4a

b

a

4a

b

b

a

a

b

4a

4b


382

RADIATION CHEMISTRY

II

determined from the intercept. The uncertainty in k± is considerably larger than in &4 . Such a plot is shown in Figure 1 for pyrene-anthracene, and in Figure 2 for pyrene-9,10-dimethylanthracene, both of which are pairs for which the back reaction rate is not negligible, and in which equilibrium is attained. The values for k and k are shown in Table I. h

8

4&

4b

Table I. Electron Transfer Rate Constants for Aromatic Radical Anions in Isopropyl Alcohol at 25°C. Back Rate Forward Rate Constant, k (M~* sec.' ) Constant, k ( M sec.' )


Donor-Accept or Pair

1

4 a

(2.1 ± 0.9) Χ 10 (3.7 ± 1.7) Χ 10

(1.8 ± 0.3) X 10° (1.3 ± 0.3) X 10°

Pyrene-anthracene Pyrene-9,10-dimethyl anthracene m-Terphenyl-p-terphenyl

(2.3 ± 0.4) Χ 10

_ 1

4 b

1

7 7

9

The value of D , in Equation 8, is taken from the almost horizontal portion of the rate curve, which represents the attainment of equilibrium between the two radical anions, at the region in which the fast portion of the decay curve becomes asymptotic to this horizontal. Such a rate curve, taken at two different sweep times, is shown in Figure 3. The un certainty in the value of D is very small since the linear portion of the curve is nearly horizontal. This linear portion is not perfectly horizontal when examined on a long time-scale simply because the rates of Reactions 2 and 3 are not completely negligible. Another observation which is indicative of the establishment of equilibrium with respect to Reaction 4 may be made in the following way. If one observes the slow decay of the two radical anions at their respective wavelengths, which in the system pyrene-anthracene would be 490 m/x for the former and 720 τημ for the latter, both anion spectra are found to be present for many tens of microseconds. Moreover, the ratio of the optical densities at these two wavelengths remains constant over this extended period. The same is found for pyrene-dimethylanthracene. e

e

In the case of m-terphenyl—p-terphenyl, the kinetics were observed for the formation of p-terphenylide anion. The rate of the back reaction is negligible at the m-terphenyl concentration used, 6 X 10" M. This may be seen from the fact that the decay of p-terphenylide anion with this concentration of m-terphenyl present, does not differ appreciably from its decay rate when only p-terphenyl is present. The decay rate of m-terphenylide anion in Reaction 2, is however, much higher than that of p-terphenylide anion. A plot of logm (D — D t ) / D against time, from the formation curves of this pair, gives a straight line from the slope of which we determine (fc4a[A ] + fc [i-PrOH]), since the decay rate 4

x

b

x

2a


25.

ARAI AND D O R F M A N

383

Aromatic Molecules

of p-terphenylide anion by Reaction 2 is negligible. A subsequent plot of this term against [ A ] gives k from the slope. The value is shown in Table I.


h

4&

Equilibrium Constant. For those cases in which the electron transfer equilibrium is overwhelmingly on the side of the acceptor anion, the rate of the back reaction, as has been pointed out (2), is negligible. In some cases where this is not true, as for diphenyl-naphthalene, the equi librium with respect to Reaction 4 may be "quenched" by the protonation of the acceptor anion in Reaction 2. In the case of pyrene-anthracene and pyrene-9,10-dimethylanthracene, where equilibration occurs, the back reaction has a measurable effect upon the electron transfer kinetics, and &4b has been determined. From these data, the equilibrium constant may also be reliably estimated since it is given by the ratio: K = fc /fc . The values obtained at 25°C. are: (1) pyrene-anthracene —86 and (2) pyrene-9,10-dimethylanthracene —35. These values for K may also be determined from the difference i n the reduction potentials of the pair as measured potentiometrically (5, 8, 9, 18). The equilibrium constant is related to the reduction potential difference, AF, by: 4a

c

4b

c

AF = RT In K

C) 9

e

Equilibrium constants of 81 and 30 for the two foregoing pairs i n tetra hydrofuran are calculated from such data, i n excellent agreement with the values from our kinetic method. 100W 5 μ S*C 1 1 'tr

—

2 μ sec 1 1 TIME

Figure 3. Decay curves of pyrenide anion in isopropyl alcohol solution containing 4.9 X 10~*M anthracene. The concentration of pyrene is 8.5 X 10~ M. The sweep time for the upper curve is S

5 /xsec./dit;., and for the lower 2 /xsec./cftt;.

Correlation with Theory. A general theory for homogeneous elec tron transfer reactions has been developed and discussed by Marcus (10, 11, 12, 13, 14 15, 16,17). The rate constant in this model, as applied to


384

RADIATION CHEMISTRY

Table II.

Correlation with Theory of Experimental Rate Constants

Donor, Acceptor Pair

AF°' (kcal./mole)

(kcal./mole)

—0.99 + 2.61 +2.01 —3.28 -2.01 -2.61 -12.2 —14.8

3.4 5.34 4.99 2.46 2.99 2.73 0.20 0.012

Diphenyl", naphthalene Anthracene", pyrene 9,10-dimethylanthracene", pyrene Diphenyl", phenanthrene Pyrene", 9,10-dimethylanthracene Pyrene', anthracene Diphenyl", pyrene Diphenyl", anthracene


II

AF*

The calculated values are based on a reorganization parameter, λ = 16 kcal./mole and upon Ζ = 1 0 M sec." . a

U

- 1

1

our systems, depends upon the dielectric behavior of the solvent and upon the difference i n reduction potentials of the donor-acceptor pair as well as upon such collision parameters as the encounter radii. The dependence upon the dielectric properties of the liquid reflects the energy of reorientation of the solvent dipoles i n the formation of the activated complex. The difference i n reduction potentials is a major contribution to the value of the exponential term in the rate constant expression, and is therefore an important parameter. The bimolecular rate constant for electron transfer, k , i n the nota tion of Marcus (10,11, 13), is given by: bi

k

M

(10)

= Ze-WRT

where AF* is the free energy of activation. Ζ is the collision number, which has been taken by Marcus to be about 1 0 M sec." , equivalent to the collision number i n the gas phase, and thus not dependent upon solvent properties. It is defined (13) as equivalent to (8rrkT/m*) r where m * is the reduced mass of the reactants and r is the distance be tween the centers in the collision complex. The free energy of activation is given (13) b y : U

_ 1

1

1/2

* =

AF

w +

±

+

4

(ID

*El i±pi +

2

2

4λ

w is the difference i n the work of bringing together the reactants and separating the products. W i t h one reactant uncharged, as in our systems, this does not involve a coulombic interaction, and w should therefore be small. Δ Ρ ° ' is the standard free energy of reaction, which is taken as the difference i n reduction potentials of the donor-acceptor pair. The re organization parameter, λ, is given b y :


25.

ARAI AND D O R F M A N

385

Aromatic Molecules

for Electron Transfer of Aromatic Molecules in Solution Experimental

Calculated kbi (Μ~' seer )

(Μ~' seer ) 1

2.6 Χ 10 2.1 Χ 10 3.7 Χ 10 6 Χ 10 1.3 Χ 10 1.8 Χ 10 5.0 Χ 10 6.4 Χ 10

1

3.0 1.2 2.2 1.6 6.5 1.0 7.2 9.8

8

7

7

8

9 9 9


9

6 c

10 10 10 10 10 10 10 10

Χ Χ Χ Χ Χ X Χ Χ

Experimental (kbi)n/(kbi)l

Calculated (hi)n/(kbi)l

1.0 0.081 0.14 2.3 5.0 6.9 19 25

1.0 0.041 0.074 5.3 2.2 3.4 240 330

8

e

e

7

7 9

8 9 10 10

a,h

Those donor-acceptor pairs not from the present work are from Reference 2. kti for the diphenyl'-naphthalene pair is taken as the reference constant.

The contribution from intramolecular vibrational reorganization, which is small for strong internal bonding, has been neglected. The effective radii for the encounter, ai and a , include i n addition to the molecular radii of the reactants, a saturated monolayer of solvent around the molecu lar anion, r is taken as a\ + D > the optical dielectric constant, is equal to the square of the refractive index, and D is the static dielectric constant, Ae is the change i n charge of the donor, which for our case is + 1 . 2

0P

8

If we take a\ = a = 5A. for all of our molecules, and hence r = 10A., we obtain λ = 16 kcal./mole from Equation 12. If w < < {λ/4 + A F ° 7 2 + ( Δ Ε ) / 4 λ } , we may obtain AF* directly from Equation 11. W e may then obtain an absolute value of k from Equation 10 if a value is assumed for the collision number, Z . Or alternatively the theory may be tested by examining the internal consistency of the calculated values, compared with our data, of the ratio (k ) J(k i)ι for a series of rate con stants relative to one reference constant, assuming, as a first approxima tion, that the collision number Ζ is the same for all our pairs, and there fore cancels out. 2

0 /

2

M

M

b

Table II shows experimental values for the rate constant from this and earlier work ( 2 ) , and calculated values for λ = 16 kcal./mole for both k (taking Ζ = 1 0 ) and (k ) J(k )ι referred to diphenyl-naphthalene. Reasonable agreement is obtained for for those cases tested for which the rate constant is significantly lower than the diffusioncontrolled value. The ratio (k ) J(k )ι also shows good agreement for these cases. bi

11

bi

M

M

M

For those cases for which the experimental rate constant corresponds to the diffusion-controlled value (diphenyl-pyrene or diphenyl-anthracene) a direct comparison of the calculated value with experiment is not


386

RADIATION CHEMISTRY II

meaningful, but must be understood on the basis of the relation (17): l/k =l/k + l/k (13) where k is the value calculated from Equation 10, and k f is the dif fusion-controlled value. Thus, for diphenyl-pyrene and diphenyl-anthracene, where k f taken as 6A. gives λ = 13 kcal./mole. The values of (k ) /(k ) calculated with either of these values for λ are not very different from the values in Table II for those cases with k < k , the important parameter being the difference in reduction potentials. oba

act

aitf

act

dif

dif

9

a( _1

ohf

Radiation Chemistry-II

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