Thermodynamic and Kinetic Properties of the Donald E. Smith Northwestern University, Evanston, IL 60201 The electrochemical cell, consisting of two or more conducting electrodes, is an extremely versatile device. Their uractical annlications are manv, including lead acid batteries Most of us fail to consider the importance of such devices until thev fail to do their iobs. Laboratories involved in research, development, food and drug assays, environmental assays, etc.. often use the 2-electrode cell for determining specific ion activities or concentrations. Such equilibrium electrochemical measnrements usually provide outputs related to -log of the chemical activity (and, therefore, concentration), snch as pH, pF, pCa, etc. Noneouilihrium electrochemical methods such as conventional polarography a t the dropping mercury electrode (DME), cyclic voltammetry, chronopotentiometry, normal pulse and differential pulse voltammetry or polarography, etc., abound in modern electrochemistry. Their uses range from highly sensitive assay work, to fundamental kinetic-mechanistic studies of electrode reactions. Normallv, 3-electrode cells are used in snch measurements. Conductimetric and conlometric measurements also are part of nonequilihrinm electrochemistry. Because of the large number of techniques based on the electrochemical cell, an undergraduate student is likely to become very confused after being exposed to both equilibrium and nonequilibrium techniques. This confusion is most easily exemplified in measnrements of conductivity and ax. voltammetry. Can one easily answer the question: "Why do conductivity and voltammetric a.c. measurements depend on different parameters?" ~~~~
Electrode .1
Working Electrode
Auxiliary Electrode*
.~
~
~
Basic Characterization of the Electrochemical Cell One mav consider two basic properties of an electrochemical origin of this ohserGable is the potential difference that develons at an electrode or membrane-solution interface. These po&ntial differences develop because of separation of charges across a "space-charge" region. Because snch potential changes cannot he measured directly, all measurements are based on the equilibrium cell potential where one uses two electrodes. Often one of the electrodes is a so-called "reference electrode." The second basic property of the electrochemical cell is a kinetic . nronertv: . " the cell i m ~ e d a n c e(or admittance). The cell impedance is the result of various rate processes such as mass transfer, heterogeneous charge transfer, adsorption, homogeneous chemical reactions, and interfacial double-layer charging. Some combination of these rate processes determines the cell impedance. This property can be subdivided into two parts, the bulk impedance and the interfacial impedance. The former involves an ohmic resistance at "normal" electric field strengths. I t arises primarily from the various processes which retard the migration of ions through an electrolyte solution nuder the influence of an electric field. The rate nrocesses associated with charee - transfer a t the electrode-solution interface represent the interfacial impedance. Here again. a subdivision into two arts is appropriate. One aspectis the double-layer capacit; (1) (do"hie-Lyer space-
Electrical representation for two and three electrode cells. (a) Ci = double layer capacity for electrode i: 4 = faradaic impedance for electrode i; R = bulk solution ohmic resistance.
= double-layer capacitance tor auxiliary, working and reference (b) CC ,C ,, electrodes, resp; Z.,Z,,Z, = faradaic impedance for auxiliary, working and reference electrodes, resp: R = bulk ohmic resistance between the auxiliary and working electrodes: 6 =ohmic resistance of reference electrode.
Table 1. Observable Physical Characteristics of the Electrochemical Cell 1. Equilibrium Cell Potential A. Arising from two different redox electrodes B. Arising from membrane electrodes 2. Cell Impedance A. Bulk Impedance-Ohmic resistance El. Interfacial Impedance 1. Faradaic Impedance 2. Double-Layer Impedance
charge region), where one witnesses virtual charge transfer across the interface, as when charging or discharging an ordinary electronic capacitor. Although differing from the electronic capacitor in several ways, it still obeys the basic capacitance equation, Z = C j o C j - 1 where j = under most conditions emuloved. . . The second interfacial characteristic is the faradaic impedance, which accommodates current bvactual charee transfer ( I ) . These are summarized in Tablei. To better understand interactions between the impedances discussed above, an approximate, hut useful, qualitative rewresentation of two- and three-electrode cells is given in the f i b r e . The hulk ohmic resistance, double-layer capacity, and faradaic impedance are represented by R , C, and Z, respectively. Subscript notation is used to designate different electrodes. I t is important to note that interfacial impedances, C and Z, are represented as parallel elements to emphasize that
a,
Volume 60
Number 4
April 1983
299
Table 2.
description of the interfarepresents an accurate cial impedance. One exception is when one or both forms of a redox couple adsorb or crystallize sufficiently t o influence the double-layer structure. Under such conditions the double-layer capacity depends on the faradaic process and vice versa, so the two components cannot he looked upon as indenendent. As implied above, the two-electrode cell in the figure is used ~rimarilvfor measuremeut of the cell potential at eauilihrium. ?he on& nonequilibrium method u k g this cell is conductivity measurements. Presently, three-electrode cells are (or should be) used for all other nonequilibrium electrocbem~cal measurements. The three-electrodes are usually referred to as the auxiliary (or counter), reference, and working electrodes, as shown in the figure. The 3-electrode cell is used in instrumentation designedto control either current, potential, or charge. Examples of these devices and explanations of their behavior are beyond the scope of this presentation. What is important is the recognition that the voltage measuring device must have a verv. high .. iuout . imoedance 11012-1014 ohms). 'I'IIII>. 11, tlw p u ~ e n t i adii . i t rlnrt Ir 1 a . 1 11w ~ ~ top oiit.. yrube mil i h r uclrkin:: ~ . l r r t n d t . . ,I III, . ~ I : I I T I I I ~ I I.1iil)li~11 I r f ~ : r < l . ~ - %,I > - a , n f ~ h t . riht r c l t r t n\x electrode is simpi;monitohg the potential (controlled current and charge) or acting as part of a negative feedback loop in a controlled potential operation. This implies that the major portion of the ohmic resistance in the reference electrode, the fritted disc or conducting membrane used to prevent mixing of the solutions in the compartments containing the reference and working electrodes, will have a negligible effect on the measurement.
.
C l a s s i f i c a t i o n 01 E x p e r i m e n t a l
Methods
In the classification of methods shown in Table 2, careful attention should he given the word "primarily." This word is used to imply that experimental objectives may he realized only approximately and that one of the aspects of the electrochemical cell which is not the objective of the measurement will influence measurement accuracy. Thus, the cell impedance influences equilibrium cell potential measurement accuracy. Also, the douhle layer charging current interferes with many methods primarily responsive to the faradaic admittance. P r i n c i p l e s of E l e c t r o c h e m i c a l M e t h o d s
It is the objective of this section to discuss a few very basic principles of electrochemical procedures and use these concepts to further explain the use of the term "primarily." Methods Primarily Responsive to the Equilibrium Cell Potential Potentiometric measurement with negligible current flow essentially negate any manifestation of the cell impedance. Assuming this is true, one can use the Nernst Equation. Knowledge of (E,,lJp and (dE,,,lldT)p allows one to compute AG, AH, AS, and K for the cell reaction. In analytical applications, usually one of the two redox systems is a reference electrode where the activity ratio is a known constant. Ion specific electrodes based on membranes are designed with two reference electrodes. One is the so-called "internal" reference electrode, which is an integral part of the ion specific electrode. The other is the "external" reference electrode. The membrane is located between the two reference electrodes. If the two reference electrodes are identical, the only contribution to the equilibrium cell potential is the ion activitv ~"to which the membrane is soecific~Otherwise.the ion activity term adds to a constant term which is easily compensated electronically. This class of electrodes represents the ~~~
~
300
Journal
of Chemical Education
Classlflcatlon of Experlrnental Methods
A. Methods Primarily Responsive to the Equilibrium Potential a) Potentiometric measurements with negligible impressed current B. Methods Primarily Responsive to the Bulk Ohmic Resistance of the Cell a) Conductivity Measurements C. Methods Primarily Responsive to the Faradaic impedance a) DC Polarography with a dropping mercury electrode (DMEP b) Linear Scan and Cyclic Voltammetry C ) Potential-step Chronoamperometry and Chronacoulometry d) Current Step Chronopotentiometry e) Normal and Differential Pulse Polarography and Voitammetry f) Square Wave Polarographyand Voltammetry g) AC Polarographyand Voltammetry h\ Second and Hiaher Harmonic AC Poiaroaraohv and Voltammetrv
tammetric Detection Methods Voltammetric Methods Based an Hydrodynamic Solution Flow, such as the Rotating Disc and Ring-Disk Electrodes m) Single Pulse Relaxation Methods, such as a single rectangular voltage pulse which often is referred to as the Double Potential Step Routine D. Methods Primarily Responsive to the Double-Layer Capacity a) Normal Differential Doublplayer Capacity Measurements with AC or Transient Signals b) Surface Temion and Potential of Zero Charge using DME c) Tensammetric Measurements E. Methods Primarily Responsive to Faradaic and Double-Layer impedance a) Coulostatic Relaxation Methods I)
Polwwlraphyrefers specillcally to use pf the DME. Measurements at stationary elect& are referred to as voltammefry, unless they involve current control.
most widely used for analysis by equilibrium cell potential measurements. We recognize that all ion specific electrodes suffer from interferences, a phenomena which is well understood in most cases. It is extremely important to emphasize that equilibrium cell notentials resnond to activities. not concentrations. The assumption that a = C (C = concentration) is a n often-used aonroximation which mav not be accurate.
one o f t h e most frequently ised of all electrodes. I t is weil known that the resistance of a glass membrane is very high, typically in the range of 10-50 megohms. In the context of the figure, essentially all of the cell impedance is represented by this high membrane resistance. In order to perform accurate measurements with such a high impedance transducer, one needs a measuring device with a much higher input impedance and a super low input current. The invention of the vacuum tube voltmeter solved this problem initially. Presently, electrometer-grade operational amplifiers will do a superior job. For examnle. if we consider the worst case resistance for a elass " membrane to be 1OS ohms and apply the input current required of a typical electrometer grade op. amp. (10-l4 A), one ohtains an I R drop across the membrane of 10-= V. At pH = 7, the voltage across the glass membrane is of the order of 0.4-0.5 V. One concludes that measurement of the cell voltage to four significant figures is possible. Typical F E T input . .. . or, . amps are approximately 1.5 orders of magnitude inferior. Nevertheless, they also are applicable to pH measurements, although the results will he less precise (3 significant figures). The above remarks illustrate that the nonequilihrium properties of an electrochemical cell can influence significantly the choice of instrumentation for equilibrium cell potential measurements. Obviously, a Poggendorff-style potentiometer is useless with a glass electrode.
Methods Primarily Responsive to Bulk Ohmic Resistance
Conductivity measurements are performed using two, quite large (1-2 cm2),chemically identical, highly roughened electrodes. Platinized wlatinnm is most commonlv used. The roughening maximizes electrode area and the reactivity, therehv. minimizing interfacial imwedances. The two electrodes are in coutact&ith the same eiectrolyte solution, so E,,n = 0. Large amplitude (1-2 V) alternating potentials are applied (e.g., 1000 Hz) which also tend to minimize the interfacial impedance and the occurrence of a net reaction. Under these conditions, effects of the hulk ohmic resistance are maximized and interfacial impedance effects are minimized. Nevertheless, precise measurements of solution resistance always involve a correction for cell capacitance, thus recognizing the existence of the interfacial impedance as a second-order effect. Methods Primarily Responsive to the Faradaic Impedance
This electrochemical technique category is extremely popular. Applications range from chemical assay work to thermodynamic and kinetic-mechanistic investigations. Assay techniques can he very sensitive with low ppb levels readily achieved. Methods used to obtain insights into kinetic, mechanistic, and thermodynamic aspects of an electrochemical reaction are many. Some are very powerful. Methods which are primarily responsive to the faradaic impedance normally use a large excess of supporting electrolyte relative to the amount of electroactive material. The effect of such concentration ratios is that ion migration through the hulk of the solution is maintainedpredominantly hy the supporting electrolyte and is essentially independent of the electroactive material. The ohmic resistance of the solution is nrimarilv determined hv the suwwortinp electrolyte .. and ion migration is not a mass transfer process as far as the electroactive snecies are concerned. ~
~
electrode-solution interface, occurs a t a late limited by the mass transfer processes. If diffusion is the rate-determining step, the diffusion layer thickness is approximately (Dtj112for planar diffusion, where D is the diffusion coefficient in cm2s-I and t is time in seconds. For time scales used in techniques listed, the diffusion layer thickness is quite small (5 X 10-4-5 X 10-2 mm), depending on the time scales used. The rate of diffusion in quiescent solutions a t a planar electrode is pronortional to iDlt)"2. This rate a t the electrode surface is proportional to the cell current. I t is also important to recognize that the rate of diffusion is time d e ~ e n d e n tbecoming : faster a t shorter times and slower a t longer times. Thus, if a wrocess vields data which are consistent with a diffusion other rate process such as the heterogeneous charge transfer step, or a coupled homogeneous chemical reaction might become partially or completely rate controlling. Only in hydrodynamic voltammetry does one deal deliberately with convective mass transfer. In this group of techniques, solution stirring usually is accomplished by rotating a planar disc IRDE) or ring disc electrode (RRDE) warallel to, hut a t a
.
rapidly reach a steady-state situation and are significantly laraer than wure diffusion currents a t the same electrode (stationary disc) measured a t time greater than, say, 1sec. A diffusion layer exists in RDE and RRDE experiments. Except a t very short times, the layer is much thinner than a t the same electrode without rotation. The diffusion layer thickness is controlled by the rotation rate, WR (rotationsisecond). For pure mass transfer controlled processes, the faradaic current magnitude varies with wn'12. Note the reciprocal square root
of time entering with both quiescent and stirred solutions for mass transfer controlled processes. I n measurements primarily responsive to the faradaic imaedance. the main interferences are caused bv the uncomgood accuracy using a well-known analog techniqne referred to as "aositive feedback iR comwensation." Here one modifies the piteutiostat so that the pioper fraction of the current signal is returned to the potentiostat input. A variety of methods exist so that the percent positive feedback can be varied, enabling one to select the appropriate amount. More sophisticated passive methods exist if one obtains data in or transforms time domain data to, the frequency domain. One may use plots of cell impedance versus frequency in the complex plane to obtain the ohmic resistance by infinite freauencv . . extra~olation. The double-layer charging response in general is a more challenging interference to eliminate (make negligible). Because the douhle-layer capacitance behaves much like a simple electronic capacitor, its contribution to the cell impedance will vary with time and frequency. In potential control methods, current due to double-layer charging will increase with de- . creasing time or with increasing frequency, and vice versa. When using potential stew or wulse methods, the double-laver capacitance. Early in the pulse or step, the charging current is very large (possibly potentiostat limited), hut decays rapidly, assuming the product RC is small (61ms). The faradaic current decays with llt112 or less, except when electroactive suecies adsorntion arocesses occur. Measurement of the faradaic current after the charging current decays to a "negligihle" value is a principle underlying all potential step, pulse, and square wave methods listed in Table 2. Clearly, it is important to minimize the RC time constant as much as possible. Positive feedback techniques contribute considerahly to this end. When one is dealing with frecluency domain data, the charging current increases with the first power of the frequency, w, while the faradaic current increases with w'I2 or less. Here again, charging current is favored relative to the faradaic current as cycle time decreases or as frequency increases. A long used and effective technique to minimize or "eliminate" contributions of the douhle-layer charging current in frequency domain data is to measure only the current component m-phase with the applied potential. The precision and accuracy of this techniqne is strongly dependent on one's ahility to minimize the RC time constant, as in the time domain measurements. Only when R C is negligible is the assumption that the charging current is orthogonal to the inphase component for all values of the faradaic current a valid one. Frequency domain cell impedance or admittance data has the advantage that the complex plane plot mentioned above for ohmic resistance determination and passive computerized compensation also is applicable to compensation of douhle-layer charging influences. The latter corrections are done using the admittance domain. ~ e t h o d primarily s responsive to the faradaic impedance are significantly influenced by equilibrium aspects of the cell reactinn. ~ssentiallvthe standard or formal ootential of the redox couple under investigation controls the location of the faradaicprocess on the potential scale. If a faradaic process is mass transfer controlled, appropriate ohservahles such as half-wave and peak potentials give quite accurate measurements (f0-5 mV) of standard (rarely) or formal (usually) potentials. Perturbations due to other rate processes can negate the foregoing statement, hut not always. ~~~~~
~
Bibliography I. , I
,,. '
I
: '?
. 1).
.. .
. ,.
,
: I I , I, . . . I . . . ' . . I . : I..,,. . . 8 .
.,,. ,
Volume GO
..'.
Number 4
I'
. ,,. .
.
:. .I.
,
1.1
4,.
1 .I#
, . , , ,
April 1983
I.. , I .
301
