J. Electroanal. Chem., 105 (1979) 43 - -51 43 © Elsevier Sequoia S.A., Lausanne - - P r in ted in The Ne the r l ands

STUDY OF THE FIRST ORDER CHEMICAL REACTION FOLLOWING CHARGE TRANSFER USING THE DOUBLE POTENTIAL-STEP CHRONOCOULOMETRIC METHOD

PART III. THE CASE WITH THE PRECEDING CHEMICAL REACTION OF ANY ORDER

K A R E L H O L U B

J. Heyrovsk2) Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, Jilskd 16, 11000 Prague 1 (Czechoslovakia)

(Received 6 th April 1979)

A B S T R A C T

The t h e o r y of the doub le po ten t i a l - s t ep c h r o n o c o u l o m e t r i c m e t h o d wi th in f in i te du ra t i on of the s econd po ten t i a l - s t ep is e x t e n d e d to more general r eac t ion scheme which inc ludes the reac t ion p reced ing the charge t r ans fe r in the first po ten t ia l - s tep . The m e t h o d for t he deter- m i n a t i o n of the first order chemica l r eac t ion fo l lowing the charge t r ans fe r is p r o p o s e d for such reac t ion scheme. The m e t h o d is based on the use of numer ica l Laplace t r a n s f o r m a t i o n of the charge- t ime of the first po ten t ia l - s tep dependence . The requi red quan t i t i e s are the measu red charges. No knowledge of the pa rame te r s of the chemica l r eac t ion p reced ing the charge t rans fe r dur ing the first po ten t i a l - s t ep is necessary.

The double potential-step chronocoulometric method has been modified [1,2] for the case of " inf ini te" duration of the'second step. This modification has some advantages [3] over the standard method with the same duration of both steps. One of them may be used for the extension of the method to the more complex reaction scheme.

In this paper a theoretical t reatment of the following scheme is given:

1st step, potential E,: chem.

A1 + A2 + . . . .

el., n 2 2nd step, potential E2: Y ~ C ~ X fast chem.

" B + B , +.. .

e l . , rt 1

C k.-_~ X chem. (A)

During the first potential-step the substance B undergoes the electrode reaction and at the same time is produced (and contingently simultaneously consumed) by a chemical reaction (or reactions). The substance C (originally not present in the solution) is generated by the electrode reaction and consumed by the first-order chemical reaction with a rate constant k.

44

During the second potential-step the concentration of the substance C is depleted by the first-order chemical reaction and by the electrode reaction. The electrode reaction should be so fast that the concentration of the substance C at the surface of the electrode could be assumed to be zero, i.e. the rate of the electrode reaction is such that the limiting diffusion current is attained.

There are no strict restrictions as far as the rate of the electrode reaction in the first step is regarded. There are also no restrictions with respect to the order and rate of the contingent chemical reaction (or reactions) by which the sub- stance B is generated.

The aim of this paper is to describe a procedure by means of which the rate constant k of the first-order chemical reaction following the charge transfer in the first step can be determined assuming only the knowledge of the measured charges of both potential-steps as a function of the duration of the first potential- step. The knowledge of the parameters of the chemical reaction preceding the electrode reaction during the first potential-step is thus not necessary.

F O R M U L A T I O N

The problem for substance C (c denoting the concentration of substance C) is formulated by an equation for diffusion towards the plane electrode:

ct = D c c x x - - k c , t E [0 , t l ] , X > 0 (1)

with the following initial and boundary conditions:

t = O , x > O : } c = O

t > 0 , x-~ oo (2)

t > O , x = O : DcCx + D b b x = 0 (3)

For the second potential-step we have:

C t = D c c x x - - kc, t ~ [0, oo] , X > 0 (4)

with the initial condition:

t = 0 : c = c ( x , t ) = c l ( x , t l ) (5)

where c~(x , t~) is the concentration of the substance C at the end of the first potential-step of duration tl. The boundary condition is:

t > 0 , x = O : c = O (6)

Let the faradaic current density during the first potential step be defined as:

il = i l ( t ) = n l F D b ( b x ) x = o (7)

and the corresponding charge:

t l

Q~ = Q~(t~) = n l q l = ; il(t) dt (8) 0

Let

i2 = i2(t) = n 2 F D c ( c x ) x = o (9)

45

be the faradaic current density in the second step. The charge that passes through the unit area of the electrode during the second (sufficiently long) potential-step is

o o

Q2 = Q 2 ( t l ) = n2q2(t,) = f i2(t) d t (!0) 0

Note that due to the definitions both i, and i2 are non-negative.

R E L A T I O N BETWEEN Q1 AND Q2

In the following we derive the relation which holds between Q, and Q2. For the derivation of the relation we shall use here the Laplace transformation method. This is not necessary, the method used in ref. 1 could be applied as well with the same result. The Laplace transform of a function f ( t ) we denote as :

f = f(s) = L ( f ( t ) } = f exp(- -s t ) f ( t ) dt o

The inverse transform of the function f we denote as:

L-' (t; f(s) } = L -1 {f} = f(t)

(11)

o r

f ( t ) - } , f - f ( t ) (12)

Equation (1) in the Laplace transform -- taking into account eqn. (2) -- can be written as:

~xx -- ~(k + s)/Dc = 0 (13)

where s is the parameter of the Laplace transform for the first step. The solution of this ordinary differential equation is:

= A (s) exp(--x [(k + s)/D c ] ,/2) (14)

Using the boundary condition (3) we find that:

A(s) = D b Dx/x/(Dc(k + s)) (15)

Equation (7) in the Laplace transform can be similarly rewritten as:

Cxx -- ~(k + p)/D¢ = ~-c,/D¢ (16)

where p is the parameter of the Laplace transform for the second step. The solution of this equation gives for the Laplace transform of D¢(cx)x=o the follow- ing relation:

D(~x)x=0 = y exp(--~x/(k +p)/D¢) c,(~, t,) d~ (17) 0

46

Now we use the following two theorems [4] for the Laplace transform:

t

(1) f f(t) dt - f/s 0

(2) Let L{f ( t ) } , L( f t ( t ) } and the limit f(t) for t + ~ exist, then

lim sL (f(t) } = ~imlf(t ) S---~O

Therefore

? = D(ex)x=o dt = lim D(ex)x=o(P) = f exp(--~ kx/-k~ c1(~, t ,) a t

p"-+0 0 0

According to (14)

el(x, tl) = L -1 (t = tl; A(s) exp(--x[(k + s)/Dc]) }

and therefore substituting (21) into (20) we rewrite (20) as

(18)

(19)

(20)

(21)

oo

f D¢cx(X : O, t) dt = L - ' ( t = t l;A(s) / exp(--~(x/~ + kx/-ff-gs)/x/~c) d~} 0 0

Using (15) we find after the integration

{ f Dccx(X =O,t) d t = L - ' t= t l ;DbDx(x=O,s ) . 0

t l

/ 0

Dbbx(x = O, t) erfcx/k(tl -- t) dt

Here we used (see e.g. refs. 4 and 5):

f(s + k) - exp(--kt) f(t)

and

[x/s(x/k + x / s ) ] - ' - exp(kt)Je~cx/-~

Taking into account the identity:

erfc x = 1 -- erf x

and (7)--(10) we derive eventually:

t l

ql( t l )--q2(t l ) = F / 0

or in the other form:

Dbbx(x = O, t) erfx/h(t~ -- t) dt

(22)

(23)

(24)

(25)

(26)

(27)

t l

q,(t,) -- q:(t l ) = j 0

t l

i,(t) erfx/k(tl -- t) dun, = f (dq,(t)/dt) erfx/h(tl t) dt

o (28)

47

D E T E R M I N A T I O N OF k

For the determination of the rate constant k we propose the following meth- od. The quantities Q~(tl) and Q2(t~) are to be known from the experiment. The ratio n~/n2 is to be known. We make separately the Laplace transforms of q l ( t l ) - - q:(t~) and q ~ ( t l ) .

From (27) and (7)--(10) it follows:

L { q l ( t l ) - - q 2 ( t l ) } = L { Q l ( t l ) - - n l Q 2 ( t l ) } / n l = F D b b x ( x = O, t l ) " x / ~ / ( s v ~ + s ) n2

and from (7) and (8):

L { q , ( t l ) } = FDbbx(x = O, tl)/S

Dividing (29) by (30) we get

P=- L{q~( t l ) - -q~( t l ) } / L ( q l ( t l ) } = [k/(k + s)] ~/2

Therefore:

k = sP2/(1 -- P:)

(29)

(30)

(31)

(32)

This is the relation that is proposed to be used. It relates the rate constant k of the first order chemical reaction (see scheme A) to the parameter s of the Laplace transform and to the ratio of the Laplace transforms of the difference of charges ql( t i ) -- q2(tl) and of the charge ql(tl).

The Laplace transforms of the charges can be obtained numerically using any suitable quadrature formula. Here we can propose e.g. the following quadrature formula:

f exp(--x) F(x ) dx = AknF(xk,~) + Rn (33) k = l 0

where Ak~ and xkn are tabulated [6]; see also Table 2. We note also that:

f(s) = f exp(--st) f ( t ) d t = exp(--x) f (x /s) dx /s o o

Therefore:

(34)

n

f(s) = s -1 ~ A k n f ( t k n ) , tkn = Xkn/S (35) k ~ l

where the abscissas Xkn and the weights Akn are those tabulated quantities.

E X A M P L E

We shall illustrate the proposed method on the results taken from ref. 3. It should be mentioned that the data of ref. 3 do not require the proposed treat- ment because presumably there is no reaction preceding the charge transfer

48

TABLE 1

Data partially published in ref. 3, Table 1, FH4

t 1/s Q 1/pC (Q 1 - - Q2 )/pC

0.21 20.5 3.2 0.53 34.5 8.2 1.02 48.5 15.8 2.04 69 30.5 3.34 90 49.5 5.0 105 65.5 9.3 143 106.5

T A B L E 2

I n t e r p o l a t e d and e x t r a p o l a t e d values of the charges f rom Table 1 in the abscissas of the four- p o i n t q u a d r a t u r e f o r m u l a wi th t he weights Ak4

k tk4 = Xk4/S QI/laC (Q1 - - Q2)/P C Ak4

1 0 .3225 25 .86 4 .962 0 .60315 2 1 .7458 63 .45 26.24 0 .35742 3 4 .5366 100.8 61 .04 0 .038888 4 9 .395 143.8 extr . 107.4 extr . 0 . 0 0 0 5 3 9

dur ing the first po ten t ia l - s tep . Never theless t he p r o p o s e d m e t h o d can be appl ied also to this case.

We use the da ta in Table 1. F r o m Table 1 we see t h a t the charges are given for t~ f r o m 0.2 to 9.3 s. In o rder to avoid the e x t r a p o l a t i o n we choose s = 1. Now we choose n in t he q u a d r a t u r e f o r m u l a n = 4. To get the value o f ql and q~ - - q2 at t he po in t s t k 4 = Xk4 (no te s = 1) we use the Lagrange i n t e rpo l a t i on p o l y n o m i a l o f degree 2. We get t he values in Table 2. We calcula te t he sums accord ing to (35) and we get Q ~ ( s = 1) ~ 42.3 and L ( Q I ( t l ) - - Q2(tl); s = 1 }~ 14.8. We have here nl = n2. App ly ing (31) we get P ~ 0.35, t h e r e f o r e f r o m (32) we ob ta in k ~ 0 .14 which is in a g r e e m e n t wi th the value k ~ 0 .14 given in ref. 3.

SPECIAL CASE

A b o v e it has been impl ic i t ly m e n t i o n e d t h a t the knowledge o f the pa ram- eters o f the chemica l r eac t ion p reced ing the charge t r ans fe r is necessary unless we use t he p r o c e d u r e o f the Laplace t r a n s f o r m a t i o n descr ibed. However , the re are s o m e special cases fo r which the t r e a t m e n t o f the measu red charges b y means o f the Laplace t r a n s f o r m is n o t necessary . Below we descr ibe s o m e o f t h e m .

Let us cons ider the fo l lowing r eac t ion scheme:

o % 1st po ten t i a l - s t ep : A ~-~ B C ~ X pa fas t

49

For a p lane e lec t rode the scheme is descr ibed b y the fo l lowing set o f equa t ions :

a t = Daaxx - - p(a - - ab ) (36)

b t = D b b x x + p(a - - ab ) (37)

t ~ O , x - > co : a = a* , b = b* (38)

t = O , x > 0

t > 0 , x = O : a x = O , b = O (39)

For D a = Db this p r o b l e m was solved [7 ] using the Laplace t r a n s f o r m m e t h o d . For D a :/= Db the p r o b l e m can also be t r ea t ed b y the Laplace t r a n s f o r m m e t h o d b u t the f inal express ions for D b b x ( x = 0) are qui te involved even in t he Laplace t r a n s f o r m .

First let us assume the special case

l < ( It < ( o 2; l==- p ( l + o) (i)

The so lu t ion of the p r o b l e m (36) - - (39) wi th (i) was ob t a ined in ref. 7 as a special case o f the f o r m u l a der ived for D a = D b = D:

x = 0: D b b x = (a* + b * ) ( D l ) l / 2 / o (40)

This so lu t ion co r re sponds to the s t a t i ona ry s ta te when a t = b t = 0. The so lu t ion of (36) - - (39) , assuming s t a t i ona ry s ta te can also be der ived

x = 0: D b b x = (a* + D b b * / D a ) [ P D b ( O + Db/Da)] I /2 /O (41)

These a p p r o x i m a t e so lu t ions are i n d e p e n d e n t o f t ime . T h e r e f o r e the charge Q l ( t l ) is t h e n d i rec t ly p r o p o r t i o n a l to the t i m e t~:

Q l ( t l ) = cons t • tl (42)

We can t ake advan tage o f the der ived re la t ion (27) and f r o m (42) we get

Q1 (n l /n2)Q~ t l

Re -= = f e r fx / r~ dr/t1 (43) Q~ o

Af te r in tegra t ion we get:

Re = 4x/~x cP(0.5, 2 . 5 , - - x ) , x =- let1 (44) 3X/Tr

where q~(a, b, x) is the c o n f l u e n t h y p e r g e o m e t r i c func t ion . This can be also r ewr i t t en to the fo rm:

Re = exp( - -x ) ( l rx ) -1/2 + (1 - - 0 .5 /x ) e r fx /~ (45)

The necessary re la t ion is t he inverse d e p e n d e n c e o f x -- kt~ on R¢ which is accessible f r o m e x p e r i m e n t . I t is given in Table 3. The relat ive e r ror A k / k = ( d x / d R ) A R / x in the d e t e r m i n a t i o n o f k can be rough ly e s t ima ted using the values o f ( d x / d R ) / x given in Table 3 and an es t ima te for the er ror AR in the d e t e r m i n a t i o n o f R .

We can cons ider the o t h e r case:

l t l < < 1 (ii)

50

TABLE3

The values of x =- k t l as a function of R c. The dependence can be used if Q1 varies linearly with time (eqn. 42). (Numbers in the parentheses (n) indicate the value of the decadic exponent by which the number should be multiplied, e.g. 1.00(--2) = 1.00 × 10 -2 .)

R c x ( d x / d R c ) / x R c x ( d x / d R c ) / X

0.005 4.41794(--5) 400.01 0.50 0.541708 4.95 0.01 1.76727(--4) 200.01 0.51 0.569059 4.90 0.02 7.07058(--4) 100.03 0.52 0.597531 4.86 0.03 1.59144(--3) 66.71 0.53 0.627179 4.82 0.04 2.83064(--3) 50.06 0.54 0.658068 4.79 0.05 4.42569(--3) 40.07 0.55 0.690262 4.76 0.06 6.37796(--3) 33.42 0.56 0.723836 4.74 0.07 8.68913(--3) 28.67 0.57 0.758868 4.72 0.08 1.13612(--2) 25.11 0.58 0.795445 4.70 0.09 1.43964(--2) 22.35 0.59 0.833660 4.69 0.10 1.77975(--2) 20.14 0.60 0.873618 4.68 0.11 2.15673(--2) 18.34 0.61 0.915432 4.67 0.12 2.57092(--2) 16.84 0.62 0.959227 4.67 0.13 3.02268(--2) 15.57 0.63 1.00514 4.68 0.14 3.51241(--2) 14.49 0.64 1.05332 4.69 0.15 4.04056(--2) 13.55 0.65 1.10395 4.70 0.16 4.60760(--2) 12.73 0.66 1.15720 4.72 0.17 5.21404(--2) 12.01 0.67 1.21328 4.75 0.18 5.86044(--2) 11.37 0.68 1.27244 4.78 0.19 6.54740(--2) 10.81 0.69 1.33494 4.81 0.20 7.27557(--2) 10.29 0.70 1.40107 4.86 0.21 8.04563(--2) 9.83 0.71 1.47118 4.91 0.22 8.85833(--2) 9.42 0.72 1.54564 4.97 0.23 9.71445(--2) 9.04 0.73 1.62491 5.04 0.24 0.106148 8.69 0.74 1.70949 5.11 ~ 0.25 0.115604 8.38 0.75 1.79996 5.20 0.26 0.125521 8.09 0.76 1.89702 5.30 0.27 0.135909 7.82 0.77 2.00146 5.42 0.28 0.146779 7.57 0.78 2.11423 5.55 0.29 0.158143 7.34 0.79 2.23646 5.70 0.30 0.170014 7.13 0.80 2.36949 5.86 0.31 0.182403 6.94 0.81 2.51498 6.06 0.32 0.195326 6.75 0.82 2.67492 6.28 0.33 0.208796 6.59 0.83 2.85178 6.53 0.34 0.222830 6.43 0.84 3.04869 6.83 0.35 0.237444 6.28 0.85 3.26958 7.17 0.36 0.252656 6.14 0.86 3.51955 7.57 0.37 0.268484 6.01 0.87 3.80530 8.05 0.38 0.284950 5.89 0.88 4.13579 8.62 0.39 0.302074 5.78 0.89 4.52333 9.31 0.40 0.319880 5.68 0.90 4.98521 10.16 0.41 0.338391 5.58 0.91 5.54655 11.22 0.42 0.357634 5.49 0.92 6.24519 12.56 0.43 0.377636 5.40 0.93 7.14073 14.32 0.44 0.398428 5.32 0.94 8.33263 16.68 0.45 0.420041 5.25 0.95 9.99985 20.00 0.46 0.442510 5.18 0.96 12.5000 25.00 0.47 0.465870 5.11 0.97 16.6667 33.33 0.48 0.490161 5.05 0.98 25.0000 50.00 0.49 0.515425 5.00 0.99 50.0000 100.00

5 1

then ( f o r D a = Db):

x = 0: D b x = (a* + b * ) [ D / ( ~ r t l ) ] l / ~ / ( 1 + a) (46)

Let:

1 << ~2 << l t l (iii)

then from the formulae derived in ref. 7 we get

x = 0: D b x = (a* + b * ) [ D / ( ~ t l ) ] 1/2 (47)

In both last cases (ii) and (iii) the gradient b x ( x = 0) varies with the inverse of the square root of the t ime tl. Thus we have:

Ql(t l) = const • ~ (48)

This relation leads to the scheme already treated in ref. 1 and

R = Q~ - - (n~/n2)Q2 = 0 . 5 ~ ) ~ p ( 0 . 5 , 2, --x) (49) QI

The dependence of x = kt~ on R is tabulated in ref. 2.

C O N C L U S I O N

The method using the Laplace t ransformation of the charges is proposed to be used for the reaction scheme A in order to determine the rate constant k. There are however some special cases, e.g. the charge in the first potential-step varies linearly with t ime or with the square root of time. For these special cases the t rea tment of data does not require the Laplace transformation.

It should be ment ioned here that the limits of the applicability of the method are of physical nature (similarly to ref. 1) so that the range of the values o f k that can be determined by the proposed method is similar to that in ref. 1, i.e. k may vary over several decades.

R E F E R E N C E S

1 K. H o l u h a n d J. Weber , J . E l e c t r o a n a l . C h e m . , 73 ( 1 9 7 6 ) 129 . 2 K. H o l u b , J . E l e c t r o a n a l . C h e m . , 6 5 ( 1 9 7 5 ) 193 . 3 J. Weber , K. H o l u b , D. H o m o l k a a n d J. P r a d ~ , J . E l ec t roana l . Chem. , 73 ( 1 9 7 6 ) 151 . 4 G .A. K o r n a n d T.M. K o r n , M a t h e m a t i c a l H a n d b o o k for Sc ient i s ts and Engineers , McGraw-Hi l l , N e w

Y o r k , 1 9 6 1 . 5 A. Erd~lyi , Tab les o f I n t e g r a l T r a n s f o r m s , Vol . I, M c G r a w Hill, N e w Y o r k , 1 9 5 4 . 6 P. R a b i n o w i t z a n d G. Weiss, Ma th . Tab les Aids C o m p u t . , 13 ( 1 9 5 9 ) 285 . 7 J . K o u t e c k ~ a n d R. Brd i~ka , Col lec t . Czech . C h e m . C o m m u n . , 1 2 ( 1 9 4 7 ) 3 3 7 .