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Study of reactant and product adsorption in double potential step and related pulse techniques
269
J. Electroanal. Chem., 263 (1989) 269-292 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Study of reactant and product adsorption in double potential step and related pulse techniques Part II. Normal puke polarography (NPP), reverse pulse polarography (RPP) and double potential step chronoamperometry Jesus Gdvez Laboratory
of Physical Chemistry, Faculty of Science, 3ooo1 Murcia (Spain)
Su-Moon Park Department
of Chemistry,
l7ze University of New Mexico, Albuquerque, NM 87131 (U.S.A.)
(Received 7 November 1988; in revised form 14 December 1988)
ABSTRACT The analytical solutions for normal pulse polarography (NPP), reverse pulse polaro$raphy (RPP), and double potential step chronoamperometry for a reversible charge transfer reaction when reactant and product are adsorbed on a plane electrode following a linear isotherm have been derived. The current-potential curves in NPP and in RPP obtained from this solution are in excellent agreement with those previously reported in the literature obtained using numerical methods. Also, our solution for the limiting current becomes equivalent to that derived by Van Leeuwen et al. (J. Electroanal. Chem., 135 (1982) 13) when the condition r’//t Q: (t’ being the pulse duration) is applied. The characteristics of the double potential step chronoamperometry response, for which no solution (analytical or numerical) had been reported previously, have also been shown. A simple method to determine the adsorption coefficients of reactant and product based on the dependence of the limiting currents in NPP and in RPP on the pulse time is discussed.
INTRODUCTION
Adsorption effects in pulse polarography are especially important [1,2], and were first indicated by Wolff and Numberg [3] in their measurements of maxima in the NPP polarograms for nitroanilines. Regarding the theory for the NPP response, this has already been treated by Flanagan et al. [4], LovriC [5-91, and Mas et al. [lo] 0022-0728/89/%03.50
0 1989 Elsevier Sequoia S.A.
270
using numerical methods. An analytical equation for the NPP limiting current was derived by Van Leeuwen et al. [ll] when the condition t '/t -=z 1 (t ’ being the pulse duration) is applied. Van Leeuwen and co-workers have also extended their approach to other types of electrodes and isotherms [12-141, although in all these cases only the limiting current was considered. All these treatments have shown that the corresponding theory is rather complex and so, no analytical solution for the current-potential (I-E) curves in NPP, even when simple conditions (plane electrode model and linear adsorption isotherm) are adopted, had been reported previously. In any case, and in spite of this shortcoming, the level of understanding of adsorption effects in NPP is quite satisfactory. In contrast, the situation in RPP or in double potential step chronoamperometry complicated with reactant and product adsorption is much less clear since in RPP only I-E curves obtained by Flanagan et al. [4] using digital simulation techniques have been reported, while in double step chronoamperometry the corresponding response, to our knowledge, has not been described yet in the literature. In Part I [15], a general analytical solution for double potential step and related pulse techniques was derived for a reversible charge transfer reaction when reactant and product are adsorbed on a plane electrode following a linear isotherm. In the present paper this equation has been applied to obtain the corresponding responses for NPP, RPP and double potential step chronoamperometry by introducing in the general solution the experimental conditions that characterize these techniques. This has allowed us to analyze and compare our predictions for the pulse responses with those reported previously. In addition, the characteristics of the new response obtained in double potential step chronoamperometry for this kind of electrode process have also been shown. This has revealed some interesting features and so, for example, a simple procedure to determine the adsorption coefficient of the reactant from measurements of the limiting current in NPP has been devised, while the adsorption coefficient of the product can also be obtained from an analogous procedure based on measurements of the limiting current in RPP. As mentioned in Part I, [15] we have adopted the plane electrode model, and this restricts the application of the present theory to types of electrodes other than planar ones. However, this simplified electrode model has the advantage that it facilitates the analysis of the corresponding responses, especially in double potential step chronoamperometry whose solutions had not been found yet.
THEORY
The general solution for double potential step chronoamperometry for a reversible charge transfer reaction when reactant and product are adsorbed on an electrode following a linear adsorption isotherm has been derived in Part I [15]. According to this solution, if t, and t’ are the pulse duration for the first (E = E,) and second (E = E2) steps, respectively, the equations for the current are (a complete list of
271
notations/definitions i(r
_
id(l)
is given in ref. 15):
e t(1 -Ye)% F(X) u + CI (0 + 40 + 7%)
(1)
1’2_
(Ye-NJ+c,)c, (e + c2)2(1 + YE*)
tc2-F1)
_
Cl-
F(Y))WX,)
qX)
(0 + 40 + YQ) _
(Ya---l)c, b+f2)0
fqX)
-
0
-
P(X))Bl
d/=B2x
+Yc2)
(2)
where id(f) and id(t’) are the diffusion currents at t = t and t = r’, i.e., id(t)
= nFA(DA/7rt)1’2cA*;
id(t’)/id(t)
= (t/t’)“’
(3)
and
F(x)=
exp(X2) c (-1)m+‘p~“~p~Xm~~1/2X m. erfc(x)
!
(4)
m=l
H(x)=
c
W>““Po...
Pm-lXmG
1
_
M. 1
m=l
F(X) p
X
(5)
2/J;; PmPm+1 = 2(m + 1)
(6)
Y = (4/b
(7)
po =
)1’2
(8)
B = KB/KA (i=1,2)
E~=~~~[(~F/RT)(E,-E~)] Bi =
1 +yc,
(i=l,2)
(9) (10)
Y(“+ri)
In these equations K, is the adsorption coefficient of the species a (a = A, B), and the arguments x, x1 and y involved in the F(x) and H(z) functions are dimensionless parameters defined by x = B,( DAt)*“/KA; y = B2( DJ’)“~/K~
x1 = B,( DAtl)“2,‘KA
(11) (12)
As has already been mentioned [15], because eqns. (1) and (2) involve only the well-known F(x) function, current-potential curves are readily obtained from these equations. Also, and since eqn. (2) is quite general, this allows us to obtain the
272
corresponding responses for pulse techniques by introducing in this expression the conditions for the different modes of pulse polarography. Thus, we obtain: (A) Normal pulse polarography
(NPP)
In NPP the potential of the first step, E,, is chosen so that no current flows at this potential. Inserting this condition (E, + cc, E, + 00) in eqn. (2) we find lNPP
_
(v--)r* (0
+ E*)(f
H(X) + YE*)
-
(la’/*B,x -
F(X))
(13)
where the argument x is now defined by (see eqns. 10 and 11 with B, + 1) x =
(D*ty*/K*
04)
and x, is defined as x by making t = t,. If no adsorption process occurs, i.e. KA = K, = 0, it follows that x and y --$ co so that F(x) = F(y) = H(X) = 1, and eqn. (13) is simplified to
(1%
‘NPP
which is the expression for the NPP current under these conditions. We wish to point out that eqn. (13) is the current response when, for 0 < t < t,, only the adsorption-controlled diffusion of the reactant occurs (see Part I). In fact, eqn. (13) could also be obtained using these particular conditions and proceeding as in Part I, although the present treatment is more general. In any case, and because eqn. (13) is valid when reactant and product are adsorbed, it provides the corresponding solutions when only adsorption of one of these species takes place. Thus, if u = 0 (i.e., K, = 0 so that only reactant is adsorbed) eqn. (13) becomes -=lNPP b(t’) +p
1 1
+yc,
H(X)
-
1 -F(X) d2B,,x
F(Y)
(16)
where y = B,( DAt’)“*/KA;
B, = (1 + yc2)/yc2
07)
If K* = 0 it follows that u + cc and only adsorption of the product is involved.
273
Under these conditions we have x --, 00 (see eqn. 14) so that F(x) Hence, eqn. (13) is simplified to lNPP
and H(X) -+ 1.
&F(Y)
-=I-
idW
(18)
where the argument y in F(y)
is given by (see eqn. 12)
y = (1 + TQ)( D*t1)1’2/K*
(19) Note that when only the product of the redox reaction is adsorbed there are no differences between the dc polarographic and NPP responses, so that eqn. (18) is identical to that obtained by Guidelli [16] in dc polarography for this kind of electrode process.
(B) Reverse pulse polarography
(RPP)
In RPP [17], the initial potential El is set on the cathodic diffusion current plateau and the potential E2 is scanned in the positive direction. Hence, the condition E, -+ - 00 (tl + 0) is met, and eqn. (2) is simplified to -=1RPP id(t’)
1 + yr,F(x) 1
t’
“* _ (ya-
(1 -F(y))H(x,)
(t 1
+F2
(0 (ua
-
+f(x)
l)ar,
-
+
(2j2
l)E2
1 -F(X)
2
H(X) (0
+
where the argument x in F(x) condition c, + 0)
c2)O
+
YC2)
-
~“~B~yux
F(Y)
(20)
and H(X) is given by (see eqns. 10 and 11 with the
x = ( DBt)1’2/K,
(21)
As previously, when no adsorption is involved we find from eqn. (20) i ““=“““(~)‘“c~(~-~~)
(22)
which is the corresponding expression for the RPP response for a reversible charge transfer reaction. When only one species is adsorbed, eqn. (20) can be simplified. Thus, for K, = u = 0 eqns. (21), (4) and (5) give x + co, F(x) = H(x) + 1, so that we find l/2
+
~-F(Y) 1 +YE2
1
(23)
In this expression the argument y in F(y) is also defined by eqn. (17). In turn, when only the product is adsorbed we have KA = 0, u --* cc and eqn. (20) becomes -=‘RPP 4l(t’)
1 +yc,F(x) 1
+YE2
I’ 1’2 (i)
-
,2(
(I-
F(y))H(xd
+ “‘1”z;:f’)
(24)
274
In this expression the argument x is also given by eqn. (21), while the argument y is defined by eqn. (19) if, in this expression, we substitute DA and KA by D, and K,, i.e., y = (1 + y~)( DBr’)“*/K,
(244
(C) Double potential step chronoamperometry The dependence of i(t) on I is obtained from eqn. (1) for z d t, and from eqn. (2) for t > I,. These equations allow us to compute the corresponding responses for any value of E, and E2 and so, for example, in the special case where the potential is stepped at E, + - rx and E, --, 00, eqns. (1) and (2) become i(tg
tl) = id=(t) =nFA(D,/at)*‘*c,*
i(t>t,)=i,(r)-nFA(D,/~f’)1’2c,*((l-F(~))(t’/t)1’2+H(~))
(25) (26)
or after normalization by the current at t = t, i(t
=-
1, ‘/* ( t i
i(t > tl) i(tl) =(~)“2-(~)1’2~(1-~~x,,($)1’z+“(x))
(27)
In these equations the argument x is given by eqn. (21) (see eqns. 10 and 11 with pi + 0), so that the response in this case depends only on the product adsorption. If no adsorption occurs it follows that x -+ cc and Q’(x) = H(x) = 1. Inserting these conditions in eqn. (26) we find i(t>
tl) = nFA (~)“2C~(++)
which is also an expression identical to the equation first derived by Kambara [18] in double potential step chronoamperometry under these conditions. RESULTS
AND DISCUSSION
(A) NPP In the first part of this section we shall discuss the characteristics of the NPP response. A detailed comparison with previous results from the literature is shown later. Figure 1 shows I-E curves when only the reactant is adsorbed ((I = 0), computed from eqn. (16) for t’ = 0.03 s, t = 1 s and different values of DA/*/K,. The corresponding curve when no adsorption is involved is also included for comparison (curve a). From Fig. 1 the following conclusions are drawn:
215
0.l
0.0
- 0.1
-0.2
-0.3
- 0.4
(E2- EW V Pig. 1. NPP: Dependence of the I-E n=1,y=1,t-ls,t’=0.03s,a=0.(D~‘2/KA)/s-1’2: (g) 0.001.
curves on D112 A / K A computed from eqn. (16) for T= 298 K, (a) co; (b) 10; (c) 1; (d) 0.5; (e) 0.1; (f) 0.01;
(a) As has been recognized [4,5,11], the limiting currents are depressed with respect to those obtained in the absence of adsorption. Note that although the i,i,-values decrease as KA increases they approach a limiting value for KA z-- 1 (see curves f and g, very strong adsorption). Thus, the i,im,NPP/id(f’)-value in Fig. 1 is 0.173 (= m) (see below). (b) The curves exhibit a maximum if the K,-values are sufficiently large (curves c-g), and as has also been reported [S], the maxima shift continuously towards more negative potentials as adsorption is stronger. However, we have found that the corresponding current, i,,, approaches a limiting value for KA z-. 1 (see curves f and g). Thus, this knit for imax,NPP/id(t’) in Fig. 1 is 1.664. The behavior of the limiting current can easily be explained if we consider that for E2 -+ - co (i.e., e2 + 0) we have y + cc, F(y) -+ 1, and eqn. (16) gives
y$y = (1 -
F(X))(
;)r’2
+ H(X)
(29)
According to this equation and recalling that F(0) = H(0) = 0 and that F(cc) = H(oo) = 1, we find ‘lim,NPP
id(t’)
_
1
for KA = 0 (no adsorption)
( c/ty2
for KA B 1 (strong adsorption)
(30)
which is in agreement with the results obtained in Fig. 1 and in the literature [5,11]. In turn, the behavior of the maxima can also be explained if we take into account that as KA increases, the potentials at which the maxima appear (Em,) shift
276
towards more negative values, so that the corresponding values of e2 become smaller. When one obtains the value of the argument y =y,,,, at E = E_x (see eqn. 17) for values of KA x==1, it follows that the increase in KA compensates the decrease in Ed, so that a constant value for ymax= 0.9 is found. In the Appendix it is shown that under these conditions we obtain
E max= E,,, + $
In
l.ll(
D*t’)1’2
(31)
KA l/2
(32) As shown in Fig. 1, eqns. (31) and (32) predict that E,,,,, shifts continuously towards more negative potentials as KA increases and that i,, becomes independent of the K,-values for KA B 1. These equations are valid only for the case of very strong adsorption and so, for example, for the data corresponding to curve (g) in Fig. 1, we find AE,,(= E,, - El,2) = -0.219 V and i,,,Jid(t’) = 1.663, which coincide with the exact values obtained from eqn. (16). Larger deviations are found as adsorption becomes weaker and so, for curve (d) we find from eqn. (31) obtained from eqn. 32 also equals 1.663 AE,,, = -0.060 V (the i_/i,(t’)-value since it does not depend on KA). The exact values are AhE,,,, = - 0.064 V and imax/id(t’) = 1.286. Equations (31) and (32) also allow us to predict the dependence of E,, and i,,, on t’ and t for a given value of KA. This dependence is shown in Table 1 where the exact values of E,, and i,,, obtained from eqn. (16) for DA/‘/K, = 10e4 sell2 are compared with the approximate ones computed from eqns. (31) and (32). For t = constant the agreement for different values of t’ is excellent and, as predicted by eqns. (31) and (32), E,_ shifts towards positive potentials while i,_/ia(t’) decreases as t’ becomes larger. We wish also to point out that for large values of t’ the maximum vanishes and eqns. (31) and (32) are no longer valid. In turn, the dependence of E,_ on t for t’ = constant (Table 1) shows that according to eqn. remains practically constant for all values of t, although for small values (31)1 E,, of t( 5 0.3 s) the E,,,,- values are more negative than those obtained with eqn. (31). Regarding the i,,- values we find that as previously for t = constant, a good agreement between eqns. (16) and (32) is obtained. We wish to emphasize that although eqns. (31) and (32) are quantitatively valid only for processes with strong adsorption, its predictions can also be applied qualitatively to processes with weaker adsorption. This is illustrated in Fig. 2 where Z-E curves for a process involving moderate adsorption (DA/‘/K, = 0.2 s-‘12) have been computed from eqn. (16) for different values of t’. As previously found for strong adsorption (see Table l), E,,,, shift towards more negative potentials and increases as t’ becomes smaller. This kind of behavior for systems involving i,, reactant adsorption has already been indicated by Van Leeuwen [2] in the NPP polarograms for the reduction of methylene blue.
211 TABLE NPP:
I Values
(1’ = constant).
of
i,/id(t’) 0:/*/K,
and
A/&,,
= 10K4 S-I/~,
(=
E,,
- E,,,)
as a function
(I = 0, y = 1. (I): computed
of
t’
(t = constant)
from eqn. (16). (II): computed
and
t
from
eqns. (31) and (32) r=1s C//s
LJM~‘)
(9
0.001
8.761
8.741
- 0.320
- 0.322
0.005
3.940
3.932
-0.300
- 0.301
0.01
2.805
2.800
- 0.291
- 0.292
0.02
2.011
2.009
- 0.283
- 0.283
0.05
1.324
1.324
- 0.272
- 0.272
0.1
1.001
1.000
- 0.265
- 0.263
- 0.272
1’ = 0.05 s
t/s
i,,/ia(O
(1)
irnpx/id(O (11)
0.2
0.768
0.750
- 0.281
0.3
0.846
0.839
- 0.277
- 0.272
0.5
1.001
1.000
- 0.274
- 0.272
1.0
1.324
1.324
- 0.272
- 0.272
2.0
1.811
1.809
- 0.271
- 0,272
5.0
2.805
2.800
- 0.271
- 0.272
10.0
3.939
3.932
- 0.270
- 0.272
20.0
5.551
5.540
- 0.270
- 0.272
Regarding the dependence of the I-E curves on t for t’ = constant, this is shown in Fig. 3 for the same value of Di”/K, used in Fig. 2. As described for strong adsorption in Table 1, i,,, increases with t while, in contrast, E,, practically does not vary, except for small values of 2 (5 0.3 s). Note, however, that under these conditions (t < 0.3 s) the maximum vanishes rapidly (see curve (a) computed with t = 0.1 s). These facts could be applied for analytical purposes and so, if we are interested it follows from eqn. (32) that the t/t’ ratio should be chosen as in enhancing i,,, large as possible. In contrast, if adsorption effects need to be minimized this ratio should be decreased. Thus, for example, if Ok/‘/K, = 5 s-l/* (very weak adsorption) no maximum is observed for t’ = 0.05 s and t = 1 s. However, a wave exhibiting a maximum with imax/id(t’) = 1.927 will be obtained if t’ = 0.001 s and t = 10 s. On the other hand, Fig. 3 also illustrates an interesting feature of the dependence of the limiting current on t, namely that the ilim,NPP-valuespass through a minimum values in Fig. 3 as t changes). This as t increases (note the sequence of the i,im,NPPfact is also in agreement with the results obtained by Van Leeuwen et al. [ll] who provided a detailed discussion of it. By considering the expression for the limiting current given above (see eqn. 29), this minimum arises because for very small values of t (i.e., for t + t’) the F(x) and H(X) functions become smaller and
278
-0.1
0.1 (EZ-
- 0.3
Eel/ V
Fig. 2. NPP: Dependence of the I-E curves on. I’ computed from qn. (16) for I = 1 s, Di’2/K,, = 0.2 s-1/z, (I = 0. C//s: (a) 0.005; (b) 0.01; (c) 0.02; (d) 0.05; (e) 0.1; (f) 0.2 (g) 0.5. Other conditions as in Fig. 1.
= Jt’Tii 4 1 if th e adsorption is not very weak and the t’-values are sufficiently low. As t increases, F(x) and H(x) increase too, although the decrease in the contribution (1 - F((x))fi prevails, and the ii,,,,,-values become smaller. In contrast, at long times, the contribution of H(x) prevails and ilim,NPPincreases with t (for r --, cc we have F(x) = H(x) + 1, so that imax,NPP/id(t’) --, 1). The NPP response when both reactant and product are adsorbed is shown in Fig. 4 where the I-E curves have been computed from eqn. (13) for DA/‘/K, = 0.2 s-1’2, t’ = 0.02 s, t = 1 s and different values of cr. The maximum depends strongly on the u-values so that the corresponding i,, -values decrease and shift toward more positive potentials as u increases. For large values of u no maximum is observed. However, and in agreement with the literature data [4,5], the limiting current does not depend on the u-values. This is easily explained from eqn. (13) since when the condition E2 --) - co is introduced in this expression we obtain again eqn. (29) which does not involve u. Hence, no information about the adsorption of the product can be obtained from the NPP limiting current. Although the NPP I-E curves could provide this information, that means that we have to deal with eqn. (13), which is much more complex than the corresponding expression for the i,,NPP/id(t’)
279
I
h
(E2- E”MV Fig. 3. NPP: Dependence of the I-E curves on t computed from eqn. (16) for t’ = 0.05 s, Di”/K, = 0.2 S-I/~, o = 0. t/s: (a) 0.1; (b) 0.3; (c) 0.5; (d) 1; (e) 2; (f) 5; (g) 10; (h) 20. Other conditions as in Fig. 1.
limiting current (eqn. 29). Fortunately, the RPP response (see below) provides an easier procedure to obtain K, from the corresponding RPP limiting current. Hence, we shall limit the discussion to the procedures to determine KA, which are shown next. The expression for the NPP limiting current is relatively simple and this allows us to devise a procedure to obtain the adsorption coefficient of the reactant. Thus, from eqn. (29) we obtain I
‘h-l
NPP-
,NPP
id(t)
l/2
=1-F(x)+
(
;
1
H(X)
(33)
where x = (DAt)‘/2/KA (see eqn. 14). A striking feature of this equation is that because x, and therefore F(x) and H(X), depend on r only, if we perform measurements of INpp by modifying 1’ with will be linear with intercept a the condition t = constant, a plot of I,,, vs. @r and slope b given by a=l-F(X)
b=H(x)
280
E _= 2 ..
0.4
C
1
t
_i
-._
I
0.2
0.0
-0.2
0;'
I
1
(E2- E?i V
2
3’
4
5
(t/t*y?
Fig. 4. NPP: Dependence of the Z-E curves on a computed from eqn. (13) for t’ = 0.02 s, I = 1 s, Dif*/KA = 0.2 s-‘fl. Q: (a) 0; (b) 0.005; (c) 0.02; (d) 0.05; (e) 0.1; (f) 1; (g) IO; (h) 100. Other conditions as in Fig. 1. Fig. S. ZNpp vs. fl 10-Z; (c) 1.
computed
from eqn. (33) for 1=1
s, DA =fOe5
cm2 s-‘.
lu,,/cm:
(a) 10S5; (b}
That these kinds of plots are effectively linear has already been suggested by Van Leeuwen et al. (see Fig. 2 in ref. 11). However, we can go further if we exploit the relationship between the F(x) and H(x) functions, i.e. H(X) = 1 - F(x)/( a’/*~) (see eqns. 4 and 5). Thus, taking into account this relationship and the x-value defined by eqn. (14) we find from eqns. (34)
KA= l-b ( 7rDAty2 l-a
(35)
an expression that allows KA to be determined easily from the slope and intercept of these plots. Note that for processes with strong adsorption F(x) and H(X) + 0, so that b -+ 0 and a --j 1. Under these conditions we obtain a straight line parallel to the abscissa with intercept = 1. In contrast, if the adsorption is very weak we have #Y(x) and H(X) + 1, and therefore, a --, 0, b --* 1. In these cases the corresponding plots
will pass through the origin with slope = 1. The upper and lower limits of the K,-values that can be obtained from eqn. (35) are determined by the fact that these plots show a slope and intercept different from 0 and 1 respectively (very strong adsorption), or from 1 and 0 (very weak adsorption). The slope and intercept of these plots are defined by eqns. (34) from which we can obtain estimative values for this limit. Thus, if we consider that plots for x = 0.05 and x = 5 differ from those obtained for strong and weak adsorption, it follows by supposing DA = 1O-5 cm2 s-l and f = 1 s that K,-values of = 6 X low2 to 6 x lop4 cm could be determined. An example illustrating these considerations is given in Fig. 5 where the corresponding plots have been computed from eqn. (33) for K, = low5 cm (weak adsorption), KA = 10e2 cm (moderate adsorption), and KA = 1 cm (strong adsorption). Finally, we wish to point out that for large values of KA these plots show only that adsorption is very strong, although in these cases we could use the dependence of Enlax on KA (eqn. 31) to determine the adsorption coefficients of the reactant if the E,,,-value is known. Comparison with literature data
As mentioned above, the NPP response complicated with adsorption processes has already been treated in the literature [4-111, so that a comparison with our results can be performed: (a) The dependence of the Z-E curves on KA and u shown in Figs. 1 and 4 is in agreement with the NPP polarograms obtained by Flanagan et al., using digital simulation techniques (see Figs. 3 and 5 in ref. 4). A quantitative comparison is not possible because these authors used a different electrode model. (b) Van Leeuwen et al. [ll] derived an analytical solution of the NPP limiting current using the condition t’/t << 1. It is readily shown that in our notation Van Leeuwen et al’s equation adopts the form +y
= (1 - P(x))(
f)l’l+
H(X*)
d
If we take into account that for t’/t e 1, t = t,, it follows that H(X) = H(xi) so that eqns. (36) and (29) are equivalent. Hence, eqn. (29) is the general solution for the NPP limiting current when there is no restriction regarding the values of t and t’. Also, the agreement between other features of the NPP response derived by Van Leeuwen et al. [2,11], and those obtained in this paper has been mentioned above. (c) The NPP response has also been described by Lovric [5] using numerical methods. In Table 2 we compared the E,,,,,- values computed by LovriC for u = 0 and different values of KA (Fig. 2 in ref. 5) with those obtained from eqns. (16) and (31) (eqn. 31 can also be applied since the case treated by LovriC corresponds to strong adsorption). The agreement between both sets of data is excellent. We wish also to point out that LovriC indicated that if KA changes by an order of magnitude, E maxchanges by 60 mV. Note that eqn. (31) provides a theoretical justification of this fact.
282 TABLE 2 Values of AI?,, ( = E,, - Et,,) as a function of KA for DA = lo-’ cm* s-‘, r = 1 s, t’ = 0.05 s, 0 = 0, y = 1. (I) From Fig. 2 in ref. 5 ‘. (II) Computed from eon. (16). (III) Computed from eqn. (31)
log ( KA /cm)
A-Lx WV
AEm, (W/V
4m.x (m/v
0
-0.183 - 0.241 -0.300 -0.360 - 0.420 - 0.481
-0.183 - 0.242 - 0.302 -0.361 - 0.420 - 0.479
- 0.183 - 0.242 - 0.301 -0.360 - 0.419 - 0.478
1 2 3 4 5
Values of i,,/ia(?‘) as a function of f/f’. KA =103 cm. (I) From Fig. 3A in ref. 5 a. (II) Computed from eqn. (16). (III) Computed from eqn. (32) t/t’
im,/id(t’)
10 20 30 40
1.000 1.338 1.590 1.850
(1)
imax/Mt’) 1.001 1.324 1.586 1.811
(11)
i,,/ia(t’)
(III)
1.000 1.324 1.585 1.809
a Data from ref. 5 contain some uncertainty since they have been taken from figures.
Values of imax/id(t’) vs. t/t’ have also been reported by LovriC (Fig. 3A in ref. 5), who suggested that the i,,,- values are the same regardless of whether the t/t’ ratio is obtained by maintaining t’ = constant and varying t, or by keeping t = constant and changing t’. These data are reproduced in Table 2 and they are compared with those obtained from our eqns. (16) and (32). Again, the agreement is excellent. Furthermore, Lovric’s suggestion mentioned above can easily be explained from eqn. (32) which shows that imax/id(t’) dep en d s only on the t/t’ ratio and not on the individual values of t and t’. In addition, this authors also suggested that for the data shown in Table 2 the relationship between imax/id(t ‘) and t/t’ seems to be linear, although this relationship is not held as t/t’ increases. Equation (32) allows us also to explain this observation since according to this equation the dependence of i,,/ia( t’) on t/t’ is not linear (although for the data given in Table 2 an approximately linear relationship is obtained), and the larger deviations should be found as t/t’ increases. The data given in Table 1, which shows values of imax/id( t’) obtained f or a wider interval of t/t '-values than those given in Table 2, confirm this fact. Finally, the NPP polarograms reported in Fig. 1 in Lovric’s paper [5] for different values of u are also in excellent agreement with the I-E curves obtained from eqn. (13) using the values of (I, KA, t and t’ given in that figure. (B) RPP Figure 6 shows RPP polarograms computed from eqn. (23) for different values of Di”/K, when only the reactant is adsorbed. Curve (a) corresponds to a process in
283
-0.99
- 0.3
-0.1 (E2 - Eel/V
0.1
Fig. 6. RPP: Dependence of the I-E curves on DA/‘/K,+, computed from eqn. (23) for f = 1 s, I’ = 0.05 s, K, = 0. (Di’2/KA)/s-“2: (a) cc; @) 1; (c) 0.1; (d) 0.01; (e) 0.001. Other conditions as in Fig. 1.
the absence of adsorption and has been included for comparison. Note that the I-E curves are similar to those obtained without adsorption. In addition, for E, + cc eqn. (23) gives i,,,,,,/i,(t’) = m - 1, which coincides with the expression for the limiting current for a diffusion process. Thus, only the shift of the waves towards more negative potentials reveals that an adsorption process is involved. This situation is similar to that observed in NPP when only product adsorption occurs, although in this case the waves shift towards more positive, instead of negative, potentials. This behavior is in agreement with the RPP polarograms obtained by Flanagan et al. using digital simulation techniques (see Fig. 4 in ref. 4). The characteristics of the I-E curves when only the product is adsorbed are shown in Fig. 7 where RPP polarograms have been computed from eqn. (24) for different values of D’,/*/K,. As in Fig. 6, curve (a) corresponds to a process without adsorption. From Fig. 7 (which is equivalent to Fig. 1 in NPP when only the reactant is adsorbed) the following conclusions are obtained: (a) The limiting currents are also depressed with respect to those obtained in the absence of adsorption. However, we found that in NPP the i,im,NPP/id(t’)-values approach a limiting value (= m) for strong adsorption (see Fig. 1 and eqn. 30).
284
_ ; I= . L ._!
-0.6
-1.0
f
e -1.8 - 0.1
0.0
0.1
0.2
g a3
0.4
(E2- E’)IV Fig. 7. RPP: I-E curves computed from eqn. (24) for I = 1 s, t’ = 0.03 s, K,+ = 0. ( D~‘2/K,)/s-‘/2: co; (b) 10; (c) 1; (d) 0.1; (e) 0.01; (f) 0.001; (g) 0.0001. Other conditions as in Fig. 1.
(a)
In contrast, in RPP the value of i,,,,,,/i,(t’) for K, s-=.1 approaches zero (see curves e-g). (b) The waves exhibit a maximum if the K,-values are sufficiently large (curves c-g), which shifts continuously towards more positive potentials as K, increases (stronger adsorption). In contrast, the corresponding current, i,,,, approaches a limiting value for K, s=- 1 (curves e-g). Thus, this limit in Fig. 7 for imax,.&id(t’) is 1.574. The behavior of the limiting current is explained from eqn. (24) which for E, + 00 (i.e., y -+ co, F(y) --j 1) gives
=F(X)(I;I)‘/2_-H(X)
*
(37)
d
According to this equation we obtain ‘lim,RPP
i,j(t’)
( t r/t)1’2
_ i
0
-
1
for K, = 0 (no adsorption) for K, z+ 1 (strong adsorption)
(38)
so that the RPP limiting current approaches that obtained for a diffusion process for weak adsorption (curve b in Fig. 7), or to zero for strong adsorption (curves e-g). The behavior of the maxima can be explained in an analogous way to that in NPP. Thus, and taking into account that Em, shifts towards more positive potentials as K, increases (i.e., the corresponding E,-values become larger), one finds that these two effects compensate each other in eqn. (24a), so that a constant
285 TABLE 3 RPP: Values of ‘;;~Ld’~a(,‘r’:,‘s!$a$ (t’ = constant). 9 from eqns. (39) and (40)
rmx A
-E,,,)
as a function of I’ (t = constant) and f
7 y =l. (I) Computed from eqn. (24). (II) Computed
1=ls
t ‘/s
L,/id(~‘)
0.001 0.005 0.01 0.02 0.05 0.1 0.2 0.3 0.5
-
(1)
8.745 3.903 2.753 1.937 1.206 0.830 0.553 0.423 0.277
i,,/i.,(t’)
(11)
- 8.755 - 3.908 - 2.156 - 1.939 - 1.207 -0.831 - 0.554 - 0.423 - 0.277
A.%, WV
AK,,,, (WV
0.320 0.300 0.291 0.282 0.270 0.261 0.252 0.247 0.241
0.320 0.299 0.290 0.281 0.270 0.261 0.252 0.247 0.240
r’=0.05s t/s
kw./kJ(t’)
0.1 0.2 0.3 0.5 1.0 2.0 5.0 10.0 20.0
-
value for y,, E max=
El,2 -
i max,RPP_
k(t’)
(I)
0.277 0.479 0.619 0.830 1.206 1.728 2.753 3.903 5.525
k,Jid(~‘) - 0.277 - 0.480 -0.619 -0.831 - 1.207 - 1.730 - 2.756 - 3.908 - 5.553
(II)
A%,, 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270
(1)/V
AK,,,
(WV
0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270
= 0.83 is obtained. Under these conditions we find (see Appendix) $
-0.277
In
1.20( Dat’)1’2 KB l/2
(40)
According to these equations E,,,, shifts continuously towards more positive potentials as K, increases (59/n mV if K, changes by an order of magnitude), while i,,, becomes independent of the K,-values for K, B 1. The I-E curves for strong adsorption in Fig. 7 are in agreement with these predictions. Equations (39) and (40) also show the dependence of E,, and i,,, on t’ and t for a given value of K,. This is illustrated in Table 3 where the values of E,,, and inI, obtained from eqn. (24) for D’,/‘/K, = lop4 sell2 have been compared with the approximate ones computed from eqns. (39) and (40). As in NPP (see Table l), imax.id(t’) for t = constant increases as t’ becomes shorter. However, contrary to the behavior of the maxima in NPP, we found that in RPP, maxima will be observed
286
even for large values of t ‘. Moreover, E,,,,, shifts in the opposite way to that in NPP as t ’ changes (i.e., towards more negative potentials as t ’ increases). If t ’ = constant Table 3 shows that E,, remains constant as t changes, so that AE, computed from eqn. (39) (A&, = 0.270 V) coincides with that obtained from eqn. (24). Note that, contrary to NPP, E,,,, does not vary even for small values of t (compare Tables 1 and 3). We wish also to point out that in both cases (t = constant and t’ = constant), the E,,and i,,- values computed from eqns. (39) and (40) are in excellent agreement with those obtained from eqn. (24). Equation (40) shows that, as in NPP (see eqn. 32), the imax/id( t’)-values do not depend on the individual values of t and t’ (see Table 3), but on the t/t’ ratio. An interesting consequence of this fact is that for strong adsorption and for the typical values of t and t’ used in RPP (i.e., t’/t e l), it follows that imax.RPPa l/t’. An
0.2T
a I -0.1
, 0.3
0.1 (EZ-
E%’
-1.81 -0.2
0.0
0.2
(E2- ET/V
Fig. 8. RPP: Dependence of the I-E curves on Z’ computed from eqn. (24) for t =l s, Dyz/KB = 0.2 s -‘/’ 1 KA = 0. r’//s: (a) 0.005; (b) 0.01; (c) 0.02; (d) 0.05; (e) 0.1; (f) 0.2; (g) 0.5. Other conditions as in Fig. 1. Fig. 9. RPP: Dependence of the I-E curves on o computed from eqn. (20) for 1’ = 0.02 s, t = 1 S, Dy2/KB = 0.2 S-I/~. O: (a) co, (b) 100; (c) 50; (d) 25; (e) 10; (f) 1; (g) 0.1; (h) 0.01. Other conditions as in Fig. 1.
I
287 TABLE 4 RPP: Dependence of i,,_/id(t’), AE,,,, ( = E,, tion. D’/’ = 0 .2 s-‘12, t’ = 0.05 s, KA = 0 B/B K
- E,,,)
and ilim/id(t’)
on t for moderate adsorp-
t/s
Ldidt’)
A-%,/V
jlim/k(t’)
0.2 0.3 0.5 1.0 2.0 5.0 10.0 20.0
- 0.400
0.080 0.079 0.078 0.078 0.078 0.078 0.078 0.078
- 0.022 - 0.042 - 0.074 -0.127 -0.194 - 0.305 - 0.403 - 0.505
-
0.532 0.722 1.034 1.418 2.041 2.584 3.157
analogous conclusion is also valid for NPP (see eqn. 32). Note that, in contrast, the diffusion current in the absence of adsorption is a l/c. Hence, strong adsorption provides a procedure for obtaining enhanced currents which could be applied for analytical purposes. Although eqns. (39) and (40) have been obtained for processes with strong adsorption, its predictions are qualitatively valid for weaker adsorption also. This is shown in Fig. 8 where RP polarograms for moderate adsorption have been obtained from eqn. (24) for different values of t’. As described for strong adsorption (see Table 3), Em, shifts towards more positive potentials and i,,, increases as t’ becomes smaller. Note that for large values of t’ (curve g) the limiting current becomes positive. This fact is discussed more extensively below. Also, the dependence of the Z-E curves on t for moderate adsorption and t’ = constant behaves as for strong adsorption, i.e., Em, remains practically constant as t changes and i,, increases with t (see Table 4). Table 4 also illustrates the dependence of the RPP limiting current on t. In this case, we found that the lli,,, a,,-values increase towards more negative values as t becomes larger, so that no minimum is observed. This behavior contrasts with that obtained in NPP where i,im,NPPpasses through a minimum as t increases (see Fig. 3). The characteristics of the RPP polarograms when both reactant and product are adsorbed are shown in Fig. 9. In this figure the Z-E curves have been obtained from eqn. (20) for D’,/*/K, = 0.2 s-l/* and different values of u. The i,,,-values decrease as u becomes lower, so that the waves will not exhibit maxima for small values of (I. This behavior is also in agreement with that obtained by Flanagan et al. using numerical methods (see Fig. 4 in ref. 4). However, the limiting current does not depend on the values of u. This is in agreement with eqn. (20) which for E, + 00 gives eqn. (37) also. In this equation the argument, x, in the F(x) and H(X) functions does not depend on u, so that only the adsorption coefficient of the product, K,, is involved. This situation is similar to that described above in NPP, although in this case the information that can be obtained from the RPP limiting
288
current is related to the product, and not the reactant. Thus, from eqn. (37) we find I RPP
(41)
=
Equation (41) is equivalent to eqn. (33) in NPP, and by following an analogous procedure to that described above we find that if t = constant, a plot of I,,, vs. fl will be linear with intercept a and slope b defined by Q= -F(x)
b=ff(x)
i
(42)
From eqns. (4) (5) (41) and (42) we obtain K, = e(~D,t)“~
(43)
and therefore, K, can be determined from the slope and intercept of the plot of I RPP vs. fi. For processes with weak adsorption we have that F(x) and H(X) --, 1, so that a -+ - 1, b -+ 1. In contrast, for strong adsorption we obtain a + 0, b + 0 so that the corresponding plot approaches the abscissa axis. The upper and lower limits of the K,-values that can be obtained from eqn. (43) are determined as previously, and coincide approximately with the interval of K,-values given for NPP. An example of these kinds of plots for weak, moderate and strong adsorption is shown in Fig. 10. Note that for values of K, B- 1 these plots show only that the adsorption is very strong, so that this situation is analogous to that described previously in NPP. In these cases we could also use the dependence of E,, on K, (eqn. 39) to determine the adsorption coefficient of the product if the +,-value is known. (C) Double potential step chronoamperometry Although eqns. (1) and (2) allow us to obtain the double chronoamperometric response for any value of E, and E2, for simplicity the characteristics of this response are shown for E, -+ - 00 and E, + co (see eqns. 25-27). Thus, in Fig. 11 we have plotted i(t)/i(t,) vs. t computed from eqns. (27) for t, = 1 s and different values of Dk’*/K,. The corresponding curve in the absence of adsorption has also been included. Note that under these conditions (E, + - 00) there are no differences between the different responses for t < t,. However, for t > t, the responses are different enough, so that depending on the K,-values the current may become more positive as t increases (see curves c and d in Fig. 11). This is shown clearly in Fig. 12 where we have represented the current for the second step, i.e. i( t > tl)/i( t,) vs. t’, for t, = 1 s. For weak adsorption the i(t > t,)-values are negative, although they are larger than those obtained in the absence of adsorption (curve b in Fig. 12). As adsorption becomes stronger the values of i(t > t,) become more positive as t increases and they pass through an ill-defined maximum (curves c and d). After the
289
Fig. 10. I,, vs. F I I 0.01; (c) 0.1.
computed from eqn. (41) for I = 1 s, Da = lo-’
cm* s-l.
Ka/cm:
(a) 0.001; (b)
Fig. 11. Double potential step chronoamperometry: dependence of i(t)/i( I,) on t computed from eqns. (27) for tI = 1 s and different values of (D~‘z/K,)/s-‘/2: (a) co; (b) 5; (c) 1; (d) 0.1.
maximum the current approaches zero slowly, although from positive values (note that in the absence of adsorption the current also approaches zero, but from negative values). For very strong adsorption the current approaches zero rapidly for all t ‘-values (curve e). We wish to point out that, in some cases, the current becomes positive after the second pulse is applied; this is due to the fact that this pulse component is the result of two contributions, one cathodic, i.e. the dc component from the first pulse which acts at all times (even after the second pulse is applied [19]), and the other one anodic (see eqns. 26 and 27). In the absence of adsorption the anodic component is always larger than the cathodic one (although their values approach one another as t increases), and therefore, the net current also approaches zero from negative values. However, when product adsorption is involved, the anodic component may become smaller than the cathodic one as t changes, so that the net current will be positive. In any case, eqn. (27) shows that at long times i(t > ti) approaches zero for all values of K,. On the other hand, the value of the ratio - i(2t,)/i(t,) can be used to analyze the conditions of a stable system [20]. This ratio in the absence of adsorption is
290 -0.6-
0.11 0
2
4
6
8
I 10
Fig. 12. Double potential step chronoamperometry: dependence of i( t > t,)/i(t,) on 1’ computed from (a) co; (b) 5; (c) 1; (d) 0.4; (e) 0.03. eqns. (27) for 1, =l s and different values of (D~‘*/K,)/s-‘I*:
independent of c, and is equal to 0.293 [20], so that deviations of these current ratios from this value have been ascribed to kinetic complications in the electrode reaction [20]. As expected, when product adsorption occurs, the value of this ratio TABLE 5 Double potential step chronoamperometry: Dependence of - i(2t,)/i(t,) = I(tl) computed from eon. + co) this ratio is equal to 0.293 and is (27) on z, and Dkfl/K,. In the absence of adsorption (Dy2/KB independent of t, (D;‘*/K,)/s-“’
I(O.5)
Co 10.0 5.0 3.0 1.0 0.5 0.3 0.1 0.05 0.01
0.293 0.240 0.1% 0.148 0.037 - 0.0015 -0.011 - 0.0088 - 0.0053 - 0.0012
I(l)
-
0.293 0.255 0.221 0.182 0.068 0.013 0.0059 0.011 0.0069 0.0017
f(2)
I(5)
I(lO)
0.293 0.266 0.240 0.210 0.104 0.037 0.0052 -0.012 - 0.0088 - 0.0023
0.293 0.275 0.259 0.238 0.153 0.079 0.032 - 0.010 -0.011 - 0.0036
0.293 0.280 0.268 0.253 0.186 0.117 0.063 -0.0046 - 0.012 - 0.0048
291
also differs from the value 0.293 corresponding to a process without adsorption. This is shown in Table 5 where the values of -i(2t,)/i(t,) have been computed from eqns. (27) for different values of DL”/K, and t,. Note that the value of this ratio depends strongly on the values of t, and K,, so that - i(2t,)/i( tr) may even become negative. These results are explained from eqns. (27) which for t = 2t, (i.e., t’ = zr) give __ 4%)
+qX)_
x=
y;
i(b)
@Dgtl)‘/’
(4)
B
According to eqn. (44) it follows that - i(2t,)/i( tl) + 0 for x + 0, passes through a minimum = -0.01189 for x = 0.22, and approaches 0.293 for x -+ cc. The data in Table 5 are in agreement with this behavior. APPENDIX
(i) Equation (31) is derived as follows: For strong adsorption and for the E-values at which the maximum, E,,, appears, we have ye, +z 1 so that the corresponding argument, y,,, in eqn. (17) becomes Ymax
=
(DAf’)1’2/yG,t
From eqns. (Al) and (9) and taking into account that y,, E max= E1,2 + z
In
l.ll(
DAt’)l” KA
641) = 0.9 we find
(AZ)
where l/2
In
E l,2=E“-z
(A3)
(ii) In turn, eqn. (32) is derived as follows: For strong adsorption the arguments x and x1 in eqn. (16) are +z 1 (see eqn. 14), although in this case and due to the fact that yc2 < 1 we find from eqns. (5) and (14)
H(x,) ---=
2( DAtl)1’2
(A4)
r’/2yc2KA
-f-f2
Inserting eqn. (Al) in eqn. (A4)
H(x,)
=-
2Y,,
11
Tl/2
YE2
( 7
1’2 )
Also, from eqns. (17), (14) and (Al) we obtain
‘,x=yc,
1
(D,t)"' K
A
646)
292
Finally, inserting eqns. (A5) and (A6) in eqn. (16), taking into account that F(x) and H(X) + 0 under these conditions, and that F( Y,,,) = 0.7283 and H(Y,,,,,) = 0.5435, we obtain eqn. (32). (iii) Equations (39) and (40) are derived in an analogous way. Thus, and taking into account that in RPP yeI B 1 at the potentials at which the maximum is obtained, eqn. (24a) becomes Ymax= yQ( D,t’)1’2/K*
(A7)
Proceeding as previously and recalling that in this case y,, = 0.83 (see text), we obtain eqn. (39). In turn, eqn. (40) is derived by considering that under these conditions l/2
2Y YE2WXJ
=
-$
3 (
and that F(y,,)
1
= 0.7046. Inserting these relationships in eqn. (24), we obtain eqn.
(40). ACKNOWLEDGEMENTS
A grateful acknowledgement is made to the Fulbright/MEC Commision for providing a fellowship to J.G. for this collaborative work, and to the U.S. Department of Energy for supporting this work (Grant DE-FG22-86PC90519). J.G. also appreciates the financial support of the Comisi6n Asesora de Investigacibn Cientifica y TCcnica (project No. 763/84). REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
H.P. van Leeuwen, J. Electroanal. Chem., 133 (1982) 201. H.P. van Leeuwen, J. ElectroanaI. Chem., 162 (1984) 67. G. Wolff and H.W. Numberg, Z. Anal. Chem., 216 (1966) 169. J.B. FIanagan, K. Takahashi and F.C. Anson, J. Electroanal. Chem., 257 (1977) 257. M. LovriC, J. ElectroanaI. Chem., 170 (1984) 143. M. Lovrif, J. ElectroanaI. Chem., 181 (1984) 35. S. Komorsky-Lovrit and M. Lovrit, J. Electroanal. Chem., 190 (1985) 1. M. Lovrif, J. Electroanal. Chem., 197 (1986) 49. M. LovriQ J. Electroanal. Chem., 223 (1987) 271. F. Mas, J. Puy, F. Sam and J. Virgili, J. Electroanal. Chem., 183 (1985) 73. H.P. van Leeuwen, M. Sluyters-Rehbach and K. Holub, J. Electroanal. Chem., 135 (1982) 13. K. Holub and H.P. van Leeuwen, J. ElectroanaI. Chem., 162 (1984) 55. K. Holub and H.P. van Leeuwen, J. Electroanal. Chem., 191 (1985) 281. H.P. van Leeuwen, M. Sluyters-Rehbach and K. Holub, J. ElectroanaI. Chem., 191 (1985) 293. J. Galvez and S.-M. Park, J. ElectroanaI. Chem., 263 (1989) 257. R. Guide& J. Electroanal. Chem., 18 (1968) 5. J. Osteryoung and E. Kirowa-Eisner, Anal. Chem., 52 (1980) 62. T. Kambara, Bull. Chem. Sot. Jpn., 27 (1974) 523. I. Ruzif and M. Sluyters-Rehbach, Anal. Chim. Acta, 99 (1978) 177. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, pp. 181, 182.
J. Electroanal. Chem., 263 (1989) 269-292 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Study of reactant and product adsorption in double potential step and related pulse techniques Part II. Normal puke polarography (NPP), reverse pulse polarography (RPP) and double potential step chronoamperometry Jesus Gdvez Laboratory
of Physical Chemistry, Faculty of Science, 3ooo1 Murcia (Spain)
Su-Moon Park Department
of Chemistry,
l7ze University of New Mexico, Albuquerque, NM 87131 (U.S.A.)
(Received 7 November 1988; in revised form 14 December 1988)
ABSTRACT The analytical solutions for normal pulse polarography (NPP), reverse pulse polaro$raphy (RPP), and double potential step chronoamperometry for a reversible charge transfer reaction when reactant and product are adsorbed on a plane electrode following a linear isotherm have been derived. The current-potential curves in NPP and in RPP obtained from this solution are in excellent agreement with those previously reported in the literature obtained using numerical methods. Also, our solution for the limiting current becomes equivalent to that derived by Van Leeuwen et al. (J. Electroanal. Chem., 135 (1982) 13) when the condition r’//t Q: (t’ being the pulse duration) is applied. The characteristics of the double potential step chronoamperometry response, for which no solution (analytical or numerical) had been reported previously, have also been shown. A simple method to determine the adsorption coefficients of reactant and product based on the dependence of the limiting currents in NPP and in RPP on the pulse time is discussed.
INTRODUCTION
Adsorption effects in pulse polarography are especially important [1,2], and were first indicated by Wolff and Numberg [3] in their measurements of maxima in the NPP polarograms for nitroanilines. Regarding the theory for the NPP response, this has already been treated by Flanagan et al. [4], LovriC [5-91, and Mas et al. [lo] 0022-0728/89/%03.50
0 1989 Elsevier Sequoia S.A.
270
using numerical methods. An analytical equation for the NPP limiting current was derived by Van Leeuwen et al. [ll] when the condition t '/t -=z 1 (t ’ being the pulse duration) is applied. Van Leeuwen and co-workers have also extended their approach to other types of electrodes and isotherms [12-141, although in all these cases only the limiting current was considered. All these treatments have shown that the corresponding theory is rather complex and so, no analytical solution for the current-potential (I-E) curves in NPP, even when simple conditions (plane electrode model and linear adsorption isotherm) are adopted, had been reported previously. In any case, and in spite of this shortcoming, the level of understanding of adsorption effects in NPP is quite satisfactory. In contrast, the situation in RPP or in double potential step chronoamperometry complicated with reactant and product adsorption is much less clear since in RPP only I-E curves obtained by Flanagan et al. [4] using digital simulation techniques have been reported, while in double step chronoamperometry the corresponding response, to our knowledge, has not been described yet in the literature. In Part I [15], a general analytical solution for double potential step and related pulse techniques was derived for a reversible charge transfer reaction when reactant and product are adsorbed on a plane electrode following a linear isotherm. In the present paper this equation has been applied to obtain the corresponding responses for NPP, RPP and double potential step chronoamperometry by introducing in the general solution the experimental conditions that characterize these techniques. This has allowed us to analyze and compare our predictions for the pulse responses with those reported previously. In addition, the characteristics of the new response obtained in double potential step chronoamperometry for this kind of electrode process have also been shown. This has revealed some interesting features and so, for example, a simple procedure to determine the adsorption coefficient of the reactant from measurements of the limiting current in NPP has been devised, while the adsorption coefficient of the product can also be obtained from an analogous procedure based on measurements of the limiting current in RPP. As mentioned in Part I, [15] we have adopted the plane electrode model, and this restricts the application of the present theory to types of electrodes other than planar ones. However, this simplified electrode model has the advantage that it facilitates the analysis of the corresponding responses, especially in double potential step chronoamperometry whose solutions had not been found yet.
THEORY
The general solution for double potential step chronoamperometry for a reversible charge transfer reaction when reactant and product are adsorbed on an electrode following a linear adsorption isotherm has been derived in Part I [15]. According to this solution, if t, and t’ are the pulse duration for the first (E = E,) and second (E = E2) steps, respectively, the equations for the current are (a complete list of
271
notations/definitions i(r
_
id(l)
is given in ref. 15):
e t(1 -Ye)% F(X) u + CI (0 + 40 + 7%)
(1)
1’2_
(Ye-NJ+c,)c, (e + c2)2(1 + YE*)
tc2-F1)
_
Cl-
F(Y))WX,)
qX)
(0 + 40 + YQ) _
(Ya---l)c, b+f2)0
fqX)
-
0
-
P(X))Bl
d/=B2x
+Yc2)
(2)
where id(f) and id(t’) are the diffusion currents at t = t and t = r’, i.e., id(t)
= nFA(DA/7rt)1’2cA*;
id(t’)/id(t)
= (t/t’)“’
(3)
and
F(x)=
exp(X2) c (-1)m+‘p~“~p~Xm~~1/2X m. erfc(x)
!
(4)
m=l
H(x)=
c
W>““Po...
Pm-lXmG
1
_
M. 1
m=l
F(X) p
X
(5)
2/J;; PmPm+1 = 2(m + 1)
(6)
Y = (4/b
(7)
po =
)1’2
(8)
B = KB/KA (i=1,2)
E~=~~~[(~F/RT)(E,-E~)] Bi =
1 +yc,
(i=l,2)
(9) (10)
Y(“+ri)
In these equations K, is the adsorption coefficient of the species a (a = A, B), and the arguments x, x1 and y involved in the F(x) and H(z) functions are dimensionless parameters defined by x = B,( DAt)*“/KA; y = B2( DJ’)“~/K~
x1 = B,( DAtl)“2,‘KA
(11) (12)
As has already been mentioned [15], because eqns. (1) and (2) involve only the well-known F(x) function, current-potential curves are readily obtained from these equations. Also, and since eqn. (2) is quite general, this allows us to obtain the
272
corresponding responses for pulse techniques by introducing in this expression the conditions for the different modes of pulse polarography. Thus, we obtain: (A) Normal pulse polarography
(NPP)
In NPP the potential of the first step, E,, is chosen so that no current flows at this potential. Inserting this condition (E, + cc, E, + 00) in eqn. (2) we find lNPP
_
(v--)r* (0
+ E*)(f
H(X) + YE*)
-
(la’/*B,x -
F(X))
(13)
where the argument x is now defined by (see eqns. 10 and 11 with B, + 1) x =
(D*ty*/K*
04)
and x, is defined as x by making t = t,. If no adsorption process occurs, i.e. KA = K, = 0, it follows that x and y --$ co so that F(x) = F(y) = H(X) = 1, and eqn. (13) is simplified to
(1%
‘NPP
which is the expression for the NPP current under these conditions. We wish to point out that eqn. (13) is the current response when, for 0 < t < t,, only the adsorption-controlled diffusion of the reactant occurs (see Part I). In fact, eqn. (13) could also be obtained using these particular conditions and proceeding as in Part I, although the present treatment is more general. In any case, and because eqn. (13) is valid when reactant and product are adsorbed, it provides the corresponding solutions when only adsorption of one of these species takes place. Thus, if u = 0 (i.e., K, = 0 so that only reactant is adsorbed) eqn. (13) becomes -=lNPP b(t’) +p
1 1
+yc,
H(X)
-
1 -F(X) d2B,,x
F(Y)
(16)
where y = B,( DAt’)“*/KA;
B, = (1 + yc2)/yc2
07)
If K* = 0 it follows that u + cc and only adsorption of the product is involved.
273
Under these conditions we have x --, 00 (see eqn. 14) so that F(x) Hence, eqn. (13) is simplified to lNPP
and H(X) -+ 1.
&F(Y)
-=I-
idW
(18)
where the argument y in F(y)
is given by (see eqn. 12)
y = (1 + TQ)( D*t1)1’2/K*
(19) Note that when only the product of the redox reaction is adsorbed there are no differences between the dc polarographic and NPP responses, so that eqn. (18) is identical to that obtained by Guidelli [16] in dc polarography for this kind of electrode process.
(B) Reverse pulse polarography
(RPP)
In RPP [17], the initial potential El is set on the cathodic diffusion current plateau and the potential E2 is scanned in the positive direction. Hence, the condition E, -+ - 00 (tl + 0) is met, and eqn. (2) is simplified to -=1RPP id(t’)
1 + yr,F(x) 1
t’
“* _ (ya-
(1 -F(y))H(x,)
(t 1
+F2
(0 (ua
-
+f(x)
l)ar,
-
+
(2j2
l)E2
1 -F(X)
2
H(X) (0
+
where the argument x in F(x) condition c, + 0)
c2)O
+
YC2)
-
~“~B~yux
F(Y)
(20)
and H(X) is given by (see eqns. 10 and 11 with the
x = ( DBt)1’2/K,
(21)
As previously, when no adsorption is involved we find from eqn. (20) i ““=“““(~)‘“c~(~-~~)
(22)
which is the corresponding expression for the RPP response for a reversible charge transfer reaction. When only one species is adsorbed, eqn. (20) can be simplified. Thus, for K, = u = 0 eqns. (21), (4) and (5) give x + co, F(x) = H(x) + 1, so that we find l/2
+
~-F(Y) 1 +YE2
1
(23)
In this expression the argument y in F(y) is also defined by eqn. (17). In turn, when only the product is adsorbed we have KA = 0, u --* cc and eqn. (20) becomes -=‘RPP 4l(t’)
1 +yc,F(x) 1
+YE2
I’ 1’2 (i)
-
,2(
(I-
F(y))H(xd
+ “‘1”z;:f’)
(24)
274
In this expression the argument x is also given by eqn. (21), while the argument y is defined by eqn. (19) if, in this expression, we substitute DA and KA by D, and K,, i.e., y = (1 + y~)( DBr’)“*/K,
(244
(C) Double potential step chronoamperometry The dependence of i(t) on I is obtained from eqn. (1) for z d t, and from eqn. (2) for t > I,. These equations allow us to compute the corresponding responses for any value of E, and E2 and so, for example, in the special case where the potential is stepped at E, + - rx and E, --, 00, eqns. (1) and (2) become i(tg
tl) = id=(t) =nFA(D,/at)*‘*c,*
i(t>t,)=i,(r)-nFA(D,/~f’)1’2c,*((l-F(~))(t’/t)1’2+H(~))
(25) (26)
or after normalization by the current at t = t, i(t
=-
1, ‘/* ( t i
i(t > tl) i(tl) =(~)“2-(~)1’2~(1-~~x,,($)1’z+“(x))
(27)
In these equations the argument x is given by eqn. (21) (see eqns. 10 and 11 with pi + 0), so that the response in this case depends only on the product adsorption. If no adsorption occurs it follows that x -+ cc and Q’(x) = H(x) = 1. Inserting these conditions in eqn. (26) we find i(t>
tl) = nFA (~)“2C~(++)
which is also an expression identical to the equation first derived by Kambara [18] in double potential step chronoamperometry under these conditions. RESULTS
AND DISCUSSION
(A) NPP In the first part of this section we shall discuss the characteristics of the NPP response. A detailed comparison with previous results from the literature is shown later. Figure 1 shows I-E curves when only the reactant is adsorbed ((I = 0), computed from eqn. (16) for t’ = 0.03 s, t = 1 s and different values of DA/*/K,. The corresponding curve when no adsorption is involved is also included for comparison (curve a). From Fig. 1 the following conclusions are drawn:
215
0.l
0.0
- 0.1
-0.2
-0.3
- 0.4
(E2- EW V Pig. 1. NPP: Dependence of the I-E n=1,y=1,t-ls,t’=0.03s,a=0.(D~‘2/KA)/s-1’2: (g) 0.001.
curves on D112 A / K A computed from eqn. (16) for T= 298 K, (a) co; (b) 10; (c) 1; (d) 0.5; (e) 0.1; (f) 0.01;
(a) As has been recognized [4,5,11], the limiting currents are depressed with respect to those obtained in the absence of adsorption. Note that although the i,i,-values decrease as KA increases they approach a limiting value for KA z-- 1 (see curves f and g, very strong adsorption). Thus, the i,im,NPP/id(f’)-value in Fig. 1 is 0.173 (= m) (see below). (b) The curves exhibit a maximum if the K,-values are sufficiently large (curves c-g), and as has also been reported [S], the maxima shift continuously towards more negative potentials as adsorption is stronger. However, we have found that the corresponding current, i,,, approaches a limiting value for KA z-. 1 (see curves f and g). Thus, this knit for imax,NPP/id(t’) in Fig. 1 is 1.664. The behavior of the limiting current can easily be explained if we consider that for E2 -+ - co (i.e., e2 + 0) we have y + cc, F(y) -+ 1, and eqn. (16) gives
y$y = (1 -
F(X))(
;)r’2
+ H(X)
(29)
According to this equation and recalling that F(0) = H(0) = 0 and that F(cc) = H(oo) = 1, we find ‘lim,NPP
id(t’)
_
1
for KA = 0 (no adsorption)
( c/ty2
for KA B 1 (strong adsorption)
(30)
which is in agreement with the results obtained in Fig. 1 and in the literature [5,11]. In turn, the behavior of the maxima can also be explained if we take into account that as KA increases, the potentials at which the maxima appear (Em,) shift
276
towards more negative values, so that the corresponding values of e2 become smaller. When one obtains the value of the argument y =y,,,, at E = E_x (see eqn. 17) for values of KA x==1, it follows that the increase in KA compensates the decrease in Ed, so that a constant value for ymax= 0.9 is found. In the Appendix it is shown that under these conditions we obtain
E max= E,,, + $
In
l.ll(
D*t’)1’2
(31)
KA l/2
(32) As shown in Fig. 1, eqns. (31) and (32) predict that E,,,,, shifts continuously towards more negative potentials as KA increases and that i,, becomes independent of the K,-values for KA B 1. These equations are valid only for the case of very strong adsorption and so, for example, for the data corresponding to curve (g) in Fig. 1, we find AE,,(= E,, - El,2) = -0.219 V and i,,,Jid(t’) = 1.663, which coincide with the exact values obtained from eqn. (16). Larger deviations are found as adsorption becomes weaker and so, for curve (d) we find from eqn. (31) obtained from eqn. 32 also equals 1.663 AE,,, = -0.060 V (the i_/i,(t’)-value since it does not depend on KA). The exact values are AhE,,,, = - 0.064 V and imax/id(t’) = 1.286. Equations (31) and (32) also allow us to predict the dependence of E,, and i,,, on t’ and t for a given value of KA. This dependence is shown in Table 1 where the exact values of E,, and i,,, obtained from eqn. (16) for DA/‘/K, = 10e4 sell2 are compared with the approximate ones computed from eqns. (31) and (32). For t = constant the agreement for different values of t’ is excellent and, as predicted by eqns. (31) and (32), E,_ shifts towards positive potentials while i,_/ia(t’) decreases as t’ becomes larger. We wish also to point out that for large values of t’ the maximum vanishes and eqns. (31) and (32) are no longer valid. In turn, the dependence of E,_ on t for t’ = constant (Table 1) shows that according to eqn. remains practically constant for all values of t, although for small values (31)1 E,, of t( 5 0.3 s) the E,,,,- values are more negative than those obtained with eqn. (31). Regarding the i,,- values we find that as previously for t = constant, a good agreement between eqns. (16) and (32) is obtained. We wish to emphasize that although eqns. (31) and (32) are quantitatively valid only for processes with strong adsorption, its predictions can also be applied qualitatively to processes with weaker adsorption. This is illustrated in Fig. 2 where Z-E curves for a process involving moderate adsorption (DA/‘/K, = 0.2 s-‘12) have been computed from eqn. (16) for different values of t’. As previously found for strong adsorption (see Table l), E,,,, shift towards more negative potentials and increases as t’ becomes smaller. This kind of behavior for systems involving i,, reactant adsorption has already been indicated by Van Leeuwen [2] in the NPP polarograms for the reduction of methylene blue.
211 TABLE NPP:
I Values
(1’ = constant).
of
i,/id(t’) 0:/*/K,
and
A/&,,
= 10K4 S-I/~,
(=
E,,
- E,,,)
as a function
(I = 0, y = 1. (I): computed
of
t’
(t = constant)
from eqn. (16). (II): computed
and
t
from
eqns. (31) and (32) r=1s C//s
LJM~‘)
(9
0.001
8.761
8.741
- 0.320
- 0.322
0.005
3.940
3.932
-0.300
- 0.301
0.01
2.805
2.800
- 0.291
- 0.292
0.02
2.011
2.009
- 0.283
- 0.283
0.05
1.324
1.324
- 0.272
- 0.272
0.1
1.001
1.000
- 0.265
- 0.263
- 0.272
1’ = 0.05 s
t/s
i,,/ia(O
(1)
irnpx/id(O (11)
0.2
0.768
0.750
- 0.281
0.3
0.846
0.839
- 0.277
- 0.272
0.5
1.001
1.000
- 0.274
- 0.272
1.0
1.324
1.324
- 0.272
- 0.272
2.0
1.811
1.809
- 0.271
- 0,272
5.0
2.805
2.800
- 0.271
- 0.272
10.0
3.939
3.932
- 0.270
- 0.272
20.0
5.551
5.540
- 0.270
- 0.272
Regarding the dependence of the I-E curves on t for t’ = constant, this is shown in Fig. 3 for the same value of Di”/K, used in Fig. 2. As described for strong adsorption in Table 1, i,,, increases with t while, in contrast, E,, practically does not vary, except for small values of 2 (5 0.3 s). Note, however, that under these conditions (t < 0.3 s) the maximum vanishes rapidly (see curve (a) computed with t = 0.1 s). These facts could be applied for analytical purposes and so, if we are interested it follows from eqn. (32) that the t/t’ ratio should be chosen as in enhancing i,,, large as possible. In contrast, if adsorption effects need to be minimized this ratio should be decreased. Thus, for example, if Ok/‘/K, = 5 s-l/* (very weak adsorption) no maximum is observed for t’ = 0.05 s and t = 1 s. However, a wave exhibiting a maximum with imax/id(t’) = 1.927 will be obtained if t’ = 0.001 s and t = 10 s. On the other hand, Fig. 3 also illustrates an interesting feature of the dependence of the limiting current on t, namely that the ilim,NPP-valuespass through a minimum values in Fig. 3 as t changes). This as t increases (note the sequence of the i,im,NPPfact is also in agreement with the results obtained by Van Leeuwen et al. [ll] who provided a detailed discussion of it. By considering the expression for the limiting current given above (see eqn. 29), this minimum arises because for very small values of t (i.e., for t + t’) the F(x) and H(X) functions become smaller and
278
-0.1
0.1 (EZ-
- 0.3
Eel/ V
Fig. 2. NPP: Dependence of the I-E curves on. I’ computed from qn. (16) for I = 1 s, Di’2/K,, = 0.2 s-1/z, (I = 0. C//s: (a) 0.005; (b) 0.01; (c) 0.02; (d) 0.05; (e) 0.1; (f) 0.2 (g) 0.5. Other conditions as in Fig. 1.
= Jt’Tii 4 1 if th e adsorption is not very weak and the t’-values are sufficiently low. As t increases, F(x) and H(x) increase too, although the decrease in the contribution (1 - F((x))fi prevails, and the ii,,,,,-values become smaller. In contrast, at long times, the contribution of H(x) prevails and ilim,NPPincreases with t (for r --, cc we have F(x) = H(x) + 1, so that imax,NPP/id(t’) --, 1). The NPP response when both reactant and product are adsorbed is shown in Fig. 4 where the I-E curves have been computed from eqn. (13) for DA/‘/K, = 0.2 s-1’2, t’ = 0.02 s, t = 1 s and different values of cr. The maximum depends strongly on the u-values so that the corresponding i,, -values decrease and shift toward more positive potentials as u increases. For large values of u no maximum is observed. However, and in agreement with the literature data [4,5], the limiting current does not depend on the u-values. This is easily explained from eqn. (13) since when the condition E2 --) - co is introduced in this expression we obtain again eqn. (29) which does not involve u. Hence, no information about the adsorption of the product can be obtained from the NPP limiting current. Although the NPP I-E curves could provide this information, that means that we have to deal with eqn. (13), which is much more complex than the corresponding expression for the i,,NPP/id(t’)
279
I
h
(E2- E”MV Fig. 3. NPP: Dependence of the I-E curves on t computed from eqn. (16) for t’ = 0.05 s, Di”/K, = 0.2 S-I/~, o = 0. t/s: (a) 0.1; (b) 0.3; (c) 0.5; (d) 1; (e) 2; (f) 5; (g) 10; (h) 20. Other conditions as in Fig. 1.
limiting current (eqn. 29). Fortunately, the RPP response (see below) provides an easier procedure to obtain K, from the corresponding RPP limiting current. Hence, we shall limit the discussion to the procedures to determine KA, which are shown next. The expression for the NPP limiting current is relatively simple and this allows us to devise a procedure to obtain the adsorption coefficient of the reactant. Thus, from eqn. (29) we obtain I
‘h-l
NPP-
,NPP
id(t)
l/2
=1-F(x)+
(
;
1
H(X)
(33)
where x = (DAt)‘/2/KA (see eqn. 14). A striking feature of this equation is that because x, and therefore F(x) and H(X), depend on r only, if we perform measurements of INpp by modifying 1’ with will be linear with intercept a the condition t = constant, a plot of I,,, vs. @r and slope b given by a=l-F(X)
b=H(x)
280
E _= 2 ..
0.4
C
1
t
_i
-._
I
0.2
0.0
-0.2
0;'
I
1
(E2- E?i V
2
3’
4
5
(t/t*y?
Fig. 4. NPP: Dependence of the Z-E curves on a computed from eqn. (13) for t’ = 0.02 s, I = 1 s, Dif*/KA = 0.2 s-‘fl. Q: (a) 0; (b) 0.005; (c) 0.02; (d) 0.05; (e) 0.1; (f) 1; (g) IO; (h) 100. Other conditions as in Fig. 1. Fig. S. ZNpp vs. fl 10-Z; (c) 1.
computed
from eqn. (33) for 1=1
s, DA =fOe5
cm2 s-‘.
lu,,/cm:
(a) 10S5; (b}
That these kinds of plots are effectively linear has already been suggested by Van Leeuwen et al. (see Fig. 2 in ref. 11). However, we can go further if we exploit the relationship between the F(x) and H(x) functions, i.e. H(X) = 1 - F(x)/( a’/*~) (see eqns. 4 and 5). Thus, taking into account this relationship and the x-value defined by eqn. (14) we find from eqns. (34)
KA= l-b ( 7rDAty2 l-a
(35)
an expression that allows KA to be determined easily from the slope and intercept of these plots. Note that for processes with strong adsorption F(x) and H(X) + 0, so that b -+ 0 and a --j 1. Under these conditions we obtain a straight line parallel to the abscissa with intercept = 1. In contrast, if the adsorption is very weak we have #Y(x) and H(X) + 1, and therefore, a --, 0, b --* 1. In these cases the corresponding plots
will pass through the origin with slope = 1. The upper and lower limits of the K,-values that can be obtained from eqn. (35) are determined by the fact that these plots show a slope and intercept different from 0 and 1 respectively (very strong adsorption), or from 1 and 0 (very weak adsorption). The slope and intercept of these plots are defined by eqns. (34) from which we can obtain estimative values for this limit. Thus, if we consider that plots for x = 0.05 and x = 5 differ from those obtained for strong and weak adsorption, it follows by supposing DA = 1O-5 cm2 s-l and f = 1 s that K,-values of = 6 X low2 to 6 x lop4 cm could be determined. An example illustrating these considerations is given in Fig. 5 where the corresponding plots have been computed from eqn. (33) for K, = low5 cm (weak adsorption), KA = 10e2 cm (moderate adsorption), and KA = 1 cm (strong adsorption). Finally, we wish to point out that for large values of KA these plots show only that adsorption is very strong, although in these cases we could use the dependence of Enlax on KA (eqn. 31) to determine the adsorption coefficients of the reactant if the E,,,-value is known. Comparison with literature data
As mentioned above, the NPP response complicated with adsorption processes has already been treated in the literature [4-111, so that a comparison with our results can be performed: (a) The dependence of the Z-E curves on KA and u shown in Figs. 1 and 4 is in agreement with the NPP polarograms obtained by Flanagan et al., using digital simulation techniques (see Figs. 3 and 5 in ref. 4). A quantitative comparison is not possible because these authors used a different electrode model. (b) Van Leeuwen et al. [ll] derived an analytical solution of the NPP limiting current using the condition t’/t << 1. It is readily shown that in our notation Van Leeuwen et al’s equation adopts the form +y
= (1 - P(x))(
f)l’l+
H(X*)
d
If we take into account that for t’/t e 1, t = t,, it follows that H(X) = H(xi) so that eqns. (36) and (29) are equivalent. Hence, eqn. (29) is the general solution for the NPP limiting current when there is no restriction regarding the values of t and t’. Also, the agreement between other features of the NPP response derived by Van Leeuwen et al. [2,11], and those obtained in this paper has been mentioned above. (c) The NPP response has also been described by Lovric [5] using numerical methods. In Table 2 we compared the E,,,,,- values computed by LovriC for u = 0 and different values of KA (Fig. 2 in ref. 5) with those obtained from eqns. (16) and (31) (eqn. 31 can also be applied since the case treated by LovriC corresponds to strong adsorption). The agreement between both sets of data is excellent. We wish also to point out that LovriC indicated that if KA changes by an order of magnitude, E maxchanges by 60 mV. Note that eqn. (31) provides a theoretical justification of this fact.
282 TABLE 2 Values of AI?,, ( = E,, - Et,,) as a function of KA for DA = lo-’ cm* s-‘, r = 1 s, t’ = 0.05 s, 0 = 0, y = 1. (I) From Fig. 2 in ref. 5 ‘. (II) Computed from eon. (16). (III) Computed from eqn. (31)
log ( KA /cm)
A-Lx WV
AEm, (W/V
4m.x (m/v
0
-0.183 - 0.241 -0.300 -0.360 - 0.420 - 0.481
-0.183 - 0.242 - 0.302 -0.361 - 0.420 - 0.479
- 0.183 - 0.242 - 0.301 -0.360 - 0.419 - 0.478
1 2 3 4 5
Values of i,,/ia(?‘) as a function of f/f’. KA =103 cm. (I) From Fig. 3A in ref. 5 a. (II) Computed from eqn. (16). (III) Computed from eqn. (32) t/t’
im,/id(t’)
10 20 30 40
1.000 1.338 1.590 1.850
(1)
imax/Mt’) 1.001 1.324 1.586 1.811
(11)
i,,/ia(t’)
(III)
1.000 1.324 1.585 1.809
a Data from ref. 5 contain some uncertainty since they have been taken from figures.
Values of imax/id(t’) vs. t/t’ have also been reported by LovriC (Fig. 3A in ref. 5), who suggested that the i,,,- values are the same regardless of whether the t/t’ ratio is obtained by maintaining t’ = constant and varying t, or by keeping t = constant and changing t’. These data are reproduced in Table 2 and they are compared with those obtained from our eqns. (16) and (32). Again, the agreement is excellent. Furthermore, Lovric’s suggestion mentioned above can easily be explained from eqn. (32) which shows that imax/id(t’) dep en d s only on the t/t’ ratio and not on the individual values of t and t’. In addition, this authors also suggested that for the data shown in Table 2 the relationship between imax/id(t ‘) and t/t’ seems to be linear, although this relationship is not held as t/t’ increases. Equation (32) allows us also to explain this observation since according to this equation the dependence of i,,/ia( t’) on t/t’ is not linear (although for the data given in Table 2 an approximately linear relationship is obtained), and the larger deviations should be found as t/t’ increases. The data given in Table 1, which shows values of imax/id( t’) obtained f or a wider interval of t/t '-values than those given in Table 2, confirm this fact. Finally, the NPP polarograms reported in Fig. 1 in Lovric’s paper [5] for different values of u are also in excellent agreement with the I-E curves obtained from eqn. (13) using the values of (I, KA, t and t’ given in that figure. (B) RPP Figure 6 shows RPP polarograms computed from eqn. (23) for different values of Di”/K, when only the reactant is adsorbed. Curve (a) corresponds to a process in
283
-0.99
- 0.3
-0.1 (E2 - Eel/V
0.1
Fig. 6. RPP: Dependence of the I-E curves on DA/‘/K,+, computed from eqn. (23) for f = 1 s, I’ = 0.05 s, K, = 0. (Di’2/KA)/s-“2: (a) cc; @) 1; (c) 0.1; (d) 0.01; (e) 0.001. Other conditions as in Fig. 1.
the absence of adsorption and has been included for comparison. Note that the I-E curves are similar to those obtained without adsorption. In addition, for E, + cc eqn. (23) gives i,,,,,,/i,(t’) = m - 1, which coincides with the expression for the limiting current for a diffusion process. Thus, only the shift of the waves towards more negative potentials reveals that an adsorption process is involved. This situation is similar to that observed in NPP when only product adsorption occurs, although in this case the waves shift towards more positive, instead of negative, potentials. This behavior is in agreement with the RPP polarograms obtained by Flanagan et al. using digital simulation techniques (see Fig. 4 in ref. 4). The characteristics of the I-E curves when only the product is adsorbed are shown in Fig. 7 where RPP polarograms have been computed from eqn. (24) for different values of D’,/*/K,. As in Fig. 6, curve (a) corresponds to a process without adsorption. From Fig. 7 (which is equivalent to Fig. 1 in NPP when only the reactant is adsorbed) the following conclusions are obtained: (a) The limiting currents are also depressed with respect to those obtained in the absence of adsorption. However, we found that in NPP the i,im,NPP/id(t’)-values approach a limiting value (= m) for strong adsorption (see Fig. 1 and eqn. 30).
284
_ ; I= . L ._!
-0.6
-1.0
f
e -1.8 - 0.1
0.0
0.1
0.2
g a3
0.4
(E2- E’)IV Fig. 7. RPP: I-E curves computed from eqn. (24) for I = 1 s, t’ = 0.03 s, K,+ = 0. ( D~‘2/K,)/s-‘/2: co; (b) 10; (c) 1; (d) 0.1; (e) 0.01; (f) 0.001; (g) 0.0001. Other conditions as in Fig. 1.
(a)
In contrast, in RPP the value of i,,,,,,/i,(t’) for K, s-=.1 approaches zero (see curves e-g). (b) The waves exhibit a maximum if the K,-values are sufficiently large (curves c-g), which shifts continuously towards more positive potentials as K, increases (stronger adsorption). In contrast, the corresponding current, i,,,, approaches a limiting value for K, s=- 1 (curves e-g). Thus, this limit in Fig. 7 for imax,.&id(t’) is 1.574. The behavior of the limiting current is explained from eqn. (24) which for E, + 00 (i.e., y -+ co, F(y) --j 1) gives
=F(X)(I;I)‘/2_-H(X)
*
(37)
d
According to this equation we obtain ‘lim,RPP
i,j(t’)
( t r/t)1’2
_ i
0
-
1
for K, = 0 (no adsorption) for K, z+ 1 (strong adsorption)
(38)
so that the RPP limiting current approaches that obtained for a diffusion process for weak adsorption (curve b in Fig. 7), or to zero for strong adsorption (curves e-g). The behavior of the maxima can be explained in an analogous way to that in NPP. Thus, and taking into account that Em, shifts towards more positive potentials as K, increases (i.e., the corresponding E,-values become larger), one finds that these two effects compensate each other in eqn. (24a), so that a constant
285 TABLE 3 RPP: Values of ‘;;~Ld’~a(,‘r’:,‘s!$a$ (t’ = constant). 9 from eqns. (39) and (40)
rmx A
-E,,,)
as a function of I’ (t = constant) and f
7 y =l. (I) Computed from eqn. (24). (II) Computed
1=ls
t ‘/s
L,/id(~‘)
0.001 0.005 0.01 0.02 0.05 0.1 0.2 0.3 0.5
-
(1)
8.745 3.903 2.753 1.937 1.206 0.830 0.553 0.423 0.277
i,,/i.,(t’)
(11)
- 8.755 - 3.908 - 2.156 - 1.939 - 1.207 -0.831 - 0.554 - 0.423 - 0.277
A.%, WV
AK,,,, (WV
0.320 0.300 0.291 0.282 0.270 0.261 0.252 0.247 0.241
0.320 0.299 0.290 0.281 0.270 0.261 0.252 0.247 0.240
r’=0.05s t/s
kw./kJ(t’)
0.1 0.2 0.3 0.5 1.0 2.0 5.0 10.0 20.0
-
value for y,, E max=
El,2 -
i max,RPP_
k(t’)
(I)
0.277 0.479 0.619 0.830 1.206 1.728 2.753 3.903 5.525
k,Jid(~‘) - 0.277 - 0.480 -0.619 -0.831 - 1.207 - 1.730 - 2.756 - 3.908 - 5.553
(II)
A%,, 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270
(1)/V
AK,,,
(WV
0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270 0.270
= 0.83 is obtained. Under these conditions we find (see Appendix) $
-0.277
In
1.20( Dat’)1’2 KB l/2
(40)
According to these equations E,,,, shifts continuously towards more positive potentials as K, increases (59/n mV if K, changes by an order of magnitude), while i,,, becomes independent of the K,-values for K, B 1. The I-E curves for strong adsorption in Fig. 7 are in agreement with these predictions. Equations (39) and (40) also show the dependence of E,, and i,,, on t’ and t for a given value of K,. This is illustrated in Table 3 where the values of E,,, and inI, obtained from eqn. (24) for D’,/‘/K, = lop4 sell2 have been compared with the approximate ones computed from eqns. (39) and (40). As in NPP (see Table l), imax.id(t’) for t = constant increases as t’ becomes shorter. However, contrary to the behavior of the maxima in NPP, we found that in RPP, maxima will be observed
286
even for large values of t ‘. Moreover, E,,,,, shifts in the opposite way to that in NPP as t ’ changes (i.e., towards more negative potentials as t ’ increases). If t ’ = constant Table 3 shows that E,, remains constant as t changes, so that AE, computed from eqn. (39) (A&, = 0.270 V) coincides with that obtained from eqn. (24). Note that, contrary to NPP, E,,,, does not vary even for small values of t (compare Tables 1 and 3). We wish also to point out that in both cases (t = constant and t’ = constant), the E,,and i,,- values computed from eqns. (39) and (40) are in excellent agreement with those obtained from eqn. (24). Equation (40) shows that, as in NPP (see eqn. 32), the imax/id( t’)-values do not depend on the individual values of t and t’ (see Table 3), but on the t/t’ ratio. An interesting consequence of this fact is that for strong adsorption and for the typical values of t and t’ used in RPP (i.e., t’/t e l), it follows that imax.RPPa l/t’. An
0.2T
a I -0.1
, 0.3
0.1 (EZ-
E%’
-1.81 -0.2
0.0
0.2
(E2- ET/V
Fig. 8. RPP: Dependence of the I-E curves on Z’ computed from eqn. (24) for t =l s, Dyz/KB = 0.2 s -‘/’ 1 KA = 0. r’//s: (a) 0.005; (b) 0.01; (c) 0.02; (d) 0.05; (e) 0.1; (f) 0.2; (g) 0.5. Other conditions as in Fig. 1. Fig. 9. RPP: Dependence of the I-E curves on o computed from eqn. (20) for 1’ = 0.02 s, t = 1 S, Dy2/KB = 0.2 S-I/~. O: (a) co, (b) 100; (c) 50; (d) 25; (e) 10; (f) 1; (g) 0.1; (h) 0.01. Other conditions as in Fig. 1.
I
287 TABLE 4 RPP: Dependence of i,,_/id(t’), AE,,,, ( = E,, tion. D’/’ = 0 .2 s-‘12, t’ = 0.05 s, KA = 0 B/B K
- E,,,)
and ilim/id(t’)
on t for moderate adsorp-
t/s
Ldidt’)
A-%,/V
jlim/k(t’)
0.2 0.3 0.5 1.0 2.0 5.0 10.0 20.0
- 0.400
0.080 0.079 0.078 0.078 0.078 0.078 0.078 0.078
- 0.022 - 0.042 - 0.074 -0.127 -0.194 - 0.305 - 0.403 - 0.505
-
0.532 0.722 1.034 1.418 2.041 2.584 3.157
analogous conclusion is also valid for NPP (see eqn. 32). Note that, in contrast, the diffusion current in the absence of adsorption is a l/c. Hence, strong adsorption provides a procedure for obtaining enhanced currents which could be applied for analytical purposes. Although eqns. (39) and (40) have been obtained for processes with strong adsorption, its predictions are qualitatively valid for weaker adsorption also. This is shown in Fig. 8 where RP polarograms for moderate adsorption have been obtained from eqn. (24) for different values of t’. As described for strong adsorption (see Table 3), Em, shifts towards more positive potentials and i,,, increases as t’ becomes smaller. Note that for large values of t’ (curve g) the limiting current becomes positive. This fact is discussed more extensively below. Also, the dependence of the Z-E curves on t for moderate adsorption and t’ = constant behaves as for strong adsorption, i.e., Em, remains practically constant as t changes and i,, increases with t (see Table 4). Table 4 also illustrates the dependence of the RPP limiting current on t. In this case, we found that the lli,,, a,,-values increase towards more negative values as t becomes larger, so that no minimum is observed. This behavior contrasts with that obtained in NPP where i,im,NPPpasses through a minimum as t increases (see Fig. 3). The characteristics of the RPP polarograms when both reactant and product are adsorbed are shown in Fig. 9. In this figure the Z-E curves have been obtained from eqn. (20) for D’,/*/K, = 0.2 s-l/* and different values of u. The i,,,-values decrease as u becomes lower, so that the waves will not exhibit maxima for small values of (I. This behavior is also in agreement with that obtained by Flanagan et al. using numerical methods (see Fig. 4 in ref. 4). However, the limiting current does not depend on the values of u. This is in agreement with eqn. (20) which for E, + 00 gives eqn. (37) also. In this equation the argument, x, in the F(x) and H(X) functions does not depend on u, so that only the adsorption coefficient of the product, K,, is involved. This situation is similar to that described above in NPP, although in this case the information that can be obtained from the RPP limiting
288
current is related to the product, and not the reactant. Thus, from eqn. (37) we find I RPP
(41)
=
Equation (41) is equivalent to eqn. (33) in NPP, and by following an analogous procedure to that described above we find that if t = constant, a plot of I,,, vs. fl will be linear with intercept a and slope b defined by Q= -F(x)
b=ff(x)
i
(42)
From eqns. (4) (5) (41) and (42) we obtain K, = e(~D,t)“~
(43)
and therefore, K, can be determined from the slope and intercept of the plot of I RPP vs. fi. For processes with weak adsorption we have that F(x) and H(X) --, 1, so that a -+ - 1, b -+ 1. In contrast, for strong adsorption we obtain a + 0, b + 0 so that the corresponding plot approaches the abscissa axis. The upper and lower limits of the K,-values that can be obtained from eqn. (43) are determined as previously, and coincide approximately with the interval of K,-values given for NPP. An example of these kinds of plots for weak, moderate and strong adsorption is shown in Fig. 10. Note that for values of K, B- 1 these plots show only that the adsorption is very strong, so that this situation is analogous to that described previously in NPP. In these cases we could also use the dependence of E,, on K, (eqn. 39) to determine the adsorption coefficient of the product if the +,-value is known. (C) Double potential step chronoamperometry Although eqns. (1) and (2) allow us to obtain the double chronoamperometric response for any value of E, and E2, for simplicity the characteristics of this response are shown for E, -+ - 00 and E, + co (see eqns. 25-27). Thus, in Fig. 11 we have plotted i(t)/i(t,) vs. t computed from eqns. (27) for t, = 1 s and different values of Dk’*/K,. The corresponding curve in the absence of adsorption has also been included. Note that under these conditions (E, + - 00) there are no differences between the different responses for t < t,. However, for t > t, the responses are different enough, so that depending on the K,-values the current may become more positive as t increases (see curves c and d in Fig. 11). This is shown clearly in Fig. 12 where we have represented the current for the second step, i.e. i( t > tl)/i( t,) vs. t’, for t, = 1 s. For weak adsorption the i(t > t,)-values are negative, although they are larger than those obtained in the absence of adsorption (curve b in Fig. 12). As adsorption becomes stronger the values of i(t > t,) become more positive as t increases and they pass through an ill-defined maximum (curves c and d). After the
289
Fig. 10. I,, vs. F I I 0.01; (c) 0.1.
computed from eqn. (41) for I = 1 s, Da = lo-’
cm* s-l.
Ka/cm:
(a) 0.001; (b)
Fig. 11. Double potential step chronoamperometry: dependence of i(t)/i( I,) on t computed from eqns. (27) for tI = 1 s and different values of (D~‘z/K,)/s-‘/2: (a) co; (b) 5; (c) 1; (d) 0.1.
maximum the current approaches zero slowly, although from positive values (note that in the absence of adsorption the current also approaches zero, but from negative values). For very strong adsorption the current approaches zero rapidly for all t ‘-values (curve e). We wish to point out that, in some cases, the current becomes positive after the second pulse is applied; this is due to the fact that this pulse component is the result of two contributions, one cathodic, i.e. the dc component from the first pulse which acts at all times (even after the second pulse is applied [19]), and the other one anodic (see eqns. 26 and 27). In the absence of adsorption the anodic component is always larger than the cathodic one (although their values approach one another as t increases), and therefore, the net current also approaches zero from negative values. However, when product adsorption is involved, the anodic component may become smaller than the cathodic one as t changes, so that the net current will be positive. In any case, eqn. (27) shows that at long times i(t > ti) approaches zero for all values of K,. On the other hand, the value of the ratio - i(2t,)/i(t,) can be used to analyze the conditions of a stable system [20]. This ratio in the absence of adsorption is
290 -0.6-
0.11 0
2
4
6
8
I 10
Fig. 12. Double potential step chronoamperometry: dependence of i( t > t,)/i(t,) on 1’ computed from (a) co; (b) 5; (c) 1; (d) 0.4; (e) 0.03. eqns. (27) for 1, =l s and different values of (D~‘*/K,)/s-‘I*:
independent of c, and is equal to 0.293 [20], so that deviations of these current ratios from this value have been ascribed to kinetic complications in the electrode reaction [20]. As expected, when product adsorption occurs, the value of this ratio TABLE 5 Double potential step chronoamperometry: Dependence of - i(2t,)/i(t,) = I(tl) computed from eon. + co) this ratio is equal to 0.293 and is (27) on z, and Dkfl/K,. In the absence of adsorption (Dy2/KB independent of t, (D;‘*/K,)/s-“’
I(O.5)
Co 10.0 5.0 3.0 1.0 0.5 0.3 0.1 0.05 0.01
0.293 0.240 0.1% 0.148 0.037 - 0.0015 -0.011 - 0.0088 - 0.0053 - 0.0012
I(l)
-
0.293 0.255 0.221 0.182 0.068 0.013 0.0059 0.011 0.0069 0.0017
f(2)
I(5)
I(lO)
0.293 0.266 0.240 0.210 0.104 0.037 0.0052 -0.012 - 0.0088 - 0.0023
0.293 0.275 0.259 0.238 0.153 0.079 0.032 - 0.010 -0.011 - 0.0036
0.293 0.280 0.268 0.253 0.186 0.117 0.063 -0.0046 - 0.012 - 0.0048
291
also differs from the value 0.293 corresponding to a process without adsorption. This is shown in Table 5 where the values of -i(2t,)/i(t,) have been computed from eqns. (27) for different values of DL”/K, and t,. Note that the value of this ratio depends strongly on the values of t, and K,, so that - i(2t,)/i( tr) may even become negative. These results are explained from eqns. (27) which for t = 2t, (i.e., t’ = zr) give __ 4%)
+qX)_
x=
y;
i(b)
@Dgtl)‘/’
(4)
B
According to eqn. (44) it follows that - i(2t,)/i( tl) + 0 for x + 0, passes through a minimum = -0.01189 for x = 0.22, and approaches 0.293 for x -+ cc. The data in Table 5 are in agreement with this behavior. APPENDIX
(i) Equation (31) is derived as follows: For strong adsorption and for the E-values at which the maximum, E,,, appears, we have ye, +z 1 so that the corresponding argument, y,,, in eqn. (17) becomes Ymax
=
(DAf’)1’2/yG,t
From eqns. (Al) and (9) and taking into account that y,, E max= E1,2 + z
In
l.ll(
DAt’)l” KA
641) = 0.9 we find
(AZ)
where l/2
In
E l,2=E“-z
(A3)
(ii) In turn, eqn. (32) is derived as follows: For strong adsorption the arguments x and x1 in eqn. (16) are +z 1 (see eqn. 14), although in this case and due to the fact that yc2 < 1 we find from eqns. (5) and (14)
H(x,) ---=
2( DAtl)1’2
(A4)
r’/2yc2KA
-f-f2
Inserting eqn. (Al) in eqn. (A4)
H(x,)
=-
2Y,,
11
Tl/2
YE2
( 7
1’2 )
Also, from eqns. (17), (14) and (Al) we obtain
‘,x=yc,
1
(D,t)"' K
A
646)
292
Finally, inserting eqns. (A5) and (A6) in eqn. (16), taking into account that F(x) and H(X) + 0 under these conditions, and that F( Y,,,) = 0.7283 and H(Y,,,,,) = 0.5435, we obtain eqn. (32). (iii) Equations (39) and (40) are derived in an analogous way. Thus, and taking into account that in RPP yeI B 1 at the potentials at which the maximum is obtained, eqn. (24a) becomes Ymax= yQ( D,t’)1’2/K*
(A7)
Proceeding as previously and recalling that in this case y,, = 0.83 (see text), we obtain eqn. (39). In turn, eqn. (40) is derived by considering that under these conditions l/2
2Y YE2WXJ
=
-$
3 (
and that F(y,,)
1
= 0.7046. Inserting these relationships in eqn. (24), we obtain eqn.
(40). ACKNOWLEDGEMENTS
A grateful acknowledgement is made to the Fulbright/MEC Commision for providing a fellowship to J.G. for this collaborative work, and to the U.S. Department of Energy for supporting this work (Grant DE-FG22-86PC90519). J.G. also appreciates the financial support of the Comisi6n Asesora de Investigacibn Cientifica y TCcnica (project No. 763/84). REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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