The J. Heyrousk?j InsiiSufe of Physical Chemisfry and Electrochemistry, Academy of +ences, Opletuloua 25, I2 0 00 Prague 1 (C.zechosI&akia)

(Received 30th June 1975)

Czechoslouak

Schwzzrz and Shak [I] developed a double_-puke chronoamperometric method for determination-of the rate constant k of a pro&Jet of the electrode reaction :

1st pulse, potential E, : A~~~~~ l3LX

2nd pulse,.potential Ez : A -s E&X

me potentials E, and E, of the electrode are assumed to be such that the electrode reaction is very fast 30 that the concentration of the substance undergoing the electrode reaction is zero at the surface of the electrode, w@ich is; in this and in the foJJowkg~schemes, expressed by the word “fast”. They gave a rather complicated soltition of the problem. A subs+nti& simpli- fictition of the~tfieory can be obtained [2] by using the chronocorrlometric method even for the more general scheme

1st pulse, potential El : A el., Is LX fast

2nd pufse; pbteki+ E2: Y Tz BE,X _ w

.- _ .-

The.condition’ for-the~s~plification is that the charge due to the second -pulse is me&red after a..suffi+entfy long time (when it reaches a c&x&ant value). -. The him of this p.aper is to-give_ a theoreticat treatment of. the following _-._ scheme. :

1st puke, potentkl El : A

2nd pulse, potential Ez: kB =C, A el..nl f n2 FC 2

fast

63

Forthefkst puke the problemisfommlated by means ofequationsfor diffusion toward plane electrode

at = Daa= 0) b, =D,,b, -kb (2)

with folIowing initial and boundary conditions

t=o,x>o I t>O,X+- I : a=a*, b=O

x = 0, t E (0, tl) : Q = 8, Daar + Dbb, = 0 (4) where a and b are the concentrations of the substances A and B correspondingly. Subscripts x, t, xx denote the partial derivatives: a, = h/ax, at = aa/a t, a, = a2a;ax? The initial concentration of the substance A is assumed to be con- stant (a*)_ The concentration ofthesubstance Bis assumed to be zero,

The solution can be obtained [2] by using the solutions of known problems only. Thus we get

Tnerefore the charge which h& passed through the unit area of the electrode duriig time & of the first pulse (due to the faradaic current) is

Ql = nlFa*2(D,tl/x)'/2 (7)

For the second puke we denote t as a time which elapses from the begin- n&g af the second p@s+ The problem is formulated by the eqn. (2) with the folIow5ng initial

.-

and boundary conditions

~f>O,c=~O~:b~o. - @I- _-I

-Fcrr.the subs- ce~+we-can .use.$he form-tiation for the Erst @_zl.se and -the :

195

solution for the first pulse which we rewrite (in the time of the second pulse) as (cf. eqn. (5))

a, = a* [zD,(t f &)]-lJa

The charge Q2 is defined as follows

(W

Q2 = (nI + n2)PDSj a,dt + nzFDb j b,dt (11) 0 0

Therefore Q2 is the charge which has passed through the unit area of the elec- trode during time t of the second pulse. Using the results of ref_ 2 and per- forming the integration of eqn. (IO) we get

where

x=k&y=kt

anti for the function R(x, y) we have

(13)

Y

R(x, y) = 1 - (4xp2 s

e- x e-E$1/2

s 1

0 mlSf2 0 77 + $ [X(X - #q] *r2 dEd77 (14)

For the case of k = 0 we get

Qz/Qx = I%, f n2)/a,] [(l f t&p2 - l] + (n&Q)[(t/t#‘” - (1 + t/ty $- l]

(12’1 The function R(x, y) converges quite fast to the limit Rfx, 00)

Wx, Y) - Wx, -1 Rb, -1

< [exp(y)(xy)-*'2 - erfc(y1/2)] g(x)

where g(x) is a decreasing function

x > 0 : c < g(x) = ~oW2) -1,(x/2) = a($, 2, -=) ( 1

~o(xl2) + Wxf2) @(+,2,-x). -

(15)

(16)

I0 and I1 are the modified Bessel functions of the first kind, ip is the confluent hypergeometric function. Thus for y = kt sufficiently great, eqn. (12) can be rewritten in the form

where

R(x+).= (;iTx j1j2 aqo.5; 2; -x)/2 (18)

0.01 o_oz 0.03 0.04 0.05 -0.96 o,g7 0.08 0.09 0.10 0_11 0.12 a.13 O-14 0.15 Oh6 0.17 O-18 0.19 0.20 -0.21 a22 0.23 0;24 0.25 0.26 .0.27 -0-28 0.29 -0.30 G-31 0.32 0.33 0.34 0.35 0.36 G-37 0.38 0.39 0.40 0.41 Q-42 0.43 &44 a.45 --0_46 A.L,.ai G.46

0.000127 o.rDO509 0.301.7-47 G.GO2G39 0_0031_88 G-004594 0.006258 0.008182 G_Ol_O367 G_G128f4 G_O15526 0.018505 0.021752 0-025272 0.029066 0.033137 G-037490 0.042126 O.G4?Cj52. 0.052269 .0.057?84 0.063600 0.069723 0_076158 <i.GS2?lG -o_os399s7- _0.097394- G.iG5138 OSP3227 O-121668 0;1-30469 t-X139640 .o_la!3189 0.159li7

-_ 6_169&64 0.18G210 0:191378 0.202-981 0.215030 02275427

20G;01 100x3 66_70 50.05 40.06 33.41 28-66 25.10 22.34 20.13 18.32 16.82 15.55 14.47 13.58 12.71 Il.99 11.35 10.78 10.26 9.80 9.38 9.00 8.66 8.34 8.05 7.78 7.53 ,750, T-08 6_S9 6.70 6.53. 6.37: 6-22 6.08 5.95 5.83 5.71, 5.60 5.50 5.41 5.32:

.5.23 5-16 -5.08 -_5.02 - a_95 -4A39

_.'.4.84.

0.53. 0.400733 0.52 0.420296 0.53 0.440620 0.54 0.461741 0.55 0.483697 0.56 0.506532 0.57 0.530291 0.58 0.555024 0.69 0.580785 0.60 G-607634. 0.61 G-635634 0.62 0.664857 0.63 0.695380 0.64 0.727287 0.65 Or760673 0.66 0.795641 0.67 0.832307 0.68 0.87079S 0.69 0.91li58 0.70 0.95354-i O_?l 0.995745 0.72 1.046156 0.73 1.096313 0.74 l-149478- 0.75 1.205957 0.76 1.266098 0.77 I.330309 0.78 1.399066 0.79 1.472929 OS0 1.552566 0.81~ 1.638776 0.82 i-732529 0.83 I-_835014 0.84 1.947707 0.85 2.072469 0.86 2.211680 0.87 2.368449 0.88 2.546922 0.89 2-752763 0.90 2.993948 O-91 3.282104 0.92 3.634837 0.93 4.oso5ss 0.94 4.667152 8-95 5.482979. 0.96 -6.7?07167 0.97 8.763273 O-98 i2.9d6647 0.99 25.3s92q1.

4.79 4.74 4.70 4.66 4.63 4.60 4.57 4.55 4.53 4.51 4.50 4.49 4.49 4.49 4.49 4.50 4.51 4.53 4.55 4.58 4.62 4.66 4.71 4.76 4.83 4.91 4.99 5.09 5.20 5.33 5.48

> 5.65 6.85 6.08 6.35 6.66 7.04 7.50 8.06 -

8.76 9.65

IO.82 12.39 14.60 17.84 22.88 31_40--

: k&27 __-9S_41- _ :

197 :

Ch=g&s v&&h are necessary for charging the double layer can be obtained by measureme@ &I the absence of the electroactive substance. These charges can be then subtracted from the measured charges and Q1 and Q&are ob- tained. Note that this correctionfor the double layer~can be done only after tile retaxatioz Grne (for an &timate see ref, 2) of the charging of the condenser of the double layer. Then we can use the eqn. (17) to find x from Table 1 and use eqn. (13); i.e_ we divide x by f, and we obtain the value of the rate constant fi. Of course the condition that y is sufficiently large hasto be ful- filled (see eqn. (15)). This can be checked; the condition corresponds to constant value of calculated r.h.s. of the eqn. (17) with respect to t at con- stant ti*

The relative error of thus calculated rate constant k can be estimated from the following expression

Akjk - ((dx/dR)/x)aR (19)

where AIZ is the error in the determination of R(v,-) from the eqn. (17); see Table I,

The merit of the proposed method for the study of the treated reaction mechanism is that the chemical reaction can not only be detected but that also its rate constant can be measured. Note that using e.g. the polarographic method-the chemical volume reaction cannot be determined. For detailed discussion of the problems connected with the proposed method see the quoted paper [2].

REFERENCES

1 W.M. S&warn and I. Shain, J. Phys. Chem., 69 (1965) 30. 2 K. Holub and J. Weber, to be published.