Frequency dependence of capacitance at the desorption potentials of organic molecules

J. Electroanal. Chem., 82 (1977) 403--412 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

403

F R E Q U E N C Y DEPENDENCE OF CAPACITANCE AT THE DESORPTION POTENTIALS OF ORGANIC MOLECULES *

ROBERT S. HANSEN and K.G. B A I K E R I K A R

Ames Laboratory-ERDA ** and Department o f Chemistry, Iowa State University, Ames, Iowa 50011 (U.S.A.) (Received 19th January 1977)

ABSTRACT A theory on kinetics of adsorption is presented which assumes that diffusion is the sole rate determining step in adsorption. The impedance of the electrical double layer is modelled as a capacitor and resistor in series and equations have been derived relating capacity to frequency. The theory has been tested using capacity data obtained at anodic desorption peak maxima of two neutral organic adsorbates, and found to agree reasonably with experimental results.

INTRODUCTION °

Levich has maintained a keen interest in the various problems confronting electrochemistry, one of them being kinetics of diffusion controlled adsorption processes [1]. The influence of adsorption of capillary active molecules at a metal-solution interface on the structure of the electrical double layer and on the rate of electrode reactions is widely known. At the mercury electrode, the adsorption of such molecules from solutions leads to characteristic differential capacity-potential curves. Adsorption of aliphatic compounds, for example, decreases the capacitance of the mercury electrode because these molecules increase the thickness of the c o m p a c t double layer and lower its dielectric constant. For a given adsorbate concentration, adsorption will be greatest (and capacity depression correspondingly greatest) near the potential of zero charge of the electrode surface. A s t h e electrode charge and electric field increase, there is an increased preference for material of high dielectric constant in the c o m p a c t double layer, leading to displacement of organic molecules by water. At extreme polarizations this displacement is complete; characteristic peaks in the capacitypolarization curves appear, both on anodic and cathodic sides, at polarizations where the rate of change of adsorption with polarization is large. The capacity values at such peaks are strongly frequency dependent. The capacity dispersion at the peaks occurs because adsorption is a rate process; adjustment to a new equilibrium following a perturbation in the polarization is hence characterized by a relaxation time. If this time is long compared to the * In honour of the 60th birthday of Benjamin G. Levich. ** Prepared for the U.S. Energy Research and Development Administration under contract No. W-7405-eng-82.

404

period of the alternating current used to measure the capacity, equilibrium will not be able to keep up with the corresponding variation in the alternating potential. The kinetics of adsorption of neutral molecules and ions at a metal-solution interface has received an increasing a m o u n t of attention in recent years [ 1--33]. Although various experimental methods involving coulostatic [22--24], rectification [25], potential step [18] and modulation polarography [28] measurements have been described, most published results have depended on direct impedance measurements. Thus the kinetics of adsorption of surface active substances at the mercury-solution interface has been studied mainly by two methods: the frequency dependence [2,4--9] or the time dependence [10--17] of the double layer capacitance. A.c. impedance measurements were first used for the determination of the kinetics of adsorption by Frumkin and Melik-Gaikazyan [2] who based on the frequency dependence of the differential capacity at the desorption peaks, derived equations that allow one to distinguish between diffusion control or activation control of adsorption from solution. The m e t h o d was generalized somewhat b y Lorenz and MSckel [4,5] to take account simultaneously of diffusion in the solution, and a slow step at the electrode surface. Berzins and Delahay [21] also presented a theoretical analysis of the adsorption kinetics under conditions where diffusion, adsorption, or both processes control the overall rate. The variation of the differential capacity with time during adsorption of a surfactant may be due, in principle, to either a slow diffusion step or a slow adsorption step. Experimental results have so far shown that for simple organic molecules, the adsorption step is generally fast and that the kinetics are entirely diffusion-controlled. Diffusion controlled adsorption of organic molecules on an initially clean electrode surface has been investigated by several workers [2,7--17]. A few reviews are also available [3,31--33]. In general, the coverage (or the surface concentration) by the adsorbate at a given time is determined by the nature of the adsorption isotherm. Thus, the time dependence of adsorbate coverage has been theoretically evaluated for the case of the linear isotherm [10,35], the Langmuir isotherm [11,29,34] and with a reasonable approximation, for any arbitrary isotherm [1,36]. In the present work, the frequency dependence of capacity at the desorption peaks has been studied from a theoretical as well as from an experimental point of view. The theory to be presented below is based on the assumption that diffusion is the sole rate determining step in adsorption and provides formulae directly relating capacity to frequency. THEORETICAL

We can write d o = ( ~ o / ~ E ) r d E + (~o/aF)E dI"

(1)

where o is the charge per unit area on the electrode at a polarization E and I" is the adsorbate surface excess. We are interested in the effect of a small alternating voltage superimposed on a steady direct current polarization Eo on electrode behavior, specifically double

405 layer capacity. Using complex notation for the a.c. polarization we have (2)

E = E o + a e i¢°t

where a is small and real. We expect to obtain our information from the magnitude and phase of the surface current density ~ = d o / d t , and have from eqn. (1) 0 = ( O o / 3 E ) r 1~ + (3o/3F)E 1~

(3)

If adsorption is diffusion-controlled, F is a function of E and c(0, t), the surface concentration of adsorbate. As Frumkin and Melik-Gaikazyan [ 2] have given a rather general treatment of this problem with and w i t h o u t activation, the treatment of diffusion controlled adsorption and its effect on surface current density can be very brief. We have 1~ = (3F/3c)o 5(0, t) + (3F/3E)o/~

(4)

where the zero subscripts indicate evaluations at Co, Eo. Fick's second law of diffusion gives 3c/3t

= D(32e/3x 2)

(5)

A solution of proper periodicity in time approaching Co as x -~ ~ is c = Co + A e- " x e i¢°t

(6)

where n = (co/D) 112 e "i14. From Fick's first law = D(3c/3x)x=o

= - - A N D ei¢°t

(7)

But values of ~: from eqn. (2) and of ~(0, t) from eqn. (6) can be substituted in eqn. (4) and the result compared to eqn. (7) to obtain A = --aico(aF/3E)o/[nD

+ ico(3F/3c)o]

(8)

Using i~ from eqn. (7) with A given by eqn. (8), and ~: from eqn. (2) in eqn. (3), using the value of n given following eqn. (6), and rationalizing we find O=ae

i¢°t

•

30

lcoI(~)r+(1-2

+

-

)~-~JE~)Ol.{ " 2"- ~g-FJE,OE]o I J i j

(9)

where b = ( c o / D ) 1 / 2 ( 3 r / 3 c ) o . If we model the a.c. admittance of I cm 2 of electrical double layer as a capacitance Cp and resistance R p in p a r a l l e l , we have d = a e i¢~t [icoCp + 1/Rp]

(10)

Hence by comparison with eqn. (9) l+~b Cp = ~-~ r

Rp

~-F E ~-E 0 1 + bx/~ + b 2

(lla)

co ~-~ ~ ~-~ o l + b x / ~ + b 2

Equations equivalent to (11a) and ( l l b ) were given by Frumkin and MelikGaikazyan [2].

406

Cp as given by eqn. ( l l a ) is readily interpreted as consisting of two capacities in parallel. The first, with capacity (~ a/~E)l., is the " t r u e " or constant coverage capacity, and is also the limiting value of Cp as b -+ ~ , i.e., as co -+ = , when the frequency is too great for appreciable changes in F to be induced by the small change in E over a period of the a.c. polarization. The second term on the right is a pseudocapacity resulting from the field-induced change in surface excess. The term (~o/~I')~(~F/~E)o is the limiting (zero frequency, i.e., b -+ 0) value of the pseudocapacity, and its coefficient accounts for the variation in pseudocapacity with frequency at fixed adsorbate concentration. Because the double layer impedance is in series with the solution resistance, it is convenient to represent the double layer impedance as a capacity Cs and resistance Rs in series, so that d[R~ + 1/icoC~] = a e i¢°t

(12)

Comparing with eqn. (10) C~ = C , [ 1 + !/(coCpR,) 2]

(13a)

R s = Rp[1 + (COCpRp)2] - 1

(13b)

with Cp and Rp given by eqns. (11a) and ( l l b ) , respectively. When the double layer-solution impedance is balanced by an external capacitor and resistor in series, the measured properties are Cs and R s plus the solution resistance. It is plain from eqn. ( l l b ) that (coRp) - 1 -+ 0 as b -+ 0 and as b -+ ~ , hence as co -+ 0 and co -+ =, while eqn. ( l l a ) shows that Cp is finite and non-zero at both limits. H e n c e (coRpCp) - 2 --> 0 as co -~ 0 and as co -+ =, and from eqns. ( l l a ) and (13a) we have C ~ = lim C~ = lim Cp = (ao/OE)r

(14a)

AC ° = lim C~-- C~ = lim Cp -- C ~ = (~o/~F)v(aF/~E)o

(14b)

CO--> 0

CO--->0

C~ is the constant coverage or " t r u e " capacity, and AC°~ is the limiting pseudocapacity. Let

x / ~ b/[1 + b~/~ + b 2 ] f - f(b) = 1 + --if-

(15a)

b g = g(b) = b +----~

(15b)

then from eqns. (13a), ( l l a ) , ( l l b ) , (14a) and (14b) g

2

Equation (16) represents the dependence of the series eapaeity on frequency (through the variable b on which the functions f and g depend). To complete the theory we need to interpret the parameters C~ , AC g, and bco-1/2 = D-112(a F/ aC)o.

407

Earlier papers [37--39] from this laboratory have evolved a m e t h o d for inferring surface excesses of organic compounds from differential double layer capacity measurements. The m e t h o d is based on the Frumkin isotherm equation O/(1 -- O) = Bc e 2~° (fixed polarization)

(17)

and the Frumkin model of the double layer as two condensers in parallel, one containing only water, the other only adsorbate, so that o = Ow(1 - - O ) + O C ' ( E - - E N )

(18)

The Frumkin model leads to ®/(1 -- O) = Boc e 2~® e-(q)/['mRT)

(19)

where E

qb=f

[ow--C'(E--EN) ]dE

o

and C = Cw(1 -- ®) + C'® +

[Ow -- C'(E --EN)] 2 O(1 -- ®) FmRT 1 -- 23®(1 -- ®)

(20)

In these expressions Ow and Cw are the charge density and capacity per unit area on the water condenser at polarization E, C' the capacity per unit area of the adsorbate condenser (assumed independent of polarization), Fm is the limiting value of the adsorbate surface excess, ® = F/Fro, EN is the shift in the potential of the electrocapillary m a x i m u m as ® changes from 0 to 1, E is the polarization referred to the electrocapillary m a x i m u m in the absence of adsorbate, B o and are constants, and c is the adsorbate concentration (which will be assumed equal to its activity). From eqns. (14a) and (20) we have immediately C~ = Cw(1 -- ®) + C'® (21) and from eqns. (14b) and (20) AcO = [Ow -- C'(E - - E N ) ] 2 ~)(1 -- ®) FmRT 1 -- 23®(1 -- ®)

(22)

Equation (22) could also be derived by exploiting eqn. (14b) and the relation, (a o/0 F)E (0 F/OE)0 = (~ o/~ F)~ (~ F/0 P)E, where p is the adsorbate chemical potential. Using eqn. (17) to evaluate (aF/Oc)0 , and expressing c in tool 1-1 we obtain b =/27r~]1/2 1000 l~m ®(1 -- ®) \D] c 1 -- 23®(1 -- ®)

(23)

We are particularly interested in the capacity maximum, and obtain the value of ® at this m a x i m u m by setting d ~:(C~ +~C °)=0.

408

The following equation for ® results: (9 = 0.500 _ 3 FmRT(Cw -- C')[1 -- 2a(9(1 -- (9)]2 2 [Ow -- C'(E -- EN)] 2 + l(gdCw [1 -- 2a(9(1 -- O)]3(PmRT) 2 dE [o w -- C'(E --EN)] 3 (9(1 -- (9)

(24)

The capacity peaks most frequently studied are those for which (9 > 0.5 at the electrocapillary maximum. The coverage at the desorption peaks is then approximately (9 = 0.50, and this value can be used in the second and third terms to give a first order correction to (9. The result, using eqn. (22) is (gmax = 0.500

3 ( 2 - - a ) C w - - C ' 2 - - a o w - - C ' ( E - - E N ) dCw 16 A C ~ + 32 (AC°~) 2 dE

(25)

The last term proved negligible in all peaks studied in the present work, while the second term was found to be about 0.21 or less. Because AC ° and b only depend on ® through the product (9(1 -- (9) which has a m a x i m u m at (9 = 0.5 the zeroth order approximation (9 = 1 is already quite good in their evaluation, so that for those peaks resulting from adsorbate concentrations such that (9 > 0.5 at the electrocapillary maximum, to very good approximation C~ = 1 3(2 -- ~) (Cw -- C') 2 ~(Cw + C') + - 16 AC °

ac o = b =

2 ( 2 - - ~) FmRT

2(2--~)c

I1

32

32

(26a)

\

1

AC°~ ] J

\ AC°~ ]

(26b)

(26c)

EXPERIMENTAL

The experimental part of this investigation was limited to the measurement of differential double layer capacity at an expanding mercury drop by employing an impedance bridge. For successful application of the theory, it is important t h a t the frequency dispersion of capacity observed for mercury in contact with base electrolyte solution be held to a minimum. This requires dropping mercury electrodes to be constructed w i t h o u t major imperfections. Special care was therefore taken to prepare these electrodes. Capillaries used for the preparation of the dropping mercury electrodes were siliconized prior to use. Only such capillaries were selected (capillary tip diameter not exceeding 0.008 cm) that furnished mercury drops which had diameters of at least 0.08 cm at the time of bridge balance. With such capillaries, the capacity dispersion for mercury in contact with 0.1 M perchloric acid base electrolyte solution at the potential of zero charge was normally n o t more than 0.3 to 0.4% and was sometimes as low as 0.1% in the frequency range 400--1400 Hz. The effect of small electrode imperfections becomes prominent at higher frequencies leading to larger capacity dispersion. In the present work, therefore, the kinetics of adsorption of neutral or-

409

ganic molecules was investigated in the above frequency region only, i.e., 400-1400 Hz. The differential capacities were measured by the bridge m e t h o d which is essentially the same at that of Grahame [40] in which, as pointed o u t later by Damaskin [41] there is a significant advantage in using the polarizing source of e.m.f, n o t across the cell b u t across a diagonal of the bridge. A Leeds and Northrup shielded ratio box (Catalog No. 1553) served as the central connecting terminal. A precision decade capacity box (1413) and a precision decade resistance box (1433-W), both General Radio Co. components, connected in series formed part of the measuring arm of the bridge. A Freed Transformer Co. (Model 1350) decade capacity box connected in parallel with the General Radio Co. capacity box was used whenever the capacity of the mercury electrode exceeded the measuring range of the latter. A Hewlett Packard oscillator (4204A) isolated from the bridge by a screened transformer supplied a test signal of 3 mV r.m.s, to the bridge. The null detector was a specially designed phase selective amplifier (lock-in amplifier) which supplied start and stop pulses corresponding to the birth of a new drop and balanced state of the bridge to the gating circuit which controlled the counter timer (Monsanto, Model 100A). The dropping mercury electrodes normally had drop times in the range of 10--12 s and the individual drops were balanced at 80--90% of their droplife. A cylindrical platinum gauze of large surface area symmetrically surrounded the dropping mercury electrode which also served as the counter electrode. The drop time of the dropping mercury electrode was registered with an accuracy of one part in thousand. The precision of capacity measurements, with 0.1 M perchloric acid as test solution, was 0.05% within a given run, 0.15% from one run to another and a b o u t 0.3% from one capillary to another over most of the polarization range provided (a) the work was carried out under conditions of scrupulous cleanliness and provided (b) the capillary characteristics were maintained the same. Capacity measurements were made with a few electrolyte solutions and the good agreement obtained between our data and the literature capacity data confirmed the reliable functioning of the impedance bridge. In the present work it was confirmed by measuring capacity per unit area at fixed a.c. frequency b u t at different times in the droplife, that the double layer capacities measured had their equilibrium values and that there was no complication due to the slow diffusion of the organic adsorbate. The saturated (NaC1 solution) calomel electrode (SCE) which served as the reference electrode was isolated from the experimental cell as discussed previously [42]. All potentials were measured correct to better than +0.2 mV using a K-3 universal potentiometer (Catalog No. 7553-5) and a d.c. null detector (Catalog No. 9834), b o t h manufactured b y Leeds and Northrup Co. Materials

Mercury supplied by the Ames Laboratory stock r o o m had been triply distilled. Prior to distillation, it had been purified b y the usual rigorous procedure. Water was quadruply distilled with the third distillation over alkaline permanganate in an all-glass assembly. The base electrolyte used in all experiments was 0.1 M perchloric acid which was prepared from Baker analyzed reagent,

410

n-pentanoic acid and n-hexanoic acid supplied by Fisher Company were high purity compounds and these were purified by simple distillation. RESULTS A N D DISCUSSION

Figures 1 and 2 present differential capacity data measured at three frequencies i.e. 400 Hz, 800 Hz and 1400 Hz for n-hexanoic acid and n-pentanoic acid from 0.1 M perchloric acid solutions. As can be seen, the frequency dispersion of capacity in the neighborhood of desorption peaks is maximum for n-hexanoic acid. Because of possible complications due to hydrogen discharge at large negative polarizations, only peaks at the positive potentials were considered to verify the theory. Capacity peaks were carefully tracked at 12 mV intervals but the maximum was located with as much accuracy as possible depending upon the sharpness of the peak. The capacity and the charge corresponding to the potential of peak maximum in the absence of adsorbate were obtained by interpolation from capacitypotential and charge-potential curves for the base electrolyte solution. The charge-potential curve was obtained by integrating the capacity-potential curve starting from the potential of electrocapillary maximum (e.c.m.), i.e., --0.477 V vs. SCE. A value of 5.6 × 10 - 6 c m 2 S- 1 was chosen as representative for the diffusion coefficient of the adsorbates. To represent the capacity at peak maximum, a and EN were treated as adjustable parameters. Fm values were taken from a previous electrocapillary investigation [42]. C' was obtained by extrapolation of capacity data at maximum adsorption [ 3].

56

56 +

48 N u-

~uE 48 u.

*÷

* t $

~ 40

o 40 + +

u 32

÷

t**

.~ 24 7-

}

o

*

32 ++

$

÷++,

÷,+÷

,$

t

~ ~6 u. ÷

8

0 -I.2

I -LO

I

-0.8

÷

+++÷++ l

-0.6

+

I

÷

+

8

I

-0.4 -0.2 POTENTfAL, E/V VERSUS SCE

I

I

0.0

0,2

÷+++÷÷+÷

0

I

-I.2

-I.0

t

I

I

I

-0.8 -0.6 -0.4 -0.2 POTENTIAL, E/V VERSUS SCE

I 0.0

I 0.2

Fig. 1. Differential capacity-potential data (measured at three frequencies, i.e., 400 Hz, 800 Hz and 1400 Hz) for n-hexanoic acid in 0 . 1 M HC104 solution. 103 c = 1 1 . 3 7 3 M. Fig. 2. Differential capacity-potential data (measured at three frequencies, i.e., 400 Hz, 800 Hz and 1400 Hz) for n-pentanoic acid in 0.1 M HC104 solution. 102 c = 4.761 M.

--0.194 --0.169 --0.125 ---0.087 --0.020

vs. SCE

Emax/V

0.385 0.404 0.426 0.438 0.452

@)max

16.694 16.015 15.182 14.736 14.414

C°°/PF cm--2

30.904 36.080 45.031 53.073 67.696

AC°/I,~F cm--2 800Hz 33.396 37.929 46.387 56.098 72.910

36.299 41.040 49.661 59.064 75.380

30.934 35.165 43.321 53.183 70.335

1400 Hz

(calculated)

400 Hz

Cmax/PF c m - 2

2.547 3.318 4.761 6.083 12.167

102 c/tool 1-1

--0.205 --0.180 --0.135 --0.105 --0.020

vs. SCE

Emax/V

0.290 0.338 0.382 0.400 0.429

18.928 17.603 16.250 15.690 15.016

@)max C¢/P F cm--2

18.437 23.849 31.863 36.974 51.331

AC° /PF cm--2

800 Hz 35.081 38.844 45.417 50.119 64.474

35.714 39.570 46.173 50.837 65.011

34.416 38.081 44.614 49.351 63.891

1400 Hz

(calculated) 400 Hz

Cmax/PF c m - 2

P a r a m e t e r s used are: 1010 P m = 4.754 tool c m - 2 , C' = 4 . 4 4 4 p F c m - 2 , (~ = 0.94, E N -- 0.385 V vs. e.c.m.

n - P e n t a n o i c acid

TABLE 2

6.083 7.927 11.373 17.440 30.941

103 c ] m o l 1-1

P a r a m e t e r s used are: 101° F m = 4.202 tool c m - 2 , C r = 4.082 p F c m - 2 , c~ = 1.08, E N = 0.369 V vs e.c.m.

n - H e x a n o i c acid

TABLE 1

39.287 43.331 48.696 52.137 60.036

400 Hz

32.978 36.791 43.367 52.741 65.967

38.258 42.118 47.515 51.128 58.953

800 Hz

37.116 40.782 46.457 49.721 58.071

1400 Hz

(observed)

35.285 39.739 46.689 56.376 69.751

1400 Hz

(observed)

800 Hz

Cmax/pF c m - 2

38.404 43.305 50.572 61.229 74.205

400 Hz

Cmax/PF c m - 2

412

Tables 1 and 2 present the agreement obtained between theory and experiment (Ema× and Cmax correspond to peak maximum). The capacity at peak maximum could be fit within a m a x i m u m uncertainty of 6.6% in the case of nhexanoic acid and 10% in the case of n-pentanoic acid. The fit in each case could have been slightly improved by selecting somewhat smaller values of ~ than shown in Tables 1 and 2, but this would have necessitated adjusting parameter EN to unrealistically high values. We did not treat ~ as a function of polarization [3] since from our previous electrocapillary study [42], it was apparent that the electrical state of interface remains unaltered as a result of the change in polarization. REFERENCES 1 B.G. Levich, B.I. K h a i k i n a n d E.D. B e l o k o l o s , Sov. E l e c t r o c h e m . , 1 0 ( 1 9 6 6 ) 1 1 3 7 . 2 A.N. F r u m k i n a n d V.I. M e l i k - G a i k a z y a n , D o k l . A k a d . N a u k S S S R , 7 7 ( 1 9 5 1 ) 8 5 5 . 3 B.B. D a m a s k i n , O . A . Petrii a n d V.V. B a t r a k o v , A d s o r p t i o n o f O r g a n i c C o m p o u n d s o n E l e c t r o d e s , P l e n u m Press, New Y o r k , 1 9 7 1 . 4 W. L o r e n z a n d F. M S c k e l , Z. E l e k t r o c h e m . , 6 0 ( 1 9 5 6 ) 5 0 7 . 5 W. L o r e n z , Z. E l e k t r o c h e m . , 6 2 ( 1 9 5 8 ) 1 9 2 . 6 J.B. H a y t e r a n d R . J . H u n t e r , J. E l e e t r o a n a l . C h e m . , 3 7 ( 1 9 7 2 ) 8 1 . 7 K . T a k a h a s h i , E l e c t r o c h i m . A c t a , 13 ( 1 9 6 8 ) 1 6 0 9 . 8 R. P a r s o n s a n d P.C. S y m o n s , T r a n s . F a r a d a y S o e . , 6 4 ( 1 9 6 8 ) 1 0 7 7 . 9 R . D . A r m s t r o n g , W.P. R a c e a n d H . R . T h i r s k , J. E l e c t r o a n a l . C h e m . , 1 6 ( 1 9 5 8 ) 5 1 7 . 1 0 P. D e l a h a y a n d I. T r a c h t e n b e r g , J. A m e r . C h e m . S o c . , 7 9 ( 1 9 5 7 ) 2 3 5 5 . 11 P. D e l a h a y a n d C.T. F i k e , J. A m e r . C h e m . 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