4483
J. Phys. Chem. 1988, 92, 4483-4490 4.00
k, and D, estimates were obtained on the basis of an assumed radius of 21 A for the cylindrical micelles:
I
D, = DBp+ Dp = 9 kq = 2.4
1.00
'
0
I
300
I
600
,
9 00
_.
.. I
X
X
lo-" m2 s-l
lo7 s-l
These results are very reasonable. The decay data gave no indication on a migration of benzophenone during time window studied. A result quoted by Russell et alez9allows an estimation of the distribution of benzophenone between water and micelle and from that the importance of migration. The distribution constant for benzophenone between octanol and water is given as KD = 1514. Equation 25 can be written
.I
1200
t i i i e Ins1
Figure 5. Fluorescence decay curves of pyrene, 2 X M, in micelles of CTAC, 40 mM. Curve 0 represents the decay without quencher, and 1-5 represent decays with 4.45 X lo4 M benzophenone as quencher. Sodium chlorate was added in decays 2-5: 34 mM in 2, 42 mM in 3, 49 mM in 4, and 64 mM in 5. Additions of 72 and 100 mM NaC103 gave curves very similar to curve 5 .
Insertion of Do = 5 X mz s-I, 4 = 0.01 (40 mM CTAC), R , = 20 A, and KD = 1500 gives T , = 1.4 X s. Since the time window utilized is about 1.4 ~ s t ,0; 7 = m ; T>O;
7=0;
ae
@=O; C = l
--=NUAT; a7
(21)
ae
E+/3C-=0 a7
a7
(22)
where Sc, Pr, and Nu are the Schmidt, Prandtl, and Nusselt numbersZof the electrolyte, respectively, and H ( 7 ) is the hydrodynamic velocity function for a RDE. The last boundary condition stipulates that the electrolyte flux is zero at the electrode surface in accordance with the fact that the supporting electrolyte itself is not electrochemically active. We have considered in this example a thermal perturbation in which for T > 0 a constant heat flux is established at the electrode surface. This mathematical relation appropriately describes the situation in our experiments in which a constant flux laser beam impinges on the nonelectrolyte side of a “thermally” thin electrode disk. In eq 22, the thermal gradient at the electrode is prescribed such that, for a given Nu, a surface temperature difference AT is attained at steady state. For aqueous 7 and Nu 0.9817. A numerical procedure electrolytes Pr involving an orthogonal collocation algorithm, similar to that discussed elsewhere3 for integrating the heat transport equations, was used to obtain a solution to the system of eq 18-22. In Figure 1, we show numerical results obtained for the perturbation in the surface concentration as a function of time, following thermal flux steps of sufficient magnitude to lead to AT of 1 or 5 OC, in a dilute solution (0.01 M) of KCl. The value p = 0.001 for this sytem was determined from the heats of transport given by Snowdon and Turner.” It is apparent that even for a surface temperature perturbation of 5 OC, which is usually higher than those employed in thermal modulation experiments on the
-
RDE, the thermodiffusion perturbation in the electrolyte concentration is less than 0.1%. Consequently, the principal contribution to the thermodiffusion potential derives from the second term in eq 9. For most inorganic salts, the value of p is commensurate with that of KCl in dilute solutions, and appreciably smaller at higher electrolyte concentrations. For these systems, the thermodiffusion potential generated at a heated RDE is obtained from eq 9 by assuming a negligible variation in the electrolyte concentration throughout the boundary layer region. The temperature coefficient of the thermodiffusion potential on a RDE, tsoret(RDE),is then given by
F
AT= 1 OC
9.998
Valdes and Miller
-
(1 1) Snowdon,P.N.;Turner, J. C . R. Trans. Faraday Soc. 1960,56, 1409.
-DA)
(23)
Since the concentration perturbation from the thermodiffusion is found to be small, the first term in eq 9 will be small in comparison to the second term. Therefore, the temperature coefficient of the thermodiffusion potential, E,in both the thermogalvanic cell and the RDE will be the same and is that given by eq 23. Moreover, the thermal coefficient of the thermodiffusion potential in a multicomponent system can be obtained from eq 16 which assumes a negligible concentration perturbation. The magnitude of the concentration perturbation from thermodiffusion will depend to a large extent on the value of p, and hence the particular electrolyte, and stirring conditions. For example, in 0.01 M KCl and a 5 OC difference between compartments in a nonconvective thermogalvanic cell, eq 13 predicts that the concentration perturbation is about 0.7%. As noted earlier, the corresponding effect for this electrolyte, but with a heated RDE, is even less (0.1%) because hydrodynamic convection in this system tends to offset the contribution from thermodiffusion. With 0.01 M KCl the value of E calculated from eq 15 is +0.0348 mV/OC and from eq 23 is +0.0342 mV/OC. Consequently, the results for E obtained with either a RDE or a thermogalvanic cell will be the same for most electrolytic systems and therefore essentially interchangeable. More importantly, however, these systems mutually provide a means for establishing an internal calibration scale for surface temperature perturbations at a RDE. Before we leave this topic, it is worth mentioning that some electrolytes, such as the tetralkylammonium salts, exhibit exceptionally large heats of transport, but primarily in dilute aqueous solutions. As an example of this, let us consider a 0.01 M solution of tetra-n-butylammonium chloride, (n-Bu),NCl. The value p = 0.01 for this system is about an order of magnitude larger than that for KC1 at the same concentration? In Figure 1 are also shown the numerical calculations for the surface concentration perturbation in this electrolyte as a function of time and AT. The concentration perturbation from thermodiffusion is larger than in 0.01 M KCI but still only amounts to less than a 1% change for a 5 OC surface temperature perturbation. Such temperature excursions are beyond those employed in thermal modulation experiments and since practical experimental conditions commonly involve much higher electrolyte concentrations, the contribution from concentration gradients on the thermodiffusion potential is negligibly small, even for salts with large heats of transport. Mass Transport Limiting Currents. Under isothermal conditions, the mass transport limiting current on a RDE was obtained by the classical Levich analysisI2 and is given by the expression iL = 0.62nFAD2/3v-1/6w1~2C b
(24) where D and C, are the diffusion coefficient and bulk concentration of the electroactive species, respectively, u is the kinematic viscosity of the electrolyte, n is the number of electrons transferred, and A is the electrode area. Under nonisothermal conditions, the limiting current will be a function of the local fluid temperature implicitly through the diffusion coefficient and kinematic viscosity of the fluid. The effect on limiting currents from a temperature perturbation of these fluid properties was analyzed previously2 by using a laser-heated RDE under steady-state conditions. The ( 12) Levich, V. G.Physicochemical Hydrodynamics; Prentice-Hall:Englewocd Cliffs, NJ, 1962.
Electrochemical Response of Rotating Disk Electrodes
0.6
I /
AT= 1 O'C
0.10 O," 0.15
0.20 0 25
0 30
0.2
oov"
00
01
02
,287 ,571 797 ,930 ,984 ,997
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4487
Levich 0
,289 ,574 ,799 ,931 ,984 ,998
03
Distance normal t o electrode/
04
1
05
-40
-5
15
20
25
30
35
40
45
50
55
?j
Figure 2. Influence of thermodiffusion on the concentration profile of electroactivespecies under limiting current conditions, for AT = 10 OC and @ = 0.01. Calculated results are shown graphically by the solid line Numerical results inserted. and the Levich profile by (0).
perturbation in the limiting current density, AiL, was shown theoretically to be directly proportional to the surface temperature perturbation, AT, and this conclusion was verified experimentally for oxidation limiting currents in the system 5 mM Fe(CN)6-3-5 m M Fe(CN)b4-1 M KCl. Under limiting current conditions, the heat and mass transport equations characterizing a nonisothermal RDE are those given by eq 18 and 19. However, the concentration, C, refers to that of the electroactive species, and the value of B corresponds to the entropy of transport for the same. At the limiting current, the concentration of the electroactive species at the electrode surface is zero and eq 22 is changed appropriately to reflect this fact. The resulting set of equations was solved numerically by the same procedure mentioned earlier. In Figure 2 we show the steady-state results obtained for the concentration profile of the electroactive species in the boundary layer region for AT = 10 OC. The value of B chosen here provides an estimate for the upper limit of thermodiffusion effects on limiting current conditions. For the purposes of comparison, p = 0 (symbols) corresponding to the Levich solution is also shown on this graph. It is clear that the influence on limiting currents is small as evidenced by the nearly superincumbent concentration profiles (see numerical results). In conclusion, unlike their effect on open circuit potentials, thermodiffusion processes are not a contributory factor to limiting currents on a RDE.
Experimental Section Nonisothermal thermogalvanic experiments were performed for the redox couple Fe(CN)6f/4- at 5 mM concentrations of the potassium salts in separate 1 M KCl and 1 M LiBr electrolytes. Reagent grade chemicals and Milli-Q water were used throughout. One part of the solution with a gold wire electrode was thermostated in a jacketed cell by using a Neslab RTE-4 circulating refrigerated bath and the second compartment, with its corresponding gold wire electrode and the same solution, was held at a constant temperature (usually -30.1 "C) throughout each experiment. The compartments were connected by a short quartz tubing bridge of the same solution. The solution temperature of the first compartment was varied between -20 and -50 OC, and the temperature in both cells was continuously monitored with a digital OMEGA 5800 dual channel glass-covered thermistor thermometer. The open circuit equilibrium potential, AU, was measured between the two gold wire electrodes with a Keithley 177 Microvolt DVM, after establishing thermal equilibrium conditions in each cell (usually about 30 min). Stability of runs was checked by inspecting the data to see if AU for a AT = 0 was also zero, and by cooling the first compartment down to confirm reproducibility. Thermogalvanic experiments with silver halide electrodes were also performed in the same way. Ag/AgCl electrodes were freshly made by anodizing a silver wire electrode for approximately 2 h at 1.5 V with respect to another silver electrode in a solution of 1 M KC1. Ag/AgBr electrodes were also prepared in a similar fashion but with the voltage between the silver electrode kept instead at 1.3 V in a 1 M LiBr solution. Thermogalvanic experiments using paired Ag/AgCl electrodes in 1 M KCl solution
T/'C
Figure 3. Open circuit potential data measured in a thermogalvanic cell as a function of temperature for (0)5 mM Fe(CN):-/+ in 1 M KC1 on Au electrodes, and ( 0 )Ag/AgCl electrodes in 1 M KC1 solution.
and paired Ag/AgBr electrodes in 1 M LiBr were conducted in order to determine the thermal coefficients of equilibrium potentials for these reference electrodes. The isothermal coefficients of the electrode potential of Fe(CN)63-/4-relative to the silversilver halide couple in a single compartment were also measured. Measurements of the equilibrium potential were then made for the redox couple Fe(CN)63-/4-at a gold electrode with respect to a Ag/AgCl electrode as a function of temperature in a common 1 M KC1 solution. Similar isothermal experiments were conducted in 1 M LiBr and with a Ag/AgBr electrode. Thermogalvanic experiments were also carried out for tetra-n-butylammonium bromide in widely varying concentrations with a pair of Ag fAgBr electrodes. Laser-heated RDE experiments were performed on a Ag/AgBr RDE in solutions of tetra-n-butylammonium bromide. Open circuit potential perturbations were measured as a function of input laser power and rotation speed for different concentrations of the tetralkyl salt. The procedure followed here is similar to that described elsewhere.2 The Ag/AgBr RDE was made by electroplating silver from a silver cyanide bath onto a gold RDE of hollow-tube configuration with the quartz support modifi~ation.~ Silver was deposited at a current density of 0.4 mA/cm2 and a rotation speed of 900 rpm for approximately 5 h. The deposit was thoroughly washed in an ultrasonic bath with concentrated N H 4 0 H for -2 h with further rinsings in Milli-Q water under sonic agitation, followed by overnight storage in Milli-Q water. This procedure is followed to eliminate any cyanide residue left behind from the electrodeposition step which might interfere with the establishment of the proper equilibrium potential for these reference electrodes. Anodization of the silver RDE into a Ag/AgBr electrode was performed at a constant current density of 0.25 mA/cm2 for approximately 40 min at 1600 rpm.
Results and Discussion The influence of thermodiffusion effects on equilibrium electrode potentials was first investigated by using the thermogalvanic cell configuration previously described. In Figure 3 are shown two sets of experimental results obtained in such a cell and corresponding to (1) 5 m M Fe(CN)63-/4-in 1 M KC1 on Au electrodes (a), and (2) Ag/AgCl electrodes also in 1 M KCl solution ( 0 ) .The open circuit electrode potential, AU, measured in each of these systems is plotted as a function of the controlled temperature in one cell, with the other compartment always maintained at a reference temperature of 30.1 OC. An excellent linear correlation is found between AU and T in both systems. The thermal coefficients of electrode potentials calculated from the slope of this data are = -1.374 mV/OC for the ferri-ferrocyanide system, and ether,,, = +0.240 mV/OC for the Ag/AgCl electrode. The intrinsic thermal coefficients of electrode potentials in each of these systems can be readily calculated from thermodynamic data.I3 The respective electrode reactions and thermal coefficients are (13) Bard, A. J., Parsons, R., Jordan, J., Eds.; Standard Potentials in Aqueous Solutions; Marcel Dekker: New York, 198.5. (14) Goodrich, J. C.; Goyan, F. M.; Morse, E. E.; Preston, R. G.; Young, M. B. J . Am. Chem. SOC.1950, 72, 4411.
4488 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
Fe(CN)63- + e- = Fe(CN)64-, AgCl
+ e- = Ag + C1-,
€Mirm= -1.623 e#&,,
mV/'C
Valdes and Miller
::I---_1\
(25)
(Ag/AgCI-A")
= +0.225 mV/OC ( 2 6 )
.
230
We note that, in the ferri-ferrocyanide system, the calculated thermodynamic thermal coefficient of the electrode potential differs significantly, ( A 0.249 mV/OC), from the measured value. In contrast, the measured and calculated values of the thermal coefficients of electrode potentials for the Ag/AgCI system differ by only 15 pV/OC, within the experimental error. Thus, these results demonstrate that the contribution from thermodiffusion effects on open circuit electrode potentials in the Ag/ AgC1-I M KCI system is negligibly small in comparison to its intrinsic thermodynamic value. However, a similar result should also have been observed in the ferri-ferrocyanide system because the supporting electrolyte used in each case was the same, Le., 1 M KCI. The thermodiffusion contribution to the thermal coefficient of electrode potential normally is determined primarily by the supporting electrolyte. In spite of the facts that the ferri-ferrocycanide system ions are multiply charged and have high individual entropies of transport, the supporting electrolyte concentration is still sufficiently high, and the net effect on potential sufficiently small, to make the %ret contribution from the redox moieties themselves of little impact. Using the data of Breck and LinI5 on the entropies of transport for these species and eq 16 for a multicomponent system, we find that the thermal coefficient for the Soret potential is +0.025 mV/OC in the ferri-ferrocyanide1 M KCI case and cannot account for the disparity (about an order of magnitude larger) between the calculated thermodynamic and experimentally measured thermal coefficients of electrode potential. We note also that this value of the thermal coefficient for the Soret potential represents an upper limit since the entropies of transport used here were at infinite dilution and are expected to decrease markedly at higher concentration^.^ These calculations suggest that the intrinsic thermodynamics of the Fe(CN)63/4- electrode reaction on gold are changed in some "chemical" way through interactions with the supporting electrolyte itself. In order to test this hypothesis, we consider the following relationship between thermal and isothermal coefficients of electrode potentials in each of these systems
> E
220
3
N
210 200
N
e&,,
=::!e
e&,,
=
et:
190
15
(15) Breck, W. G.; Lin,
J. Trans. Faraday SOC.1965, 61, 2223.
25
35
30
40
45
55
50
Figure 4. Open circuit potential data measured between a An/AaC1 and Au electrode in an isothermal cell as a function of temperature fo; 5 mM Fe(CN)63-/4-in 1 M KC1. 20
,, 10
I\
: 2 E
-20: -30 '
-4OV 15
20
25
35
30
40
50
45
55
T/OC
Figure 5. Open circuit potential data measured in a thermogalvanic cell as a function of temperature for: (0)5 mM Fe(CN)6)-Ie in 1 M LiBr on Au electrodes, and ( 0 )Ag/AgBr electrodes in 1 M LiBr solution. 400 I
380
, 3,0 E
3
1
1
I
\ - 1 740mV/OC
360.
350
.
340
t
5mM Fe(CN),-3/-4 ' M LB,
330
15
20
25
30
35
40
45
50
55
T/'C
Figure 6. Open circuit potential data measured between a Ag/AgBr and Au electrode in an isothermal cell as a function of temperature for 5 mM Fe(CN)2-/4- in 1 M LiBr.
diffusion to the thermal coefficient of electrode potential in the ferri-ferrocyanide system is indeed small, just as in the Ag/AgCl system ( 15 pV/OC), and as required by the consistent set of numerical results obtained between thermal and isothermal coefficients measured here. Consequently, these results support the hypothesis that the difference between the measured and thermodynamically calculated thermal coefficient of electrode potential in the ferri-ferrocyanide system is attributable not to thermodiffusion but instead to factors involving the interaction of the supporting electrolyte with the equilibrium state of this electrode reaction. In Figure 5 are shown thermogalvanic cell data for the redox couple Fe(CN)63-/k in 1 M LiBr on Au electrodes (0) and for Ag/AgBr electrodes in a 1 M LiBr electrolyte (0).Both sets of experimental data exhibit excellent linear correlations for the range of temperatures studied. The calculated thermal coefficient of electrode potential for the ferri-ferrocyanide system in 1 M LiBr is -1.324 mV/OC, close to that found previously in a 1 M KCl solution. For the Ag/AgBr electrode system, the value of +0.418 mV/OC compares favorably with the value determined thermodynamically, +0.378 mV/OC. The small difference of -40 pV/OC is commensurate with the quantity found earlier for Ag/AgCI electrodes in 1 M KCI solutions and provides an estimate N
Equation 29 reveals that the difference between thermal coefficients of electrode potentials for any two electrode reactions conducted separately, but with the same supporting electrolyte, will be equivalent to an isothermal experiment for which the temperature coefficient of the potential is measured between these two electrodes in the same electrolyte as before. In Figure 4 we show experimental data obtained in such an isothermal cell for the open circuit potential measured between Ag/AgCI and Au electrodes in 5 m M Fe(CN)63-/4--1 M KCl, as a function of the cell temperature. The isothermal coefficient of electrode potential calculated from a linear regression of the data is -1 A37 mV/OC and compares remarkably well with the difference between the thermal coefficients of electrode potentials previously found -1.614 mV/OC, (-1.374 - 0.240 mV/OC). These results demonstrate internal consistency between isothermal and thermal experimental data. Moreover, it suggests that the contribution from thermo-
20
T/OC
+ 0.871 mV/OC + eSoret + 0.871 mV/OC + eSoret
where ethm and eiso refer to thermal and isothermal coefficients, respectively, and the thermodiffusion contribution, eSoret, is the same for each system because of a common 1 M KCI supporting electrolyte. Subtraction of eq 28 from eq 27 leads to the following useful relationship between thermal and isothermal coefficients of electrode potentials
-1 637 mV/'C
Electrochemical Response of Rotating Disk Electrodes
- l o r ' 15 20
"
25
30
"
35
40
'
45
"
50
55
I 60
T/OC
Figure 7. Open circuit potential data measured between Ag/AgBr electrodes in a thermogalvanic cell as a function of temperature for ( 0 ) 0.01 M ( ~ - B U ) ~ and N B (0) ~ 0.10 M ( ~ - B U ) ~ N B ~ . 1.60 0 - 1
0
25
50
75
100
125
150
175
200
q,,/mW
Figure 8. Open circuit potential data measured between a laser-heated Ag/AgBr RDE and a Ag/AgBr wire electrode in the same solution as a function of laser output power, qIn,and at 900 rpm.
on the magnitude of thermodiffusion effects for two salts which are of the type commonly used as supporting electrolytes in electrochemical systems. In Figure 6 are shown isothermal experimental data on the open circuit potential measured between Ag/AgBr and Au electrodes in 5 m M Fe(CN)63-/"--1 M LiBr solution as a function of the cell temperature. The isothermal coefficient of electrode potential calculated from this data is -1.740 mV/OC and compares very well with the difference between the thermal coefficients measured separately for each of these electrodes -1.742 mV/"C, (-1.324 - 0.418 mV/OC). These results once again establish internal consistency between thermal and isothermal experiments. Thermogalvanic data are shown in Figure 7 for the open circuit potential measured by using Ag/AgBr electrodes in 0.01 and 0.10 M aqueous solutions of tetra-n-butylammonium bromide. The thermal coefficients of the electrode potential calculated from the slope of these data are +0.653 and +0.459 mV/OC, for the 0.01 and 0.10 M solutions, respectively. Similar results (+0.674 mV/OC) have been obtained by Goodrich et al.14 for a 0.01 M solution. The thermal coefficient of the 0.10 M solution calculated from the thermodynamic entropy change of the Ag/AgBr electrode reaction and the Nernst equation is +0.576 mV/OC. The contribution from thermodiffusion on the thermal coefficient of electrode potential in this electrolyte (-77 rV/OC) is consistent with a higher entropy of transport reported in the literature for this salt.g Thermogalvanic experiments on Ag/AgCl electrodes using the chloride form of the tetralkyl salt also give similar results. In Figure 8 are shown experimental data obtained on a laser-heated RDE for the open circuit electrode potential, AU, measured between a thermally modulated Ag/AgBr RDE and a Ag/AgBr wire electrode in the same solution. AU was measured as a function of the laser output power, q,,,, in various concentrations of tetra-n-butylammonium bromide at 900 rpm. A linear correlation was obtained for AU vs qInfor each electrolyte concentration studied and all correctly extrapolate to zero at no input laser power. On a laser-heated RDE, ql,, is directly proportional to the temperature difference manifested between the electrode surface and the bulk of the solution, at a given rotation speed. The slope of this data is therefore related to the thermal coefficient
The Journal of Physical Chemistry, Vol, 92, No. 15, 1988 4489 of electrode potential as measured in a thermogalvanic cell. An absolute comparison of experimental data obtained on a RDE and that measured in a thermogalvanic cell requires the temperature at the surface of the RDE to be known. The actual value of this temperature was shown to be influenced by thermal losses in the system.2 However, this difficulty can be easily circumvented here by comparing instead the ratio of the slope, (dAU/dq,,), at two different electrolyte concentrations with the corresponding ratio of ether,,, obtained from the thermogalvanic data shown in Figure 7. In this manner, the effect of thermal loss necessarily cancels out since experimental data on the RDE were all obtained at the same rotation speed. The calculated value for the ratio of slopes corresponding to the 0.10 M over 0.01 M concentration data on the RDE is 0.68 (dimensionless) and is in good agreement with the corresponding value of 0.70 obtained from thermogalvanic cell data. These results are of considerable importance in establishing a working correspondence between the temperature and electrode potential on a laser-heated RDE system and are applicable to any electrochemical system.
Summary and Conclusions The influence of thermodiffusion processes at a thermally modulated rotating disk electrode and in a thermogalvanic cell were analyzed theoretically by using the concept of the entropy of transport. We find that thermodiffusion generally adds only a small contribution to the thermal coefficient of the equilibrium potential from that calculated by the intrinsic thermodynamic change associated with the electrode reaction. These results are corroborated by experimental data obtained in a thermogalvanic cell for the redox couple Fe(CN)6*/e in various halide supporting electrolytes in conjunction with those for silver halide electrodes. Experiments performed on a laser-heated RDE are fully consistent with thermogalvanic data and establish an important correspondence between temperature and electrode potential measured on the RDE. This kind of relationship is essential for a comprehensive analysis of thermal modulation processes on the RDE, and for the use of this technique as an analytical tool for probing electrochemical processes in the temperature domain. Acknowledgment. We acknowledge instructive discussions with Dr. Stephen W. Feldberg about the temporal behavior of the Soret effect.
Glossary C Ci
Di
F
H
dimensionless concentration concentration of species i, mol/cm3 diffusion coefficient of species i, cm2/s Faraday constant, 96484.56 C/mol hydrodynamic function for axial velocity component on RDE
T T, AT AU
current density, mA/cm2 limiting current density flux of species i, mol/(cm2.s) flux of electrolyte dimensionless Nusselt number separation distance between compartments in a thermogalvanic cell dimensionless Prandtl number laser output power, mW dimensionless Schmidt number partial molar entropy, eu/mol of the ith species entropy of transport, eu transported entropy thermodynamic entropy change of electrode reaction nonthermodynamicentropy change time, s absolute temperature, K temperature of bulk electrolyte temperature difference between nonisothermal regions, K open circuit potential measured in nonisothermal cell, mV
Zi
charge of ionic species i
i !L !i IC
Nu L
Pr 9in
sc
Si
ss
as a* I
Greek Letters A @
equivalent conductance of electrolyte, mho.cm2/equiv electric potential, mV
4490
B t
11 K
PI
v
J. Phys. Chem. 1988,92, 4490-4498 entropy of transport of the electrolyte temperature coefficient of the thermodiffusion potential, mV/OC dimensionless coordinate normal to electrode surface ionic conductivity of solution, mho/cm electrochemical potential energy of species i kinematic viscosity of electrolyte, cm2/s
W
B 7
disk rotational frequency, rad/s temperature difference, T - T, dimensionless time
Registry No. Fe(CN):-, 13408-62-3; Fe(CN),&, 13408-63-4; KCI, 7447-40-7; LiBr, 7550-35-8; Ag, 7440-22-4; AgCI, 7783-90-6; AgBr, 7785-23-1; Au, 7440-57-5; (n-Bu)4NBr,1643-19-2.
Electric and Nonelectric Free Energy of Nonionic-Ionic Micelles Hiroshi Maeda Department of Chemistry, Faculty of Science, Nagoya University, 464 Nagoya, Japan (Received: December 9, 1987) An approach to the statistical thermodynamics of nonionic-ionic micelle solutions is presented with a special reference to the relationship between small systems and macroscopic systems. Three relations describing the potentiometric titration, the area bounded by the titration curves, and the differential stability of micelles are derived. The effect of variable aggregation number m is dual: a direct effect through m and an indirect effect through nonelectric and electric free energies. The direct effect on the area and the differential stability is explicitly given, while it is absent in the potentiometric titration. Accordingly, the same potentiometric equation holds for both micelles and linear polyelectrolytes. Application to dodecyldimethylamine oxide solutions showed that the direct effects were significant. Dissection of the electric and the nonelectric contributions, which was first possible after subtracting the direct effects of m,was achieved with the aid of approximate calculation of the electric interaction based on the equivalent sphere model. The dependence of the nonelectric contributions on the degree of ionization of micelles was consistent with the interpretation based on the indirect effect of m plus some other particular contribution.
Introduction A particular variable in the thermodynamics of micelle solutions is the aggregation number, which varies depending on environmental factors such as temperature, pressure, and ionic strength. It varies also with pH in the case of nonionic-ionic micelles. There are several differences between linear polyelectrolytes and ionic micelles; the number of ionizable sites is fixed in the former but is variable for the latter as a result of the exchange of monomer between solution phase and micelles. Further, monomers in a polymer chain are covalently connected with each other and hence distinguishable, while monomers in a micelle exchange their positions and are hence indistinguishable. It is the aim of the present study to analyze the thermodynamic properties of ionic micelles, in particular their potentiometric titrations, and to extract the nonelectric contribution from experimental data. In section I, statistical thermodynamics of ionic micelle solutions are given partly following the analysis given by Aranow,' with special emphasis on the connection between macroscopic thermodynamics and that of small systems2 In section 11, the chemical potential of micelles is derived. In section 111, thermodynamic analysis of the titration curves is presented. In section IV, the differential stability of ionic micelles is derived. In section V, the analyses given in sections I11 and IV are applied to the experimental data on dodecyldimethylamine oxide (DDAO). In the general treatment given in sections I-IV, amine oxides are considered as an example of ionic micelles. The ionization of amine oxide is schematically written as follows: DH+ = D + H+ (1) Here D and D H represent uncharged (deprotonated) and protonated species, respectively. Intrinsic dissociation constants of reaction 1 are denoted as K,and KM for monomers and micelles, respectively. I. Statistical Thermodynamics of Nonionic-Ionic Micelle Solutions Let us consider an ensemble specified by a set of variables T (temperature), P (pressure), No (number of solvent molecules), (1) Aranow, R. H . J . Phys. Chem. 1963, 67, 556.
(2) Hill, T. L.Sfatistical Thermodynamics of Small Sysfems;Benjamin: New York, 1963; Vol. I, 11.
N, (number of uni-univalent salt molecules, M'X-), ND' (the total number of surfactant molecules), and pH (chemical potential of hydrogen ions). Since macroscopic properties of the system are completely determined by the set of these variables, average values of the number of free monomers NI and of their degree of ionization aIare uniquely determined. Assumption 1: Fluctuations of Nl and a1are ignored. According to assumption 1, fluctuation of the chemical potential pDof un-ionized surfactant D is also neglected, since it is uniquely related to Nl and a Ias follows: . MD = PL,*(T,P,N~/NO) + kT In [(I - al)Nl/No] (2) The number of surfactant molecules constituting micelles ND and the number of hydrogen ions bound to them NH are related to their total amount as follows: ND = ND'
- Nl
NH
= NHt
- alN1
(3) Let us divide the ensemble into subensembles specified by a set of micelle size distribution (v,,,), v, representing the number of m-mers: ND = xmv, m
(4)
The partition functions of these subensembles are denoted as rM and are related to the partition function of the original ensemble as follows:
r~= Cr,(T,P,No,Ns,N,,Nlal,(V,J,~H)
(5)
where the summation extends over all possible sets of size distribution. Each rM is related to another partition function A,,,(T,P,{N}) through rM = C C e x P ( C C n m + H / k T ) A,,,( TYJ",Ns,N1 ,Nlal,(vmI,(nm,J) m q
(6) Here the summation is taken over possible values of n,, the number of hydrogen ions bound to the 7th m-mer. The partition function A,, is expressed as eq 7 in terms of kinetic energy Ukl" and potential energy U of the system: A,,, = l e - P v / k Td V l e x p ( - ( V
+ vkl")/kTJ (dp] (dq)/h3'"
0022-3654/88/2092-4490$01.50/00 1988 American Chemical Society
(7)
