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Hydrodynamic voltammetry at channel electrodes
J Electroanal Chem, 79 ( 1 9 7 7 ) 4 9 - - 7 8
49
© Elsevmr S e q u o i a S A , L a u s a n n e - - P r i n t e d in T h e N e t h e r l a n d s
HYDRODYNAMIC VOLTAMMETRY AT CHANNEL ELECTRODES PART II. THEORY OF FIRST-ORDER KINETIC COLLECTION EFFICIENCIES
K O I C H I A O K I , KOICHI T O K U D A a n d H I R O A K I M A T S U D A *
Department o f Electronic Chemistry, Tokyo Institute o f Technology, Ookayama, Meguro-ku, Tokyo (Japan) (Received 9 t h J u l y 1 9 7 6 )
ABSTRACT T h e t h e o r y of first-order kinetic c o l l e c t i o n efficmncms at t h e d o u b l e c h a n n e l e l e c t r o d e is d e v e l o p e d for t h e following t w o s c h e m e s : (I) A ± nle ~ B (at t h e g e n e r a t o r e l e c t r o d e ) , B ~ P (m s o l u t i o n ) , B ± n2e ~ Y (at the d e t e c t o r e l e c t r o d e ) , (II) A + nle ~ B, B ~ A , B ± n2e-~ Y The e x a c t e x p r e s s i o n s for t h e k i n e t m c o l l e c t i o n efficmncms are o b t a i n e d as a s c e n d i n g a n d a s y m p t o t i c series w i t h respect t o t h e kinetic p a r a m e t e r F u r t h e r , a p p r o x i m a t e f o r m u l a e m e x p o n e n t i a l f o r m s are given, w h m h h o l d w i t h i n a n e r r o r o f a b o u t 2% for c o n v e n t i o n a l elect r o d e g e o m e t r y Finally, t h e validity o f t h e a p p r o x i m a t e p r o c e d u r e , w h i c h has b e e n used previously t o o b t a i n t h e kinetic c o l l e c t i o n efficmncies for fast h o m o g e n e o u s r e a c t i o n s , is discussed m c o m p a r i s o n w i t h t h e p r e s e n t t h e o r y .
INTRODUCTION
In the study of electrode reactions, it is of great importance to detect the reaction products or intermediates produced by charge transfer processes. For this purpose, in 1959, Frumkin et al. [1] first presented an elegant electrochemical technique, which is called the rotating ring-disk electrode (RRDE). The intermediate produced on the disk electrode (generator) diffuses into the solution, where it is carried downstream by convection to the ring electrode (detector). If the intermediate decomposes homogeneously in the solution, then the current at the detector electrode or the collection efficiency will be reduced. Thus, the measurements of the collection efficiency allow us to evaluate the rate constant of the homogeneous reaction. Another electrolysis arrangement, i.e., the double channel electrode, which seems to have almost the same potentialities as the RRDE has, was presented in 1965 by Gerischer et al. [2]. The double channel electrode comprises two neighboring rectangular electrodes located flush with one of the surfaces of a rectangular channel, through which solution flows. Though the RRDE has been extensively examined both theoretically and experimentally by a n u m b e r of investigators for various reaction schemes [3], relatively little study has been devoted to the double channel electrode [2,4--6]. For the case in which the charge * To w h o m c o r r e s p o n d e n c e s h o u l d be addressed.
50 transfer 1S f o l l o w e d b y a first-order h o m o g e n e o u s r e a c t m n (e.c. m e c h a m s m ) , Gerischer et al. [2] derived an e q u a t i o n for the c o l l e c t i o n e f f l c m n c y , b u t it is only a q u a h t a t i v e one. A f t e r w a r d s Braun [5] a p p h e d the m a t h e m a t i c a l a p p r o a c h used b y A l b e r y and B r u c k e n s t e i n [7] t o the d o u b l e channel e l e c t r o d e to o b t a i n e q u a t i o n s for the kinetic c o l l e c t i o n e f f i c m n c y f o r the e.c. m e c h a n i s m with a fast, first-order reaction. As Braun h i m s e l f p o i n t e d out, however, these equations c o n t a i n some a m b i g u i t y which m a y be a t t r i b u t e d t o the a p p r o x i m a t i o n utilized in the inverse Laplace t r a n s f o r m a t i o n b y A l b e r y and B r u c k e n s t e l n . In this paper, we shall a t t e m p t a t h e o r e t i c a l a p p r o a c h to the v o l t a m m e t r y at the d o u b l e channel e l e c t r o d e and derive the t h e o r e t m a l e q u a t i o n s for the firstorder kinetm c o l l e c t i o n efficiencies, b y utilizing the m a t h e m a t i c a l t r e a t m e n t similar to t h a t used in a previous p a p e r [8] (which is r e f e r r e d to as Part I). FORMAL EXPRESSIONS FOR THE COLLECTION EFFICIENCIES We consider the following t w o r e a c t i o n schemes: A- nle ~ B Scheme I :
(at
B ~ P (in
the g e n e r a t o r e l e c t r o d e )
(1)
solution)
(2) (3) (1')
•B + - n 2 e - ~ Y ( a t the d e t e c t o r e l e c t r o d e ) A + - n l e - ~ B (at the g e n e r a t o r e l e c t r o d e ) k
S c h e m e II:
1
(2')
B ~ A (in solution) ± n2e ~ Y (at the d e t e c t o r e l e c t r o d e )
where k is the T h e specms Y any e l e c t r o d e tion o f B, t h e 3 Um aC b y~x=D-
(3')
rate c o n s t a n t o f the (pseudo-) first-order h o m o g e n e o u s reaction. m a y be identical with A and we assume t h a t P does n o t u n d e r g o r e a c t i o n at the d e t e c t o r e l e c t r o d e . L e t c r e p r e s e n t the c o n c e n t r a i n t e r m e d i a t e . T h e n we have the c o n v e c t i v e diffusion e q u a t i o n : a2c
(4)
ay 2 - - k c
where Um is the m e a n flow v e l o c i t y and b the half height of the channel. T h e c o n f i g u r a t i o n and the related p a r a m e t e r s o f the d o u b l e channel e l e c t r o d e are described in a previous p a p e r [6]. In o r d e r to set up eqn. (4), the same assumptions as used in Part I are made. T h e b o u n d a r y c o n d i t i o n s for eqn. (4) are given as follows: c=O
forx=0,
e-~ O
forx>
y>
0
0, y - ~ o
for 0 ~ x ~ Xl, y D(Oc/Oy)
=
O (generator)
0
for x I < x < x2, Y = 0 (gap)
i2/n2F
for x2 ~ x ~< x3, Y = 0 ( d e t e c t o r )
(5)
51
where il and i2 are t h e c u r r e n t densities at t h e g e n e r a t o r and the d e t e c t o r electrodes, respectively. T h e signs o f currents are t a k e n so as t o be positive, w h e n the e l e c t r o d e r e a c t i o n s (1) and (3) p r o c e e d f r o m left t o right. F u r t h e r , w h e n the p o t e n t i a l applied to the d e t e c t o r e l e c t r o d e is assumed t o c o r r e s p o n d t o the limiting c u r r e n t plateau for t h e process (3), the following c o n d i t i o n should be added: c = 0
for x2 ~< x ~< x3, y = 0
(detector electrode)
(6)
I n t r o d u c i n g new variables: = A 3 x / i x 1 and T = ( k / D ) I / 2 y
(7)
with A = (k/D1/3)l/2(bXl/3
Um) 1/3
(8)
we o b t a i n (9)
T ( i ) C / i ) ~ ) = ( } 2 C / ~ T 2 --- C
with the b o u n d a r y c o n d i t i o n s : c=0
forT>0,~=0
(10)
c-~0
f o r T - ~ oo, ~ > 0
(11)
= hl(~) =- ( k D ) - l / 2 ( i l / n i F )
f o r t = 0, 0 ~< ~ ~< A ~ f o r t = 0, A a < ~ < ( 1
~c/~T = h ( } ) ~ = 0
(12) + 5)A 3
(13)
/
[ = h2(~) = ( k D ) - l / 2 ( 1 2 / n 2 F )
f o r ~ = 0, (1 + 5)A 3 ~< } ~< (1 + 5 + u)A 3
(14)
where 5 = (x2 - - - X l ) / X l and p = (x 3 - - x 2 ) / X l
(15)
T h e c o n d i t i o n (6) can be t r a n s f o r m e d t o c=0
forT=0,(l+5)A
3~<~< (1+5
+p)A 3
(16)
Taking t h e Laplace t r a n s f o r m s with respect t o ~, the b o u n d a r y value p r o b l e m described b y eqns. (9)--(14) can be r e d u c e d to TSC = d2c-/dT 2 - - c-
(17)
with K -~ 0
for T -~
d c / d T = h-(s) f o r ~ = 0
(18)
(19)
T h r o u g h o u t this paper, we d e n o t e the Laplace t r a n s f o r m of a f u n c t i o n ¢(~) as ~(s) or £~b, w h e r e s is the t r a n s f o r m e d variable. T h e solution o f eqn. (17) with eqn. (18) has the f o r m : b- = const. Ai(sl/3 T + s - 2 / 3 )
52
where Ai denotes the Airy function [10]. Substituting the above equation into eqn. (19) leads to (c-)~=o = --f-o(S) h-(s)
(20)
where f-o(S) = --
Ai(s-2/3) sl/3 Ai'(s-2/a)
(21)
The inverse Laplace transform of eqn. (20) is (c)~=o = -- f
h(t)fo(~ -- t) dt
(22)
0
where fo(~) is the reverse Laplace transform of eqn. (21) and its expression in the form of ascending and asymptotic series with respect to ~ is given in Appendix I. Introducing the condition (16) into eqn. (22) and taking into consMeration the boundary conditmns (12)--(14), we obtain for ~ >~ (1 + 5)A a A3
f
h2(t)fo( ~ - t )
dt = -
(1+6)A 3
f
hl(t) f o ( ~ - - t ) d t
(23)
0
This is the Volterra integral equation of first kind with respect to the u n k n o w n function h2(~). Solution of this integral equation is greatly facilitated by introducing the function g(~), which was used in Part I to derive the equation of catalytic currents and is defined in the Laplace transform as follows:
~(s) =--s -~la Ai'(s-2/a)/M(s -2/a)
(24)
In Appendix I, the expression of g(~) is Dven in the form of ascending and asymptotm series with respect to ~. The function g(~) fulfills the relation:
f fo(t)g(~ -- t) dt = f fo(~ -- t)g(t) d t = 1 0
(25)
0
for any value of ~, since g(s)fo(s) = 1/s, as can rea&ly be seen from eqns. (21) and (24). Utilization of the function g(~) allows us to obtain the solution of eqn. (23). Writing u instead of ~ in eqn. (23), multiplying both sides of the resulting equation by g(~ - - u ) , then integrating with respect to u from (1 + 6)A a to ~, and differentiating with respect to ~, we obtain d ~ h2(})=--d~f
A3
g(}--u) f
(1 +6 )A 3
hl(t)fo(u--t)dtdu
0
where the following calculation was used:
d
(
~
u
-- u) ( ]
(3I + 5d ) A
(1+5)& 3
- - t) dt du
(26)
53 d
¢
= ,4--~ {"
h2(t)
(1+6)A 3
d = d--~ (f+
¢
[" ro(U --t)g(~--u) du dt t
h2(t) dt = h2(~) 6 )A 3
Noting eqns. (12) and (14), we obtain for the current density at the detector electrode _ n 2
/2(~)
d
A3
~
nl d~ f
kl +,5 )/\3
g(~---u) f
il(t)fo(U -- t) dt du
(27)
0
The total currents flowing through the generator and the detector electrodes are given by x1
I1 = w f
,\3
a-3 f
tl(X) d x = W X l
0
il(t )dt
(28)
0
and x3
I2=w f
i2(x) dx
x2
n2 nl
(1+6 +,a)A 3 WXI n -3
,\3
g([1 + 6 + p] A a -- u) J
¢ (1+5)A t pA 3
n2
tl(t) fo(U-- t) dt du
0 A3
il(t)fo([1 + 6 + p]A a - - t - - u) dt du 0
(29)
0
respectively, where w is the c o m m o n width of the generator and the detector electrodes. Thus, the collection efficiency N ls given by 22_
rt2
~A3
\3
A3
N = I 1 nl f
g(u) f
h(t)fo([l+5+p]Aa--t--u)dtdu/ f
0
o
o
h ( t ) dt (30)
So far, we have not imposed any restriction on il. Thus, the collection efficiency can be calculated from eqn. (30) for an arbitrary current density distribution at the generator electrode. However, what we can control externally is n o t the current density, but the total current flowing through the entire electrode surface. Therefore, except for the electrodes with a uniformly accessible surface, such as the rotating disk electrodes [3] and the stagnation-point electrodes [9], it is quite difficult to control the current density distribution at the electrode surface. On the other hand, under the conventional voltammetric condmons, in which the solution contains a large excess of supporting electrolyte and the dimensions of working electrode are very small, the electrode potential may be assumed to be constant over the entire electrode surface. Thus, it ls desirable, in
54 general, to employ the condition of controlled-potential electrolysis. On the ground of this consideration, we shall here evaluate the collectmn efflcmncy for the followmg three cases: (A) In scheme I, the potential of the generator electrode is assumed to be set on the hmiting current regmn of the electrode process (1). In this case, the current density h ( x ) can be expressed by [6]
(31)
il(X) = const, x -1/3
When the electrode process (1) proceeds reversibly, the current density distnbutmn of eqn. (31) holds for any given potential along the current-potential curve. (B) In scheme II, the same electrolysis condition as m case (A) is employed. As shown m Part I [8], the current density ll(x) is given by /I(X) = const,
g(AaX/Xl)
(32)
(C) In scheme I, the current density l l ( X ) lS assumed to be com pl et el y controlled by the kmetms of the electrode process (1). In this case, Zl(X) is expressed by /I(X) =
const.
(33)
This situation can be obtained, when the electrode process (1) is totally irreversible and the potential of the generator electrode is set on the f o o t of t he currentpotential curve. Substituting eqns. (31)--(33) into eqn. (30), we obtain for the cases (A), (B) and (C) pA 3
A3
fo([1 + 5 + p ] A 3 --- t - u ) t
3 nI
0
1/3 dt du
(34)
0
/22 ~ / k 3
\3
Ncat=h~a J
g(u) f
0
0
:\3
fo([l+5+/alA3--t--u)g(t)dtdu/
f
g(t)
dt
(35)
0
and p\3
172
\~
glll : __ ]•--3 f
g(u) f
0
0
/21
fo([1 + 5 + p] A a -- t - u) dt du
(36)
respectively. SERIES EXPANSIONS FOR NUMERICAL CALCULATIONS Since the expressions for the collection efflcmncms, eqns. (34)- t 3 6 ) , contain double integrals, it is difficult to proceed further with the analytic evaluatmn of N's. Hence, we expand eqns. (34)--(36) into ascending and asymptotm series for small and large values of A, respectively. Then, we can readily evaluate them by means of numerical calculations.
55
Ascending serws First, consider the most complicated case of eqn. (35), because it is possible to express eqns. (34) and (36) as the special cases of eqn. (35). The ascending series of f0(~) and g(~) are obtained in Appendix I as eqns. (I-5) and (I-8), respectively. Thus, substituting them into eqn. (35), we obtain, after performing some calculations and rearranging terms,
nl
=
k=O F
['
F
o
[,
J
(37) where the coefhcmnts p's and q's are defined by eqns. (I-4) and (I-7), respectively, and T~jk is given by /zA 3
T,,k = f o
A3
u(2J-1)/af
((l+5+p)Aa--t--ub(2'-2)/at(2k-1)/adtdu
(38)
o
The double integral in T, jk can be obtained as follows: substituting t = ((1 + 5 + p)A 3 - - u } v into eqn. (38) yields p 43 T,j k = f u(21-1)(3{( 1 + ~ + ]A)a 3 - - u } 20+k)/3 0
1+5 +p)Aa_u
3
'
3
where B is the incomplete beta function, defined by
B(t; a,
t b) = f va-1 (1 -- v) b-1 dv o
(40)
Expanding the incomplete beta function into the ascending series [10], and transforming the integral variable by u = ((1 + 5 + p)A 3 } v, we obtain + n (1 + ~ + p)2o+j)/3
F
n
T,I k = A20+j+k+l)
n=0
× B
+5 +g '
M( ~ 2 + 2- k
3
'
3
+n ) F (2~)
n
(41)
Introducing eqn. (41) into eqn. (37) and performing some rearrangements allows us to write the final result: No+~F(5) ~ / ~ __,~1__ \,5,/ m = 1 [ I , I , k ) O
p,q, qkA~klA am
Ncat = - -
1 + [' ( 5 ) , ~ 1
{ q m / [ ' (2-2~m3+~5)} A 2m
]
-
(42)
56
with
"/
No = n~ ~2/3G(1/5) + G(/J/6) -- (1 + ~ + p)2/3G 1
A1jk-
;, B 1 + 6 + p
(
"
'
)}
(
~ { r (2-3~2~+ n) (1 + (~ + p)20+J)/3---n
3
'
3
n
)1
(44)
where the functmn G is defined by [4,11] 31/2 ?
-
d~
_
In 4~
+
tan_ ~ 2 01/3 31/2
(1 + 01/3) 3
+ --
(45)
4
As shown m Appendix VI, (n2/n 1)F(5/3)Aoo 0 is equal to the diffusion-controlled collection efficiency, No. Hence, in eqn. (42) N o is introduced into the numerator instead of (n2/nl)F(5/3)Aoo o. Next, we treat the case (A). Taking into conslderat~on that the current density il at the generator electrode for the case (A) is given by the first term of the ascending series for g(~) [8], the expressmn of Nr~,, can be obtained by setting the subscript k = 0 in the numerator of eqn. (42) and omitting the sum in its denominator. Thus, we have l+l=m
N~e, = N0 +n2- F ( 5 ) ?/1
~
(t)lqjAuo} A 2m
~
(46)
m=l 1,130
Finally, consider the case (C), m which the u m f o r m accessibility at the generator electrode is satisfied. As shown in Appendix VII, the expression of N ~ can be gwen by 1+ J= 171]
N m = N 0 '+ ~ mn2= l 2 { 2 ~ ~,j~o p l q j A I J ½ } A 2m
(47)
with
No =n~ nl
(I+5+p)G
P
-J6
--(5+p)G
+~-
[(1+
--51/3 ]
(48)
where N~) is the diffusion-controlled collection efficiency corresponding to the umform current density dlstribui~ion at the generator electrode and Alj 21 is given by eqn. (44) m which the subscript k is formally set equal to 1/2.
57
Asymptotic series In the following, we shall derive the asymptotic expressions for N's, which can be used conveniently for large values of A. Exchanging the order o f integration in eqn. (30), we obtain A3
N
n2
pA 3
A3
=n-1 f
/l(t)f
g(U)fo([1 + 8 + p ] a a - - t - - u ) dudt / f
0
0
0
il(t) dt
(49)
The asymptotic expression for the inner integral in the numerator of eqn. (49) is given in Appendix III. From eqn. (III-12), it follows A3
N ~ I n2 {f
i l ( t ) [ f l ( [ l + 5 ] A a - - t ) + ~ ( g 2 _*n ( P A 3 ) f n ( [ l + 5 ] A a - - t )
0
n=l A3
--an--lg~--n([1 + ~ + p ] A 3 - - t ) } ]
dt}/f
il(t ) dt
(50)
o where the coefficients a~ are tabulated for the first 11 terms in Table 3 in Appendix I and the functions fn and g~ are given below (see eqns. 54, 55 and 57). In order to proceed further, the explicit expression of il(x) should be introduced into the above equation. For the case (A), eqn. (31) can be substituted for il(x). Then, the following two types of integrals appear in eqn. (50): A3
U, = f
t-1/a f , ( [ 1 + 6]A 3 -- t) dt
(51)
t -1/3 g~_n([1 + 5 + p]A a -- t) dt
(52)
0 A3
0
These two integrals are evaluated in Appendices IV and V, respectively. From eqn. (IV-7), we have ~ o F ( J + 1) A - a , - 1 fn+,+1(5A3) 1 2 _ __ F(g)rn+213([1+ 5 ] A 3) Un-- = _F(1/3)
(53)
with n--1
= Z; (-1), :1=0
fn(})
an-j-1 ~ J!
(--1) nF(1/3) ~ + 31/aF(2/3)
~(2J+3n--2)/3
~--oPJ = F(~j+n+½)
(for small values of ~) -f~(~)
f~(~) = 1 ~ 7r]= 0
(54)
(for large values of ~) b~{2/(3 ~)l/2}-v+j+l K-v+j+l(4
~1/2/31/2)
(55)
58 t
t
where the coefficients aj and bj are t a b u l a t e d in Table 3 and K~, is the m o d i f i e d Bessel f u n c t i o n o f the second kind [ 1 0 ] . F r o m eqns. (V-8) and (V-9), U~ is given as
{g~ n([6
Un~-Ag~_n([(5 + p ] A 3 ) - ( A + -32 g 2 _ n ( [ 1
+
6 + ~]
+
td A3)) 2 2/3)g~_n([6 + p ] A 3)
A 3 -- to)
t~/3
(56)
with b,{2/(3 })1/2}
g~(}) = 1 ~
v+,+l K_,+j+I(4 }1/2/31/2)
(57)
7r j= 0
to = ((1 + 4811 + 6 + p] A3) 1/2 -- 1 } / 2 4
(58)
where the coefficients bj are given in Table 3. T h e n , a f t e r p e r f o r m i n g the integratlon in the d e n o m i n a t o r o f eqn. (50), we o b t a i n the following a s y m p t o t i c expression for N~,.: N~ev
n2 n-~ 23 A-2{UI + ~
[g*2-n(~tA3)Un--a~a-IVn]}
(59)
n=l
For the case (B), il(x) is given by eqn. (32). Thus, if eqn. (32) IS introduced into eqn. (50), the following three types of integrals appear: A3
Vn=f o
g(t)fn([1 + 61A 3 - - t ) dt
(60)
g(t)g~_n([1 + 6 + p] A 3 -- t) dt
(61)
g(t) dt
(62)
A3
v =fo
A3
S
=f
o T h e a s y m p t o t i c expressions o f the first t w o integrals are derived In A p p e n d i c e s III and V, respectively, and the last integral was evaluated in Part I [8]. We have
gn ~ fn+l(6A3) ~ - ~ a'n--~gj*+l([1 + 6] A3) j=l
+E
3 * 3 )--an+j--I ' (fn+j(6A)gz-j(A g2--j([1 + 6]A3)}
(63)
j=1
Vn -
(1 +g~(A3)}2{g2_n([6 +p]A3)} 2
(64)
* 3 --go(A )g2_n([6 + p]A3) + {I + gl(A3)}gl_n([6 + p]A3)
S ~ A 3 + 0 . 2 5 - - g ~ ( A 3)
(65)
59 Using the above three asymptotic expansions, we obtain the following asymptotic expression for Neat:
(66)
Neat _ n~ S - 1 (Vl + ~ [ g~ n(pA3)Vn __ an--1Vn]} /21 n=I
For the case (C), we introduce eqn. (33) into eqn. (50) and p e r f o r m the indlcated integrations to obtain
Mr, . n2. A_. 3(f2(aA3) f2([1+5]A3)+ ~ . . nl
n=l
* ~)[f.+i(~A 3) g2_.(~A (67)
. f n + l ( [ l. + ~]A3)] .
. ~ . an--l[g~--n([8 + p ] A 3) n=l
g3_n([1 * + 5 +p]AS)]}
where the relations (II-1) and (I-23) were used.
RESULTS AND DISCUSSION
Nu rnerzcal calculattons The values o f the collection efficiencies N for the three cases were calculated numerically on a digital c o m p u t e r HITAC 8700 as functions of the kinetic parameter A ~ for various combinations of the values of the geometry parameters of double electrode, 5 (0--1.0) and p (0.25--3.0), where eqns. (46) and (59); (42) and (66); and (47) and (67) were used for the cases (A), (B) and (C), respectively. Equations (46), (42) and (47), which are expressed in ascending series, would give, in principle, precise values of N for any value of A 2, if there were no physical limitations of fixed-word-size m e m o r y and no fixed-decimalpoint restriction. In practice, however, such limitations may lead to t runcat i on and rounding of individual values, which action may cause errors in the result of computation. Thus, for the values of A 2 up to 2.0 (sometimes up to 3.0 or more for special combinations of the values of 8 and p), eqns. (46), (42) and (47) were used to obtain the values of N with considerable accuracy (with errors less than 1 0 - 6 ) , where the n u m b e r of terms summed up In the series was less than 50. F o r large values of A 2 (greater than 1.5), eqns. (59), (66) and (67), which are expressed in asymptotic series, were used. The summation in the series usually included the first four terms. The values of N for the intermediate region of A 2 can be calculated by using bot h the ascending and the asymptotic series. Both results are in agreoment within a m a x i m u m difference of 2%. Figures I and 2 indicate typical results of numerical calculations. In Fig. 1, the variation of the ratio, Nrev/No, with A 2 for the case (A) is given for p = 1.0 and for a series of values of 5. Figure 2 shows the dependence of Neat~No on A 2 for the case (B) for the same electrode geometries as in Fig. 1. Comparison of Figs. 1 and 2 indicates that the following inequality holds in general:
Neat > Nrev
(68)
For the case (C), results of numerical calculations indicated that the values of
N,~/No are almost equal to the corresponding ones ofNeat/N o for any values of
60
i
I°I
lO
08
05
O6
O6
2
~04
Z
O4
02
0 0
I
2
3
0
4
1
A2
2
3
4
A2
Fig 1 VarlaDons of t h e ratio, N r e v / N 0 , with A 2 for the case (A) for various g e o m e t r y param=(a) 00,(b) 005,(e) 02,(d)05and(e)10 eters of d o u b l e e l e c t r o d e t ' = 1 0 a n d b Fig 2 V a r i a t i o n s of t h e ratio, N c a t / N O, with A 2 for t h e case (B) for various ~ e o m e t r y parameters of d o u b l e e l e c t r o d e p = 1 0 a n d 5 = ( a ) 0 0 , ( b ) 0 05, (e) 0 2, (d) 0 5 a n d (e) 1 0 .
6 , / l a n d A 2. T h e d i f f e r e n c e s b e t w e e n these values b e c a m e greatest w h e n A 2 was a b o u t u n i t y and t h e y were less t h a n 2%. Thus, we can c o n c l u d e t h a t
NI~/N~ ~ Neat/No.
(69)
Approximate equatmns for N I t is desirable to derive simple a p p r o x i m a t e e q u a t i o n s which r e p r e s e n t with r e a s o n a b l e a c c u r a c y the N - - A 2 r e l a t i o n f o r given regions o f the g e o m e t r y p a r a m eters, 6 and/~. Detailed i n s p e c t i o n o f the results o f n u m e r i c a l c a l c u l a t i o n s suggests t h a t the N - - A 2 r e l a t i o n can be e x p r e s s e d b y
N/No = exp(- -~A 2 +/~A 4 - - 3,A 6) f o r 1.0 ~> N/No >~ 0.08
(70)
and
N / N o = exp(--c~' + ~ ' A 2 - - 3/A 4)
f o r 0.1 ~> N / N o >~ 0 . 0 0 2
(71)
respectively. T h e c o e f f i c i e n t s a, ~, etc. can be given b y p o l y n o m i n a l s o f the g e o m e t r y p a r a m e t e r s o f d o u b l e e l e c t r o d e , as follows" 3
.3
C~ = E
E C~m,n 6 m g n m=O n=O
(72) I ~r
w h e r e c~ stands for a,/J, 7, a ,
and 7
r.
61 F o r t h e c a s e ( A ) , w h e r e N/No in e q n s . ( 7 0 ) a n d ( 7 1 ) is Ntis~No, t h e v a l u e s o f am ,n, t3m,n, e t c . a r e t a b u l a t e d in T a b l e 1. E q u a t i o n ( 7 0 ) is v a l i d f o r 0 . 2 5 ~ p ~ 2 . 0 a n d 0 . 0 ~< 5 ~< 1 . 0 w i t h i n a m a x i m u m e r r o r o f 2%. I t s h o u l d b e n o t e d t h a t if t h e r e g i o n o f 8 is n a r r o w e d s u c h as 0 . 2 ~< 5 ~< 1.0, e q n . ( 7 0 ) h o l d s in w i d e r r a n g e , i.e., 1 . 0 >~ Nrev/No >~ 0 . 0 2 , w i t h i n t h e s a m e e r r o r . O n t h e o t h e r h a n d , e q n . ( 7 1 ) h o l d s f o r 0 . 7 ~< p ~< 3 . 0 a n d 0 . 0 5 ~< 8 ~< 0 . 7 w i t h i n a n e r r o r o f 2%. F o r t h e c a s e (B), in w h i c h N/No c o r r e s p o n d s t o N c a t / / N o , e q n . ( 7 0 ) h o l d s f o r 0 . 2 5 ~< p ~< 2 . 0 a n d 0 . 0 ~< 5 ~< 1 . 0 a n d e q n . ( 7 1 ) f o r 0 . 7 ~< p ~< 3 . 0 a n d 0 . 0 5 ~< 5 ~< 0 . 7 , b o t h w i t h i n a n e r r o r o f 3% ( u s u a l l y less t h a n 2%). T h e v a l u e s o f t h e coe f f i c i e n t s a m , n , tim,n, e t c . f o r t h i s c a s e a r e g i v e n m T a b l e 2. A s c a n b e s e e n f r o m e q n . ( 6 9 ) , t h e v a l u e s ofN, rJNo f o r t h e c a s e (C) a r e e q u a l t o t h o s e o f Ncat/No w i t h i n a m a x i m u m e r r o r o f 2%. T h u s , t h e s a m e a p p r o x i m a t e e q u a t i o n s as f o r Ncat//N0 c a n b e u s e d f o r N~rr/N'o. Since the logarithms of eqns. (70) and (71) are cubic and quadratic equations w i t h r e s p e c t t o A 2, r e s p e c t i v e l y , t h e v a l u e s o f A c a n e a s i l y b e c a l c u l a t e d f r o m t h e m e a s u r e d v a l u e s o f N , a n d in t u r n t h e v a l u e s o f t h e r a t e c o n s t a n t k c a n b e o b t a i n e d b y t h e u s e o f e q n . (8).
Effect of the length of detector electrode I n g e n e r a l , i t is e x p e c t e d t h a t t h e l o n g e r t h e d e t e c t o r e l e c t r o d e , t h e h i g h e r t h e c o l l e c t i o n e f f i c i e n c y , if o t h e r e x p e r i m e n t a l v a r i a b l e s a r e k e p t c o n s t a n t . W e e x a m me the effect of the length of detector electrode on the collection efficiency. S i n c e a s i m i l a r t r e n d o f t h e e f f e c t c a n b e e x p e c t e d f o r t h e t h r e e cases, w e c o n s i d e r h e r e t h e s i m p l e s t c a s e (C) in s o m e d e t a i l . L e t t i n g p -~ ~ m e q n . ( 6 7 ) , w e
TABLE 1 The values of the coefflcmnts used in eqn (72) for Nrev/N 0 (case A) .
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m n
O~,n,n
/3m,n
')'m,n
O~na,n
~'m,n
'~nl ,n
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3
1.0191E0 4.7604E-1 --1 9917E-1 4 0072E-2 2 6344E0 --1 6944E0 1 1264E0 --2 3247E-1 --2 3079E0 2 9096E0 --1.8598E0 3.4469E-1 1 1972E0 --1 6474E0 1 0080E0 --17168E-1
2 0224E-1 --34945E-2 5 7600E-2 --1.1940E-2 --2 2514E-1 2 1858E-1 --1.9006"E-1 3 8814E-2 6 7679E-1 --4 5329E-1 2 9534E-1 --3 4239E-2 --3.5639E-1 2.5321E-1 --1 3319E-1 20130E-3
2 3139E-2 --14887E-2 I 2265E-2 --18591E-3 --2 9141E-2 5 5107E-2 --2 5154E-2 1 3668E-3 6 2330E-2 --1 8852E-2 --4 6552E-2 2 9681E-2 --2 2546E-2 --1 6230E-2 5 8548E-2 --28266E-2
--9.9994E-2 60931E-1 --2 3869E-1 3 3748E-2 1 6482E0 --2.3220E0 6 6628E-1 --5.3692E-2 --4 6772E0 5 8737E0 --2 2224E0 2 5628E-1 4 1372E0 --5 2472E0 2 3546E0 [-3 2398E-1
--8 6814E-1 --1 8720E-1 9 4952E-2 --16921E-2 --2 1414E0 1 2392E0 --1 2994E0 2 8266E-1 2 3822E0 --5 0082E0 4 ]354E0 --8 6295E-1 --1 6415E0 4 3979E0 --3 3752E0 69319E-1
--6.1625E-2 --63848E-3 2 3591E-3 --48508E-4 1.4544E-1 --1 8966E-2 --5 9076E-2 1 6780E-2 --4 4010E-1 --1 8386E-2 1 5123E-1 --4 2241E-2 3 3496E-1 --1 5051E-2 --7 8071E-2 24822E-2
0 1 2 3 0 1 2 3 0 1 2 3 0 I 2 3 .
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62 TABLE 2 The values o f the c o e f h c l e n t s used m eqn (72) for N c a t / N 0 (case B) and for
N~r/No
(case C)
m n
C~m,n
/~m ,n
"/m,n
121m,n
~m,n
~m,n
0 0 0 1 0 2 0 3
8 9953E<1 5.8226E-1 --27649E-1 60111E-2
2 2 6 --3
0833E-1 2754E-2 2753E-2 1880E-2
2 7627E-2 1 8513E-2 --1.2062E-2 2 3084E-3
--4 2700E-1 1 8085E0 --1 1029E0 20463E-1
--8 8241E-1 4 2964E-1 --36093E-1 7 4217E-2
--7 5533E-2 9 5692E-2 --7 1734E-2 14200E-2
1 1 1 1
0 1 2 3
2 --2 2 --6
7982E0 3925E0 1791E0 3045E-1
4 --3 --4 1
8814E-2 0285E-1 0137E-3 0113E-1
6.5984E-2 --3 0181E-1 3 2668E-1 --1 0165E-1
5 -1 1 --2
4209E0 8539E1 2946E1 5214E0
--7 --7 5 --1
0979E-1 0335E0 1297E0 0235E0
4 --1 1 --1
6030E-1 4525E0 0161E0 9853E-1
2 0 2 1 2 2 2 3
--2 4 --4 1
3579E0 2897E0 4093E0 3686E0
1 6 --5 1
5484E-1 2286E-1 3587E-1 2120E-1
--1 6 --1 4
0567E-1 8875E-1 0127E0 0746E-1
--1 4 --3 6
0541E1 5240E~ 4104E1 8595E0
1 1 --1 2
1715E0 3961E1 1953E1 5287E0
--1 3 --2 5
0200E0 5963E0 7013E0 4285E-1
3 3 3 3
1 --2 2 --8
1499E0 3873E0 5480E0 0934E-1
--3 5 --4 1
3409E-1 9335E-1 3625E-1 3555E-1
--1 1 I --9
0004E-1 8496E-1 0100E-1 1527E-2
4 --3 2 --5
9643E0 1034E1 5182E1 2129E0
--3 --7 7 --1
0551E0 0294E0 6457E0 7356E0
5.101~E-1 --2 4697E0 1.9958E0 --4.1234E-1
0 1 2 3
obtain ( N ~ ) u _ ~ = n~ A - 3 {f2(SAa) __ f2([ ] + (~] A3)}
(73)
/'/1
In Fig. 3, the values ofN, rr/(Nlrr)~_~ are p l o t t e d against p for 5 = 0.5 and for several values o f A 2. This Figure shows t h a t the ratio Nlr~/(Nt~)~-~ is increased with increasing values o f p and b e c o m e s a l m o s t c o n s t a n t and close t o u n i t y for p greater t h a n 1.5. This fact indmates t h a t if p / > 1.5 the c o l l e c t i o n e f f l c m n c y hardly d e p e n d s on p for large values of A, especially for A 2 ~> 2.0. F o r the cases (A) and (B) a l m o s t the same b e h a v i o r prevails as for the case (C) described above.
Range of measurable rate constants N o w , we estimate the range o f rate c o n s t a n t s which can be m e a s u r e d b y means o f the d o u b l e channel electrode. The d i m e n s i o n s o f the cell w h m h is usually e m p l o y e d m our e x p e r i m e n t s are: w = 0.5 cm, b = 0.03 cm, xl = 0.02 cm, x2 - - X l = 0.01 cm (f = 0.5) and x 3 - - x 2 = 0.02 cm (p = 1.0). A s s u m i n g t h a t the d e t e c t a b l e limit o f N ~ , is 2 × 10 - 3 for fast reactions and t h a t N~ev m u s t be smaller t h a n 0.9 No for slow reactions, we have f r o m eqns. (70) and (71) t h e following c o n d i t i o n : 0.05 < A 2 < 3 T a k i n g t y p i c a l value o f D o f 1 X 10 -~ c m 2 s - 1 , we o b t a i n f r o m eqn. (8) 0.3
2 / 3 < 2 0 ( c m -2/3s 1/3)
Since the m e a n flow v e l o c i t y Um can be v a n e d easily b e t w e e n 5 and 5 0 0 cm s- 1 ,
63
~05 z
0
1
2
Fig 3. Variations of the ratm, A 2 are given on the curves
3
Nlrr/(N~rr)
p ,o~
with p for various values of A 2 The values of
we have 1 S- 1 < k < 10 3 S- 1 . O f course, the u p p e r limit o f k can be raised b y m a k i n g Xl (length o f g e n e r a t o r electrode), x2 - - x l (length o f gap) and b (halfhmght o f channel) smaller, and conversely the l o w e r limit is r e d u c e d b y increasing Xl, x2 - - X l and b. When the i n t e r m e d i a t e reacts with a specms Z the c o n c e n t r a t i o n o f which can be varied so as to satisfy the c o n d i t i o n o f pseudo-first-order r e a c t i o n , t h e n k = k' C z
,~z shou:d Ee in the fo!low[ng range: 1M>cz>
10 - a M
so t e a t we have for the s e c o n d - o r d e r rate c o n s t a p t k' l M - -~s - 1 < le' < 1¢ 5 7If- 1 s--1. In conclusion, it can be said t h a t the range o f measurable rate c o n s t a n t s at the d o u b l e ehat:nel e l e c t r o d e is c o m p a r a b l e to th,~; o f R R D E .
Approximr,te procedure for fast homogeneous reactzons In o r d e r to solve a b o u n d a r y value p r o b l e m for e l e c t r o d e preces,;es coupled with a fast l~orr,.ogeneot~s reaction, an a p p r o x i m a t e procedure~ b ~ e d o n the ass u m p t i o n t h a t the thickness o f the r e a c t i o n layer is m u c h t h i n n e r t h a n t h a t o f the diffusion layer, is o f t e n e m p l o y e d . In p o l a r o g r a p h y , this a p p r o x i m a t i o n corresponds t o the neglect o f the t i m e - d e p e n d e n t t e r m in the r e a c t i o n layer, i.e., the so-called steady-state a p p r o x i m a t i o n , whereas in h y d r o d y n a m i c v o l t a m m e t r y dealing with the s t a t i o n a r y state it c o r r e s p o n d s t o the omission o f the convection t e r m in the r e a c t i o n layer. This a p p r o x i m a t i o n has been applied t o the convective diffumon e q u a t i o n at the g e n e r a t o r e l e c t r o d e b y A l b e r y and Bruekenstein [7] and by B r a u n [5] to o b t a i n the expressions for the c o l l e c t i o n e f f i c i e n c y at the R R D E and the d o u b l e channel e l e c t r o d e , respectively. In this section, we apply this a p p r o x i m a t e p r o c e d u r e to the p r e s e p t p r o b l e m and discuss its validity. T h e solution o f eq~. (9) w i t E o u t the c o n v e c t i o n term, t o g e t h e r wtth the con-
64 dltions (10)--(12), 1S readily obtained to be c =---hi(}) e x p ( ~ ) Thus, at the end of the generator electrode or at } = A a, c = --hi (A a) exp(--~q)
(74)
Now, the problem to be solved is eqn. (9) with the boundary cohdittons (11~, (13), (14), (16) and (74). Writing }1 = } -- A a and introducing the function: = exp(~r~) + C / h l ( A a)
(75)
we have ~
'/~ ~ 1
~2~ -
~T]2
~
(76)
with ~=0
for}l=0,
~0
f o r } l > 0, r? ~ oo
~_~ + 1 =
h(}l)
[=0
hl(A3) I - h2(}) hl(A 3)
r?> 0
for0~<}l<~A3,7?=0
(77)
for 5A 3 ~< }1 <~ (5 +/~)A 3, r? = 0
(78)
Taking the Laplace transforms with respect to }1 and solving the resulting boundary value problem by using the procedure similar to that used to obtain eqn. (20) from eqns. (17)--(19), we obtam
Sl--il-(~)}f-o(Sl)
(';)~=o=/1 h(sl)
(79)
where sl is the transformed variable of ~l. From eqns. (75) and (79), we obtain m view of eqn. (II-8) (b-)~=0 = --hl(A a) fl(S1) --fi(S1) f-o(81)
(80)
The inverse Laplace transform of eqn. (80) is
E1 (c)~ =0 = --hl(A a) f l ( } l ) -- jf h(t) fo(~l -- t) dt 0
(81)
Introducing the conditions (16), (77) and (78) into eqn. (81) and using the same procedure as used to obtain eqn. (26), we have d g(~l -- t) fl(t) dt h2(}1) = --hl(A3) d ~ l f s\ 3
(82)
Hence, taking into account eqns. (12) and (14) and integrating eqn. (82) over
65
the surface of the detector electrode, we obtain for the total c u r r e n t / 2 ,tt ,\ 3
12 = (n2/nl) w x l A - 3 i l ( A 3) f
g(t) fl([6 + p ] A 3 -- t) dt
(83)
o
In view of eqn. (III-1), eqn. (83) can be rewritten as
Iz = ( n 2 / n l ) w x l A - 3 i l ( A 3) V I ( p A 3 ; [6
+ p ] A 3)
(84)
Thus, it follows from eqns. (28) and (84), that A3
N = n2
n~ il(Aa)VI(PAa; [8 + p]Aa)/ f
i l ( t ) dt
(85)
0
Since we are concerned with large reaction rate, we only consider the asymptotic expression of eqn. (85). For such a case, as can easily be derived from eqn. (III-13), VI(pAa; [8 + p ] A 3) is expressed as V I ( p A 3 ; [(~ + p ] A 3) = f2(SA 3) -- g2([5 + p ] A 3) +
* 3 ' * {fn+l(SA3 )g2_n(PA )--ang2_n([5 +p]A3)}
C
(86)
n = 1
Substituting eqns. (31) -(33) into eqn. (85) yields Nrev -
(n2/nl)(2/3)A-3VI(pA3; [5
Ncat ~
(n2/nl)(1
+ p]A 3)
+ g l ( A 3 ) } S - 1 V I ( p A 3 ; [5 + / d ] A 3)
N~r ~ (n2/nl)A - 3 VI(pA3; [5 + p] A 3)
(87)
(88) (89)
where the asymptotm series of S is given by eqn. (65).
1
2
3
4
Az
Fig 4 Comparmon of the approximate soluhons with the exact ones for two sets of geometric parameters ( 1 ) p = 2 0,6 = 0 1,(2) t2= 2 0 , 6 = 0 5 (R) log(Nrev/No) ( - ) exact curves, calculated from eqns (46) and (59), ( - ) approximate curves, calculated from eqn. (87) (C) log(Ncat/No). ( - ) calculated from eqns (42) and (66), ( ) calculated from eqn (88). (B) Calculated from Braun's eqn (90)
66 In Fig. 4, typical plots o f log(Nro~/No) and log(N¢~t/No) agamst 1'\2 calculated f r o m eqns. (87) and (88) are s h o w n for 5 = 0.1, p = 2 0 and 8 = 0 5, p = 2.0, t o g e t h e r with the e x a c t values whmh were calculated f r o m eqns. {46), (59), (42) and (66). As can be e x p e c t e d f r o m eqn. (69), the values o f log(Nl,,/N()) o b t a i n e d f r o m eqn. (89) are n u m e r i c a l l y in g o o d a g r e e m e n t with those of log(Nc~t/N0) f r o m eqn. (88). We can see f r o m Fig. 4 t h a t m the cases (B) and (C) t h e a p p r o x > m a t e values a p p r o a c h a s y m p t o t m a l l y to the c o r r e s p o n d m g e x a c t curves w~th mcreasing values of A 2, while m the case (A) t h e y a p p r o a c h to the curves, which are b y a b o u t 10% l o w e r t h a n the e x a c t ones. This fact mdmates t h a t thin approximate p r o c e d u r e m a y be used o n l y w h e n a u n i f o r m c u r r e n t density d i s t r i b u t i o n is o b t a i n e d at the g e n e r a t o r e l e c t r o d e . I n c l u d e d m FN. 4 for c o m p a r i s o n are plots o b t a i n e d f r o m the e q u a t i o n derived b y Braun [ 5 ] : N~
~ (n2/n,)(1/6)A
a exp(----4 6A a)
(90)
As can be seen f r o m Fig. 4, Braun's results are u n s a t i s f a c t o r y for any values o f A 2"
APPENDIX I
In tins a p p e n d i x the reverse Laplace t r a n s f o r m s of the following t w o functions are derived" s - " f-o(S)
--
=
A;(s 2 , 3 ) _ _ _ sl/a+" Ai' ( s - 2 /3 )
(M)
and s-"g(s) =
-
-
Af(s-'~/a) s2/a +, Al(s-2/a)
(I-2)
where v is zero or any positive n u m b e r . Since these t w o f u n c t i o n s have singular points in the same region as for the f u n c t i o n ~-.(s) and f u r t h e r t h e y satisfy the c o n d i t i o n s of J o r d a n ' s l e m m a , the inverse t r a n s f o r m s can be o b t a i n e d b y the integration along the p a t h BCDEFG, which was s h o w n in Fig. 7 of Part I [8]. A s c e n d i n g serws F o r small Is1-2/3, eqn. (I-l) can be d e v e l o p e d to the following series: s_~,Fo(S ) _
F(1/3) ~ p~ 8--23/3 31/a F ( 2 / 3 ) j = o
,'
1/3
(i-3)
with j--1
P0 = 1, P l = ~1 -- X1, P~ = oz~ -- X~ -- ~
k=l
Pk Xj--k
(I-4)
where c% and X~ are d e f i n e d b y eqn. (A3) o f Part I. Following the p r o c e d u r e
67 described in the Appendix of Part I, we have ~--1
~21/3+v--2/3
{ S - - V f o ( S ) } --
r(1/3) ~ p, 3 1 / a F ( 2 / 3 ) , =o r ( § j + , + ~-)
(I-5)
where £ - 1 denotes the operator of inverse Laplace transformation. Similarly, s - " ~(s) is developed as follows: s-" g(s) -
31/3 F(2/3) g, 2_a q, s -2'la -v-2/3 P(1/3) ,=o
(>6)
with j--1
qo = 1, ql
=
X1
CO1, q3 = Xj -- cos -- ~
--
k=l
qk COj--k
(I-7)
Hence,
£ - - 1 {S- V g ( s ) } --
The
31/3F(2/3) ~ o ~2,13+,-1/a F(1/3) q' -P(~j + , + 5) - 2 2-
(i-8)
first 11 values o f p j and qj are listed in Table 3.
A s y m p t o t i c series
Applying eqns. ( A 8 ) - - ( A l l ) of Part I to s - " -[o(S), we obtain the following asymptotic forms along each path:
s - " fo(s) -
_,[C2 1 CID2 + C2D1 ] -e 4/asj s /~22+2 D~
(BC)
s - " C2/D2
(CD)
s -~ C 1/191
(DE)
s - " C2/D2
(EF)
s
_vfC2 /D~2
i 2
C1D2 + C2D1 D2
}
e 4/as_
(I-9)
(FG)
where 1 is the imaginary unit and C1, Ca, D1 and De are defined by eqns. (A13) and (A14) of Part I. Substituting eqn. (I-9) Into the inversion formula (A1), we have
BCDEFG
- 2rri
D
~s - D T d s + f
DE
s'--D~Tds+f
I£FG
e'~ ds s" D2
68
o
I
I
I
~
o
o
0
~
I
I
I
I
I
I
~
~
~
TT??TTTT?
E
I
I
i
I
I
c,.) ¢J
I co
I
]
I
r
!
69
FG
D~
S"
We d e n o t e the first and the second terms on the right-hand side of eqn. (I-10) by T1 and Te, respectively. Since the path DE is the semi-circle of radius p with its center at the origin and C~, C2, D1 and De approach u n i t y as p -~ 0, we have
f
(eS~/sv)(Cl/D1)ds = f (e~/s')(C2/D2) ds
DE
DE
Taking into a c c o u n t the above relation and i n t r o d u c i n g the following series representation: C~ = ~ a~ z ~ ([arg zl < u / 2 ) De j=o
(I-11)
we have ,
T 1 = E (-1), 3= 0
1
e st~ s j-'' ds a, ~ 1 1 f BCDEFG
The integrals in the above equation correspond to Hankel's integral representation of G a m m a function. Thus, T1 = ~
(--1)' a I ~ v - - I - - 1 / F ( P - - J)
(I-12)
1=0
When v is a positive integer n, the s u m m a t i o n in eqn. (I-12) is carried o u t f r o m j = 0 to n -- 1. For Te, by introducing the series expansion:
C1D2 + C2DI- ~ D~
b~ z 3 ( I a r g z l < u / 2 )
(I-13)
~=o
and p u t t i n g s = r e x p ( - ~ Q along the path BC and s = r exp(ui) along FG, respectively, we obtain T2 - cos PT~ ~ bj 2 ~ j=o
e
'~ -4/3r Fj--u
dr = cos p~ f;(})
(I-14)
0
w~th
=1 E
bI
K_~+j~ 1 ~ 3 ~ -
]
(I-15)
where K~, is the m o d i f i e d Bessel f u n c t i o n of the second kind of order p. Note t h a t f~ satisfies the following relation:
f ° f~(t) dt
:
f*~+~(~)
(I-16)
70 F r o m eqns. (I-10), (I-12) and (I-14), it follows t h a t ,
£-~ (s-'?0(s)} ~ G
}r--j--1
(--1) 1 a, F(u --]~ + cos uTr f2(})
(I-17)
1=0
If u is a positive integer n, eqn. (I-17) can be reduced to n--1 Z--1{8--n/o(8)}
~
G i
:0
, ~n--1--1 (-1) 1 a I - - - -
+ (--1)nf,~(~)
(I-18)
F(n---j) ,
The first 11 values of the coefficmnts a~ and b I are tabulated in Table 3. For the f u n c t i o n s -~ g-(s), a similar procedure as given above leads to oo
~c--l{8--Vg(S)} ~ G
j=0
(---1) ] 121 r ( p - - J
+ 1) + cos
pTr g~,+l(~ )
(I-19)
with
i -''+1+ 1 K_,+j+1 [4 \_3 ~}~/2 _ _ ]1 3 })1/2/
2 g~(}) = 1--1o~ bj (i -7r =
(I-20)
where a I and bj are the coefficients of the following series:
(D2/C2) = ~
a I z I (largzl<~r/2)
(I-21)
= 7_/ bj z J ( l a r g z l < l r / 2 ) 1=o
(I-22)
1=0
and ~1D2 + C2D1 C~
respectively. The first 11 values of aj and bj are listed in Table 3. We have further the following relation for g,*,
/g,*,(t) dt = gz[+l(~)
(I-23)
When u is a positive integer n, eqn. (I-19) is t r a n s f o r m e d as follows:
£--l(s--ng(s)} ~
(--1)1 al F ( n - - j + 1) + ( - 1
gn+l(~)
(I-24)
APPENDIX II In this appendix, we derive the two relations, which will be used m the subsequent appendices. (1) We define the f u n c t i o n fn(~) (n = 1,2,3, ...) by
fn(~) = /
f n - l ( t ) dt
(II-1)
71 where the f u n c t i o n fo(~) is the inverse Laplace t r a n s f o r m of fo(s) which is given by eqn. (I-l) with v = 0. Then, we show t h a t the f u n c t i o n fn(~) can be expressed by n--1 fn(~) = ~ (---1) } J=O
j!
+ (-1) n
{s-nyo(S)}
t
fn(0) = an-1
(II-2) (II-3)
where £ - 1 {s-nTo(S)} is given by eqns. (I-5) and (I-18) in the f o r m of the ascending and the a s y m p t o t i c series, respectively. For small Isl (larg si
1=0
(--1)~ a, s'
(II-4)
Making use of the initial-value t h e o r e m for Laplace t r a n s f o r m a t i o n , we have lim ~
y fo(t)
Thus, ?
(n-5)
dt = lim fo(s) = ao
0
s~O
f o ( t ) d t exists and
0
(n-6)
f l ( 0 ) ----? ~ /c0(t ) d r - - a 0 o
Then fl(~) =
?
fo(t) dt --
0
j
f o ( t ) d t = a'o - -
0
j
fo(t) dt
(II-7)
0
The Laplace t r a n s f o r m a t i o n of eqn. (II-7) and the use of eqn. (II-4) yields f l ( S ) = a 0' 8- 1 - - 8 - 1 #o(S) ~ ~ (---1)' 1=o
a >' , s
J
~II-8~
Applying the initial-value t h e o r e m again, it is easily shown t h a t f2(0) exists and is equal to a~. Now, we assume t h a t fn(S) = a,~--i 8 - 1 - - 8 - 1 } n _ l ( S ) !
fn(0) = an--1
(II-9)
(II-10)
and
~,,(s)
~3 (-1)'
1=0
an+,
sj
(II-11)
72 then, by the initial-value t h e o r e m we have hm f ~
fn(t) dt
=
0
hm fn(S)
=
(II-12)
an
s---O
Thus, we have m vmw of eqns. (II-1) and (II-12) fn+l(0)
! f,(t) dt= %
=
(II-13)
0
Hence, fn+l(}) =
fn(t) dt
fn(t) dt-0
=
an -- ; fn(t) dt
0
(II-14)
0
The Laplace t r a n s f o r m of eqn. (II-14) is (II-15)
Fn+l(S) = an s- 1 - - s --1 [,(s) Substituting eqn. (II-11) into eqn. (II-15) ymlds E (---1) I arn+l+l s ~ J=O
f-n+l(S)
(11-16)
Therefore, it can be concluded by means of m a t h e m a t i c a l i n d u c t i o n t h a t eqns. (II-9)--(II-11) hold for any pomtwe integer. Successive use of eqn. (II-9) leads to n
1
f-n(8) = E l=0
(---1) I an--i '
1 $ 1--1 + ( _ _ l ) n s--n f O ( S )
(II-17)
The inverse t r a n s f o r m of eqn. (11-17) is eqn. (II-2). (2) Next, we derive the following relahon:
5
{aforn:0
l=o
alan
l
0forn=l,2,3
(II-18) ....
Multiplying eqn. (I-II) by eqn. (I-21) gwes
~l=O
Hence
(1
--
a o a o' ) - -
~ n=l
I=O
a l a n' _ l
)
Zn:O
The c o n d i t i o n t h a t the above e q u a h o n should be vahd for any value of z, yields eqn. (II-18).
73 APPENDIX III
In this appendix, the a s y m p t o t i c series of the following integral is derived: X
Vn(X; ~) = f
g(t) fn(~ -- t) dt (n = 0, 1, 2, ...)
(III-1)
o
where X and ~ are large positive n u m b e r s and ~ > X. We write Vn(X; ~) = r 1 -- r 2
(III-2)
where
TI= 7 g(t) fn(~ -- t) dt
(III-3)
0
and
T2 = f g(t) [n(~ -- t) dt
(III-4)
As T1 is the c o n v o l u t i o n integral, it can be written as (III-5)
T1 = £--1 { g ( s ) fn(8)}
For n = 0, T1 is simply equal to 1, since g(s)fo(s) = s-1. For n/> 1, substituting eqn. (II-17) for fn(S) in eqn. (III-5) yields n--1
( _ l ) j a~-j--] s --j-1 g-(s) + (--1)n8 - n - l }
T] = £ - 1 { ~
(III-6)
3=0
Performing the inverse t r a n s f o r m a t i o n of eqn. (III-6) and using eqn. (I-24) leads to n--1
T1
--
(1+ 1
]=oD (--1)'
a'n_]_l/k~= o
~l+l_k
(--1)k ak 0 + l - - k ) !
*
(--1)'
]
~n
g]+2(~)J + (--1) n n~ (iii-7)
Changing the order of the s u m m a t i o n gives n
T1
ZJ ( - 1 ) '-1 ]=0
~J n - - ] ~ k=O
t
~
r
an_l_ka k +
a n
n--1 t
,
~n
- - ~ a n - - , - 1 g,+2(}) + ( - - 1 ) n ~ j=0
(III-8)
By use of eqn. (II-18), it can be seen t h a t the first term on the right-hand side is reduced to ( - 1 ) n-1 ~n/n!. Hence, we have n--1
T1 -a'.-
~
l=O
a~'~_,__1 gj+2(~)
(III-9)
~4 Next, let us derive the e x p r e s a o n for T2. Substituting the a s y m p t o t m equation of g(~), 1.e., eqn. (I-24) with n = 0, into eqn. (III-4) leads to
T2 - f
(III-lO)
fn(~-- t) dt + f gift) fn(~ -- t) dt
X
X
Taking into a c c o u n t eqns. (I-23), (II-1) and (11-10) and repeating integration of eqn. (III-10) by parts gives T2 . an . . f n. + l .( ~ - - X )
{ f n + , ( ~ - - X ) g2* ,( X )---a.+,_~ '
~
g:._,(~)}
(III-11)
F r o m eqns. (III-2), (III-9) and (III-11), we have for n = 0 Vo(X; ~) - fl(~ -- X) + ~ { fj(~ -- X) g~ ,(X) -- a I 1 g 2 - - , ( ~ ) ~
(III-12)
and for n ~> 1 Ffn+,(~ -- X) g ; , (X) -- a'.+, ~ g2-,(~)}
V~(X; ~) - f.+~(~ -- X) + ~ ,=1 n--1
-- ~
a'n_,_~ gJ*+2(~)
(III-13)
1=0
APPENDIX IV This appendix gives the a s y m p t o t i c equation of the integral X
Un= / o
t 1 / a f n ( ~ _ _ t ) dt
(IV-l)
where 0 < < X < ~. By means of arguments mmflar to those given m A p p e n d i x III, we have for T1
T1 = /
t -1/a f.(~ -- t) d t = F(2/3) ~ - 1 ~s-2/3 fn(8)}
0
n-1 = F( 2 ) £--1{E
(IV-2) (--1)' a ; ~ _ , _ 1 S--' 5/3 + (___l)n s - n - 2 / 3 ~O(S))
1=0
Using eqn. (I-17), we obtain p
n--1
(--1)' an---'-~ },+2/3 + ~ (__l)n+,__ _ a, _ _ ~n--, 1/3 ,=0 P(J + 5) j=o F(n - j + 2)
T 1 ~ F(2) ~
-- (1/2) f2+2/3(~) = F ( ~ ) { ~( ,
a,
-1) ~+'
p(n-j
r ~n--,--1 __ 1
+ -~)
f~+2/a(~)
}
)
0v-3)
75 For
T 2,
T2 = f t -1/a fn(~ -- t) dt, x
(IV-4)
we obtain after repeating integration by parts T2 ~ ~
F(j -- n + !)a (a~ ~n-~-I/a __/j+l(~ -- X) X n-~-I/3 }
J= n
(IV-5)
F(1)
Inspection shows t h a t the first term on the right-hand side of eqn. (IV-3) is equal to the first t e r m of eqn. (IV-5), because of the relation: P(j - - n + 1 / 3 ) F ( n - - j + 2/3) = (2 7r/31/2)( - -1) ~-)
(IV-6)
Thus, we have f r o m eqns. (IV-3) and (IV-5) U~(X; ~) ~ ~
.p(j . . . . + ½) fn+~+i(~ --- X) X -J-1[3 r(½)
APPENDIX
-- ~ 1
2 f ~+2/3(~) . r(5)
(IV-V)
V
The followmg two integrals are evaluated by means of the a s y m p t o t i c technique for indefinite integration [12] u n d e r the c o n d i t i o n 0 < < X < }: X
U~(X; ~) = f
t - l / 3 g ~ _ ~ ( ~ -- t) dt
(V-l)
g(t)g*2_n( ~ -- t) dt,
(V-2)
0
and X
V'(X; ~) = f 0
where the f u n c t i o n g~(~) is defined by eqn. (I-20). According to this t e c h m q u e , the a s y m p t o t i c expression of the integral can be related to the ratio of the integrand to its derivative. First, we evaluate the f u n c t i o n U;~(X; ~). The m t e g r a n d is t h e n 01 = t-1/3 g2--n(~ -- t)
(V-3)
Hence (dO1/dt)/O
= {gl-n(
--
-
t)} - ( 1 / 3 t)
(V-4)
Taking m t o consideration t h a t g~ (~) is given by eqn. (I-20) and the a s y m p t o t i c form of the modified Bessel f u n c t i o n K,,(z) is expressed by [10] K~,(z) ~ (7r/2
Z) -1/2
exp(--z) ( l + (4 u2 __ 1)/8 z + ...}
(V-5)
we have (d01/dt)/0~ ~ 2(3(~ -- t)} - 1 / 2 -- (1/3 t)
(V-6)
76
Equation (V-6) shows t h a t (dO1/dt)/O , >~ 0 for t X to where to = {--1 + (1 + 48 ~)1/'~}/24.
(V-7)
Hence, for 0 < t < to, using the a s y m p t o t i c technique for (dO~/dt)/01 < 0 [12], we have to
f
01(t) dt
(3/2) to2/3 gz-,,(~
-
-
to)
iV-S)
o
On the other hand, for t o < t, utilizing the a s y m p t o t i c technique for (dO1/dt)/ 01 > 0 [12], we obtain x
f Oa(t)d t
~ [{O~(t)}Z/Ol(t)]to
x
tO
.
.
.
.
{g2--n(} -- X)} e
- -
(V-9)
X l / 3 g { _ n { ~ - - to) - - 3--1 X - 2 / 3 g 2 _ n ( } - - to)
Therefore, U'n(X; }) is given by the sum of eqns. (V-8) and (V-9). However, the term given by eqn. (V-8) m a y be often neglected, since t o is relatively small in the region u n d e r consideration, if X > > to. The f u n c t i o n Vn(X; }) can be evaluated m a similar way. Let
05 = g(t) g2-n(} -- t)
(V-10)
then we have d0z/0 dt:/'
2
= g{~n(~_ - - t) g2--n(} Z i;--
g~)(t) 1 + g~ (t)
2 {3(} - - t)} 1/2 > 0
(V-11)
Hence, applying the a s y m p t o t i c technique for (dOe/dt)/02 > 0, V,',(X; }) can be given as {1 + g{(X)} 2 {g;_ n(~
V~(X; }) ~ -
--g;(X)g~_
--- X ) }
2
n(~ - - X ) + {1 + g~ (X)} g[
(V-12) n(} --- X )
APPENDIX VI
This appendix shows t h a t ( n 2 / n l ) F ( 5 / 3 ) A o o o is equal to the diffusion-controlled collection e f h c l e n c y , N o. F r o m a comparison of eqn. (44) wath eqn. (41), it follows Aoo o = {F(1/3) F ( 2 / 3 ) 2 A 2}
1Tooo
(VI-1)
Transforming the variable in eqn. (38) by t = {(1 + 6 + p ) A 3 -- u}v/(1 + v), we obtain
Too o = f
u --11:) f
o
o
v 1/3(1 + v)
dv du
77
= F(1/3)F(2/3) f o
u-1/3 G
(
"
(5 + p)A 3 --
)
du
(VI-2)
Following the same procedure as given in the appendix of ref. 4, Tooo can be reduced to Tooo = ~ F(~) V(~)A
p
G
+ G
-- (1 + 5 + p)2/3 G 5(1 +P5 + p) (VI-3)
Thus it follows
. r(5)A°°°=n~~11
g2/3G
+G
--(l+5+p)2/3G
(.)} 5(1+5 +p)
(vi.4,
The right-hand side of eqn. (VI-4) is equal to No, which was obtained in ref. 4 as eqn. (40). APPENDIX VII Thxs appendix describes the derivation of eqns. (47) and (48). Substituting eqns. (I-5) and (I-8) into eqn. (36) yields N,~
n2 A_ 3 p, q, nl m=Ot,,j~>o F ( ( 2 i + 1 } / 3 ) F ( { 2 j + 2}/3)
=
IIA 3
A3
× f
u (2j-1)/a f
0
0
{(1 + 5 + g)A a - t - - u} (2'-2)/3 dt du
(VII-l)
By reference to eqn. (38), the double integral in eqn. (VII-l) can be written as T,~½. A comparison of eqn. (41) with eqn. (44) ymlds T ' = A2('+o+aF
F
A
'
(VII-2)
Hence, we obtain l+]=in ,
n2
N,,~ = N O + - - ~ ( ~ p, q~ A~,_~ } A 2m nl m = l 1 , ] ~ 0
(VII-3)
where p A3
,\ 3
_n2 31/2 A_3 f No = n--l Aoo½ nl 27c
u-1/3 f
0
0
,
rl2
{(1 + 5 + p ) A 3 - - t - - u) -2/3 d t d u
(VII-4)
which is the diffusion-controlled collection efficiency for the uniform current density distribution at the generator electrode. Performing the inner integration
78 in eqn. (VII-4) leads to
N;-
n2 33/2 f x nl 2 v
u-1/3{[(1 + 6 +p)A 3 - - - U ]
1/3 - - [ ( 5
0
"1- ~ / ) A 3 - - / g ] 1 / 3 }
du (VII-5)
Substituting u = (1 + 6 + p ) A a v and u = (~ + p)Aav into the first and the second term of eqn. (VII-5), respectively, and using the following equality:
( - 1 ~ ) ~/3 dt = 3~-a-/~2YG (~-)~~_ + ~2/3(1 _ ~)1/3,
(VII-6)
0
we can easily derive eqn. (48).
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49
© Elsevmr S e q u o i a S A , L a u s a n n e - - P r i n t e d in T h e N e t h e r l a n d s
HYDRODYNAMIC VOLTAMMETRY AT CHANNEL ELECTRODES PART II. THEORY OF FIRST-ORDER KINETIC COLLECTION EFFICIENCIES
K O I C H I A O K I , KOICHI T O K U D A a n d H I R O A K I M A T S U D A *
Department o f Electronic Chemistry, Tokyo Institute o f Technology, Ookayama, Meguro-ku, Tokyo (Japan) (Received 9 t h J u l y 1 9 7 6 )
ABSTRACT T h e t h e o r y of first-order kinetic c o l l e c t i o n efficmncms at t h e d o u b l e c h a n n e l e l e c t r o d e is d e v e l o p e d for t h e following t w o s c h e m e s : (I) A ± nle ~ B (at t h e g e n e r a t o r e l e c t r o d e ) , B ~ P (m s o l u t i o n ) , B ± n2e ~ Y (at the d e t e c t o r e l e c t r o d e ) , (II) A + nle ~ B, B ~ A , B ± n2e-~ Y The e x a c t e x p r e s s i o n s for t h e k i n e t m c o l l e c t i o n efficmncms are o b t a i n e d as a s c e n d i n g a n d a s y m p t o t i c series w i t h respect t o t h e kinetic p a r a m e t e r F u r t h e r , a p p r o x i m a t e f o r m u l a e m e x p o n e n t i a l f o r m s are given, w h m h h o l d w i t h i n a n e r r o r o f a b o u t 2% for c o n v e n t i o n a l elect r o d e g e o m e t r y Finally, t h e validity o f t h e a p p r o x i m a t e p r o c e d u r e , w h i c h has b e e n used previously t o o b t a i n t h e kinetic c o l l e c t i o n efficmncies for fast h o m o g e n e o u s r e a c t i o n s , is discussed m c o m p a r i s o n w i t h t h e p r e s e n t t h e o r y .
INTRODUCTION
In the study of electrode reactions, it is of great importance to detect the reaction products or intermediates produced by charge transfer processes. For this purpose, in 1959, Frumkin et al. [1] first presented an elegant electrochemical technique, which is called the rotating ring-disk electrode (RRDE). The intermediate produced on the disk electrode (generator) diffuses into the solution, where it is carried downstream by convection to the ring electrode (detector). If the intermediate decomposes homogeneously in the solution, then the current at the detector electrode or the collection efficiency will be reduced. Thus, the measurements of the collection efficiency allow us to evaluate the rate constant of the homogeneous reaction. Another electrolysis arrangement, i.e., the double channel electrode, which seems to have almost the same potentialities as the RRDE has, was presented in 1965 by Gerischer et al. [2]. The double channel electrode comprises two neighboring rectangular electrodes located flush with one of the surfaces of a rectangular channel, through which solution flows. Though the RRDE has been extensively examined both theoretically and experimentally by a n u m b e r of investigators for various reaction schemes [3], relatively little study has been devoted to the double channel electrode [2,4--6]. For the case in which the charge * To w h o m c o r r e s p o n d e n c e s h o u l d be addressed.
50 transfer 1S f o l l o w e d b y a first-order h o m o g e n e o u s r e a c t m n (e.c. m e c h a m s m ) , Gerischer et al. [2] derived an e q u a t i o n for the c o l l e c t i o n e f f l c m n c y , b u t it is only a q u a h t a t i v e one. A f t e r w a r d s Braun [5] a p p h e d the m a t h e m a t i c a l a p p r o a c h used b y A l b e r y and B r u c k e n s t e i n [7] t o the d o u b l e channel e l e c t r o d e to o b t a i n e q u a t i o n s for the kinetic c o l l e c t i o n e f f i c m n c y f o r the e.c. m e c h a n i s m with a fast, first-order reaction. As Braun h i m s e l f p o i n t e d out, however, these equations c o n t a i n some a m b i g u i t y which m a y be a t t r i b u t e d t o the a p p r o x i m a t i o n utilized in the inverse Laplace t r a n s f o r m a t i o n b y A l b e r y and B r u c k e n s t e l n . In this paper, we shall a t t e m p t a t h e o r e t i c a l a p p r o a c h to the v o l t a m m e t r y at the d o u b l e channel e l e c t r o d e and derive the t h e o r e t m a l e q u a t i o n s for the firstorder kinetm c o l l e c t i o n efficiencies, b y utilizing the m a t h e m a t i c a l t r e a t m e n t similar to t h a t used in a previous p a p e r [8] (which is r e f e r r e d to as Part I). FORMAL EXPRESSIONS FOR THE COLLECTION EFFICIENCIES We consider the following t w o r e a c t i o n schemes: A- nle ~ B Scheme I :
(at
B ~ P (in
the g e n e r a t o r e l e c t r o d e )
(1)
solution)
(2) (3) (1')
•B + - n 2 e - ~ Y ( a t the d e t e c t o r e l e c t r o d e ) A + - n l e - ~ B (at the g e n e r a t o r e l e c t r o d e ) k
S c h e m e II:
1
(2')
B ~ A (in solution) ± n2e ~ Y (at the d e t e c t o r e l e c t r o d e )
where k is the T h e specms Y any e l e c t r o d e tion o f B, t h e 3 Um aC b y~x=D-
(3')
rate c o n s t a n t o f the (pseudo-) first-order h o m o g e n e o u s reaction. m a y be identical with A and we assume t h a t P does n o t u n d e r g o r e a c t i o n at the d e t e c t o r e l e c t r o d e . L e t c r e p r e s e n t the c o n c e n t r a i n t e r m e d i a t e . T h e n we have the c o n v e c t i v e diffusion e q u a t i o n : a2c
(4)
ay 2 - - k c
where Um is the m e a n flow v e l o c i t y and b the half height of the channel. T h e c o n f i g u r a t i o n and the related p a r a m e t e r s o f the d o u b l e channel e l e c t r o d e are described in a previous p a p e r [6]. In o r d e r to set up eqn. (4), the same assumptions as used in Part I are made. T h e b o u n d a r y c o n d i t i o n s for eqn. (4) are given as follows: c=O
forx=0,
e-~ O
forx>
y>
0
0, y - ~ o
for 0 ~ x ~ Xl, y D(Oc/Oy)
=
O (generator)
0
for x I < x < x2, Y = 0 (gap)
i2/n2F
for x2 ~ x ~< x3, Y = 0 ( d e t e c t o r )
(5)
51
where il and i2 are t h e c u r r e n t densities at t h e g e n e r a t o r and the d e t e c t o r electrodes, respectively. T h e signs o f currents are t a k e n so as t o be positive, w h e n the e l e c t r o d e r e a c t i o n s (1) and (3) p r o c e e d f r o m left t o right. F u r t h e r , w h e n the p o t e n t i a l applied to the d e t e c t o r e l e c t r o d e is assumed t o c o r r e s p o n d t o the limiting c u r r e n t plateau for t h e process (3), the following c o n d i t i o n should be added: c = 0
for x2 ~< x ~< x3, y = 0
(detector electrode)
(6)
I n t r o d u c i n g new variables: = A 3 x / i x 1 and T = ( k / D ) I / 2 y
(7)
with A = (k/D1/3)l/2(bXl/3
Um) 1/3
(8)
we o b t a i n (9)
T ( i ) C / i ) ~ ) = ( } 2 C / ~ T 2 --- C
with the b o u n d a r y c o n d i t i o n s : c=0
forT>0,~=0
(10)
c-~0
f o r T - ~ oo, ~ > 0
(11)
= hl(~) =- ( k D ) - l / 2 ( i l / n i F )
f o r t = 0, 0 ~< ~ ~< A ~ f o r t = 0, A a < ~ < ( 1
~c/~T = h ( } ) ~ = 0
(12) + 5)A 3
(13)
/
[ = h2(~) = ( k D ) - l / 2 ( 1 2 / n 2 F )
f o r ~ = 0, (1 + 5)A 3 ~< } ~< (1 + 5 + u)A 3
(14)
where 5 = (x2 - - - X l ) / X l and p = (x 3 - - x 2 ) / X l
(15)
T h e c o n d i t i o n (6) can be t r a n s f o r m e d t o c=0
forT=0,(l+5)A
3~<~< (1+5
+p)A 3
(16)
Taking t h e Laplace t r a n s f o r m s with respect t o ~, the b o u n d a r y value p r o b l e m described b y eqns. (9)--(14) can be r e d u c e d to TSC = d2c-/dT 2 - - c-
(17)
with K -~ 0
for T -~
d c / d T = h-(s) f o r ~ = 0
(18)
(19)
T h r o u g h o u t this paper, we d e n o t e the Laplace t r a n s f o r m of a f u n c t i o n ¢(~) as ~(s) or £~b, w h e r e s is the t r a n s f o r m e d variable. T h e solution o f eqn. (17) with eqn. (18) has the f o r m : b- = const. Ai(sl/3 T + s - 2 / 3 )
52
where Ai denotes the Airy function [10]. Substituting the above equation into eqn. (19) leads to (c-)~=o = --f-o(S) h-(s)
(20)
where f-o(S) = --
Ai(s-2/3) sl/3 Ai'(s-2/a)
(21)
The inverse Laplace transform of eqn. (20) is (c)~=o = -- f
h(t)fo(~ -- t) dt
(22)
0
where fo(~) is the reverse Laplace transform of eqn. (21) and its expression in the form of ascending and asymptotic series with respect to ~ is given in Appendix I. Introducing the condition (16) into eqn. (22) and taking into consMeration the boundary conditmns (12)--(14), we obtain for ~ >~ (1 + 5)A a A3
f
h2(t)fo( ~ - t )
dt = -
(1+6)A 3
f
hl(t) f o ( ~ - - t ) d t
(23)
0
This is the Volterra integral equation of first kind with respect to the u n k n o w n function h2(~). Solution of this integral equation is greatly facilitated by introducing the function g(~), which was used in Part I to derive the equation of catalytic currents and is defined in the Laplace transform as follows:
~(s) =--s -~la Ai'(s-2/a)/M(s -2/a)
(24)
In Appendix I, the expression of g(~) is Dven in the form of ascending and asymptotm series with respect to ~. The function g(~) fulfills the relation:
f fo(t)g(~ -- t) dt = f fo(~ -- t)g(t) d t = 1 0
(25)
0
for any value of ~, since g(s)fo(s) = 1/s, as can rea&ly be seen from eqns. (21) and (24). Utilization of the function g(~) allows us to obtain the solution of eqn. (23). Writing u instead of ~ in eqn. (23), multiplying both sides of the resulting equation by g(~ - - u ) , then integrating with respect to u from (1 + 6)A a to ~, and differentiating with respect to ~, we obtain d ~ h2(})=--d~f
A3
g(}--u) f
(1 +6 )A 3
hl(t)fo(u--t)dtdu
0
where the following calculation was used:
d
(
~
u
-- u) ( ]
(3I + 5d ) A
(1+5)& 3
- - t) dt du
(26)
53 d
¢
= ,4--~ {"
h2(t)
(1+6)A 3
d = d--~ (f+
¢
[" ro(U --t)g(~--u) du dt t
h2(t) dt = h2(~) 6 )A 3
Noting eqns. (12) and (14), we obtain for the current density at the detector electrode _ n 2
/2(~)
d
A3
~
nl d~ f
kl +,5 )/\3
g(~---u) f
il(t)fo(U -- t) dt du
(27)
0
The total currents flowing through the generator and the detector electrodes are given by x1
I1 = w f
,\3
a-3 f
tl(X) d x = W X l
0
il(t )dt
(28)
0
and x3
I2=w f
i2(x) dx
x2
n2 nl
(1+6 +,a)A 3 WXI n -3
,\3
g([1 + 6 + p] A a -- u) J
¢ (1+5)A t pA 3
n2
tl(t) fo(U-- t) dt du
0 A3
il(t)fo([1 + 6 + p]A a - - t - - u) dt du 0
(29)
0
respectively, where w is the c o m m o n width of the generator and the detector electrodes. Thus, the collection efficiency N ls given by 22_
rt2
~A3
\3
A3
N = I 1 nl f
g(u) f
h(t)fo([l+5+p]Aa--t--u)dtdu/ f
0
o
o
h ( t ) dt (30)
So far, we have not imposed any restriction on il. Thus, the collection efficiency can be calculated from eqn. (30) for an arbitrary current density distribution at the generator electrode. However, what we can control externally is n o t the current density, but the total current flowing through the entire electrode surface. Therefore, except for the electrodes with a uniformly accessible surface, such as the rotating disk electrodes [3] and the stagnation-point electrodes [9], it is quite difficult to control the current density distribution at the electrode surface. On the other hand, under the conventional voltammetric condmons, in which the solution contains a large excess of supporting electrolyte and the dimensions of working electrode are very small, the electrode potential may be assumed to be constant over the entire electrode surface. Thus, it ls desirable, in
54 general, to employ the condition of controlled-potential electrolysis. On the ground of this consideration, we shall here evaluate the collectmn efflcmncy for the followmg three cases: (A) In scheme I, the potential of the generator electrode is assumed to be set on the hmiting current regmn of the electrode process (1). In this case, the current density h ( x ) can be expressed by [6]
(31)
il(X) = const, x -1/3
When the electrode process (1) proceeds reversibly, the current density distnbutmn of eqn. (31) holds for any given potential along the current-potential curve. (B) In scheme II, the same electrolysis condition as m case (A) is employed. As shown m Part I [8], the current density ll(x) is given by /I(X) = const,
g(AaX/Xl)
(32)
(C) In scheme I, the current density l l ( X ) lS assumed to be com pl et el y controlled by the kmetms of the electrode process (1). In this case, Zl(X) is expressed by /I(X) =
const.
(33)
This situation can be obtained, when the electrode process (1) is totally irreversible and the potential of the generator electrode is set on the f o o t of t he currentpotential curve. Substituting eqns. (31)--(33) into eqn. (30), we obtain for the cases (A), (B) and (C) pA 3
A3
fo([1 + 5 + p ] A 3 --- t - u ) t
3 nI
0
1/3 dt du
(34)
0
/22 ~ / k 3
\3
Ncat=h~a J
g(u) f
0
0
:\3
fo([l+5+/alA3--t--u)g(t)dtdu/
f
g(t)
dt
(35)
0
and p\3
172
\~
glll : __ ]•--3 f
g(u) f
0
0
/21
fo([1 + 5 + p] A a -- t - u) dt du
(36)
respectively. SERIES EXPANSIONS FOR NUMERICAL CALCULATIONS Since the expressions for the collection efflcmncms, eqns. (34)- t 3 6 ) , contain double integrals, it is difficult to proceed further with the analytic evaluatmn of N's. Hence, we expand eqns. (34)--(36) into ascending and asymptotm series for small and large values of A, respectively. Then, we can readily evaluate them by means of numerical calculations.
55
Ascending serws First, consider the most complicated case of eqn. (35), because it is possible to express eqns. (34) and (36) as the special cases of eqn. (35). The ascending series of f0(~) and g(~) are obtained in Appendix I as eqns. (I-5) and (I-8), respectively. Thus, substituting them into eqn. (35), we obtain, after performing some calculations and rearranging terms,
nl
=
k=O F
['
F
o
[,
J
(37) where the coefhcmnts p's and q's are defined by eqns. (I-4) and (I-7), respectively, and T~jk is given by /zA 3
T,,k = f o
A3
u(2J-1)/af
((l+5+p)Aa--t--ub(2'-2)/at(2k-1)/adtdu
(38)
o
The double integral in T, jk can be obtained as follows: substituting t = ((1 + 5 + p)A 3 - - u } v into eqn. (38) yields p 43 T,j k = f u(21-1)(3{( 1 + ~ + ]A)a 3 - - u } 20+k)/3 0
1+5 +p)Aa_u
3
'
3
where B is the incomplete beta function, defined by
B(t; a,
t b) = f va-1 (1 -- v) b-1 dv o
(40)
Expanding the incomplete beta function into the ascending series [10], and transforming the integral variable by u = ((1 + 5 + p)A 3 } v, we obtain + n (1 + ~ + p)2o+j)/3
F
n
T,I k = A20+j+k+l)
n=0
× B
+5 +g '
M( ~ 2 + 2- k
3
'
3
+n ) F (2~)
n
(41)
Introducing eqn. (41) into eqn. (37) and performing some rearrangements allows us to write the final result: No+~F(5) ~ / ~ __,~1__ \,5,/ m = 1 [ I , I , k ) O
p,q, qkA~klA am
Ncat = - -
1 + [' ( 5 ) , ~ 1
{ q m / [ ' (2-2~m3+~5)} A 2m
]
-
(42)
56
with
"/
No = n~ ~2/3G(1/5) + G(/J/6) -- (1 + ~ + p)2/3G 1
A1jk-
;, B 1 + 6 + p
(
"
'
)}
(
~ { r (2-3~2~+ n) (1 + (~ + p)20+J)/3---n
3
'
3
n
)1
(44)
where the functmn G is defined by [4,11] 31/2 ?
-
d~
_
In 4~
+
tan_ ~ 2 01/3 31/2
(1 + 01/3) 3
+ --
(45)
4
As shown m Appendix VI, (n2/n 1)F(5/3)Aoo 0 is equal to the diffusion-controlled collection efficiency, No. Hence, in eqn. (42) N o is introduced into the numerator instead of (n2/nl)F(5/3)Aoo o. Next, we treat the case (A). Taking into conslderat~on that the current density il at the generator electrode for the case (A) is given by the first term of the ascending series for g(~) [8], the expressmn of Nr~,, can be obtained by setting the subscript k = 0 in the numerator of eqn. (42) and omitting the sum in its denominator. Thus, we have l+l=m
N~e, = N0 +n2- F ( 5 ) ?/1
~
(t)lqjAuo} A 2m
~
(46)
m=l 1,130
Finally, consider the case (C), m which the u m f o r m accessibility at the generator electrode is satisfied. As shown in Appendix VII, the expression of N ~ can be gwen by 1+ J= 171]
N m = N 0 '+ ~ mn2= l 2 { 2 ~ ~,j~o p l q j A I J ½ } A 2m
(47)
with
No =n~ nl
(I+5+p)G
P
-J6
--(5+p)G
+~-
[(1+
--51/3 ]
(48)
where N~) is the diffusion-controlled collection efficiency corresponding to the umform current density dlstribui~ion at the generator electrode and Alj 21 is given by eqn. (44) m which the subscript k is formally set equal to 1/2.
57
Asymptotic series In the following, we shall derive the asymptotic expressions for N's, which can be used conveniently for large values of A. Exchanging the order o f integration in eqn. (30), we obtain A3
N
n2
pA 3
A3
=n-1 f
/l(t)f
g(U)fo([1 + 8 + p ] a a - - t - - u ) dudt / f
0
0
0
il(t) dt
(49)
The asymptotic expression for the inner integral in the numerator of eqn. (49) is given in Appendix III. From eqn. (III-12), it follows A3
N ~ I n2 {f
i l ( t ) [ f l ( [ l + 5 ] A a - - t ) + ~ ( g 2 _*n ( P A 3 ) f n ( [ l + 5 ] A a - - t )
0
n=l A3
--an--lg~--n([1 + ~ + p ] A 3 - - t ) } ]
dt}/f
il(t ) dt
(50)
o where the coefficients a~ are tabulated for the first 11 terms in Table 3 in Appendix I and the functions fn and g~ are given below (see eqns. 54, 55 and 57). In order to proceed further, the explicit expression of il(x) should be introduced into the above equation. For the case (A), eqn. (31) can be substituted for il(x). Then, the following two types of integrals appear in eqn. (50): A3
U, = f
t-1/a f , ( [ 1 + 6]A 3 -- t) dt
(51)
t -1/3 g~_n([1 + 5 + p]A a -- t) dt
(52)
0 A3
0
These two integrals are evaluated in Appendices IV and V, respectively. From eqn. (IV-7), we have ~ o F ( J + 1) A - a , - 1 fn+,+1(5A3) 1 2 _ __ F(g)rn+213([1+ 5 ] A 3) Un-- = _F(1/3)
(53)
with n--1
= Z; (-1), :1=0
fn(})
an-j-1 ~ J!
(--1) nF(1/3) ~ + 31/aF(2/3)
~(2J+3n--2)/3
~--oPJ = F(~j+n+½)
(for small values of ~) -f~(~)
f~(~) = 1 ~ 7r]= 0
(54)
(for large values of ~) b~{2/(3 ~)l/2}-v+j+l K-v+j+l(4
~1/2/31/2)
(55)
58 t
t
where the coefficients aj and bj are t a b u l a t e d in Table 3 and K~, is the m o d i f i e d Bessel f u n c t i o n o f the second kind [ 1 0 ] . F r o m eqns. (V-8) and (V-9), U~ is given as
{g~ n([6
Un~-Ag~_n([(5 + p ] A 3 ) - ( A + -32 g 2 _ n ( [ 1
+
6 + ~]
+
td A3)) 2 2/3)g~_n([6 + p ] A 3)
A 3 -- to)
t~/3
(56)
with b,{2/(3 })1/2}
g~(}) = 1 ~
v+,+l K_,+j+I(4 }1/2/31/2)
(57)
7r j= 0
to = ((1 + 4811 + 6 + p] A3) 1/2 -- 1 } / 2 4
(58)
where the coefficients bj are given in Table 3. T h e n , a f t e r p e r f o r m i n g the integratlon in the d e n o m i n a t o r o f eqn. (50), we o b t a i n the following a s y m p t o t i c expression for N~,.: N~ev
n2 n-~ 23 A-2{UI + ~
[g*2-n(~tA3)Un--a~a-IVn]}
(59)
n=l
For the case (B), il(x) is given by eqn. (32). Thus, if eqn. (32) IS introduced into eqn. (50), the following three types of integrals appear: A3
Vn=f o
g(t)fn([1 + 61A 3 - - t ) dt
(60)
g(t)g~_n([1 + 6 + p] A 3 -- t) dt
(61)
g(t) dt
(62)
A3
v =fo
A3
S
=f
o T h e a s y m p t o t i c expressions o f the first t w o integrals are derived In A p p e n d i c e s III and V, respectively, and the last integral was evaluated in Part I [8]. We have
gn ~ fn+l(6A3) ~ - ~ a'n--~gj*+l([1 + 6] A3) j=l
+E
3 * 3 )--an+j--I ' (fn+j(6A)gz-j(A g2--j([1 + 6]A3)}
(63)
j=1
Vn -
(1 +g~(A3)}2{g2_n([6 +p]A3)} 2
(64)
* 3 --go(A )g2_n([6 + p]A3) + {I + gl(A3)}gl_n([6 + p]A3)
S ~ A 3 + 0 . 2 5 - - g ~ ( A 3)
(65)
59 Using the above three asymptotic expansions, we obtain the following asymptotic expression for Neat:
(66)
Neat _ n~ S - 1 (Vl + ~ [ g~ n(pA3)Vn __ an--1Vn]} /21 n=I
For the case (C), we introduce eqn. (33) into eqn. (50) and p e r f o r m the indlcated integrations to obtain
Mr, . n2. A_. 3(f2(aA3) f2([1+5]A3)+ ~ . . nl
n=l
* ~)[f.+i(~A 3) g2_.(~A (67)
. f n + l ( [ l. + ~]A3)] .
. ~ . an--l[g~--n([8 + p ] A 3) n=l
g3_n([1 * + 5 +p]AS)]}
where the relations (II-1) and (I-23) were used.
RESULTS AND DISCUSSION
Nu rnerzcal calculattons The values o f the collection efficiencies N for the three cases were calculated numerically on a digital c o m p u t e r HITAC 8700 as functions of the kinetic parameter A ~ for various combinations of the values of the geometry parameters of double electrode, 5 (0--1.0) and p (0.25--3.0), where eqns. (46) and (59); (42) and (66); and (47) and (67) were used for the cases (A), (B) and (C), respectively. Equations (46), (42) and (47), which are expressed in ascending series, would give, in principle, precise values of N for any value of A 2, if there were no physical limitations of fixed-word-size m e m o r y and no fixed-decimalpoint restriction. In practice, however, such limitations may lead to t runcat i on and rounding of individual values, which action may cause errors in the result of computation. Thus, for the values of A 2 up to 2.0 (sometimes up to 3.0 or more for special combinations of the values of 8 and p), eqns. (46), (42) and (47) were used to obtain the values of N with considerable accuracy (with errors less than 1 0 - 6 ) , where the n u m b e r of terms summed up In the series was less than 50. F o r large values of A 2 (greater than 1.5), eqns. (59), (66) and (67), which are expressed in asymptotic series, were used. The summation in the series usually included the first four terms. The values of N for the intermediate region of A 2 can be calculated by using bot h the ascending and the asymptotic series. Both results are in agreoment within a m a x i m u m difference of 2%. Figures I and 2 indicate typical results of numerical calculations. In Fig. 1, the variation of the ratio, Nrev/No, with A 2 for the case (A) is given for p = 1.0 and for a series of values of 5. Figure 2 shows the dependence of Neat~No on A 2 for the case (B) for the same electrode geometries as in Fig. 1. Comparison of Figs. 1 and 2 indicates that the following inequality holds in general:
Neat > Nrev
(68)
For the case (C), results of numerical calculations indicated that the values of
N,~/No are almost equal to the corresponding ones ofNeat/N o for any values of
60
i
I°I
lO
08
05
O6
O6
2
~04
Z
O4
02
0 0
I
2
3
0
4
1
A2
2
3
4
A2
Fig 1 VarlaDons of t h e ratio, N r e v / N 0 , with A 2 for the case (A) for various g e o m e t r y param=(a) 00,(b) 005,(e) 02,(d)05and(e)10 eters of d o u b l e e l e c t r o d e t ' = 1 0 a n d b Fig 2 V a r i a t i o n s of t h e ratio, N c a t / N O, with A 2 for t h e case (B) for various ~ e o m e t r y parameters of d o u b l e e l e c t r o d e p = 1 0 a n d 5 = ( a ) 0 0 , ( b ) 0 05, (e) 0 2, (d) 0 5 a n d (e) 1 0 .
6 , / l a n d A 2. T h e d i f f e r e n c e s b e t w e e n these values b e c a m e greatest w h e n A 2 was a b o u t u n i t y and t h e y were less t h a n 2%. Thus, we can c o n c l u d e t h a t
NI~/N~ ~ Neat/No.
(69)
Approximate equatmns for N I t is desirable to derive simple a p p r o x i m a t e e q u a t i o n s which r e p r e s e n t with r e a s o n a b l e a c c u r a c y the N - - A 2 r e l a t i o n f o r given regions o f the g e o m e t r y p a r a m eters, 6 and/~. Detailed i n s p e c t i o n o f the results o f n u m e r i c a l c a l c u l a t i o n s suggests t h a t the N - - A 2 r e l a t i o n can be e x p r e s s e d b y
N/No = exp(- -~A 2 +/~A 4 - - 3,A 6) f o r 1.0 ~> N/No >~ 0.08
(70)
and
N / N o = exp(--c~' + ~ ' A 2 - - 3/A 4)
f o r 0.1 ~> N / N o >~ 0 . 0 0 2
(71)
respectively. T h e c o e f f i c i e n t s a, ~, etc. can be given b y p o l y n o m i n a l s o f the g e o m e t r y p a r a m e t e r s o f d o u b l e e l e c t r o d e , as follows" 3
.3
C~ = E
E C~m,n 6 m g n m=O n=O
(72) I ~r
w h e r e c~ stands for a,/J, 7, a ,
and 7
r.
61 F o r t h e c a s e ( A ) , w h e r e N/No in e q n s . ( 7 0 ) a n d ( 7 1 ) is Ntis~No, t h e v a l u e s o f am ,n, t3m,n, e t c . a r e t a b u l a t e d in T a b l e 1. E q u a t i o n ( 7 0 ) is v a l i d f o r 0 . 2 5 ~ p ~ 2 . 0 a n d 0 . 0 ~< 5 ~< 1 . 0 w i t h i n a m a x i m u m e r r o r o f 2%. I t s h o u l d b e n o t e d t h a t if t h e r e g i o n o f 8 is n a r r o w e d s u c h as 0 . 2 ~< 5 ~< 1.0, e q n . ( 7 0 ) h o l d s in w i d e r r a n g e , i.e., 1 . 0 >~ Nrev/No >~ 0 . 0 2 , w i t h i n t h e s a m e e r r o r . O n t h e o t h e r h a n d , e q n . ( 7 1 ) h o l d s f o r 0 . 7 ~< p ~< 3 . 0 a n d 0 . 0 5 ~< 8 ~< 0 . 7 w i t h i n a n e r r o r o f 2%. F o r t h e c a s e (B), in w h i c h N/No c o r r e s p o n d s t o N c a t / / N o , e q n . ( 7 0 ) h o l d s f o r 0 . 2 5 ~< p ~< 2 . 0 a n d 0 . 0 ~< 5 ~< 1 . 0 a n d e q n . ( 7 1 ) f o r 0 . 7 ~< p ~< 3 . 0 a n d 0 . 0 5 ~< 5 ~< 0 . 7 , b o t h w i t h i n a n e r r o r o f 3% ( u s u a l l y less t h a n 2%). T h e v a l u e s o f t h e coe f f i c i e n t s a m , n , tim,n, e t c . f o r t h i s c a s e a r e g i v e n m T a b l e 2. A s c a n b e s e e n f r o m e q n . ( 6 9 ) , t h e v a l u e s ofN, rJNo f o r t h e c a s e (C) a r e e q u a l t o t h o s e o f Ncat/No w i t h i n a m a x i m u m e r r o r o f 2%. T h u s , t h e s a m e a p p r o x i m a t e e q u a t i o n s as f o r Ncat//N0 c a n b e u s e d f o r N~rr/N'o. Since the logarithms of eqns. (70) and (71) are cubic and quadratic equations w i t h r e s p e c t t o A 2, r e s p e c t i v e l y , t h e v a l u e s o f A c a n e a s i l y b e c a l c u l a t e d f r o m t h e m e a s u r e d v a l u e s o f N , a n d in t u r n t h e v a l u e s o f t h e r a t e c o n s t a n t k c a n b e o b t a i n e d b y t h e u s e o f e q n . (8).
Effect of the length of detector electrode I n g e n e r a l , i t is e x p e c t e d t h a t t h e l o n g e r t h e d e t e c t o r e l e c t r o d e , t h e h i g h e r t h e c o l l e c t i o n e f f i c i e n c y , if o t h e r e x p e r i m e n t a l v a r i a b l e s a r e k e p t c o n s t a n t . W e e x a m me the effect of the length of detector electrode on the collection efficiency. S i n c e a s i m i l a r t r e n d o f t h e e f f e c t c a n b e e x p e c t e d f o r t h e t h r e e cases, w e c o n s i d e r h e r e t h e s i m p l e s t c a s e (C) in s o m e d e t a i l . L e t t i n g p -~ ~ m e q n . ( 6 7 ) , w e
TABLE 1 The values of the coefflcmnts used in eqn (72) for Nrev/N 0 (case A) .
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m n
O~,n,n
/3m,n
')'m,n
O~na,n
~'m,n
'~nl ,n
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3
1.0191E0 4.7604E-1 --1 9917E-1 4 0072E-2 2 6344E0 --1 6944E0 1 1264E0 --2 3247E-1 --2 3079E0 2 9096E0 --1.8598E0 3.4469E-1 1 1972E0 --1 6474E0 1 0080E0 --17168E-1
2 0224E-1 --34945E-2 5 7600E-2 --1.1940E-2 --2 2514E-1 2 1858E-1 --1.9006"E-1 3 8814E-2 6 7679E-1 --4 5329E-1 2 9534E-1 --3 4239E-2 --3.5639E-1 2.5321E-1 --1 3319E-1 20130E-3
2 3139E-2 --14887E-2 I 2265E-2 --18591E-3 --2 9141E-2 5 5107E-2 --2 5154E-2 1 3668E-3 6 2330E-2 --1 8852E-2 --4 6552E-2 2 9681E-2 --2 2546E-2 --1 6230E-2 5 8548E-2 --28266E-2
--9.9994E-2 60931E-1 --2 3869E-1 3 3748E-2 1 6482E0 --2.3220E0 6 6628E-1 --5.3692E-2 --4 6772E0 5 8737E0 --2 2224E0 2 5628E-1 4 1372E0 --5 2472E0 2 3546E0 [-3 2398E-1
--8 6814E-1 --1 8720E-1 9 4952E-2 --16921E-2 --2 1414E0 1 2392E0 --1 2994E0 2 8266E-1 2 3822E0 --5 0082E0 4 ]354E0 --8 6295E-1 --1 6415E0 4 3979E0 --3 3752E0 69319E-1
--6.1625E-2 --63848E-3 2 3591E-3 --48508E-4 1.4544E-1 --1 8966E-2 --5 9076E-2 1 6780E-2 --4 4010E-1 --1 8386E-2 1 5123E-1 --4 2241E-2 3 3496E-1 --1 5051E-2 --7 8071E-2 24822E-2
0 1 2 3 0 1 2 3 0 1 2 3 0 I 2 3 .
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62 TABLE 2 The values o f the c o e f h c l e n t s used m eqn (72) for N c a t / N 0 (case B) and for
N~r/No
(case C)
m n
C~m,n
/~m ,n
"/m,n
121m,n
~m,n
~m,n
0 0 0 1 0 2 0 3
8 9953E<1 5.8226E-1 --27649E-1 60111E-2
2 2 6 --3
0833E-1 2754E-2 2753E-2 1880E-2
2 7627E-2 1 8513E-2 --1.2062E-2 2 3084E-3
--4 2700E-1 1 8085E0 --1 1029E0 20463E-1
--8 8241E-1 4 2964E-1 --36093E-1 7 4217E-2
--7 5533E-2 9 5692E-2 --7 1734E-2 14200E-2
1 1 1 1
0 1 2 3
2 --2 2 --6
7982E0 3925E0 1791E0 3045E-1
4 --3 --4 1
8814E-2 0285E-1 0137E-3 0113E-1
6.5984E-2 --3 0181E-1 3 2668E-1 --1 0165E-1
5 -1 1 --2
4209E0 8539E1 2946E1 5214E0
--7 --7 5 --1
0979E-1 0335E0 1297E0 0235E0
4 --1 1 --1
6030E-1 4525E0 0161E0 9853E-1
2 0 2 1 2 2 2 3
--2 4 --4 1
3579E0 2897E0 4093E0 3686E0
1 6 --5 1
5484E-1 2286E-1 3587E-1 2120E-1
--1 6 --1 4
0567E-1 8875E-1 0127E0 0746E-1
--1 4 --3 6
0541E1 5240E~ 4104E1 8595E0
1 1 --1 2
1715E0 3961E1 1953E1 5287E0
--1 3 --2 5
0200E0 5963E0 7013E0 4285E-1
3 3 3 3
1 --2 2 --8
1499E0 3873E0 5480E0 0934E-1
--3 5 --4 1
3409E-1 9335E-1 3625E-1 3555E-1
--1 1 I --9
0004E-1 8496E-1 0100E-1 1527E-2
4 --3 2 --5
9643E0 1034E1 5182E1 2129E0
--3 --7 7 --1
0551E0 0294E0 6457E0 7356E0
5.101~E-1 --2 4697E0 1.9958E0 --4.1234E-1
0 1 2 3
obtain ( N ~ ) u _ ~ = n~ A - 3 {f2(SAa) __ f2([ ] + (~] A3)}
(73)
/'/1
In Fig. 3, the values ofN, rr/(Nlrr)~_~ are p l o t t e d against p for 5 = 0.5 and for several values o f A 2. This Figure shows t h a t the ratio Nlr~/(Nt~)~-~ is increased with increasing values o f p and b e c o m e s a l m o s t c o n s t a n t and close t o u n i t y for p greater t h a n 1.5. This fact indmates t h a t if p / > 1.5 the c o l l e c t i o n e f f l c m n c y hardly d e p e n d s on p for large values of A, especially for A 2 ~> 2.0. F o r the cases (A) and (B) a l m o s t the same b e h a v i o r prevails as for the case (C) described above.
Range of measurable rate constants N o w , we estimate the range o f rate c o n s t a n t s which can be m e a s u r e d b y means o f the d o u b l e channel electrode. The d i m e n s i o n s o f the cell w h m h is usually e m p l o y e d m our e x p e r i m e n t s are: w = 0.5 cm, b = 0.03 cm, xl = 0.02 cm, x2 - - X l = 0.01 cm (f = 0.5) and x 3 - - x 2 = 0.02 cm (p = 1.0). A s s u m i n g t h a t the d e t e c t a b l e limit o f N ~ , is 2 × 10 - 3 for fast reactions and t h a t N~ev m u s t be smaller t h a n 0.9 No for slow reactions, we have f r o m eqns. (70) and (71) t h e following c o n d i t i o n : 0.05 < A 2 < 3 T a k i n g t y p i c a l value o f D o f 1 X 10 -~ c m 2 s - 1 , we o b t a i n f r o m eqn. (8) 0.3
2 / 3 < 2 0 ( c m -2/3s 1/3)
Since the m e a n flow v e l o c i t y Um can be v a n e d easily b e t w e e n 5 and 5 0 0 cm s- 1 ,
63
~05 z
0
1
2
Fig 3. Variations of the ratm, A 2 are given on the curves
3
Nlrr/(N~rr)
p ,o~
with p for various values of A 2 The values of
we have 1 S- 1 < k < 10 3 S- 1 . O f course, the u p p e r limit o f k can be raised b y m a k i n g Xl (length o f g e n e r a t o r electrode), x2 - - x l (length o f gap) and b (halfhmght o f channel) smaller, and conversely the l o w e r limit is r e d u c e d b y increasing Xl, x2 - - X l and b. When the i n t e r m e d i a t e reacts with a specms Z the c o n c e n t r a t i o n o f which can be varied so as to satisfy the c o n d i t i o n o f pseudo-first-order r e a c t i o n , t h e n k = k' C z
,~z shou:d Ee in the fo!low[ng range: 1M>cz>
10 - a M
so t e a t we have for the s e c o n d - o r d e r rate c o n s t a p t k' l M - -~s - 1 < le' < 1¢ 5 7If- 1 s--1. In conclusion, it can be said t h a t the range o f measurable rate c o n s t a n t s at the d o u b l e ehat:nel e l e c t r o d e is c o m p a r a b l e to th,~; o f R R D E .
Approximr,te procedure for fast homogeneous reactzons In o r d e r to solve a b o u n d a r y value p r o b l e m for e l e c t r o d e preces,;es coupled with a fast l~orr,.ogeneot~s reaction, an a p p r o x i m a t e procedure~ b ~ e d o n the ass u m p t i o n t h a t the thickness o f the r e a c t i o n layer is m u c h t h i n n e r t h a n t h a t o f the diffusion layer, is o f t e n e m p l o y e d . In p o l a r o g r a p h y , this a p p r o x i m a t i o n corresponds t o the neglect o f the t i m e - d e p e n d e n t t e r m in the r e a c t i o n layer, i.e., the so-called steady-state a p p r o x i m a t i o n , whereas in h y d r o d y n a m i c v o l t a m m e t r y dealing with the s t a t i o n a r y state it c o r r e s p o n d s t o the omission o f the convection t e r m in the r e a c t i o n layer. This a p p r o x i m a t i o n has been applied t o the convective diffumon e q u a t i o n at the g e n e r a t o r e l e c t r o d e b y A l b e r y and Bruekenstein [7] and by B r a u n [5] to o b t a i n the expressions for the c o l l e c t i o n e f f i c i e n c y at the R R D E and the d o u b l e channel e l e c t r o d e , respectively. In this section, we apply this a p p r o x i m a t e p r o c e d u r e to the p r e s e p t p r o b l e m and discuss its validity. T h e solution o f eq~. (9) w i t E o u t the c o n v e c t i o n term, t o g e t h e r wtth the con-
64 dltions (10)--(12), 1S readily obtained to be c =---hi(}) e x p ( ~ ) Thus, at the end of the generator electrode or at } = A a, c = --hi (A a) exp(--~q)
(74)
Now, the problem to be solved is eqn. (9) with the boundary cohdittons (11~, (13), (14), (16) and (74). Writing }1 = } -- A a and introducing the function: = exp(~r~) + C / h l ( A a)
(75)
we have ~
'/~ ~ 1
~2~ -
~T]2
~
(76)
with ~=0
for}l=0,
~0
f o r } l > 0, r? ~ oo
~_~ + 1 =
h(}l)
[=0
hl(A3) I - h2(}) hl(A 3)
r?> 0
for0~<}l<~A3,7?=0
(77)
for 5A 3 ~< }1 <~ (5 +/~)A 3, r? = 0
(78)
Taking the Laplace transforms with respect to }1 and solving the resulting boundary value problem by using the procedure similar to that used to obtain eqn. (20) from eqns. (17)--(19), we obtam
Sl--il-(~)}f-o(Sl)
(';)~=o=/1 h(sl)
(79)
where sl is the transformed variable of ~l. From eqns. (75) and (79), we obtain m view of eqn. (II-8) (b-)~=0 = --hl(A a) fl(S1) --fi(S1) f-o(81)
(80)
The inverse Laplace transform of eqn. (80) is
E1 (c)~ =0 = --hl(A a) f l ( } l ) -- jf h(t) fo(~l -- t) dt 0
(81)
Introducing the conditions (16), (77) and (78) into eqn. (81) and using the same procedure as used to obtain eqn. (26), we have d g(~l -- t) fl(t) dt h2(}1) = --hl(A3) d ~ l f s\ 3
(82)
Hence, taking into account eqns. (12) and (14) and integrating eqn. (82) over
65
the surface of the detector electrode, we obtain for the total c u r r e n t / 2 ,tt ,\ 3
12 = (n2/nl) w x l A - 3 i l ( A 3) f
g(t) fl([6 + p ] A 3 -- t) dt
(83)
o
In view of eqn. (III-1), eqn. (83) can be rewritten as
Iz = ( n 2 / n l ) w x l A - 3 i l ( A 3) V I ( p A 3 ; [6
+ p ] A 3)
(84)
Thus, it follows from eqns. (28) and (84), that A3
N = n2
n~ il(Aa)VI(PAa; [8 + p]Aa)/ f
i l ( t ) dt
(85)
0
Since we are concerned with large reaction rate, we only consider the asymptotic expression of eqn. (85). For such a case, as can easily be derived from eqn. (III-13), VI(pAa; [8 + p ] A 3) is expressed as V I ( p A 3 ; [(~ + p ] A 3) = f2(SA 3) -- g2([5 + p ] A 3) +
* 3 ' * {fn+l(SA3 )g2_n(PA )--ang2_n([5 +p]A3)}
C
(86)
n = 1
Substituting eqns. (31) -(33) into eqn. (85) yields Nrev -
(n2/nl)(2/3)A-3VI(pA3; [5
Ncat ~
(n2/nl)(1
+ p]A 3)
+ g l ( A 3 ) } S - 1 V I ( p A 3 ; [5 + / d ] A 3)
N~r ~ (n2/nl)A - 3 VI(pA3; [5 + p] A 3)
(87)
(88) (89)
where the asymptotm series of S is given by eqn. (65).
1
2
3
4
Az
Fig 4 Comparmon of the approximate soluhons with the exact ones for two sets of geometric parameters ( 1 ) p = 2 0,6 = 0 1,(2) t2= 2 0 , 6 = 0 5 (R) log(Nrev/No) ( - ) exact curves, calculated from eqns (46) and (59), ( - ) approximate curves, calculated from eqn. (87) (C) log(Ncat/No). ( - ) calculated from eqns (42) and (66), ( ) calculated from eqn (88). (B) Calculated from Braun's eqn (90)
66 In Fig. 4, typical plots o f log(Nro~/No) and log(N¢~t/No) agamst 1'\2 calculated f r o m eqns. (87) and (88) are s h o w n for 5 = 0.1, p = 2 0 and 8 = 0 5, p = 2.0, t o g e t h e r with the e x a c t values whmh were calculated f r o m eqns. {46), (59), (42) and (66). As can be e x p e c t e d f r o m eqn. (69), the values o f log(Nl,,/N()) o b t a i n e d f r o m eqn. (89) are n u m e r i c a l l y in g o o d a g r e e m e n t with those of log(Nc~t/N0) f r o m eqn. (88). We can see f r o m Fig. 4 t h a t m the cases (B) and (C) t h e a p p r o x > m a t e values a p p r o a c h a s y m p t o t m a l l y to the c o r r e s p o n d m g e x a c t curves w~th mcreasing values of A 2, while m the case (A) t h e y a p p r o a c h to the curves, which are b y a b o u t 10% l o w e r t h a n the e x a c t ones. This fact mdmates t h a t thin approximate p r o c e d u r e m a y be used o n l y w h e n a u n i f o r m c u r r e n t density d i s t r i b u t i o n is o b t a i n e d at the g e n e r a t o r e l e c t r o d e . I n c l u d e d m FN. 4 for c o m p a r i s o n are plots o b t a i n e d f r o m the e q u a t i o n derived b y Braun [ 5 ] : N~
~ (n2/n,)(1/6)A
a exp(----4 6A a)
(90)
As can be seen f r o m Fig. 4, Braun's results are u n s a t i s f a c t o r y for any values o f A 2"
APPENDIX I
In tins a p p e n d i x the reverse Laplace t r a n s f o r m s of the following t w o functions are derived" s - " f-o(S)
--
=
A;(s 2 , 3 ) _ _ _ sl/a+" Ai' ( s - 2 /3 )
(M)
and s-"g(s) =
-
-
Af(s-'~/a) s2/a +, Al(s-2/a)
(I-2)
where v is zero or any positive n u m b e r . Since these t w o f u n c t i o n s have singular points in the same region as for the f u n c t i o n ~-.(s) and f u r t h e r t h e y satisfy the c o n d i t i o n s of J o r d a n ' s l e m m a , the inverse t r a n s f o r m s can be o b t a i n e d b y the integration along the p a t h BCDEFG, which was s h o w n in Fig. 7 of Part I [8]. A s c e n d i n g serws F o r small Is1-2/3, eqn. (I-l) can be d e v e l o p e d to the following series: s_~,Fo(S ) _
F(1/3) ~ p~ 8--23/3 31/a F ( 2 / 3 ) j = o
,'
1/3
(i-3)
with j--1
P0 = 1, P l = ~1 -- X1, P~ = oz~ -- X~ -- ~
k=l
Pk Xj--k
(I-4)
where c% and X~ are d e f i n e d b y eqn. (A3) o f Part I. Following the p r o c e d u r e
67 described in the Appendix of Part I, we have ~--1
~21/3+v--2/3
{ S - - V f o ( S ) } --
r(1/3) ~ p, 3 1 / a F ( 2 / 3 ) , =o r ( § j + , + ~-)
(I-5)
where £ - 1 denotes the operator of inverse Laplace transformation. Similarly, s - " ~(s) is developed as follows: s-" g(s) -
31/3 F(2/3) g, 2_a q, s -2'la -v-2/3 P(1/3) ,=o
(>6)
with j--1
qo = 1, ql
=
X1
CO1, q3 = Xj -- cos -- ~
--
k=l
qk COj--k
(I-7)
Hence,
£ - - 1 {S- V g ( s ) } --
The
31/3F(2/3) ~ o ~2,13+,-1/a F(1/3) q' -P(~j + , + 5) - 2 2-
(i-8)
first 11 values o f p j and qj are listed in Table 3.
A s y m p t o t i c series
Applying eqns. ( A 8 ) - - ( A l l ) of Part I to s - " -[o(S), we obtain the following asymptotic forms along each path:
s - " fo(s) -
_,[C2 1 CID2 + C2D1 ] -e 4/asj s /~22+2 D~
(BC)
s - " C2/D2
(CD)
s -~ C 1/191
(DE)
s - " C2/D2
(EF)
s
_vfC2 /D~2
i 2
C1D2 + C2D1 D2
}
e 4/as_
(I-9)
(FG)
where 1 is the imaginary unit and C1, Ca, D1 and De are defined by eqns. (A13) and (A14) of Part I. Substituting eqn. (I-9) Into the inversion formula (A1), we have
BCDEFG
- 2rri
D
~s - D T d s + f
DE
s'--D~Tds+f
I£FG
e'~ ds s" D2
68
o
I
I
I
~
o
o
0
~
I
I
I
I
I
I
~
~
~
TT??TTTT?
E
I
I
i
I
I
c,.) ¢J
I co
I
]
I
r
!
69
FG
D~
S"
We d e n o t e the first and the second terms on the right-hand side of eqn. (I-10) by T1 and Te, respectively. Since the path DE is the semi-circle of radius p with its center at the origin and C~, C2, D1 and De approach u n i t y as p -~ 0, we have
f
(eS~/sv)(Cl/D1)ds = f (e~/s')(C2/D2) ds
DE
DE
Taking into a c c o u n t the above relation and i n t r o d u c i n g the following series representation: C~ = ~ a~ z ~ ([arg zl < u / 2 ) De j=o
(I-11)
we have ,
T 1 = E (-1), 3= 0
1
e st~ s j-'' ds a, ~ 1 1 f BCDEFG
The integrals in the above equation correspond to Hankel's integral representation of G a m m a function. Thus, T1 = ~
(--1)' a I ~ v - - I - - 1 / F ( P - - J)
(I-12)
1=0
When v is a positive integer n, the s u m m a t i o n in eqn. (I-12) is carried o u t f r o m j = 0 to n -- 1. For Te, by introducing the series expansion:
C1D2 + C2DI- ~ D~
b~ z 3 ( I a r g z l < u / 2 )
(I-13)
~=o
and p u t t i n g s = r e x p ( - ~ Q along the path BC and s = r exp(ui) along FG, respectively, we obtain T2 - cos PT~ ~ bj 2 ~ j=o
e
'~ -4/3r Fj--u
dr = cos p~ f;(})
(I-14)
0
w~th
=1 E
bI
K_~+j~ 1 ~ 3 ~ -
]
(I-15)
where K~, is the m o d i f i e d Bessel f u n c t i o n of the second kind of order p. Note t h a t f~ satisfies the following relation:
f ° f~(t) dt
:
f*~+~(~)
(I-16)
70 F r o m eqns. (I-10), (I-12) and (I-14), it follows t h a t ,
£-~ (s-'?0(s)} ~ G
}r--j--1
(--1) 1 a, F(u --]~ + cos uTr f2(})
(I-17)
1=0
If u is a positive integer n, eqn. (I-17) can be reduced to n--1 Z--1{8--n/o(8)}
~
G i
:0
, ~n--1--1 (-1) 1 a I - - - -
+ (--1)nf,~(~)
(I-18)
F(n---j) ,
The first 11 values of the coefficmnts a~ and b I are tabulated in Table 3. For the f u n c t i o n s -~ g-(s), a similar procedure as given above leads to oo
~c--l{8--Vg(S)} ~ G
j=0
(---1) ] 121 r ( p - - J
+ 1) + cos
pTr g~,+l(~ )
(I-19)
with
i -''+1+ 1 K_,+j+1 [4 \_3 ~}~/2 _ _ ]1 3 })1/2/
2 g~(}) = 1--1o~ bj (i -7r =
(I-20)
where a I and bj are the coefficients of the following series:
(D2/C2) = ~
a I z I (largzl<~r/2)
(I-21)
= 7_/ bj z J ( l a r g z l < l r / 2 ) 1=o
(I-22)
1=0
and ~1D2 + C2D1 C~
respectively. The first 11 values of aj and bj are listed in Table 3. We have further the following relation for g,*,
/g,*,(t) dt = gz[+l(~)
(I-23)
When u is a positive integer n, eqn. (I-19) is t r a n s f o r m e d as follows:
£--l(s--ng(s)} ~
(--1)1 al F ( n - - j + 1) + ( - 1
gn+l(~)
(I-24)
APPENDIX II In this appendix, we derive the two relations, which will be used m the subsequent appendices. (1) We define the f u n c t i o n fn(~) (n = 1,2,3, ...) by
fn(~) = /
f n - l ( t ) dt
(II-1)
71 where the f u n c t i o n fo(~) is the inverse Laplace t r a n s f o r m of fo(s) which is given by eqn. (I-l) with v = 0. Then, we show t h a t the f u n c t i o n fn(~) can be expressed by n--1 fn(~) = ~ (---1) } J=O
j!
+ (-1) n
{s-nyo(S)}
t
fn(0) = an-1
(II-2) (II-3)
where £ - 1 {s-nTo(S)} is given by eqns. (I-5) and (I-18) in the f o r m of the ascending and the a s y m p t o t i c series, respectively. For small Isl (larg si
1=0
(--1)~ a, s'
(II-4)
Making use of the initial-value t h e o r e m for Laplace t r a n s f o r m a t i o n , we have lim ~
y fo(t)
Thus, ?
(n-5)
dt = lim fo(s) = ao
0
s~O
f o ( t ) d t exists and
0
(n-6)
f l ( 0 ) ----? ~ /c0(t ) d r - - a 0 o
Then fl(~) =
?
fo(t) dt --
0
j
f o ( t ) d t = a'o - -
0
j
fo(t) dt
(II-7)
0
The Laplace t r a n s f o r m a t i o n of eqn. (II-7) and the use of eqn. (II-4) yields f l ( S ) = a 0' 8- 1 - - 8 - 1 #o(S) ~ ~ (---1)' 1=o
a >' , s
J
~II-8~
Applying the initial-value t h e o r e m again, it is easily shown t h a t f2(0) exists and is equal to a~. Now, we assume t h a t fn(S) = a,~--i 8 - 1 - - 8 - 1 } n _ l ( S ) !
fn(0) = an--1
(II-9)
(II-10)
and
~,,(s)
~3 (-1)'
1=0
an+,
sj
(II-11)
72 then, by the initial-value t h e o r e m we have hm f ~
fn(t) dt
=
0
hm fn(S)
=
(II-12)
an
s---O
Thus, we have m vmw of eqns. (II-1) and (II-12) fn+l(0)
! f,(t) dt= %
=
(II-13)
0
Hence, fn+l(}) =
fn(t) dt
fn(t) dt-0
=
an -- ; fn(t) dt
0
(II-14)
0
The Laplace t r a n s f o r m of eqn. (II-14) is (II-15)
Fn+l(S) = an s- 1 - - s --1 [,(s) Substituting eqn. (II-11) into eqn. (II-15) ymlds E (---1) I arn+l+l s ~ J=O
f-n+l(S)
(11-16)
Therefore, it can be concluded by means of m a t h e m a t i c a l i n d u c t i o n t h a t eqns. (II-9)--(II-11) hold for any pomtwe integer. Successive use of eqn. (II-9) leads to n
1
f-n(8) = E l=0
(---1) I an--i '
1 $ 1--1 + ( _ _ l ) n s--n f O ( S )
(II-17)
The inverse t r a n s f o r m of eqn. (11-17) is eqn. (II-2). (2) Next, we derive the following relahon:
5
{aforn:0
l=o
alan
l
0forn=l,2,3
(II-18) ....
Multiplying eqn. (I-II) by eqn. (I-21) gwes
~l=O
Hence
(1
--
a o a o' ) - -
~ n=l
I=O
a l a n' _ l
)
Zn:O
The c o n d i t i o n t h a t the above e q u a h o n should be vahd for any value of z, yields eqn. (II-18).
73 APPENDIX III
In this appendix, the a s y m p t o t i c series of the following integral is derived: X
Vn(X; ~) = f
g(t) fn(~ -- t) dt (n = 0, 1, 2, ...)
(III-1)
o
where X and ~ are large positive n u m b e r s and ~ > X. We write Vn(X; ~) = r 1 -- r 2
(III-2)
where
TI= 7 g(t) fn(~ -- t) dt
(III-3)
0
and
T2 = f g(t) [n(~ -- t) dt
(III-4)
As T1 is the c o n v o l u t i o n integral, it can be written as (III-5)
T1 = £--1 { g ( s ) fn(8)}
For n = 0, T1 is simply equal to 1, since g(s)fo(s) = s-1. For n/> 1, substituting eqn. (II-17) for fn(S) in eqn. (III-5) yields n--1
( _ l ) j a~-j--] s --j-1 g-(s) + (--1)n8 - n - l }
T] = £ - 1 { ~
(III-6)
3=0
Performing the inverse t r a n s f o r m a t i o n of eqn. (III-6) and using eqn. (I-24) leads to n--1
T1
--
(1+ 1
]=oD (--1)'
a'n_]_l/k~= o
~l+l_k
(--1)k ak 0 + l - - k ) !
*
(--1)'
]
~n
g]+2(~)J + (--1) n n~ (iii-7)
Changing the order of the s u m m a t i o n gives n
T1
ZJ ( - 1 ) '-1 ]=0
~J n - - ] ~ k=O
t
~
r
an_l_ka k +
a n
n--1 t
,
~n
- - ~ a n - - , - 1 g,+2(}) + ( - - 1 ) n ~ j=0
(III-8)
By use of eqn. (II-18), it can be seen t h a t the first term on the right-hand side is reduced to ( - 1 ) n-1 ~n/n!. Hence, we have n--1
T1 -a'.-
~
l=O
a~'~_,__1 gj+2(~)
(III-9)
~4 Next, let us derive the e x p r e s a o n for T2. Substituting the a s y m p t o t m equation of g(~), 1.e., eqn. (I-24) with n = 0, into eqn. (III-4) leads to
T2 - f
(III-lO)
fn(~-- t) dt + f gift) fn(~ -- t) dt
X
X
Taking into a c c o u n t eqns. (I-23), (II-1) and (11-10) and repeating integration of eqn. (III-10) by parts gives T2 . an . . f n. + l .( ~ - - X )
{ f n + , ( ~ - - X ) g2* ,( X )---a.+,_~ '
~
g:._,(~)}
(III-11)
F r o m eqns. (III-2), (III-9) and (III-11), we have for n = 0 Vo(X; ~) - fl(~ -- X) + ~ { fj(~ -- X) g~ ,(X) -- a I 1 g 2 - - , ( ~ ) ~
(III-12)
and for n ~> 1 Ffn+,(~ -- X) g ; , (X) -- a'.+, ~ g2-,(~)}
V~(X; ~) - f.+~(~ -- X) + ~ ,=1 n--1
-- ~
a'n_,_~ gJ*+2(~)
(III-13)
1=0
APPENDIX IV This appendix gives the a s y m p t o t i c equation of the integral X
Un= / o
t 1 / a f n ( ~ _ _ t ) dt
(IV-l)
where 0 < < X < ~. By means of arguments mmflar to those given m A p p e n d i x III, we have for T1
T1 = /
t -1/a f.(~ -- t) d t = F(2/3) ~ - 1 ~s-2/3 fn(8)}
0
n-1 = F( 2 ) £--1{E
(IV-2) (--1)' a ; ~ _ , _ 1 S--' 5/3 + (___l)n s - n - 2 / 3 ~O(S))
1=0
Using eqn. (I-17), we obtain p
n--1
(--1)' an---'-~ },+2/3 + ~ (__l)n+,__ _ a, _ _ ~n--, 1/3 ,=0 P(J + 5) j=o F(n - j + 2)
T 1 ~ F(2) ~
-- (1/2) f2+2/3(~) = F ( ~ ) { ~( ,
a,
-1) ~+'
p(n-j
r ~n--,--1 __ 1
+ -~)
f~+2/a(~)
}
)
0v-3)
75 For
T 2,
T2 = f t -1/a fn(~ -- t) dt, x
(IV-4)
we obtain after repeating integration by parts T2 ~ ~
F(j -- n + !)a (a~ ~n-~-I/a __/j+l(~ -- X) X n-~-I/3 }
J= n
(IV-5)
F(1)
Inspection shows t h a t the first term on the right-hand side of eqn. (IV-3) is equal to the first t e r m of eqn. (IV-5), because of the relation: P(j - - n + 1 / 3 ) F ( n - - j + 2/3) = (2 7r/31/2)( - -1) ~-)
(IV-6)
Thus, we have f r o m eqns. (IV-3) and (IV-5) U~(X; ~) ~ ~
.p(j . . . . + ½) fn+~+i(~ --- X) X -J-1[3 r(½)
APPENDIX
-- ~ 1
2 f ~+2/3(~) . r(5)
(IV-V)
V
The followmg two integrals are evaluated by means of the a s y m p t o t i c technique for indefinite integration [12] u n d e r the c o n d i t i o n 0 < < X < }: X
U~(X; ~) = f
t - l / 3 g ~ _ ~ ( ~ -- t) dt
(V-l)
g(t)g*2_n( ~ -- t) dt,
(V-2)
0
and X
V'(X; ~) = f 0
where the f u n c t i o n g~(~) is defined by eqn. (I-20). According to this t e c h m q u e , the a s y m p t o t i c expression of the integral can be related to the ratio of the integrand to its derivative. First, we evaluate the f u n c t i o n U;~(X; ~). The m t e g r a n d is t h e n 01 = t-1/3 g2--n(~ -- t)
(V-3)
Hence (dO1/dt)/O
= {gl-n(
--
-
t)} - ( 1 / 3 t)
(V-4)
Taking m t o consideration t h a t g~ (~) is given by eqn. (I-20) and the a s y m p t o t i c form of the modified Bessel f u n c t i o n K,,(z) is expressed by [10] K~,(z) ~ (7r/2
Z) -1/2
exp(--z) ( l + (4 u2 __ 1)/8 z + ...}
(V-5)
we have (d01/dt)/0~ ~ 2(3(~ -- t)} - 1 / 2 -- (1/3 t)
(V-6)
76
Equation (V-6) shows t h a t (dO1/dt)/O , >~ 0 for t X to where to = {--1 + (1 + 48 ~)1/'~}/24.
(V-7)
Hence, for 0 < t < to, using the a s y m p t o t i c technique for (dO~/dt)/01 < 0 [12], we have to
f
01(t) dt
(3/2) to2/3 gz-,,(~
-
-
to)
iV-S)
o
On the other hand, for t o < t, utilizing the a s y m p t o t i c technique for (dO1/dt)/ 01 > 0 [12], we obtain x
f Oa(t)d t
~ [{O~(t)}Z/Ol(t)]to
x
tO
.
.
.
.
{g2--n(} -- X)} e
- -
(V-9)
X l / 3 g { _ n { ~ - - to) - - 3--1 X - 2 / 3 g 2 _ n ( } - - to)
Therefore, U'n(X; }) is given by the sum of eqns. (V-8) and (V-9). However, the term given by eqn. (V-8) m a y be often neglected, since t o is relatively small in the region u n d e r consideration, if X > > to. The f u n c t i o n Vn(X; }) can be evaluated m a similar way. Let
05 = g(t) g2-n(} -- t)
(V-10)
then we have d0z/0 dt:/'
2
= g{~n(~_ - - t) g2--n(} Z i;--
g~)(t) 1 + g~ (t)
2 {3(} - - t)} 1/2 > 0
(V-11)
Hence, applying the a s y m p t o t i c technique for (dOe/dt)/02 > 0, V,',(X; }) can be given as {1 + g{(X)} 2 {g;_ n(~
V~(X; }) ~ -
--g;(X)g~_
--- X ) }
2
n(~ - - X ) + {1 + g~ (X)} g[
(V-12) n(} --- X )
APPENDIX VI
This appendix shows t h a t ( n 2 / n l ) F ( 5 / 3 ) A o o o is equal to the diffusion-controlled collection e f h c l e n c y , N o. F r o m a comparison of eqn. (44) wath eqn. (41), it follows Aoo o = {F(1/3) F ( 2 / 3 ) 2 A 2}
1Tooo
(VI-1)
Transforming the variable in eqn. (38) by t = {(1 + 6 + p ) A 3 -- u}v/(1 + v), we obtain
Too o = f
u --11:) f
o
o
v 1/3(1 + v)
dv du
77
= F(1/3)F(2/3) f o
u-1/3 G
(
"
(5 + p)A 3 --
)
du
(VI-2)
Following the same procedure as given in the appendix of ref. 4, Tooo can be reduced to Tooo = ~ F(~) V(~)A
p
G
+ G
-- (1 + 5 + p)2/3 G 5(1 +P5 + p) (VI-3)
Thus it follows
. r(5)A°°°=n~~11
g2/3G
+G
--(l+5+p)2/3G
(.)} 5(1+5 +p)
(vi.4,
The right-hand side of eqn. (VI-4) is equal to No, which was obtained in ref. 4 as eqn. (40). APPENDIX VII Thxs appendix describes the derivation of eqns. (47) and (48). Substituting eqns. (I-5) and (I-8) into eqn. (36) yields N,~
n2 A_ 3 p, q, nl m=Ot,,j~>o F ( ( 2 i + 1 } / 3 ) F ( { 2 j + 2}/3)
=
IIA 3
A3
× f
u (2j-1)/a f
0
0
{(1 + 5 + g)A a - t - - u} (2'-2)/3 dt du
(VII-l)
By reference to eqn. (38), the double integral in eqn. (VII-l) can be written as T,~½. A comparison of eqn. (41) with eqn. (44) ymlds T ' = A2('+o+aF
F
A
'
(VII-2)
Hence, we obtain l+]=in ,
n2
N,,~ = N O + - - ~ ( ~ p, q~ A~,_~ } A 2m nl m = l 1 , ] ~ 0
(VII-3)
where p A3
,\ 3
_n2 31/2 A_3 f No = n--l Aoo½ nl 27c
u-1/3 f
0
0
,
rl2
{(1 + 5 + p ) A 3 - - t - - u) -2/3 d t d u
(VII-4)
which is the diffusion-controlled collection efficiency for the uniform current density distribution at the generator electrode. Performing the inner integration
78 in eqn. (VII-4) leads to
N;-
n2 33/2 f x nl 2 v
u-1/3{[(1 + 6 +p)A 3 - - - U ]
1/3 - - [ ( 5
0
"1- ~ / ) A 3 - - / g ] 1 / 3 }
du (VII-5)
Substituting u = (1 + 6 + p ) A a v and u = (~ + p)Aav into the first and the second term of eqn. (VII-5), respectively, and using the following equality:
( - 1 ~ ) ~/3 dt = 3~-a-/~2YG (~-)~~_ + ~2/3(1 _ ~)1/3,
(VII-6)
0
we can easily derive eqn. (48).
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