6642
J. Phys. Chem. 1991, 95.6642-6641
repulsions, which are much stronger than Bi-Bi dipole repulsions, cause the Cs, to be much more uniformly spread than Bin. As a consequence of both these effects, Cs, will more efficiently block platinum sites needed for hydrogen abstraction. These, we feel, are the dominant reasons that Cs, poisons the dehydrogenation of cyclopentene and cyclohexane faster than Bi,, when compared on a Yper adatom" basis. Of course, there are other differences between Cs, and Bi, due to the stronger electronic effects of Cs,, which will also affect the dehydrogenation probability. For example, the stabilization of H, by Cs,, and the consequent increase in the dehydrogenation rate constant, will cause Cs, to poison dehydrogenation less effectively than Bi,. This effect is clearly less important than the effect of the lateral distribution of adspecies brought about by hydrogen43 attractive lateral interactions and Cs-Cs repulsions. For cyclopentene and cyclohexane, the decrease in the desorption rate constant due to Cs, will also cause Cs, to poison dehydrogenation less effectively than Bi,. Again, this is clearly not as important as the effect of the nonrandom lateral distribution due to hydrocarbon-Cs attractions. Only in the case of benzene is dehydrogenation poisoned a t similar rates by Cs, and Bi,. However, even this is due to a fortuitous cancellation of effects ( 2 ) and (3) above. That is, while the presence of Cs, causes ( 2 ) an increase in kd&, due to a destabilization of Ha, this is nearly balanced by (3) an increase in kdcsdue to electrostatic destabilization of C6D6, by Cs,. V. Conclusions The coadsorption of cesium with benzene, cyclohexane, and cyclopentene on the Pt( 111) surface yielded a poisoning of the
dehydrogenation pathway of these hydrocarbon molecules, with a corresponding enhancement in the amount of molecular desorption. This net result evolves from a balancing of three effects: (1) an ensemble effect in which cesium adatoms sterically poison platinum surface sites required to abstract hydrogen atoms from adsorbed hydrocarbon molecules, ( 2 ) an increase in the rate constant for dehydrogenation (kdeh)resulting from the electronic stabilization of adsorbed hydrogen by cesium adatoms, and (3) an electronically induced change in the rate constant for desorption (kb) with cesium coverage. We have shown that cesium poisons the dehydrogenation of the hydrocarbons studied in this paper, mainly as a result of the ensemble effect. However, the electrostatic field created by adsorbed cesium adatoms did have a noticeable influence on the overall probability for dehydrogenation. For cyclopentene and cyclohexane adsorbed to cesium-precovered Pt( 111) surfaces, attractive interactions between the alkali metal and the hydrocarbons were obvious. It is proposed that these adspecies therefore cluster, poisoning the dehydrogenation pathway via ensemble effects faster than that expected for the poisoning that would result from a purely random distribution of adspecies across the surface. For benzene coadsorbed to cesium-precovered Pt(l11) surfaces, effects ( 2 ) and (3) above appear to approximately cancel, thus resulting in a poisoning of the dehydrogenation pathway at a similar rate as that expected for poisoning resulting from purely ensemble effects.
Acknowledgment. We acknowledge the National Science Foundation for support of this research. We also acknowledge Steven Pauls and Richard Madison for their assistance in data collection.
Ultrasonic Reiaxatlon and Electrochemical Measurements on Monomer/Miceile Exchange in Dodecylpyridinium Bromide Solutions Containing Added Sodium Bromide W. A. Wan-Badhi, R. Palepu, D. M. Bloor, D. G.Hall, and E. Wyn-Jones* Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4WT, U.K. (Received: November I , 1990) The ultrasonic relaxations associated with the perturbation of the monomer/micelle equilibrium in dodecylpyridinium bromide (DPBr) have been measured in the presence of lo4 and lo-' mol dm-' sodium bromide (NaBr). The emf of a specially constructed dodecylpyridinium cationic surfactant membrane electrode relative to, successively, a bromide ion, sodium ion, and also a liquid junction electrode has been measured, and the monomer concentration of DP+ in the micellar range and the effective degree of micellar dissociation have been evaluated. The combined dynamic and equilibrium data have been used to investigate the kinetics of the monomer/micelle exchange process by using different versions and modifications of the so-called fast relaxation time equation of Aniansson and Wall and also an also an alternative approach to investigate kinetics-a phenomenological treatment. The scope and limitations of these approaches to derive new micellar parameters are discussed. From the concentration dependence of the amplitude of the ultrasonic relaxation the volume change associated with the monomer/micelle equilibria has been estimated. In addition density measurements and light scattering measurements were taken.
Introduction In aqueous surfactant solutions containing micelles there is a dynamic exchange process occurring between surfactant monomers in bulk solution and aggregated surfactant monomers in the micelle. In chemical relaxation studies the perturbation of this equilibrium leads to the so-called fast relaxation time r1.I+ As (1) Aniansson, E. A. G.; Wall, S. N. J . Phys. Chem. 1974, 78, 1024. (2) Aniansson. E. A. 0.;Wall, S.N. J . Phys. Chem. 1975, 79, 857. (3) Aniansson, E. A. 0.;Wall, S. N.; Almgren, M.; Hoffmann, H.; Kicbmann. 1.; Ulbricht. W. J.; Zana, R.; Lang, J.; Tondre, C. J . Chem. Phys. 1976. 80, 905.
(4) Chemical and Biological Applications of Relaxation Spectrometry; Wyn-Jones, E. Ed.; D. Reidel Publishing Co.: Dordrecht, The Netherlands, 1975; pp 132-264.
0022-3654/91/2095-6642$02.50/0
a result of a comprehensive treatment of micellization kinetics
by Annianson and Wall1-' it was claimed that the analytical relaxation expression for the concentration dependence of 1/ T I contained new information on the micellar systems. Strictly speaking the Annianson and Wall theory only applies to nonionic surfactants? whereas the vast majority of the experimental dataCd refer to ionic surfactants. In order to allow for this omission an additional term involving counterion binding has been introduced to the original relaxation Finally a further modi(5) a n a , R. Surfactant Solutions: New Methods of Investigation; Zana, R., Ed.;Marcel Dekker: New York, 1987. (6) Gormally, J.; Wyn-Jones, E. Molecular Inieraciions; Ratajczak, H., Orville-Thomas, W. J., Ed.; John Wiley and Sons, Ltd.: New York, 1981, Vol. 2, p 143.
0 1991 American Chemical Society
Monomer/Micelle Exchange in Dodecylpyridinium Bromide
The Journal of Physical Chemistry, Vol. 95, NO. 17, 1991 6643
fication9 taking into account the ionic strength of the solution has been proposed, the starting point of this development beiig a linear phenomenological treatment. In the applications of all these treatments to evaluate the "new" micellar parameters, the prerequisite experimental informations required are the relaxation data at different overall surfactant concentrations and equilibrium data involving monomer surfactant concentrations in the intermicellar region and also the amount of micellar counterion binding commonly referred to as the degree of micelle dissociation a. A survey of the literature reveals that with the exception of the work in refs 7 and 8 and a recent communication from this laboratoryL0 the only information available on micellar systems is the relaxation data. As a result, in the absence of the equilibrium data is normal practice to assume that in the micellar range the monomer surfactant concentration is constantM and assumed to be equal to the critical micellar concentration (cmc) of the surfactant and to ignore any effects due to counterion binding. As has been recently demonstrated with surfactant selective electrodes,l1-l3 this approximation is very often incorrect and the monomer surfactant concentration is found to decrease significantly with increasing surfactant in the micellar region. In order to assess the scope and limitations of the usefulness of the relaxation data in providing new information on micellar systems, we have carried out a systematic and thorough study of the cationic surfactant dodecylpyridinium bromide (DPBr) at two different concentrations of sodium bromide (NaBr), using ultrasonic relaxation and emf data involving the surfactant selective electrode, a sodium ion electrode, and a bromide ion electrode. In addition, density and light scattering data were measured. These data have been used to investigate the various modifications of the fast relaxation time equation. Whereas the original version of the Annianson and Wall equation was derived by actually solving, in an ingenious the differential equations associated with the multistep micelle formation from monomers, a separate generalized treatment of association kinetics now exists-the general phenomenological approach, which was developed in this 1aborat0ry.I~ The experimental data obtained in this work are also used to compare the same micellar information derived from these two independent approaches. Experimental hocedure a. Emf Data and Evaluation of Effective Degree of Micellar Dissociation, u. A surfactant membrane electrode selective to the dodecylpyridinium cation (DP+) was constructed by using a procedure that has been described in previous publi~ations.'O-"~ In all the experiments that were carried out the aqueous DPBr solutions were doped with a small but constant amount of sodium bromide (NaBr). In these solutions the free concentration of the sodium ions is constant because the cationic surfactant and sodium ions carry the same charge. (This can be confirmed by using a calomel electrode as a reference.) When micellization occurs some of the bromide ions in solution will start binding onto the cationic micelles. In the electrochemical experiments on these micellar solutions, the emf of the surfactant electrode was measured relative to a commercial sodium electrode (Kent, EIL), and from these data the emf of the surfactant electrode can be plotted against surfactant concentration. During the experiment simultaneous emf measurements were also taken of the surfactant electrode
(7) Lcssner, E.; Teubner, M.; Kahlweit, M. J . Phys. Chem. 1981,85, 1529. ( 8 ) Elvingson, C.; Wall, S. N. J. Colloid Interlace Sci. 1988, 121, 414. (9) Hall, D. G. J . Chem. Soc., Faraday Trans. 2 1981, 77, 1973. (IO) Gharibi, H.; Takisawa, N.; Brown, P.; Thomason, M. A.; Painter, D. M.;Bloor, D. M.; Hall, D. 0.;Wyn-Jones, E. J. Chem.Soc.,Faraday Trans. 1991, 87, 707. ( 1 1 ) Painter, D. M.; Hall, D. G.; Wyn-Jones, E. J . Chem. SOC.,Faraday Trans. I 1988,84, 773. (12) Takisawa, N.; Brown, P.; Bloor, D. M.; Hall, D. G.; Wyn-Jones, E. J . Chem. Soc., Faraday Trans. I 1989,85, 2099. (13) Palepu, R.; Hall, D. G.; Wyn-Jones, E. J. Chem.Soc., Faraday Trans. 1990,86, 1535. (14) Hall, D. G.; Gormally, J.; Wyn-Jones, E. J . Chem. Soc., Faraday Trans. 2 1983, 79, 643.
1% T
Figure 1. Plot of emf of surfactant electrode as a function of DPBr concentration: ( 0 ) IO4 mol dm-' NaBr; (+) lo-' mol dm-) NaBr.
-19
4 I9
I
-IS
I7
-16
IS
.I4
13
11
I I
I
09
lo(lwd
Figure 2. Determination of a for DPBr using eq 1. Plot of log (mly*) vs log (mzrr): ( 0 ) lo4 mol dm-3 NaBr; (+) 10-I mol dm-' NaBr.
relative to a commercial bromide electrode (Kent, EIL); the purpose of this experiment was to estimate the amount of counterion binding. Typical plots of the emf of the surfactant electrode as a function of DPBr concentrations are shown in Figure 1. The main features of the plots are as follows: (i) There is a distinct break a t the critical micellar concentration (cmc). (ii) In the concentration range when DPBr exists as monomer in solution (below the cmc), the emf data show almost perfect Nernstian behavior. (iii) Once micelles are formed the monomer surfactant tends to decrease with increasing surfactant concentration at very low sodium bromide concentration but tends to level off with surfactant concentration as the sodium bromide increases. From these data it is possible to evaluate the monomer surfactant concentration m 1in the micellar region. In the micellar range the monomer Br- ion concentration mz and the mean activity coefficient Ti can be evaluated from the monomer surfactant concentration m land the emf of the surfactant electrode relative to the bromide electrode. The procedure used for the estimation of these parameters has been described in a previous publication." Once MI,mz,y+, and the cmc of DPBr in the presence of different but constant amounts of sodium bromide are known, it is possible to evaluate the effective degree of micellar dissociation a by using two independent m e t h o d ~ . ' ~ * ' ~ J ~ The first is by use of eq 1 log (m,r+) = K - (1 - a) log
hrt)
(1)
where K is a constant. A plot of log (mly+) against log (mzra) can be constructed from the values of MI, Y+, and m2 derived for all salt concentrations and is a straight line of slope -( 1 - a)as shown in Figure 2. The a value obtained for this plot was 0.21. (15) Hall, D. G. J . Chem. Soc., Faraday Trans 11981, 177, 1121. (16) Hall, D. G. Aggregation Processes in Solurion; Wyn-Jones, E., Gormally, J., Eds.; Elsevier: Amsterdam, 1983; Vol. 26, p 7.
6644 The Journal of Physical Chemistry, Vol. 95, No. 17, I991
Wan-Badhi et al.
1.15
Q.7
a
-
\
I I O t
I
0.8 O
0.1
02
03
OL
OS
L6
0.8
0.7
0.9
b l ( 35) Figure 3. Determination of a for DPBr using dependence of cmc on salt concentration (q2).
Second, from the dependence of the cmc on salt concentration, it has been shown that log [ c m ~ r ~ / c m c ~ = r+~] -(I - a) log [(cmc + m3)yh/cmc0yh0] (2) where m3 is the concentration of added salt and the superscript 0 refers to zero NaBr concentration. From this experiment a plot of the left side of eq 2 against log [(cmc + m3)yh/cmc0yh0] is a straight line of slope equal to -( 1 - a) as shown in Figure 3, giving an a value of 0.20. Finally, we have camed out a separate emf experiment on DPBr solutions by measuring the emf of both the surfactant electrode and bromide electrode against a double junction type reference electrode supplied by Orion Research (Cambridge, MA) and represented by AglAgC11KCl(sat)(KN03 or N H 4 N 0 3 From these data, we can use the well-documented charge phase separation model" to evaluate a by using the equation E + / S I = constant - (1 - a ) E - / S 2 (3) where E+ and E- refer respectively to the emf's of the surfactant and bromide electrode measured relative to the double junction electrode and SIand Sz are the slopes of the respective emf concentration plots in the monomer surfactant region. From the plot of ,!?+/SI against E 4 S 2 shown in Figure 4 an a value of 0.21 was found. b. Ultrasonic Relaxation Experiments. Ultrasonic absorption and velocity measurements were taken with the Eggers resonance m e t h ~ d , which ~ ~ J ~covers the frequency range 0.2-20 MHz, and a conventional pulse method over the range 15-90 MHz. The ultrasonic measurements were carried out in micellar solutions of DPBr in the presence of lo4 and also 10-1 mol dm4 NaBr. In all cases, a well-defined single-relaxation process was observed and the absorption data were fitted to the single-relaxation equation
./f
(4) = A / [ 1 + V/Ll21+ f3 where a is the sound absorption coefficient at frequencyf, A is a relaxation amplitude parameter,f, is the relaxation frequency, and B represents contributions to a/f that are independent of frequency. The observed relaxation is attributed to the perturbation of the monomer/micelle equilibrium process in DPBr, and the pertinent relaxation parameters are T~ the fast relaxation time and p, the maximum absorption per wavelength unit. These are given by 71 = 1/2TA pm
= !M~fc
(5) ~
~
~ ~ _ _ _ _ _
(17) Macda, R.; Satakc, I. Bull. Chcm. Soc. Jpn. 1984, 57, 2396. (18) Eggets, F. Acousrica 1968, 19, 3223. (19) Eggers, F.;Funk, J. L.; Richmann, K. H. Rw. Sci. Insrrum. 1976, 42, 361.
im
I
im
I IO
im
I W
140
IJO
ita
ldhn?
Figure 4. Determination of a for DPBr using charge phase separation model (q3).
where v is the measured sound velocity. c. Density Measurements. The densities of various solutions of DPBr in both the monomer and micellar range were measured at 298 K by using a commercial density meter (DMA 55; A. Paar, Austria). d. Light Scattering Measurements. The aggregation number of micellar DPBr in 10" and lo-' mol dm-3 NaBr was found to be 70 and 80, respectively. The values were kindly evaluated by Dr. D. Attwood (University of Manchester, England), using light scattering measurements.
Analysis of Data We now turn our attention to the analysis of the concentration dependence of the fast relaxation time r1 by making use of the relaxation equation (6), which has been derived for the monomer/micelle exchange process by Aniansson and WalL3 where C is the total surfactant concentration, n the micellar aggregation number, k- the rate constant for the dissociation of a monomer from the micelle, and u the width of the micelle distribution. It should be noted that this equation was derived basically for nonionic micelles but has been applied exclusively for ionic surfactants. An inspection of literature on micellar kinetics3" reveals that all investigators have plotted I/rl against (C- ml)/ml and used the slope and intercept of the straight line plots to evaluate k-/n and k-/a2. In most micellar systems n is known; thus, k- and u can be found. With the exception of ref 7, 8, and 10 all the reported analyses of eq 6 have been carried out by assuming that ml is constant in the micellar region and equal to the cmc of the surfactant. This is an approximation that is normally used in micellar systems, especially in the absence of data on the monomer concentration in the micellar region. As a result of progress with the surfactant selective electrodes, we have demonstrated in the previous section that it is possible to measure m l in the micellar region. For the micellar solution of DPBr in the presence of lo4 mol dm-) NaBr, the respective 1/71 plots against (C - ml)/ml, with m1 = constant = cmc and the actual mi values measured by using the electrode, are shown in Figure 5a,b, respectively. The values of the respective intercept, k-/a-, and slope, k-/n, are in Table I. The significant feature is that the difference in the respective slopes is almost an order of magnitude. This clearly suggests that the assumption made by many previous authors in the analysis of relaxation data, viz. m, = constant = cmc leads to large errors in the micelle parameters. As we have stated previously, eq 6 applies to nonionic micelles. In order to account for the degree of micellar dissociation cy for an ionic micelle, eq 6 has been extendedls to give I/r, = k-/d
+ ( k - / n ) [ l + (1 -
+ ao)]a
(7)
where a = (C - ml)/ml,. This equation was derived on the basis of Aniansson and Wall kinetics. Hall? starting from a linear phenomenological approach, introduced a further correction for ionic micelles, and the following
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6645
Monomer/Micelle Exchange in Dodecylpyridinium Bromide 16
14
12
4
2
I
I
I
I
I
I
I
5.
7
6 5 E
Q
P
4
2 1
0
0.00
I I
I I
I I
I
I
I
,
1
I
1
I
I
5 .OO
10.00
15.00
20.00
25.00
30.00
35.00
40.00
Coeficient of Y x 10)
Fipn 6. Plot of I/rl vs coefficient of k-/n for DPBr in IO-’ mol dm” NaBr: (+) a; (A)b; (0) c; (m) d. See text for comments a-d.
equation (8) can be derived from an extensive treatment of micellar kinetics: I/rl = k - / u 2
+ ( k - / n ) [ l + ( 1 - CX)~/(I+ aa) + 2y’ml]a (8)
where 7’ = -[0.587/b(1 + L J ) ~ ] b. = (ml + m3)1/Z, and m3 is the concentration of added salt. It should be noted here that eq 8, in the form expressed above, is not in the original paper. Here it is stated in a form suitable for the graphical analysis of the relaxation data and is derived by algebraic manipulation of eq 23 in the original paper.9 By use of the relaxation and electrode data for micellar DPBr in the presence of IO4 NaBr, it is possible to use eqs 7 and 8 by plotting 1 / r I against the respective concentration functions and obtain linear plots giving further refined estimates of &-/a2and
k-/n, as shown in Figure 5c,d with the data again listed in Table I. Once the electrode data have been taken into account, there is very little difference in the slopes and intercepts of the respective plots using the different corrections/additions in the relaxation equations, (6),(7), and (8). The large differences only occur when ml = cmc. The next question to address is how these differences can be avoided or under what conditions one can use the assumptions ml = cmc = constant in the analysis of the fast relaxation time. The clue to this question is in Figure 1, where the emf data for DPBr are shown for the case of 10-I mol dm-3 NaBr. In the presence of this relatively high salt concentration, the monomer surfactant concentration is constant and equal to the cmc in the micellar range. The respective relaxation plots corresponding to Figure 5 a-d are shown in Figure 6. In the above treatment we have examined all the options currently available to investigate the kinetics of monomer/micelle
Wan-Badhi et al.
6646 The Journal of Physical Chemistry, Vol. 9.5, No. 17, 1991 TABLE I: Kinetic Panmetem for D o d e c ~ b ~ r i dBromide i d ~ ~ ~ (DPBrI measd method of anal. param" value/106b value/106c cq6 k-/n 0.9 f 0.05 0.2 f 0.01 m, = const = cmc k-ld -0.3 (max = 0.1) -0.1 (max = 0.3) k64 4 17*1 eg6 k-/n 0.1 f 0.005 0.2 0.01 ml from emf data keto2 1.4 -0.03 (max = 0.2) k8 f 0.04 16 i 0.08 4 7 k-/n 0.1 f 0.005 0.2 0.01 k-/a2 1.2 -0.2 (max = 0.1) k8 0.04 15 f 0.8 eq8 k-/n 0.1 f 0.005 0.2 f 0.05 k-/a2 1.2 -0.2 (max = 0.1) k7 0.4 15 f 0.8 eg 15 k-/n 0.1 0.005 0.2 0.01 k9 f 0.5 17 0.08
"T
*
*
*
*
oValuca of k-/d may only be considered to be estimates. For DPBr in
lo-'mol dm-' NaBr. CForDPBr in lo-' mol dm-) NaBr.
exchange in DPBr solutions, using relaxation and emf data. The boundary conditions imposed on the analysis depend on how much experimental information is available. Up to now the majority of work reported on this subject has been confined to experimental data from chemical relaxation exwriments. in which case the data were analyzed by using eq 6 with ml = cmc = constant in the micellar range. Although this type of analysis seems to be consistent in the sence that the relaxation equation can account for the experimental data (see Figure 5a), the resulting micellar parameters derived from the slopes and intercept of the straight line are almost an order of magnitude different from the values obtained when mI has been properly accounted for in the micellar region. The merits of using eqs 7 and 8 to take into account the properties of ionic micelles are questionable in the sence that once mI is taken into account in eq 6 further use of a and y' terms does not really give rise to any further advantages. If we take into account the errors in the experimental relaxation time, which are of the order f58, then the slopes of the linear plot leading to k-/n are not very different. However, the intercepts can only lead to estimates of u. These are found by using an extrapolation procedure that results in values of k-/u2 being extremely small and sometimes even negative when a linear least-mean-square analysis is applied to the data. If however we take into account the experimental errors in the l/sl values, then the upper limits of the k-/u2 values are always positive. It is not however possible to make any sensible comments about the magnitude of u in these circumstances. This situation has been highlighted if we read between the lines of the original attempts to analyze the relaxation d a h 3 We consider that for the micellar solutions of DPBr in 10-4 mol dm-3 NaBr the u values are at best estimates, whereas no conclusions may be made about u from the data for DPBr in lo-' mol dm-3 NaBr. The overall conclusions from these experiments is that if a perspective or overall impression of micelle kinetics is required, reasonably accurate rate constants can be obtained from relaxation experiments only if the measurements are carried out in solutions containing excess (10-I mol dm-3) of added salt. At this ionic strength it becomes very difficult to estimate u. On the other hand, if the monomer concentrations are known in the micellar region, then for surfactant solutions containing extremely small amount of added salt (- lo4 mol dm-') it is possible to obtain k- and an approximate value of u. Amplitude Analysis The expression for the concentration dependence of the relaxation amplitude parameter the maximum absorption per wavelength unit p, arising from the perturbation of the monomer/micelle equilibrium has been derived by TeubneP and takes the form of the following expression: P m = (?.~o~/2R'I?((A~'m1(a2/n)a/[1 + (u2/n)all ( 9 ) where p is the density of the solution and AV the volume change (20)Teubner, M.J . Phys. Chem. 1979.83, 2917.
OM 0
t
I
*.Id I I
Figure 7. Plot of p, vs u according to eq 9: (0)lo-' mol dm-3 NaBr; ( 4 ) lo-' mol dm-3 NaBr. Solid lines are theoretical, taken from data in Table 11. TABLE II: Amplitude h l y r i s for Dodeylpyridiaium Bromide (DPBr) m e a d Param' valueb valueC method of anal. ea9 106AV.m3 mol-l 11 & 2 7f2 -
I
eq 10
u7/n 106AV, m3 mol-I
0.1
4
10f3
9 f 3
a Values of u2/nmay only be considered to be estimates. For DPBr in IO-' mol dm-3 NaBr. CForDPBr in lo-' mol dm" NaBr.
associated with the process. All the other terms in the above equation have been defined previously. As a result of the kinetic treatment described above, we do not consider it necessary to introduce additional terms due to micellar dissociation and ionic strength terms in the above equation. Experimentally p,, p , 0, and a are known, and it is possible to analyze the concentration dependence of the amplitude data with a least-mean-square procedure using a two parameter fit in u2/n and AV. The data are presented in Figure 7, where the solid lines represent the result of the fitting procedure and the points are experimental. The resulting values of these micellar parameters are in Table 11. The noteworthy features concerning the above fitting procedure involving the concentration dependence of pm are that once micellization occurs pmincreases rapidly with increasing concentration and reaches a maximum, and then the further concentration dependence is dominated entirely by the ml term on the right side of eq 9, i.e. in the monomer concentration that decreases with increasing concentration of DPBr in the presence of lp mol dm" NaBr and is constant in the 10-1mol dm-' NaBr micellar solutions. The goodness of fit displayed in Figure 7 shows the consistency between the p, and mI values. The highest value of pmdepends exclusively on the AV value and once this has been fixed, a range of values of u2/nvarying by as much as one order of magnitude are acceptable. This again means that only order of magnitude estimates of u are possible. It is also possible to estimate A V values from density measurements2 by using eq 10 where M Iis the molecular weight of A V = -[(Mi + C*Z)/(P*)~I [ ( a p / a c ~ ) m o n- ( a p / a c ~ ) m i c l (10)
the solvent; C, = ( N z / N I )where N 2 is the number of moles of surfactant, N1the number of moles of solvent, and M 2the molecular weight of surfactant, with the superscript (*) referring to measurement taken at the cmc. (ap/aC2)with the appropriate subscripts are the respective slopes of the plots of density vs C2 in the monomer and micellar regions. The values of AVobtained by using this method are also listed in Table 11. Phenomenological Approach In this laboratory we have developed a novel phenomenological treatment to handle and interpret relaxation data on complex (21) Benjamin, L. J . P h p . Chem. 1966, 70,3790.
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6647
Monomer/Micelle Exchange in Dodecylpyridinium Bromide "T nm
t
/
in brackets on the right side follows from the amplitude analysis of eq 9 described above. In these circumstances we can evaluate the equilibrium forward (= backward) rate RI associated with the aggregation process described in (11). Once the equilibrium rate has been evaluated in this manner, we can then revert to conventional kinetics in the sense that, if we consider the backward rate process Rb involving the dissociation of monomers from the micelles, then one expects this to be a first-order rate process depending on the concentration of the micelles. Thus
R I = Rb = ( k - / n ) ( C - mi) l
0.M
o
a
u)
m
a (Cad x
im
IM
110
IM
xa
im
Id I d dm.7
(c-
Figure 8. plot of backward rate (Rb)vs micellar concentration m,) according to cq 15: ( 0 ) 1 P mol dm-' NaBr; (+) IO-' mol dm" NaBr.
aggregating systems." Although the treatment has been successfully used on a number of different aggregating phenomena,11Jz.2z,z3 we have only oncez4been in a position, until now, to compare the kinetic data from this general approach with a relaxation evaluation like (6),which has been derived by using the conventional relaxation approach, albeit in an ingenious way. In the following section we describe the phenomenological approach and compare the kinetic data with those derived from the treatment of eq 6 using the emf data. In general, the prerequisites for the application of the general phenomenological treatment are as follows: (i) The process being perturbed by the sound wave can be described by a general one-step equilibrium. (ii) Ultrasonic relaxation arising from the perturbation of such an equilibrium is a well-defined single process. (iii) Equilibrium data are available concerning the concentrations of monomer and aggregated species. In the case of the monomer/micelle equilibrium all the above conditions are met, and the equilibrium can be defined as "free" monomer f "micellar" monomer
(1 1)
In these circumstances, the phenomenological equations of interest to the analysis of ultrasonic relaxation data are 1/ 7 , = ( R t / R r ) ( W W , pm
dAV)~u~/2(aA/a€)c
(12) (13)
where A is the reaction affinity, ( is the extent of the reaction, c denotes that the derivatives are evaluated in a closed system. R I is the forward (= backward) rate of the process. Combining eq 12 with eq 13 we obtain pm/rl
= [(dAv)z/2Rr)~u21RI
(14)
In the last equation, p m and 71 on the left side are known for the ultrasonic relaxation experiment, and the thermodynamic term (22) Kelly, G.; Takisawa, N.; Bloor, D. M.; Hall, D. G.; Wyn-Jones, E. J . Chem. Sa.,Faraday Trans. I 1989,85,4321. (23) Gormally, J.; Natarajan, N.; Attwood. D.; Gibson,J.; Hall, D. G.; Wyn-Jones, E.J . Chem. Soc., Faraday Tram. 2 1984,80,243. (24) Jones, P.;Tiddy, G.J. T.; Wyn-Jones, E. J. Chcm. Soc., Faraday Trans. 2 1981.83, 2135.
(15)
The plots of Rb evaluated from the procedure described above against (C - m l ) for DPBr in the presence of different amounts of sodium bromide are shown in Figure 8. In both cases, they are straight lines passing through the origin clearly showing that eq 15 is appropriate, and the values of k- are in Table I. It is very encouraging to note that these k- values agree well from those derived by using the relaxation equations (6)-(8). Conclusions We consider that the work described in this communication represents the most comprehensive treatment to date of the analysis of the fast relaxation time for micellar kinetics. The following noteworthy features were observed: (1) In the absence of equilibrium measurements of monomer surfactant in the micellar range, the analysis of the Aniansson and Wall equation (6) where ml = cmc leads to micellar parameters that are unacceptable unless the measurements are carried out at high salt concentration (-lo-' mol dm-3). (2) The refinements of the original Aniansson and Wall equation to take into account micellar dissociation and the ionic strength of the solution are included in eqs 7 and 8, respectively. However, when the relaxation and emf data are analyzed by using these equations, it is questionable whether the values of the resulting micellar parameters justify such detailed modifications and extra experiments. (3) If only relaxation experiments are available to investigate this phenomenon, then it is advisable to carry out the measurements in reasonably concentrated salt solutions, in which case the assumption ml = cmc is justified. (4) Even when circumstances are favorable, only an order of magnitude estimate of u can be obtained. ( 5 ) Once the monomer concentration has been accounted for in the relaxation equations (6)-(8), the dissociation rate constant k- is fairly constant and agrees very well with the value found from the general phenomenological treatment. (6) In the amplitude analysis the AV values agree well with those found from density measurements. (7) ml decreases in the micellar range. The forward rate coefficient must therefore vary with concentration since k+/k= l/ml.
Acknowledgment. W.A.W.-B. thanks the Malaysian Government and the Agriculture University of Malaysia for a postgraduate studentship. R.P. thanks the authorities of the University College of Cape Breton, Sydney, Canada for granting sabbatical leave. Support by the SERC is also gratefully acknowledged.
