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Reorganization energy of the medium in homogeneous and electrode reactions
INTRODUqIOti
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The frequency dispersion of the dielectric permittivity and the differences ‘in the behaviour of quantum and classical modes result in an essentially quantum b& haviour of a part of the orientation polarization, so that it does’ iiot make any contribution to the reorganization energy [2]. This must beallowed for by-introducing into eqns. (1) and (2) a correction factor less than unity. On the basis: of ‘the experimental data on the frequency dependence of E for water 141, this factor was calculated in ref. 5 to be close to 0.8. In the case of particles of the same size (a, = u1 = a) approaching each other- to the point of immediate contact (R = 2a), or an ion in contact with metal surface ( R’ = a), eqns. (1) and (2) are simplified: Es = N,( A& E,=N,(Ae)“(l/r,-
I/r,
- 1,&)1/4a
0.a) (2a)
l/c,jl/2a
It will be seen that in this case the reorganization energy for a homogeneous reaction is twice as large as that for an electrode reaction with a particle of the same radius. Hence, it follows that for reactions with zero free energies of the elementary act (homogeneous isotope exchange between oxidized and reduced forms, an electrode redox reaction at the equilibrium potential), the activation free energies differ by a factor of two. To the first approximation, this rule holds true for reactions of some large complex ions [6]. It is of interest to consider the applicability of eqns; (1) and (2) to reactions of proton donors, both with respect to the absolute value of Es and the ratio between Es for homogeneous and heterogeneous processes. As is known, in crystals the H,O’ ion is isomorphic to K’ and NH,), so that its crystallographic radius can be taken to be = 0.15 nm (radius of Kf according to Gourary-Adrian). Substituting a = 0.15 nm into eqns. (la) and (2a), we obtain the values of Es equal to 126 kJ mol-’ and 252 kJ mol-’ for the electrode and the homogeneous reactions respectively (&hen corrected for- the frequency dispersion, 100 and 200 kJ mol-‘). The value of Es for an electrode reaction can be estimated from the data-on the activation energy at the transition point from ordinary to barrierless discharge [7]. At this point the true activation energy is equal to E, (see Fig.‘l). In’ this case: it should be taken into account that at the transition point from ordinary to barrierless discharge it is the ideal activation energies ( W) which become equal and not the real ones (A) IS]. The relation between them is as follows: : W=A+aTAS=A+aT(~SHZ+SH,O-SH1O+-Sc)
:
-
(3)
Here S are the standard entropies of the species mentioned in the-,btrlk -of the respective phases. Moreover, for comparison with calculation by eqns (!) and (2), we need not the activation energy but the activation freti energy. For barrierless discharge, .. :. -,. AG’=W-TASP=W-T(~H,-+~H20,-~H,0’-~e)~
.-
_.I,.
where the subscript s refers to adsorbed partic!es (more &cisely,
:.:-: :-.’ ..:._ :
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.t_o.P;&ick% in the
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Fig--l. &heme of potential curv& at the potential of transition from ordi&y energies of relevant states are marked with horizontal lines.
to barrierless discharge. The
states immediately before or after the elementary act of proton transfer). Substituting eqn. (3) into eqn. (4) and taking into account that for a barrierless discharge a =_ 1, we obtain for the ideal activation_ free energy at the transition points to barrierless discharge: S,, f ASaH o+c-(5) 3 : Here AS, are the standard entropies_of .adsorption: B.elow are the numerical values of
AG+.;
= A’ f- ?-(+S&Y
the:entropy components:- Su, 2 130.6J K-t mol-t [9],_A.SiH0 = 4.8 J K-’ mol-‘, .determined from the- temperature dependence. of --the-differekce of interfacial tensions,.mercury-water and mercury-air [lo], Sk; = 7.9 J K.-t m~l~t.calculated from the vibration .frkquency~estimatesof .&orb@ ,EJ;.A&,‘b; F 2l,J,K-!. mol-‘, estimatedas one-half the entropy of desolvation of,. th.< isomor@ic ion K’ [ 111. -_~ Substitu~on of these’valuek’gives. : -A&+_* =;4*:$;21;8
kJ.&o+ ;_
I ..
-.
:
1.
.- .’
.~
:.,_. __(sa)
,- : &uations. (3)-;‘(S) i+kkin:.the true activatio~.ene~~es:‘~~~ A; whereasapparent -.act&+on ~~energ@_s :aredeter+$+d ~~~e~_~~nt~~;_~f~ring..b~.: the..ener& of.!.ion.
transfer.._fr& solution_bulk to .@ cosi+on- in@rich ;t;s~disct;~g~~.,~~_s’ energy_. _-_., :. 1 consists of the~e]e+ostat+(*, ~)~:~d.non-Coulom6:(~~=)-~niporients_ -:,-..:: :
: .-:
10
Figure2 shows a scheme of polarization curves in the region of the transit+ from ordinary to barrierless discharge for two different solutions corres$onding to different +i potential values. The activation energies being measured at the t&rsition points differ by the difference of overpotentials E,* = E: - Aq*F.. For calculating E, we are interested in the quantity E* - &F. At the transition point of curve.2 it is equal to ET -+,(2)F=
E: -&(l)F+A#,F-Aq*F
where A+, is the shift of 4, in the negative direction for curve? cpmpared to curve I. As the slopes of Tafel plots for ordinary and barrierless discharge differ (at 4, = constant) by a factor of two, then as can be seen from Fig.2, we have A+, = Aq*! i.e. ET - +,(2)F= E; -+,(I)F. Thus, the difference of interest to us does not depend on our choice of the experimental curve. The extrapolated Tafel plot for an ordinary discharge in 1 M acid intersects the plot for the banierless discharge at the overpotential 0.21 V, which corresponds to the activation energy 75 kJ mol-’ [IO]. In 1 M acid the +, potential can be estimated as being = -0.07 V, whence E* - #, F- 82 W mol-‘. The non-Coulomb energy component can be related to two effects. First, ion incorporation into the Helmholtz layer is unfavourable owing to the decrease of the effective dielectric constant-this effect can amount to several kJ mol-’ [12]. Second, for sufficiently effective tunnelling an approach to a less than equilibrium distance is necessary. For this purpose work against the repulsion forces is to be expended, which can also be estimated as being several kJ mol-’ [13]. It is difficult
t!$
Fig. 2. Scheme of polarization curKS in the transition region from ordinary to barrierless discharge.
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:.. : :. _. :td es&a& -. nC.accu&ely, _but:it -i&‘po$sible to give a:tent&k e&&&of .its’lo+vei. ‘ht bh-t&.‘&+su&icbn t&it the c&icentration~‘of .di&hz&ing b&O %-ions ap+oach..ing- the.electrode. to-the point of -imi+ediate contact .Shotild:be.&all; &~~z@mp&d: to : ,,~~i~~&qcent~&i&i~i& Ibe &ey Hehnholtz &me [ 141..::This st&llne$ sti&d remain tiheri::&he& poten~al~sliifts strongiy~~iri:. the -n~~ative_.di_recti~n,;wi?ich leads to :an increase lof t@ H,C) * concentration. in..:& -muei .layer by several orders; I&-us ’ ~aSSli& &tit- the &o&it of~H,Ot -in.the.inner_l~y~r does not eXc&d~l% of their~total amount- dti the ~ele&ode surface, .even -if ;the._&&entration: in the first monolayer increases by three orders (#p s’l80 mv). Then E,$ +.2.3&T. X 5 =i29 kJ mol-‘i
: On the o&r .harid;it 4sunlikely that E nC_would greatly -exceed this v&e: In -fact, the ,measur&i:real_a~p*entLctivation knergi&in the case of ~ordinarydischarge are as loti as 42 kJ mol-‘;. This quantityincludes the adsorption~energy, i.e. E,,=.+ a&&‘. enk@y jn 1 M acid is 23 kJ mol-‘, so that not If-E,, = 29 kJmol_l, the-ad&ptioti much. remains for the activation b&i& prober. So it would be expected that the upper limit of Emc could .ntit .differ greatly, f&m the above estimate, and.-tie may. assllme En& = 29 kJ mol-‘.
’
From the above estimates the true real -activation energy at $A* = E* - $;FEn, - 53 kJ mol- ‘.-According to eqn. @a), the corresponding value of the standard free reorganization energy is - 75 &J mol-*. Another possible method -of ,determining AGs* is to halve the difference of. energies (overpotentials) -bet&en the transition points -from barrierless to ordinary and from.ordikry to activatjonless, dischar&(see Fig. 3): in this method there is n6 need to introduce corkctjons for transition from real -to ideal and from app-nt. to
true quantities, as these corrections for both points are virtually similar and mutually compensate& It would be difficuh to use this method for electrode reactions, since. no transition from o&nary to activationless hydroxonium,ion discharge has been experinientally;observed’ as yet.- Moreover, difficult%s:arise here associated with a probtible. i&reaseof the~pre-exporiential, connected:with a graduaVdecrease of thetunnelhng disttice due to-a closer approach of the ion under the action of the rising electrode -charge [1.1$5]; Estimates of probabte ~limiting current. densities of ‘an activationless.discharge in ref. I1 &e the values of g? (overpotential at the transition point. ib. t& actkationless: process -in 1 M acid) of about 119-2.I-V, G&i& -correspot& to the re&g a&ration- energy 84-92. kJ mol -.I, i-e_ -Close enough to: that
: L... . reporuzdabovc. ... -=. .- : : ~. rJ -_. : -Thus, ‘different methods.of estimating &of the hydroxo&mion-dischakgc give a ~zduq- of ki&o& 80 Id mok?. %+.s estimate,-is qu$e close:,to.: that ob&ied~--by caiculation~-with~~eqn.(la), especially when. *o*ng for:the. &rmittivity- fr$&+cy
dispersioh,dorrectio6..(!00,kJ -mole’ j. .It- should be. *oted -that ~~e,c&ulat&l &lu& bg &il le!% byl6kJ I&I,‘, if the effect-of excluditig the proper ti@ume df ions is t&&i .aca%u&qf $ :ihe .theory [.lB].Some. decrease .of. .the klcuiated .valui-.may .be
&ill
a&:~es&ct&j~+.+ r;esliltof-the.spatial.dispersion.of .e_-:.:: I -.... .I .; .L-.- -_ :. :‘: ._I ~i.~w consider the que&kof ihe reorganiza tion energy in._~orn?geneous.proton~ .tr$sfer. .J$&rgelbody .of. 6gperi&eti@ !$,d#nc_e _prim~aiil~:. obtaikd: by Ei$n. [ii] ‘: .-...- :. +a,tik~tl+,dep&nd&g >on--d p& &e $t$idard .free energy: 6f &e process+ 66 reaction._... .’ __ ._._’ ., ._ .-_
i
ES
AL
WI
EaFP,
Fig. 3. Scheme of potential curves for activatiortlek and barrierless processes. .The energy levels of reactants up to the point of their mutual approach to the distance optimum for reaction are marked with horizontal lines.
passes from the diffusion-limited process rate region to that of usual kinetics, and ~ further to the region of the kinetics limited by difftision of products. In principle, ES could be determined from the difference of energies c&responding to the boundaries of activationlcss and banierless regions (Fig. 3) [7]_ -It is difficult, however, to find the exact position of these boundaries from the data- on homogeneous kinetics, since near these regions. the diffusion rate becomes the determining factor. For this reason we adopted the following. procedure for determining ES. As can be shown by means of a general.formula,-taking account:both of the diffusion rate and the rate of proton transfer proper [17];at the.rate equal to half the limiting diffusion value, the reaction rate -constant is equal .to that- .of diffusion. Let us assume that under these conditions not only are the ‘rate consttints equal but also the free activation energies of these two processes, i.e.. that :the’ activation energy of the reaction of interest to us is .Y 12 k.I x&l Y ‘. .If wc know, the free activation energy E #, by means of ‘Marcus’s eqn; (6) we can find .&if the value of the true free energy of the elementary act AJ, is known -. ~.
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E * = (Es + A Jo)2/4Es We will
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consider the reaction serieswith at.least one-uncharged reactant. In .t&
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.b;65l;r&&+&~. ~,&t&~ol&ules; :&de+&& b&-&&&&&~e~~of.the’ re&ng&,$l& :‘.~~-.l;o;m’;lhie:_i’ela~v~:acidifieslof .sutistances.,iniroI~~~.~~titlay:be positi~e,:as..~~!l,-as &g&e_&~, &~~&&~,i&& &ij;& _diffe&w : ii_. smai!;: f& _th~_~t&- &&be&of ,,l&i&g$& ,b&& :rqnG& .&s&n~.~~The &&i&&i &r&:. i&s.&ry,~fo~. .the closerapproachlof-reac!ant.t~ the-~o&m~. t.umielling.dist~nti was .&timated.in.:ref: 13 to b:e &~‘smah’a.$:&t0;8.._~-~~~l~.~~.-T;oml&~&$ &tiop_gt -kuch a- !&all. q&&oh w&d prob.ably,bjz be$ond th& ac&ra&df:&r cal&ul~tilins &d h&z& x&.&nit the last term-
_.:_ ‘_:I.. :_-, 1 -. .:_;-..:y:_ .--:_: .‘_r.:.-,’:.; I-‘_ 1’.I. _,.- .r :. _;,_-,‘: .:_ ‘1 .: :I : [email protected] :tij@i$~$ ~<& of &I are.- experimentally -accessible. :Tht%y.were atimat@ :-f@m-:Eigen’s:d&:[-171 $l.,2;3RT:hpK, .the ,_Ap’K.va+k: c&responding to E f=. 12 !k.I ti$Y’, :nti; truk’ free: &ergy A& differ.+ froni the Yvalue A$ by. the in *G;:(&j.
.difference of- -_ the- &&,&for’ the initial reactants -a+ the &d&ts.The small nonCoulombic. contiibut~onk for both of. theni are $ractically equal; so- this component can& -out.,When the.reatitant,are ~uncharg&l there.is- an- electrostatic.attraction of the charged product& Th&.energy of -this-attia$ion .E,-gti depends .Gb&ntially on the. distanck between two j0n.s.’We have,‘&ried out .the Cal&la&on. in two versions, takingI~c~U,-=52-andi6~-mo~~!;..~e& va&s correspon~~to Iargk (-0.45 MI) and small (-O.I>..mn) i&k radii. (the-.&oice of .tlieseradii is substantiated jater). The figure%in Tab!e .l :,ark for: the f+It -version, for the second one -they are --6.7 W mol-’ -higher: I&j..cak@ated’ ES ark in the range ~65.3-87.~ kJ mol-~‘; the- more pro&&l&
~tefi~~is~:‘7’0-80
kJ mol-.‘_. ..i. :. -:
.._
.-
-‘-
@H.&d’@@ groups. are~cl&Sin’size~ tdthe I-&O? ion, so that in their &se rkorganiiation~ener~es sinkjrtd th&calcuiated above would be expected. Iti fact, however;-’ ih& @culatiofi :auqording::to e&i- -,(2a)’‘]proved..td .differ :greatly from The
experiment, ex&ding it by.~a-factor of almost :ihiee. .It 1i<: important to em@hasize that. t& &es&n hereis notonly of.a verylarge quantitative-difference but-also of.a iualitative-one: E;for~aih&ogeneous rea+on @ovea to be‘pra&ally equzkto~that~ for- a heterogeneous rektjon, though it followa from. the thkory that it- should be .’ _
_
14
twice as large. Consequently, the situation for proton-transfer reactions .differs significantly from that for simple redox reactions, for which the theory agrees with experiment, not only qualitatively but also semiquantitatively. Unlike simple electron-transfer reactions, proton transfer includes the sub-barrier. transition of the heavy particle-the proton. For this reason, a significant mutual approach of the partners is necessary to facilitate proton tunnehing. Estimates of optimum tunnelling distances were given in ref. 13. For proton donor discharge on metals. the tunnelling distance is estimated to be - 0.04 nm. It is clear that with the H,O i ion radius 0.15 nm and the length of the O-H bond 0.1 nm, as assumed by us, taking account of the mutual approach of reactants will not significantly affect the charge-transfer distance and hence the reorganization energy. Therefore, this factor should not markedly influence the estimate of Es for an electrode reaction. Regarding the homogeneous reactions of OH acids with 0- or N-bases, as was shown in ref. 13, for these partners bound by hydrogen bonds and therefore
significant!y drawn together, the optimum tunnelling distance differs little from the equilibrium distance, so that this factor cannot cause a marked decrease of E;. Another difference between electrons and proton-transfer reactions is that in the former the interaction with the medium of fairly large complex ions not forming hydrogen bonds can. to a good approximation, be described in terms of a simple dielectric formalism. In proton-transfer reactions, however, the charge is centred on a particle or group with a rather small radius, which, moreover, can form strong hydrogen bonds with solvent_ A small ion radius can lead to a decrease of the reorganization energy, both due to dielectric saturation near the ion and because of the influence of the spatial dispersion of the dielectric constant. The hydrogen bonds of ion with the nearest water molecules act in the same direction. Let us consider, for example, a scheme of proton transfer from a H,O + ion to acceptor A (Fig.4). In accordance with the current concepts. the H,O + ion forms three strong hydrogen bonds with the water molecules, so that a H,O,i particle exists in solution. The fourth nearest to the H,O’ water molecule (ion surroundings are terahedral) does not form a hydrogen bond with it, undergoing orientation only under the action of the ion-dipole
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Fig. 4. Scheme of hydrogen bonds around the H,O + ion: (a) before proton transrer to acceptor A: (b)
nfter proton transfer.
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~15
i&era&ion (Fig. 4a). Following proton transfer- to acceptor (it is achieved by a shift -of two’protons indicated b;j, arrows), the orientation of three water mo&ules b&&l with the .-central molecule by. hydrogen bonds- remains ‘u&hanged, practically. at equilibrium, i.e. there is ‘no reorganization of this-part.of the first salvation sphere of. F&If. Only.the fourth water molecule is in a non-optimum orientation: it-would be more advantageous for it to .form a hydrogen bond (Fig. 4b). However, it should be taken into account, that in liquid water far from all possible hydrogen bonds are .realized (at room temperature about half..of them), so that an-adjacent position.of two H,O molecules not : bound by a. hydrogen bond .and belongiug to differentclusters is a. rule rather thti an exception. On the whole, it may be concluded that during proton transfer from the H30 c ion (and other species, to a large degree included in the net of hydrogen bonds of water) the first solvation sphere undergoes reorganization only slightly. Proceeding from this reasoning, let us take as the ion radius that of the first
solvation sphere a = 0.45 nm (cI~,~ + +2anZo). Calculation with such a radius gives ES 7 67 kJ mol-’ (not.allowing for E dispersion, 84 kJ mol-‘), with correction for the excluded volume, ES = 63 kJ mol - ‘. Thus, our model yields a result in a quite good agreement with experiment, and this fact is a good argument in favour of the popular concept that proton transfer follows a bridge mechanism with the participation of intermediate water molecules. It should- be._ noted that estimation of the. size of HsO+. from the rate of its diffusion-limited reactions also leads to the conclusion that the reacting particle incorporates the first solvation shell. The calculation has been carried out-for reactions of the H30f ion with a species of similar structure and shape.. In fact, the experimental data refer to qeries of organic molecules of the type RX and RXH (with the charge + 1, 0 and - I), for which a low polarity radical R occupies a part of the nearest and the second-nearest salvation sphere. This fact must decrease the reorganization energy somewhat, and with allowance for this effect, our model gives a somewhat too low value of ES. Precisely this result was to be expected, since in calculation-the first solvation sphere is taken to be completely rigid and unreorganizable; whereas some slight reorganization must occur (see above, the discussion -on the behaviour of ‘the fourth water molecule) It was
and this makes a certain contribution to ES. ; demonstrated above that .calculation under the assumption that a = 0.15 nm.gives reasonable agreement with experiment for an electrode reaction, whereas for a homogerieous
reaction it is necessary to take into account the rigidity of- the solvation sh_ell.Quite n&mally, a question arises why this factor has no effect in the case of the H,O f _idn discharge at- the electrode. When a hydroxonium ion undergoes discharge it is not bound with the-electrode by a- system of such strong.hydrogen bonds as exists in-the water bulk between I-I,0 + _a+ _+ei proton: acceptor. Moreover, during discharge at an el&trode, -the entre of gravity of a pdsitive chargemust be l&lized in a’ position most f&o&able from. .me .viewpoint of elect@atic& .i.e. .closer.:‘to :_the. ~ele+ode~ .As.sho&‘_for example,. tin. Fig. 5a. One would. hardly expect. that. after..proton- detachment iand.
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Fig. 5. Scheme of hydrogen bonds of the H,O •Fion near the electrode: (a) ‘before pr&ondischarge
&d
transfer; (b) after proton transfer. -_-.
adsorbed hydrogen formation, the water molecule formed ‘from the H,O +-. ion_.. should retain its former orientation, since the interaction kith the electrode -a-nd primarily with the double-layer field till significantly affect its equilibrium-~oriehtation. But the change of the orientation of this molecule will result in reorientation of its neighbours (one of such conceivable schemes is shown on .Fig. 5b). ;All. &is. means that near the electrode the H,OC ion has a practically reorganizable first s&atiori sphere. Such factors as a change of the liquid water structure near the interface also
act in this direction. It is expedient to mention here that- the ion penetration into the first water monolayer with a low dielectric permittivity would cause some_ikkase of the reorganization energy [ 181. This penetration also means some stibkantiai d&tiu& tion of the first solvation shell. Thus, the investigation of the reorganization energies of .homogeneous and heterogeneous proton-transfer reactions made in this paper points to the :very important role of the first solvation sphere structure and to the necessity of.further theoretical analysis of this problem at the microscopic level. REFERENCES
1 A.N. Frumkin. Z. Phys. Chem., A160 (1932) 116. 2 R.R. Dogonadze and A.hM. Kuznetsov, Fiicheskaya its), Itogi Nauki. Vol. 2. VINITI, Moscow, 1973. 3 R.A. Marcus, J. Chem. Phys., 24 (1956) 966.
Khirniya Kinetika, (Physical Chemistry, $+et-
4 R. Saxton. Proe. Roy. Sot.. A213 (1552) 473. 5 M.A. Vorotyntsev, R.R. Dogonaclze and A.M; Kuzrtetsov, Dokl. Akad.~Nauk; S.S.S.R., 155.(157C) ; : .. : 1135. .:, 6 R.A. Marcus. Ann. Rev. Phys. Chem., 15 (1964) 155. 7 L.I. Krishtalik, Electrokhimiya, 15 (1975) 1505; Sovremennye aspekty elektro~mic~esr;oy.kineiiki: Materials IV Frumkinskikh chteniy (Modem.Aspects of Electrochemical Kine&s_MatetjaIs.of i\! _.-.. Frumkin Lectures). Metsniereba. Tbilisi, 1980, p. 33. _. s L-1. Krishtalik, Elektrokhimiya. 5 (1969) 3. -. 9 Selected Values of Chemical lIxrmodynamic Properties, Natidnal .B&eau of Standards, CipqlarSCl$~’ : Part I, 1961. ._ : : :_i _ _..I.’ ,.. 10 L.I. Krishtalik. Elektrokhimiya. 2 (1566) 1176. 11 L-1. Krishtalik, Elektrodnye reaktsii. Mekh&i&n elcmentornogo_akta.(Electrode-~~~tio~s~ ,j&$$& _:. nism of th& Elementary Act). Nat&& MO&W, i679;.Ch. IV; -.. ‘:- ?’ ” -, -.,y,. I ,: .. 12 M.A. Vorotyntsev, VS. Krylov and L.1. Krishtabk. ~~ektrokhimiy~. 15 (1979) 7&k -‘. -. .;, :..’ I:“-? -il’. .. .-- :._ ~.
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8
The frequency dispersion of the dielectric permittivity and the differences ‘in the behaviour of quantum and classical modes result in an essentially quantum b& haviour of a part of the orientation polarization, so that it does’ iiot make any contribution to the reorganization energy [2]. This must beallowed for by-introducing into eqns. (1) and (2) a correction factor less than unity. On the basis: of ‘the experimental data on the frequency dependence of E for water 141, this factor was calculated in ref. 5 to be close to 0.8. In the case of particles of the same size (a, = u1 = a) approaching each other- to the point of immediate contact (R = 2a), or an ion in contact with metal surface ( R’ = a), eqns. (1) and (2) are simplified: Es = N,( A& E,=N,(Ae)“(l/r,-
I/r,
- 1,&)1/4a
0.a) (2a)
l/c,jl/2a
It will be seen that in this case the reorganization energy for a homogeneous reaction is twice as large as that for an electrode reaction with a particle of the same radius. Hence, it follows that for reactions with zero free energies of the elementary act (homogeneous isotope exchange between oxidized and reduced forms, an electrode redox reaction at the equilibrium potential), the activation free energies differ by a factor of two. To the first approximation, this rule holds true for reactions of some large complex ions [6]. It is of interest to consider the applicability of eqns; (1) and (2) to reactions of proton donors, both with respect to the absolute value of Es and the ratio between Es for homogeneous and heterogeneous processes. As is known, in crystals the H,O’ ion is isomorphic to K’ and NH,), so that its crystallographic radius can be taken to be = 0.15 nm (radius of Kf according to Gourary-Adrian). Substituting a = 0.15 nm into eqns. (la) and (2a), we obtain the values of Es equal to 126 kJ mol-’ and 252 kJ mol-’ for the electrode and the homogeneous reactions respectively (&hen corrected for- the frequency dispersion, 100 and 200 kJ mol-‘). The value of Es for an electrode reaction can be estimated from the data-on the activation energy at the transition point from ordinary to barrierless discharge [7]. At this point the true activation energy is equal to E, (see Fig.‘l). In’ this case: it should be taken into account that at the transition point from ordinary to barrierless discharge it is the ideal activation energies ( W) which become equal and not the real ones (A) IS]. The relation between them is as follows: : W=A+aTAS=A+aT(~SHZ+SH,O-SH1O+-Sc)
:
-
(3)
Here S are the standard entropies of the species mentioned in the-,btrlk -of the respective phases. Moreover, for comparison with calculation by eqns (!) and (2), we need not the activation energy but the activation freti energy. For barrierless discharge, .. :. -,. AG’=W-TASP=W-T(~H,-+~H20,-~H,0’-~e)~
.-
_.I,.
where the subscript s refers to adsorbed partic!es (more &cisely,
:.:-: :-.’ ..:._ :
@I
.t_o.P;&ick% in the
: :
‘. .
:
.. . -:. p,
..
:
Fig--l. &heme of potential curv& at the potential of transition from ordi&y energies of relevant states are marked with horizontal lines.
to barrierless discharge. The
states immediately before or after the elementary act of proton transfer). Substituting eqn. (3) into eqn. (4) and taking into account that for a barrierless discharge a =_ 1, we obtain for the ideal activation_ free energy at the transition points to barrierless discharge: S,, f ASaH o+c-(5) 3 : Here AS, are the standard entropies_of .adsorption: B.elow are the numerical values of
AG+.;
= A’ f- ?-(+S&Y
the:entropy components:- Su, 2 130.6J K-t mol-t [9],_A.SiH0 = 4.8 J K-’ mol-‘, .determined from the- temperature dependence. of --the-differekce of interfacial tensions,.mercury-water and mercury-air [lo], Sk; = 7.9 J K.-t m~l~t.calculated from the vibration .frkquency~estimatesof .&orb@ ,EJ;.A&,‘b; F 2l,J,K-!. mol-‘, estimatedas one-half the entropy of desolvation of,. th.< isomor@ic ion K’ [ 111. -_~ Substitu~on of these’valuek’gives. : -A&+_* =;4*:$;21;8
kJ.&o+ ;_
I ..
-.
:
1.
.- .’
.~
:.,_. __(sa)
,- : &uations. (3)-;‘(S) i+kkin:.the true activatio~.ene~~es:‘~~~ A; whereasapparent -.act&+on ~~energ@_s :aredeter+$+d ~~~e~_~~nt~~;_~f~ring..b~.: the..ener& of.!.ion.
transfer.._fr& solution_bulk to .@ cosi+on- in@rich ;t;s~disct;~g~~.,~~_s’ energy_. _-_., :. 1 consists of the~e]e+ostat+(*, ~)~:~d.non-Coulom6:(~~=)-~niporients_ -:,-..:: :
: .-:
10
Figure2 shows a scheme of polarization curves in the region of the transit+ from ordinary to barrierless discharge for two different solutions corres$onding to different +i potential values. The activation energies being measured at the t&rsition points differ by the difference of overpotentials E,* = E: - Aq*F.. For calculating E, we are interested in the quantity E* - &F. At the transition point of curve.2 it is equal to ET -+,(2)F=
E: -&(l)F+A#,F-Aq*F
where A+, is the shift of 4, in the negative direction for curve? cpmpared to curve I. As the slopes of Tafel plots for ordinary and barrierless discharge differ (at 4, = constant) by a factor of two, then as can be seen from Fig.2, we have A+, = Aq*! i.e. ET - +,(2)F= E; -+,(I)F. Thus, the difference of interest to us does not depend on our choice of the experimental curve. The extrapolated Tafel plot for an ordinary discharge in 1 M acid intersects the plot for the banierless discharge at the overpotential 0.21 V, which corresponds to the activation energy 75 kJ mol-’ [IO]. In 1 M acid the +, potential can be estimated as being = -0.07 V, whence E* - #, F- 82 W mol-‘. The non-Coulomb energy component can be related to two effects. First, ion incorporation into the Helmholtz layer is unfavourable owing to the decrease of the effective dielectric constant-this effect can amount to several kJ mol-’ [12]. Second, for sufficiently effective tunnelling an approach to a less than equilibrium distance is necessary. For this purpose work against the repulsion forces is to be expended, which can also be estimated as being several kJ mol-’ [13]. It is difficult
t!$
Fig. 2. Scheme of polarization curKS in the transition region from ordinary to barrierless discharge.
..~ ,,,
:
-._‘,i.
_~
._.
-.:-
:-
--::
:
.-.I
~.
::_.
.~ :-.,
I
,, :.
:.
. .:
:
_.
!!
:.. : :. _. :td es&a& -. nC.accu&ely, _but:it -i&‘po$sible to give a:tent&k e&&&of .its’lo+vei. ‘ht bh-t&.‘&+su&icbn t&it the c&icentration~‘of .di&hz&ing b&O %-ions ap+oach..ing- the.electrode. to-the point of -imi+ediate contact .Shotild:be.&all; &~~z@mp&d: to : ,,~~i~~&qcent~&i&i~i& Ibe &ey Hehnholtz &me [ 141..::This st&llne$ sti&d remain tiheri::&he& poten~al~sliifts strongiy~~iri:. the -n~~ative_.di_recti~n,;wi?ich leads to :an increase lof t@ H,C) * concentration. in..:& -muei .layer by several orders; I&-us ’ ~aSSli& &tit- the &o&it of~H,Ot -in.the.inner_l~y~r does not eXc&d~l% of their~total amount- dti the ~ele&ode surface, .even -if ;the._&&entration: in the first monolayer increases by three orders (#p s’l80 mv). Then E,$ +.2.3&T. X 5 =i29 kJ mol-‘i
: On the o&r .harid;it 4sunlikely that E nC_would greatly -exceed this v&e: In -fact, the ,measur&i:real_a~p*entLctivation knergi&in the case of ~ordinarydischarge are as loti as 42 kJ mol-‘;. This quantityincludes the adsorption~energy, i.e. E,,=.+ a&&‘. enk@y jn 1 M acid is 23 kJ mol-‘, so that not If-E,, = 29 kJmol_l, the-ad&ptioti much. remains for the activation b&i& prober. So it would be expected that the upper limit of Emc could .ntit .differ greatly, f&m the above estimate, and.-tie may. assllme En& = 29 kJ mol-‘.
’
From the above estimates the true real -activation energy at $A* = E* - $;FEn, - 53 kJ mol- ‘.-According to eqn. @a), the corresponding value of the standard free reorganization energy is - 75 &J mol-*. Another possible method -of ,determining AGs* is to halve the difference of. energies (overpotentials) -bet&en the transition points -from barrierless to ordinary and from.ordikry to activatjonless, dischar&(see Fig. 3): in this method there is n6 need to introduce corkctjons for transition from real -to ideal and from app-nt. to
true quantities, as these corrections for both points are virtually similar and mutually compensate& It would be difficuh to use this method for electrode reactions, since. no transition from o&nary to activationless hydroxonium,ion discharge has been experinientally;observed’ as yet.- Moreover, difficult%s:arise here associated with a probtible. i&reaseof the~pre-exporiential, connected:with a graduaVdecrease of thetunnelhng disttice due to-a closer approach of the ion under the action of the rising electrode -charge [1.1$5]; Estimates of probabte ~limiting current. densities of ‘an activationless.discharge in ref. I1 &e the values of g? (overpotential at the transition point. ib. t& actkationless: process -in 1 M acid) of about 119-2.I-V, G&i& -correspot& to the re&g a&ration- energy 84-92. kJ mol -.I, i-e_ -Close enough to: that
: L... . reporuzdabovc. ... -=. .- : : ~. rJ -_. : -Thus, ‘different methods.of estimating &of the hydroxo&mion-dischakgc give a ~zduq- of ki&o& 80 Id mok?. %+.s estimate,-is qu$e close:,to.: that ob&ied~--by caiculation~-with~~eqn.(la), especially when. *o*ng for:the. &rmittivity- fr$&+cy
dispersioh,dorrectio6..(!00,kJ -mole’ j. .It- should be. *oted -that ~~e,c&ulat&l &lu& bg &il le!% byl6kJ I&I,‘, if the effect-of excluditig the proper ti@ume df ions is t&&i .aca%u&qf $ :ihe .theory [.lB].Some. decrease .of. .the klcuiated .valui-.may .be
&ill
a&:~es&ct&j~+.+ r;esliltof-the.spatial.dispersion.of .e_-:.:: I -.... .I .; .L-.- -_ :. :‘: ._I ~i.~w consider the que&kof ihe reorganiza tion energy in._~orn?geneous.proton~ .tr$sfer. .J$&rgelbody .of. 6gperi&eti@ !$,d#nc_e _prim~aiil~:. obtaikd: by Ei$n. [ii] ‘: .-...- :. +a,tik~tl+,dep&nd&g >on--d p& &e $t$idard .free energy: 6f &e process+ 66 reaction._... .’ __ ._._’ ., ._ .-_
i
ES
AL
WI
EaFP,
Fig. 3. Scheme of potential curves for activatiortlek and barrierless processes. .The energy levels of reactants up to the point of their mutual approach to the distance optimum for reaction are marked with horizontal lines.
passes from the diffusion-limited process rate region to that of usual kinetics, and ~ further to the region of the kinetics limited by difftision of products. In principle, ES could be determined from the difference of energies c&responding to the boundaries of activationlcss and banierless regions (Fig. 3) [7]_ -It is difficult, however, to find the exact position of these boundaries from the data- on homogeneous kinetics, since near these regions. the diffusion rate becomes the determining factor. For this reason we adopted the following. procedure for determining ES. As can be shown by means of a general.formula,-taking account:both of the diffusion rate and the rate of proton transfer proper [17];at the.rate equal to half the limiting diffusion value, the reaction rate -constant is equal .to that- .of diffusion. Let us assume that under these conditions not only are the ‘rate consttints equal but also the free activation energies of these two processes, i.e.. that :the’ activation energy of the reaction of interest to us is .Y 12 k.I x&l Y ‘. .If wc know, the free activation energy E #, by means of ‘Marcus’s eqn; (6) we can find .&if the value of the true free energy of the elementary act AJ, is known -. ~.
.._.
E * = (Es + A Jo)2/4Es We will
+ Eapp
.
1 .._ ..:(6)
consider the reaction serieswith at.least one-uncharged reactant. In .t&
.:,. _YL
:.
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.;
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..--.I.. ). -.
~$&;d~.
;: g;&+tti
L_ ._,
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,-
;
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,--
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.:‘... -y.. ._ _;..;.y..-:.... -1.1 _.:
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__.
: y. -
-_;..1:13.: ..._
. .
n_l;lclucae_~::~~~trost~~i~~~_~~~.~~~~~_~e .noi;-Cdulor_nbiclp.~~f~: .j’ .:..___ ~,_nx~~~~&‘i~@h~d -.m~.akather,-$rong .h&l&&j b.o;id%is- small,] The:
.-.‘eli~~~~cif-_.,app~~a~~.to..~~ ~~libi;iumlpo~~tio~~~~~~~~~~~~b~~t~e~iffe;en~.~f-.j the .h~~~~en-.bond~:~-o;f. rea_cta& to&& &her&d’ to water.m&&&$&d~
of
.b;65l;r&&+&~. ~,&t&~ol&ules; :&de+&& b&-&&&&&~e~~of.the’ re&ng&,$l& :‘.~~-.l;o;m’;lhie:_i’ela~v~:acidifieslof .sutistances.,iniroI~~~.~~titlay:be positi~e,:as..~~!l,-as &g&e_&~, &~~&&~,i&& &ij;& _diffe&w : ii_. smai!;: f& _th~_~t&- &&be&of ,,l&i&g$& ,b&& :rqnG& .&s&n~.~~The &&i&&i &r&:. i&s.&ry,~fo~. .the closerapproachlof-reac!ant.t~ the-~o&m~. t.umielling.dist~nti was .&timated.in.:ref: 13 to b:e &~‘smah’a.$:&t0;8.._~-~~~l~.~~.-T;oml&~&$ &tiop_gt -kuch a- !&all. q&&oh w&d prob.ably,bjz be$ond th& ac&ra&df:&r cal&ul~tilins &d h&z& x&.&nit the last term-
_.:_ ‘_:I.. :_-, 1 -. .:_;-..:y:_ .--:_: .‘_r.:.-,’:.; I-‘_ 1’.I. _,.- .r :. _;,_-,‘: .:_ ‘1 .: :I : [email protected] :tij@i$~$ ~<& of &I are.- experimentally -accessible. :Tht%y.were atimat@ :-f@m-:Eigen’s:d&:[-171 $l.,2;3RT:hpK, .the ,_Ap’K.va+k: c&responding to E f=. 12 !k.I ti$Y’, :nti; truk’ free: &ergy A& differ.+ froni the Yvalue A$ by. the in *G;:(&j.
.difference of- -_ the- &&,&for’ the initial reactants -a+ the &d&ts.The small nonCoulombic. contiibut~onk for both of. theni are $ractically equal; so- this component can& -out.,When the.reatitant,are ~uncharg&l there.is- an- electrostatic.attraction of the charged product& Th&.energy of -this-attia$ion .E,-gti depends .Gb&ntially on the. distanck between two j0n.s.’We have,‘&ried out .the Cal&la&on. in two versions, takingI~c~U,-=52-andi6~-mo~~!;..~e& va&s correspon~~to Iargk (-0.45 MI) and small (-O.I>..mn) i&k radii. (the-.&oice of .tlieseradii is substantiated jater). The figure%in Tab!e .l :,ark for: the f+It -version, for the second one -they are --6.7 W mol-’ -higher: I&j..cak@ated’ ES ark in the range ~65.3-87.~ kJ mol-~‘; the- more pro&&l&
~tefi~~is~:‘7’0-80
kJ mol-.‘_. ..i. :. -:
.._
.-
-‘-
@H.&d’@@ groups. are~cl&Sin’size~ tdthe I-&O? ion, so that in their &se rkorganiiation~ener~es sinkjrtd th&calcuiated above would be expected. Iti fact, however;-’ ih& @culatiofi :auqording::to e&i- -,(2a)’‘]proved..td .differ :greatly from The
experiment, ex&ding it by.~a-factor of almost :ihiee. .It 1i<: important to em@hasize that. t& &es&n hereis notonly of.a verylarge quantitative-difference but-also of.a iualitative-one: E;for~aih&ogeneous rea+on @ovea to be‘pra&ally equzkto~that~ for- a heterogeneous rektjon, though it followa from. the thkory that it- should be .’ _
_
14
twice as large. Consequently, the situation for proton-transfer reactions .differs significantly from that for simple redox reactions, for which the theory agrees with experiment, not only qualitatively but also semiquantitatively. Unlike simple electron-transfer reactions, proton transfer includes the sub-barrier. transition of the heavy particle-the proton. For this reason, a significant mutual approach of the partners is necessary to facilitate proton tunnehing. Estimates of optimum tunnelling distances were given in ref. 13. For proton donor discharge on metals. the tunnelling distance is estimated to be - 0.04 nm. It is clear that with the H,O i ion radius 0.15 nm and the length of the O-H bond 0.1 nm, as assumed by us, taking account of the mutual approach of reactants will not significantly affect the charge-transfer distance and hence the reorganization energy. Therefore, this factor should not markedly influence the estimate of Es for an electrode reaction. Regarding the homogeneous reactions of OH acids with 0- or N-bases, as was shown in ref. 13, for these partners bound by hydrogen bonds and therefore
significant!y drawn together, the optimum tunnelling distance differs little from the equilibrium distance, so that this factor cannot cause a marked decrease of E;. Another difference between electrons and proton-transfer reactions is that in the former the interaction with the medium of fairly large complex ions not forming hydrogen bonds can. to a good approximation, be described in terms of a simple dielectric formalism. In proton-transfer reactions, however, the charge is centred on a particle or group with a rather small radius, which, moreover, can form strong hydrogen bonds with solvent_ A small ion radius can lead to a decrease of the reorganization energy, both due to dielectric saturation near the ion and because of the influence of the spatial dispersion of the dielectric constant. The hydrogen bonds of ion with the nearest water molecules act in the same direction. Let us consider, for example, a scheme of proton transfer from a H,O + ion to acceptor A (Fig.4). In accordance with the current concepts. the H,O + ion forms three strong hydrogen bonds with the water molecules, so that a H,O,i particle exists in solution. The fourth nearest to the H,O’ water molecule (ion surroundings are terahedral) does not form a hydrogen bond with it, undergoing orientation only under the action of the ion-dipole
“..% ‘=‘H
3’ ‘M_
““H,o*-- “0 .H’ :-
,.-- ’ H“LA .H'
:
A.-=
%...
__H'
-_ --H,o/-A :
;0.-.
‘H
‘0
..H’
.H’ _--
l ..,, ‘A
(a)
(b)
Fig. 4. Scheme of hydrogen bonds around the H,O + ion: (a) before proton transrer to acceptor A: (b)
nfter proton transfer.
_-
~~
-,..
.-.
~15
i&era&ion (Fig. 4a). Following proton transfer- to acceptor (it is achieved by a shift -of two’protons indicated b;j, arrows), the orientation of three water mo&ules b&&l with the .-central molecule by. hydrogen bonds- remains ‘u&hanged, practically. at equilibrium, i.e. there is ‘no reorganization of this-part.of the first salvation sphere of. F&If. Only.the fourth water molecule is in a non-optimum orientation: it-would be more advantageous for it to .form a hydrogen bond (Fig. 4b). However, it should be taken into account, that in liquid water far from all possible hydrogen bonds are .realized (at room temperature about half..of them), so that an-adjacent position.of two H,O molecules not : bound by a. hydrogen bond .and belongiug to differentclusters is a. rule rather thti an exception. On the whole, it may be concluded that during proton transfer from the H30 c ion (and other species, to a large degree included in the net of hydrogen bonds of water) the first solvation sphere undergoes reorganization only slightly. Proceeding from this reasoning, let us take as the ion radius that of the first
solvation sphere a = 0.45 nm (cI~,~ + +2anZo). Calculation with such a radius gives ES 7 67 kJ mol-’ (not.allowing for E dispersion, 84 kJ mol-‘), with correction for the excluded volume, ES = 63 kJ mol - ‘. Thus, our model yields a result in a quite good agreement with experiment, and this fact is a good argument in favour of the popular concept that proton transfer follows a bridge mechanism with the participation of intermediate water molecules. It should- be._ noted that estimation of the. size of HsO+. from the rate of its diffusion-limited reactions also leads to the conclusion that the reacting particle incorporates the first solvation shell. The calculation has been carried out-for reactions of the H30f ion with a species of similar structure and shape.. In fact, the experimental data refer to qeries of organic molecules of the type RX and RXH (with the charge + 1, 0 and - I), for which a low polarity radical R occupies a part of the nearest and the second-nearest salvation sphere. This fact must decrease the reorganization energy somewhat, and with allowance for this effect, our model gives a somewhat too low value of ES. Precisely this result was to be expected, since in calculation-the first solvation sphere is taken to be completely rigid and unreorganizable; whereas some slight reorganization must occur (see above, the discussion -on the behaviour of ‘the fourth water molecule) It was
and this makes a certain contribution to ES. ; demonstrated above that .calculation under the assumption that a = 0.15 nm.gives reasonable agreement with experiment for an electrode reaction, whereas for a homogerieous
reaction it is necessary to take into account the rigidity of- the solvation sh_ell.Quite n&mally, a question arises why this factor has no effect in the case of the H,O f _idn discharge at- the electrode. When a hydroxonium ion undergoes discharge it is not bound with the-electrode by a- system of such strong.hydrogen bonds as exists in-the water bulk between I-I,0 + _a+ _+ei proton: acceptor. Moreover, during discharge at an el&trode, -the entre of gravity of a pdsitive chargemust be l&lized in a’ position most f&o&able from. .me .viewpoint of elect@atic& .i.e. .closer.:‘to :_the. ~ele+ode~ .As.sho&‘_for example,. tin. Fig. 5a. One would. hardly expect. that. after..proton- detachment iand.
:
A
I
H-0
R’~----“~.
'"-_./
“\ H
..:
“,o;__I”No
.-:. :
.-.:
,.;$j. _.-:._:.;;.I:._::~-._:;;
.-
.I.
,_-.
-.,
_.-
~.
H’
6
: .::. : : l_.,‘,.‘;_._’
:_ 1. .:.
t .H ’ --‘-H ~0
‘H
a
~‘..
:..
.;.
:_;
._ ..’ -
Fig. 5. Scheme of hydrogen bonds of the H,O •Fion near the electrode: (a) ‘before pr&ondischarge
&d
transfer; (b) after proton transfer. -_-.
adsorbed hydrogen formation, the water molecule formed ‘from the H,O +-. ion_.. should retain its former orientation, since the interaction kith the electrode -a-nd primarily with the double-layer field till significantly affect its equilibrium-~oriehtation. But the change of the orientation of this molecule will result in reorientation of its neighbours (one of such conceivable schemes is shown on .Fig. 5b). ;All. &is. means that near the electrode the H,OC ion has a practically reorganizable first s&atiori sphere. Such factors as a change of the liquid water structure near the interface also
act in this direction. It is expedient to mention here that- the ion penetration into the first water monolayer with a low dielectric permittivity would cause some_ikkase of the reorganization energy [ 181. This penetration also means some stibkantiai d&tiu& tion of the first solvation shell. Thus, the investigation of the reorganization energies of .homogeneous and heterogeneous proton-transfer reactions made in this paper points to the :very important role of the first solvation sphere structure and to the necessity of.further theoretical analysis of this problem at the microscopic level. REFERENCES
1 A.N. Frumkin. Z. Phys. Chem., A160 (1932) 116. 2 R.R. Dogonadze and A.hM. Kuznetsov, Fiicheskaya its), Itogi Nauki. Vol. 2. VINITI, Moscow, 1973. 3 R.A. Marcus, J. Chem. Phys., 24 (1956) 966.
Khirniya Kinetika, (Physical Chemistry, $+et-
4 R. Saxton. Proe. Roy. Sot.. A213 (1552) 473. 5 M.A. Vorotyntsev, R.R. Dogonaclze and A.M; Kuzrtetsov, Dokl. Akad.~Nauk; S.S.S.R., 155.(157C) ; : .. : 1135. .:, 6 R.A. Marcus. Ann. Rev. Phys. Chem., 15 (1964) 155. 7 L.I. Krishtalik, Electrokhimiya, 15 (1975) 1505; Sovremennye aspekty elektro~mic~esr;oy.kineiiki: Materials IV Frumkinskikh chteniy (Modem.Aspects of Electrochemical Kine&s_MatetjaIs.of i\! _.-.. Frumkin Lectures). Metsniereba. Tbilisi, 1980, p. 33. _. s L-1. Krishtalik, Elektrokhimiya. 5 (1969) 3. -. 9 Selected Values of Chemical lIxrmodynamic Properties, Natidnal .B&eau of Standards, CipqlarSCl$~’ : Part I, 1961. ._ : : :_i _ _..I.’ ,.. 10 L.I. Krishtalik. Elektrokhimiya. 2 (1566) 1176. 11 L-1. Krishtalik, Elektrodnye reaktsii. Mekh&i&n elcmentornogo_akta.(Electrode-~~~tio~s~ ,j&$$& _:. nism of th& Elementary Act). Nat&& MO&W, i679;.Ch. IV; -.. ‘:- ?’ ” -, -.,y,. I ,: .. 12 M.A. Vorotyntsev, VS. Krylov and L.1. Krishtabk. ~~ektrokhimiy~. 15 (1979) 7&k -‘. -. .;, :..’ I:“-? -il’. .. .-- :._ ~.
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~:.:14.:~En~:-.Etekt~y~6 (1970)!l|~65;-l.iEl~'ctro~/alLChem.i"35.(1972) 15"/.: !:-:...~: -../• i~/-l~'~Eige~-/~gew.~Chem~,75.(1963)'~489.
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