J . Phys. Chem. 1989, 93, 7275-7280 A similar explanation may be given for curves A'B' C'D', P'Q, and R'S' obtained for the precipitation of the red species. The regions A'B' and P'Q' represent one stable state due to KI in agar-agar gel, and the regions C'D' and R'S' represent another stable state due to K2Hg14. A detailed theoretical understanding of the results awaits further study. Acknowledgment. We are grateful to Prof. R. P. Rastogi,
1215
Vice-chancellor, BHU, for his help and inspiration, Prof. S. Giri, Head of the Chemistry Department, for providing laboratory facilities, and Dr.P. S.Pandey, Director, Academic Staff College, University of Gorakhpur, for encouragement. Thanks are due to CSIR, New Delhi, for financial support to Dr. Anal Pushkarna. We are thankful to Sudha Chand for rendering help at various stages in the experimental work. Registry No. Hg12, 7774-29-0.
Thermal Modulation Voltammetry Response of Reversible Redox Systems: Theory and Experiment J. L. Valdes and B. Miller* AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received: November 28, 1988; In Final Form: February 14, 1989)
Thermal modulation voltammetry (TMV) at a rotating disk electrode (RDE) is a new technique developed for understanding electrochemical processes in the temperature domain. A theoretical analysis of this method and its experimental verification are conducted here for cases of reversible (Nernstian) electrode reactions. When both the oxidized and reduced forms of the electroactive species are present, the thermally modulated component of the current can exhibit a well-defined peak near the equilibrium electrode potential as the potential is scanned between the limiting current plateaus. The existence and magnitude of this characteristic peak are shown to be a sensitive function of the thermodynamic entropy change (ASo)of the electrode reaction and also to de end upon the thermal properties of the solution. TMV experiments on two separate redox systems, Fe(CN)6*/C and Fez+ 3+, confirm theoretical predictions. Results are shown to be consistent with steady-state measurements at different temperatures in a thermostated cell.
P
Introduction
In recent the steady-state and frequency-dependent behavior of mass transport limited currents and open circuit potentials at a thermally modulated rotating disk electrode (RDE) has been examined. In these cases the electrochemical response was found to be determined by the prevailing surface temperature perturbation AT. To gain insight into electrode reactions through the thermal parameters, it is necessary to broaden the treatment to cover the response of currents over the entire electrode potential region of interest. In this work, we present a theoretical analysis and an experimental study of thermal modulation voltammetry (TMV) at a RDE for reversible (Nernstian) electrode reactions. In a TMV experiment, the electrode temperature is modulated at a given frequency while the electrode potential is scanned in the region of interest and the corresponding cell current modulation is extracted. The modulated current response will depend upon the temperature coefficientsof the many variables which contribute to the electron-transfer process; they may be grouped as thermodynamic, kinetic, and mass transport. In a purely reversible system (reactions not limited by kinetics), we will show that the limiting current (mass transport) and standard electrode potential (thermodynamics) are two such quantities whose disparate thermal sensitivities govern the net response. In this work we develop the theoretical model necessary to describe the complete voltammetric response of a thermally modulated reversible reaction. This understanding allows obtaining thermodynamic reaction entropies from a TMV experiment. The theoretical predictions derived in this paper have been experimentally tested by conducting TMV measurements on the redox system, Fe(CN)63-/e, in either 1 M KCl or 0.4 M Sr(N03)2 electrolytes, under conditions approaching reversibility. AddiMiller, B. J . Elecfrochem. SOC.1983, 130, 1639. Valdes, J. L.; Miller, B. J . Phys. Chem. 1988, 92, 4483. (3) Valdes, J. L.; Miller, B. J . Electrochem. SOC.1988, 135, 2223.
(1) (2)
tionally, steady-state current-voltage curves have been taken at two temperatures in a thermostated cell and their differences obtained to test the "zero thermal modulation frequency" analogue of the TMV experiment. TMV scans were also performed on the system has Fe2+/3+system in 1 M HClO,; although the slower electron transfer than that of Fe(CN)63-/4-, the reaction entropy of the reaction is comparable in magnitude, but opposite in sign, to that of Fe(CN)63-/4-, and thus provides an additional test of theory developed here. Theory
A purely reversible redox system of the general form Ox
+ ne- = Red
(1)
is described by the Nernst equation
U = UO
+ -1nRT
Gx
nF
Cscd
cd
where and Cox refer to the concentration of the reduced and oxidized species at the electrode surface, respectively, and UO is the standard electrode potential of the reaction. For a system with defined mass transport conditions, G
X
y = l + & '
i
(3)
'Lox
and (4)
where i is the current (positive for anodic and negative for cathodic), il,+ and il,oxare the absolute values of the anodic and cathodic limiting currents, respectively, and bulk species concentrations are denoted by a superscript b. At a RDE the anodic
0022-3654/89/2093-7275$01 .50/0 0 1989 American Chemical Society
7276
The Journal of Physical Chemistry, Vof. 93, No. 20, 1989
and cathodic limiting currents are given by the respective Levich expressions4
4.5
Valdes and Miller
,
I
Substituting eq 3 and 4 into the Nernst equation and solving for the current yields -1.51 -10
where 4 is a dimensionless potential, defined by 4 3 (nF/RT)(U = (Dd/D0x)2/3, is assumed to be independent of temperature from the StokesEinstein relationship. Differentiation of the current in eq 7 with respect to temperature at constant potential gives
- V), and the mass diffusivity ratio parameter, y
'
'
'
'
'
'
'
'
'
-8
-6
-4
-2
0
2
4
6
8
I
10
~F(u-u')/RT
I
1.5 I
a-7
(8) where ASo = nF(avO/aT)is the standard thermodynamic entropy change of the reaction. The limiting currents are implicit functions of temperature through the diffusion coefficient and kinematic viscosity, themselves usually exponential functions of the temperature. The partial derivatives with respect to temperature of the anodic and cathodic limiting currents are thus given by,2 respectively, (9)
and
ELed and E": are the activation energies associated with mass transport processes for the reduced and oxidized species, respectively. The activation energies EL" and E"; account for the temperature dependence of both the diffusion coefficients and the kinematic viscosity as manifested in the measured limiting currents. Substituting eq 9 and 10 into eq 8, assuming small temperature perturbations, and normalizing the thermal response with respect to the anodic limiting current perturbation Ail,red,yields the following dimensionless equation for the current perturbation Ai at any given potential,
(e)=
[1 - exp(-4)1 - 4 1 + 7 ) exp(-4) (11) [1 + Y exp(-+)l [ I + y exp(-4)I2
The dimensionless parameters u and u
c
are defined as
= TASo/E$
(12)
= Eg/E$
(13)
and c
u is a ratio of the entropic energy of reaction to the energy of activation for a mass transport limited process. In a purely reversible electrode reaction, this ratio governs the current response of the system to a thermal perturbation since y and t will both generally not be far from unity. For aqueous solutions, E D is on the order of a few kcal/mol. The standard entropy change for electrode reactions is related
(4) Levich, V. G. Physicochemicnl Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962.
-4.5 -10
-8
-6
-4
-2
0
2
4
6
8
10
~F(u-u')/RT
Figure 1. Dimensionless thermal current response curves for a reversible redox reaction for negative (a, top) and positive (b, bottom) values of u; dashed curve corresponds to the dimensionless unmodulated currentpotential response ( i / i , , r d ) .
thermodynamically to the thermal coefficient of electrode potentiaL5 From an extensive compilation by de Bethune,6 the temperature coefficients of electrode potential lie between about f2.4 mV/K. The corresponding range of values for the standard entropy change ASo is therefore about f55 cal/(mol K). In Figure 1 we show plots of the dimensionless thermal current reponse, Ai/Ail,rd, as a function of the dimensionless potential 4 for y = 1, c = 1, and different values and signs of CT. The unmodulated dimensionless current, i/il,rd,is also shown for comparison. The thermally modulated curves exhibit well-defined peaks near the standard potential V and are symmetrical with respect to the sign of the value of u. The dimensionless peak value in the modulated current has a sign opposite to that of the standard entropy of reaction. Note that, as the absolute value of u decreases, the peak height also decreases and this feature eventually dis1. Additionally, for the case of c = 1 the appears as 1. modulated and unmodulated dimensionless traces become superincumbent as ASo 0, as gleaned by inspection of eq 7 and 11. The occurrence of a peak in the modulated current response curve is attributed to the difference in the relative thermal sensitivities of the standard electrode potential and the limiting current. Consider the dimensionless current-potential curves shown in Figure 2 for a reversible reaction at two different temperatures and for ASo C 0. For this system, the standard potential UO shifts to more negative values at the higher temperature ( T , + AT). The anodic and cathodic limiting currents both increase in absolute value with increasing temperature. Subtracting one curve from the other yields a thermal current response curve with a positive peak located near V .On the other hand, when ASo > 0 the standard potential shifts with increasing temperature to more positive potentials, as shown in Figure 3. Subtraction of the current-potential curves yields a mirror image of the thermal current response shown before with a peak near V but a value opposite in sign to that of ASo (or u ) . The location of the peak in the thermally modulated current response curve on the potential axis is obtained by differentiation
-
-
( 5 ) Bard, A. J., Parsons, R., Jordan, J., Eds. Standard Potentials in Aqueous Solutions; Marcel Dekker: New York, 1985. (6) de Bethune, A. J.; Licht, T. S.; Swendeman, N. J . Electrochem. SOC. 1959, 106, 616.
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989 1271
TMV Response of Reversible Redox Systems
I
-
1 '-'
0.75
1
-0.75
-200
/ -100
100
0
U/mV
VI
200
Inspection of eq 16 reveals that a peak in the thermal response curve occurs when IuI > 1, as previously demonstrated from the theoretical current response plots. For values of 1. >> 1 the peak potential occurs at the standard electrode potential U",Le., 4 = 0. At this point the dimensionless current response (Ai/Ai,,rd)P = - u/2. In a TMV experiment, the measured peak current can be used along with eq 15 or eq 17 to obtain a value of u and thereby the thermodynamic entropy change of the electrode reaction under study.
reference
-0.50 -1 S O
-200
-100
0
100
200
U/mV vm reference
Figure 2. (a, top) Current-potential curves for a reversible system at two temperatures (AT = 5 K), ASo = -46.1 cal/(mol K) and ED = 3 kcal/mol. (b, bottom) Thermal current response obtained by subtraction of current-potential curves shown in (a). 1.25
0.75 h
Ge
s
0.25
-0.25 -0.75 -1.25 I -200
0.00
-3.00
0
1
1
-200
I
1 -100
U/mV
"-
VI
100
200
100
200
reference
I
I
/
I -100
0
U/mV vn reference
Figure 3. (a, top) Current-potential curves for a reversible system at two temperatures (AT = 5 K), ASo = +46.1 cal/(mol K) and ED = 3 kcal/mol. (b, bottom) Thermal current response obtained by subtraction of current-potential curves shown in (a).
of eq 11 with respect to 4. The potential at which the peak occurs, @ , is given by @' = In
(q[
[
UY(1 + 7 ) - r(t + 7 )
41
+ 7) + (e + 7)
]
(14)
The corresponding value of the peak at @ is
=- 241
4 d
-
are taken to be equal (y l ) , and the ratio of limiting current l ) , the peak activation energies is assumed to be unity (t potential and current expressions simplify to
7-1
+ r)(e- 7 ) + 4 4 1
+ + 7)
1
+ + 7)2
(15)
If the diffusion coefficients of the oxidized and reduced species
Experimental Section TMV experiments were conducted on two separate redox systems with large thermodynamic entropies of reaction of opposite sign; 1 mM concentrations each of K4Fe(CN)6and K3Fe(CN)6 in either 1 M KCl or 0.4 M S r ( N 0 3 ) 2as supporting electrolyte, and 1 mM concentrations each of FeZ+/Fe3+,from the perchlorate salts, in 1 M HC104. All chemicals were reagent grade and solutions were prepared with 18 M Q c m resistivity Milli-Q water. The laser beam assembly and electronic components used in conducting TMV experiments are identical with those described previ~usly.~A Pine RDE-3 potentiostat was used to scan the electrochemical potential between the anodic and cathodic limiting current regions while the electrode temperature was modulated by laser beam heating at constant frequency (usually 5-20 Hz). The signal containing the modulated response in the current Ai is fed into a Wavetek/Rockland 352 band-pass filter set to have Hi/Lo cutoff frequencies at one-half and twice the thermal modulation frequency. The amplified signal (usually 1OOX) is then fed into a lock-in amplifier and the modulated current is recorded on a Hewlett-Packard 7090A recording system and later uploaded to a computer. TMV experiments were conducted using a three-electrode configuration consisting of a gold wire counter electrode, a saturated calomel electrode (SCE) as the reference, and a 3 mm diameter gold thermal R D E of either the quartz cylinder3 or air-backed design,'q2 primarily the former, unless otherwise noted. A jacketed glass cell connected to a Neslab RTE-4 recirculating refrigerated bath provided control of the electrolyte temperature at 25.0 f 0.1 OC for TMV experiments. Nonisothermal steady-state experiments not employing the laser beam assembly were also performed for comparison with TMV results. One comparment of the nonisothermal cell contained a gold wire counter electrode and a gold R D E as the working electrode. The reference electrode resided in a second compartment connected to the first by a short glass tube. Each compartment was maintained at the desired temperature by a separate recirculating bath. Steady-state current-voltage curves were obtained at two different temperatures while maintaining the temperature of the reference compartment constant. In this manner no correction is required for the thermal coefficient of the potential for the reference electrode. A digital point-by-point subtraction of these curves on the computer, at common potentials, affords a steady-state finite temperature difference current-voltage curve that may be compared to the analogous differential curve acquired in a TMV experiment. Diffusion coefficients and activation energies for mass transport were also measured by using this arrangement. Results and Discussion In Figure 4 are shown TMV and current-voltage curves for the redox couple 1 mM Fe(CN)6*/& in 1 M KCl at a gold thermal RDE under a potential scan rate of 0.83 mV/s. The disk rotation speed, thermal modulation frequency, and input laser power are
7278
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989 4.0
b
h
1.0
9
Valdes and Miller
,
1
-2.0 0.00
0.10
0.20 U/volts
VI
0.30 SCE
0.40
Figure 4. Experimental TMV and current-voltage (dashed) curves for 1 mM Fe(CN)&/& in 1 M KC1 at a gold thermal RDE scan rate = 0.83 mV/s, disk rotation = 2500 rpm, modulation frequency = 20 Hz,and laser power = 188 mW.
respectively 2500 rpm, 20 Hz, and 188 mW. The measured semilogarithmic slope from the current-voltage curve, correcting for ohmic losses, is 64.6 mV/decade. The curves have been normalized to their respective limiting anodic values for direct comparison with the theoretical results. The TMV curve exhibits a well-defined peak on the anodic side near the reversible potential (zero current), and a plateau region at both the anodic and cathodic limiting currents. These results are consistent with the theoretical model presented earlier for a redox reaction having ASo C 0 and lul > 1. For the ferri-ferrocyanide redox system in 1 M KCI,ASo= -33.67 cal/(mol K)and EL* = 1.46 kcal/mol (see ref 2); thus u = -6.88. Activation energies for mass transport of the electroactive species, E$ or Eo:, were determined through an Arrhenius plot of the respective limiting currents measured at two temperatures ( A T 5 "C) in the steady-state nonisothermal experiments. E"$ for this system was found to be 1.12 kcal/mol; thus t = 0.80. Substituting these system parameters into eq 15 along with y = 0.88 yields a theoretical value of (Ai/Ail,rd)P = 3.78, in good agreement with the experimentally measured value of 3.65. The simplified expression for (Ai/Ail+)P given in eq 17, which assumes y = e = 1, yields a value of 3.51 for a difference of less than 7% incurred in the approximation. The apparent concordance between calculated and measured T M V peak ratios obtained for the ferri-ferrocyanide system in the experiment of Figure 4 is deceptive as we have found experimentally that the peak ratio is a function of disk rotation speed. The theoretical result of eq 11 in fact suggests that the ratio of thermally modulated currents (Ai/Ail,d)P ought to be a constant, independent of disk rotation speed, modulation frequency, and input laser power. The peak ratio should only be determined by the system parameters u, y, and t, and principally by u. In order to understand the origins of this w-dependent behavior, eq 11 evaluated at peak conditions is rewritten in the following form,
-
(e)= From our previous study on the steady-state behavior of a thermally modulated RDE? we have theoretically demonstrated and experimentally verified that the temperature perturbation A T at the electrode surface, as measured through both open circuit potentials and limiting currents, is an inverse square root function of the rotation speed (w-llz). From eq 5 and 9 we find that, since Ail,rd,which is ildAT, is in theory i,,rd wl/z and AT independent of rotation speed.z In spite of this theoretical independence, experimental factors, namely the heat losses in a laser-heated RDE system, cause Ail,d to be a function of w1/2, which we were able to modeL2 With this factor included, it necessarily follows that the overall functionality of the square-
-
-
0.801 mM Fe(CN)6-3/-4
0.60 .
4
KC' A
0.40 .
a 0.20
.
n nn -._0.01
P
A
0.03
0.05
A
0.07
y-1/2/Tm-l/2
Figure 6. Dependence of the TMV peak on rotation speed for the ferri-ferrocyanide and 1M KCI system; symbols as in Figure 5 .
bracketed quantity in the second term in eq 18 with respect to In contrast, the first term in eq 18 is a rotation speed is constant at the peak potential and independent of rotation speed. Consequently, a plot of (&/bw)P vs is expected to be linear with a slope proportional to the curly-bracketed term in eq 18, m corresponding to the first term in this and an intercept at w equation which can be alternatively expressed by
-
When y =
t
= 1, the intercept is given by
The results given in eq 19 and 20 suggest that extracting thermodynamic information for reversible electrode reactions in a TMV experiment is possible, even when thermal losses are present in the system, by an extrapolation to infinite rotation speed (CO-I/~ 0). In Figure 5 are plotted the experimentally determined peak ratios as a function of w - [ / * for three different sets of data, including the set compiled from TMV curves such as that shown in Figure 4. These data were acquired at modulation frequencies of 5 and 20 Hz, and input laser powers of 126-128 and 188 mW. In each case a good linear correlation is found (the inverse values m), of the plotted intercepts are equal to 4.5, 1.7, and 2.2 a t w but the spread indicates the relatively large error involved in such an extrapolation. In practice, we find that the thermally modulated current at the peak (Ai)p is a relatively insensitive function of the disk rotation speed, the change in the ratio is attributable to Ai,, being a strong function of w, as previously d e m ~ n s t r a t e d . ~ Multiplication of eq 18 by AilJd affords theoretical justification for this experimental result since the second term in the equation dominates at the peak potential and the product of il,rdAT is independent of w. In Figure 6 are plotted peak thermally modulated currents in dimensional units as a function of w-1/2. Note
-
-
The Journal of Physical Chemistry, Vol. 93, No. 20, 1989 7279
TMV Response of Reversible Redox Systems 6.0
b
3.0 2.0
I
I
t
-2.0
0.0
1.50
.
1.00
.
Peak-I .57
h
0.1
0.2
-1.50
0.4
0.3
I
0.0
0.1
0.2
0.4
0.3
I
0.5
U/volts v8 SCE
U/volts vs SCE
Figure 7. Experimental TMV and current-voltage (dashed) curves for 1 mM Fe(CN)63-/C in 0.4 M Sr(N03)2at a gold thermal RDE, w = 900
rpm, other conditions as in Figure 4. 7.0 1
Figure 9. Dimensionless thermal current response curve digitally con-
structed from current-voltage curves obtained at temperatures of 24.5 and 28.9 OC, and for ferri-ferrocyanide Sr(N03)2system.
I
1.5
I
1
1 .o h
e
0
b h
0
2 2 0.0 y 0.00
0.01
0.02
0.03
0.04
0.0 -0.5 -1.0 -1.5 -2.0 -2.5
I
0.05
0.2
#-1/2/"-1/2
Figure 8. Dimensionless TMV peak ratios vs cyanide and 0.4 M Sr(N03)*system.
0.5
0.3
0.4
0.5 U/volta
for the ferri-ferro-
that the current responses obtained for the 20 H z data (A)are higher because these experiments were performed at larger input laser powers using the air-backed thermal RDE which has greater absorptivity for laser light from a CuO back surface coating. The experimental results shown here are in accord with theory and are also consistent with the observed rotation speed dependence of (Ai/Ail,rd)P. In a reversible electrode reaction, the modulated current response, as observed in a TMV experiment, has been shown to depend fundamentally on both a thermodynamic quantity, ASo, and a mass transport energy of activation, ED TMV experiments were conducted for the redox couple 1 m M Fe(CN)63-/4- in 0.4 ,Xand M S T ( N O ~ )as~ ,a supporting electrolyte with different P E$ than 1 M KCI, in order to explore the mass transport dependence of the modulated current response. Activation energies for mass transport were measured experimentally in this system by the procedure noted earlier.3 The values for the oxidized and reduced species, E"; = 2.96 kcal/mol and E$ = 3.68 kcal/mol, respectively, are higher by more than a factor of 2 than those of the 1 M KC1 system. The dimensionlessparameters for the nitrate electrolyte are u = -2.73, y = 0.88, and t = 0.80. Note that, although u is notably different, the values for y and, in particular, t, are the same as in the KCI supporting electrolyte. Substituting these values into eq 15 yields a peak ratio of (Ai/Ail,rd)P= 1.66 at a dimensionless potential of qV' = 0.55. In Figure 7 are shown TMV and current-voltage curves for this system acquired under the same experimental conditions as in Figure 4, but at 900 rpm. A peak is observed in the TMV curve near zero current and the = 4.52. In Figure 8 are plotted measured peak ratio is (Ai/Ail,rcd)P the measured peak ratios vs u - I / ~ . The extrapolated value of the intercept is found to be 0.21, consonant with the calculated value of 0.34 using eq 19. Steady-state thermal measurements without the use of laser beam heating of the electrode were also performed on this system in order to check their self-consistency with the results obtained in a TMV experiment. Steady-state current-voltage curves were taken at two temperatures, 24.5 and 28.9 "C, while maintaining the reference electrode compartment always at 28.9 'C. The data were then digitally processed to obtain a differential curve simulating, in effect, a TMV experiment at zero modulation frequency. The constructed thermally "modulated" current-voltage
VI
0.6
0.7
0.8
SCE
Figure 10. Experimental TMV and current-voltage curves for 1 mM Fe"/'+ in 1 M HCIO,, at a gold thermal RDE, scan rate = 3 mV/s, disk rotation = 900 rpm, modulation frequency = 3 Hz, and laser power = 152 mW.
curve for the ferri-ferrocyanide/Sr(N03)2 system is shown in FiguYe 9 for w = 1600 rpm. The constructed curve exhibits the same characteristic features as obtained in a real-time TMV experiment. The measured peak ratio is 1.57, close to the predicted value of 1.66 obtained from the differential analysis. A high degree of reproducibility of individual curves is required for this approach because of the subtraction of two "large" numbers for successive scans and the period required for thermal equilibrium with conventional cells and thermostats. In general it is much more difficult to perform repeatedly than TMV experiments. Additionally, the ability to modulate the electrode temperature at frequencies upwards of 20 Hz allows for higher potential scan rates and greater speed in a TMV experiment. As a final test of the theoretical model for TMV we consider experiments on a redox couple with a positive standard entropy of reaction, 1 mM Fe2+/3+in 1 M HC104, at a gold thermal RDE. In Figure 10 are shown TMV and current-voltage curves for this system scanned at 3 mV/s with disk rotation speed, thermal modulation frequency, and input laser power of 900 rpm, 3 Hz, and 152 mW, respectively. The TMV curve exhibits plateau regions at both limiting currents and the peak is negative with respect to the anodic limiting current modulation Ail,rd,in accordance with the theory which predicts a modulated current response of this type for ASo > 0. The standard entropy of reaction for this system is AS' = +47.48 cal/(mol K): and the measured mass transport activation energies are = 4.81 kcal/mol, and E"; = 4.22 kcal/mol. The dimensionless parameters for this system are thus u = +2.95, y = 0.86, and t = 0.88, for a calculated peak ratio of (Ai/Aiw)P = -1.77. The experimentally measured values for the peak ratio are -2.48, -2.10, and -1.69 at rotation speeds of 400, 900, and 1600 rpm, respectively. We should point out, however, that the measured semilogarithmic slope in this system, accounting for ohmic losses, is 75.4 mV/decade and indicative of an appreciable departure from ideal reversible conditions. In contrast, a slope of 63.2 mV/decade was measured for ferri-ferrocyanide in 0.4 M Sr(N03)2as supporting electrolyte. Consequently, although the experimental current response curves are in good qualitative concordance with theoretical predictions for the ferric-ferrous system, quantitative comparison between
7280 The Journal of Physical Chemistry, Vol. 93, No. 20, 1989
theory and experiment awaits consideration of the influence of electrochemical kinetics. It may be mentioned that an extrapolation to infinite rotation speeds is not the only means of extracting thermodynamic information (Le., ASo)from experimental TMV data that exhibit w-dependent current peak ratios. Alternatively, the peak and limiting current perturbations may be used in eq 18, along with the extant surface temperature perturbation, AT, in order to calculate the thermodynamic entropy of the reaction. It was previously that AT can be experimentally determined for any set of thermal modulation conditions by measuring potential perturbations at open circuit. Hence, for a given electrolyte it is possible to obtain an accurate and remarkably reproducible calibration scale for the temperature perturbation in a TMV experiment. Common aqueous electrolytes will have quite similar calibrations. In this manner, temperature-dependent phenomena can be studied under well-defined and measured conditions in a TMV experiment. Summary and Conclusions
A theoretical analysis and an experimental study of TMV at a RDE for Nernstian-type redox electrode reactions show that the thermally modulated component of the current exhibits a peak near the standard potential of the reaction whose magnitude depends primarily on the parameter u, the ratio of the thermodynamic entropy of reaction to the activation energy for mass transport. TMV-RDE experiments conducted on two systems with widely different thermodynamic entropies of reaction are basically consistent with our theoretical findings. The departure from a predicted constant dimensionless peak ratio in the modulated current response is ascribed to the dependence of Ai, on rotation speed as a result of thermal losses in the RDE system? In contrast, the unnormalized peak value of the thermally modulated current, AP, is found experimentally to be a constant with rotation speed, as theory requires. In purely reversible systems, as we have demonstrated, the peak ratio is determined by the ratio of a thermodynamic quantity (reaction entropy) to a mass transport one (activation energies of convective diffusion). It can be anticipated theoretically that TMV in irreversible systems will depend on a kinetic parameter (enthalpy of activation) relative to the mass transport quantity. This consequence is under investigation along with the under-
Valdes and Miller standing of systems, such as metal electrodeposition, where a differential reponse to local temperature perturbations of two or more reactions leads to interesting selectivity and patterning possibilities. Glossary A C D ED
F i
Ai il Ai,
Aso T AT
U
electrode surface area, cm2 concentration of species, mol/cm3 diffusion coefficient of species, cm2/s activation energies for mass transport, kcal/mol Faraday constant, 96484.56 C/mol current density, mA/cm2 current perturbation, mA/cm2 mass transport limiting current density, mA/cm2 limiting current perturbation, mA/cm2 standard thermodynamic entropy change of electrode reaction, cal/(mol K) absolute temperature, K temperature difference between electrode surface and bulk of solution, K electrode potential, V
Greek Letters dimensionless ratio of activation energies for mass transport, eq 13 dimensionless mass diffusivity ratio (Dd/Dox)2/3 Y Y kinematic viscosity of electrolyte, cm2/s w disk rotational frequency, rad/s dimensionless electrode potential, (nF/RT)(U- V ) 4 d dimensionless ratio of reaction entropy to mass transport activation energy eq 12 t
Superscripts S electrode surface b bulk of solution P peak conditions Subscripts red reduced species ox oxidized species Registry No. Fe(CN),"-, 13408-63-4; Fe(CN);-, 7439-89-6.
13408-62-3; Fe,
(7) Von Gutfeld, R. J.; Gelchinski, M. H.; Romankiw, L. T.; Vigliotti, D. R. Appl. Phys. Lett. 1983, 132, 2576.
