adsorption of reactant at stationary electrode

Effects of Adsorption of Electroactive Species in

Robert €3. Wopschall and Irving Shah Chemistry Department, University of Wisconsin, Madison, Wis.

The theory of stationary electrode polarography for single scan and cyclic triangular wave experiments has been extended to describe electrochemical systems where either the reactant, the product, or both the reactant and product are adsorbed at the electrode- solution interface (Langmuir isotherm). A numerical method was applied to solve the integral equations obtained from the boundary value problems, and extensive data were calculated which made it possible to characterize quantitatively the adsorption parameters of the system. Correlation of theoretical and experimental parameters made it possible to develop diag- nostic criteria sa that unknown systems could be characterized by studying the variation of peak shape and peak current as a function of scan rate and bulk concentration

SINCE THE FIRST electroanalytical study of adsorption by Brdicka ( I , 2) using conventional polarography, similar con- cepts have been applied to other methods, such as chronopo- tentiometry (3, 4), electrolysis with constant potential (5, 6)) ac polarography (7), and stationary electrode polarography (chronoamperometry with linear potential scan) (8,9). These have provided versatile tools for the study of adsorption, and although theoretical and experimental studies have been carried out for several of these newer methods, only qualitative studies [with the exception of a method for determining the amount of adsorbed reactant (IO)] have been conducted using stationary electrode polarography.

In the presence of weakly adsorbed material, stationary electrode polarograms may exhibit enhancement of the peak currents ( IO, I I ) because of electron transfer involving the adsorbed material at nearly the same potential as the “normal” electron transfer (Figure 1 , curves A and B). On the other hand, if the product or reactant is strongly adsorbed, a separate adsorption peak may occur prior to or after the normal peak (9) (Figure 1, curves C and D) in analogy to the polarographic prewave ( I , 2) . One of the first studies of strongly adsorbed materials using stationary electrode polarography (single scan) was carried out by Mirri and Favero (8) on the prepeak which is observed with methylene blue. More recently Kemula, Kublik, and Axt (9) carried out qualitative studies on methylene blue under conditions where both ad-

(1) R. Brdicka, Collection Czech. Chem. Commun., 12, 522 (1947). (2) I. M. Kolthoff and J. J. Lingane. “Polarography,” 2nd ed.,

Interscience, New York, 1952, Vol. I, p. 256. (3) W. Lorenz, Z . Elektrochem., 59, 730 (1955). (4) W. H. Reinmuth, ANAL. CHEM., 33, 322 (1961). (5) F. C. Anson, Ibid., 38, 54 (1966). (6) J. H. Christie, G. Lauer, and R. A. Osteryoung, J . Electround.

(7) B. Breyer and H. H. Bauer, “Alternating Current Polarography

(8) A. M. Mirri and P. Favero, Ric. Sci., 28, 2307 (1958). (9) W. Kemula. Z. Kublik, and A. Axt, Roczniki Chem., 35, 1009

(10) R. A. Osteryoung, 6. Lauer, and F. C. Anson, J . Electrochem.

(11) C. A. Streuli and W. D. Cooke, ANAL, CHEM., 26,963 (1954).

Chenz., 7,60 (1964).

and Tensammetry,” Interscience, New York, 1963, p. 69.

(1961).

SOC., 110,926 (1963).

, I I

C

0.1 0.0 -0.1

I I

D

I I

0.1 0.0 -0.1

Figure 1. Theoretical stationary electrode polarograrns for cases involving adsorption

A, reactant adsorbed weakly ; E , product adsorbed weakly ; C, reactant adsorbed strongly; D, product adsorbed strongly. Dashed lines indicate behavior for the uncomplicated Nern- stian charge transfer

sorption and diffusion waves were observed for both cathodic and anodic scans. Hartley and Wilson (12) in a recent study of flavin mononucleotide observed similar adsorption prepeaks on mercury electrodes. These qualitative studies all indicated that additional theoretical work would be necessary to provide a quantitative basis for understanding the effect of the adsorption of electroactive materials in stationary electrode polarography.

To proceed with the study, it was necessary to describe the adsorption with an isotherm, and while there are several which could have been employed (13), a Langmuir isotherm was used because it embodies the general characteristics of more accurate isotherms (i.e,, limits adsorption to a fixed amount of material per unit area and reduces to Henry’s law at low concentrations) but is sufficiently simple to be handled conve- niently. The mechanism which was considered assumed a reversible charge transfer with both product and reactant adsorbed, and in equilibrium with the dissolved species:

(12) A. M. Hartley and G. S . Wilson, Ibid., 38,681 (1966). (13) R. Parsons, Proc. Roy. SOC. (London), A261,79 (1961).

15 14 e ANALYTICAL CHEMISTRY

OeOh * OadS (1)

0 + ne- a R (11)

Rsoh Rads (111)

The rates of the adsorption-desorption processes were assumed to be sufficiently fast. While this is undoubtedly an important limitation of some adsorption studies (14, IS), especially when the potential scan rate is high, it probably can be neglected for properly selected ranges of the experimental parameters. The boundary value problem was written in terms of the initial charge transfer being a reduction, occurring on a cathodic scan. However, the results are also applicable to cases in which an initial oxidation occurs on an anodic scan, by changing the appropriate signs.

BOUNDARY VALUE PROBLEM

For a reversible reduction with adsorption of both reactant 0 and product R (Equations I-III), the boundary value problem for stationary electrode polarography at a plane electrode x is:

bCo/bt = Do(b2Co/bxz) (1)

bCR/bt = DR(b2CR/bX2) (2 )

t 0, x 2 0: CO = C O W , C R = CR* (=a) (3)

(4)

t 2 0, x --f : CO * cO*, CR * 0 (5 )

(7)

(8)

(9)

TO = TO*, r R = rR* (=o)

t > 0, x = 0: Co/cR = exp[(nF/RT) ( E - E")] (6 )

D ~ ( ~ c ~ / ~ x ) - bro/at = - D R ( ~ C R / ~ X ) + brR/a t

To = ro'Co/(Ko + Co)

r R = rRSCR/(KR f C R )

where t is the time, x is the distance from the electrode-solution interface, Co and CR are the solution concentrations of 0 and R, Co* and CR* are the initial bulk concentrations, and Do and DR are the diffusion coefficients. E is the potential of the electrode and E" is the formal reduction potential for Reaction 11. To and rR are the surface concentrations (in moles per unit area), Pos and r R s are the saturation values at maximum surface coverage, and To* and r R * are the initial surface concentrations. The terms KO and KR which appear in the Langmuir adsorption isotherms (Equations 8 and 9) for species 0 and R, are related to the free energies of adsorption, AG, by

-In K = AG/RT (10)

While the free energies of adsorption are normally dependent on solvent, electrode surface, temperature, the presence of other adsorbable materials, electrode potential, and possibly many other variables, all of these except electrode potential are held sufficiently constant in stationary electrode polarography so that to a good approximation KO and KR can be considered dependent only on potential.

The problem is essentially the same as the case without adsorption (16) except the relation between the fluxes of 0 and R (Equation 7) includes terms related to the rate of change of the surface concentrations of adsorbed materials.

(14) W. Lorenz and F. Mockel, 2. Elektrochem., 60, 507 (1956). (15) Ibid., p. 939. (16) R. S. Nicholson and I. Shah, ANAL. CHmf., 36, 706 (1964).

For stationary electrode polarography (including the cyclic experiment), the potential is a triangular function of time with the direction of scan reversed when t = A:

Here, v is the rate of potential scan, Et is the initial potential, and Ex is the potential when the scan is reversed.

As in previous cases (16), the form of Equation 11 precludes direct application of Laplace transform techniques to this boundary value problem. Instead, the solution involves conversion of Equations 1-5 into the corresponding integral equations relating the fluxes to the concentrations. This transformation is carried out by taking the Laplace transform of the partial differential equations, solving for the transform of the surface concentrations in terms of the transforms of the surface fluxes, and then applying the convolution theorem to obtain:

fR(t) = D R @ ~ R / ~ X )

The total current is given by

i = nFA& - bJ?o/bt) = -nFA(fR - br&t)

where A is the area of the electrode. Combining Equations 6 and 11,

c ~ / c R = es(t) where

8 = exp[(nF/Rr> (Ea - E")]

exP(--O, o < t < x i exp(at - 2aX), A 5 t < 2X S(t) =

and

a = nFv/RT

The boundary conditions (Equation 17, and Equation 7 after integration) can be combined with Equations 8, 9, 12, and 13 to eliminate the concentration terms. The resulting integral equations can be made dimensionless by a change in variable,

VOL. 39, NO. 13, NOVEMBER 1967 151 5

where

In previous electroanalytical studies of adsorption in which the Langmuir isotherm has been applied, the isotherm parameters KO and KR were assumed to be independent of potential. However, if the simple form of the Langmuir isotherm is used in the boundary value problem above, it is not possible to account for the shape of the adsorption prepeaks observed in many cases, For example, the 1,l '-diethyl-4, 4 '-dipyridinium ion undergoes a one-electron reversible reduction and exhibits an extremely sharp adsorption prepeak (10-15 mV half peak width) (17) . Systems such as this probably obey some isotherm more complex than the simple form of the Langmuir isotherm (13). However, the Langmuir isotherm has advantages because of the relative ease of handling it mathematically, and it appeared reasonable to consider the modified form in which the term related to the free energy of adsorption, KO or KR, is potential dependent. As discussed by Parsons (18), the free energy of adsorption, AG, can be expressed by an expansion in the potential field X near the electrode surface:

-(RT)ln(K) = AGO + CYX + PXZ + y X 3 + . . . = AG (30)

where 4G0 is the free energy of adsorption in the absence of a field. If one approximates the field X by ( E - E,"), where E," is a standard state dependent reference potential at which AG is not influenced by the electrode potential (i.e., AG = AGO), and assumes that only the linear term of the expansion is significant (for a discussion of the applicability of the approximation, see References 18 and 19), then

KR = K~'exp[( - URnF/RT) (E - &/2)] (31)

(17) W. M. Schwarz, Ph.D. thesis, University of Wisconsin, Mddi-

(18) R. Parsons, J . Electroanal. Chem., 5 , 397 (1963). (19) Ibid., 7, 136 (1964).

son, Wis., 1961.

where

KR' = KR'~XP[- (~R~F/RT) (E112 - E,")] (32)

(33)

UR = a/& (34)

KR" = exp(-AG"/AT)

and

Thus, uR determines the magnitude of the influence of the field. This potential dependence becomes important only for the case of strong adsorption. In this work, extensive theoretical calculations were carried out only for the case in which the product is strongly adsorbed, and thus, to simplify the treatment, Equation 24 has been written for the case in which only KR is potential dependent. For all cases in which the product is not strongly adsorbed, KR was considered to be constant, with UR = 0. Allowing KR to be potential dependent makes it convenient to define

(35)

and to use PR' as a parameter rather than PR, noting that PRt

The adsorption isotherms, Equations 8 and 9, assume the product and reactant are each adsorbed independently, How- ever, assuming only monomolecular coverage of the electrode, the fraction of the surface actually covered by adsorbed molecules at saturation is related to the effective areas of the molecules. This requires that the isotherms (Equations 8 and 9) be interdependent, and although they can be modified appropriately, it is mathematically convenient not to impose this condition. If only one species is adsorbed, the problem does not arise. Also, if Henry's law holds, it is assumed that the surface is only fractionally covered and again the interdependence of the isotherms is irrelevant. The only case considered in this work which does not satisfy the above re- striction is that where both product and reactant are adsorbed with Langmuir behavior. However, this case was considered only to obtain a qualitative description, and the results are sufficiently accurate for this purpose.

Using the numerical method described previously (16, 20) the integral equations were solved simultaneously to obtain theoretical stationary electrode polarograms for selected values of PO, PR (or PR'), UR, po, and (FR. For use in the theoretical calculations, each of these was considered an independent variable. For interpretation and correlation of the stationary electrode polarograms with theory, the curves were characterized in terms of the experimental parameters, such as scan rate, bulk concentration, etc. Unfortunately, the calculation parameters are to some extent interdependent, and thus it was not possible to obtain explicit relations in most cases. For comparison of these results with previous work (119, a current function analogous to that for the uncomplicated reversible case was calculated, and using this current function, #(at), the current is defined by

-+ PR as LTR --t 0.

i(t) = [nFACo*vz-a]$(at) (36)

THEORETICAL CORRELATIONS

The characteristics of the theoretical stationary electrode polarograms depend markedly on the strength of the adsorption and on which of the materials is adsorbed (product, reactant, or both), A complete analysis would be complex, and

(20) R. H. Wopschall, Ph.D. thesis, University of Wisconsin, Madison, Wis., 1967.

1 5 7 6 a ANALYTICAL CHEMISTRY

0.E

z + 0 z LL

I- 2 W E U 3 0

0

3 0 . 4

0.C

-0.1 I I I I

0.0 - 0.1 (E-E,,,)n, V

Figure 2. Stationary electrode polarograms with reactant weakly adsorbed-Variation with scan rate

Corresponding to relative scan rates, c, of 2500,100, and 1 Pop0 = 0.01. A, Po = 5.0; B,Po = 1.0; C, Po = 0.1.

is beyond the scope of this work. Therefore, seven simplified cases are presented where the strength of the adsorption is limited to strong or weak (the intermediate ranges being ignored) and where either product, reactant, or both can be adsorbed. Because they are frequently encountered, two of the more important cascs are presented in considerable detail: the weak adsorption of reactant and the strong adsorption of product. Also included is a discussion of the limiting, low concentration (Henry’s law) behavior and a summary of diag- nostic criteria for determining the presence of and type of adsorption.

The results of the theoretical calculations are all expressed in terms of the current function +(at) defined in Equation 36. Experimentally the current function is determined by i/nFACo* d z a which is presumed to be known. In most cases comparisons were made with the current function for the uncomplicated, reversible charge transfer. [For this uncomplicated case, the cathodic peak current function [$(at),], has a value of 0.446, and is not a function of concentration or scan rate. When measured from a base line constructed from the appropriate cathodic decay, the anodic peak current function also has this value (Id).] However, in some cases direct comparisons cannot be made, because experimentally one may not be able to determine [$(at)], (Le., A, DO, CO*, etc., may not be known). For such cases, ratios of current func- tions are presented wherever feasible.

The types of adsorption described as “strong” or “weak” can be more closely defined by the consideration that the strength of the adsorption is directly related to the free energy of adsorption, AG-Le., the isotherm parameter, K (Equation IO). In general, a small value of K indicates strong adsorption, while a large value of K indicates weak adsorption. By examining the isotherm and making some reasonable as-

0.01 I I I 0.1 I IO (RELATIVE SCAN RATE)“‘

Figure 3. Ratio of cathodic peak current function for weak adsorption to that for uncomplicated Nernstian charge transfer as a function of scan rate

A , reactant adsorbed, Pop0 = 1.0; B, Product adsorbed, PR9R = 1.0. Units shown on abscissa are actually PO for curve A, and PR for curve B

sumptions, one can determine approximately when each will predominate, It is generally accepted that prepeaks, as with methylene blue, are due to the strong adsorption of product. Because of the symmetry of the prepeaks, at the maximum rR = TES/2 (this was shown to be the case for the model chosen) and from Equation 9, CR = KR. Also, since the prepeak is sufficiently separated from the normal diffusion peak, the potential must be anodic of El,z by 120/n mV or more, im- plying that CR _< 0.01 CO*. Thus, for strong adsorption it is required that CO*/KR 2 lo2.

Weak adsorption generally implies that a separate stationary electrode polarographic peak cannot be observed, and that the diffusion-controlled and adsorption-controlled peaks occur at nearly the same potential, Therefore, as above, it can be assumed that when T R = rRs/2, the potential must be fairly close to or cathodic of Eli2, and that CR 2 C O * / ~ . Thus, when CO*/KR _< 2, the adsorption can be considered weak. The intermediate range, which has not been considered, would be for 2 < CO”/KR < 100. Since P R ~ R = CO*/KR (when DO = DR), the strength of the adsorption can be related to this quantity or, analogously, to P o ~ o .

In the boundary value problem presented above, it was assumed that the initial potential is sufficiently anodic so that the polarograms are independent of the initial potential. Experimentally, this usually require; the initial potential to be at least 200/n mV anodic of the first peak, whether because of adsorption or diffusion. In cases where reactant is adsorbed, it was assumed that the experiment starts at equilibrium, Le.,

To” = TosCo*/(Ko + Co*) (37) and since the experiment starts at sufficiently anodic potentials, rR* = 0.

WEAK ADSORPTION OF REACTANT

The weak adsorption of reactant leads generally to an enhancement of the cathodic peak current and, on the reverse scan, to a smaller increase in the anodic peak current, as shown in Figure 1, curve A . This case is particularly important because in the use of stationary electrode polarography for analysis, the adsorption of reactant can cause significant errors. Also, the shape and height of the polarograms are often used to determine the reduction mechanism, and distor- tion due to adsorption may cause incorrect interpretations.

VOL. 39, NO. 13, NOVEMBER 1967 e 1517

I I I

0.1 I IO 100 (RELATIVE SCAN RATE)”*

Figure 4. Ratio of anodic to cathodic peak current function for weak adsorption as a function of scan rate with

( E X - Eliz)n = 180 mV

A, Reactant adsorbed, Pop0 = 1.0; B, Product adsorbed, P R ~ R = 1.0. Units shown on abscissa are actually PO for curve A , and PR for curve B

Dependence on Scan Rate. The scan rate is probably the most important experimental parameter for differentiating between the effects due to adsorbed reactant and those due to material arriving at the electrode by diffusion. This is because the adsorbed material constitutes, at constant To*, a fixed amount of material and charge flow, while the amount of material diffusing to the electrode surface is time dependent. At sufficiently fast scan rates, the amount of diffusing material is small relative to the amount of adsorbed material reacting at the electrode surface, while at slow scan rates the reverse is true. Therefore, an increase in scan rate must cause the current function to increase, as shown in Figure 2. Curve C is essentially the same as the uncomplicated reversible case, while curves A and B illustrate enhancement of both the cathodic and anodic peaks. This behavior is summarized in Figure 3, where the ratio of the cathodic current in the presence of weak adsorption to the cathodic current for the uncomplicated reversible case is plotted as a function of L; for a typical value of Pope.

In addition to affecting the cathodic peak height, the scan rate also affects the anodic peak height. The increase in the anodic peak comes from the material initially adsorbed on the electrode surface, which after reduction begins to diffuse away from the electrode surface. If the scan rate is large, and if the switching potential is not too far past the peak, little of this material escapes before the reverse scan. Under these circumstances, both the anodic and cathodic peaks involve contributions to the current from the adsorbed material, giving a constant value for the ratio of anodic to cathodic peak currents (ip)a/(ip)c, but which usually is not unity, as would be expected for the uncomplicated reversible case. This behavior is summarized in Figure 4. At lower scan rates, the portion of the current dependent on adsorption becomes negligible, and one obtains the same results as for an uncomplicated reversible case where (ip),/(i,)c is unity.

The decreased relative contribution to the current by adsorbed materials at slow scan rates is general and holds for all systems. Thus in cases where it is useful to eliminate the effect of adsorption as a factor, it may be possible to do so by using slow scan rates, or by extrapolating experimental data to low scan rates,

While most systems (including all first-order kinetic cases) demonstrate only linear concentration dependence, all adsorption systems show a

Dependence on Concentration.

A

-2 -I 0 I

log (Po

Figure 5. Ratio of cathodic peak current function with adsorption, to that for uncomplicated Nernstian charge transfer- as a function of bulk concentration (i.e., P O )

Po = 3.0. A, Low concentration limiting behavior (Henry’s law); B, Cangmuir behavior. Dashed line indicates behavior for uncomplicated Nernstian charge transfer. Henry’s law value for current function can be obtained from Equation 49. Note that (00 is a direct function of CO* (Equation 27)

marked nonlinear influence on the current with variations in the bulk concentration, CO*. As the concentration is increased, the surface of the electrode ultimately becomes nearly saturated and the total charge flowing during reduction due to adsorbed material becomes constant, while the total charge due to diffusing material continues to increase with concentration, This causes the fraction of the peak current due to adsorbed material to decrease with increasing concentration, and the current function approaches that of the uncomplicated reversible case. This is another important and general rule for cases involving adsorption: as the bulk concentration is increased, adsorption must have a lesser relative influence on the stationary electrode polarograms. In the present case of weakly adsorbed reactant, for example, the dependence on concentration is shown in Figure 5 . For high concentrations, the behavior approaches that of an uncomplicated reversible reduction, as expected, As the concentration is decreased, the ratio increases and finally approaches that of the limiting low concentration case, where the behavior becomes independent of concentration. As discussed previously, an increase in scan rate increases the current function and a complete treatment involves a family of curves corresponding to curve B, Figure 5 , for various values of PO. Changing PO, however, only affects the limiting value of curve B at low values of Co*, and to a good approximation this limiting value can be determined from the low concentration value for [#(at)],, given below in the discussion of the low concentration limiting behavior. Then curve B can be expanded to extend from the lower limit to the new upper limit for [+(at)],, for any value of Po.

In Figure 5 the concentration must be varied over nearly three orders of magnitude to observe this transition from the limiting low concentration behavior to that for an uncomplicated reduction. Experimentally, it may be difficult to obtain either the high or low concentrations necessary for the construction of the entire plot, and this may cause some dif- ficulty in determining the isotherm parameters.

Dependence on Free Energy of Adsorption. With the adsorption of reactant sufficiently weak, corresponding to a small or even negative free energy of adsorption, a normal, uncomplicated reversible polarogram is observed (Figure 2, curve C ) . As the free energy of adsorption increases, PO in-

151 8 * ANALYTICAL CHEMISTRY

creases, the adsorbed reactant is more difficult to reduce, and the reduction occurs at a more cathodic potential. This in turn causes the cathodic peak to be increased and shifted cathodically (curves A and B) with an increase in AGO or PO. If the free energy is further increased, the adsorbed material is sufficiently difficult to reduce so that a separated peak, or postpeak, is observed for the reduction process (curve C, Figure 1). Additional increase in the free energy of adsorption will continue to shift the postpeak cathodically.

Dependence on I'd. With all else constant, an increase in the amount of material initially adsorbed, r O * , causes the total charge flow during reduction to increase, while the charge due to diffusion remains essentially unchanged. This causes a corresponding increase in the peak current. If equilibrium is established before the potentials can begins, To* must be directly proportional to Tos (Equation g), and thus, the cathodic peaks increase with increasing ros. Because the scan rate and Pos are interdependent (Equation 25), Figures 2 and 3 give the dependence of stationary electrode polarograms with the parameter ros.

Dependence of Anodic Scan on Switching Potential, Ex. In general the anodic peak is enhanced in those cases where, after reduction, material which was initially adsorbed does not have sufficient time to diffuse into the bulk of the solution. However, as the time that elapses between the cathodic and anodic peaks increases, more material is lost, giving less increase to the anodic peak. Because this elapsed time is dependent on Ex, the switching potential has a significant influence on the anodic scan. Because of this, a quantitative treatment of the anodic behavior in cyclic experiments would require an extensive presentation of numerical data for specific experimental conditions. This did not seem to be of sufficient practical value for inclusion here, although such calculations can be carried out as desired using the computer programs (20).

bb7EAK ADSORPTION OF PRODUCT

The weak adsorption of product is not as important as the weak adsorption of reactant except in cyclic experiments, because the reverse scan is influenced most by weak adsorption of product. Unfortunately, the reverse scan is also greatly dependent on switching potential. Thus, only a brief, qualitative summary of the behavior exhibited in this case is presented.

Dependence on Scan Rate. The weak adsorption of product has little effect on the cathodic scan (Figure 6). With increase in scan rate, the cathodic peak shifts anodically, much like the case of a succeeding chemical reaction (16), but exhibits a decrease rather than an increase in peak height. Be- cause the adsorption of product requires the electrode reaction to provide material for both the diffusion process and the adsorption process, or more material than necessary in the absence of adsorption, the peak occurs at more anodic potentials, in analogy with the succeeding chemical reaction. However, while a succeeding chemical reaction consumes the product and causes a Nernstian shift in the equilibrium which results in a higher peak, in the case of adsorbed product the concentration of product near the electrode is slightly enhanced because of adsorption, and this results in a lower cathodic peak (the maximum decrease being less than 8%). The dependence of the cathodic peak current function on scan rate is shown in Figure 3.

On the other hand, the curves in Figure 6 indicate that the anodic peak increases markedly with scan rate and becomes more symmetrical. This is because the product formed on

0 -0.1 0. I

(E-E,,,)n, V

Figure 6. Stationary electrode polarograms with product weakly adsorbed-variation with scan rate

PR = 0.1. Relative scan rates are 4 X lo4, 2.5 X los, and 1

P R ~ R = 0.01. A , PR = 20.0; B, PR = 5.0; C,

the cathodic sweep accumulates on the electrode surface and is reoxidized on the subsequent anodic scan, enhancing the anodic current. As expected, an increase in either scan rate or r R s , both leading to an increase in PR, leads to an increased anodic peak current function for the same reasons that the cathodic peak is increased as PO is increased when the reactant is weakly adsorbed. The dependence of the ratio of the anodic to the cathodic peak currents ( ip)a / ( ip)c as a function of scan rate is shown in Figure 4, curve B, for representative values of PR, (PR and Ex, Because the cathodic peak is nearly constant, the increase with scan rate reflects almost entirely the increase in the anodic peak.

WEAK ADSORPTION OF BOTH REACTANT AND PRODUCT

The case of weak adsorption of both reactant and product is more complex than either of the previous cases, because of the number of variables and the interdependence of the adsorption isotherms. Thus, the numerical calculations become extensive and make it impossible to present the behavior of such systems in convenient form. This is not in- tended to imply that this case is unimportant experimentally; in fact, the opposite is probably true.

STRONG ADSORPTION OF PRODUCT

When the electrode reaction involves the strong adsorption of the product, the stationary electrode polarogram exhibits an adsorption peak anodic of the normal difFusion peak (Fig- ure 1, curve 0).

VOL. 39, NO. 13, NOVEMBER 1967 1519

As before, the theoretical stationary electrode polarograms depend on a set of parameters which contain the physically significant quantities PR' and (PR (Equations 35 and 28). The potential dependence of the modified Langmuir isotherm is included in KR' given by Equations 32 and 33 :

KR' = exp[-(AG"/RT) - (UR~F/RT) (Ell2 - E,")] (38)

where UR gives the potential dependence of the free energy of adsorption as expressed by Equations 30 and 34. The polarograms were characterized in terms of the isotherm parameters uR, AG", and r R S , and the experimentally variable parameters, scan rate and concentration, even though the numerical calculations were carried out in terms of PR' and p ~ ,

To make it possible to determine the isotherm parameters from experimental polarograms, it was necessary to choose a minimum of three quantities which could be obtained from the experimental measurements. The selection of these measured quantities was based on the criteria that they should vary markedly with changes in the isotherm parameters, scan rate, or concentration, but be easily and accurately measured. After an analysis of many polarograms, it was concluded that the three quantities which appeared most adequate were the adsorption half-peak width, AE,, (Le., the width of the adsorption peak at one half the maximum current), the ratio of the adsorption prepeak height to the diffusion peak height, p , and the separation of these peaks, AE,.

Qualitatively, it was found that the adsorption peak width, AErr7, depends primarily on the isotherm parameter uR, while the position of the prepeak is primarily dependent on the free energy of adsorption in the absence of a potential field, AGO. Since AGO is related to PR' through KR', PR' is the parameter used in the numerical calculations which influences the peak position. The ratio of the adsorption peak height to the diffusion peak height was found to vary with both the value of uR and the bulk concentration. This is because OR changes the shape, causing a change in the adsorption peak height, while the concentration increases the diffusion peak but not the adsorption peak. Even though the effect of each isotherm parameter can be qualitatively separated from the others, there is considerable interaction and in general the dependence of the theoretical polarograms on the various parameters could not be determined explicitly in any convenient form.

To make the problem amenable to analysis, it was necessary to place restrictions on the values of the various parameters. In general, the numerical calculations were carried out such that the individual contributions to the total current from the adsorption process and the diffusion process could be eval- uated separately. Thus, restricting the system to strong adsorption required that P R ' p R be greater than about lo2, which implies a certain minimum value for the peak separation, AE,. In practice it was found that for a peak separation of greater than 10/n mV, a well defined adsorption peak was obtained. While the adsorption half-peak width could be varied over a very wide range of values (determined by uR), experimentally observed widths have been roughly between 8 / n and 90/n mV, and therefore theoretical polarograms were limited to prepeaks in this range. Thus a maximum limit on uRF/RT of about 0.4 mV-l was set with the minimum value being zero. The third parameter, the ratio of adsorption peak height to diffusion peak height, can have almost any value experimentally. However, when p becomes very much larger than unity, the diffusion peak becomes small and difficult to measure as it appears after the adsorption peak. For p less than unity, simple relationships between the parameters

I I I I

300 200 100 0 -100 (E-Elll)n, mV

Figure 7. Variation of stationary electrode polarograms with product strongly adsorbed as a function of free energy of adsorption, AGO

p~ = 1.0; A , PR' = 2.5 X lo6; B, PR' = 2.5 X lo6; Cy PR' = 2.5 X 10'; D, PR' = 2.5 X 10*. Free energies of adsorption with (4rRadijd?roR) = 1 are 29.4,24.8,20.2, and 15.6 kcal/mole, respectively

URF/RT = 0.05 mV-1.

could be obtained, while for larger values the relationships were complicated. For this reason the value of p usually was restricted to values less than unity. Although the correlations reported below hold rigorously only for results within the limits set above, estimates of the behavior outside these limits are reasonably accurate,

For this particular model, if the diffusion peaks (both cathodic and anodic) are subtracted from the total current, the adsorption peaks are symmetrical about the zero current axis. Therefore, no information can be obtained from the anodic scan other than that which is available from the cathodic, and only the cathodic scan is discussed below,

Dependence of Peak Separation AEp on Free Energy of Adsorption AGO. The free energy of adsorption in the absence of an applied field, AGO, causes a shift in the potential of the adsorption prepeak but does not have a significant influence on the shape or height of either the adsorption peak or diffusion peak. As the free energy of adsorption is increased, the energy required for reduction to the adsorbed state is decreased and the reduction potential need not be as cathodic. Therefore the peak separation increases as shown in Figure 7. Curve D appears slightly higher at the adsorption peak than curves A , B, and C, but if the foot of the diffusion peak is subtracted, curve D is approximately the same as the others.

While the peak separation, AE,, is determined primarily by the value of AGO, the bulk concentration also has an effect. Therefore, it is not possible to use measured values of AE, to determine AGO (or PR') directly.

Dependence of Adsorption Prepeak on UR. The potential dependent Langmuir isotherm was selected so that an increase in the parameter uR would cause the free energy of adsorption to change more rapidly with potential. If UR is sufficiently large, the free energy goes from a very low value to a very high value over a narrow potential range. This, in turn, causes the reduction to adsorbed product to take place over a cor- respondingly narrow potential range and gives a sharp adsorption prepeak. Typical stationary electrode polarograms which illustrate the dependence of the adsorption prepeak width, AE,., on UR are shown in Figure 8.

If all isotherm parameters except UR are held constant, UR changes only the shape of the prepeak but does not change its position. To show this for the range of AE, and p under consideration, the value of the surface concentration, rR, at the

1520 e ANALYTICAL CHEMISTRY

I

300 200 too 0 -100 (E-E,,Jn, mV

Figure 8. Effect of variations in uR on shape of adsorption prepeak for strong adsorption of product

0.05 mV-1; C, 0.0 mV-* PR = 1.6; KR' = 8.4 X lo2. uRF/RTequdS: A,0.20mV-'; B,

adsorption peak potential, E,, can be approximated by r R * / 2 because of the prepeak symmetry (Figures 7 and 8). From Equation 9, this requires that K R = CR. Since at these potentials CR remains small, CO = co*. From these approximations one can obtain, by combination of Equations 6, 31, and 32.

CO*IKR" = exp[-(uRnF/RT) (Ep - E,") - (nF/RT) (E, - EI:P)I (39)

Evaluating the constant term CO*/KR' at UR = 0 and substi- tuting this value back into Equation 39, one can solve for uR to obtain: U R ( E ~ - E,') = 0 or Ep = Eao when CO* is the standard state reference concentration. Therefore, with all other variables constant, the potential at which the adsorption peak occurs is independent of aR.

Although the adsorption peak width is determined primarily by uR, the parameter pa also has a minor influence on the peak width, as illustrated in Figure 9. This dependence is most important for values of pR less than unity, which are seldom encountered, where the bulk concentration is low, or FR 'Ala rge . If the bulk concentration is low, the diffusion process cannot provide material at a rate sufficiently

60 c

-2 - I log (cRF/RT)

Figure 10. Adsorption prepeak width, AEw, for stationary electrode polarograms where product is strongly adsorbed, as a function of UR F/RT

(OR = 10.0

Figure 9. Variation of nAE, as a function of p~ for various values of URF/RT with product strongly adsorbed

URF/RT equals: A , 0.01 mV-l; B, 0.05 mV-1; C, 0.20 mV-1; D, 0.40 mV-l

fast to maintain the surface concentration in equilibrium with the bulk solution concentration. This causes the peak to be broadened because it becomes partially diffusion controlled. Similarly, an increased scan rate or an increased need for material as reflected in a larger value of r R s would cause the adsorption peak to be more dependent on diffusion.

However, to a good approximation, for pR greater than 2.0, and certainly for p~ greater than 10, nAEw is dependent only on uR. Therefore, one can usually determine UR from the adsorption peak width using Figure 10 which gives the value of nAEpv as a function of UR for a sufficiently large value of p~ to make results independent of (FR. Data necessary to construct a more accurate graph are given in Table I.

Correlation between AEa, AGO, and UR. As shown in Figure 7, the free energy of adsorption has a significant influence on the potential at which the adsorption peak appears. However, the magnitude of the influence depends on the value of UR, and an explicit relation for this can be obtained by considering the limiting behavior at both large and small values of UR. This provides both a method of compar- ing two systems in order to obtain an estimate of the relative free energies of adsorption, and a convenient relation for calculating the free energy of adsorption.

For two systems which are identical except for the free energies of adsorption (which can be expressed in terms of KR~' and K R ~ ' ) the difference in the free energies of adsorption results in different peak potential separations, AE,, and AE,, (Figure 7). For UR small or zero, Equation 39 can be ex-

Table I. Adsorption Prepeak Width, AEru, for Stationary Electrode Polarograms Where Product Is Strongly Adsorbed,

as a Function of CRFIRT P R = 10.0

URFIRT, mV-l AEw, mV 0.01 71 0.03 55 0.05 40 0.10 25 0.20 14.5 0.30 9.9 0.40 7.5

VOL. 39, NO. 13, NOVEMBER 1967 1521

I 1

I I I I I J 200 100 0 -100

(E-E,,,)n, rnV

Figure 11. Effect of parameter r R 8 & on theoretical stationary electrode polarograms for case in which product is strongly adsorbed

PR = 1.25; C, VR = 5.00. For curves A, B, and C, if value of r R s is constant, the relative values of scan rates are 64:16:1. If scan rate is held constant, relative values of

URF/RT = 0.05 mv-'; P R ' 9 R = 2.5 x lo6. A , VR = 0.625; 8,

are 8:4:1

pressed for both KR,' and sultant equations gives

and the difference in the re-

(RT/nF)ln(KR,'/KRa') = AE,, - AEp2 (40)

Thus, a 10-fold decrease in KR', corresponding to a 4.6 kcal/ mole increase in AGO, would result in the prepeak appearing at potentials about 60/n mV more anodic. However, if UR

is relatively large [Le., 4.G" << URTZF(E~,~ - ED)], then Equa- tions 32 and 33 lead to

(RT/uRnF)ln(KR, '/KR~') = AEp, - AEp2 (41)

Again there is a resultant anodic shift for an increased free energy of adsorption, but the magnitude is less than 60/n mV. Considering the limiting forms above, an expression was found relating AE,, KR', and UR which reduced to the correct expression for both large and small values of UR, and which also fit theoretical results for intermediate values of UR :

(RT/nF)[uR $- eXp(- UR)lln(KR, '/KR~') =

AEm - AEpz (42)

Equation 42 was tested over the range of values of UR used in this work (1.0 to 10) and the calculated values of peak separation agreed with the results of the numerical calculations within 2 % over the entire range.

By restricting p to values less than unity, the condition has been imposed on the system that the concentration is high enough to provide complete coverage of the surface, and thus, the value of TRS determines the total amount of material adsorbed. This is reflected in the theoretical stationary electrode polarograms (Figure 11) where r R s in effect determines the total area under the adsorption prepeak. Here the product PR'PR is held constant, and if the scan rate is held constant, the variable pR can be used to determine the influence of r R 8 on the polarograms. In these polarograms the relative values of rna were selected to be 8:4:1 for curves A , B, and C, respectively. As expected, the areas under these adsorption peaks are also in the approximate ratios of 8 :4 : 1 .

The diffusion peak decreases as r R s is increased, because during the adsorption process, electroactive material is re-

Dependence on r R s .

- too 200 IO0 0 (E-Elle)n, mV

Figure 12. Effect of variation of bulk concentration Co* on stationary electrode polarograms for a reaction in which product is strongly adsorbed URF/RT = 0.05 mV-l; PR' = 1.0 X lo6. 2.0; C, 9~ = 8.0. Relative bulk concentrations are 1:4:16

A , VR = 0.5; 8, PR =

moved from the solution in the vicinity of the electrode surface, causing the diffusion peak to be lowered and shifted slightly cathodic. Because the calculations presented here were restricted to conditions where the adsorption peak is no !arger than the diffusion peak ( p less than unity) the diffusion peak is only slightly affected by the value of FnS in this range. However, if the value of I?%' becomes large enough, the adsorption prepeak can become so large that the diffusion peak vanishes and the adsorption peak becomes completely diffusion controlled.

Dependence of Polarograms on Scan Rate. Although the effect of varying the scan rate is identical to the effect of changes in the maximum amount of material adsorbed (because the two parameters always appear together as rRs& as discussed in the preceding section), it is important to discuss this erect of scan rate separately because this is the principal experimental variable which can be controlled separately over a wide range. As the scan rate increases, the ratio of the adsorption peak height to the diffusion peak height increases. This behavior is shown in Figure 11 where only the scan rate is varied if r R s is held constant.

At low scan rates the behavior approaches that of an uncomplicated reversible reduction (Figure 11, curve C). As the scan rate is increased, the adsorption peak appears and increases with only a small decrease in the diffusion peak current function (curves A and B) . With further increase of the scan rate, the adsorption peak current function increases further, and the diffusion peak current function decreases until only one peak appears at the potential of the adsorption peak.

A small cathodic shift of the adsorption peak with increased scan rate also can be seen in Figure 11. This shift is only 13/n mV per 100-fold change in L' for uRF/RT = 0.01 mV-l, and becomes even smaller for increasing values of UR, There is also a slight cathodic shift in the diffusion peak with increasing scan rate which is due to the partial depletion of reactant during the adsorption process.

Dependence of Polarograms on Concentration. As the bulk concentration, Co*, is increased, PR increases, leading to a relative decrease in the height of the adsorption peak and an anodic shift in its potential. This behavior is shown in Figure 12, where all other parameters are held constant. The current function for the adsorption peak decreases, whereas the current function for the diffusion peak changes

1522 0 ANALYTICAL CHEMISTRY

only slightly with concentration. (The absolute current for the adsorption prepeak remains reasonably constant over this range.) The anodic shift of the prepeak occurs because at all potentials the concentration of product is also increased, and the surface concentration of adsorbed product approaches its saturation value at an earlier point in the scan. This can be expressed more quantitatively by rearranging Equation 39 and solving for E, to obtain

[RT,(UR + l)FZF]ln(CO*/KRo) =

[(UREGO + E1/2)/(6R + 1)1 - E p (43)

Evaluating at two values of concentration, (CO*)~, and (CO*),, but holding the standard state constant, gives:

(44)

where E,, and Ep2 are the respective peak potentials measured from Eliz. Thus, a decrease in concentration requires a cathodic shift in the peak potential, as indicated in Figure 12, and the magnitude of the shift depends on uR.

Diffusion-Controlled Adsorption Peak. At sufficiently fast scan rates, high values of r R s , or low concentrations, the diffusion process may not be able to provide the material necessary to maintain equilibrium between the surface concentration and the bulk concentration of reactant during the adsorption peak, Under these conditions the adsorption prepeak becomes diffusion controlled, but still appears anodic of the potential at which the normal diffusion peak would occur at slow scan rates or high concentrations. This limiting behavior is included in the curves shown in Figure 13, where the dependence of the stationary electrode polarograms over a wide range of scan rates is illustrated. Curve A shows the case where the adsorption prepeak is small relative to the diffusion peak. As the scan rate is increased, the current function for the adsorption peak undergoes a relative increase (curves B and C) while the diffusion peak current function decreases. At very high scan rates the adsorption peak current function reaches its diffusion controlled maximum and the “diffusion” peak is absent (curve 0). This dependence on scan rate is analogous to the normal DME polarographic dependence on concentration, where at very low concentrations only a single wave appears which is diffusion controlled and shifted anodically because of adsorption.

Theoretical Correlations of Experimental Measurements. In an experimental investigation, the measured quantities are AE,, A&-, and p, (and their variation with L; and CO*) and these must be correlated with the calculation parameters PR’

[RT,(uR + l)nFlln[(C0*)~l(C0*)*1 = E,, - Epl

Table 11. Values of p q ~ as a Function of aRFIRTfor Stationary Electrode Polarograms with the Product Strongly Adsorbed

0.01 0.315 0.03 0.446 0.05 0.568 0.10 0.893 0.20 1.526 0.30 2.208 0.40 2.997

uRF/RT, mV-l P‘fR

and p~ in order to determine the isotherm parameters. The first step involves making appropriate measurements to determine whether the system fits the theory developed here. Using a cyclic experiment in initial investigations, the absence of an anodic adsorption peak, or the presence of more than one, would indicate marked deviations from the theory. Deviations would also be indicated if the anodic and cathodic adsorption peaks occurred at different potentials or were markedly different in magnitude.

After initial qualitative investigations indicate the experimental system fits the theoretical model, the adsorption prepeak-diffusion peak ratio, p, would be plotted as a function of d i / C ~ * for a range of concentrations and scan rates. Theo- retical studies showed that for the theoretical model, p varies linearly with pR-1, with the slope dependent only on uR.

The value of the slope as a function of r~ is given in Table 11. Since pR-l is proportional to V$C0*, the experimental plot should yield a straight line over a range of concentrations and scan rates of several orders of magnitude.

A second plot which would be made is the ratio of the diffusion peak current function to the current function for the uncomplicated reversible reduction (in practice, the function i,/Co*.\/; can be used, and is usually easier to obtain) as a function of C O * / ~ ; ’ Qualitatively, the diffusion peak current function decreases with decreased concentration or increased scan rate due to depletion of reducible material. This decrease is not dependent on PR‘ and is only slightly dependent on UR, leaving it almost entirely determined by uR (Figure 14). Thus, the plot of i , / C O * d v ~ s a function of C , * / d ; should agree in shape with Figure 14 if the experimental system fits the theoretical model.

~

0.2 0 - 0.2 (E-E,,,)n, V

Stationary electrode polarograms with product Figure 13. strongly adsorbed-variation with scan rate

Relative scan rates: A , 1; E , 25; C, 625; D, 2500

2

Figure 14. Ratio of diffusion peak current function, for case in which product is strongly adsorbed, to diffusion peak current function for uncomplicated reversible case

URFIRT equals: A, 0.20 mV-l; B, 0.05 mV-1

VOL. 39, NO. 13, NOVEMBER 1967 e 1523

Table III. Standard Peak Separation, &,, as a Function of URFIRT

(see Equation 45) for PR’ = lolo, $R = 5.0 URFIRT, mV-l ABPj mV

0.01 531 0.03 384 0.05 303 0.10 204 0.20 131 0.30 100 0.40 83

Then the value of uR can be calculated for each stationary electrode polarogram. (While calculation of the isotherms is possible from a single polarogram, it is assumed that polarograms at various values of CO* and zi have been obtained.) This is accomplished by first making an initial estimate of UR

from 4Epr using Figure 10 (Table I). From this and the value of p , an estimate of oR can be made using results presented in Table II. Successive approximations (usually the second will be accurate enough) provide values of UR.

From the average value of UR, the theoretical value of p p ~ (the slope of the plot of p against y-1) is obtained (Table XI), and from the plot of p GS. d</Co* , the experimental slope is obtained. From these, the quantity ~ T D o R T / ~ F / ~ ~ R ~ can be calculated and for each polarogram, (FR can be determined.

There are two ways in which r R s can be obtained. First, it can be obtained from the values of d a D o R T / n F j 4 r ~ ~ , provided DO is known. However, this method is subject to a large error. A second method, involving the measurement (by integration) of the total charge under the adsorption prepeak provides a more accurate value. Because the area under the prepeak is proportional to r R S when the adsorption is strong, this technique can provide values accurate to a few per cent if charging current can be accounted for and if the peaks are reasonably well separated.

To determine PR‘, Equations 42 and 44 can be used, re- placing KR’ by PR‘ and Co* by yR (this change was shown to be correct by making measurements directly on the theoretical polarograms). The resultant equation then gives the dependence of AE, as a function of q ~ , PR’, and OR:

[RT/(uR -k l )nF] ln (p~ /@~) f (RT/nF>[uR + exp(- uR)I Inh’ jPR’I =

AE, - Ae,/n (45)

where co*, and PR’ are arbitrarily selected “standard” values for Co* and PR‘, respectively. A “standard” value for the peak separation, ABp, i s then obtained, which is dependent on gR, co*, and pR’, Values of AB, are tabulated in Table III. Once pR and rR are known, PR‘ can then be determined from Equation 45. If the value of rRs has been determined, and the other parameters contained in PR’ are known, AGO can then be determined.

STRONG ADSORPTION OF REACTANT

If ?he reactant is strongly adsorbed, a postpeak is observed, as was shown in Figure 1, curve C. This case was not treated because of its similarity to the case of strong adsorption of product. For strong adsorption of either product or reactant, it is possible to separate the effects of the adsorption peak from those of the diffusion peak. Thus, it is possible to use the results presented above for the strong adsorption of product to

I I I I I

0. 2 0 - 0.2 (E-E,,Jn, V

Figure 15. Stationary electrode polarograrns with product strongly adsorbed and reactant weakly adsorbed at various scan rates

Pop0 = Co*/Ko = 0.2. Relative scan rates: A, 1; B , 2 5 ; C,625

obtain semiquantitative results for the strong adsorption of reactant, and many of the results discussed above can be applied directly.

STRONG ADSORPTION OF PRODUCT WITH WEAK ADSORPTION OF REACTANT

Because of the number of parameters involved, the case where both product and reactant are adsorbed has not been fully investigated. Only a qualitative description has been considered for the case where the reactant is weakly adsorbed and the product is strongly adsorbed, which is most probably the situation with the methylene blue system at high concentrations. Qualitatively, the theoretical behavior is a combination of these two types of adsorption considered separately, and the polarograms exhibit both a prepeak and enhancement of ?he diffusion peak.

The mathematical model used to obtain theoretical stationary electrode polarograms for this case, as mentioned previously, is no? adequate because it does not allow for interdependence of r‘o and r R . Therefore, it would not be expected that the theoretical results would accurately describe experimental systems. However, because the reactant is assumed to be weakly adsorbed, the value of To should never become very large and the model should be accurate enough to give good qualitative results.

At low concentrations, this case reduces to the case where only the product is adsorbed. Therefore, if a system exhibits weak adsorption of reactant as well as strong adsorption of product, the theory discussed in detail above can be applied at low concentrations. This can be shown by considering specific values of the parameters where the weak adsorption of reactant must be considered. For example, i f P R = lo:, PO = 10, and OR = (FO = 1.0, then for a 100-fold decrease in concentration, if p is to be maintained less than unity, there must be a corresponding decrease in scan rate by a factor of about lo4 , This causes changes in the various parameters to give PR = lo3, PO = lon1, and p~ = PO = 1.0. As Figures 2 and 7 indicate, for PO less than about 1.0 there is little enhancement of the diffusion peak, while for a value of PR equal to l o 3 an adsorption prepeak is still observed. Thus, as the concentration is decreased, the effect of the weak adsorption should decrease and ultimately become negligible.

The effect on the theoretical polarograms of a 10-fold change in concentration is shown more clearly by a comparison of Figures 15 and 16. In effect, the only change is the concentration, Le., the parameter Poqo = CO*/KO (which is propor-

1524 * ANALYTICAL CHEMISTRY

z 0.4

s z c 0

3 L L

F- z cr. (3e 3 0

l l 1 0.2

0.0 I I I I I I ]

0.2 0 - 0.2 (E -Edn , V

Figure 16. Stationary electrode polarograms with product strongly adsorbed and reactant weakly adsorbed at various scan rates

Pop0 = Co*/Ko = 2.0. Relative scan rates: A , 1; B , 2 5 ; C, 625

tional to PRyR), is changed. At the lower concentration (Figure 15), the diffusion peak current function decreases with increased scan rate, similar to the behavior observed when only the product is adsorbed, as shown in Figure 14. How- ever, if the concentration is increased 10-fold (Figure 16), the behavior is markedly different for the same scan rates. The most obvious differences are that at the increased concentration the peak separation, AE,, is increased, and the current fgnction for both peaks increases with scan rate, whereas only the prepeak current function increases at lower concentrations. If the scan rate is increased further, there is ultimately a decrease in the current function for the second peak. This is because, as the scan rate becomes large, the reactant initially adsorbed and the diffusion process combined cannot provide the material required for the strong adsorption of the product. Thus, all the adsorbed reactant is reduced in the potential region of the prepeak, and the second “diffusion” peak vanishes just as when only the product was adsorbed. This behavior is shown in Figure 17 (which is analogous to Figure 14) for various relative bulk concentrations. Curve D is the case where reactant is not adsorbed. As the concentration is increased, there is an increase in the current function for all values of po. Curve A shows the increase followed by a decrease in the current function as the scan rate is increased. The change in behavior in changing the concentration 4-fold (curve A to curve C) indicates how important the concentration is, once it exceeds some minimum value. Because of this. it is possible to see a complete change in behavior with a 10-fold change in concentration for such systems.

The relation between p and qo-1 is also significantly changed by the adsorption of reactant. For the adsorption of only the product, this was a linear relation with slope dependent on CR. The adsorption peak current increases approximately linearly with scan rate while the diffusion peak current increases approximately as the square root of the scan rate, Therefore p continues to increase with scan rate until there is no diffusion peak and p becomes undefined. When the reactant is also adsorbed, the above description still holds at low scan rates because the adsorbed material does not contribute significantly to the diffusion peak. However, as the scan rate becomes large, the diffusion peak becomes almost entirely adsorption controlled, and both peaks increase almost linearly with scan rate, giving a nearly constant value for p. Figure 18 shows typical behavior as a function of scan

- I 0 I

log 4 0

Figure 17. Behavior of stationary electrode polarographic peak current function for case in which reactant is weakly adsorbed and product is strongly adsorbed-Ratio of peak current function with adsorption to peak current function for uncomplicated reversible case as a function of yo for various relative bulk concentrations

PR = (1.42 X 103) PO; PR = 1 . 2 6 ~ ~ . Relative concentrations A, 4 (Pop0 = 2); B, 2 (Pop0 = 1); C, 1 (Po90 = 0.5); D , reactant not adsorbed, UR-= 0. The abscissa is really a function of scan rate (i.e., po 0 l j d c ) since concentrations are held constant for each curve

1.0

Q

0.5

0 4 8 $4

Figure 18. Ratio of adsorption peak to diffusion peak, p , as a function of parameter p0-I for stationary electrode polarograms where reactant is weakly adsorbed and product is strongly adsorbed

For each curve Pop0 and P R ~ R are held constant; therefore the abscissa is proportional to & PR = (1.42 X lo-3)Po; p~ = 1.26~0. Popoequals: A, 0.0 (i.e., only product adsorbed); B, 0.5; C, 1.0; D , 2.0; E, 8.0

VOL. 39, NO. 1 3 , NOVEMBER 1967 e 1525

Table IV. Ratio of Cathodic Current Function for Henry’s Law Adsorption of Reactant to Current Function for Uncom- plicated Reversible Case as a Function of Adsorption Param-

eter PO Po [\L(ar)1,/0.446 Po [$(ai)1,/0.446

0.01 1.004 4.0 2.068 0.1 1.024 5.0 2.357 0.2 1.046 6.0 2.646 0 .3 1.069 7.0 2.937 0.4 1.091 8.0 3.229 0.5 1.116 9.0 3.522 0 .6 1.140 10.0 3.816 0 .7 1.163 12.0 4.403 0 . 8 1.190 14.0 4.992 0 .9 1.214 16.0 5 I584 1.0 1.239 20.0 6.760 2.0 1.506 30.0 9.713 3.0 1.784 50.0 15.62

rate (here, q0-l is proportional to -<;-since Pop0 is held constant) for several relative bulk concentrations. As the bulk concentration is increased, the amount of adsorbed reactant increases, therefore decreasing the limiting value for p . This limiting value may not be reached experimentally because the two peaks often blend into one and the diffusion peak cannot be measured.

LOW CQNCENTRATIOK LIMITING BEHAVIOR-HENRY ’S LAW

Regardless of the adsorption isotherm which a material obeys, at sufficiently low concentrations, almost all materials obey Henry’s law, Le., the surface concentration is proportional to the concentration of the material in solution. The Langmuir isotherm exhibits such behavior, for if the concentration is low enough, c << K, and Equations 8 and 9 reduce to

r = (r8!mc (46)

or Henry’s law where Fs/K is the proportionality factor. condition is generally satisfied for the reactant if

This

Pop0 = Co*/Ko << 1 (47)

Similarly, since CR 5 yeo*, Equation 46 holds for the product if

(48) P R ~ R = CO”!KR << 1

For Henry’s law behavior there are two important changes from Langmuir behavior. First, the current is a linear function of the bulk concentration, because the amount of adsorption is proportional to concentration. If the bulk concentration is increased, the adsorption is increased propor- tionately, leading to the same relative increase in the current due to diffusing species and adsorbed species. As a result, the values of p0 and pR, which are proportional to concentration, approach zero for those cases where Equations 47 and 48 are valid. Thus, if Equations 8 and 9 are replaced by the appropriate Henry’s law expressions, the resulting integral equations do not contain (PO and (PR.

The other difference is that when the adsorption can be described by a Henry’s law isotherm, a prepeak (or postpeak) cannot be observed. For example, using the Henry’s law isotherm for the case where the product is adsorbed, the surface concentration of product is proportional to the solution concentration at the electrode surface, and since CR(X = 0)

continually increases during the cathodic scan, r R increases with no upper limit. For Langmuir behavior there is an upper limit, Le., I’R must remain less than r R s , and as r R approaches r R g the current producing adsorbed material must fall to nearly zero. Thus, when a prepeak is present, the surface is covered and the current decreases before the normal reduction potential is reached. On the other hand, if the bulk concentration is sufficiently decreased the behavior approaches Henry’s law and the prepeak shifts toward cathodic potentials (Equation 43) and becomes superimposed with the diffusion peak. Thus, there can be no significant difference in the form of the polarograms for different strengths of adsorption when a Henry’s law isotherm is applicable, in con- trast to the behavior observed for cases where the Langmuir isotherm can be used.

Reactant Adsorbed. If only the reactant is adsorbed, on the cathodic scan there exists at the electrode surface more material than in the absence of adsorption. The reduction of this material causes the current function to be higher at the peak than in the uncomplicated reversible case, just as in the case of weak Langmuir adsorption shown in Figure 2. This increase in the current function is dependent only on the value of Po, and can be summarized most con- veniently as the ratio of the peak current function in the presence of adsorption to that in the absence of adsorption as a function of PO (Table TV). For values of Po greater than about 20 this current function ratio, [$(af)],/0.446, becomes a linear function of Po:

[$(~~t)],/0.446 = 0.851 + 0.2954Po (49)

For small values of Po, or slow scan rates, the current function in the absence of adsorption can be obtained. Then from data measured at higher scan rates, the current function ratio can be calculated and used to determine PO. Referring back to the case of weak Langmuir adsorption of reactant, this value of the current function is the maximum value which can be reached for this particular value of Po. Therefore, from the results at low concentrations presented here, one can determine the maximum value of curve B, Figure 5.

While the cathodic scan is easily described, the current function for the anodic scan is, as before, dependent on switching potential, and consequently a complete description becomes complex and was not considered here.

The results presented in Figures 3 and 4 are essentially unchanged from the case of weak Langmuir adsorption, and can be used to help determine the adsorption behavior. The only important difference is that if the concentrations are sufficiently low, the behavior will be independent of concentration.

Product, or Reactant and Product Adsorbed. For these cases the behavior is very similar to weak Langmuir adsorption, except there is no concentration dependence. Figures 3, 4, and 6 qualitatively describe the behavior observed.

DIAGNQSTIC CRITERIA

When considering the applications of stationary electrode polarography, it is useful to have criteria to determine if adsorption is really present, to determine the extent to which it is present, and to determine the type of adsorption involved. Generally only the concentration and scan rate can be varied (although changing solvent composition or electrode surfaces may influence values of r8, K, and u). For this reason all figures presented above are given in terms of scan rate or concentration except where this would be confusing or restrictive.

1526 ANALYTICAL CHEMISTRY

Determination of Presence of Adsorption. In cases where a prepeak or postpeak appears, the presence of adsorption will usually be obvious. In other cases the presence of adsorption may not be readily apparent because the adsorption and diffusion processes occur simultaneously and give rise to charge transfer at the same potential. To detect the presexe of adsorption in these cases, it generally requires that the adsorption process contribute significantly to the total current at the scan rates and concentrations studied. Under these conditions there are three quick tests which can be applied. The first, which is strictly qualitative, is to observe the general shape of the current-potential curves. If they show more symmetry than for the uncomplicated Nernstian charge transfer, or increase in symmetry with increased scan rate or decreased concentrations (Figures 2 and 6), there is reason to suspect adsorption.

The other two methods involve quantitative measurement and cannot be applied with certainty if other processes (Le., kinetic effects, coupled chemical reactions, etc.) are present. Varying the scan rate, adsorption processes generally cause the experimental equivalent of the current function, i,lCo*& to increase rapidly with increasing scan rate, while i,/cCo* may remain nearly constant. Very few other processes are likely to show similar behavior (16).

While i,/Co* is constant for many processes: when adsorption is present an increase in i,; CO* with decreasing concentration usually is observed, possibly leveling off at some constant value for low concentrations, analogous to Figure 5 . How- ever, caution should be employed when using concentration dependence to detect adsorption as one may be in a range where this is not an important factor.

Determination of Type of Adsorption. If weak adsorption is known to be present, plots of the type shown in Figures 3, 4, and 5 can be used to determine whether reactant or product is adsorbed, and which isotherm is most likely valid. For

The last method involves concentration dependence.

only one species adsorbed, the behavior is easily characterized, while if both are adsorbed, intermediate behavior will be observed and it may be necessary to resort to different electrochemical techniques or additional theoretical calculations,

If the adsorption is strong, the position of the adsorption peak can be used to determine if it is the product or reactant which is adsorbed. However, if both species are adsorbed with significantly different values of the free energy of adsorption AGO, it may be necessary to make plots analogous to Figures 17 and 18 to determine the presence of the less strongly adsorbed material. If both species are strongly adsorbed, the behavior will be about the same as if they were both weakly adsorbed, and neither a prepeak or postpeak will be observed, This is because with both strongly adsorbed, the surface is completely covered at all times and the ratio of the surface concentrations, r o / r R , is Nernstian. (The isotherms considered, Equations 8 and 9, do not assume any interdependence of r~ and ro, which must exist, and therefore do not adequately describe this case.) Thus, for the adsorption process, the current function becomes independent of mass transfer, while the portion of the current function dependent on diffusion is exactly as for an uncomplicated reversible charge transfer. Addition of these two current func- tions then gives the total current-potential behavior.

These results indicate that a wide range of effects can be observed when adsorption occurs in stationary electrode polarography, and that adsorption must always be considered when using the method in analysis and in studies of electrode kinet- ics.

RECEIVED for review March 24, 1967. Accepted August 18, 1967. Presented in part at the IVth International Congress on Polarography, Prague, July 1966. During the 1965-66 academic year, R. H. Wopschall held a Public Health Service Fellowship. Work also supported by the National Science Foundation under Grant No. GP 3907.

Adsorption Characteristics of the Methylene Blue System Us i n g Stationary Elect rode Po I a ro gra p h y Robert H. Wopschall and Irving Shain Chemistry Department, Unicersitj’ of Wisconsin, Madison, Wis.

The reduction of methylene blue was studied to test the theory of stationary electrode polarography for the case in which the product of the electrode reaction is strongly adsorbed. A brief investigation of the mechanism of the electrode reaction, using both aqueous ethanol and aprotic solvent systems, indicated that the reduction probably proceeds through successive one- electron charge transfers, with a very rapid reversible protonation interposed between the charge transfers. The intermediate appears to be more easily reduced @.Eo o 51 mV) than the methylene blue. In spite of this experimental deviation from the model used in the theoretical calculations, i t was possible to make empirical calculations which permitted correlation of the experimental results with theory, and the isotherm parameters for the adsorbed product (leuco methylene blue) were calculated. Satisfactory agree- ment with polarographic results was obtained for methylene blue concentrations below 0.2mM, but at higher concentrations, the adsorption of methylene blue itself becomes important, and only qualitative comparisons were made with theory.

To VERIFY the theory of stationary electrode polarography (1) for systems exhibiting the strong adsorption of the product, and to evaluate the applicability of the theoretical correlations, the reduction of methylene blue was studied. The methylene blue system was selected because it had been studied extensively, and direct comparison with Brdicka’s ( 2 ) polarographic results was possible. In stationary electrode polarography, methylene blue exhibits a typical adsorption prepeak on the cathodic scan, characteristic of a system in which the product of the electrode reaction is strongly adsorbed (Figure 1). Using stationary electrode polarography, the system was studied by Mirri and Favero (3), and by Kemula, Kublik, and Axt (4). Mirri and Favero

(1) R. H. Wopschall and I. Shain. ANAL. CHEM.. 39, 1514 (1967). (2) R. Brdicka. Collection Czech. Cliem. Commun., 12, 522 (1947). (3) A. M. Mirri and P. Favero, Ric. Sci., 28, 2307 (1958). (4) W. Kemula, Z . Kublik, and A. Axt, Roczniki Chem., 35, 1009

(1961).

VOL. 39, NO. 13, NOVEMBER 1967 0 1527

adsorption of reactant at stationary electrode

Documents