2426
J. Phys. Chem. 1994,98, 2426-2432
Thermodynamic Relationships Important for Interpreting Apparent Formal Potentials and Apparent Reaction Entropies of Redox Couples in Permselective Media Jody Redepenning,' Harmon M. Tunison, and Jayson Moy Department of Chemistry, University of Nebraska, Lincoln, Nebraska 68588-0304 Received: September 13, 1993; In Final Form: December 3, 1993"
Apparent formal potentials are measured for O ~ ( b p y ) 3 ~ + exchanged /~+ in Nafion films on electrodes. The apparent formal potentials in the films are compared with the formal potential for O ~ ( b p y ) 3 ~ + in / ~ solution + using a description that assumes O ~ ( b p y ) 3 ~and + O ~ ( b p y ) 3 equilibrate ~+ with sodium ions in the bathing electrolyte solution. Good agreement is obtained between the experimentally determined values of the apparent formal potentials and the values expected using the equilibrium model. Apparent reaction entropies are also measured for O ~ ( b p y ) 3 ~ +exchanged /~+ in Nafion. Once the entropy of ion transport is taken into account, it appears that the reaction entropy of O ~ ( b p y ) 3 ~ + in / ~ +Nafion is similar to that observed in aqueous solutions. The environment in the vicinity of O ~ ( b p y ) 3 ~in+Nafion appears to be very similar to the environment in the vicinity of Os(bpy)3*+ in Nafion. The results also demonstrate that there is a large entropic contribution to the free energy of exchange, which is consistent with the so-called hydrophobic effects observed by other workers.
Introduction The formal potentials of redox couples in polymers on electrodes are frequently found to differ from the values observed for those same redox couples at bare electrodes. The concentration of supporting electrolyte in solution is known to be one factor that influences the apparent formal potential of redox probes in polymers, and this influence has been described in terms of the contribution of the Donnan potential at the polymer/solution interface to the overall cell potentia1.l Work of Szentirmay et al. is noteworthy because their measurements of ion-exchange selectivity coefficients demonstrate that the driving force for incorporating redox probes in Nafion can be very large.2 A description that completely accounts for differences between formal potentials observed for redox couples in ion-exchange polymers and those observed in solution should include the influence of supporting electrolyte concentration on Donnan potentials, and it should also include the large free energies associated with incorporating the electroactive species into the polymer. A description that accounts for both of these influences is the focus of the work described below. Also considered below are the reaction entropies of redox probes in ion-exchange polymers. The entropy difference (reaction entropy) between oxidized and reduced forms of a redox couple is known to be influenced by ionic charge and solvent environment. The reaction entropy, which can be determined by measuring the temperature dependence of the formal potential in a nonisothermal cell,3 has been used to examine a variety of systems in solution ranging from transition-metal complexes in water to waterinsoluble organic molecules in aprotic media.4-6 Reaction entropies have also been used to investigate the environment in the electrochemical double layer using surface-attached species7 and to examine the environment near redox probes in ion-exchange polymers.1.*-lO It has been suggested that ion transport coupled to electron transfer can influence apparent reaction entropies measured for cationic redox probes in ion exchangers.'* We consider here a relatively simple system which demonstrates the importance of ionic contributions to reaction entropies when the transport number of one electrolyte ion approaches unity. In these studies apparent reaction entropies are measured as a exchanged in function of electrolyte activity for O~(bpy)~3+/2+ Nafion films. The results are consistent with the expectation
* To whom correspondence should be addressed.
Abstract published in Advance ACS Abstracts, February 1, 1994.
0022-3654/94/2098-2426$04.50/0
that the apparent reaction entropy should be influenced in a welldescribed manner by electrolyte activity and by the ion-exchange entropies for both oxidation states of the redox couple. Experimental Section Materials. Os(bpy)3Iz was prepared according to the procedure of Creutz et al.II The iodide salt was converted to the chloride on a Rexyn anion-exchange column. A 5 wt % solution of Nafion 1100 (Aldrich, Lot No. 02915MP) and NaCl (Mallinckrodt) were used as purchased. Apparatus and Procedures. All electrochemical measurements were performed in a nonisothermal cell with a thermally jacketed compartment for the working and auxiliary electrodes, and a separate thermally jacketed compartment for the reference electrode. A PAR Model 273 potentiostat, an IBM XTcomputer, anda GraphtecModel WX1200x-yrecorder wereused toaquire all electrochemical data. Apparent formal potentials were assumed to be the average of the anodic and cathodic peak potentials obtained from cyclic voltammograms run a t 10 mV/s. Voltammetric data were analyzed using a modified version of Headstart. Nafion coatings were prepared by applying 20 NL of Nafion solution to 0.44 cm2 platinum electrodes mounted in an inverted Pine Instruments Model ASRP rotator. After rotating the electrode at 3000 rpm for 60 s, the resulting films were immersed for 10 min in a NaCl solution and then for 20 min in a 1 mM aqueous solution of Os(bpy)&lz. Electrodes were then conditioned by cycling the potential between 0.0 and 1.OV at 10 mV/s while being immersed in one of the electrolyte solutions to be used in subsequent measurements. The electrodes were maintained at 35 "C throughout this procedure. One hour was typically required toreach a steady state. Upon recording a voltammogram at 35 OC, the temperature was lowered in 10 OC increments to 5 "C. Voltammograms were recorded at each increment after equilibrating the polymer with the electrolyte for approximately 30 min. The temperature was then raised in 10 OC increments to 35 OC, and voltammograms were again recorded at each increment to establish that there was no hysteresis. To minimize uncertainty associated with film preparation, entropy measurements were performed using the same electrode in electrolyte solutions at five different concentrations. When new films were cast for each electrolyte solution used, greater scatter was observed for the apparent reaction entropies. At 35 OC when electrolyte concentrations were 0.2 m and greater, 0 1994 American Chemical Society
Redox Couples in Permselective Media
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2427
O~(bpy)~3+/2+ exchanged out of the films on the time scale of the experiments. At electrolyte concentrations less than 0.005 m, uncompensated resistance caused unacceptable uncertainty in apparent formal potentials. Consequently, although a greater concentration range would have decreased the uncertainty in the apparent reaction entropies, we were limited to electrolyte concentrations between 0.005 and 0.100 m.
Results and Discussion Equation 1 gives the half-cell potential of a bare electrode that is in equilibrium with a solution containing both the oxidized and reduced forms of a redox couple. In eq 1 sox(,) and U R ~ ( , )are the RT E=Eos+-ln-
nF
'Ox(s) 'Red(,)
expand the equilibrium constants given in eqs 4 and 5 as is shown in eqs 8 and 9. Because the activity coefficients shown in eqs 8
and 9 are unknown and difficult to acquire, one frequently sees these equilibrium constants approximated using concentrations or mole fractions instead of activities. Such an approximation is given in eqs 10 and 11. The activity of supporting electrolyte
activities of the oxidized and reduced forms of the redox couple dissolved in solution. Equations 2 and 3 are valid if both the ox;+ Red:-')+
+ zc;
= ox;+
+ zc,+
+ (z - 1 ) C l = Red:-)'+
+ (z - 1)C,+
(2)
(3)
oxidized and reduced forms of the redox couple are cationic and if they are free to equilibrate between an electrolyte solution and a permselective cation exchanger. C+ in eqs 2 and 3 denotes supporting electrolyte cations. The subscripts s and p denote ions found in either the solution or the polymer phases, respectively. The equilibrium constants that describe the partitioning of Oxz+ and Red(z-')+ into the ion exchanger are given by eqs 4 and 5.
in solution has been retained in both of these equations because it can be known at all concentrations for many different electrolytes in water. Note that these approximations are given the symbols and KEd' to distinguish them from the thermodynamically well-defined quantities in eqs 8 and 9. Also note that the relationships between eqs 8 and 10 and between eqs 9 and 11 are given by eqs 12 and 13:
e
(4)
Solving for aoxg)and U R ~ ( , )in eqs 4 and 5 and then substituting the results into eq 1 gives the following relation for a coated electrode at equilibrium with a bathing solution:
Unfortunately, we have not been able to use eq 6 to evaluate our experimental data. The problem stems from the fact that activity coefficients in the polymer are not known. Because the activity coefficients are not known, the equilibrium constants defined in eqs 4 and 5 are not known, and hence eq 6 has limited utility. A useful formalism for evaluating the experimental results still exists, however. That formalism is described below. The electrode potential for a redox couple in solution can be given by
E = EO',
RT [Ox], +lnnF [Red],
where
Next, by rearranging eq 10 to solve for [Ox], and by rearranging eq 11 to solve for [Red],, eqs 14 and 15 are obtained:
[Red], =
Substituting eqs 14 and 15 into eq 7a gives
E = EO',
T [OX] + RInnF [Red], R T KEd' R T RT nF In- e - nF In [C'], + nF In uc(,) +
(16a)
If E is measured when [OxIp = [Red],, then (Eo'p)app=
EO',
[Red] [C+];-~)KR~' C
[EO!,
RT +nF In-
RT
-nF In
[C+Ip] +
R T yox(s) = Eo, + -lnnF TRcd(s)
and where E O ' , is the formal potential for the couple in solution. Ultimately we seek a relationship between the electrode potential and the amounts of Ox, Red, and supporting electrolyte exchanged in the polymer. To obtain this relationship, it is necessary to
where (Eo'p)app is defined as the apparent formal potential for the redoxcouplein thepolymer. Touseeq 16 tocomparetheapparent formal potential in the polymer with the formal potential in solution, it is necessary to know the concentration of exchange
2428 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
sites in the polymer. If the concentration of exchange sites is not known, [Ox] [Red],, and [C+], are not known, and hence, KEd' and cannot be known. The question of what the concentration of exchange sites actually is in our films is one we have not attempted to establish quantitatively. Although the bulk concentration of sites in Nafion is not difficult to estimate, the concentration of exchange sites in the hydrophilic domains of Nafion is more difficult to ascertain. Whitely et al. have measured the concentration of exchange sites in the hydrophilic domains of Nafion using ultramicroelectrodes,12but we have not established that our films are sufficiently similar to the films of Whitely et al. to warrant using their values for the concentration of sites. We have found that it is not necessary to know theconcentration of exchange sites in Nafion to reconcile differences between the apparent formal potential measured for redox couples in Nafion and those measured for the same redox couples in solution. Presumably, the rationale we apply to Nafion may also apply to other permselective media. It is sufficient to measure the amounts of Ox, Red, and supporting electrolyte cations in the Nafion films to reconcile differences between and EO',. Even if the concentration of sites is not known, it is still possible to say that [CilP= qt[S], where [CJ is defined as the concentration of species i in the polymer, where qi is the ionic molar fraction of species i (Le., mol of i/equiv of sites)" and where [SI is the concentration of sites. Thus, [Ox], = tloX[S],[Red], = 1)Rd[S]r and [C+], = qC[S]. By substituting these definitions of [Ox],, [Red],, and [C'], into eqs 14 and 15 and by substituting the results into eq 7a, the following relation is obtained:
@
Redepenning et al. into the film are taken into account. Additionally, the earlier work examined the influence of concentration on the apparent formal potential rather than using activities as has been done here. Finally, changing liquid junction potentials between the electrolyte solution in the working electrode compartment and the saturated electrolyte solution in the reference compartment were not accounted for by Naegeli et al. Because neither changing activity coefficients nor changing liquid junction potentials was taken into account and because these effects partially cancel each other in the cells used, the slopes obtained for plots of the apparent formal potential versus the log of the electrolyte concentration were closer to 59 mV than might have been expected by these authors. The experiments described below use known mean ionic activity coefficients for the supporting electrolyte, and any ambiguities which are introduced by liquid junction potentials, which change as the electrolyte activity is changed, have been eliminated . Factors Influencing the Apparent Formal Potential Electrolyte Activity. One way of measuring the influence of electrolyte activity on ( E O ' , ) is to measure the potential of the working electrode (when q h = QRd) versus a SSCE in a series of solutions with different electrolyte activities. Unfortunately, such measurements necessitate the introduction of liquid junction potentials, E,, that change as the electrolyte activity changes. So it is necessary to correct the cell potentials for the changing liquid junction potentials. We used the cell represented by eq 23 to HgOlHg C1 Jsat.NaCIlXm NaCIIAgClIAgO 1
RT RT -lnqc+-lnac(,) nF nF
(17)
where
and where
If E is measured when vox = q R d , then
= EO',
RT @"" R T +a l n p - -In qc nF C
where is defined as the apparent formal potential for the redox couple in the polymer. As can be seen in eq 22, should change by 59.2 mV for each factor of 10 by which the activity of the electrolyte changes in solution. A similar dependence of the apparent formal potential on electrolyte was observed by Naegeli et al.la Some important distinctions should be made between the workof Naegeli et al. and the work presented here, however. First of all, eqs 17 and 22 provide a more comprehensive description of the apparent formal potential because the ion-exchange equilibria for partitioning Ox and Red
22
3
4
5
(23)
determine the liquid junction potentials. For this cell differences between measured cell potentials and those calculated using the Nernst equation (along with known sodium chloride activities) can be attributed to the liquid junction potential across interface number three in eq 23. When E, is determined in this manner and is plotted versus log UC($), a straight line with a slope of -1 3.1 f 0.3 mV is obtained. This value is in good agreement with the value of -12.3 mV expected if the liquid junction potentials are calculated14 making the assumption that the liquid junction is a simple'type 1" junction15and that the transport numbers of Na+ and C1- at infinite dilution, 0.396 and 0.604,16 are used to approximate the true values of the transport numbers. If the values of E, measured for the cell represented by eq 23 are used to correct the cell potentials measured for Os(bpy)33+/2+ in Nafionversus SSCE, then for O~(bpy),3+/~+ in Nafion can be determined. The influence of electrolyte activity on the apparent formal potential of O~(bpy)3~+/*+ exchanged in Nafion is shown in Figure 1. In this figure (Eo'p)spp is determined when qa(III)= qa(11)for sodium chloride concentrations ranging from 0.005 to 0.1 m. In Figure 1 a plot of versus log ( I N ~ C I is fit by the solid line to give a slope of 55.8 f 0.8 mV. This value is consistent with the value predicted by eq 22 for highly permselective films. The fact that the slope is slightly less than the ideal value of 59.2 mV suggests that the Nafion film deviates slightly from ideal permselectivity or that the activity coefficients in the polymer are changing to some extent as the electrolyte activity in solution is changed. The following measurement was made to verify the results obtained versus SSCE. Instead of using an SSCE as the reference electrode and going through the trouble of correcting for the changing liquidjunction potentials, a Ag/AgCl reference electrode was exposed directly to the same electrolyte solution to which the Nafion-coated working electrode was exposed. Although this means that the potential of the reference electrode also changes as the electrolyte activity changes, this scheme is advantageous because it eliminates all liquid junction potentials. The changing reference potential does not introduce any interpretive difficulties because that change is well-known. If the mean ionic activity
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2429
Redox Couples in Permselective Media 0.65
i
-
leo0
c 0
8 3
1
0.60 -
-
0.40 1
J/r---== A
il A 0.20
0 -2.5
-1.5
-2
-1
loea
Figure 1. Solid line: linear least-squares fit to values of (Eo’P)aPP (open squares) measured for Os(bpy)j3+/*+in Nafion as a function of electrolyte activity; slope = 55.8 0.8 mV. Dashed line: electrolyte dependence of (EO’ ) calculated using independently determined values of E”’,, K N ~ ~KN,~(II)”, ( ~ ~and~ TC. ; ~Filled triangle: formal potential of Os(bpy)~~+/Z+ in aqueous 0.050 m NaC1. All potentials are reported versus SSCE after accounting for changing liquid junction potentials. coefficients are used to calculate anion and cation activities in a 1:l electrolyte, then when the half-cell described by eq 22 is measured versus Ag/AgCl, the cell potential is given by eq 24,where the standard potential for the Ag/AgCl electrode, Eoref,
is 0.2223 V. As can be seen in eq 24, E,II should increase by 118.4 mV for each factor of 10 by which the sodium chloride activity increases. The reference electrode should shift by -59.2 mV while (Eo’p)app shifts by 59.2 mV. If EEcllis measured when qa(111) qa(11) for concentrations ranging from 0.010 to 0.50 m, a plot of Effillvs log aqs)gives a straight line with a slope of 116.4 f 1.7 mV. Thus, if one assumes that contribution of the reference electrode to this slope is exactly 59.2 mV, the contribution of the working electrode is 57.2 f 1.7 mV. This agrees with the 55.8 f 0.8 mV slope obtained versus SSCE. The results of measurements made versus Ag/AgCl are not included in Figure l to keep it from becoming needlessly convoluted. Determination of F S and qc. We now turn to they intercept of the plot shown in Figure 1, i.e., to the quantity in brackets in eq 22. (Note that the quantity in brackets in eq 22 is equivalent to the quantity in brackets in eq 16b.) Eo’#for O~(bpy)3~+/’+ was measured in 0.050 m NaCl and found to be +0.635 V. qc is estimated as follows. All measurements were performed on films in which the loading level of O ~ ( b p y ) 3 ~ +was / ~ +near its maximum value. Anson et al. have shown that if a film is loaded to high levels with Os(bpy)32+, transferred to a solution that is free of osmium, and then cycled between oxidized and reduced forms, the maximum loading level is one in which all of the exchange sites are occupied by O~(bpy)~3+ when the film is in the oxidized form. For the same film in its fully reduced form, two-thirds of the exchange sites are occupied by Os(bpy)32+ and one-third by supporting electrolyte c0unterions.1~Consequently, because E,ll was measured in our experiments when l o x= 7 ~ 4qc , is assumed to be one-sixth. and To get reasonable estimates Determination of of K N ~ ~ (it~was ~ )necessary ”, to choose [ O ~ ( b p y ) 3 ~and + ] ~[Na+Is so that qa(11)in the film was significantly less than one-third. If ~a(11) is greater than one-third, then greater than two-thirds of the available exchange sites are occupied by Os(II), and upon @*’I.
! 10 15
Figure 2. Time dependence for loading of Os(bpy)3’+ (open squares)
and Os(bpy)32+(filled squares) into Nafion. The dashed line shows the maximum mole fraction of exchange sites (Le., z/3) that can be occupied by Os(bpy)32+and be determined coulometrically.
oxidation of the film some of the exchanged osmium complex must be expelled to maintain electroneutrality. In practice it was necessary for us to keep qm(11)near 0.25 to avoid a large percent uncertainty in the value of q~~ when qa(11) approached one-third. Because of the large magnitude of KN,~(II)” it was necessary to use Os(bpy)32+ concentrations on the order of 1 X lo-* M. Because it was not convenient to measure [ O s ( b ~ y ) 3 ~ + ] ~ at theselow levels-themolar absorptivity ofthe bipyridineligands is not large enough to make UV absorption measurements useful-the volume of solution used during the exchange experiments was chosen to be large enough such that any change of [ O ~ ( b p y ) ~ 2during + ] ~ the uptake of Os(bpy)32+ into the Nafion film was small. Additionally, the loading solutions were stirred to ensure that uptake of O ~ ( b p y ) ~into ~ +the film was not rate limited by mass transfer to the polymer surface. Under these conditions the rate of uptake was found to be influenced by [Os( b ~ y ) ~ 2 +[Na+],, ] ~ , and by the temperature at which the exchange was performed. Efforts to measure KN~@(III)” by monitoring the uptake of Os( b ~ y ) ~ from 3 + aqueous solutions containing sodium chloride were hindered by spontaneous oxidation of water and concomitant reduction of Os(bpy)33+ to Os(bpy)32+. Although this redox reaction is relatively slow at neutral pH,’* it occurs rapidly compared to the rate of uptake of O~(bpy)3~+ from the dilute solutionsof Os(bpy)s3+required to measure KN~&(III)”. Attempts to monitor O s ( b ~ y ) 3 ~uptake + while the loading solutions were electrolyzed to maintain the complex as Os(bpy)s3+ also failed. Os(bpy)32+was the predominant species present in the films when the loading was performed in this manner. O~(bpy)3~+ was successfully loaded into films when the electrodes were immersed in Os(bpy)3*+ solutions while the potential was poised approximately 150 mV positive of the apparent formal potential of the O ~ ( b p y ) ~ ~ +couple / 3 + in the films. Because both the rate of Os( b ~ y ) ~ 3uptake + and the rate of the spontaneous reduction of O ~ ( b p y ) ~ to 3 +O ~ ( b p y ) , ~by+ water are slow compared to the rate of electron transfer at the polymer/solution interface, under these conditions the film is free to equilibrate with a solution in which the O~(bpy)~’+concentration near the polymer surface is essentially equal to the concentration of O~(bpy)3~+ in the bulk of the loading solution. The time dependence of O~(bpy)3~+ uptake from aqueous solutions containing 2.1 X 10-8 m O ~ ( b p y ) 3 and ~ + 0.050 m NaCl is shown in Figure 2. After correcting the solutionconcentrations for the amount of metal complex exchanged into the Nafion, KN~@(II)” and KN~@(III)’’ were found to be 5.4 X 104 and 7.5 X 104, respectively. The standard deviations for these equilibrium constants, which are estimated to be f 15%, were determined by calculating the standard deviation for the scatter in the ionic
Redepenning et al.
2430 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
molar fractions measured for the osmium complex in the film when the film had reached equilibrium with the loading solution. 0The standard deviations for the ionic molar fractions of sodium ions in the film were then determined, and these standard 8 -20 deviations wereused in eqs 18and 19 toobtain standard deviations for KN,WW”and KN,a(III)”. Having determined E”,, WNa, KN,~(*I)”, and KNaa(””,it iS possible to substitute these values into eq 22 to calculate the cell potentials as a function of electrolyte activity. The results of these calculations are plotted as the dashed line in Figure 1. The apparent formal potentials measured as a function of electrolyte activity are in good agreement with those determined by substituting measured values of EO’,, WN,, KN,&(II)’ and , KNa08(*”)’ into eq 22. For this particular electrode the offset between the lines is only 5 mV a t an electrolyte concentration of 0.050 m.To -80-8 -5 -4 -3 obtain a more statistically informative comparison, the apparent Inn formal potentials of 10 different Nafion-coated electrodes Figure 3. Open squares: apparent reaction entropies for O~(bpy)3~+/*+ saturated with O ~ ( b p y ) 3 ~ +were / ~ + measured in 0.050 m NaCl. in Nafion as a function of electrolyte activity. Solid line: linear IeastThe average of these ten measurements was +OS855 V and the squares fit, slope = 8.8 1.6 J K-I mol-I,y intercept = -6.7 6.7 J K-I standarddeviation wasO.OO24V. According toeq22, theapparent mol-’. Filled triangle: reaction entropy for O~(bpy)3~+/~+ in 0.05 m NaCI. formal potential in 0.050 m NaCl should be +0.590 f 0.005 V given the 15% standard deviations in KN,~(~I)” and KN,@(III)”. we measured K~~a(I11)” and K~~a(11)” as defined in eqs 18 and Thus, the apparent formal potentials measured as a function of 19-we cannot use eqs 25a and 25b a t this point in our argument electrolyte activity are indistinguishable from those determined to interpret the apparent reaction entropies in Nafion. What we by substituting known quantities of EO’,, VN,, &@(I1)”, KN,~(~~~)”, can compare is the temperature dependence of the apparent formal and a ~ ( ~into ) , eq 22. It appears that the description developed potential with the temperature dependence of the terms on the in eqs 7-22 is a valid means of interpreting the differences between right-hand side of eq 22. This gives eq 26a, which can be written formal potentials of redox probes in solution and the apparent as eq 26b, where (AScOX)O” and (AScRd)O’’ are associated with formal potentials of those same redox couples in permselective media.
1:
7 *
Evaluation of Apparent Reaction Entropies Having established the utility of using eqs 7-22 to interpret the apparent formal potentials of redox probes in permselective media, it should be possible to use the temperature dependence of these equations to interpret the temperature dependence of apparent formal potentials measured for O ~ ( b p y ) 3 ~ +in/ ~Nafion. + As is shown in the following, the derivative of eq 16 with respect to temperature gives the apparent reaction entropy, in the polymer:
In eq 25b ASo’,,n,, is the reaction entropy of O ~ ( b p y ) 3 ~ +in /~+ (ao’rm,p)app
= ao’,xn,s
+
(e)o’
- (@lo’ R In [C’], + R In u,(~) (25b)
solution, and (ASEd)O’ are the ion-exchange entropies for partitioning the oxidized and reduced forms of the redox couple into the polymer. We assume that the activity coefficients are constant. The activity coefficients of the electrolyte in solution do have a small temperature dependence, but that dependence is negligible compared to some of the other terms on the righthand side of eq 25b, and it is also much smaller than the uncertainty in our measured values of the apparent reaction entropics. Earlier in this discussion we were able to show that it is not necessary to know the concentration of exchange sites in Nafion in order to account for the differences in the formal potential of a redox couple in solution and the apparent formal potential of that same redox couple in Nafion. Because we did not include the concentration of sites in our development-recall that we did not measure KN,~(III)‘ and KN~@(II)’ as defined in eqs 10 and 1 1, but
(ao’rxn,p)app
= a o ’ r x n , s + (@“)O”-
(@)O”-
R In vC
+ R In aC(,)(26b)
the temperaturedependence of Kcw‘and KcRd”. Therelationship of (AScox)O”and (AScRd)O” to (AScoX)O’and(AScRd)O’ will be discussed in more detail later. First we use eq 26b to interpret the experimental results obtained by measuring the temperature dependence of the apparent formal potentials of Os(bpy)~3+/2+ in Nafion as a function of electrolyte activity. If the resulting apparent reaction entropies are plotted versus In ac(,), eq 26b suggests that a straight line with a slope equal to the ideal gas constant and an intercept of {AS,m,,o’ (AScRd)O”- AS COX)^" - R In qc] should be obtained. Such a plot is shown for three different electrodes in Figure 3. The slope reported was determined from the best fit to the pooled data for all three electrodes while making no distinction as to which data points came from which electrodes. Although this work will not long be remembered for the accuracy or the precision with which the ideal gas constant has been determined, the data do show that the apparent reaction entropy has the expected electrolyte dependence. The slope of the plot is within experimental error of the known value. The uncertainty in the individual values of the apparent reaction entropy is f(5-10) J K-I mol-’, which is consistent with the uncertainty of f l eu that Weaver estimated for measuring reaction entropies of redox probes in solution.& Although the data in Figure 3 show a large amount of scatter, it appears that it may be difficult to improve on these measurements because of the inherent limitations of measuring the reaction entropies in this manner. Still, the experimental data clearly indicate that the entropy of transport of counterions can play an important role in determining the apparent reaction entropy, and,
+
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2431
Redox Couples in Permselective Media
as willbeshown below, a t low supporting electrolyteconcentrations h(AS,,) is -50 J K-I mol-’; of this total -20 J K-I mol-’ is the apparent reaction entropy in the polymer may actually be attributed to ((AScRd)O‘ - (ASc0x)O’) and the remainder is dominated by the entropy change associated with transporting attributed to the entropy change associated with ion transport. counterions.’a-9b Given the small value of -20 J K-I mol-’ obtained for the difference To evaluate further the usefulness of eq 26b for interpreting between the entropies of exchange, a change in oxidation state the apparent reaction entropy in the polymer, we now consider of this particular redox probe apparently does not produce a large the intercept of the plot shown in Figure 3. We desire to compare degree of reorganization of the polymer or of the solvent. One the value of this intercept with the values of ASm,,O’, (AScO”)””, might argue that this is expected given the small reaction entropy (AScRd)O”and R In qc measured independently. In principle measured for Os(bpy)33+/2+ in water. When Os(bpy)33+ is ( A S C O ~ ) ~and ” (AScR“)O”can be determined by measuring the dissolved in water, changing the oxidation state from O~(bpy)3~+ temperature dependence of KN,OS(II1Y and KN,@(II)”.In practice, to O ~ ( b p y ) 3 ~apparently + does not induce significant changes in because the changes in KNa&(III)”and K N , ~ ( I I )are ” small over the the water structure around the complex. Likewise, when Ostemperature range accessible to us and because of the uncertainty ( b ~ y ) 3 is ~+ exchanged in Nafion, our results suggest that changing in KN,&(III)”and KN,@(II)”,the entropy changes determined by the oxidation state from O~(bpy)~3+ to O ~ ( b p y ) ~apparently ~+ measuring the temperature dependence of these quantities are induces only small changes in the water/polymer structure. Some only qualitativelyuseful. We find that (ASc&)O”and (hScRd)O” of the difference between (AScRd)O’and (AScO”)”’is due to the are roughly 70 f 40 J K-1 mol-’. In both cases it is clear that much process of distributing multiply charged ions over exchange sites of the free energy of exchange is associated with processes that with unit charge. Cruickshank and Meares have shown that if increase the overall entropy of the system, but the uncertainties N / 2 divalent cations are randomly distributed among Nexchange in these values are so large that it is not useful to compare them sites, then the increase in entropy per equivalent of sites is 5.8 to the apparent reaction entropies using eq 26b. As an alternative J K-I m01-l.’~ For N / 3 trivalent cations randomly distributed to this approach, although no information concerning the among N exchange sites, the increase in entropy per equivalent magnitudes of the entropies of exchange is obtained, it is possible of sites is 8.8 J K-1 mol-’. So approximately 3 J K-1 mol-’ of the to use the intercept of the plot in Figure 3 to ascertain the difference difference between (AS$“)”’ and (ASCOX))”’ can be attributed between the entropy of exchange for O ~ ( b p y ) 3 ~and + that for to differences in the configurational entropy of the two systems. Os(bpy)33+. The difference in these entropies of exchange can We attribute the remainder of the difference between (AScRd)O’ then be compared to the apparent reaction entropies to get an and ( A S C ~ ~to)the ~ ’expectation that a slightly larger entropy indication of the environmental changes which occur in the of exchange should beobserved for Os(bpy)33+than for O~(bpy)3~+ membrane in conjunction with the redox proecess. because the former more effectively cross-links the polymer while Knowing that the intercept of the plot in Figure 3 is -6.7 f expelling more water to increase the entropy of the system. 6.7 J K-I mol-’, one can determine ((AScRd)O’’ - ( ~ S C ~ ”if ) ~ ” } We end with a short discussion of the large free energies (large qc and ASrxn,so‘ are known. We take vc to be one-sixth for the selectivity coefficients) observed for partitioning ions with same reasons it was taken to be one-sixth when examining eq 22. structures similar to O~(bpy)3~+/2+ into Nafion. Exchange of ASrxn,so’ was measured in 0.05 m NaCl and was found to be 0 solution ions with O~(bpy)33+/~+ appears to be entropically driven f 5 J K-1 mol-’. Thus, ((ASCR“)””- ( A S C ~ ) ~is”approximately ) in both oxidation states. Entropy increases have long been known -22 f 7 J K-1 mol-’. Next, eqs 25b and 26b can be combined to contribute significantly to the free energies of exchange for to give univalentaivalent ion-exchange in non-cross-linked polymers such as sulfonated polystyrene20921 and Nafion.22 Two excellent { ( A S y ) O ’ - (AS?)O’) = { ( h s p y ’ -( A S ? ) O ” ] thermodynamic discussions concerning exchange of univalent ions R In [C?], - R In qC (27) with divalent metal ions are available.19~23 When sodium ions in Nafion are exchanged for multivalent ions such as O~(bpy)3~+1~+, So, if an estimate of the concentration of supporting electrolyte the entropic contributions to the free energy of exchange is much cations in the polymer is available, ((AcRd)O’ can larger than that observed when, for example, sodium ions are be estimated. Whitely et al. have reported that the density of exchanged with barium ions.22 Our results are consistent with 1100E W Nafion in the sodium form is 1.35 g/cm3.’2 By assuming the so-called hydrophobic interactions that are believed to lead that no clustering of exchange sites occurs and that the sites are to the large equilibrium constants seen when hydrophobic ions distributed evenly throughout the polymer, a lower limit of 1.2 are partitioned into Nafion. However, as is known to be the case M is placed on the concentration of exchange sites. This for similar systems, our data also demonstrate that the term establishes a lower limit of [C+], = 0.2 M, because for these “hydrophobic interaction” is somewhat of a misnomer.24 There experimental conditions one-sixth of the exchange sites are are no identifiable enthalpic interactions which dominate the occupied by sodium ions when [O~(bpy)3~+], = [ O s ( b ~ y ) 3 ~ + ] ~ exchange process. No bonds are made or broken and there are Having values of [C+],, qc, and { ( A S C ~ ~ ) O ” (AScO”)O”}, we probably not any strong van der Waals attractions between the estimate {(AScRd)O’ - ( A S C ~ ~ )to ~ ’be ) -20 J K-1 mol-’ (--5 hydrophobic bipyridine rings and the Teflon-like hydrophobic eu). How does this help one interpret the differences between regions of Nafion. Thus, it is not surprising that the enthalpies the apparent reaction entropies in the polymer and the reaction of exchange appear to be small. The phenomena which lead to entropy in solution? By making the substitution in eq 25b that the large equilibrium constants are as much solvent-based as A(AS,,,) = {(ASrxn,po’)app - ASrxn,so’),the following relationship they are Nafion-based. In many respects the exchange process is obtained: seems analogous to micelle formation in aqueous solutions. There is a relatively large entropy increase associated with proceeding - R In [C’], A(ASrxn) = {(ASEd)O’from a system with hydrophobic molecules surrounded by highly ordered water molecules to a system in which the hydrophobic R In %(s) (28) molecules have formed micelles and the water has become more disordered.25 Clint et a1.26b and Schick26a have shown that It is evident from this relationship that ((AScRd)”’ - (AScoX)O’) enthalpies of micelle formation can actually be positive in some includes those contributions to A(ASrxn)that are not associated cases, yet the free energies of micelle formation remain large. It with ion transport. Once the contributions of ion transport to the seems likely that there is a relatively large entropy increase apparent reaction entropy in the polymer are taken into account, all other factors that contribute to the reaction entropy of associated with proceeding from a system in which hydrophobic O~(bpy)3~+/2+ in Nafion are small. For example in 0.020 m NaCl metal complexes (such as O~(bpy)~2+) are surrounded by highly
+
+
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The Journal of Physical Chemistry, Vol. 98, No. 9, 1994
ordered water molecules in an aqueous bathing solution to a system in which the hydrophobic metal ions are aggregated in hydrophobic Nafion domains and the water is free to become more disordered. Likewise, for the Nafion itself, there are some entropic advantages to expelling water and minimizing the surface area between the hydrophobic and hydrophilic domains. In many respects this process may be analogous to an emulsion in a separatory funnel that eventually separates to form a well-defined two-phase system in which hydrophobic solutes are partitioned intoan organic phase to produce a more disordered aqueous phase. In conclusion, the equilibrium description developed above appears to be consistent with the differences between the formal potentials observed for redox couples in solution and the apparent formal potentials of those same redox couples in permselective media. Additionally, the description lends itself to interpreting apparent reaction entropies measured for redox probes in permselective media. The above description does not provide information concerning nonequilibrium conditions or about how equilibrium is reached, but we feel that this equilibrium description and recent descriptions of charge propagation, solvent transport, and ion t r a n ~ p o r t ~are ~ -leading ~~ to a more sophisticated and more comprehensive understanding of electrochemical phenomena in thin films on electrodes. Acknowledgment. This research was supported by the donors of the Petroleum Research Fund and by NSF EPSCoR Cooperative Agreement OSR-9255225. Discussions with Colby Foss and Fred Anson are also a pleasure to acknowledge. References and Notes (1) (a) Naegeli, R.; Redepenning, J.; Anson, F. C. J. Phys. Chem. 1986, 90,6227. (b) Redepenning, J.; Anson, F. C. J. Phys. Chem. 1987,91,4549. (2) Szentirmay, M. N.; Martin, C. R. Anal. Chem. 1984, 56, 1898. (3) Yee, E. L.; Cave, R. J.; Guyer, K. L.; Weaver, M. J. J. Am. Chem. SOC.1979, 101, 1131. (4) (a) Yee, E. L.; Cave, R. J.; Kendall, L. G.; Tyma, P. D.; Weaver, M. J. J. Am. Chem. SOC.1979,101, 1131. (b) Yee, E. L.; Weaver, M. J. Inorg. Chem. 1980, 19, 1077. (c) Sahami, S.;Weaver, M. J. J. Solution Chem. 1981,10, 199. (d) Hupp, J. T.; Weaver, M. J. Inorg. Chem. 1984,23,3639. (5) (a) Jaworski, J. S.Polyhedron 1987, 6, 2151. (b) Jaworski, J. S. Electrochim. Acta 1988, 33, 717. (c) Jaworski, J. S.J. Electroanab Chem. 1989, 260, 327. (6) Nagaoka, T.; Yoshino, T.; Okazaki, S.J. Elecrroanal. Chem. 1988, 242, 323. (7) Hupp, J. T.; Weaver, M. J. J . Electrochem. SOC.1984, 131, 619. (8) Tsou, Y.-M.; Anson, F. C. J. Electrochem. SOC.1984, 131, 595. (9) (a) Lieber,C.M.;Schmidt,M.H.;Lewis,N.S. J. Phys. Chem. 1986, 90,1002. (b) Schmidt, M. H.; Lewis, N. S.J. Phys. Chem. 1988,92,2018. (10) Oyama, N.; Oshaka, T.; Ushirogouchi, T.; Sanpei, S.;Nakamura, S. Bull. Chem. SOC.Jpn. 1988, 61, 3103. (11) Creutz, C.; Netzel, T. L.; Okumura, M.; Sutin, N. J. J. Am. Chem. SOC.1980, 102, 1309. (12) Whitely, L. D.; Martin, C. R. J. Phys. Chem. 1989, 93, 4650. (1 3) Duncan, J. F. Proc. R. SOC.London 1952, A21 4, 344.
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