3535
POTEXTIOSTATIC CURREST-TIMECURVES
Influence of Heterogeneous Chemical Reactions upon Potentiostatic Current-Time Curves
by Roland0 Guidelli Institute of Analytical Chemistry, University of Florence, Florence, Iraly
(Received M a y 2, 1968)
The theoretical current-time curves at constant potential on a plane electrode have been derived for the case of a reversible or totally irreversible electrode process coupled with a preceding, following, or parallel firstorder heterogeneous chemical reaction. In the present treatment it is assumed that the reacting species are adsorbed at the electrode surface according to linear isotherms. The expressions for the theoretical i-t curves in the absence of adsorption can easily be derived from the more general equations corresponding to linear adsorption. The comparison between these equations and the corresponding expressions relative to a homogeneous chemical reaction show that the traditional polarographic techniques can allow a distinction between homogeneous and heterogeneous reactions only for values of the “formal” reaction-layer thickness, M , quite large compared with molecular dimensions ( p > low4to 5 X cm). While the electrode processes combined with homogeneous chemical reactions have been widely studied through the use of several electrochemical techniques, less emphasis has been placed upon the study of heterogeneous chemical reactions coupled with electrode processes.’ This fact may probably be ascribed to the difficulties encountered in determining the mechanism of an electrode process unambiguously, when this involves one or more chemical reactions taking place only among species adsorbed at the electrode surface. I n fact, under these conditions, even excluding the existence of a slow adsorption or desorption step, the nature of the adsorption isotherms, the possible dependence of the adsorption parameters upon the applied potential, as well as the character of the double layer and the surface heterogeneity, may have a remarkable influence on the experimental current. On the other hand, the distinction between homogeneous and heterogeneous chemical reactions becomes quite problematic when the rate of a supposedly homogeneous chemical reaction is so high that the corresponding thermodynamic affinity is different from zero only in a layer of solution around the electrode (reaction layer) having a thickness comparable with that of the diffuse double layer. Thus many protonation reactions of anions, A-, of weak acids preceding the electroreduction of the undissociated form, HA, which have been studied polarographically are so fast that their original ascription to the class of homogeneous chemical reactions has been debated recently.2-4 Owing to the difficulties encountered in the study of electrode processes complicated by adsorption of the reactants and by heterogeneous chemical reactions, some simplifying assumptions are usually made in their investigation. Thus it has often been assumed that the coverage of the electrode by reactants and products is sufficiently low to allow the use of linear adsorption
isotherms.5-* Furthermore, the constancy of adsorption coefficients with respect to potential over the range of potentials explored has frequently been post~lated.~~~~~~~ The present paper considers the theoretical currenttime curves a t a constant potential on a stationary plane electrode for the case of a “reversible” or “totally ire-
reversible” charge-transfer process A B coupled with a heterogeneous chemical reaction. I n order to obtain analytical solutions of the problems under study, only first-order and pseudo-first-order chemical reactions will be considered. I n the following treatment it will be assumed that reactants and products are adsorbed according to linear isotherms and that the corresponding adsorption coefficients do not change with time at constant potential. The constancy of adsorption coefficients with respect to potential is not required. The various cases may be expressed by the two general schemes
C
+ XX2ZA + yY
(14
e-
A Z B
Ob)
e-
A Z B
(1) K. J. Vetter, 2. Phys. Chem. (Leipzig), 194, 199 (1950). (2) J. Koryta, Rev. Polarog. (Kyoto), 13, 1 (1965). (3) S. G. Mairanovskii and L. I. Lischeta, Isv. Akad. N a u k SSSR, Otd. Khim. N a u k , 1984 (1962). (4) S. G. Mairanovskii, E. D. Belokolos, V. P. Gul’tyai, and L. I. Lischeta, Elektrokhimiya, 2, 693 (1960). (5) G. C. Barker and J. A. Bolzan, 2. Anal. Chem., 216, 215 (1966). (6) B. Kastening, H. Gartmann, and 1,. Holleck, Electrochim. Acta, 9 , 741 (1964). (7) K. Holub and J. Koryta, Collect. Czech. Chem. Commun., 30,3785 (1965).
Volume ‘72*Number 10 October 1968
ROLANDO GUIDELLI
3536 Reaction l a represents the heterogeneous generation of the electroactive species A from the electroinactive species C (preceding reaction). X and Y are other electroinactive reagents and products and 1 is a positive stoichiometric coefficient. It will be postulated that the bulk concentration of X is sufficiently high with respect to the bulk concentration of C and that reaction l a is sufficiently slow for the surface concentration of X to remain constant during the electrolysis. Under these circumstances the rate of reaction l a is given by
- -drc- - Prc dt where Fa is the surface concentration of C. Reaction 2b expresses the heterogeneous regeneration of the depolarizer A from the product B of the chargetransfer process 2a, provided that 1 is positive. More particularly the regeneration of depolarizer is partial or total according to whether I is lower or equal to unity. When 1 equals zero, reaction 2b expresses the inactivation of the product B (subsequent reaction). For the species X and Y contained in eq 2b, the same considerations hold which were previously made in connection with eq la. Thus the rate of reaction 2b is given by
Ka,Kb, and K, are the concentrations in the bulk of the solution, the surface concentrations, the diffusion coefficients, and the adsorption coefficients of A, B, and C, respectively. Equation 8 expresses the fact that the sum of the quantities of A and B which are adsorbed at the electrode surface per unit of time and per unit area of the electrode equals the sum of the fluxes of A and B toward the electrode plus the amount of A produced per unit area in the same unit of time as a result of reaction la. Obviously the current is expressed by the equ& tion
Passing to the Laplace transforms of eq 3-10, rearranging terms, and carrying out the inverse transformation partially, one has
-i- - Ob*
OKa + K b
nFA
X
(Kay - D:'z)eY2t erfc(y$/')]
+
lpK,c*
OK,
Preceding Reaction (Scheme 1) Case of a Nernstian Electron Transfer. The potentiostatic current-time curves on a stationary plane electrode for the case under examination are obtained by solving the following system of differential equations
(4)
- a*
P
+
(Kb - ?)eYzt
+ Kb X
erf~(yt'~')] 2Xy -
p
-
y2
X
+
+
where y = (OD,"' Dbl/')/(OKa Kb) and X = Dc"'/2Kc. The separation of the Laplace transform on the right in eq 11 allows the following inverse transform to be obtained
for the initial and boundary conditions
(12)
(7)
where a!
= h(l
+ 5);
p
=
X(1 - 5);
E
=
d1 - ( p / h 2 )
(13) When 0 + m , r b ---+ 0, and furthermore y Da'/'//Ka. Thus from eq 11 and 12, it immediately follows that ---+
(9)
where Fa = K,a(O, t ) , r b = Kbb(0, t), and Tc = Kcc(O,t ) . Here a*, b*, and c*, I?, r b , and r,, Da, Db, and D,, and The Journal of Physical Chemistry
(8) R. Guidelli, L. Nucci, and G. Raspi, unpublished data. (9) R. Guidelli, J . Electroanal. Chem., in press.
3537
POTENTIOSTATIC CURRENT-TIME CURVES 1 .o
0.8 Equation 14 expresses a current solely controlled by the diffusion toward the electrode of the quantity of B present in the solution (b*) before the electrolysis. In 0.6 fact, it is apparent that under these conditions 4 the heterogeneous reaction l a does not affect the current. On the contrary, when 8 --t 0 and con0.4 sequently y -* DI,'/~/K~,,eq 11 and 12 show that the potentiostatic current consists of a diffusional contribution due to the quantity of A present in the solution 0.2 (a*) before the electrolysis ( ~ F A D a l / ' ~ * / , l / * ~ l / ' ) plus a Contribution due to the quantity of A which originates from the heterogeneous chemical reaction la. 6 10 15 20 Obviously the latter contribution depends on the rate constant, p , of reaction l a and tends to I times the difFigure 1. Variations of + with ht'/2. The various curves fusion-limiting current of C ( i e . , lnFAD,'/'c*/n'/'t'/') correspond t o the following values of i : (1) 0.996, (2) 0,990, when p + a . From eq 13 it is manifest that eq 12 (3) 0.986, (4) 0.980, (5) 0.970, (6) 0.960, (7) 0.940, (8) 0,910, holds only for 4KC2p# D, and furthermore that ( is (9) 0.870, (10) 0.750, (11) 0.010, (12) i, (13) 2i. real or imaginary according t o whether 4KC2pis lower or higher than D,. A method suitable for the numerical calculation of the current when the i = i(t)equa1 shows a series of curves expressing 9 as a function of tion contains complex parameters is outlined in the At'/' for different values of 6. It can be seen that 9 = Appendix. For the sake of brevity, the case 4KC2p= 1fort = 0, independent of the value of E, and approaches D, will be neglected. The expression of the current is zero when Xtl/' + a. The function @ may be replaced strongly simplified when it is assumed that the diffusion by a simpler expression both when At'/' or -it tends to coefficients and the adsorption coefficients for A, B, infinity and when h2 >> u. I n the latter case (Az >> and C are equal: Da = Db = D, = D and Ka = Kb = u), one has K , = K . Under these conditions the current is given by 4 = 1/1 - (u/X2) E 1 - ( 4 2 x 2 ) E 1 a* lKpc* ~i -- _Ob*_,'/zt'/2D'"_ + ___ so that l + O X nFA 1+8 E exp[(u/2X)2t] erfc [(u/2~)t"'] (17)
+
where a=X(1 E), fl = X ( l - E), X = D1"/2K, and 6 = dl - (,>/A2). Equation 15 shows more clearly than eq 11 and 12 that the potentiostatic current consists of a diffusional contribution due to the presence of the electroactive species A and B in the bulk of the solution plus a contribution due to the generation of depolarizer. As the function
'
+ t)2t]erfclX(1 + t ) t ' / ' ]
- exp[X2(1
2t
=
Vl -
@
2X/a'/"t'/'
(18)
I n the case under study, X = D'/'/2K and u = p. Consequently, when p + a , -it + a , and, in view of eq 18, eq 15 becomes
(16)
occurs frequently in the following calculations, it is convenient to consider some of its properties. I n general
'
When Xtl/' or -9 tends to infinity, the error-function complement (erfc(x)) may be replaced by the first term (exp( - x 2 ) / r ' / ' x ) of its well-known series expansion for high values of the argument. By rearranging one obtains
(U/XZ)
where u is a parameter taking different forms according to the diffusional problem under investigation. Figure
Equation 19 shows that the contribution to the potentiostatic current due to the heterogeneous reaction l a becomes purely diffusional if the rate of such a reaction is sufficiently high. It is interesting to consider the form taken by eq 15 when the adsorption coefficient K is so low that D/ 4K2 >> p . Under these conditions, taking eq 17 into account, eq 15 becomes Volume 78, Number 10 October 1968
3538
ROLANDO GUIDELLI
LE!?!? exp[ (K P) t ] 1
+e
erfc( $$/.>
(20)
The above expression for the current, which constitutes a limiting case of eq 15 for low values of K, represents the rigorous solution of a boundary-value problem analogous to that previously considered, with the difference that the diffusion coefficients of the various species are assumed equal and the amounts of A, B, and C accumulating a t the electrode surface are neglected (dr,/dt = drb/dt = dr,/dt = 0 in conditions 7 and 8). Also, in the following boundary-value problems it will be easy to pass from the more general solution to a simplified relation corresponding to the absence of adsorption. I n fact it will be sufficient to equalize all diffusion and adsorption coefficients, subsequently replacing @ by the approximate expression 17. Putting a* = b* = 0 and 1 = 1 into eq 20 and letting 0 tend to 0, one has
where Kp represents the rate constant for the heterogeneous reaction l a provided the concentration of C for z = 0 is expressed by c(0, t) instead of Po. It can be easily seen that eq 21 is identical with the well-known expressionl0 (eq 22) of a limiting current controlled by a prek ceding homogeneous chemical reaction C +A under the assumption that the equilibrium C ;Jt A is greatly in favor of C, provided we put Kp/D119= JC1’zKequi11/2Kequil
is the constant of the homogeneous equilibrium C e A. It follows that if the species A, B, and C are adsorbed in a low degree, it is difficult to ascertain whether the preceding chemical reaction is homogeneous or heterogeneous. When the protonation of anions of weak acids, HA, is considered, the existence of a homogeneous chemical reaction cannot be logically excluded. However, the possible adsorption of anions A- from the solution layer contiguous to the electrode and their successive surface protonation and electroreduction may in practice minimize the shift of the homogeneous reaction AH + ;Jt HA from equilibrium. Under these circumstances the potentiostatic current could be almost exclusively controlled by the rate of the heterogeneous reaction. Case of a Slow Electron Transfer. If the charge transfer is totally irreversible, it is not necessary to consider the diffusional expression 4,as B does not exert
+
The Journal of Physical Chemistry
any influence upon the potentiostatic current. The solution of the diffusional expressions 3 and 5 must be carried out taking into account the initial and boundary conditions
“*}for {xz +> O , t = O
a= c = c*
w,t>O
dr,/dt = D , ( ~ c / ~ x ) , =o pro dr,/dt
=
D,(ba/bz)x=o- ktr,
+ lpr,
(23) (24) (25)
where
ra = K,U(O,t);
rc = K,C(O,t )
(26)
kf is the rate constant for the forward charge-transfer process and eq 25 expresses the fact that the quantity of A accumulating per unit area of the electrode and per unit of time equals the flux of A toward the electrode minus the amount of A exchanging electrons with the electrode (ktr,) plus the quantity of A originated from reaction l a (ie., ZpI’,). An analogous problem was solved by Holub and Koryta2r7under the simplifying assumption that the bulk concentration of A equals zero (a* = 0) while its surface concentration reaches a steady-state value since the beginning of the electrolysis dr,/dt
=
0;
(ba/bz),=o = 0
Furthermore, the above authors used the initial conditions
rc = o (x = 0, t = 0) c = c* (x > 0, t = 0)
(27)
noting that under these circumstances the theoretical current-time curve exhibits a maximum which would not be observed in the absence of adsorption (ie., for dr,/dt = 0). Koryta2 suggested the detection of this maximum as a valid criterion for the verification of adsorption but it should be noted that the presence of such a maximum is connected with the initial condition 27 which cannot be easily realized. I n fact the electrode surface should come into contact with the liquid instantaneously at time t = 0 in order to adsorb only the very small amount of C: contained within the infinitesimal layer of solution contiguous to the electrode before other C reaches the electrode in consequence of diffusion. On the contrary, the initial condition employed here (r, = K,c* for t = 0) may be satisfied in practice even if reaction l a proceeds on the electrode surface before the electrolysis. I n fact a rapid laminar flow of the solution around the electrode immediately before the electrolysis causes the surface concentrations of C and A to reach their equilibrium values with respect to the corresponding bulk concentrations ( i e . , K,c* and K,u*). Such a “washing” of the electrode can be (10) J. Kouteck9 and R. BrdiElca, Collect. Czech. Chem. Commun., 12, 337 (1947).
3539
POTENTIOSTATIC CURRENT-TIME CURVES achieved for instance with the cell of the electrode with periodical renewal of the diffusion layer (dlpre) described by Coazi, Raspi, and Nucci.'l The potentiostatic current lor the case under examination is given by
i/nFA = kfr,
i =
Kkfa*[
nFA
1 - 4'
'Kpkrc*
kf -
Passing to the Laplace transforms of eq 3, 5, 24, 25, and 28, one obtains
[
exp(p2t) erfc(bt1'9) -
- E exp(a2t) erfc(ai/S) -
+ +
+
P
+
2'
" exp(p'2t) erfc(p't1'2) 1 - E' 2t'
where a
where X, = Da'/'//2K, and A, = D,'/'/2Kc. The righthand side of eq 29 can be separated in order to obtain the inverse transform
i ~nFA
-
(Kakfa*- ZKopc*) X 1 - Ea exp(Pa2t) erfc(pat'") - -X 2ta
1+
exp(aa2t)erfc(aatllZ)
1+
2t
Kakfa* - lKcpc* - kfKaa* - kfL-' s(s 2Xas1/2 kf) nFA _i _ -
exp(a'2) erfc(a't''')
2E'
(28)
exp(p'2t) erfc(p't1'2) -
2t'
1Kckfp2c*X
=
X(l
exp ( a 9 ) erfc (a't'l')]
+ E ) , p = X(l - t ) , a'
p'
=
X ( l - t'), X
E'
=
v1 - (kf/X2).
=
D'/'/2K, $.
=
=
dl - (p/X2), and
~I
{ [
e r f c Kkf [st
]
lKPkfC* (K P ) exp -t erfc #/' kf - P [KP
]+ ]-
exp[ yt]erfc[ *Kkf t
where
=
4C. (kf/Xa2);
_ _ _ _ ~ fo
=
4 1
-
(p/Xc2)
+
~ ( 1 t'),
The first term on the right-hand side of eq 32 represents the contribution to the current due to diffusion, adsorption, and charge transfer of the amount of A present in the solution (a*) before the electrolysis. The second term, which depends on both the values of kf and p, is due to the diffusion of C toward the electrode followed by adsorption, surface reaction with formation of A, and consequent charge transfer. This is more manifest if we consider the limiting case corresponding to low values of the adsorption coefficient K (D/4K2 >> p and D/4Kz >> k f ) . Under these conditions, in view of eq 17, eq 32 becomes i = K h a * enp[---gt] (Kh) nFA
(a
(32)
(31)
From eq 31 it is obvious that and are either real or imaginary according to whether Xa2 and Xo2 are higher or lower than kc and p, respectively. In the latter case, eq 30 may be given a form more suitable for the numerical calculation of the current (see the Appendix). If D, = Dc = D and Ka = KO= K, eq 30 becomes
I}
(33)
where the first term on the right is the well-known expression of the potentiostatic current for a totally irreversible electrode process in the absence of adsorption. Kkf and Kp are the rate constants for the forward charge-transfer process and for the heterogeneous chemical reaction la, respectively, provided that the concentrations of A and C at the electrode surface are expressed by a(0, t ) and c(0, t ) . Letting kf tend to infinity in eq 32 and taking eq 18 into account one has eq 34.
(11) D. Cozzi, G. Raspi, and
L. Nucci, J . Electroanal. Chem., 12, 36
(1966).
Volume 73, Number 10 October 1968
3540
ROLANDO GUIDELLI
+
+
where X = (8Da'/* Db1/')/2(8K, Kb). If the method of partial fractions for separating the term within the parentheses in eq 39 is used, the inverse transformation yields
where il is the limiting current. Obviously eq 34 may also be obtained from eq 15 by letting 8 tend to zero. 1 exp(p2t) erfc(pt'l') - -
Parallel or Subsequent Reaction (Scheme 2) Case of a Nernstian Charge-Transjer Process. The current at constant potential for the case under examination can be obtained by solving the differential expressions 3 and 4 for the initial and boundary conditions a b
= =
ra/Fb
(dr,/dO
a* x 2 0,t = 0 b*} for x + 0 0 , t > 0
{
8'
Or
a(0, t)/b(O, t )
+ (drb/dt) = Da(da/dx),=o + Db(db/bx),=o + ( I
=
+ 2H+ + 2e-
HzOz
'/zO,
8
-.&(db/dX),,o
+
(36)
where a
(37)
d 1 (1 -1)Kbp/X2(8Ka Kb). If we assume that D, = Db = D and K , = Kb = K , eq 40 becomes
A(l
=
4)) p
+
- 1)prb
HzOz
+ H20
In the general case, the equation for the potentiostatic current is obtained by noting that i/nFA
x
(35)
Condition 37 expresses the fact that the sum of the quantities of A and B adsorbed per unit area of the electrode and per unit of time equals the sum of the fluxes of A and B toward the electrode plus a term representing the change in the sum of these quantities produced in consequence of the heterogeneous reaction 2b. Obviously when the total regeneration of depolarizer is considered ( I = l ) , the latter term vanishes as the heterogeneous chemical conversion of B into A does not exert any influence on the fluxes of A and B toward the electrode. The case 1 = 1will be considered separately. The reduction of 0, to HzO in XaOH on platinized platinum*,12constitutes a case of a reversible chargetransfer process coupled with the partial regeneration of the depolarizer (where Z = "2)
O2
2x4
+ (drb/dt) +
prb
i
-=
nFA
= X(1
- t ) , and 4
=
+
Kp-
O f 1
(e
where now X
+
=
1)2
(a*
+ b*) X
D1/'/2Kand
The second term on the right-hand side in eq 41 constitutes a purely diffusional contribution, while the first term on the right-hand side represents a contribution due to the regeneration of A (1 > 0) or to the inactivation of B (1 = 0). It is interesting to consider the particular case when only B is present in the bulk of the solution (a* = 0). Under these conditions the following expression for the limiting current is obtained, when 8-0
(38)
Passing to the Laplace transforms of eq 3, 4, and 35-37 and rearranging terms, one has, in view of eq 38
(42)
./c+
where = D'/'/2K and ( = [(Z - l ) p / X 2 ] . Equation 42 shows that besides a diffusion-limiting cur-
(39) The Journal of Physical Chemistry
(12) G. Raspi and L. Nucci, Ric. Sci., 37, 509 (1967).
POTENTIOSTATIC CURRENT-TIME CURVES
354 1
rent (-nFAD'/'b*/rr'/'t'/') attainable for 8 - t , an oppositelimiting current due to the heterogeneous chemical generation of A. from B and the parallel electrochemical transformation of A into B is to be expected when e + 0. The presence of a "pseudocathanodic" voltammetric curve of this kind may be observed with HzOz on platinized p l a t i n ~ m . 8 ~Obviously ~~ if the product B of the electrode reactionis simply inactivated ( I = 0), no limiting current is observed for 8 + 0, as appears from eq 42 by putting 1 = 0. Equation 41 represents as usual the starting point for the obtainment of the potentiostatic current in the absence of adsorption (dr,/dt = drb/dt = 0). Thus for sufficiently low values of K (4(1 - I ) K2p lo), eq 43, which expresses the current for an electrode process coupled with a heterogeneous inactivation reaction when I = 0, becomes
where & = Kp/D'/'B. For sufficiently high values of p or t, the first term on the right-hand side can be neglected with respect to the second
i
~-
nFA
- D1"a*pl exp(p12t)erfc(plt'")
(46)
It can be seen that eq 45 and 46 are identical, provided we put k'Ia = K~/D'/'.
IA
+ yY
A+B
(2-11)
f
Scheme 2-1 expresses the case of a simultaneous electroreduction (or electrooxidation) and chemical inactivation of B (under these conditions it is immaterial whether B originates A or electroinactive products different from A in consequence of the chemical reaction, as the species A does not exert any influence upon the current). The present problem can be solved by considering the differential expression 4 together with the conditions
drh/dt
nFA
(2-1)
or in the opposite direction with respect to the heterogeneous chemical reaction
b
Under the assumption that the rate constant, k, of the homogeneous inactivation reaction is >7 and that 0 > 10, eq 44 takes the simplified form
+ yY
BkfA A
+
(43)
1A
=
=
i
x>O,t=O x:+ t >0
b*
=JJ
Db(db/dX)z=O - k f r b - p r b
The above boundary-value problem is perfectly identical with the problem of an irreversible charge-transfer process characterized by a rate constant (kf p) and complicated by the linear adsorption of depolarizer. Taking into account that in the present case the current is given by
+
i
-=
nFA
kfrb
in view of ref 9, one has
i
-nFA - K&b*[
exp(p2t) erfc@t"*) -
+
where CY = X(l t ) , = X ( l - t ) , X = Dh1/'/2Kb, and t = d1 - [ ( p + k f ) / ~ 2 ] , The potentiostatic currenttime curves expressed by the above equation have the same shape as the curves for a simple irreversible electrode process with linear adsorption of depolarizer9 but differ from these latter for their relative positions with respect to the potential axis. Concerning scheme 2-11, for the sake of simplicity only the case of D, = Db = D and K , = KI, = K will be considered. The instantaneous current at constant potential is obtained by solving the boundary-value problem (13) D. H. Kern, J. -4mer. Chem. Soc., 76, 1011 (1954).
Volume 72,Number 10 October 1968
ROLANDO GUIDELLI
3542 ba/bt = Dd2a/bx2;
a b
=
=
'b**}
for
{xx+1 0 , t
where Fa
=
=
- pFb
kfra+ lprb
= Kb(0, t).
rb
i/nFA
=
kJ',
D(ba/bx),,o -
Ka(0, t ) and
=
(50)
Noting that
(51)
kfF,
nFA
+ k f ){ [ ( h a *
Kkf
X(P
Zpb*)
2(1 - l)KkfKpa* ( K P Kkd(1 - X )
+
(49)
the use of the Laplace transforms leads to the expression
i __ -
- Z K P ~ *+
0 a ,t > 0
dFb/dt = D(bb/bx),,o dr,/dt
[Kkfa*
db/dt = Db2b/dx2 (47)
+
erfc[
(KP
Ix
+ K2D'/Z kf)(l -
Total Regeneration os the Depolaiixer (Scheme d for 1 = 1 ) . If the charge transfer is Kernstian, the current can be determined by solving the system of differential expressions 3 and 4 for the initial and boundary conditions 35 and 36. I n the present case, condition 37 becomes (dF,/dt)
+ (dh/dt)
=
D,(da/bx),=o
+ Db(bb/bx),=o
The above boundary condition shows that the total heterogeneous regeneration does not exert any influence upon the flux of the electroactive species toward the electrode. Therefore, the diffusional problem under examination is identical with that relative to a reversible electrode process complicated by linear a d s o r p t i ~ nwith , ~ the only difference that in the present case the expression for the current is given by eq 38. Consequently, on the basis of ref 9 we have
- i- -
nFd
+
+ +
(8Ka
+ Kb)'
-+ D,'/'Kb2 --
8Db'/'Ka2
,'/2t1/2
where a+ = X(l a+),p+ = X(1 - w+), ar- = X(l w-), B- = X(l - w-),. . X = D'/'/2K,. w+, = 4 1 - rL(1 x.-,) , . . w- = dl - +(l - x ) , = 2K2(p k f ) / D , and x = dl - [4(1- Z ) p k , / ( p k f ) ' ] . If De I'Hospital's theorem is applied when k tends to infinity, eq 52 yields
+ +
i
lim nFA - = lKp(a*
Ef+m
+
+ b*)[ T e x p ( P z t )erfc@t"') +
I
-
where 1 = D'/'/2K and $. = dl [(I - l ) p / X ' ] . Equation 53 can also be obtained from eq 41 by putting 8 = 0. The limiting expression of the current intensity for low values of the adsorption coefficient K can be derived from eq 52, taking eq 17 into account
- inFA
-
Kk t X ~ ( K P Kkd
+
+
which holds also in the absence of adsorption. Let us now consider the case of an irreversible chargetransfer process with total regeneration of the depolarizer. The respective current can be obtained by solving the system of differential 'expressions 47 for conditions 48-50 with 1 = 1. In view of eq 51, the use of the Laplace transforms yields the result
_i _--Kkfa* - Kkf X nFA
The Journal of Phyeical Chemistry
+
where = (OD,'/' Db'/')/(BK, Kb). The first term on the right-hand side in eq 54 represents a purely diffusional contribution, while the second term expresses the contribution to the current due to the total regeneration of A. When D, = Db = D and K , = KI, = K , eq 54 takes the simplified form
POTENTIOSTATIC CURRENT-TIME CURVES
3543
Performing the inverse transformation, we have
(57)
+ +
where CY = X ( l E), p = X ( l - E), X = D’/’/2K, and E = dl - [(#o ki)/X2]. When lcf tends to infinity, taking eq 18 into account, one obtains the following expression for the limiting current
which holds also in the absence of adsorption. Equation 58 shows that the contribution of the total regeneration of depolarizer to the limiting current is constant. An expression analogous to eq 58 was derived by 1 I ~ I n t y r e . l ~ For low values of the adsorption coefficient (4K2(p kf)
+ u,
itz),f(t) is
sin [(ui - X,Z)JYI
> ut, one has - u,)-"'exp(-hit) sinh [(X,z -
~,)~/*t]
The computation of the integrals contained in eq 63 and 64 may be performed n~rnerically.~
in Molten Alkali Metal Nitrates
by R. F. Bartholomew and D. W. Donigian Research and Development Laboratories, Corning Glass Works, Corning, New York 14830
(Received M a y 6 , 1968)
The kinetics of the reaction between NOz gas and iodide dissolved in molten KN03 and equimolar KNOr NaS03 were investigated using an emf technique. The reaction was found to be first order with respect to iodide for a given partial pressure and flow rate of NOZ. The effectof flow rate and partial pressure of NO2 on the kinetics was determined. The postulated mechanism NO2 I- -+ I/ZIZ NOZ- and NOZ- NOz e KO3KO was found to explain the experimental findings when the rate-controlling process was assumed to be the rate of transfer of KO2 across the gas-melt interface. This reaction is discussed in the light of the oxidation of halides in acidic nitrate melts.
+
+
Reactions taking place in molten salts as solvents have been neglected until recent years.l In particular, studies of the rates of such reactions have received little attention. The main interest in this area has centered around the investigations carried out by Duke and coworkersz on acid-base reactions in molten alkali metal nitrates. I n all these systems, the reactions were postulated as proceeding through the nitronium ion, NOz+, which arises from the self-dissociation of the nitrate ion according to the equation NO3- Ft NO%+ 02-. Recently, Zambonin and Jordan3 have shown that the O z - ion cannot exist in nitrate melts in any appreciable concentration because of the reaction NOa02- + NO%- 02-. Topol, et aZ., have questioned the existence of KOZ- as an intermediate in acid-base react i o n ~ . They ~ concluded that the behavior of dissolved NO2 in the melt is equivalent to that attributed to NOz+ in acidified melts. They showed that bubbling NOz
+
+
+
+
+
through a nitrate melt containing bromide or iodide ions resulted in the production of halogen and NO. The over-all reaction was written
where X is Br or I. The work reported in this paper was undertaken in order to determine the mechanism of the above reaction by measuring its rate using an electrochemical technique. Molten KN03 and equimolar KN03-NaN03were used as solvents. (1) W. Sundermeyer, Angew. Chem. Int. Ed. Engl., 4, 222 (1965). (2) F. R. Duke in "Fused Salts," B. R. Sundheim, Ed., McGraw-Hill Book Co., Inc., New York, N. Y., 1964, p 409. (3) F. G . Zambonin and J. Jordan, J. Amer. Chem. Soc,, 89, 6365 (1967). (4) L. E. Topol, R. A. Osteryoung, and J. H. Christie, J . Phys. Chem., 70, 2857 (1966). Volume 72,Number 10 October 1968
