Studies on Alternating Current Electrolysis IV. Mathematical Treatment of Reversible Electron Transfer with Alternating Voltage Control and Distorted Current

A mathemat ica l t r ea tmen t is developed which yields equations re la t ing fa rada ic current , voltage, and t ime when an a l te rna t ing vol tage is appl ied to an e lect rolyt ic cell composed of a p lane and aux i l i a ry electrodes immersed in a solut ion containing in i t ia l ly suppor t ing e lec t ro ly te and only r eve r s ib ly oxidizable or reduc ib le species. Both ox idant and r educ tan t a re t aken to be soluble, and specific adsorpt ion is assumed to be absent. The vol tage across tha t b ranch of the equiva lent circui t th rough which only fa rada ic cu r ren t flows is assumed to be per iodic wi th fixed ampl i tude and wi th or wi thout an addi t ional direct appl ied vol tage component; the resu l t an t cur ren t is distorted. Diffusion control led kinet ics is postulated, and i t is assumed tha t equ i l ib r ium is essent ia l ly es tabl ished at the e lect rode surface. The equations deve loped show tha t a "s teady state" (i.e., a per iodic s tate) is quickly a t tained, y ie ld diagnost ic tests of use in es tabl ishing the revers ib le mechanism, make i t possible to de te rmine the s tandard potent ial , and finally y ie ld for the per iodic state a re la t ion be tween fa rada ic cu r ren t and time. These resul ts are then general ized so as to include systems in which the revers ib le e lect rochemical s tep is fol lowed by a sufficiently slow secondary reac t ion step. One diagnost ic resul t of in teres t in the l a t t e r connection is tha t the mean fa rada ic cur ren t vanishes in the per iodic state, regardless of the amp l i tude or of the shape of the appl ied per iodic potential , when the fo l low-up react ion occurs to a negl igible extent . The p r o b l e m of d e v e l o p i n g a m a t h e m a t i c a l d e s c r ip t ion of t he c h e m i c a l effects of a l t e r n a t i n g c u r r en t on r e v e r s i b l e r e d o x sys t ems has b e e n a t t a c k e d w i th v a r y i n g deg rees of success b y s e v e r a l i n v e s t i ga to r s (1 -12) sub j ec t to t h e fo l lowing r e s t r i c t ions : ( a ) u n d i s t o r t e d vo l t age a n d c u r r e n t waves , a n d / o r (b) bo th c o m p o n e n t s of t he r e d o x s y s t e m m u s t e i t he r b e p r e s e n t in t h e so lu t ion i n i t i a l l y or p r e s e n t a t the e l ec t rode ' s su r f ace as t h e r e s u l t of s u p e r i m posed d i r ec t cu r r en t . Berz ins and D e l a h a y (8) a s s u m e d w h a t in t he c lass i f ica t ion of D e l a h a y , Senda , and Weis (17) m a y be ca l l ed a l t e r n a t i n g v o l t a g e (A.V.) con t ro l [i.e., con t ro l l ed s inuso ida l p o t e n t i a l d i f fe rence across t h a t b r a n c h of t he e q u i v a l e n t c i r cui t for one e l ec t rode t h r o u g h w h i c h on ly f a r a d a i c ( f ) c u r r e n t is f lowing] . T h e y d e r i v e d an e q u a t i o n for t he i n s t a n t a n e o u s v a l u e of t he ( d i s t o r t e d ) f a r a da ic c u r r e n t as a func t ion of t ime. This e q u a t i o n con ta ined t r a n s i e n t t e r m s and a " s t e a d y s t a t e " (or, be t t e r , a " p e r i o d i c s t a t e " ) t e rm. P l a n e e l ec t rodes w e r e invo lved . R e c e n t l y M a t s u d a and D e l a h a y (13) e x t e n d e d t h e A.V. con t ro l p r o b l e m to d e r i v e an e q u a t i o n for the t r a n s i e n t c u r r e n t p r o d u c e d b y f a r a d a i c rec t i f i ca t ion a f t e r t he p e r i o d i c c o m p o n e n t h a d r e a c h e d a s t e a d y s ta te . T h e i r e q u a t i o n was r e s t r i c t e d to s m a l l va lue s of t he A.V. and to so lu t ions i n i t i a l l y cont a i n ing bo th o x i d a n t and r e d u c t a n t . 1 A l f r e d P . S l o a n F e l l o w ; N.S .F . S e n i o r P o s t D o c t o r a l F e l l o w a t I n s t i t u t e o f M a t h e m a t i c a l Sc i ences , N e w Y o r k U n i v e r s i t y (1960-61). K o u t e c k y (14) , a s s u m i n g A.V. cont ro l , d e r i v e d an equa t ion for the i n s t a n t a n e o u s v a l u e of t he f a r a da ic c u r r e n t as a func t ion of t i m e for t h e g e n e r a l case of a pe r iod i c vo l t a ge of op t iona l w a v e shape and a p p l i e d i t to t h e spec ia l cases of s ine waves , s q u a r e waves , a n d t r i a n g u l a r waves . His equa t i ons w e r e l a rge ly , a l t h o u g h no t en t i r e ly , conce rned w i t h t h e condi t ions c h a r a c t e r i s t i c of p o l a r o g r a p h y (i.e., d r o p p i n g e l e c t r o d e ) , b u t his w o r k a n d an in i t i a l po r t i on of t he p r e s e n t w o r k a r e r e l a t ed . W e sha l l be conce rned w i t h s t a t i o n a r y e lec t rodes , w i t h a t r e a t m e n t for ( in some w a y s ) s l i g h t l y m o r e g e n e r a l sys tems , w i th a d e t a i l e d t r e a t m e n t of t he s t a t e m e n t of the pe r iod i c s ta te , and w i t h dev i s ing a m e t h o d of d e t e r m i n i n g t h e s t a n d a r d p o t e n t i a l a n d o t h e r p r o p e r t i e s (m and n b e l o w ) of a r e v e r s i b l e e l e c t r o chemica l s tep in an o t h e r w i s e i r r e v e r s i b l e process . In the p r e s e n t p a p e r w e sha l l d iscuss a s y s t e m u n d e r g o i n g A.V. e l ec t ro lys i s w i t h or w i t h o u t s u p e r imposed d i r ec t a p p l i e d vo l t a ge for the case e m p l o y ing vo l t a ge c on t ro l w h e n the i n i t i a l c o n c e n t r a t i o n of e i t h e r o x i d a n t or r e d u c t a n t is zero. C lea r ly , l a r g e A.V. a m p l i t u d e s m a y be n e c e s s a r y to b r i n g conc e n t r a t i o n s a t t he e l ec t rode su r f ace in to t he po i sed r eg ion w h e r e r e a s o n a b l e a c c u r a c y in p o t e n t i a l c o n c e n t r a t i o n r e l a t i o n s h i p s m a y be ach ieved . ~ This case is of on ly l i m i t e d p r a c t i c a l i n t e r e s t for A c e r t a i n e x p e r i m e n t a l d i f f i cu l ty w h i c h , i t h a s b e e n s u g g e s t e d [p. 313 of r e f . (17) ] , cou ld c o m p l i c a t e i n t e r p r e t a t i o n of t h e m o r e u s u a l ( low a m p l i t u d e ) A. V. c on t r o l s y s t e m s , is a b s e n t h e r e b e c a u s e of the d i f ferent m e t h o d u s e d to i n t e r p r e t t h e r e su l t s .