Theoretical Calculations of the Separation Factors in the Hydrogen Evolution Reaction for the Slow Electrochemical Desorption Mechanism

Vol. 111, No. 7 S E P A R A T I O N F A C T O R S I N H E V O L U T I O N 853

Ca r lmichae l , and K. T h o r m a l i n g a m , and Messrs . D. B. M a t t h e w s a n d R. H a y n e s for h e l p f u l t h e o r e t i c a l d iscuss ions . M a t h e m a t i c a l a s s i s t ance f rom Mr. C. V a s e e k a r a n and p a r t of t he c o m p u t e r fac i l i t i es a f - f o r d e d b y Dr. L. Nan i s a r e also g r e a t l y a p p r e c i a t e d .

Manuscr ip t rece ived Aug. 5, 1963; rev ised m a n u - scr ip t rece ived Dec. 11, 1963. This paper is f rom work car r ied out in pa r t i a l ful f i l lment of r equ i rements for the degree of Ph.D.

Any discussion of this pape r wi l l appea r in a Discus- sion Section to be publ i shed in the June 1965 JOURNAL.

REFERENCES 1. T. Kei i and T. Kodera , J. Research Inst. Cat., 5,

105 (1957). 2. B. E. Conway, Proc. Roy. Soc., A247, 400 (1958). 3. T. Kodera and T. Saito, J. Research Inst. Cat., 7, 5

(1959). 4. G. Okamoto, J. Horiut i , and K. Hirota , Sci. Papers

Inst. Phys. Chem. Research (Tokyo), 29, 223 (1936).

5. J. Horiut i , T. Keii, and K. Hirota, J. Research Inst. Cat., 2, 1 (1951).

6. J. Hor iu t i and T. Nakamura , ibid., 2, 73 (1951). 7. J. O'M. Bockris, "Modern Aspects of E lec t rochem-

is t ry," vol. I, chap. IV, But te rwor ths , London (1954).

8. B. E. Conway, Proc. Roy. Soc., A256, 128 (1960). 9. B. E. Conway, "Transact ions Sympos ium on Elec-

t rode Processes," chap. 15, The Elect rochemical Society, John Wi ley & Sons, New York (1961).

10. S. Glasstone, K. J. Laidler , and H. Eyring, "The Theory of Rate Processes," McGraw Hil l Book Co., New York (1941).

11. C. E. H. Bawn and G. Ogden, Trans. Faraday Soc., 30, 432 (1934).

12. B. E. Conway, Can. J. Chem., 37, 178 (1959). 13. H. S. Johnston, "Advances in Chemical Physics,"

vol. III, p. 131, In tersc ience Publ ishers , New York (1961).

14. R. Parsons and J. O'M. Bockris, Trans. Faraday Soc., 47, 914 (1951).

15. B. E. Conway, J. O'M. Bockris, and B. Lovrecek, "Proceedings C.I.T.C.E., S ix th Meeting," p. 207, But te rwor ths , London (1955).

16. L. Melander , " Isotope Effects on React ion Rates," Ronald Press, New York (1960).

17. S. Sr inivasan, Thesis, Univers i ty of Pennsy lvan ia (1963).

18. M. Ikus ima and S. Azakami , J. Chem. Soc. Japan, 59, 40 (1938).

19. J. Hor iut i and G. Okamoto, Bull. Chem. Soc. Japan, 10, 503 (1935).

20. O. Sepal l and S. G. Mason, Can. J. Chem., 38, 2024 (1960).

21. G. Brown, Refer red to in ref. (20). 22. W. S. Benedict, N. Gailer , and E. K. P lyer , J. Chem.

Phys., 24, 1139 (1956). 23. W. F. Libby, ibid., 11, 101 (1943). 24. G. Herzberg, " Inf ra Red and Raman Spect ra of

Polya tomic Molecules," p. 173, Van Nostrand, New York (1945).

25. F. H. Westheimer , Chem. Rev., 61, 265 (1961). 26. L. Paul ing, "The Na ture of the Chemical Bond,"

Cornel l Univers i ty Press, New York (1962). 27. St. G. Christov, Electrochim. Acta, 4, 194 (1961). 28. J. O'M. Bockr is and S. Sr inivasan, To be publ i shed

in Electrochem. Acta. 29. J. O'M. Bockris, M. A. V. Devanathan , and K.

Miiller, Proc. Roy. Soc., A274, 55 (1963). 30. H. S. Johns ton and D. Rapp, J. Am. Chem. Soc., 83,

1 (1961). 31. R. P. Bell, Trans. Faraday Soc., 55, 1 (1959). 32. A. Fa rkas and L. Farkas , J. Chem. Phys., 2, 468

(1934). 33. J. O'M. Bockris and S. Srinivasan, This Journal, 111,

853 (1964).

Theoretical Calculations of the Separation Factors in the Hydrogen Evolution Reaction for the Slow Electrochemical

Desorption Mechanism John O'M. Bockris and S. Srinivasan

The Electrochemistry Laboratory, The University o~ Pennsylvania, Philadelphia, Pennsylvania

ABSTRACT

Separa t ion factors (S) for the slow e lec t rochemical desorpt ion mechan i sm at low and in te rmed ia te overpotent ia l s a re ca lcula ted for Ni. The S va lues a re d is t inc t ly separa ted f rom those for a l l o ther mechanisms. The resul ts a re con- s is tent wi th var ia t ion of percen tage coulombic ene rgy of a l l interact ions. The separa t ion factor me thod alone cannot be used to d is t inguish be tween e i ther of the r a t e - d e t e r m i n i n g steps in a l inked discharge e lec t rochemical desorpt ion mechanism. A knowledge of the degree of coverage is also requi red .

Ca lcu la t ions of e l e c t ro ly t i c s e p a r a t i o n fac to rs a r e of v a l u e in e x a m i n a t i o n of m e c h a n i s m of h y d r o g e n evo lu t i on and a r e d i scussed e l s e w h e r e (1 ) . Of the m e c h a n i s m s cogent in th is d iscuss ion , t he e l e c t r o - c h e m i c a l d e s o r p t i o n m e c h a n i s m is one of i m p o r t a n c e . The t h e o r e t i c a l va lues , a s soc ia t ed w i t h th is m e c h a n - ism, a r e c a l c u l a t e d in th is pape r .

In p r e v i o u s ca lcu la t ions of the s e p a r a t i o n f ac to r for th i s m e c h a n i s m (2) , t he r a t e of on ly one of the two p a t h s for HD evo lu t i on

HsO + + eo q- MD-> HD + HeO + M [la]

H2DO + eo q- MH-+ HD + H20 + M [ib]

was compared with the rate of Ha evolution accord-

ing to H30 + - } - e o + M H - > H 2 + H 2 0 + M [2]

T h a t HI) e v o l u t i o n could occur b y the two p a r a l l e l p a t h s [ l a ] a n d [ l b ] was cons ide red b y the a u t h o r of t he p r e v i o u s p a p e r , who conc luded h o w e v e r t ha t HD e v o l u t i o n b y s tep [ l b ] w o u l d l e a d to s e p a r a t i o n f ac -

) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 129.81.226.78Downloaded on 2014-09-05 to IP

http://ecsdl.org/site/terms_use

854

tors less than unity, which is in conflict with experi- mental observations. Since the total quanti ty of HD in the electrolytic gas is evolved according to the two parallel paths, it is necessary to compare the rates of both paths simultaneously with the rate of path [2]. Fur ther refinements on the previous calculations are possible through inclusion of the zero point energy differences of the isotopic activated complexes (which were assumed sufficiently short lived that their vibrational frequencies do not contribute to the activation energy), and of tunneling corrections [as there is still some question as to the conditions under which these contribute significantly to the separation factors (3, 4) ].

For mechanisms involving adsorbed species in the rate determining step, the ratio of adsorbed atomic hydrogen to deuterium (0n/SD) is a part icularly im- portant quantity. In the previous separation factor calculations, this ratio was worked out for a s teady- state situation, allowing for isotopic differences of free energies of activation in the forward and back- ward directions of the discharge step. The result of assuming that 0H/~D is approximately equal to the ratio of activities of the isotopic ions HsO + and H2DO + was also considered. Since an isotope effect exists in the pr imary discharge step, this latter as- sumption is not correct.

Present Calculations Slow Electrochemical Desorption at Low and

Intermediate Overpotentials

Expression ]or separation ]actor.--Under these conditions, the discharge steps may be considered to be in equilibrium. They are represented by

HsO + + eo + M ~ MH + H20 [3]

H2DO + + eo + M~-- MD + H20 [4]

and are followed by [1], [2] and

HsO + + eo + MH-~ H2 + H20 + M [5]

If an2ot > > aHDO1, the separation factor is given by

1 ( C ~ ) aHDol SD = - ~ g aH201

[6]

where (CH/CD)g is the ratio of atomic concentrations of H to D in the electrolytic gas. During electrolysis, (CH/CD) g may be expressed as

( C_~D ) = 2iS,H2 "~ i8,HD [7]

g ~3,HD

where iS,He is the electrochemical desorption current according to path [5] and is,HD is the total current for HD evolution according to the two parallel paths [1] and [2]. Since i3,H2 > > i8,HD, SD is given by

aHDOI i3,H2 s~ = - - - - [8]

aH2ol i3,HD

The currents is,H2 and iS,HD may be represented as

is,He = ks,n2 anso+ aMH e -BVF/RT [9]

i3,HD : iS,H,D "~ i3,D,H [10]

JOURNAL OF THE ELECTROCHEMICAL SOCIETY

where

July 1964

a i3,H~D = kS,H,D--~- H2DO + aMH e -~vF/RT [11]

iS,D,H = ka,D,H aH30+ alViD e -~vF/RT [12]

iS,D,H and ia,n,D are the partial currents according to paths [1] and [2], respectively; k's are the re- spective rate constants; aMn and aMD are the activities of the adsorbed hydrogen and deuter ium atoms respectively, on the surface; and V is the metal solution potential difference.

It follows from Eq. [9] to [12] that

iS,HD i3,H,D -~ iS,D,H

i3,H2 i3,H2

1 kS~H,D --~- aH2DO + kS,D,H

-~-. , kS,H2 aH30 + ~8,H2

[13]

FH fH,D # fHsO +

rI) fH2 # JH2DO +

r H ~CD,H# + - - I~D 5H2 ~/~

--gH30 +/RT FH fH,D :2~ e

_ _ .

F D fH2 ?~ -~ +/RT H2DO

e

aMD

aMH

1 aH2DO +

3

all30 +

fMH aMH

fMD aMD

[14]

[15]

and

where

iS,HD ( YH,D# + fD,H~ ] FH

i3,H2 \ ]H27~ ]H2 r / FD

It follows from Eq. [8] and [17] that

S D = - - F H InDOg KD [ :f~'D# + J~D'n# ] -1 [18] FD fH2Og k fH2 # fH2 #

Equation [18] may also be expressed in the form

1 1 1 - - - - + ~ [19]

SD SD,E1 SD,E2

--p. /RT H2Og

e

-I~ /RT HDOg

e [17]

SD,E1 ~ " - - rH ]H2 # :fHDOg KD [20] FD ~H,D # fH20g

SD,E2 ~ - I~H fH2 ~ fHDOg KD [21] FD fD,H # fH20g

r, Y, ~ with the appropriate suffices are the tunneling factor, partit ion function, and chemical potential, respectively, of the indicated isotopes.

By considering the equilibrium reactions [3] and [4] and also between the isotopic oxonium ions and water molecules in the liquid phase and then in the gaseous phase (5), Eq. [16] may be writ ten as

- a /RT MH

r H )CD,H# e + - - [16]

r D fH2:2~- --p. /RT MD

e



Vol. 111, No. 7 S E P A R A T I O N F A C T O R S I N H E V O L U T I O N 855

<--V~-2--> <--V2-3-~ M . . . . . . . . . . . . H . . . . . . . . . . . . 14 r . . . . . . . . . . . . OI-I2 1 2 3 4 < V1-3 > < V~4 >

Fig. 1. Activated complex for slow electrochemical desorption mechanism.

KD is t he e q u i l i b r i u m cons t an t for the r e a c t i o n

H201 -~ HDOg ~-- HfOg -t- HDO1 [22]

The H / T s e p a r a t i o n f ac to r for t he fas t d i s c h a r g e - s low e l e c t r o c h e m i c a l d e s o r p t i o n m e c h a n i s m a t low and i n t e r m e d i a t e o v e r p o t e n t i a l s is g iven b y an e x - p ress ion , s i m i l a r to [19] .

Numerical Calculations of ~eparation Factors

Product oI equil ibrium constant (KD or KT) and partition ]unction ratio of isotopic water molecules in the gas phase (]HDOg/]gfOg o r /HWOg/fH2Og).--In a p r e v i o u s p a p e r (3 ) , i t was shown

fHDOg KD �9 - - - - 63 .36 [23]

fH2Og

K T . fHWOg _ 3 1 6 . 4 8 [24] ]H20g

Partit ion ]unction ratios of isotopic activated complexes (]H,D=J-/fH2 ~ , fH,T:~4=/]H2:2(:, ID,H~/fH2 =fl, and fW,H~/fH2 r m a y be a s s u m e d t h a t t he a c t i v a t e d c o m p l e x has a conf igura t ion in w h i c h the cen t e r s of the m e t a l a tom, h y d r o g e n a tom, p a r t i a l l y n e u t r a l i z e d h y d r o g e n ion and w a t e r m o l e c u l e a r e co l l inear , as s h o w n in Fig . 1.

F o r t he ca l cu l a t i on of ]H,D:2-L//IH2 :'~-, i so topic s u b s t i - t u t ion occurs a t p o s i t i o n 3 in Fig . 1, w h e r e a s for c a l cu l a t i ng fD,H:J=/]H2 =fi, i so topic subs t i t u t i on is a t pos i t ion 2. The a c t i v a t e d c o m p l e x e s m a y be r e g a r d e d as immob i l e , u n d e r w h i c h cond i t ions t h e i r t r a n s l a - t i ona l p a r t i t i o n func t ion r a t i o is un i ty . The c o n t r i - bu t i on to t he i so tope effect due to r e s t r i c t e d r o t a t i o n abou t t he two axes , m u t u a l l y p e r p e n d i c u l a r to t he axis of t he m o l e c u l e is ins igni f icant , due to t he h e a v y end a toms. Thus, t he p a r t i t i o n func t ion ra t ios of t he i sotopic a c t i v a t e d c o m p l e x e s r e d u c e to

5 ]H2 ~ IIi s inh (hvi /2kT) H.D

- - - - [ 2 5 ]

]H,D:2-4 5 1] i s inh (h~,/2kT) H2 5

fH2 ~ H i s inh (h , i /2kT) D.H [26]

~D,H=2~ - - 5 Hi s inh (hvl /2kT) He

To d e t e r m i n e the coo rd ina t e s of the cen t r a l a toms (or i ts i so topes ) in t he a c t i v a t e d complex , the p o t e n - t i a l e n e r g y of t he s y s t e m (V) was t r e a t e d as a f ou r a t o m p r o b l e m (6) . V is g iven b y

V ~ Kl2 § Kf~ q- K~4 -~ Kl~

- - ~ { ( ~ - B ) 2 + ( B - ~ , ) ~ + ( ~ , - ~ ) ~} [27]

w h e r e = J12 ~- J34 [28]

fl =: Jl~ [29]

T = J23 [ 3 0 ] and

Vij = Kij -~- J,j [ 31 ]

acco rd ing to t he H e i t l e r - L o n d o n me thod . V u is the p o t e n t i a l e n e r g y due to i n t e r a c t i o n s b e t w e e n a toms i and j w i t h v a r y i n g i n t e r n u c l e a r d i s t ance ; Kij and J,j a r e t he r e s p e c t i v e cou lombic and e x c h a n g e con- t r i b u t i o n s to the to ta l ene rgy .

The M-HeO and H - H z O i n t e r a c t i o n s w e r e n e g - l ec ted s ince t h e i r r e s p e c t i v e i n t e r n u c l e a r d i s t ances a r e r e l a t i v e l y f a r a p a r t and also be c a use these i n t e r - ac t ions a r e c o n s i d e r a b l y less t h a n the o the r s w h e r e s t rong c h e m i c a l b o n d i n g exis ts .

As in t he m e t h o d of E y r i n g et al., t he r e f e r e n c e s t a t e for t he e n e r g y was t a k e n as t h a t of t he s e p a - r a t e d a toms 2H + M + HeO [HeO is t r e a t e d as a p seudo a t o m ( 7 ) ] . The s u m of the energ ies , (V12 + V34), r e p r e s e n t the p o t e n t i a l e n e r g y of t he in t i a l s ta te as a f u n c t i o n of t he M - H and H + - O H e d i s - tances . F o r th is ca lcu la t ion , the r e s p e c t i v e Morse func t ions w e r e used.

The p o t e n t i a l e n e r g y (Vie + V34) was e x p r e s s e d as a func t ion of t he d i s t ance b e t w e e n the H + ion (pos i t ion 1) a n d H20 m o l e c u l e and also b e t w e e n the m e t a l a t o m and H a t o m (pos i t ion 2) b y c o n s i d e r - ing the fo l l owing B o r n - H a b e r cycles (8) .

2H + HeO + M--> H+ + HeO + M + H + eo ( I ) [32] H + + H f O + H + M W e o

-> Ha O+ ~- eM ~- H + M (D12-- 4) [33]

H3 0 + -~ eM "~- H ~- M--> H 3 0 + -[- eM ~- M H (D34) [34]

A d d i n g

2H + H20 + M--> H30 + + eM ~- M H (Vie + V34) [35]

V12 + V~4 = I + D12 - - ~ -}- D34 [36] w i t h

D12 = D12~ - - e-a l f ( r l f - r12~ 2 ' - ILl [37] and

D34 ---- Va4 Dg4O[e-2a34(rg4-r34 ~ - - 2e-a34(r34-r34 ~ [38]

L a n d D12 ~ a r e t he h e a t of so lva t ion of a p r o t o n and the p r o t o n aff ini ty for a w a t e r m o l e c u l e in the gas phase , r e spe c t i ve ly , as used in t he s low d i s - c h a r g e m e c h a n i s m ca lcu la t ions (5) . Vea is t he e n - e r g y of the f inal s ta te , Ha, as a func t ion of the H - H d i s t ance and is g iven b y the Morse func t ion of t he f inal s ta te . I n t h e a bse nc e of spec t roscop ic d a t a for t he M - H + in t e r ac t i on , t he Morse f u n c t i o n of M - H was used for t he V13 ene rgy . (Spec t ro scop ic d a t a for H g - H and H g - H + a re p r a c t i c a l l y t he same . ) As in t he s low d i s c h a r g e m e c h a n i s m ca lcu la t ion , the h y - d r o g e n ion is a s s u m e d to be an ion, w h e r e i n t e r - ac t ions w i t h t he w a t e r m o l e c u l e a r e conce rned and as an a t o m for i n t e r a c t i o n w i t h the o t h e r h y d r o g e n a toms. H o w e v e r , t he e n e r g y of t he e l e c t r o n has been accoun ted for t h r o u g h the chosen r e f e r e n c e s ta te . F u r t h e r , w h e n the h y d r o g e n ion is suff ic ient ly s e p a - r a t e d f r o m the w a t e r molecu le , t he H + - O H e i n t e r - ac t ion is s m a l l and the H - H e n e r g y is p r e d o m i n a n t and i t is l i k e l y t h a t in th is pos i t ion the h y d r o g e n



856 JOURNAL OF THE ELECTROCHEMICAL SOCIETY July 1964

Table I. Physical constants in calculation of potential energy of system (V)

I n t e r a c t i o n

P a r a m e t e r H-H N i - H H+-0H~

D ~ kcal mole -~ 109.52 60.00 187.00 a A -1 1.94 1.60 1.375 re A 0.74 1.48 1.05

ion is fu l ly neu t ra l i zed . Thus, the a s sumpt ion is not se r ious ly in er ror .

As in the s low r e c o m b i n a t i o n m e c h a n i s m ca l cu l a - t ion (9) , t he m e t a l a t o m was cons ide red as fixed. In addi t ion, the pos i t ion of the w a t e r molecu le , w i t h respec t to the m e t a l a tom, was also cons idered to be f ixed for purposes of s impl i fy ing the ca lcu la t ion of v i b r a t i o n a l f requenc ies . The e r ro r i n v o l v e d in the ca lcu la t ion of the v i b r a t i o n a l f r equenc ies , due to the l a t t e r assumpt ion , is smal l since the w a t e r mo lecu l e is h e a v y c o m p a r e d to the h y d r o g e n or d e u t e r i u m a toms and also because the mot ion of the w a t e r mo lecu l e is r e s t r i c t ed due to h y d r o g e n bond ing w i t h s u r r o u n d i n g w a t e r molecules .

The po ten t i a l e n e r g y of the sys t em was t h e n ca l - cu la ted as a func t ion of the coord ina tes of the two h y d r o g e n a toms w i t h respec t to the m e t a l a tom using a compute r . F o u r ca lcu la t ions w e r e ca r r i ed out for nickel . The constants , used in the ca lcu la t ion of V, a re shown in Tab le I. In the first two calcula t ions , an Ni -HeO dis tance of 3.5A was used, as in the ca l cu - la t ion of C o n w a y and Bockr i s (8) . No saddle point was ob ta ined w h e n the to ta l e n e r g y was a s sumed to be 100%coulombic for a l l i n t e rac t ions nor for the case of 20% cou lombic e n e r g y for the N i -H , H + - O H e in te rac t ions and 15% coulombic for the H - H i n t e r - action. The resu l t s of these two ca lcu la t ions i nd i - cate t ha t the e l e c t rochemica l desorp t ion m e c h a n i s m can t ake p lace r ead i l y for the a s sumed d is tance be - t w e e n the Ni a t o m and w a t e r molecule .

In the t h i r d and f o u r t h calcula t ions , the d i s tance b e t w e e n t h e Ni a t o m and w a t e r mo lecu l e was t aken as 5.4A. This d i s tance fo l lows f r o m the mode l of the double l aye r accord ing to Bockris , D e v a n a t h a n , and Mii l le r (10) in w h i c h a l a y e r of w a t e r mo lecu le s separa tes the m e t a l su r face f r o m the first l aye r of ions. In the t h i rd ca lcula t ion , the to ta l e n e r g y of the sys t em was a s sumed to be coulombic. In the four th ca lcu la t ion 20% of the H+-OH~, N i - H in t e rac t ions and 15% of the H - H in te rac t ions w e r e a s sumed to be coulombic . Sadd le points w e r e ob ta ined in the two calcula t ions . L e t the coord ina tes of the H a tom (po- s i t ion 2) and H ion (pos i t ion 3) at the saddle poin t be (xo,D,D) and (yo,O,O). (The axis of the ac t iva t ed c omplex is the x axis, the y and the z axis a re the two axes m u t u a l l y p e r p e n d i c u l a r to the x ax is ) .

The p o t e n t i a l e n e r g y in the ne ighbo rhood of the saddle po in t m a y be e x p a n d e d by a Tay lo r ser ies and is g i v e n by

1 V = Vo + -- ~:~ a ~ ( x -- Xo) ~

2 x

+ ax~(X-- Xo) (~ - - ~o) + a~y~? + az~Z~ [39]

Since the a c t i v a t e d com plex is l inear , the o ther cross t e r m s vanish. The t e r m s axx . . . . a~ w e r e ob- t a ined by a d e t e r m i n a t i o n of the co r r e spond ing sec- ond d e r i v a t i v e s ( O ~ V / a x 2 . . . . O~V/OzO~) of the po- t en t i a l func t ion at t he saddle point . T h e y w e r e t h e n used in the secular equa t ions

ax~ - - 4miX, a ~ : 0 [40] a~, a ~ - - 4m2~

a y y - - 4miX, a ~ a~ , a~, - - 4m2~, : 0 [41]

a~z - - 4talk, a~ az~, a ~ - 4 m ~ : 0 [42]

w h e r e k : 4~r2~ 2, m~ and m2 are the masses of h y d r o - gen (or its i sotopes) a t posi t ions 2 and 3, r e s pec - t ive ly . One of the f r equenc ie s is imag ina ry . The bend ing f r equenc i e s are doub ly d e g e n e r a t e due to the cy l indr ica l s y m m e t r y of the po ten t i a l func t ion a round the saddle point . The resu l t s of the two ca lcu la t ions inc lud ing pa r t i t i on func t ion ra t ios of the ac t i va t ed c o m p l e x e s are s u m m a r i z e d in Tab l e II.

The f o u r t h ca lcu la t ion g a v e an ac t i va t i on e n e r g y ( E r cons ide rab ly h ighe r t h a n the third. The e x - p e r i m e n t a l l y o b s e r v e d E r for the h y d r o g e n e v o l u - t ion r eac t ion on Ni in acid solu t ions (11) is 7 kca l mole - ' and this f igure is also m u c h less t h a n the

Table II. Force constants (a's) vibrational frequencies (co's) partition function ratios (fH2~/fH,D~, etc.) of isotopic activated

complexes for the slow electrochemical desorption mechanism

P e r c e n t a g e c o u l o m b i e e n e r g y f o r N i - H , H + - O H e a n d H - H i n t e r a c t i o n s

(p) a n d c a l c u l a t i o n n u m b e r i n p a r e n t h e s e s

p N i - H = p N I - H ~ pH+-OH 2 ~ pH+-OH~ = 20

Parameter p~_~ = 100 3 pH-~ = 15 4

r12~/~.* 2.40 2.74 r23r A* 1.88 1.58 r34~/~* 1.12 1.08 E r kcal mole - l * 21.1 50.8 axx kcal mole -1 A -2 --109.52 --191.39 ayy kcal mole -1 A -2 39.47 22.94 azz kcal mole -1 A -2 39.47 22.94 a$~ kcal mole -1 A -2 448.84 298.18 a , , kcal mole -1 A -2 61.10 27.96 a~ kcal mole -1 A -2 61.10 27.96 ax~ kcal mole -1 A -2 70.84 152.68 ay~ kcal mole -1 A -2 --22.07 --14.65 az~ kcal mole -1 A -2 --22.07 --14.65 ~o for Ni - - - H - - - H a 588i, 1156, 468, 829i, 999, 343,

- - - OH2 (cm -1) 274, 468, 274 176, 343, 176 co f o r N i - - - H - - - D r 584i, 823, 387, 806i, 726,295,

- - - OH2 (am -1) 234, 387, 234 145, 295, 145 co f o r N i - - - D - - - H r 418i, 1151,440, 600i, 975,310,

- - - OH2 (cm -1) 206, 440, 206 137, 310, 137 co f o r N i - - - H - - - T e 571i, 675, 366, 794i, 596,286,

- - - OH2 (cm -1) 202, 366, 202 124, 286, 124 c o f o r N i - - - T - - - H e 341i, 1149,433, 496i, 964,301,

- - - OH2 (cm -1) 171, 433, 171 116, 301, 116 fH2~/fH,D~ 0.1864 0.2359 fH2~/fD,H=~ 0.4411 0.4411 fH2~/fH,T# 0.08048 0.1077 fH2~/fW,Hr 0.2828 0.2845

* r l2 , r~s a n d vs4 a r e t h e N i - H , H - H a n d H - O H 2 i n t e r -

nuclear distances in the activated complex; E is the activation energy.



Vol. 111, No. 7 S E P A R A T I O N F A C T O R S IN H E V O L U T I O N

Table III. Separation factors for the slow electrochemical desorption mechanism at low and intermediate overpotentials

857

S e p a r a t i o n f a c t o r s S e p a r a t i o n f a c t o r s C a l c u l a t i o n P e r c e n t a g e c o u l o m b i c e n e r g y (p) S e x c l u d i n g t u n n e l i n g c o r r e c t i o n s i n c l u d i n g t u n n e l i n g c o r r e c t i o n s

No. pN i - H pH+-OH2 pH-H SD, El* SD,E2* SD = SD,E l SD,B 2 SD

3 100 100 100 11.8 27.9 8.3 13.0 30.7 9.1 4 20 20 15 14.9 27.9 9.7 16.4 30.7 10.7

:-~T, ]~1 * ST, E2* ~T * ST, E 1 ~'T, E 2 ST

3 100 100 100 27.9 89.5 19.8 29.5 103.8 23.0 4 20 20 15 34.1 90.0 24.7 39.5 104.4 28.7

value obta ined in the th i rd calculation. The p a r t i - t ion funct ion rat ios in the two calculat ions are, however, not significantly different. The value obtained in the th i rd calculat ion is p re fe rab le since E~ in this calculat ion is closer to the exper imenta l ac t ivat ion energy. I t may be expected tha t the correct p a r t i - t ion function rat ios of the ac t iva ted complexes may be s l ight ly less than the values of calculat ion 3. The isotope effect is g rea te r when the isotopic subs t i tu - t ion is a t posit ion 2 r a the r than posit ion 3 (see Fig. 1).

Tunneling ]actor ratios (I~It/I~D and FH/I~T).--The tunnel ing factor rat ios were obtained for hydrogen ion or atom (or its isotopes) t ransfe r distances of 2.13A [5 .40- - (1.48 + 1.05 -t- 0.74)]. Calculat ions for both parabol ic and Eckar t ba r r i e r s (5, 12) were in good agreement . FH/rD is 1.09 for a parabol ic and 1.10 for an Eckar t bar r ie r , the corresponding r~/ rw ratios being 1.13 and 1.16, respect ively.

Separation Factors

Separation iactors excluding tunneling corrections (SD* and Sw*). - -Using Eq. [19] to [21], [23], [24] and the da ta in Table II, SD* and ST* were calculated and are given in Table III. The values obtained in calculat ion 3 are closer to the correct values, since the act ivat ion energy obta ined in this calculat ion is closer to the exper imenta l act ivat ion energy than that of calculat ion 4.

Separation 5actors including tunneling corrections (SD and ST).---The separa t ion factors including t un - neling corrections are also given in Table III. I t is probable tha t the tunnel ing factors are somewhat overes t imated in v iew of the low ba r r i e r height at an overpoten t ia l of 0.4v on Ni and thus the sepa ra - t ion factors, excluding tunnel ing correct ions are the p re fe r red values for this mechanism.

Table IV. Separation factors for the slow electrochemical desorption mechanism at all overpotentials

S e p a r a t i o n S e p a r a t i o n f a c t o r s f a c t o r s

P e r c e n t a g e e x c l u d i n g including c o u l o m b i c t u n n e l i n g t u n n e l i n g e n e r g y (p) c o r r e c t i o n s correct ions

O v e r p o t e n t i a l r e g i o n pNS-H pPI+-OH 2 pvi-Pi SD* ST* SD ST

Low and intermediate 100 100 100 8.3 19.8 9.1 23.0

Low and intermediate 20 20 15 9.7 24 .7 10.7 28.7

High 100 100 100 3.4 5.4 4.1 7.0 High 20 20 15 3.6 5.7 4.4 7.5

Slow Electrochemical Desorption Mechanism at High Overpotentials

Expression for separation factor.--At high overpotent ials , the reverse currents of the discharge steps 3 and 4 m a y be neglected. The separa t ion factor expression for this case was worked out p rev i - ously (5) as

1 1 1 - - - - - - + [ 4 3 ]

SD 2SD.1 SD,E1

where SD1 is the separa t ion factor expression for the discharge step and is given by

1 ]H r JHDOg S D , 1 = - ~ - " f D r ]H2Og " K D [44]

This equation holds for a l inked d i scharge-e lec t ro - chemical desorpt ion mechanism. However, at high overpotent ia ls , if the coverage of the e lectrode with adsorbed atomic hydrogen (8) is low, the mechanism is ra te de te rmined by the discharge step, whereas if 8 is high, the e lect rochemical desorpt ion step is r a t e - determining. Equat ion [43] was obta ined in the genera l case but holds equa l ly wel l for the two special cases of low and high degree of coverage, the de te rmina t ion of which dist inguishes the a l te rna te r a t e -de t e rmin ing steps.

Numerical values ef separation 5actors.--The nu- mer ica l values of separa t ion factors were worked out prev ious ly and are given in Table IV, along wi th the S values for the slow electrochemical desorpt ion mechanism at low and in te rmedia te overpotent ia ls .

Conclusions At low and in te rmedia te overpotent ia ls , when the

discharge step can be t r ea ted as in equi l ibr ium, the ca lcula ted S values are d is t inc t ly separa ted from the corresponding values for all other mechanisms.

The theore t ica l separa t ion factors, using a 100% coulombic energy for all interact ions, are closer to the t rue S values, since the exper imenta l act ivat ion energy is closer to the calcula ted act ivat ion energy for this case than for the case where the coulombic energy is t aken at 20% for the Ni -H and H+-OH2 bonds and 15% for the H - H bond. It is not possible to reduce the calcula ted values any fur ther , since the separa t ion factors in both the p resen t ly cal - cula ted cases are not much different even though the ca lcula ted act ivat ion energies differ by as much as 30 kca l mole - I .

At high overpotent ia ls the separa t ion factor is given by that for a l inked d ischarge-e lec t rochemica l



858 J O U R N A L O F T H E E L E C T R O C H E M I C A L S O C I E T Y J u l y 1964

Acknowledgments F i n a n c i a l suppor t f rom the A e r o n a u t i c a l Sys tems

Command , Ai r Force Sys t ems Command , Un i t ed Sta tes Ai r Force, Cont rac t 33(657)-8823, is g ra t e - fu l ly acknowledged. The au thors also wish to t h a n k Drs. E. R. N i xon a nd K. T h a r m a l i n g a m for he lp fu l theore t ica l discussions. Ma thema t i ca l ass is tance f rom Mr. C. V a s e e k a r a n and compute r faci l i t ies af - forded b y the Physics D e p a r t m e n t are g rea t ly ap - preciated.

Manuscript received Aug. 9, 1963; revised m a n u - script received Dec. 11, 1963. This paper is from work carried out in part ial fulf i l lment of requ i rement for the degree of Ph.D.

Any discussion of this paper will appear in a Discus- sion Section to be published in the June 1965 JOURNAL.

REFERENCES 1. J. O'M. Bockris and S. Srinivasan, Electrochim.

Acta, in press. 2. B. E. Conway, Proc. Roy. Soc., A247, 400 (1958). 3. J. O'M. Bockris and S. Srinivasan, This Journal, 111,

858 (1964). 4. B. E. Conway, Can. J. Chem., 37, 178 (1959). 5. J. O'M. Bockris and S. Srinivasan, This Journal,

l U , 858 (1964). 6. S. Glasstone, K. J. Laidler, and H. Eyring, "The

Theory of Rate Processes," p. 121, McGraw Hill Book Co., New York (1941).

7. R. Parsons and J. O'M. Bockris, Trans. Faraday Soc., 47, 914 (1951).

8. B. E. Conway and J. O'M. Bockris, Can. J. Chem., 35, 1124 (1957).

9. J. O'M. Bockris and S. Srinivasan, This Journal, 111, 844 (1964).

10. J. O'M. Bockris, M. A. V. Devanathan, and K. Miiller, Proc. Roy. Soc., A274, 55 (1963).

11. J. O'M. Bockris and E. C. Potter, J. Chem. Phys., 20, 614 (1952).

12. S. Srinivasan, Ph.D. Thesis, Univers i ty of Pennsy l - vania (1963).

13. J. Bigeleisen and M. Wolfsberg, in "Advances in Chemical Physics," vol. I, chap. II, Interscience Publications, New York (1960).

desorp t ion mechan i sm. Thus, the separa t ion factor me thod canno t be used to d i s t ingu i sh b e t w e e n a r a t e - d e t e r m i n i n g d ischarge and a r a t e - d e t e r m i n i n g e lec t rochemica l desorp t ion m e c h a n i s m at high over - potent ia ls . The on ly w a y of d i s t ingu i sh ing b e t w e e n these two m e c h a n i s m s is by a d e t e r m i n a t i o n of the degree of coverage of adsorbed h y d r o g e n on the electrode, since the coverage is low for a s low discharge m e c h a n i s m and is h igh for a slow elec- t rochemica l desorp t ion mechan i sm.

Some aspects of separation factor ca lcu la t ions . - I t m a y be though t tha t there are inaccurac ies in such ca lcula t ions [ref. (5, 9) and this pape r ] . The s ta t i s - t ical mechan i ca l t r e a t m e n t of reac t ion ra tes is qu i te sat isfactory. The on ly doub t arises in the ca lcu la t ion of v ib r a t i ona l f requenc ies of t r ans i t i on s tate which is vital . Defects of ca lcu la t ion are those of the semi - empi r i ca l me thod of E y r i n g et al. The H e f t i e r - L o n - don t r e a t m e n t for H bonds is used in the p re sen t ca l - cula t ions as well . The inaccuracies of the ca lcu la - t ions are m u c h less t h a n is thought , as has been shown b y a va r i a t i on of pa rame te r s : metal , M- M distance, pe rcen tage coulombic energy. In an y case, m a n y defects are p re sen t in absolu te ca lcu la t ion of reac t ion rates, t hey are p a r t i c u l a r l y usefu l in con- j u n c t i o n wi th isotope effects (3) .

The separa t ion factors for the slow e lec t rochemical desorp t ion m e c h a n i s m at low and i n t e r m e d i a t e overpo ten t ia l s and the slow discharge m e c h a n i s m are d i s t inc t ly separated, m a i n l y due to the h igh rea l s t re tch ing f requenc ies i n the ac t iva ted complex of the slow discharge step. Thus, the separa t ion factor me thod has p roved usefu l in d i s t i ngu i sh ing b e t w e e n these two m e c h a n i s m s on a n u m b e r of meta l s (1) in which cases other methods (e.g., s to ichiometr ic n u m b e r s , va r i a t i on of degree of coverage wi th po- t en t i a l ) are inapp l i cab le (12).

Theoretical Calculations of the Separation Factors in the Hydrogen Evolution Reaction for the Slow

Recombination Mechanism John O'M. Bockris and S. Srinivasan

The Electrochemistry Laboratory, The University of Pennsylvania, Philadelphia, Pennsylvania

ABSTRACT

The theoretical calculations of H/D and H / T separat ion factors have been carried out on the metals Ni and Pt, following the l ines of earlier calculations by Okamoto et al. The effect of var ia t ions in the me ta l -me ta l in te rnuc lear distances and the coulombic exchange energy ratio on the separat ion factor was small. Tunne l ing corrections were also made. The calculated H / T separation factor on Pt is in agreement with the exper imenta l ly determined value. The theoretically predicted separation factors for a slow molecular hydrogen dif- fusion mechanism are not dist inctly separated from the values for a slow re- combinat ion mechanism. Separat ion factor determinat ions on the p la t inum group of metals or the tempera ture coefficients of separation factors on these metals should be useful to dist inguish be tween the two mechanisms.

Ear l i e r ca lcu la t ions (1-3) of the separa t ion fac- P t surfaces were used. Recent ove rpo ten t i a l meas - tors for the slow r e c o m b i n a t i o n m e c h a n i s m are con- u r e m e n t s in Ni (4) showed tha t the most closely t rad ic tory . In the o r ig ina l ca lcu la t ions (1, 2), the packed p l ane has the smal les t overvol tage a nd it is longer of the two possible M-M dis tances for Ni and hence more appropr i a t e to use the closer M- M dis-



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Theoretical Calculations of the Separation Factors in the Hydrogen Evolution Reaction for the Slow Electrochemical Desorption Mechanism