STANLEY
LLOYD11ILLF:K
Solutions to the differential equations for the combination of diffusion and first-order reaction a t a plane electrcde are given, ,ind applied with several approximations to the dropping mercury electrode The agreement of the equations u i t h the data of Kolthoff and Parry for the kinetic currents of ferric ion in t h e presence of hydrogen peroxide is good The effective reaction layer thickness is discussed, and values of this thickness of to cni 'ire 5honn t o lie more significant n i certain cases than the previously assumed value of 10 cin
I n a recent paper by Kolthoff and Parry' the investigation of the catalytic polarographic currents of ferric iron in the presence of hydrogen peroxide was described. In the interpretation of their data, they used the treatment of Hrdicka and U-iesner' which involves the estimation of the effective thickness of the reaction layer. \Vhen the reaction layer was assumed to be lo-' cni. thick, the rate constant for the reaction oi ferrous ion with hydrogen peroxide was calculated to be 10' times larger than the Lnoww value. In this paper a different and more rigorous interpretatioii of this type of kinetic effect will be given, arid the factors entering into the estimation (?ithe reac+ioii layer thickness will be discussed. IVhen ferric ion is presciit iri ;L sci1utic;n of hydrogen peroxide arid a sufficiently positive potential S.C.E.J is applied to the droppin:: electrode such that neither hydrogen peroxide nor reducetl :it the electrode, the electrode
action the change in concentration of A with time is
Siiiiilarly for I3 we have
'l'he change in concentration of B with time due to chemical reaction is negative because B is depleted by the chemical reaction. At a plane electrode the following initial and boundary conditions will hold. ?.'
2
.?'
= (t
t =
II
il
IJ
/I
=
A =
-40,U
= lJ
u
-4 and U are the concentrations in moles per liter, A Ois the concentration in the body of the solution, and D is the diffusion coeificient in sec.-]. AUtlioughin general D Aand D J are ~ not equal, in most cases the diffusion coefficients are approximately equal. '1'0 solve these differential equations i:i, .i ~= i',.': !I; would be quite difficult because of the boundary coriditiuns unless the assumption that D A and D B ;iri!l in the solution tiear the electrode t,he reaction is are eyual is made. Furthermore a steady state $- H2(-l> -- 211 - = 2Fc'a + 21320 (,IT, solution will give us the approximate influence of The rate law for this reaction (neglecting side D.4 and U g on the current when they are not equal. reactions a t i d ea talysisj has been show1 by 1-arious These equations can be solved, then, by means of investigators to be the Laplace Transformation3 and also by the usual methods of differential e q ~ a t i o n s . The ~ ~ ~solution d(Fe '.3j;ldt = k , ( F i . L ? ) ~ H 2 0 2 : ior .4 is It' the peroxide is in excess, the concentratioti of r i: c the peroxide will im t change appreciably (luring the rr;icticm. 'i'he i)erositle coricentratiori can then bi: absGirbccl into the rate constant such that k = k , ( H a O z )aiid , tlil rt.:tction will appear tci he ;t firstorder reaction in ferrous. I n the general case, corresponding to cq. 1, the t.lcJctrodereaction is and for B is Ai -1= 13 !ILI) ,
.
and i n the solutioii iie:ir thc electrotlr, corrcspotiding to eel. I1
5
=
. l o - .i
(
3)
where
/k
.I (,I \-,I '1.11~e1i:trigc iii concentration of species A with time tlue t o tltiusion at a plane electrode is given by Fick's second lam of diffusion as aillbt = D A ~ A 3.?. This equation holds for every point in the solution. The change in concentration of A with time due to chemical reaction is dA/dt = KB. 1;
--f
This also holds for every point in the solution. Therefore, fur the cornbinaeion of diffusion and re-
Khen k = 0, which corresponds to pure diffusion, we have from eq. 2, A = A0 erf(x/2&i) which is / the same result as has been previously found for (31 D. M. Kern, Ph.D. Dissertation, Univ. of California, 19.19. The solution of this particular problem ia given as far a s eq. 8. (4) I. M , Kolthoff and S. E. Khalafalla (personal communication) have solved these equations and obtained the same results for equations 2 and G, but they used 3 different method for approximating the current. 1.51 I{. S . (:arsl;iw ; i n d J ( 2 , J d p y e r , "Conduction of Heat in Solids *' O x f o r d ITniv. I ' r ~ s ,I!) I 7 I I 1 .rnd ?70, give solutions to eriiixtioris iiitiil:ir
to t h c at>nve.
POLAROGRAPHIC CURRENTS FROM DIFFUSION AND REACTION
Aug. 20, 1952
pure diffusion.6 When t -+ 03 ,we obtain from eq. 2 A = Ao (1 e - f l D X ) (4) and from eq. 3 B -= A @ - ~ D X (51 This ineans that a steady state is set up. The gradient of A a t x = 0 is given by (dA/dx),,o = A0 d/K7D (Ckt/-\/J;kt erf ( d k t ) ) (6) \\;hen k = 0 we have from eq. 6
-
+
(ail /dx),-o
= Ao/dDTt
(7)
which is the same as for pure diffusion.6 As t becomes large, eq. 6 tends rapidly to A O ~ . The current a t the rapidly obtained steady state is given by i
= nF~D(drl/dx),-o =
nFsflkAo
(8)
4131
trode. The area of the dropping electrode is s = 0.85 m'latt'/awhere m is the rate of flow of mercury in grams/sec., and t is the time in seconds. I n the case of pure diffusion there is a distortion of the diffusion gradient due to the expanding drop which increases the gradient by a factor of f13.7 This factor will be assumed for the present to hold for the case of diffusion and reaction also. This is not rigorous, and a correction will be made later on the basis of the experimental results. The current as a function of time is obtained by combining eqs. 6 and 8 to give i = 0 . 8 5 m 3 nFAo m2/8t'/6dDT (e-kt erf
+ da
