(COX1RIHUT1O"I FROM T17F (\OA T f S CITF#ICAL 12.4BORATORV,
LCJUISIAUA STATE
UNIVERSITY
1
Theory of Alternating Polarographic Currents--Case of Reversible Waves
This is a theoretical treatment of the method involving the superposition of an alternating voltage of low amplitude t o the voltage applied to a polarographic cell. It is shown that the alternating current a t the half-wave potential is proportional t o the concentration of reducible substance and to the square root of the frequency of the alternating voltage. Theoretical results are conipnrrd with experimeiit,il tl:itn Conclurionr J\ t o the application of the method in derivative polarography in
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Introduction lluller and co-workers' were tlic first to clernonstrate the possibility of carrying out polarographic measurements by superposing an alternating voltage of low amplitude to the voltage applied to n polarographic cell. It was shown by hfuller, et al., that half-wave potentials could be measured very accurately by this Attempts to determine the concentration of reducible substance by measuring the alternating current through the polarographic cell were made by MacX1eavy3 m d by Sample.' These authors pointed out that the alternating current exhibits a maximum a t the half-wave potential; they also observed that the dternating current at is proportional to the concentration of reducible substance. Extensive data on the application of the method, however, mere reported neither by XacAleavy nor by Sample Recently, Breyer, Cutman and Hacobianj rnacle utnidn, ,ind c~~nsequcritlv the prese n t studv w ~ d buiidrrtAcii
Derivation of Current Boundary Condition.-\Ve consider the reversible polarographic reduction of a substance Ox to Red, and we assume f h a t substance Red is soluble either in the aqueous phase or in mercury. Let C ( x , t ) be the function representing the concentration of reducible substance in terms of the distance .2: from the electrode and the time t elapsed since the beginning of the electrolysis. I t is shown in the classical theory of polarography t h a t the relationship between the concentration of reducible substance at the electrode surface, Tio,t), arid the potential E of the dropping mercury elect rod t i,
where El/,is the half-wave potential, COthe bulk concentration of substance Os, and the other notations are convent i ~ n a l . ~ OIf potential E is not very different from the halfwave potential (see below), equation (1) can be simplified by writing C(o,t) as a function of E and by expanding the exponential function thus obtained. By retaining the first t n o terms i n the series, one has
Assume now that potential E varies periodically according t o the function E = E1 i2 1: sin ut (3) where E' is the amplitude of a small alternating voltage superp o d to the voltage applied t o the cell; w is related to the frequency f of the a1tern:iting voltage by the eqiiation w =
+
a7r"f.
By combining ( 2 ) and (3) one obtains the boundary condition for 11hich the diffu4on problem should be solved.
mli
Variations of Concentration.-Since
the variations of po-
tential are very rapid in comparison with the rate of the diffusion phenomenon at the electrode, it is possible to apply the differential equation of linear diffusion to the c a s o f the
dropping mercury electrode. T o solve the boundary .i-:iluc problem, it is convenient t o introduce the functioii I I tlcfnctl hy the equation il(rn,ii = C f N , t ) - (C";,., (.j) I n terms of this new function the initial and boundary cotitlitions a r c :IS follons: i t f x , o ) = 0 , and zt(o,tj = co I 1 I; in wt (we equation (4i 1 . The solutioii of the dif-
a
3l
r
fui;ion prohiem for this boundary condition can be obtained, for example, hy the Laplace transforination" and the subwquent use of the inversion integral.'* In this manner one obt:iins." after returning to the function C(x,f 1, the conccntr:ition as n function of timc 1 ant1 tli?tancc x from thc i
IZ??
..
..
1101 I. 1 1 K o l t h o f f x n d J . J 1.ingane. " P o l a r o g r a p h y , " I n t e r w r n w I'uhlishers, Inr , St.\\- Y o r k S . \-. l 9 % l , pp. 143--144, ?;ate t h a t t h e Atnerican c o n v e n t i o n of electrode signs I - tired here i n o r d e r t o rnake t h e c u r r e n t increase with m o r e poqitive p o t e n t i a l < . i l l ' H. S . Carslaw a n d J. C. J a e g e r , " C o n d u c t i o n of H e a t i n Solid- ' ' Oxford T.niveriity P r r s i . Oulord 1917 p 2>\i ( 1 2 ) 1:or 3 detailed d i s c u 4 o n of t h e m e t h o d ser R . V . C h u r c h i l l . " h l o t l ~ r n O p e r a t i o n a l \ l a l h e r n a t i c i In Engineering." S l c G r a w - H i l l Book Cu , Xycw Y o r k . S . 17.,I!)U, pp. 128-178. R e i r r c n r e % t u m o n o xr!iplt\ r i n I h i - wibierr xri. i:ivrn i n t l i i c t p u t l m n k .
A h T E R N i T T N G POL.iROC,R.\PTTIC CURRENTS :
Sov. 20, 19,52
Thus, the alternating term in the equation for
electrode. C ( x , t ) is
C(x,t) =
Lo \
--
2 )
R @Tv e x p . [ -
1-
x
(&)’”] sin [ w f - x (&)’”l/(6)
Equation (6) is a n approximate solution of the present boundary value problem. Indeed, the initial condition was written as u ( v , o i = 0 or C ( x , o ) = C0/2, \Thereas actually the concentration varies from Co/2 a t x = 0 t o Co for x = m . However, the use of an approximate initial condition is justified, because the gradient of concentration corresponding t o the alternating current is much larger than the gradient corresponding t o the direct electrolysis current at the half-wave potential. Therefore, the influence of the alternoting current does not extend in a region of the solution where the concentration is markedly different from Co/Z. Equation (6) was derived by assuming that the potential is not very different from E’/?, ; . e . , t h a t C(a,t) is not very different from C0/2. Therefore, one concludes from (6) that the following condition should be satisfied nF - a
