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Stoichiometric numbers and component currents
Electroanalytical Chemistry and Interfacial Electrochemistry, 53 (1974) 187-204
~: ElsevierScquioa S.A.. Lau~nne
187
Printedin The Netherlands
S T O I C H I O M E T R I C N U M B E R S AND C O M P O N E N T C U R R E N T S
JOIIN C. REEVE Chemistry Department A, Buildin,q 207, The Technical University of Denmark. 2800 Lyn.qby (Denmark)
(Received 12th September 1973; in revised form 7th February 1974)
INTRODUCTION Principally in connection with the hydrogen evolution reaction, the quantity called stoichiometric number has been much discussed ever since it was first introduced by Horiuti and lkusima 1. A number of limitations of the earlier treatments were indicated by Parsons 2, who also introduced an essentially equivalent, but more tangible definition. Since that time a number of discussions have appeared 3-5, in which the authors present conflicting interpretations; Parsons 6 has also pointed out an error in one treatment 4. However, mistakes in the above works 3-5 have passed unnoticed or without comment for many years, and a general derivation has not been given. Krishtalik 7 has pointed out that in a fairly recent text-book 8 a clear distinction is not drawn between actual theoretical values and apparent values obtained by extrapolation procedures. Because of the variety of approaches involving classical chemical kinetics. transition-state theory and irreversible thermodynamics, and also because of the number of criticisms and unchallenged errors traceable back to often virtually inaccessible publications, the reader is almost Unable to find the essential relevant facts and loses sight of the essentially simple kinetic principles involved. As Vetter 9 has stated, the quantity is of rather limited diagnostic value for mechanism determination. He pointed out that the stoichiometric number as usually estimated by extrapolation from large overpotentials can have a v e r y w i d e range of values 10r many mechanisms and that these ranges usually overlap extensively; also, precise estimates could not usually be made. The use of labelled atoms could in theory give estimates of stoichiometric numbers (actual effective values) but has been little used; Frumkin 3 has pointed out that in the important case of hydrogen electrode studies, a special difficulty arises because of the relatively large isotope effect. At present stoichiometric number determination has consequently been of very limited value, but such determinations are potentially valuable and the quantity and the concepts surrounding it are of considerable theoretical importance. As the results given in this paper differ in many cases from those given in some much quoted works, it will be appropriate and helpful to pin-point the causes of these divergencies. This is not intended as an exhaustive review but it will be helpful to first summarize the various definitions and interpretations which have been attached to the term stoichiometric number.
188
J.C. REEVE
THE BACKGROUND In the works of F r u m k i n a and Makrides '*'5 the original a p p r o a c h of Horiuti and Ikusima t was developed and the stoichiometric n u m b e r (r) defined in terms of the ratio of the c o m p o n e n t currents* of the reaction in question
i_ --
v
where i+ and i_ are the forward and reverse c o m p o n e n t currents, respectively, ~ is the overpotential-.-positive for the forward (anodic or cathodic) c o m p o n e n t , and measured in units of F/RT in some of the quoted w o r k s - - a n d n is the total n u m b e r of electrons transferred for one complete stoichiometric reaction as written. It is clear that n is to some extent arbitrary depending on how the overall equation is written, but as stressed by Makrides 4 the ratio n/r will be independent of the m o d e of writing the chemical equation. However, it is not inappropriate to mention here that v as defined by relation (1) cannot be stated to be always a constant (at a given potential and temperature) except when the mechanism is simple and adsorption behaviour idealized** (discussed below). An alternative definition (the n u m b e r of times the "rate determining" (r.d.) reaction must occur for one complete occurrence of the stoichiometric reaction as written) introduced by Parsons 2, but intended for the case of a single r.d.-step, leads t o ( l ) provided the forward and backward overall rate constants are independent of the concentrations of reactants and products~°; r/ is the overpotential as measured, without any F r u m k i n correction tl or similar environmental correctiont°. The dependence of the rate constants on the concentration (and consequent failure o f ( l ) with Parsons' interpretation of v ) c a n be shown to arise in at least two ways: (a) T w o or more steps of differing stoichlometric n u m b e r (Horiuti's t2 later definition*** or that of Parsons) being simultaneously r.d. Thus if the v's for two r.d.-steps 1 and 2 are 2 and 1, respectively, then it follows from eqn. (72) of the Appendix that the ratio of the overall rate constants is given by
(k+/k_)=klk2X/k_tk_2
(cf. eqn. 1)
where X is the concentration of the intermediate. (b) The adsorption of involved species being non-Langmuirian. This is because reaction rates can no longer be written simply as a p r o d u c t of a rate constant and a coverage, but involve an extra exponential factor, e x p [ - ~ t g ( 0 ) ] , when the product o f the step is adsorbed in a n o n - L a n g m u i r i a n manner. ,q(0) is a function (usually taken to be linear for intermediate values of 0) of the coverage 0, and ~t * The concept of component current will be considered in detail later, when an important oversight in its application s 5 to heterogeneous processes will be pin-pointed and discussed. ** It is relevant to note that (1) will still be valid in the presence of parallel reactions provided v and reactants and products are the same in each case and i_ and i_ are taken to be the summed components' 5. *** The stoichiometric number of a reaction step is the number of times that step (not necessarily r.d.) must occur for one overall act of the reaction. A conceivable reaction step. which is actually uninvolved in the considered reaction, has a stoichiometric number of zero: it may have a finite value for some alternative reaction path.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
189
is a symmetry factor describing the relative influence of the non-Langmuirian adsorption on the free energy of activation (-~RT,q(O)) for the reaction step and on the free energy of the adsorbed species ( - RTg(O)); the effect has been reviewed by Gileadi and Conway 13. Irrespective of the number of steps involved in the r.d.-process, such influences will cancel from the numerator and denominator of the ratio (i+/i_), so that (1) is still valid if the function g(O) is the same for all the adsorbed species and the c~'s are equal to 0.5 (the effects will always tend to cancel), and if reactions between intermediate species are not involved in non-equilibrium steps. As explained below, if reaction between intermediates is involved in r.d.-steps, the significance of v is already complicated. Equation (V)of the work of Makrides 4 would thus appear to be inapplicable in the general case examined, unless K is no longer considered to be an equilibrium constant and the expression simply regarded as a definition. Hence, the associated deductions are apparently of no greater generality than the others given there. This type of analysis is not possible if there exist concentration (coverage) gradients with diffusion of reacting intermediates across the surface; relation (1) is therefore more likely to be valid as i( = i + - i _ ) ~ 0 * . From this discussion it may be concluded that the more modern and tangible definitions of stoichiometric number by Horiuti ~2 and Parsons z often give rise to the relationship expressed in (1), but that this expression can give rise to a very complicated significance for stoichiometric number when used for the purpose of definition. However, simple interpretations of experimental estimates of stoichiometric number are likely to be invalidated (at least partially) by non-ideality. However, as will appear later, many of the difficulties arise from consideration of conditions where finite net currents are passing (7 :/: 0). Differentiation of (1) leads to l/Ib+ l+ 1/Ib-I = (n/v)(F/2.303 R T )
(2)
where b+ =t3E/~? logloi+, b_ =tgE/t? log~oi_ and E represents electrode potential. h+, b_ and v are values for a particular potential. Writing the net current as i= i+ - i _ , differentiating with respect to q and writing i+ =i_ =io at q = 0 1/Ib+ l+ 1/tb- I = (1/2.303 )( ~ i/,3q )n= o / io
(3)
and with (2) this gives
vn : o = (nF/R T)( ,?,/~3i) , = o io
(4)
This is the most used equation for evaluation of stoichiometric numbers. However, its use has been interpreted in two ways with regard to the value to be used for io. Clearly, the value intended here is the actual exchange current at r/= 0, as could (in theory) be found at q = 0 by the use of labelled atoms; this might be called the theoretical value. Of greater practical importance is the value of v * A d d e d in revision: This situation is indicated as likely if a plot of [ql vs. log(ia/i¢) is approximately a straight line through the origin and independent of concentrations as i ~ 0 . i, and ic are the observed currents at small values of + r / a n d --r/, respectively. See ref. 30, eqn. (5).
190
J.C. REEVE
obtained by use of the io obtained from extrapolation of plots of log(component current) vs. potential at high values of r/ back to r/=0. O f course, the plot of log(current) vs. potential will not be linear in the transition range (where more than one step is r.d.), when the separately r.d.-steps of the sequence give rise to Tafel behaviours with differing slope (=~3E/g', log i)la'ls*; linear extrapolation will lead to an apparent i0 3"16' 17 and a corresponding apparent stoichiometric number (i* and i,*, respectively). The influences of heterogeneity, for example, causing currents associated with the presence of a single r.d.-step to have non-linear Tafel plots, may render values of i* and v* found by linear extrapolation of even more remote significance. Under the heading of stoichiometric numbers, Vetter 9 considers only the quantity v* and uses a modification of eqn. (4) for definition:
v* = ( n F/R T)( Orl/?;i). o i*
(5)
=
EXAMINATION OF SOME EARLIER RESULTS The example of the simple Volmer-Horiuti or Volmer-Heyrovsk2~ (abbreviated V - H ) mechanism of hydrogen evolution l
e+M+H
+ .
'HM
(6)
-1 2
e+H++HM
-
' H2+M
(7)
-2
in which vt = v 2 = 1 and M represents a site for adsorption, will now be used to draw attention to divergencies between results given below and those in two much quoted works 3'5. The divergencies apparently have a c o m m o n origin t 7. t s. Langmuirian behaviour was assumed and the various rates (s) written as 3"5 s,
= kl(1-O)**
(8)
s-i=k-tO
(9)
s2 = k2 0
(10)
s-2= k-2(l-0)**
(11)
k l . 0 k 2 , o = k_ 1,0k-2,0
(12)
so that
where the suffix zero indicates equilibrium. In the steady state, the forward rate is given by ( s t - s _ forward current is given by 2F(st - s _ l). Hence O=(kl +k-2)/(kl+k-l+k2+k_2)
l)=(s2-s_2);
the
(13)
* It may be noted that thc quantity r defined in ref. 15 is essentially the same as the reciprocal of the quantity ? used by Frumkin and others and defined below. ** k~ and k_ 2 contain factors representing concentrations of reactants and products, respectively. at the electrode.
STOICHIOMETRIC NUMBERS AND COMPONENI CURRENTS
191
Defining 'io.t = Fk_ 1.o0o;
io.2 = Fk2.oOo
(14)
a quantity ~, was defined 3 by 7 = io. t/io,2 = k - 1.o/k2.o In the quoted works 3"5 io,t and/oat were written for io.~ and io.2 and included in the rate constant. The forward component current (i+) was presumably derived s-2 to zero. This component current was evidently considered to be two parts: that corresponding to that part of s~ which did not return given by sl - - -$2 -
(15)
the factor F by equating made up of via s_ 1 and
(16)
s-t+s2
and the forward current corresponding to s2 through step 2. The value of i+ was in fact simply stated to be
i+ = F
[
Sls2
s_~+s2+
sz
]
(17)
which was credited to Lukovtsev 17'1s (currents associated with successive steps are additive and only as a special case equal). Similarly, setting sa = 0
s-zs-1 ] i _ = F l . ~ - ~ - _ i + s-1
(18)
Actually, the corresponding equations for i_ (ref. 3, eqn. 8 and ref. 5, eqn. 19) were written in a form similar to (18) but with the second term s_ i replaced by s-2; many of the subsequent equations were in error on this account apart from the oversight discussed below. This involves the fact that the O's appropriate to (17) are not the same as the O's appropriate to (18); also (17) is only valid in conjunction with (8)-(11) when i_ ,~ i+ and (18) is only valid in conjunction with (8)-(11) when i, ~ i_. Thus, use of(17) and (8)-(11) to estimate io (a situation requiring i+ = i _ ) in terms of io.1 and io.2 leads to a false expression (ref. 3, eqn. 12). As shown later, recognition of lack of compatibility between (17) and (8)-.{ 11) leads to the well-known result (eqn. 32). The expressions (17) and (18) for i+ and i_ were not separately expanded (in refs. 3 and 5) using (8){11), but i+/i_ was first expressed in terms of the s's and the ratio expanded using a single value of 0 giving (incorrectly) v = 1 + ~/. That the value of i+/i_, v, etc., obtained by this method using a single value of 0 and using the correct expression (18) for i_ is also in error, can be readily appreciated from the fact that the calculated value of i+/i_ does not then equal unity at ~t=0 (unless k-2=kl). When (8)-{ 11) are accepted, in the steady-state O = ( 1 - O ) ( k t + k _ 2 ) ( k _ , +k2) -I
(19)
192
J.C. REEVE
By setting O = ( l - O ) k f f k _ l + k 2 ) -l (eqn. 19 when k~>>k-2), the value obtained for i. is i+ = 2F(I - O)kl k2(k_, + k2) -I
(20)
The value obtained for i_ by setting O = ( i - O ) k _ 2 ( k _ l + k 2 ) -~ (eqn. 19 when k-2 >>kl) is i- = 2 F ( 1 - 0 ) k _ l k _ 2 ( k _ l +k2) -l
(21)
Although the parent eqns. (17) and (18) are not generally (even separately) valid when combined with (8)-(! 1), (20) and (21) are generally valid. This fortuitous situation arises from a repeated lack of distinction between the O's involved during the above substitutions. The ratio i+/i_ from (20) and (21) is also correct i+/i_ = k l k 2 / k - 1 k - 2 = exp(2Fq/RT)
(22)
and v= 1 irrespective of the relative rates of the steps of the process (as pointed out by Parsons6); this was not the result in the quoted works 3s (or in the earlier work of Makrides 4, but due to another oversight indicated by Parsons6). Another example of such a concealed error and made in a work of the present author is mentioned in the first Appendix; the.error involves the equations for component currents. Closer examination of the basis of the method for finding the component currents (i+ and i_) reveals that the values are obtained by considering hypothetical situations (with s_ 2 and sl, respectively, set equal to zero) so that the values of 0 referred to are often also hypothetical; in fact, no less than two values of 0 are involved in connection with these estimates of i+ and i_. These are the actual value 0 given by (13), and the value 0+ corresponding to the situation where the product concentration is set equal to zero; the corresponding value 0- for the backward component is shown below to be equal to ( 0 - 0 + ) . The basis of setting the product concentration (or s_ 2) equal to zero can be seen by considering completely or purely la,belled reactants (i.e. consisting of only labelled reactants) forming labelled products which are instantly removed (to keep product concentrations at the electrode unaffected by the progress of reaction and the product at the electrode completely, free of labelled species). The forward component rate is immediately seen to be equal to the coverage by labelled intermediate multiplied by the second forward, rate constant under the actual conditions. However, the coverage by labelled intermediate (0+, in both the real and the hypothetical situations) can be seen to be less than 0 as given by (13) (i.e. the coverage by unlabelled intermediate and the coverage by labelled intermediate taken together). Thus although (8)-(11) are generally true with a single value of 0 as given by (13), the forward and backward component currents are not then given by (17) and (18). When (17) and (18) are to be applied, the rates of the steps must be redefined. When si and s_ 2 are considered, the uncovered area in the real situation is required, for it is clearly immaterial, whether the coverage is by labelled or unlabelled intermediate. For estimation of s_ 1 and s2, it is 0+ which is required when estimating i+ and 0_ when estimating i_.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
193
The following scheme may prove helpful: (forword component) ×*
(AOobsen,) A.~
m~ obsent~
(23)
(backward component)
An asterisk indicates a labelled species, and a circle an unlabelled species. The upper path represents the change of A* to products (the forward component), and the lower path the reverse component. The actual coverage of the electrode by intermediate species X is 0; by X* it is, 0* and by X ° it is 0 ° so that 0 = 0:' + 0". The area of available (.or uncovered) surface for A or B is (1 - 0). The forward component current is k20* (i.e. k20+ is the usual symbol of this work) and the backward component k_ 10° (i.e. k_ tO-). All the equations given apply to the particular component current, even if the system is undergoing a net reaction in the opposite direction. Hence, returning to the use of eqn. (17) and the expressions (8)-{ 11) for the various rates (Langmuirian behaviour is still assumed)
S-2= k - 2 ( 1 - 0 )
(8) (11)
s- l = k- 1 0+
(24)
sl
=kl(l-0)
but S2
= k2 O+
(25)
(so that ( 12) and (15) are, however, still valid). In order to assess 0+, the steady-state principle is applied to it in the real coverage situation. The rate of loss of 0+ is O+(k_ ~+/<.2) and its rate of formation is ( l - 0 ) k ~ (no 0+ can be formed from products since s-2 is zero for them; the available or uncovered area in the actual situation is 1 - 0 and not 1 - 0 ÷ ) . Hence O+ = (l - O)k,/(k_, +k2)
(26)
and using (13) to substitute for O,
O+ _ k,/(k, +k_, + k 2 + k - 2 )
(27)
This is not the same as (13) with k-2 set equal to zero, since quite likely k-2 is not much less than kl under the actual conditions and when 0 is given by (13). For example, at the equilibrium potential and when io.1 =io.2, kl = k - 2 . Then, while 0 is given by (13), 0+ is given by eqn. (27) with k_2=kl and is not given by eqn. (13)(with k_ 2 ~ k l or with k_ 2 = kl); io (the component current i+ under the actual conditions) is given by k20+, not k20. This is the source of error in the expression for io in ref. 3, eqn. 12 (correct relation is eqn. 32). When 0 is not small, i+ and i_, and also 0+ and 0_, are not independent as in the homogeneous case, but are coupled through their dependence on 0 (eqns.
194
J.c. REEVE
(29) and (26), respectively, for the cases of i+ and 0+). The factor ( 1 - 0 ) involved in the expression for the net current may provide an alternative explanation for some of the anomalies observed, particularly at low overpotentials, and interpreted earlier as an influence of surface diffusion. The ratio (i+/i_) will not be influenced as both components are proportional to ( 1 - 0). Actually, 0 is a maximum va'lue of coverage more accurately regarded as a measure of the concentration near the electrode; the actual coverage can be written pO where 0~
O_ = k- z/(k, +k_ , +k2 + k - 2)
(28)
0 = 0 + + 0 _ (as it must since the same arguments may be applied to the unlabeiled sequence) and (O+/O-)=kl/k_z. In the present case of the V-H mechanism, (17) becomes on expansion (using eqns. 8, 11, 24 and 25)
[k,(1-O)
i, =l_-k--~+k~ + k20+
]F
(29)
and because of proportionality between 0+ and ( 1 - 0 ) as given by (26), eqn. (20) is recovered. Rewriting (20) as i+ = 2/'(1 -O)k,k20+(k20+ + k _ , 0 + ) - '
(30)
we find
io = 2io.lio.2/(io., +io.z) or (el: eqn. 47) 2/io = 1/io., + l/io.z
(31) (32)
(The divergency between this result and those in refs. 3 and 5 arises from the lack of distinction there between 0 and 0+.) The statement 3 repeated by Makrides 5 that io can be large while io.1 is small is seen to be incorrect for the steady state. It might be stressed here that (17) and (20) are not really equivalent, since with proper consideration of the values of 0 and 0+ at any instant.and as appropriate to (8), (11), ( 2 4 ) a n d (25), eqn. (17) is always valid (for the V-H mechanism) while (20) only applies in the steady state. In the case of the Volmer-Tafel (V-T) mechanism 1
e+M+H + .
' HM
(33)
-1 2
2HM .
' Hz+2M
(34)
-2
we have for Langmuirian conditions
i+ =
2Fk, k 2 O+ (I - 0 ) 2kzO+ + k - I
(35)
STOICHIOMETRI(" NUMBERS AND COMPONENT CURRENTS
i_
2Fk_tk_ 2(l-O) 2 k_t + 2k20_
i+ i_
klk 2 (}+(k_lW2k20_) k_tk_ 2 (l-O)(k_t+2k20+)
195
(36*)
(37)
2k202 +k_tO+ = k t ( l - O )
(38)
2k202_ + k _ , 0_ = 2k_2(I - 0 ) 2
(39)
and and the whole problem of c o m p o n e n t currents for this ideal case is even much more complicated than implied by the calculations of Makrides s, which eventually seem to apply, however, as i ~ 0 (see eqn. 55). Finally, it should be pointed out that m a n y of the expressions in the quoted works are correct due to eventual cancellation of the errors (e.g. eqns. 27 and 28 of Makrides 5 relating to the V - H mechanism). In the derivation (based on irreversible thermodynamics) given by C o n w a y ~9 for the simple V - H mechanism, it is not stated that the quantity considered is the apparent stoichiometric number, v*. This is presumably the point made by Krishtalik ~. This explains the apparent disagreement with the statement above (and that of Parsons 6) that v is always unity for the V - H mechanism. C o n w a y gave:
~'T = 1+I,
(40)
v~ = 1 + ? - t
(41)
and for extrapolations io.1 and io.2 (corresponding determination" by steps 1 and 2, respectively);
~/= io. 1rio. 2
to
currents
for and
"rate(42)
C o r r e s p o n d i n g c o m m e n t s apply to the earlier work ot'Makrides 4, in which this type of a p p r o a c h was used. However, the fact is concealed by considering only the situations outside the transitional range in all cases except that labelled 4b. The deductions of 4b were questioned by Parsons 6 and retracted by Makrides 6. It is interesting to note that Makrides in his later paper s considered both v and v*," but apparently returned to his earlier views regarding v for the V - H mechanism, thus overriding his former acceptance of the statement of Parsons 6. C o n w a y 2° stated that eqns. (40) and (41) were each only applicable in certain ranges, and that the stoichiometric n u m b e r (actually v*, as explained above) should pass t h r o u g h a fiat maximum of v * = 2 at ";= 1 and tend to unity as ), becomes very great or very small. However, both equations are apparently correct * The factor (1 - 0)2, rather than ( 1- 0) in (36) involves the supposition that adsorbed H or some other intermediate of (34) is highly mobile, so that equilibrium is established over the surface. If, however, H or any adsorbed intermediate is not at all surface mobile, the factor would be (1 - 0), and the reactive areas would have to involve adjacent sites so that vt = vx = 1. It is important to note in this connection that this distinction is not apparent in the Christiansen-type derivations of Appendix 1, because the k's are differently defined when v is not the same for all steps. In the derivations in the A'ppendix some k's involve terms in 0 in this case and as required by (70); this leads to the factor ( I - 0 ) being involved in all cases for this Christiansen-type treatment.
196
J.c. REEVE
.o
// /
~ ...f
i/lv/ /~ / ii" i / / / i # ~ /: // R
/
~
io.~ ~ ~ ~ (n.y.> n_v.) .
f~-
" lo., (niv,--niv,)
L,3-"
/../7 /
/
ii
/
i~ "0
Fig. 1 (diagrammatic). Tafel plots showing the relationships of the various currents (defined in the text) under limiting conditions to the exchange current (i0) in the transition range (i.e. during transfer of the status of"rate-determining" from one step to another). The currents for limiting behaviours are for convenience shown as giving rise to linear Tafel plots. R is only found at the same value of overpotential (q) as P (i.e. at Q) when nt vl = n2 v2.7' = lb. 1/i'o,2 = 1 at P, only if vl = v2 (see cqn. 45).
for all values of 7, and cannot "blend" their significance (as do i0.1 and io,2 yielding io, for example). However, the normal practical situation will correspond to that where (40) is applicable i.e. referring to Fig. 1, io obtained as io.1 by extrapolation from high to low q; v* will change from unity towards infinity and through 2 (at the potential where ~= 1) as q decreases. A SIMPLE, BUT G E N E R A L ANALYSIS
It is relevant to note that theoretical stoichiometric numbers are usually considered in terms of the later definition of Horiuti t2, the definition of Parsons 2 or eqn. (4), none of which involves consideration of actual component currents or departure from equilibrium. Those examples of estimation of v from estimates of (i+/i_) usually involve the condition q--*0 as the final step in the analysis (ref. 3, eqn. 2 or ref. 4, eqn. 3). It is not surprising therefore that such considerations for heterogeneous systems have resulted in a considerable and unnecessary complication of the matter, and a difficult literature as exemplified above. Also, the method of Frumkin 3 and Makrides 5 involves the suppositions that the behaviour is Langmuirian and the electrode quite homogeneous. No such suppositions are made in the derivations given here (since only the situation i = 0 is considered), except at the end of Appendix I, where a component current is again considered. Application of irreversible thermodynamics as used in the earlier publication of Makrides 4 and the book of Conway 14 helps in some respects; although interesting, it is unnecessary here and, in the author's view, offers no special advantage (of. Appendix 1).
S'FOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
197
It is useful to have also defined for the general case the quantities
i'o.i = io.1 n/nl and i'0,2 : io,2n/n2
(43)
iT = i~.1/vl and i~ -- i'0.2/v2
(44)
and nt and n2 are the numbers of electrons transferred in steps i and 2, respectively, for one act of the corresponding basic step. E nj vj = n :# E nj (in general). It should be noted that only when vt =v2, are i~.l and i'o.2 the values of io when the steps are separately rate determining; when vt :# v2, the corresponding values of io are iT and i~. Here, and in many subsequent parts of the text, reference to Fig. l may be of considerable help. We also define
7* = ( i*/i~ ) = 7(v2/vx)
(45)
and when vl =v2, ~'*=7. It should be noted that in this work io.2, for example, never appears alone, but as (io.2/n2)=So.2F; thus, although io.2 may be zero, the relevant quantity ( i o . 2 / n 2 ) is defined.
( 1 ) Theoretical stoichiometric numbers, v Consider a process of two steps of stoichiometric numbers vt and v2*. Using the earlier defined quantities, it is shown in Appendix 1 that l/so = vt/So. 1 +
v2/So.2
(46)
or
+(n2/i0.2)~'~
n/io = ( n l / i o . t ) ~ ' l
(47)
From (43), (44) and (45) i
)
-i'°"., -- nJ°-' / nt°,z -- !9±[/ /O,lZ /O.2
nl
/
n2
nl /
n2
SO, I
(48)
S0,2
so that the y of works defining it in terms of rates of chemical reactions is identical with that defined here**. From (47) and (48)
n/io = (vl + v27) n~/io.t
(49)
n/io = ( v2 + v x/7' n2/io.2
(50)
and For two steps with vl = v 2 = v, from (47)
n/io = ( nx /i,,. , + nz/io.z)v (51) and for two steps of stoich~ometric number, v l and v z, respectively, but effective * vt and v2 may have a c o m m o n factor >1 which simply represents v for this two step r.d.-proccss. ** It is important to note that 7 is not generally defined in the literature, since it is only defined in terms of currents, which are not defined in the general case. Thus the io.tt in Makrides' discussion 5 of the V - T mechanism does not correspond to any of the currents mentioned here; however, his 7 for the V - T mechanism corresponds to the 7* defined here, and his equation for the V T mechanism is c o n ~ q u e n t l y incorrect with y as defined by him.
198
J.C. REEVE
overall stoichiometric number v we can write n/io =(,h v,/io.l + n2 v2/io.2) = ( nt /io.1 + n21io.2)v
(52)
From (48) and (52) we have "rio = v( 1 + 7)(",/io.1 )
(53)
Comparison of (49) and (53) yields with (45)
vlWv2~f vl v2 (I +"~*) (54) l+y v2+v~7* For the V -H mechanism, vl = v2 = I and from (54) it is seen that v= 1 for all values of 7 (as stated by Parsons6; cf. refs. 3, 4, 5 and 8). For the V T mechanism, v~ = 2 and v2 = 1 so that from (54) ,, = (~ + 2)/(~, + 1)
(55)
as concluded by Makrides 5 (but see footnote to eqn. (48)). Hence, v will change from 2 to 1 through 1.5 as 7 increases from zero to infinity through !, and from 2 to 1 through 4/3 as "?* increases from zero to infinity through !. ( 2 ) Apparent or extrapolated stoichiometric numbers, v* From (52) and (43)
! io
= n ~ (v, + v~,) = , 1 (vl + v,ot) nio. l io. 1
,
--.,
(56)
and
,(v
(57)
2 +
io 1o.2 From (4), (5) and (44) v*=vi* io
v i°'l 1 vl io
(5'8)
and .!
v * = v i * = v/°"2 1 io v2 io SO that from (56) and (58)
(59)
(60)
v~' = L (v,+v27) YI
and from (57) and (59) V~ = -V2
(61)
V2 +
Hence, substituting for v from (54), (60) yields with (45) (v,+v2~) ~
v,v~(l+7*) ~
vl(1 +7)
v2+vl~*
and (61) yields
(62)
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
(vl+vz?) z
v (l +
199
vlvz(l+7*) 2
+
(63)
For the V-H mechanism (v~ = v z = 1) we have (as already discussed) v* = 1+ 7
(4o)
v~ = l+-f -~
(41)
and For the V-T mechanism (v~ =2, v 2 = 1) vT - (2+7)2 - 2(1+?*)z 2(1 +7) I +27*
(64)
and (2+7) 2 2(I + 7 " ) z v* - (1 + ?)? - (1 + 2"t*)7"
(65)
when )',~ 1, 7= 1 or 7 >>1, then v~' =2, 9/4 or ~,/2, respectively, and v~ =4/7, 9/2 or 1, respectively. For the same values of ~,* as for 7 above, the values of v~' and vy are 2, 8/3 or ),* and 2/7", 8/3 or 1, respectively. Of course, should extrapolation to find i* be made from part of a transition range where the behaviour were not really Tafelian, the value of i* substituted in (5) would be reduced from the actual i* towards io; the estimated value of v* would be smaller and lie between the real v* and v. If 4.1 does not follow a linear Tafel law, i~' estimated by linear extrapolation can be larger than i'o.1/vl and vl* greater than indicated by (62). The arguments presented here regarding v will not, however, be affected by log(io. ~), etc. being non-linear functions of potential. This treatment is easily extended to the case of three steps of differing v and similar rate, but this is probably only of value as an academic exercise. Considering the case of parallel (instead of the above consecutive case) reaction paths, the curve of log(current) vs. potential becomes convex to the potential axis instead of concave 14"~s; however, it is readily verified that v will still be greater than unity, since in this case the component current terms, instead of their reciprocals, are additive. In contrast, v* will be less than v and hence possibly less than 1. Makrides 4 considered "two competing reaction mechanisms", but it is clear that he was in fact considering two successive steps (of comparable exchange rate) together constituting the overall reaction. The possibility of stoichiometric numbers being less than unity for chain mechanisms has been indicated by Horiuti 21 and Makrides 4. Apart from the above connection wit.h stoichiometric numbers the real practical significance of component currents lies in their relationship to exchange currents in general and also the rate of isotopic exchange. ACKNOWLEDGEMENTS
The author wishes to thank Dr. G. Bech-Nieisen and Dr. J. Ulstrup for valuable discussions. SUMMARY
The original equation of Horiuti and Ikusima for stoichiometric number can
200
J.C. REEVE
lead to difficulties when applied to heterogeneous systems; for heterogeneous systems in particular the later definition of Horiuti and the definition of Parsons are to be preferred as much more tangible and uncomplicated. Errors in the literature demonstrate that when considering component currents much greater care must be exercised in the case of heterogeneous processes than in the case of the more familiar homogeneous ones. A new general expression is given for the theoretical stoichiometric number (v) of a process with two rate-determining-steps; similar general expressions are given for extrapolated or apparent stoichiometric numbers (v*). Various conflicting earlier statements concerning the values for v and v* are examined. Two partly new ways of possibly estimating the theoretical quantity are indicated in Appendix 2. Parallel reactions can give rise to values of v* (but not v) less than unity. APPENDICES
(1). Derivation oj" equations (46) and (47)* and comments For a two step homogeneous process for which v t = v 2 = l , it is readily verified by applying the steady-state assumption to the intermediate concentrations and equating the product concentrations to zero that
l/s+ = l/kx +k- x,/klk2
(66)
Although consideration of a two-step process is probably adequate for most purposes, the general case will be considered. In general for a multistep process and when vj = 1
1
~__t k_ak_2...k,_~j_l ~
s+
j=t
(67)
klk2...ko-t)kj
where j = l defines the final step 22.*. As pointed out by Christiansen 22, this leads to the result j=t 1 j=t 1 1 .=Z Xo jkj -=Z So
j
1
,
.1 1 So,j
(68)
where Xo,~ is the equilibrium concentration of X~, and So and So.j are the exchangerates of the overall process and the jth step, respectively. For a sequence 1
A .
2
' X 1 ; X'I . -I
l
" X 2 ; ... ; X~_ t . -2
• B
(69)
-I
'* This can also be achieved using the irreversible thermodynamics principle of proportionality between rate and affinity, and the additivity of the affinities of consecutive steps giving the overall affinity. However, the problem of identifying the appropriate part of the proportionality factor with the stoichiometric number (irrespective of heterogeneity and coverage) in complicated cases, involves considerations such as those below (t;f. ref. 4). *~' Although homogeneous kinetics were assumed by Christiansen 22. cqn. (67) is valid for Langmuirian behaviour, if the left-hand side of (67) is replaced by (1-O)/s~ ; all the individual coverages are eliminated when vi= 1.0 is the total coverage.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
201
the requirement of vj= 1 for cqn. (67) to be valid entails that eqn. (69) is written so that X~ is identical with Xj and that the k's are defined by
Xjk¢j+l)=s+¢j+l) and Xjk-j-- s-j
(70)
When the stoichiometric numbers of the steps differ, the k's must be defined to keep Xj and Xj identical. Thus for the simple two-step process A-*X, 2X--*X2 (vl =2, v2 = 1; V-T mechanism type), eqn. (66) becomes
l/s+ = l/kl +k_ t/kt k2 )(1"
(71)
in which the k's refer to the steps (written with X~=Xt) 2(A~X) and 2X~Xz. With the k's normally defined for the unit separate steps, (71) becomes (~f. eqn. 35)
l/s+ = vt/kl +k_l/klk2 Xa*
(72)
In the general case, (68) becomes 1
So
j= t
vj
j'= t SO,j
(73)
where So,j refers to the unit jth step of the process which occurs vj times when the overall process occurs once. For the heterogeneous process with species undergoing non-Langmuirian adsorption, eqns. (67) and (72) will also involve complicated functions of the coverage by the various species involved. However, it is seen that at equilibrium this will present no problem, since the new coverage terms (as well as (1 -0)), will bc included in the terms s0.~ of (68) and (73). These considerations and eqn. (73) are required before irreversible thermodynamics can be applied in derivations regarding r (~f. the first factor on the right-hand side of eqn. (1) of Parsons 6) as opposed to v*. Considering the general case with several intermediates has an additional advantage in the situation when the system is not far from equilibrium. The risk of identifying 0 of the term ( 1 - 0)with 0+ (or even 0 for any intermediate) is avoided. Two terms of (73) which become significant in a given potential range give rise to the behaviour illustrated in Fig. 1, and when both terms are simultaneously significant the transition behaviour is observed. In the ideal situation, the limiting behaviours of Fig. 1 (and their extrapolations) will be linear. If the behaviour is non-Langmuirian or otherwise non-ideal, the plots of log(i0, t), log(i~).t), etc. vs. potential in Fig. l will be non-linear extrapolations of the limiting behaviours. Although this will not affect the arguments regarding v presented in the preceding sections, v]' will be affected by the linear extrapolations of iT not being coincident with iT=i'o.a/v in the transition range. It is interesting and relevant to note that (20) may be freed of 0 by means of(13) (or i+ written as 2Fk20+) giving 2F 1 k_, (, k-z) i+-k, +k-~-; + k z + k ~ 2 * k, must be multiplied by (1 -O) in the Langmuirian case (see previousfootnote).
(74)
202
J.c. REEVE
This involves firstly one of two assumptions, (i) reactants and ~,roducts d'6 not block the electrode but simply make use of the uncovered area,(ii) reactants do not block the electrode and the product species exhibit a constant coverage, and secondly, the assumptions that the electrode is uniform, the adsorption Langmuirian and desorption negligible. Equation (74) involves the sum of four terms and may be compared with the two term result for the simple homogeneous case
2F/i+ = l/kt +k_ ~/k~ k2
(75)
and with (20) rearranged
2F/i+ = ( 1 / k l + k_ l / k , k 2 ) / ( 1 - 0 )
(20')
in an earlier publication of the author an error (equivalent to that of Frumkin 3 and Makrides 5 and referred to earlier) was made in a concealed way by writing the current i+ as 2Fk20 (instead of 2Fk20+) and using the result to eliminate 0 from (20). The result in that case was the same as (74), except that the second term in the bracket (representing a normally unimportant influence of the back reaction, through the influence on the coverage 0_) was absent; the first term in the bracket of(74) represents the rate when coverage by 0+ is complete. In the ideal case of equal overall transfer coefficients 23.1° for the two steps (~a and ~2, respectively), the terms of(74) will combine in pairs, so that (74) and (75) are of the same form. Also, the special situation of limiting Tafel slopes being in the ratio of 2:1 can arise when the second step is potential independent and ~1=0.5, and apparently also when coverage effects are of major importance, the first step is potential independent and c~2=0.5. However, consideration of eqn. (20') reveals that the final terms in the bracket of (74) can only be large when 0 is large and Langmuirian behaviour unlikely, although 0+ might be small. An example of when the effect would be important would be when 0+ --~0_ ~ ½, e.q. near equilibrium and when io. t = i0.2 and kl = k_ t. At high anodic 7, 0+--,0 and the final term in the bracket is unimportant, while at high cathodic r/, the effect is unlikely to be of much interest since i+ ,~ i_. If the smaller limiting b-value or Tafel law (b=OE/O Iogl0i) were observed, the ideal value would then be about 60 mV at 25°C*; such a Tafel law for both anodic and cathodic behaviours would be most readily explained by the above model with a chemical step between two charge-transfer steps, or a single charge transfer involving two electrons (possible 24, but unlikely25).
(2). Modifications of methods to determine v The possible use of eqn. (2) to determine v was indicated by Parsons 2 but (as explained earlier) determinations at i = 0 are often to be preferred. Recently, Oldham and Mansfeld 29 have indicated that the sum ( I b + l - l + l b _ l -a) can be * Although not previously suggested, the anodic Tafel laws of ca. 60 mV and ca. 120 mV combined with the cathodic Tafel law of ca. 30 mV for the Bi/Bi3' electrode in HCIO4 (ref. 26) would be neatly explained by this type of mechanism. Anodic: Bi~Bi÷---,(Bi*) * or Bi-,Bi* (giving ca. 60 mV or ca. 120 mV Tafel laws, respectively), and cathodic Bi3+~(Bi*)*---,Bi" (giving a ca. 30 mV Tafcl law). (Bi*)* is some modified form or state of Bi(l). Either type of explanation may be responsible for the behaviours of the Cd/Cd z" electrode in K2SO4 solution2~, and the Cu/Cu 2÷ electrode (unactivated) in HCIO4 solutionsts'28. Although c.e.c, mechanisms are well known for cathodic deposition in such systems, e.c.e, mechanisms do not seem to have been suggested previously (except in refs. 15 and 28). The single arrows indicate the r.d.-steps.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
203
determined from the asymmetry of curves of current vs. potential in the region o f zero current for corroding (mixed-potential) systems. However, it has been pointed out 3° that asymmetry is principally determined by ( I b + l - l - l b - } - l ) (c]: Faradaic rectification) so that such measurements would have to be very precise; it was also pointed out that the exchange current (corrosion current) would need to be large so that the method would be more appropriate for redox systems with high exchange currents; O l d h a m and Mansfeld are publishing a table 3~ to assist in rapid analysis of such curvcs. Potter 32 has proposed a m e t h o d of finding v by observing small departures (due to the reverse reaction) of the current from the extrapolated value. This, of course, involves determination of the quantity at finite current. The derivation apparently involved a p p r o x i m a t i o n of expanded exponential functions, and this is not necessary. Expressing the net current (i) as the forward c o m p o n e n t (extrapolated) current ( i , ) minus the backward c o m p o n e n t (i_) and substituting for i_/i+ from e q n . ( l ) inverted, it is readily found (without approximation) that (cJ: the calculations of Allen and Hickling 33) Iog~o i + - i
r
- 2.303RT
!.)
v r/
(76)
Hence, for all practical values of the current ratio, a plot of the left-hand side of (76) vs. q should give a straight line t h r o u g h the origin yielding n/v. This m c t h o d gives the possibility of examining v over a range of potentials. It is worthwhile to note that although an extrapolation procedure (for i+) is used, the method yields v and not the apparent value, v*. REFERENCES I J. Horiuti and M. lkusima. Proc. Imp. Acad. (Tokyo), 15 (1939) 39. 2 R. Parsons. Trans. Faraday Sot., 47 (1951) 1332. 3 A. N. Frumkin, Proc. Acad. Sci. USSR, Phys. Chem. Sect. (English Transl.), 119 (1 6) (1958) 179. 4 A. C. Makridcs. J. Electrochem. Sot.. 104(1957) 677. 5 A. C. Makrides, J. Electrochem. Sot'., 109 (1962) 256. 6 R. Parsons, J. Electrochem. Soc., 105 (1958) 366. 7 L. I. Krishtalik, Sor. Electrochem., 2 (1966) 582: Elektrokhimiya, 2 (1966) 624. 8 B. E. Conway. Theory and Principles ~[ Electrode Processes, Ronald Press, New York, 1965. 9 K. J. Vetter. Electrochemical Kinetics, Academic Press, New York, 1967, p. 591. 10 R. Parsons, Croat. Chem. Acta, 42 (1970) 281 (in English). 11 A. N. l-'rumkin, Z. Physik. Chem.. 164(1933) 121. 12 J. Horiuti. d. Res. Inst. Catal., Hokkaido Unit., 1 (1948) 8. 13 E. Gileadi and B. E. Conway in J. O'M. Bockris and B. E. Conway (Eds.), Modern Aspects of Electrochemistry, No. 3, Butterworths, London, 1964, p. 347. 14 B. E. Conway, Theory and Principles of Electrode Proce.sses, Ronald Press, New York, 1965, p. 110. 15 J. C. Rccve, Collect. Czech. Chem. Commun.. 36 (1971) 757 (in English). 16 P. D. Lukovtsev and S. D. Levina. Zh. Fiz. Khim.. 21 (1947) 599. 17 P. D. Lukovtsev, Dissertation, L. la. Karpov Phys. Chem. Inst., 1940. 18 P. D. Lukovtsev, Zh. Fiz. Khim., 21 (1947) 589. 19 B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press. New York, 1965, p. 268. 20 B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press, New York, 1965, p. 271. 21 J. Horiuti. Proc. Jap. Acad.. 29/1953) 160. 22 J. A. Christiansen, Adran. Catal., 5 (1953) 311; Z. Physik. Chem. (Franlqfurt), B33 (1936) 145; Z. Phy.sik. Chem ( Franklilrt ), B37 (1937) 374.
204 23 24 25 26 27 28 29 30 31 32 33
J . c . REEVE
H. 14. Bauer, J. Electroanal. Chem., 16 (1968) 419. D. C. Grahamc, Annu. Ree. Phys. Chem., 6 (1955) 337. B. E. Conway and J. O'M. Bockris, Proc. Roy. Soc. (London), A248 (1958) 394. M. S. Grilikhes, B. S. Krasikov and N. S. Solotskaya, Soy. Electrochem., 5 (1969) 799; Elektrokhimiya, 5 (1969) 859. W. Lorenz, Z. Elektrochem., 58 (1954) 912. J. C. Reeve. Extended Abstracts oJ" 23rd Meeting of I.S.E. Stockholm, 1972, p. 227. K. B. Oldham and F. Mansfeld, Extended Abstracts of 23rd Meeting of I.S.E. Stockholm, 1972, p. 109. J. C. Reeve and G. Bech-Nielsen, Corros. Sci., 13 (1973) 351. K. B. Oldham and F. Mansfeld, Corros. Sci., 13 (1973) 811. E. C. Potter, Thesis. University of London, 1950. P. L. Allen and A. Hickling, Trans. Faraday Soc., 53 (1957) 1626.
~: ElsevierScquioa S.A.. Lau~nne
187
Printedin The Netherlands
S T O I C H I O M E T R I C N U M B E R S AND C O M P O N E N T C U R R E N T S
JOIIN C. REEVE Chemistry Department A, Buildin,q 207, The Technical University of Denmark. 2800 Lyn.qby (Denmark)
(Received 12th September 1973; in revised form 7th February 1974)
INTRODUCTION Principally in connection with the hydrogen evolution reaction, the quantity called stoichiometric number has been much discussed ever since it was first introduced by Horiuti and lkusima 1. A number of limitations of the earlier treatments were indicated by Parsons 2, who also introduced an essentially equivalent, but more tangible definition. Since that time a number of discussions have appeared 3-5, in which the authors present conflicting interpretations; Parsons 6 has also pointed out an error in one treatment 4. However, mistakes in the above works 3-5 have passed unnoticed or without comment for many years, and a general derivation has not been given. Krishtalik 7 has pointed out that in a fairly recent text-book 8 a clear distinction is not drawn between actual theoretical values and apparent values obtained by extrapolation procedures. Because of the variety of approaches involving classical chemical kinetics. transition-state theory and irreversible thermodynamics, and also because of the number of criticisms and unchallenged errors traceable back to often virtually inaccessible publications, the reader is almost Unable to find the essential relevant facts and loses sight of the essentially simple kinetic principles involved. As Vetter 9 has stated, the quantity is of rather limited diagnostic value for mechanism determination. He pointed out that the stoichiometric number as usually estimated by extrapolation from large overpotentials can have a v e r y w i d e range of values 10r many mechanisms and that these ranges usually overlap extensively; also, precise estimates could not usually be made. The use of labelled atoms could in theory give estimates of stoichiometric numbers (actual effective values) but has been little used; Frumkin 3 has pointed out that in the important case of hydrogen electrode studies, a special difficulty arises because of the relatively large isotope effect. At present stoichiometric number determination has consequently been of very limited value, but such determinations are potentially valuable and the quantity and the concepts surrounding it are of considerable theoretical importance. As the results given in this paper differ in many cases from those given in some much quoted works, it will be appropriate and helpful to pin-point the causes of these divergencies. This is not intended as an exhaustive review but it will be helpful to first summarize the various definitions and interpretations which have been attached to the term stoichiometric number.
188
J.C. REEVE
THE BACKGROUND In the works of F r u m k i n a and Makrides '*'5 the original a p p r o a c h of Horiuti and Ikusima t was developed and the stoichiometric n u m b e r (r) defined in terms of the ratio of the c o m p o n e n t currents* of the reaction in question
i_ --
v
where i+ and i_ are the forward and reverse c o m p o n e n t currents, respectively, ~ is the overpotential-.-positive for the forward (anodic or cathodic) c o m p o n e n t , and measured in units of F/RT in some of the quoted w o r k s - - a n d n is the total n u m b e r of electrons transferred for one complete stoichiometric reaction as written. It is clear that n is to some extent arbitrary depending on how the overall equation is written, but as stressed by Makrides 4 the ratio n/r will be independent of the m o d e of writing the chemical equation. However, it is not inappropriate to mention here that v as defined by relation (1) cannot be stated to be always a constant (at a given potential and temperature) except when the mechanism is simple and adsorption behaviour idealized** (discussed below). An alternative definition (the n u m b e r of times the "rate determining" (r.d.) reaction must occur for one complete occurrence of the stoichiometric reaction as written) introduced by Parsons 2, but intended for the case of a single r.d.-step, leads t o ( l ) provided the forward and backward overall rate constants are independent of the concentrations of reactants and products~°; r/ is the overpotential as measured, without any F r u m k i n correction tl or similar environmental correctiont°. The dependence of the rate constants on the concentration (and consequent failure o f ( l ) with Parsons' interpretation of v ) c a n be shown to arise in at least two ways: (a) T w o or more steps of differing stoichlometric n u m b e r (Horiuti's t2 later definition*** or that of Parsons) being simultaneously r.d. Thus if the v's for two r.d.-steps 1 and 2 are 2 and 1, respectively, then it follows from eqn. (72) of the Appendix that the ratio of the overall rate constants is given by
(k+/k_)=klk2X/k_tk_2
(cf. eqn. 1)
where X is the concentration of the intermediate. (b) The adsorption of involved species being non-Langmuirian. This is because reaction rates can no longer be written simply as a p r o d u c t of a rate constant and a coverage, but involve an extra exponential factor, e x p [ - ~ t g ( 0 ) ] , when the product o f the step is adsorbed in a n o n - L a n g m u i r i a n manner. ,q(0) is a function (usually taken to be linear for intermediate values of 0) of the coverage 0, and ~t * The concept of component current will be considered in detail later, when an important oversight in its application s 5 to heterogeneous processes will be pin-pointed and discussed. ** It is relevant to note that (1) will still be valid in the presence of parallel reactions provided v and reactants and products are the same in each case and i_ and i_ are taken to be the summed components' 5. *** The stoichiometric number of a reaction step is the number of times that step (not necessarily r.d.) must occur for one overall act of the reaction. A conceivable reaction step. which is actually uninvolved in the considered reaction, has a stoichiometric number of zero: it may have a finite value for some alternative reaction path.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
189
is a symmetry factor describing the relative influence of the non-Langmuirian adsorption on the free energy of activation (-~RT,q(O)) for the reaction step and on the free energy of the adsorbed species ( - RTg(O)); the effect has been reviewed by Gileadi and Conway 13. Irrespective of the number of steps involved in the r.d.-process, such influences will cancel from the numerator and denominator of the ratio (i+/i_), so that (1) is still valid if the function g(O) is the same for all the adsorbed species and the c~'s are equal to 0.5 (the effects will always tend to cancel), and if reactions between intermediate species are not involved in non-equilibrium steps. As explained below, if reaction between intermediates is involved in r.d.-steps, the significance of v is already complicated. Equation (V)of the work of Makrides 4 would thus appear to be inapplicable in the general case examined, unless K is no longer considered to be an equilibrium constant and the expression simply regarded as a definition. Hence, the associated deductions are apparently of no greater generality than the others given there. This type of analysis is not possible if there exist concentration (coverage) gradients with diffusion of reacting intermediates across the surface; relation (1) is therefore more likely to be valid as i( = i + - i _ ) ~ 0 * . From this discussion it may be concluded that the more modern and tangible definitions of stoichiometric number by Horiuti ~2 and Parsons z often give rise to the relationship expressed in (1), but that this expression can give rise to a very complicated significance for stoichiometric number when used for the purpose of definition. However, simple interpretations of experimental estimates of stoichiometric number are likely to be invalidated (at least partially) by non-ideality. However, as will appear later, many of the difficulties arise from consideration of conditions where finite net currents are passing (7 :/: 0). Differentiation of (1) leads to l/Ib+ l+ 1/Ib-I = (n/v)(F/2.303 R T )
(2)
where b+ =t3E/~? logloi+, b_ =tgE/t? log~oi_ and E represents electrode potential. h+, b_ and v are values for a particular potential. Writing the net current as i= i+ - i _ , differentiating with respect to q and writing i+ =i_ =io at q = 0 1/Ib+ l+ 1/tb- I = (1/2.303 )( ~ i/,3q )n= o / io
(3)
and with (2) this gives
vn : o = (nF/R T)( ,?,/~3i) , = o io
(4)
This is the most used equation for evaluation of stoichiometric numbers. However, its use has been interpreted in two ways with regard to the value to be used for io. Clearly, the value intended here is the actual exchange current at r/= 0, as could (in theory) be found at q = 0 by the use of labelled atoms; this might be called the theoretical value. Of greater practical importance is the value of v * A d d e d in revision: This situation is indicated as likely if a plot of [ql vs. log(ia/i¢) is approximately a straight line through the origin and independent of concentrations as i ~ 0 . i, and ic are the observed currents at small values of + r / a n d --r/, respectively. See ref. 30, eqn. (5).
190
J.C. REEVE
obtained by use of the io obtained from extrapolation of plots of log(component current) vs. potential at high values of r/ back to r/=0. O f course, the plot of log(current) vs. potential will not be linear in the transition range (where more than one step is r.d.), when the separately r.d.-steps of the sequence give rise to Tafel behaviours with differing slope (=~3E/g', log i)la'ls*; linear extrapolation will lead to an apparent i0 3"16' 17 and a corresponding apparent stoichiometric number (i* and i,*, respectively). The influences of heterogeneity, for example, causing currents associated with the presence of a single r.d.-step to have non-linear Tafel plots, may render values of i* and v* found by linear extrapolation of even more remote significance. Under the heading of stoichiometric numbers, Vetter 9 considers only the quantity v* and uses a modification of eqn. (4) for definition:
v* = ( n F/R T)( Orl/?;i). o i*
(5)
=
EXAMINATION OF SOME EARLIER RESULTS The example of the simple Volmer-Horiuti or Volmer-Heyrovsk2~ (abbreviated V - H ) mechanism of hydrogen evolution l
e+M+H
+ .
'HM
(6)
-1 2
e+H++HM
-
' H2+M
(7)
-2
in which vt = v 2 = 1 and M represents a site for adsorption, will now be used to draw attention to divergencies between results given below and those in two much quoted works 3'5. The divergencies apparently have a c o m m o n origin t 7. t s. Langmuirian behaviour was assumed and the various rates (s) written as 3"5 s,
= kl(1-O)**
(8)
s-i=k-tO
(9)
s2 = k2 0
(10)
s-2= k-2(l-0)**
(11)
k l . 0 k 2 , o = k_ 1,0k-2,0
(12)
so that
where the suffix zero indicates equilibrium. In the steady state, the forward rate is given by ( s t - s _ forward current is given by 2F(st - s _ l). Hence O=(kl +k-2)/(kl+k-l+k2+k_2)
l)=(s2-s_2);
the
(13)
* It may be noted that thc quantity r defined in ref. 15 is essentially the same as the reciprocal of the quantity ? used by Frumkin and others and defined below. ** k~ and k_ 2 contain factors representing concentrations of reactants and products, respectively. at the electrode.
STOICHIOMETRIC NUMBERS AND COMPONENI CURRENTS
191
Defining 'io.t = Fk_ 1.o0o;
io.2 = Fk2.oOo
(14)
a quantity ~, was defined 3 by 7 = io. t/io,2 = k - 1.o/k2.o In the quoted works 3"5 io,t and/oat were written for io.~ and io.2 and included in the rate constant. The forward component current (i+) was presumably derived s-2 to zero. This component current was evidently considered to be two parts: that corresponding to that part of s~ which did not return given by sl - - -$2 -
(15)
the factor F by equating made up of via s_ 1 and
(16)
s-t+s2
and the forward current corresponding to s2 through step 2. The value of i+ was in fact simply stated to be
i+ = F
[
Sls2
s_~+s2+
sz
]
(17)
which was credited to Lukovtsev 17'1s (currents associated with successive steps are additive and only as a special case equal). Similarly, setting sa = 0
s-zs-1 ] i _ = F l . ~ - ~ - _ i + s-1
(18)
Actually, the corresponding equations for i_ (ref. 3, eqn. 8 and ref. 5, eqn. 19) were written in a form similar to (18) but with the second term s_ i replaced by s-2; many of the subsequent equations were in error on this account apart from the oversight discussed below. This involves the fact that the O's appropriate to (17) are not the same as the O's appropriate to (18); also (17) is only valid in conjunction with (8)-(11) when i_ ,~ i+ and (18) is only valid in conjunction with (8)-(11) when i, ~ i_. Thus, use of(17) and (8)-(11) to estimate io (a situation requiring i+ = i _ ) in terms of io.1 and io.2 leads to a false expression (ref. 3, eqn. 12). As shown later, recognition of lack of compatibility between (17) and (8)-.{ 11) leads to the well-known result (eqn. 32). The expressions (17) and (18) for i+ and i_ were not separately expanded (in refs. 3 and 5) using (8){11), but i+/i_ was first expressed in terms of the s's and the ratio expanded using a single value of 0 giving (incorrectly) v = 1 + ~/. That the value of i+/i_, v, etc., obtained by this method using a single value of 0 and using the correct expression (18) for i_ is also in error, can be readily appreciated from the fact that the calculated value of i+/i_ does not then equal unity at ~t=0 (unless k-2=kl). When (8)-{ 11) are accepted, in the steady-state O = ( 1 - O ) ( k t + k _ 2 ) ( k _ , +k2) -I
(19)
192
J.C. REEVE
By setting O = ( l - O ) k f f k _ l + k 2 ) -l (eqn. 19 when k~>>k-2), the value obtained for i. is i+ = 2F(I - O)kl k2(k_, + k2) -I
(20)
The value obtained for i_ by setting O = ( i - O ) k _ 2 ( k _ l + k 2 ) -~ (eqn. 19 when k-2 >>kl) is i- = 2 F ( 1 - 0 ) k _ l k _ 2 ( k _ l +k2) -l
(21)
Although the parent eqns. (17) and (18) are not generally (even separately) valid when combined with (8)-(! 1), (20) and (21) are generally valid. This fortuitous situation arises from a repeated lack of distinction between the O's involved during the above substitutions. The ratio i+/i_ from (20) and (21) is also correct i+/i_ = k l k 2 / k - 1 k - 2 = exp(2Fq/RT)
(22)
and v= 1 irrespective of the relative rates of the steps of the process (as pointed out by Parsons6); this was not the result in the quoted works 3s (or in the earlier work of Makrides 4, but due to another oversight indicated by Parsons6). Another example of such a concealed error and made in a work of the present author is mentioned in the first Appendix; the.error involves the equations for component currents. Closer examination of the basis of the method for finding the component currents (i+ and i_) reveals that the values are obtained by considering hypothetical situations (with s_ 2 and sl, respectively, set equal to zero) so that the values of 0 referred to are often also hypothetical; in fact, no less than two values of 0 are involved in connection with these estimates of i+ and i_. These are the actual value 0 given by (13), and the value 0+ corresponding to the situation where the product concentration is set equal to zero; the corresponding value 0- for the backward component is shown below to be equal to ( 0 - 0 + ) . The basis of setting the product concentration (or s_ 2) equal to zero can be seen by considering completely or purely la,belled reactants (i.e. consisting of only labelled reactants) forming labelled products which are instantly removed (to keep product concentrations at the electrode unaffected by the progress of reaction and the product at the electrode completely, free of labelled species). The forward component rate is immediately seen to be equal to the coverage by labelled intermediate multiplied by the second forward, rate constant under the actual conditions. However, the coverage by labelled intermediate (0+, in both the real and the hypothetical situations) can be seen to be less than 0 as given by (13) (i.e. the coverage by unlabelled intermediate and the coverage by labelled intermediate taken together). Thus although (8)-(11) are generally true with a single value of 0 as given by (13), the forward and backward component currents are not then given by (17) and (18). When (17) and (18) are to be applied, the rates of the steps must be redefined. When si and s_ 2 are considered, the uncovered area in the real situation is required, for it is clearly immaterial, whether the coverage is by labelled or unlabelled intermediate. For estimation of s_ 1 and s2, it is 0+ which is required when estimating i+ and 0_ when estimating i_.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
193
The following scheme may prove helpful: (forword component) ×*
(AOobsen,) A.~
m~ obsent~
(23)
(backward component)
An asterisk indicates a labelled species, and a circle an unlabelled species. The upper path represents the change of A* to products (the forward component), and the lower path the reverse component. The actual coverage of the electrode by intermediate species X is 0; by X* it is, 0* and by X ° it is 0 ° so that 0 = 0:' + 0". The area of available (.or uncovered) surface for A or B is (1 - 0). The forward component current is k20* (i.e. k20+ is the usual symbol of this work) and the backward component k_ 10° (i.e. k_ tO-). All the equations given apply to the particular component current, even if the system is undergoing a net reaction in the opposite direction. Hence, returning to the use of eqn. (17) and the expressions (8)-{ 11) for the various rates (Langmuirian behaviour is still assumed)
S-2= k - 2 ( 1 - 0 )
(8) (11)
s- l = k- 1 0+
(24)
sl
=kl(l-0)
but S2
= k2 O+
(25)
(so that ( 12) and (15) are, however, still valid). In order to assess 0+, the steady-state principle is applied to it in the real coverage situation. The rate of loss of 0+ is O+(k_ ~+/<.2) and its rate of formation is ( l - 0 ) k ~ (no 0+ can be formed from products since s-2 is zero for them; the available or uncovered area in the actual situation is 1 - 0 and not 1 - 0 ÷ ) . Hence O+ = (l - O)k,/(k_, +k2)
(26)
and using (13) to substitute for O,
O+ _ k,/(k, +k_, + k 2 + k - 2 )
(27)
This is not the same as (13) with k-2 set equal to zero, since quite likely k-2 is not much less than kl under the actual conditions and when 0 is given by (13). For example, at the equilibrium potential and when io.1 =io.2, kl = k - 2 . Then, while 0 is given by (13), 0+ is given by eqn. (27) with k_2=kl and is not given by eqn. (13)(with k_ 2 ~ k l or with k_ 2 = kl); io (the component current i+ under the actual conditions) is given by k20+, not k20. This is the source of error in the expression for io in ref. 3, eqn. 12 (correct relation is eqn. 32). When 0 is not small, i+ and i_, and also 0+ and 0_, are not independent as in the homogeneous case, but are coupled through their dependence on 0 (eqns.
194
J.c. REEVE
(29) and (26), respectively, for the cases of i+ and 0+). The factor ( 1 - 0 ) involved in the expression for the net current may provide an alternative explanation for some of the anomalies observed, particularly at low overpotentials, and interpreted earlier as an influence of surface diffusion. The ratio (i+/i_) will not be influenced as both components are proportional to ( 1 - 0). Actually, 0 is a maximum va'lue of coverage more accurately regarded as a measure of the concentration near the electrode; the actual coverage can be written pO where 0~
O_ = k- z/(k, +k_ , +k2 + k - 2)
(28)
0 = 0 + + 0 _ (as it must since the same arguments may be applied to the unlabeiled sequence) and (O+/O-)=kl/k_z. In the present case of the V-H mechanism, (17) becomes on expansion (using eqns. 8, 11, 24 and 25)
[k,(1-O)
i, =l_-k--~+k~ + k20+
]F
(29)
and because of proportionality between 0+ and ( 1 - 0 ) as given by (26), eqn. (20) is recovered. Rewriting (20) as i+ = 2/'(1 -O)k,k20+(k20+ + k _ , 0 + ) - '
(30)
we find
io = 2io.lio.2/(io., +io.z) or (el: eqn. 47) 2/io = 1/io., + l/io.z
(31) (32)
(The divergency between this result and those in refs. 3 and 5 arises from the lack of distinction there between 0 and 0+.) The statement 3 repeated by Makrides 5 that io can be large while io.1 is small is seen to be incorrect for the steady state. It might be stressed here that (17) and (20) are not really equivalent, since with proper consideration of the values of 0 and 0+ at any instant.and as appropriate to (8), (11), ( 2 4 ) a n d (25), eqn. (17) is always valid (for the V-H mechanism) while (20) only applies in the steady state. In the case of the Volmer-Tafel (V-T) mechanism 1
e+M+H + .
' HM
(33)
-1 2
2HM .
' Hz+2M
(34)
-2
we have for Langmuirian conditions
i+ =
2Fk, k 2 O+ (I - 0 ) 2kzO+ + k - I
(35)
STOICHIOMETRI(" NUMBERS AND COMPONENT CURRENTS
i_
2Fk_tk_ 2(l-O) 2 k_t + 2k20_
i+ i_
klk 2 (}+(k_lW2k20_) k_tk_ 2 (l-O)(k_t+2k20+)
195
(36*)
(37)
2k202 +k_tO+ = k t ( l - O )
(38)
2k202_ + k _ , 0_ = 2k_2(I - 0 ) 2
(39)
and and the whole problem of c o m p o n e n t currents for this ideal case is even much more complicated than implied by the calculations of Makrides s, which eventually seem to apply, however, as i ~ 0 (see eqn. 55). Finally, it should be pointed out that m a n y of the expressions in the quoted works are correct due to eventual cancellation of the errors (e.g. eqns. 27 and 28 of Makrides 5 relating to the V - H mechanism). In the derivation (based on irreversible thermodynamics) given by C o n w a y ~9 for the simple V - H mechanism, it is not stated that the quantity considered is the apparent stoichiometric number, v*. This is presumably the point made by Krishtalik ~. This explains the apparent disagreement with the statement above (and that of Parsons 6) that v is always unity for the V - H mechanism. C o n w a y gave:
~'T = 1+I,
(40)
v~ = 1 + ? - t
(41)
and for extrapolations io.1 and io.2 (corresponding determination" by steps 1 and 2, respectively);
~/= io. 1rio. 2
to
currents
for and
"rate(42)
C o r r e s p o n d i n g c o m m e n t s apply to the earlier work ot'Makrides 4, in which this type of a p p r o a c h was used. However, the fact is concealed by considering only the situations outside the transitional range in all cases except that labelled 4b. The deductions of 4b were questioned by Parsons 6 and retracted by Makrides 6. It is interesting to note that Makrides in his later paper s considered both v and v*," but apparently returned to his earlier views regarding v for the V - H mechanism, thus overriding his former acceptance of the statement of Parsons 6. C o n w a y 2° stated that eqns. (40) and (41) were each only applicable in certain ranges, and that the stoichiometric n u m b e r (actually v*, as explained above) should pass t h r o u g h a fiat maximum of v * = 2 at ";= 1 and tend to unity as ), becomes very great or very small. However, both equations are apparently correct * The factor (1 - 0)2, rather than ( 1- 0) in (36) involves the supposition that adsorbed H or some other intermediate of (34) is highly mobile, so that equilibrium is established over the surface. If, however, H or any adsorbed intermediate is not at all surface mobile, the factor would be (1 - 0), and the reactive areas would have to involve adjacent sites so that vt = vx = 1. It is important to note in this connection that this distinction is not apparent in the Christiansen-type derivations of Appendix 1, because the k's are differently defined when v is not the same for all steps. In the derivations in the A'ppendix some k's involve terms in 0 in this case and as required by (70); this leads to the factor ( I - 0 ) being involved in all cases for this Christiansen-type treatment.
196
J.c. REEVE
.o
// /
~ ...f
i/lv/ /~ / ii" i / / / i # ~ /: // R
/
~
io.~ ~ ~ ~ (n.y.> n_v.) .
f~-
" lo., (niv,--niv,)
L,3-"
/../7 /
/
ii
/
i~ "0
Fig. 1 (diagrammatic). Tafel plots showing the relationships of the various currents (defined in the text) under limiting conditions to the exchange current (i0) in the transition range (i.e. during transfer of the status of"rate-determining" from one step to another). The currents for limiting behaviours are for convenience shown as giving rise to linear Tafel plots. R is only found at the same value of overpotential (q) as P (i.e. at Q) when nt vl = n2 v2.7' = lb. 1/i'o,2 = 1 at P, only if vl = v2 (see cqn. 45).
for all values of 7, and cannot "blend" their significance (as do i0.1 and io,2 yielding io, for example). However, the normal practical situation will correspond to that where (40) is applicable i.e. referring to Fig. 1, io obtained as io.1 by extrapolation from high to low q; v* will change from unity towards infinity and through 2 (at the potential where ~= 1) as q decreases. A SIMPLE, BUT G E N E R A L ANALYSIS
It is relevant to note that theoretical stoichiometric numbers are usually considered in terms of the later definition of Horiuti t2, the definition of Parsons 2 or eqn. (4), none of which involves consideration of actual component currents or departure from equilibrium. Those examples of estimation of v from estimates of (i+/i_) usually involve the condition q--*0 as the final step in the analysis (ref. 3, eqn. 2 or ref. 4, eqn. 3). It is not surprising therefore that such considerations for heterogeneous systems have resulted in a considerable and unnecessary complication of the matter, and a difficult literature as exemplified above. Also, the method of Frumkin 3 and Makrides 5 involves the suppositions that the behaviour is Langmuirian and the electrode quite homogeneous. No such suppositions are made in the derivations given here (since only the situation i = 0 is considered), except at the end of Appendix I, where a component current is again considered. Application of irreversible thermodynamics as used in the earlier publication of Makrides 4 and the book of Conway 14 helps in some respects; although interesting, it is unnecessary here and, in the author's view, offers no special advantage (of. Appendix 1).
S'FOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
197
It is useful to have also defined for the general case the quantities
i'o.i = io.1 n/nl and i'0,2 : io,2n/n2
(43)
iT = i~.1/vl and i~ -- i'0.2/v2
(44)
and nt and n2 are the numbers of electrons transferred in steps i and 2, respectively, for one act of the corresponding basic step. E nj vj = n :# E nj (in general). It should be noted that only when vt =v2, are i~.l and i'o.2 the values of io when the steps are separately rate determining; when vt :# v2, the corresponding values of io are iT and i~. Here, and in many subsequent parts of the text, reference to Fig. l may be of considerable help. We also define
7* = ( i*/i~ ) = 7(v2/vx)
(45)
and when vl =v2, ~'*=7. It should be noted that in this work io.2, for example, never appears alone, but as (io.2/n2)=So.2F; thus, although io.2 may be zero, the relevant quantity ( i o . 2 / n 2 ) is defined.
( 1 ) Theoretical stoichiometric numbers, v Consider a process of two steps of stoichiometric numbers vt and v2*. Using the earlier defined quantities, it is shown in Appendix 1 that l/so = vt/So. 1 +
v2/So.2
(46)
or
+(n2/i0.2)~'~
n/io = ( n l / i o . t ) ~ ' l
(47)
From (43), (44) and (45) i
)
-i'°"., -- nJ°-' / nt°,z -- !9±[/ /O,lZ /O.2
nl
/
n2
nl /
n2
SO, I
(48)
S0,2
so that the y of works defining it in terms of rates of chemical reactions is identical with that defined here**. From (47) and (48)
n/io = (vl + v27) n~/io.t
(49)
n/io = ( v2 + v x/7' n2/io.2
(50)
and For two steps with vl = v 2 = v, from (47)
n/io = ( nx /i,,. , + nz/io.z)v (51) and for two steps of stoich~ometric number, v l and v z, respectively, but effective * vt and v2 may have a c o m m o n factor >1 which simply represents v for this two step r.d.-proccss. ** It is important to note that 7 is not generally defined in the literature, since it is only defined in terms of currents, which are not defined in the general case. Thus the io.tt in Makrides' discussion 5 of the V - T mechanism does not correspond to any of the currents mentioned here; however, his 7 for the V - T mechanism corresponds to the 7* defined here, and his equation for the V T mechanism is c o n ~ q u e n t l y incorrect with y as defined by him.
198
J.C. REEVE
overall stoichiometric number v we can write n/io =(,h v,/io.l + n2 v2/io.2) = ( nt /io.1 + n21io.2)v
(52)
From (48) and (52) we have "rio = v( 1 + 7)(",/io.1 )
(53)
Comparison of (49) and (53) yields with (45)
vlWv2~f vl v2 (I +"~*) (54) l+y v2+v~7* For the V -H mechanism, vl = v2 = I and from (54) it is seen that v= 1 for all values of 7 (as stated by Parsons6; cf. refs. 3, 4, 5 and 8). For the V T mechanism, v~ = 2 and v2 = 1 so that from (54) ,, = (~ + 2)/(~, + 1)
(55)
as concluded by Makrides 5 (but see footnote to eqn. (48)). Hence, v will change from 2 to 1 through 1.5 as 7 increases from zero to infinity through !, and from 2 to 1 through 4/3 as "?* increases from zero to infinity through !. ( 2 ) Apparent or extrapolated stoichiometric numbers, v* From (52) and (43)
! io
= n ~ (v, + v~,) = , 1 (vl + v,ot) nio. l io. 1
,
--.,
(56)
and
,(v
(57)
2 +
io 1o.2 From (4), (5) and (44) v*=vi* io
v i°'l 1 vl io
(5'8)
and .!
v * = v i * = v/°"2 1 io v2 io SO that from (56) and (58)
(59)
(60)
v~' = L (v,+v27) YI
and from (57) and (59) V~ = -V2
(61)
V2 +
Hence, substituting for v from (54), (60) yields with (45) (v,+v2~) ~
v,v~(l+7*) ~
vl(1 +7)
v2+vl~*
and (61) yields
(62)
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
(vl+vz?) z
v (l +
199
vlvz(l+7*) 2
+
(63)
For the V-H mechanism (v~ = v z = 1) we have (as already discussed) v* = 1+ 7
(4o)
v~ = l+-f -~
(41)
and For the V-T mechanism (v~ =2, v 2 = 1) vT - (2+7)2 - 2(1+?*)z 2(1 +7) I +27*
(64)
and (2+7) 2 2(I + 7 " ) z v* - (1 + ?)? - (1 + 2"t*)7"
(65)
when )',~ 1, 7= 1 or 7 >>1, then v~' =2, 9/4 or ~,/2, respectively, and v~ =4/7, 9/2 or 1, respectively. For the same values of ~,* as for 7 above, the values of v~' and vy are 2, 8/3 or ),* and 2/7", 8/3 or 1, respectively. Of course, should extrapolation to find i* be made from part of a transition range where the behaviour were not really Tafelian, the value of i* substituted in (5) would be reduced from the actual i* towards io; the estimated value of v* would be smaller and lie between the real v* and v. If 4.1 does not follow a linear Tafel law, i~' estimated by linear extrapolation can be larger than i'o.1/vl and vl* greater than indicated by (62). The arguments presented here regarding v will not, however, be affected by log(io. ~), etc. being non-linear functions of potential. This treatment is easily extended to the case of three steps of differing v and similar rate, but this is probably only of value as an academic exercise. Considering the case of parallel (instead of the above consecutive case) reaction paths, the curve of log(current) vs. potential becomes convex to the potential axis instead of concave 14"~s; however, it is readily verified that v will still be greater than unity, since in this case the component current terms, instead of their reciprocals, are additive. In contrast, v* will be less than v and hence possibly less than 1. Makrides 4 considered "two competing reaction mechanisms", but it is clear that he was in fact considering two successive steps (of comparable exchange rate) together constituting the overall reaction. The possibility of stoichiometric numbers being less than unity for chain mechanisms has been indicated by Horiuti 21 and Makrides 4. Apart from the above connection wit.h stoichiometric numbers the real practical significance of component currents lies in their relationship to exchange currents in general and also the rate of isotopic exchange. ACKNOWLEDGEMENTS
The author wishes to thank Dr. G. Bech-Nieisen and Dr. J. Ulstrup for valuable discussions. SUMMARY
The original equation of Horiuti and Ikusima for stoichiometric number can
200
J.C. REEVE
lead to difficulties when applied to heterogeneous systems; for heterogeneous systems in particular the later definition of Horiuti and the definition of Parsons are to be preferred as much more tangible and uncomplicated. Errors in the literature demonstrate that when considering component currents much greater care must be exercised in the case of heterogeneous processes than in the case of the more familiar homogeneous ones. A new general expression is given for the theoretical stoichiometric number (v) of a process with two rate-determining-steps; similar general expressions are given for extrapolated or apparent stoichiometric numbers (v*). Various conflicting earlier statements concerning the values for v and v* are examined. Two partly new ways of possibly estimating the theoretical quantity are indicated in Appendix 2. Parallel reactions can give rise to values of v* (but not v) less than unity. APPENDICES
(1). Derivation oj" equations (46) and (47)* and comments For a two step homogeneous process for which v t = v 2 = l , it is readily verified by applying the steady-state assumption to the intermediate concentrations and equating the product concentrations to zero that
l/s+ = l/kx +k- x,/klk2
(66)
Although consideration of a two-step process is probably adequate for most purposes, the general case will be considered. In general for a multistep process and when vj = 1
1
~__t k_ak_2...k,_~j_l ~
s+
j=t
(67)
klk2...ko-t)kj
where j = l defines the final step 22.*. As pointed out by Christiansen 22, this leads to the result j=t 1 j=t 1 1 .=Z Xo jkj -=Z So
j
1
,
.1 1 So,j
(68)
where Xo,~ is the equilibrium concentration of X~, and So and So.j are the exchangerates of the overall process and the jth step, respectively. For a sequence 1
A .
2
' X 1 ; X'I . -I
l
" X 2 ; ... ; X~_ t . -2
• B
(69)
-I
'* This can also be achieved using the irreversible thermodynamics principle of proportionality between rate and affinity, and the additivity of the affinities of consecutive steps giving the overall affinity. However, the problem of identifying the appropriate part of the proportionality factor with the stoichiometric number (irrespective of heterogeneity and coverage) in complicated cases, involves considerations such as those below (t;f. ref. 4). *~' Although homogeneous kinetics were assumed by Christiansen 22. cqn. (67) is valid for Langmuirian behaviour, if the left-hand side of (67) is replaced by (1-O)/s~ ; all the individual coverages are eliminated when vi= 1.0 is the total coverage.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
201
the requirement of vj= 1 for cqn. (67) to be valid entails that eqn. (69) is written so that X~ is identical with Xj and that the k's are defined by
Xjk¢j+l)=s+¢j+l) and Xjk-j-- s-j
(70)
When the stoichiometric numbers of the steps differ, the k's must be defined to keep Xj and Xj identical. Thus for the simple two-step process A-*X, 2X--*X2 (vl =2, v2 = 1; V-T mechanism type), eqn. (66) becomes
l/s+ = l/kl +k_ t/kt k2 )(1"
(71)
in which the k's refer to the steps (written with X~=Xt) 2(A~X) and 2X~Xz. With the k's normally defined for the unit separate steps, (71) becomes (~f. eqn. 35)
l/s+ = vt/kl +k_l/klk2 Xa*
(72)
In the general case, (68) becomes 1
So
j= t
vj
j'= t SO,j
(73)
where So,j refers to the unit jth step of the process which occurs vj times when the overall process occurs once. For the heterogeneous process with species undergoing non-Langmuirian adsorption, eqns. (67) and (72) will also involve complicated functions of the coverage by the various species involved. However, it is seen that at equilibrium this will present no problem, since the new coverage terms (as well as (1 -0)), will bc included in the terms s0.~ of (68) and (73). These considerations and eqn. (73) are required before irreversible thermodynamics can be applied in derivations regarding r (~f. the first factor on the right-hand side of eqn. (1) of Parsons 6) as opposed to v*. Considering the general case with several intermediates has an additional advantage in the situation when the system is not far from equilibrium. The risk of identifying 0 of the term ( 1 - 0)with 0+ (or even 0 for any intermediate) is avoided. Two terms of (73) which become significant in a given potential range give rise to the behaviour illustrated in Fig. 1, and when both terms are simultaneously significant the transition behaviour is observed. In the ideal situation, the limiting behaviours of Fig. 1 (and their extrapolations) will be linear. If the behaviour is non-Langmuirian or otherwise non-ideal, the plots of log(i0, t), log(i~).t), etc. vs. potential in Fig. l will be non-linear extrapolations of the limiting behaviours. Although this will not affect the arguments regarding v presented in the preceding sections, v]' will be affected by the linear extrapolations of iT not being coincident with iT=i'o.a/v in the transition range. It is interesting and relevant to note that (20) may be freed of 0 by means of(13) (or i+ written as 2Fk20+) giving 2F 1 k_, (, k-z) i+-k, +k-~-; + k z + k ~ 2 * k, must be multiplied by (1 -O) in the Langmuirian case (see previousfootnote).
(74)
202
J.c. REEVE
This involves firstly one of two assumptions, (i) reactants and ~,roducts d'6 not block the electrode but simply make use of the uncovered area,(ii) reactants do not block the electrode and the product species exhibit a constant coverage, and secondly, the assumptions that the electrode is uniform, the adsorption Langmuirian and desorption negligible. Equation (74) involves the sum of four terms and may be compared with the two term result for the simple homogeneous case
2F/i+ = l/kt +k_ ~/k~ k2
(75)
and with (20) rearranged
2F/i+ = ( 1 / k l + k_ l / k , k 2 ) / ( 1 - 0 )
(20')
in an earlier publication of the author an error (equivalent to that of Frumkin 3 and Makrides 5 and referred to earlier) was made in a concealed way by writing the current i+ as 2Fk20 (instead of 2Fk20+) and using the result to eliminate 0 from (20). The result in that case was the same as (74), except that the second term in the bracket (representing a normally unimportant influence of the back reaction, through the influence on the coverage 0_) was absent; the first term in the bracket of(74) represents the rate when coverage by 0+ is complete. In the ideal case of equal overall transfer coefficients 23.1° for the two steps (~a and ~2, respectively), the terms of(74) will combine in pairs, so that (74) and (75) are of the same form. Also, the special situation of limiting Tafel slopes being in the ratio of 2:1 can arise when the second step is potential independent and ~1=0.5, and apparently also when coverage effects are of major importance, the first step is potential independent and c~2=0.5. However, consideration of eqn. (20') reveals that the final terms in the bracket of (74) can only be large when 0 is large and Langmuirian behaviour unlikely, although 0+ might be small. An example of when the effect would be important would be when 0+ --~0_ ~ ½, e.q. near equilibrium and when io. t = i0.2 and kl = k_ t. At high anodic 7, 0+--,0 and the final term in the bracket is unimportant, while at high cathodic r/, the effect is unlikely to be of much interest since i+ ,~ i_. If the smaller limiting b-value or Tafel law (b=OE/O Iogl0i) were observed, the ideal value would then be about 60 mV at 25°C*; such a Tafel law for both anodic and cathodic behaviours would be most readily explained by the above model with a chemical step between two charge-transfer steps, or a single charge transfer involving two electrons (possible 24, but unlikely25).
(2). Modifications of methods to determine v The possible use of eqn. (2) to determine v was indicated by Parsons 2 but (as explained earlier) determinations at i = 0 are often to be preferred. Recently, Oldham and Mansfeld 29 have indicated that the sum ( I b + l - l + l b _ l -a) can be * Although not previously suggested, the anodic Tafel laws of ca. 60 mV and ca. 120 mV combined with the cathodic Tafel law of ca. 30 mV for the Bi/Bi3' electrode in HCIO4 (ref. 26) would be neatly explained by this type of mechanism. Anodic: Bi~Bi÷---,(Bi*) * or Bi-,Bi* (giving ca. 60 mV or ca. 120 mV Tafel laws, respectively), and cathodic Bi3+~(Bi*)*---,Bi" (giving a ca. 30 mV Tafcl law). (Bi*)* is some modified form or state of Bi(l). Either type of explanation may be responsible for the behaviours of the Cd/Cd z" electrode in K2SO4 solution2~, and the Cu/Cu 2÷ electrode (unactivated) in HCIO4 solutionsts'28. Although c.e.c, mechanisms are well known for cathodic deposition in such systems, e.c.e, mechanisms do not seem to have been suggested previously (except in refs. 15 and 28). The single arrows indicate the r.d.-steps.
STOICHIOMETRIC NUMBERS AND COMPONENT CURRENTS
203
determined from the asymmetry of curves of current vs. potential in the region o f zero current for corroding (mixed-potential) systems. However, it has been pointed out 3° that asymmetry is principally determined by ( I b + l - l - l b - } - l ) (c]: Faradaic rectification) so that such measurements would have to be very precise; it was also pointed out that the exchange current (corrosion current) would need to be large so that the method would be more appropriate for redox systems with high exchange currents; O l d h a m and Mansfeld are publishing a table 3~ to assist in rapid analysis of such curvcs. Potter 32 has proposed a m e t h o d of finding v by observing small departures (due to the reverse reaction) of the current from the extrapolated value. This, of course, involves determination of the quantity at finite current. The derivation apparently involved a p p r o x i m a t i o n of expanded exponential functions, and this is not necessary. Expressing the net current (i) as the forward c o m p o n e n t (extrapolated) current ( i , ) minus the backward c o m p o n e n t (i_) and substituting for i_/i+ from e q n . ( l ) inverted, it is readily found (without approximation) that (cJ: the calculations of Allen and Hickling 33) Iog~o i + - i
r
- 2.303RT
!.)
v r/
(76)
Hence, for all practical values of the current ratio, a plot of the left-hand side of (76) vs. q should give a straight line t h r o u g h the origin yielding n/v. This m c t h o d gives the possibility of examining v over a range of potentials. It is worthwhile to note that although an extrapolation procedure (for i+) is used, the method yields v and not the apparent value, v*. REFERENCES I J. Horiuti and M. lkusima. Proc. Imp. Acad. (Tokyo), 15 (1939) 39. 2 R. Parsons. Trans. Faraday Sot., 47 (1951) 1332. 3 A. N. Frumkin, Proc. Acad. Sci. USSR, Phys. Chem. Sect. (English Transl.), 119 (1 6) (1958) 179. 4 A. C. Makridcs. J. Electrochem. Sot.. 104(1957) 677. 5 A. C. Makrides, J. Electrochem. Sot'., 109 (1962) 256. 6 R. Parsons, J. Electrochem. Soc., 105 (1958) 366. 7 L. I. Krishtalik, Sor. Electrochem., 2 (1966) 582: Elektrokhimiya, 2 (1966) 624. 8 B. E. Conway. Theory and Principles ~[ Electrode Processes, Ronald Press, New York, 1965. 9 K. J. Vetter. Electrochemical Kinetics, Academic Press, New York, 1967, p. 591. 10 R. Parsons, Croat. Chem. Acta, 42 (1970) 281 (in English). 11 A. N. l-'rumkin, Z. Physik. Chem.. 164(1933) 121. 12 J. Horiuti. d. Res. Inst. Catal., Hokkaido Unit., 1 (1948) 8. 13 E. Gileadi and B. E. Conway in J. O'M. Bockris and B. E. Conway (Eds.), Modern Aspects of Electrochemistry, No. 3, Butterworths, London, 1964, p. 347. 14 B. E. Conway, Theory and Principles of Electrode Proce.sses, Ronald Press, New York, 1965, p. 110. 15 J. C. Rccve, Collect. Czech. Chem. Commun.. 36 (1971) 757 (in English). 16 P. D. Lukovtsev and S. D. Levina. Zh. Fiz. Khim.. 21 (1947) 599. 17 P. D. Lukovtsev, Dissertation, L. la. Karpov Phys. Chem. Inst., 1940. 18 P. D. Lukovtsev, Zh. Fiz. Khim., 21 (1947) 589. 19 B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press. New York, 1965, p. 268. 20 B. E. Conway, Theory and Principles of Electrode Processes, Ronald Press, New York, 1965, p. 271. 21 J. Horiuti. Proc. Jap. Acad.. 29/1953) 160. 22 J. A. Christiansen, Adran. Catal., 5 (1953) 311; Z. Physik. Chem. (Franlqfurt), B33 (1936) 145; Z. Phy.sik. Chem ( Franklilrt ), B37 (1937) 374.
204 23 24 25 26 27 28 29 30 31 32 33
J . c . REEVE
H. 14. Bauer, J. Electroanal. Chem., 16 (1968) 419. D. C. Grahamc, Annu. Ree. Phys. Chem., 6 (1955) 337. B. E. Conway and J. O'M. Bockris, Proc. Roy. Soc. (London), A248 (1958) 394. M. S. Grilikhes, B. S. Krasikov and N. S. Solotskaya, Soy. Electrochem., 5 (1969) 799; Elektrokhimiya, 5 (1969) 859. W. Lorenz, Z. Elektrochem., 58 (1954) 912. J. C. Reeve. Extended Abstracts oJ" 23rd Meeting of I.S.E. Stockholm, 1972, p. 227. K. B. Oldham and F. Mansfeld, Extended Abstracts of 23rd Meeting of I.S.E. Stockholm, 1972, p. 109. J. C. Reeve and G. Bech-Nielsen, Corros. Sci., 13 (1973) 351. K. B. Oldham and F. Mansfeld, Corros. Sci., 13 (1973) 811. E. C. Potter, Thesis. University of London, 1950. P. L. Allen and A. Hickling, Trans. Faraday Soc., 53 (1957) 1626.