The influence of perchlorates on the fluorescence quenching of 9,10-dichloroanthracene by bromide salts in acetone

The influence of perchlorates on the fluorescence quenching of 9,10-dichloroanthracene by bromide salts in acetone

7 May 1999 Chemical Physics Letters 304 Ž1999. 309–316 The influence of perchlorates on the fluorescence quenching of 9,10-dichloroanthracene by bro...

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7 May 1999

Chemical Physics Letters 304 Ž1999. 309–316

The influence of perchlorates on the fluorescence quenching of 9,10-dichloroanthracene by bromide salts in acetone Marek Mac ) , Bogdan Tokarczyk Faculty of Chemistry, Jagiellonian UniÕersity, Ingardena 3, 30-060 Krakow, ´ Poland Received 4 March 1999

Abstract Fluorescence quenching of 9,10-dichloroanthracene by lithium bromide and tetra-n-butylammonium bromide in acetone has been investigated in the presence of perchlorate salts. In the presence of LiBr, the Stern–Volmer ŽS–V. plots exhibit downward curvatures indicating that two species are responsible for the quenching process, namely free bromide anions and lithium bromide ion pairs. The addition of a perchlorate salt modifies the S–V dependencies due to the influence of perchlorates on the degree of lithium bromide dissociation. The association constant of lithium bromide has been determined by conductivity measurements and it agrees well with the estimates made from the fluorescence quenching measurements. The mechanism of fluorescence quenching by lithium bromide is discussed on the basis of electron transfer and the heavy-atom effect. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The fluorescence quenching of aromatic molecules by inorganic anions such as iodide, bromide, and thiocyanide in polar solvents Žwater, acetonitrile, and alcohols. has been the subject of many investigations w1–10x. It has been found that the fluorescence quenching mechanism is electron transfer from the anions to the excited aromatic molecules. Originally, the electron transfer mechanism was confirmed only by the dependence of the quenching rate constants on the oxidation potentials of the anions w1–6x. In some cases, the presence of anion radicals of the aromatics formed by electron transfer fluorescence

)

Corresponding author. E-mail: [email protected]

quenching was observed in flash photolysis experiments w7,10x. It has been found w8x that in acetonitrile the Stern–Volmer ŽS–V. dependencies in many cases showed upward curvatures. This effect was explained as the result of a nonequilibrium distribution of the quencher molecules around the excited fluorophore. It should be noted that positive curvatures were found to exist in systems where the electron transfer process is highly exothermic. From the nonlinear S–V dependencies, the intrinsic electron rate constants were estimated and compared with those obtained from the fluorescence decay functions. Less attention has been paid to the fluorescence quenching processes in the systems containing aromatic molecules and inorganic salts in less polar solvents, where dissolution of the salts is still possible w10x. In such solvents, dissociation of the salts is not complete and they exist in the form of free ions and ionic

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 3 3 4 - 6

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M. Mac, B. Tokarczykr Chemical Physics Letters 304 (1999) 309–316

pairs. A question arises: are the ions and ionic pairs equally effective quenching agents? The aim of this Letter is to investigate the fluorescence quenching process of 9,10-dichloroanthracene ŽDClA. by lithium bromide ŽLiBr. and tetra-n-butylammonium bromide ŽTBABr. in acetone in the presence of perchlorate salts. Two kinds of perchlorates were used in the experiments, namely lithium perchlorate and tetra-n-butylammonium perchlorate ŽTBAP.. Both salts are readily soluble in acetone, and are known to be noninteracting with excited molecules lacking a dipole moment. Previously, we have investigated the influence of lithium perchlorate on the fluorescence of bianthryl ŽBA. in 2-methyltetrahydrofuran ŽMTHF. w11x. We found a significant influence of LiClO4 on the lifetimes and band shapes of the charge transfer fluorescence of BA, having a large dipole moment. This may be explained as the effect of increasing dielectric constant upon addition of LiClO4 to the MTHF. In the systems investigated here, the effect of perchlorate salts on the fluorescence lifetimes of DClA is negligible, meaning that neither the fluorescence lifetimes nor the fluorescence quantum yields of DClA in acetone are influenced by the addition of perchlorate salts. Remarkable changes were observed, however, on the addition of the bromide salts.

2. Experimental Acetone ŽAldrich., lithium perchlorate ŽLiClO4 . ŽAldrich ., tetra-n-butylammonium perchlorate ŽTBAP. ŽFluka., lithium bromide ŽLiBr. ŽAldrich., and 9,10-dichloroanthracene ŽDClA. ŽFluka. were used as received. Tetra-n-butylammonium bromide ŽTBABr. was crystallised from ethyl acetate. The solutions containing DClA and LiClO4 or TBAP as background electrolytes and LiBr or TBABr as the quenchers were degassed by freeze–pump–thawing cycles. The fluorescence decays were recorded using the time-correlated single-photon counting technique at 450 nm. For excitation, the 337 nm line of a nitrogen flash lamp was used Žpulse width ; 1.2 ns.. We found that the fluorescence decays were single exponential in all cases and the lifetimes were not dependent on the observation wavelength.

The conductivity measurements were performed using a Radelkis conductivity meter, type OK-102r1. For the minimisation processes, the MINUITS procedure from the CERN library was applied w12x. 2.1. Stern–Volmer relations The quenching of excited DClA by lithium bromide may be described by the Stern–Volmer equation having the form: t0 X Y s 1 q K SV a c q K SV Ž1ya . c , Ž 1. tc where c is the analytical salt concentration, a c is the concentration of free bromide anions and Ž1 y a . c corresponds to the ion pair concentration. Generally, we assume that both species, i.e. free bromide anions and ionic pairs, possess different quenching properties. Therefore, the Stern–Volmer constants X Y and K SV differ from each other. The degree of K SV dissociation a of LiBr is related to its association constant K ass according to the formula: K ass s

1ya

a 2c

Pg ,

Ž 2.

where Pg is the overall activity coefficient and may be described as w13x: gpi gpp Pg s . Ž 3. 2 g ip g DH The terms g ip , gpi , and gpp represent the activity coefficients connected to the interactions of free ions with the ionic pairs Žg ip ., the ionic pairs with free ions Žgpi ., and the ionic pairs with other ionic pairs Žgpp ., whereas g DH is the Debye–Huckel activity ¨ coefficient. The latter quantity may be described as: log g DH s y

A'a c 1 q Bq'a c

.

Ž 4.

In the presence of the inert salt of concentration cX , Eqs. Ž2. and Ž4. convert into the following: K ass s

1ya c a 2 q a cX

log g DH s

Pg ,

A'a c q cX 1 q Bq'a c q cX

Ž 5. .

Ž 6.

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311

3. Results

Fig. 1. Stern–Volmer plots for fluorescence quenching of 9,10-dichloroanthracene by lithium bromide Žsolid circles. and tetra-nbutylammonium bromide Žopen circles. in acetone at 293 K in the absence of perchlorate salts.

Compilation of Eqs. Ž1. and Ž5. gives Eq. Ž7.:

t0 tc

X s 1 q K SV c

9,10-dichloroanthracene ŽDClA. is known as an red efficient electron acceptor Žin acetonitrile E1r2 s y1.55 " 0.1 eV vs. SCE w14x.. Therefore, the fluorescence of DClA may be affected by electron donors such as inorganic ions meaning that fluorescence lifetimes and quantum yields decrease with increasing salt concentration. The S–V dependencies constructed from the fluorescence lifetimes of DClA for the systems: DClA q LiBr and DClAq TBABr in acetone are presented in Fig. 1. Tetra-n-butylammonium bromide is a much efficient quencher than lithium bromide. Moreover, the S–V dependence shows a downward curvature at higher LiBr concentration, whereas when TBABr is used as the quencher the S–V dependence is perfectly linear up to 0.1 M of TBABr with the slope equal to 82.80 " 4.9 My1 . In the presence of lithium perchlorate, the efficiency of the quenching by LiBr is lowered. The fluorescence lifetimes of the DClA with and without the added perchlorates remain the same within exper-

2

(

X q X 2 q K ass Ž Pg .

ž

Y q K SV 1y

y1

c

2

(

X q X 2 q K ass Ž Pg .

y1

c

/

c,

Ž 7. where X s K ass Ž Pg .

y1 X

c q1 .

Ž 7a .

Eq. Ž7. must be solved iteratively. In the absence of perchlorates at low quencher concentration, the slope of the S–V dependencies is X equal to K SV . Alternatively, at high quencher conY centration, the slope should equal K SV . The interactions between the ion pairs and the ion or other ion pairs are also important w13x, especially in solvents of low polarity where the interactions of dipolar nondissociated molecules and ions are significant. Here we consider the case when the contributions from the interactions between the ionic pair and the ions were negligible, i.e. Pg s gy2 DH .

Fig. 2. Stern–Volmer dependencies for fluorescence quenching of 9,10-dichloroanthracene by lithium bromide in the absence of perchlorate salts Ž0., in the presence of 0.05 M lithium perchlorate Ža., in the presence of 0.2 M LiClO4 Žb., in the presence of 0.05 X M tetra-n-butylammonium perchlorate, TBAP Ža . and in the X presence of 0.2 M TBAP Žb ..

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M. Mac, B. Tokarczykr Chemical Physics Letters 304 (1999) 309–316

imental error Žt 0 s 9.1 " 0.5 ns.. In the presence of 0.2 M LiClO4 , the S–V dependence becomes linear with the slope equal to 7.08 " 0.60 My1 . In the presence of tetra-n-butylammonium perchlorate as background electrolyte we observe the opposite trend. The fluorescence lifetimes are shorter than those in the absence of the salt. Moreover, the negative curvature does not disappear in the presence of TBAP up to 0.2 M. The S–V dependencies are presented in Fig. 2.

possess different quenching abilities, the bromide anions are better electron donors than the nondissociated ion pairs. Intuitively, it is easier to remove an electron from the free bromide anion than from an ionic pair in which the bromide anion is screened by the positively charged lithium cation. Let us assume that the quenching process may be described by Eq. Ž1., and assume also that only the interactions between the ions contribute to the overall activity coefficient Pg , i.e. Pg s Žg DH .y2 . We fitted Eqs. Ž1., Ž6. and Ž7. to the experimental data X with three minimisation parameters, namely K SV , Y K SV , and K ass . The results are presented in Table 1. Validation of the method applied here may be confirmed by comparison with the results obtained by applying other methods. For the estimation of the association constants of the bromide salts, we have

4. Discussion The two effects may be explained as follows. The quencher ŽLiBr. occurs in the acetone solutions as free ions and ion pairs. The two quenching species

Table 1 Physico-chemical properties characterising the investigated systems ŽDClAq bromide salts in acetone. Property

Abbreviation and unit

Value

Fluorescence lifetime of DClA

t 0 Žns.

9.1 " 0.5

Reduction potential of DClA vs. SCE

red Ž . E1r 2 V

y1.55 " 0.1

Stern–Volmer constant for the quenching by Br

X K SV

75.58 " 6.8 ŽLiBr. 82.80 " 4.9 ŽTBABr.

Quenching rate constant ŽBry .

X k q Ž10 9 My1 sy1 .

8.3 " 1.24 ŽLiBr. 9.09 " 0.64 ŽTBABr. 8.63 c

Stern–Volmer constant for the quenching by LiBr

K SV ŽMy1 .

5.00 " 0.48

Quenching rate constant ŽLiBr.

Y kq

Association constant of bromides

K ass ŽMy1 .

1882.8 " 250 a 1708.0 " 200 b 96.8 " 16.4 b ŽTBABr.

Limiting equivalent conductivity of bromides

L0 Ž Vy1 cm2 moly1 .

153.7 " 13.1 Ž181.0 d . ŽLiBr. 139.7 " 8.3 ŽTBABr.

y

y1 .

ŽM

9

y1

Ž10 M

y1 .

s

0.54 " 0.079 0.0756 c

Estimated values of the association constant of LiBr and TBABr are 1778 and 100.5 My1 , respectively. The estimations were made using Eqs. Ž10a., Ž10b. and Ž11. with the parameters: radius of lithium cation Ž rA s 0.06 nm., radius of bromide anion Ž r B s 0.195 nm., radius of TBA cation Ž rA s 0.5 nm., distance of closest approach of LiBr Ž a s 0.255 nm., distance of closest approach of TBABr Ž a s 0.5 nm.. a Estimated from the Stern–Volmer analysis. b Estimated from the electrochemical measurements. c Calculated by solving the diffusion equation with the Jortner formula as the source function, according to the procedure described in detail in Ref. w6x, with the parameters: DGet s y0.2 eV ŽBry. , DGet s 0.07 eV ŽLiBr., radius of the donor ŽBry and LiBr. s 0.35 nm, radius of the acceptor ŽDClA. s 0.45 nm, distance of closest approachs 0.56 nm, electronic coupling matrix element at the encounter distances 0.01 eV, mutual diffusion coefficients 4 = 10y5 cm2 sy1 , static dielectric constant of acetones 21. d Taken from Ref. w15x.

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313

used the conductivity measurements of the salts in acetone. The conductance of the solutions of LiBr and TBABr were measured in acetone at 208C. The results are presented in Fig. 3 Žinsert.. The method of determination of the association constant on the basis of conductivity measurements is described by Bocris and Reddy w16x. The method is briefly outlined below. Assuming that the activity coefficient of the ion pair is equal to unity, since neutral ion pairs are not involved in the interactions with the ions, the following equation becomes valid: Z

1 s

L

L0

L cg 2 DH

K ass

q

Ž L0 .

2

Z

,

Ž 8.

where

½

Zs1yz 1yz Ž . . . .

1r2 y1r2 y1r2

5

Ž 8a .

Fig. 3. Plots of Eq. Ž8. for lithium bromide Žopen circles. and tetra-n-butylammonium bromide Žsolid circles. in acetone at 293 K Žsee text for details.. Insert: the dependence of equivalent conductivity of solutions of TBABr Žopen rectangles. and LiBr Žsolid diamonds. on the square of molar concentration.

and

zs

Ž A q B L0 . c1r2L1r2 Ž L0 .

3r2

,

Ž 8b .

where L and L0 are the equivalent conductivity and the limiting equivalent conductivity Žat salt concentration equal 0., respectively. The parameters A and B are the Debye– Huckel–Onsager parameters and in the case of a ¨ symmetrical electrolyte Ž zA s z B . are described by the following equations:

As

Bs

zA e 0 900ph

ž

8p zA z B e 02

1r2

6 Ž 1 q '2 . DkT

1r2

/ ž / ž / ž /

DkT

zA2 e 02'2

NA

1000

8p zA z B e 02 DkT

1r2

.

NA

1000

Ž 8c . 1r2

,

Ž 8d . where zA e 0 and z B e 0 are the charges of ion A and B, respectively, h is the viscosity coefficient, D is the static dielectric constant, F, NA , and k are the Faraday, Avogadro, and Boltzmann constants, re-

spectively. The dependence described by Eq. Ž8. is presented in Fig. 3. Table 1 contains the association constants of lithium bromide and tetra-n-butylammonium bromide obtained from the conductivity measurements. The agreement between the fluorometric electrochemical methods of estimation of the association constants of LiBr is quite satisfactory.To verify our conclusions, we performed the fluorescence quenching measurements of DClA using tetra-n-butylammonium bromide ŽTBABr.. This salt dissociates much better than LiBr in polar solvents, therefore we should expect a linear salt concentration dependence of the relative fluorescence lifetimes. Indeed, the S–V dependence is perfectly linear in this case Žcf. Fig. 1.. From Table 1, we can see that the quenching abilities of LiBr are significantly lower than that of Bry. Let us assume that the thermodynamic scheme presented below is valid. Thus, we can establish a relationship between DGx , i.e. free enthalpy of oxidation of LiBr pair, and the free enthalpy of oxidation of the bromide anions Ž DG 3 ., the free enthalpy of dissociation of the LiBr molecule Ž DG 2 ., and the

M. Mac, B. Tokarczykr Chemical Physics Letters 304 (1999) 309–316

314

free enthalpy of dissociation of LiBr radical into Liq and Br. Ž DG1 ..

When rA s r B s 0.5a, the dissociation constant K i Ž i s 1 or 2. may be expressed by the ŽFuoss. equation w13,16x:

Kis

Hence DGx , representing the oxidation potential of nondissociated LiBr can be calculated as: DGx s DG 2 q DG 3 y DG 1 s DG 3 q RT ln

K1 K2

. Ž 9.

As already mentioned, the value of K 2 has been determined previously using two different methods. The value of K 2 may be also evaluated in the way described below. Let us consider the following process: kd

AB | A q B . ka

The association Ž k a . and dissociation Ž k d . rate constants are described by the following equations w17x: ka s

2 kTNA

rA

rB

q

rB

rA

/

1

=

`

a

Ha

kd s

ž

3h

2q

kT 2 ph a

ž

2

1

1 q

ra

rB

`

a

Ž 10a.

,

Ž 10b.

/

exp Ž zA z B e 2rDakT .

=

,

ry2 exp Ž zA z B e 2rDrkT . d r

y2

Ha r

2

exp Ž zA z B e rDrkT . d r

where rA and r B are the radii of the reactants A and B, respectively, and a is the distance of closest approach of the reactants Žmay be assumed to be the sum of rA and r B ..

kd ka

3 s

4 p NA a3

exp Ž z A z B e 2rDakT . .

Ž 11 .

Using Eq. Ž11., we have calculated the association constants Ž K 2 .y1 of LiBr and TBABr assuming that the radii of the lithium cation, TBA cation, and bromine anion are equal to 0.06, 0.5, and 0.195 nm, respectively w18x. Assuming that the encounter distance is equal to the sum of radii of ions we have calculated the association constants for LiBr and TBABr. For the first salt, the accordance of the theoretical prediction and the observed value is excellent, the measured value is equal to 1882 " 250 My1 Žfrom S–V analysis., or 1708 " 200 My1 Žfrom electrochemical measurements. whereas the theoretically predicted value is equal to 1778 My1 . For TBABr the association constant Ž K 2 .y1 , calculated under the same assumption, is equal to 42.3 My1 , i.e. is about twice as small as the observed one. A good agreement between the calculated and observed association constants has been achieved making the assumption that the encounter distance is much smaller than the sum of the crystallographic radii of the ions Ž r TBA qs 0.5 nm, r Brys 0.195 nm, and a s 0.5 nm.. This indicates some penetration of the bromide anion between the bulky tetrabutyl groups in the molecule. Assuming that the radii of lithium cations, TBA cations, and bromine radicals are the same as those for ion pairs, we calculated the dissociation constants of LiBr and TBABr radicals. The values of the dissociation constants of LiBr and TBABr radicals Ž K 1 . are equal to 24 and 2.3 M, respectively. Thus, the value of DGx is greater by ; 0.27 eV than DG 3 for LiBr, while in the case of TBABr it is greater by 0.13 eV. Consequently, the donating property of LiBr is weaker than that of TBABr and much weaker than that of the free bromide anion. As previously mentioned, the fluorescence quenching rate constant of DClA by free X bromide anions in acetone Ž K SV rt 0 s 9.1 = 10 9 sy1 y1 . M attains a value that is smaller than that characteristic for diffusion control, indicating that the quenching process is dominantly controlled by the

M. Mac, B. Tokarczykr Chemical Physics Letters 304 (1999) 309–316

electron transfer from the bromide anion to the excited DClA molecule. However, in bimolecular systems the overall reaction rate constant may be limited by the diffusion of the reactants to the distance at which the reaction occurs w6x. To calculate the fluorescence quenching rate constant, we solved the diffusion equation with the source term being the Jortner equation for the electron transfer rate constant w19x at three DGet values 1, namely for DGet s y0.2, y0.07, and q0.07 eV, which correspond roughly to the electron transfer process from bromide anions, tetra-n-butylammonium bromide ion pairs, and lithium bromide ion pairs to the excited DClA molecule, respectively. The ratio of the fluorescence quenching rate constant for the quenching by bromide anions and LiqBry pairs exceeds 100 ŽTable 1., i.e. it is significantly larger than that X Y calculated from the K SV and K SV values. This means that the electron transfer is not exclusively responsible for the fluorescence quenching of DClA by nondissociated LiqBry molecules. An additional quenching mechanism may be an enhancement of the intersystem crossing ŽS 1 ™ T1 . occurring in the DClA molecule induced by ‘heavy’ bromine atom in LiqBry. Similar effects have been reported previously w5,6x. It has been found that fluorescence quenching rate constants of several aromatic molecules quenched by inorganic anions in methanol–ethanol mixtures are much higher than the predicted values calculated making the assumption that only electron transfer is the exclusive mechanism of the quenching. This effect is observed by the positive DG values and in the presence of ‘heavy’ atom quenchers such as iodide and bromide anions. For example, the fluorescence quenching rate constant of fluoranthene by bromide is equal 7.2 = 10 7 My1 sy1 , whereas the theory for electron transfer predicts a value of ; 10 5 My1 sy1 . In a previous paper w5x, we also investigated the activation energies of the fluorescence quenching of aromatic molecules by halide anions in an ethanol–methanol mixture. In the exothermic region of the electron transfer, the

1 ox Ž . red Ž . DGet sy E00 y E1r2 A q E1r2 D , where E00 , ox Ž . E1r 2 D are the energy of the first singlet state of

red Ž . E1r2 A , and the acceptor ŽA., the reduction potential of A, and the oxidation potential of the donor ŽD., respectively.

315

activation energies essentially resemble the value typical for activation energy of the diffusion Ž; 13 kJrmol.. With increasing DGet we observed that the activation energy of the fluorescence quenching rate constants initially increases and further monotonically decreases. This indicates the importance of another, barrierless deactivation pathway. Similar effects have been found by Carrigan et al. w20x for the fluorescence quenching of 5,6-benzoquinoline by bromide anions in aqueous solutions, where the rate constants showed a little variation with temperature. These effects may be attributed to the enhanced intersystem crossing due to a heavy atom. Further evidence for the importance of the heavy-atom effect in the fluorescence quenching processes has been pointed out by Steiner and co-workers w21x for the systems containing oxonine Žfluorescer. quenched by monohalogenated benzenes, toluenes, and anisoles in methanol. By exchanging the donor molecules they covered a broad region of the DGet for primary electron transfer reaction Ž DGet s y0.85 " 1.4 eV.. For very positive values of DGet , they found relatively large quenching rate constants of the order of Ž1–2. = 10 8 My1 sy1 in the case of bromine-containing donors and Ž6–9. = 10 8 My1 sy1 for iodine-containing donors. Moreover, they observed that the population of the oxonine triplet state takes place only in the presence of donors containing a heavy atom. The authors postulated that the external heavy-atom effect plays an important role in the quenching process for systems where the electron transfer process is thermodynamically less favourable. The same situation does appear in our system, the external heavy-atom effect is the additional mechanism of fluorescence quenching of DClA by nondissociated LiqBry.

5. Conclusions We have investigated the fluorescence quenching of 9,10–dichloroanthracene by two bromide salts in acetone in the presence of lithium perchlorate and tetra-n-butylammonium perchlorate. In the absence of perchlorates, the Stern–Volmer dependence exhibits a downward curvature for LiBr as the quencher, whereas for TBABr the S–V relationship is perfectly

316

M. Mac, B. Tokarczykr Chemical Physics Letters 304 (1999) 309–316

linear. The addition of perchlorates having a common cation with the quencher causes an increase in the fluorescence lifetimes and at 0.2 M LiClO4 a straightening of the S–V plots. For tetra-n-butylammonium perchlorate, the effect is opposite. These effects may be explained in terms of the influence of the inert salts on the dissociation equilibrium of lithium bromide in acetone. The association constants of LiBr and the S–V constants for the fluorescence quenching of DClA by free bromide anions and nondissociated LiBr molecules have been estimated from the S–V analysis. The association constants were compared with those obtained from the conductivity measurements as well as with those evaluated using Fuoss theory. It has been found that the agreement between the results obtained from these two different experimental methods are quite reasonable and that they agree well with the predictions based on the Fuoss equation for the stability constant of the ionic pairs. The quenching abilities of the free bromide anions and ionic pairs were discussed on the basis of electron transfer theory. It seems that the quenching of DClA by nondissociated LiBr molecules is of mixed character, with electron transfer and an external heavy-atom effect both being involved in the quenching process.

Acknowledgements We thank Dr. P. Nowak and Dr. J. Czapkiewicz for the gift of tetra-n-butylammonium salts, Dr. P. Milart for purification of TBABr, and Dr. W. Jarzeba for discussions and support. The authors are also indebted Dr. A.M. Turek for editorial comments.

This work was supported by State Committee for Scientific Research Žgrant No. 3 T09A 096 12.. References w1x A.R. Watkins, J. Phys. Chem. 72 Ž1973. 1207. w2x A.R. Watkins, J. Phys. Chem. 78 Ž1974. 1885, 2555. w3x H. Shizuka, T. Saito, T. Morita, Chem. Phys. Lett. 56 Ž1978. 519. w4x H. Shizuka, M. Nakamura, T. Morita, J. Phys. Chem. 84 Ž1980. 989. w5x J. Najbar, M. Mac, J. Chem. Soc., Faraday Trans. 87 Ž1991. 1523. w6x M. Mac, J. Najbar, D. Phillips, T.A. Smith, J. Chem. Soc., Faraday Trans. 88 Ž1992. 3001. w7x M. Mac, J. Wirz, J. Najbar, Helv. Chim. Acta 76 Ž1993. 1319. w8x M. Mac, J. Wirz, Chem. Phys. Lett. 211 Ž1993. 20. w9x M. Mac, J. Najbar, J. Wirz, J. Photochem. Photobiol. A: Chem. 88 Ž1995. 93. w10x T.J. Kemp, L.J.A. Martins, J. Chem. Soc., Faraday Trans. 77 Ž1981. 1425. w11x M. Mac, P. Millart, P. Kwiatkowski, B. Tokarczyk, J. Lumin. 81 Ž1999. 199. w12x F. James, M. Ross, Comput. Phys. Chem. 20 Ž1980. 29. w13x S. Petrucci, E.M. Eyring, J. Phys. Chem. 95 Ž1991. 1731. w14x M. Mac, unpublished results. w15x Landolt–Bornstein: Zahlenwerte und Funktionen aus Physik, ¨ Chemie, Astronomie, Geophysik und Technik, vol. 7, Springer, Berlin, 1959, p. 667. w16x J.O’M. Bocris, A.K.N. Reddy, in: Modern Electrochemistry, vol. 1, ch. 4, Plenum, New York, 1973. w17x C.D. Clark, M.Z. Hoffman, Coord. Chem. Rev. 159 Ž1997. 359. w18x N. Sarkar, K. Das, D. Nath, K. Bhattacharyya, Chem. Phys. Lett. 218 Ž1994. 492. w19x J. Jortner, J. Chem. Phys. 64 Ž1976. 2090. w20x S. Carrigan, S. Doucette, C. Jones, C.J. Marzzatto, A.M. Halpern, J. Photochem. Photobiol. A: Chem. 99 Ž1996. 29. w21x R.E. Foll, ¨ H.E.A. Kramer, U.E. Steiner, J. Phys. Chem. 94 Ž1990. 2476.