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Theory of proton-transfer reactions. The influence of adiabaticity on free energy relations
._--
--
.
..
Received 5 July 1983; in final form 13 February 1984
The influence of_adiabaticity on free energy relations of proton-transfer reactions is investigated within the framework of the quantum-statistical mechanical rate the&y_ It.is found that there_may be both for the Rrcmsted plot and for the free energy dependence of -the kinetic isotope -effect qualitative differences between the non-adiabatic and the adiabatic case in the exergonic region.
f+&,(R)
1. Introduction_Rate:equilibrium relationships have fascinated chemists over the years. They were often proposed in a phenomenological way and should thus; for a more detailed understanding, be derived from a rate theory. We feel’ that the quantum-statistical mechanical rate: theory is especially -suitable for such a task, the reason being that the free energy of reaction is explicitly involved in the rate expression *. The corresponding theory for proton-transfer reactions was developed in ref. [2]; for. a review of the topic see-ref. 131..According to ref_:[2] and applying the model of a linear complex, see fig.ll; the rate constant for a proton-transfer reaction -of the type (charges omitted) AHtB&H+A
ill
is given by (L T H, D)
k~.=~~.~:exp[-(E:“-EpL,/k.TI.n: RT exp[_-_U(R)j&$] y$&)1.(2j ..: .._ . - 1): _. .:_ ,- .:_ f& g&&;f &&& is~,dt:,b~~i~_ei~~~~o~~~ in + x J-dR -.
*-tie
rate expression of ~&&i&ion state tbe&y.There are hotieverattempts to hrtrodu& &-alj;?i&l~ relationsbet&en thd :free -. _ _ I -:.1 .energg& of a&.&ion and re&tion illi -.
_,
C‘_. p- :-- ->_::I_ __‘
I-.
:- -.
may. be expressed as
X
exp [
1
C 2ny,‘;,( R‘) k=Q
xexp(-[E,+AF+(E;‘-E;L) _(E:‘L_EOL)J2/4~~ksT). .
(3)
In eq. (3) it was tacitly assumed that the interaction .between- the reacting .moiecules is equal both i.n _the_i&ial.and fihal states.. For-o,,, @_‘,I?, and U(R). the iufluence .of .iibrational ‘excitation and except for U(R) also-.the R dependence is negle&ed_ Further approximations inherent in eqs. (2) and (3) are described’in refs. [2,3].- Z, is’ the vibrational pkitibn fun&on,- JZiTL-and ErL are the vibrational &iefgie& iii the t and f&l _-. initial states, m ~arid n -the vibratio.nal qua&n ;mumbers. -es is the solvent reorgamzation~ energy; 4 E t+ free _.i. _..’ -
I80
J. Siihnel. K Gustau / Free energy relations of proron - transfer reactions
energy of reaction, (I( R j the interaction potential between the reactants. mrrr the effective medium frequency, u’ the tunneiling factor for the qkanturn part of the medium poiarization. Xmk( R) is the transmission coefficient given by XL(R)
= {
1 - exp[ -2rYk,(R)]}
x (1 -$ The Landau-Zener pressed as
exp[-2~y~t;,,(R)]}-‘. parameter
(4)
YA,,( R) may be ex-
(5)
I’, is the electron the Frank-Condon the R dependence
resonance integral and S,_!_!,,,(R) factor. As indicated in eq. (5) of Vc is neglected. For
2,iry,“, > 1
(6)
the transmission coefficient is unity transition). On the other hand, for
(adiabatic
zrry,L;, < 1
(7)
the transmission L XP," =
coefficient
is given by
L
4CY”,,
(8)
(non-adiabatic transitions). The integration in eq. (2) can be avoided. as the integrand displays a sharp maximum at some value Rf;,, (transfer distance). Thus the values of the integrals in eq. (2) may be replaced by the values of the integrands at R!kk,=Z;‘CC x exi[
exp[-(ErL-EFL)/kDT] : U( Rf;,,)/k,T]
x Y?ti(R!in)e7n;,(R~wJThe reaction
(9)
volume IJ~, may be written as
uL mn = 4rr( Rf;,,)?AR.
(10)
It should be noted that eqs. (2), (3) and (9) rest on the assumption of independent transitions be-
tween all possible pairs of vibrational states. This assumption is no longer justified, if the splitting of the potential curves (A&,) near the intersection point and the energy of the proton-stretching vibration are of comparable .magnitkde [3]. According to refs. [2] and [4] A&,, can be approximately calculated as A E,,, = 2V,S~,<‘. For & =45 pm and FH= 3000 cm-’ (S,“)? has the value of 1.07 x lo-‘. With the maximum value V, = 2000 cm- * used in our paper this gives AE,,, = 42.8 cm-‘. Thus we feel that the assumption of independent vibrational transitions can be applied for V, < 2000 cm-‘, see also ref. [4]_ The free energy relations of proton-transfer rate constants (Bransted plots) were studied from the point of view of quantum-statistical mechanical rate theory in ref. [5] for ground state reactions and in ref. [6] fdr excited electronic states. It was shown that this theory can reproduce most of the experimental findings. The adiabaticity effect was not investigated in refs. [5,6]. In ref. [4] the influence of adiabaticity and anharmonicity on the dependence of the kinetic isotope effect (KIE) on the free energy of the reaction was studied assuming transfer distances R,,,, independent of the isotope. It was found that the inclusion of vibration anharmonicity and proton transition adiabaticity has no significant effect on the shape of the free energy relations for the KIE. Recently the intermolecular interaction potential was explicitly taken into account using model potentials of the parabolic and Lennard-Jones [7,8] and of the Morse type [9]. It could be shown that the usually applied assumption of isotope-independent transfer distances is not justified, if realistic interaction potentials are used. However, in refs. [7-91 free energy relations were not studied. Now the question is, do the results of ref. [4] have to be modified if the assumption of equal transfer distances is relaxed. Thus the objective of this paper is to investigate the influence of adiabaticity on the free energy relations of the proton-transfer rate constant (Bransted plot) and of the KIE (kH/kD). It will be shown that the shape of the free energy relations may be qualitatively different for non-adiabatic and adiabatic transitions in the exergonic region.
)
AL
_- -_ _-
_
I_
.
-The- transfer distan’&zs Rk,(ri,,)can be de-termined by the first derivative of- the-integrand in eq. (2) with-respect to -R or i-_ Neglecting the~slight dependence of the transfer distance bn vibrational excitation only Rh or & has to be calculated. The corresponding expression reads (_& = 27r-j&) exp[-&(r)i (2 exp[&(r)] du(r) -k,Tdr
-l}{exp[_&(~)] + -= 2 r+k,
In the non-adiabatic -d&,(r) x&i,
-d&g(;)
dr
dr
-1)
0
(11)
-
limit eq. (11) simplifies to
2 -- dU(r)-= k,Tdr+rtk,
0 .-
(12)
(Rh=&+k Q; k, = rAf, + ~BH; see fig. 1). For a practical application of eqs. (11) and (12) expressions for the r-dependent terms U and S&, have to be introduced. Unfortunately, in most cases it is not possible to arrive at an analytical relation for &_ Only for harmonic Franck-Condon factors, a parabolic interaction potential and a non-adiabatic reaction is an analytical expression obtained, see eq. (9) of ref. [7]. Thus eq. (10) has to be solved numerically_ All the equations given in refs. [7,8] for harmonic and Morse vibrational wavefunctions, and parabolic and Lennard-Jones interaction potentials in the non-adiabatic limit are special cases of eq. (12) and can thus be derived from this equation. In this paper the Franck-Condon factors are calculated in the harmonic approximation llO]_ For modelling the intermolecular interaction a Lennard-Jones potential is applied.
03) In all calculations the Lennard-Jones parameters c = 30 cm-’ and u = 300 pm were used. They should be viewed & effective values as the quantitative description of the interaction of two large molecules in a. polar medium by a Lennard:Jones potential &ur only be of a very approximate-nature. Fortunately, the ~shapes of -the free energy relations are only slightly dependent on the values
~of- the -potential p&ra~meters.:On- the-other- hand; I the-individual -~~_ues.o~:rate-coastants .ork&iic isotope effects-are very sensitive to-the parameters of- the. LentiardJones potential (especially to_:&), .. After having ‘determined the transfer distances r&j or.Rb the -rate:constants can be calculated using eqs. (3)-(5),.(9) and (10). As far as only relative rate constants-and. no..solvent isotope. effects are calculated exp( -- a’) and AR can be assumed to be 1 and 1 pm respectively_: Both terms are assumed to be independent of the isotope, see ref. [2]. This means that they cancel out in the equation for the determination of -the. KIE. All calculations were carried out for T = 300 K and aerr = 10” s-l_ Th e sum of the bond distances <+u and rn.u, see fig. 1. was chosen as kc = 204 pm. Finally, it is assumed in all cases that- the proton-stretching frequencies in the initial and final states are equal. The value of 2, is equal to unity for T= 300 K: 3. Results and discussion In this paper rate-equilibrium relationships of the rate constant k and of the kinetic isotope effect k,,/k, are investigated_ Bronsted and Pedersen proposed the following relation logk=culogK+C
04)
(r is the Bronsted coefficient and C is a constant) [Ill. It states that there is a linear correlation between the logarithms of the rate constants k and equilibrium constants K for a series of closely related proton transfer reactions. However, Bronsted himself has already pointed out that a may vary between 1 and 0 in passing from strongly endergonic to strongly exergonic reactions_ This would result in a curved Brensted plot provided a sufficiently large range of pIC values is- studied. There are a lot of experimental data which confirm this fact. Nevertheless linear correlations over extended pK ranges are also known. A compilation of the corresponding examples isgiven in ref; [12]. Hence this -subject is still being .debated.- More: over, results_on react&series *th a values‘larger than unity- or’ negative were -reported, for a review of -the older literature .and a new example see ref. [13]. .-
IS2
J. SiiJmel.K
Gustao / Free energy reialions of proron -framfer
Westheimer was the first who put forward the idea that the kinetic isotope effect (KIE) should exhibit a maximum for a symmetrical transition state [14]. Using the Hammond postulate and the Leffler principle, both of them interrelating the free energy of activation with the free energy of reaction, it was concluded that the KIE should display a maximum for thermoneutral reactions. There is now experimental evidence for this phenomenon [15]. However. also in this case there are examples of nearly constant X-,/X-, values over larger pK ranges [ 161.
3. I. The Brornted relatiotz We have performed a large number of calculations on free energy relations varying the parameters CH (proton-stretching wavenumber used for the calculation of Z, and Er”. EFL). Es and V,. Typical results are presented in fig. 2. Let us first consider the non-adiabatic limit (Y, = 10 cm-‘). In all cases the expected curvature characterized by a decreasing Bronsted coefficient in passing
Fig. 2. Calculated Bransted plots (solid he: V, = 10 cm-‘. cm-‘. a = 300 pm. 5, = 3OGOcm-‘, P, = 2-‘fli;,).
reactions
from the endergonic to the exergonic -region is. found. With an increasing vakk of the solvent reorganization energy. ES the- B&&d plots_ become more and more flattened_ One should notice here that the range ‘under study (-10000 cm-!
dashed line: V, = 2000 cm-‘;
T=
300 KS rAH + reH = 2~
pm, c = 30
(54.3) 61.7 (53.1) 60.7 i52.3) 60.3 (51.7)
3000 6000 !Noo 3ooo
ICOO 3Ooa 6000 9000
10
1500
1000 3000 6000 9000
3000
1000 3000 6000 9000
_
(0.7296) 0.8032 (0.6874) 0.7839 (0.6631) 0.7577 (0.6616)
46.4 (39.7) 45.4 (39.1) 44.8 (38.6) 44.6
(0.5305) 0.5937 (0.4551) 0.5611 (0.4260) 0.5231
(38.3)
(0.4100)
,:s, 46.4 (41.9) 46.5 (41.9) 46.5 (41.9)
0.6368
~.
(q.9998) l.oooo (0.9993) 0.9999 (0.9987) 0.9998 (0.9984)
(l.ooOo) 1_oocm (1.0000)
-l.o060.‘l.O$Kl
(1.oooo)
(l.ooooj
0.0039 (O.OillS) 0.0022 (0.0010) 0.0015 (0.0007) 0.0012 (0.0006)
0.0186 (0.0103) 0.0108 (0.0059) 0.0075 (0.0042) 0.0061
0.0441 (0.0283) 0.0258 (0.0165) 0.0181 (0.0117) O.Ol.@
(0.0034)
(0.00?6)
37.8 (33.7) 37.8
0.0008 (0.0003) .0.0004
0.0051 (0.0026) 0.0029
0.0163 (0$098) o.oq94
(33.7) 37.8 (33.7) 37.8
(O.ObO2) 0.0003 (0.0001) 0.0002
(O.OOlSj 0.0021 (0.0010) 0.0017
(0.0054) 0.0067 0.0038) 0.0054
(33.7)
(0.0001)
(0.0008)
(0.0031)
ing to proton transfer_ to a certain base family. However, in order to prove this suggestion- the different a! vahres and especially the monotonous decrease in passing- from the endergonic to the exergonic region have to be explained. Finally it has to be,mentioned that the flattening out of the Bronsted plot induced:by diffusion is completely different from the same effect being a result of the Marcus or quantum-statistical rate theory:for the exergonic region. .One can expect to find this latter effect far below the diffusion~limit. .Alf the result_!:presented so far .can also be
-:
(l.OOOOj-
0.99ss (0.9940) 0.9968 (0.9827) 0.9941 (0.9740) 0.9897 (0.9678)
-
-_-
,::E,
..
1.0000 (l.ooooj
(K-E!, 1.0000
-
-_
obtained within the framework of the Marcus theory. On- the other hand, the following results are typical for the quantu&statistical mechanic_$ rate theory: For small values of 4; an @Hating strut- _ ture is found. This effect-is well-known from other_investigatior?s. on electron-transfer . /IS] and proton-transfer reactions [19]. .-q:’ trgsfer distan&es arc. nearly. iridependent of .&see Jable 1. The effekt of_va$ng CH iS not shown in,fig_ 2. For ti, _T,1SOt) cm-’ larger ~transfer distances are :ob: mined as_ compared to jiH-~-3.00Q cm-‘,, table 1.: This leads to ‘a decrease- of -the rate _co$t_%nts by : -.
_
184
J. Siiinel. K Gustav J Free enerm relatiolzs o/proton -transfer reacrions
about two orders of magnitude_ The shape of the Bronsted plot however is not changed. A very interesting effect can be observed if the value of the electron resonance integral is increased up to V, = 2000 cm-‘. Whereas in the endergonic region the shape of the plots for V, = 2000 cm-’ and V, = 10 cm-’ is very similar, the rate constants in the exergonic region reach a maximum at -E, and decrease finally for V, = 2000 cm-‘. An analogous phenomenon is predicted by the Marcus theory for electron-transfer reactions 1201 but not for proton-transfer reactions [21.22]. This result can be understood if one bears in mind that by increasing the electron resonance integral up to V, = 2000 cm-’ most of rhe transitions between the individual vibrational levels will become adiabatic (.X$, = 1). From table 1 it can be seen that all values of the transmission coefficient are between 0.5 and 1. The different shape of the Bronsted plots for V, = 10 cm-’ and for r/,= 2000 cm-’ can be explained by analyzing the a E dependence of the rate expression. eqs. (3) and (9). The last term of eq. (3) gives “subbands“ ‘with maxima at AE=
-E,-(E;‘L-E:)L-)+(En’L-E~L)_
(15)
Provided that. the other terms being dependent on vibrational excitation are for the moment neglected. all these maxima have equal “intensity”. The only terms which can affect the probability of transitions between individual vibrational states and thus determine the shape of the Bronsted plot are. within rhe approximations adopted here. the Bohzmann term. the transmission coefficient and the term exp(-I3;,:2ay,f;)). An analysis of our calculations has shown that in the endergonic region only m ---, 0 vibrational transitions and in the exergonic region only 0 + tl transitions have to be considered_ In the endergonic region the increase of the rate constant in passing from large to small hE values is mainly governed by the increase in the Boltzmann term. The transmission coefficient remains constant in the adiabatic limit or decreases slightly in the non-adiabatic case and the term exp( -Cz:A2ny,,,,) disappears for n = 0.
In the exergonic region we come across a completely different situation. Now the Boltzmann term remains- constant and. the shape ok- the Bronsted plot is detekined by the two other terms. It seems worth noting, that this result should be essential for an analysis of the temperature dependence of free energy relations. The values of the two’ terms mentioned are dependent on adiabaticity. In the non-adiabatic limit only the transmission coefficient has to be taken into account. Its dependence on vibrational excitation is illustrated by the following example. For CH = 3000 cm-’ . r,, = 37.8 pm, Es = 1000 cm-’ and V, = 10 cm-’ it increases from .X$j = 0.0008 to Jib,H = 0.0730 and decreases then the value X& being 0.0079. The transitions m = 0 --, n = 6 and m = 0 + tz = 12 correspond to ?SE values of - 18 000 and - 36000 cm-‘. Thus in the region under study (-10000 cm-’ chE< 10000 cm-‘) the ultimate decrease is not observed_ On the other hand, for V, = 2000 cm-’ first the transmission coefficient increases up to unity and remains then constant in passing to higher vibrational states. But the Landau-Zener parameter increases further. resulting in the reduction of the rate constant by the term exp(-C;1::2?;y,f;,). In this way the decrease in the rate constant in the exergonic region for adiabatic transitions can be explained_ For smaller proton-stretching frequencies this behaviour is more pronounced as a larger number of higher excited vibrational states contributes to the rate constant in the same h E region. As the adiabatic Bronsted plot has its maximum at a E = -Es the decrease of the rate constant occurs for systems with higher Es values in the highly exergonic region. not so easy amenable to experimental studies. In other words. for higher Es values there are to be expected no or only minor differences in the hE region usually under study, see fig. 2. Upon examination of the experimental data it turns out that there is at present no indication of an ultimate decrease in the rate constkt in the highly exergonic region. Because of the difficulties in obtaining reliable data in this region (diffusion control. fast reactions) it. seems however unjustified to use this as an argument against adiabaticity in proton-transfer reactions. Furthermore it should be pointed out that a few of the possible explana-
_. J. SiihneC-K~G~+~tau~/~F~ek&v~
tions suggested.; for.. the. -absence. of-.a- decka+~g electron-tiansfer rate yconstant- in -the Marctis- invetted r&&tiay also apply to proton iliansftif: -_ 3.2.. The free energy dependence of rhe kinetic -. , isorope effect. The results of the calculations are shown in fig. 3. For Y, = 10. cm” the expected behaviour is observed. Small values of ES result in an oscilIatory structure of the free energy plot. The curve is symmetric about the maximum at AE = 0, its curvature being dependent on ~5,. In passing to V, = 2000 cm-’ there are mainly two effects. The KIE at A,?Z= 0 is reduced for the larger V, value and the. whole curve is flattened. The plot of k,/k, versus AE is n9 longer symmetric about AE=O. These effects can be explained in the following way. In the vicinity of AE= 0 mainly the O-O transition contributes to the rate constant. As can be seen from table 1 the ratio *z/X$ is larger for the smaller V, value. On the other hand the term exp{-[U(Rg)U(Rk)]/k,T) is for V,= IO cm-’ smaller than for V, = 2000 cm-’ due to
-
.-
~~~~fi~~‘~~p~~tron,t-~~~~~i~~~ : -‘.
-_ I. 1 .- ;...- _ -. _ .. ^
I
...a
=6ooo A&i9
values the transfer distances had to bei+&ased
to
calculate the correct value of the KIE. This is completely analogous to the result obtained ifi this paper. It appears that the marked asymmetry of the (kH/kD) - AE plot is closely connected with the fact that different transfer distance&for I-I and D were used, see fig. 4_ For V, = 10 cm-’ both for equal and different transfer distances a symmetric plot is calculated_ In the case of V, = 2000 cm-’ a slight asymmetry can be seen for equal transfer distances, but this asymmetry becomes very pronounced for different transfer distances. As equal transfer distances were assumed in ref. [4] no marked differences between the adiabatic and the non-adiabatic case could be n&iced.
As in the case of the Brwsted
plot there is a
scarcity of experimental data in the exergonic region. To the best of our knowledge the most
Es- 3000 mi’
6000 AE@d]
E,.9000cnC~
!w kD
. -8000
‘. 185 -_
order to reproduce a given experimental value ofthe isotope effect. It was found that for larger v,
-6000
-8000
-_-
bh_~1; iThe_ ~t~a~smissi~n.-~~~f~i~~~~~~~~tis,6~ .tJie more dronounced eff&t ieid&g-;o theoverafl & crease of- tl&I&E in thk ad&datic c&e. .-In r&f. [4] the transfer &s&es were fitted in
kD
E;6000 cd
2..
the dmr&&a .;li~~~~~“ce.~of_‘:~~~~~r-.~ddbi~~~,ta;- ;
!T!!’
r(H' kD
.. -.. .-
. . . '?O" AEl$r)~
Fig. 3. Cakulated free energy relations of the kinetic isotope effect (the same-data as in fig. 2 were used).
186
J. Sihei,
V,=lO cm-
K. Gustav / Free energy relations of proton - wansfer readions
-R:,,= !w
-6000
Rk
# Rfa
--- R:,
%I
.- Es_ The rate constant decreases for A E- -z JZtix is very weak in the non-adiabatic &nit but rather pronounced in the adiabatic case. As for the kinetic isotope effect we found evidence contrary to the conclusion drawn in ref. [4] that there should be no marked influence of adiabaticity on-the corresponding free energy relations. According to our results the (kH/kD) - AE plot is no longer syrnmetric about AE = 0 in the adiabatic case. The reason for the different results can be found in the assumption of equal transfer distances for H and D made in ret [4] and relaxed in this paper. In ref[7] it was shown that this assumption is not justified for realistic interaction potentials between the reactants_ The qualitative differences of the shapes of the free energy relations may represent a criterion for the adiabaticity of proton-transfer reactions_ Unfortunately there is still a marked scarcity of experimental data in the exergonic region.
*
6000
A E [cc’]
References Fig_ 4. Calculated free energy relations of the kinetic isotope effect for Rk = Rg and R& f Rk (E, = 6000 cm-‘: other parameters
as in fig. 2).
[II J.R. Murdoch and D.E. tMagnoli. J. Am. Chem. Sot. 104 (19SZ) 3792.
PI E.D. German, pronounced
asymmetry
can
be
Found
for the reaction acetate with bases [23]. However. it present it is not possible to decide if from a symmetric curve is within experimental scatter or not. ....
(kH/kD)
-
AE
plot
in
the
of nitroethylis felt that at this deviation the limits of
4. Conclusions Systematic investigations on the free energy relations of the proton-transfer rate constant (Brcmsted plot) and of the kinetic isotope effect were performed utilizing the quantum-statistical mechanical rate theory. It was found that the Bronsted relation displays a qualitatively different behaviour for small and large values of the electron resonance integral (non-adiabatic or adiabatic transitions) in the exergonic region (AE c -ES). In the adiabatic limit the maximum rate constant occurs at AE,,,, = -Es. whereas in the non-adiabatic case the maximum of the plot is at AE,, <
A.M. Kuznetsov and R.R. Dogonadze. J. Chem. Sot. Faraday Trans. II 76 (1980) 1128. 131 J. Ulstrup, Charge transfer processes in condensed media (Springer. Berlin. 1979). [41 E.D. German and A.M. Kuznetsov. J. Chem. Sot. Faraday Trans. 177(1951)397. R.R. Dogonadze, E.D. German, A.M. 151 V.G. Levich. Kuznetsov and Yu.1. Kharkats. Electrochim. Acta 15 (1970) 3.53. Chem. 324 (1982) 71. 161 J. Sihnel and K. Gustav. J. P&t. [71 J. Siihnel and K. Gustav. Chem. Phys. 70 (1982) 109. 181 J. Siihnel and K. Gustav, Wiss. Z. Friedrich-SchilIer-IJnivJma. Math.-Naturwiss. Reihe 31 (1952) 1113. A E-D. German and A.M. Kuznetsov. J. Chem. Sot. Faraday Trans. II 77 (1981) 2203. 1101 C. Manneback, Physica 27 (1951) 1001. Bronsted and K. Pedersen, Z. ‘Physik. Chem. [l*l J.N. Stoechiom. Verwandtschaftsl. 108 (1924) 185. [121 EC. Bordwell and D.L. Hughes, J. Org. Chem. 47 (1982) 3224. r131 J.R. Murdoch. J-A. Bryson, D-F_ McMillen man, J. Am. Chrm. Sot. 104 (1982) 600.
and J.I. Bmu-
[I41 F-H. Westheimer. Chem. Rev. 61 (1961) 265. reactions..eds. I151 R-4. More O’FerralI. in: Proton-transfer EF. Caldin and V. Gold. (Chapman and HalLLondon. 1975) p. 201. [161 B.C. Challis and EM. Miller. J. ChemI Sot_ Perkin T&sII (1972) X618.
J_ Siihnet, K Gurtm
/ Free energy reiations ofpratm -transfer recu&nu
1171RF. Be& The proton in chemistry,2nd Ed., (Chapman and HalI, London. 1973). flS] J. Wlsnup and J. Jortner.J. Chem. Phys- 63 (1975) 4358. [19] N. Bruniche-OIsen-andJ_ Us&up. J. Chem Sot- Faraday Trans. I 75 (1979) 205. [2Oj R.A. Mrucus and P. Sides. J. Phys:Chem. 86 (1982) 622.
187
l23] R.A. Marcus, 1. Phys, Cbem. 72 (1968) 891. [221 A-0. Cohen and R-k Marcus..J. Phys; Chem. 72 (1968) 4249. f23] F.G. Bordwell and WI Boyle Jr., J. Am Chem. !Zoc_97 (1975) 3447.
--
.
..
Received 5 July 1983; in final form 13 February 1984
The influence of_adiabaticity on free energy relations of proton-transfer reactions is investigated within the framework of the quantum-statistical mechanical rate the&y_ It.is found that there_may be both for the Rrcmsted plot and for the free energy dependence of -the kinetic isotope -effect qualitative differences between the non-adiabatic and the adiabatic case in the exergonic region.
f+&,(R)
1. Introduction_Rate:equilibrium relationships have fascinated chemists over the years. They were often proposed in a phenomenological way and should thus; for a more detailed understanding, be derived from a rate theory. We feel’ that the quantum-statistical mechanical rate: theory is especially -suitable for such a task, the reason being that the free energy of reaction is explicitly involved in the rate expression *. The corresponding theory for proton-transfer reactions was developed in ref. [2]; for. a review of the topic see-ref. 131..According to ref_:[2] and applying the model of a linear complex, see fig.ll; the rate constant for a proton-transfer reaction -of the type (charges omitted) AHtB&H+A
ill
is given by (L T H, D)
k~.=~~.~:exp[-(E:“-EpL,/k.TI.n: RT exp[_-_U(R)j&$] y$&)1.(2j ..: .._ . - 1): _. .:_ ,- .:_ f& g&&;f &&& is~,dt:,b~~i~_ei~~~~o~~~ in + x J-dR -.
*-tie
rate expression of ~&&i&ion state tbe&y.There are hotieverattempts to hrtrodu& &-alj;?i&l~ relationsbet&en thd :free -. _ _ I -:.1 .energg& of a&.&ion and re&tion illi -.
_,
C‘_. p- :-- ->_::I_ __‘
I-.
:- -.
may. be expressed as
X
exp [
1
C 2ny,‘;,( R‘) k=Q
xexp(-[E,+AF+(E;‘-E;L) _(E:‘L_EOL)J2/4~~ksT). .
(3)
In eq. (3) it was tacitly assumed that the interaction .between- the reacting .moiecules is equal both i.n _the_i&ial.and fihal states.. For-o,,, @_‘,I?, and U(R). the iufluence .of .iibrational ‘excitation and except for U(R) also-.the R dependence is negle&ed_ Further approximations inherent in eqs. (2) and (3) are described’in refs. [2,3].- Z, is’ the vibrational pkitibn fun&on,- JZiTL-and ErL are the vibrational &iefgie& iii the t and f&l _-. initial states, m ~arid n -the vibratio.nal qua&n ;mumbers. -es is the solvent reorgamzation~ energy; 4 E t+ free _.i. _..’ -
I80
J. Siihnel. K Gustau / Free energy relations of proron - transfer reactions
energy of reaction, (I( R j the interaction potential between the reactants. mrrr the effective medium frequency, u’ the tunneiling factor for the qkanturn part of the medium poiarization. Xmk( R) is the transmission coefficient given by XL(R)
= {
1 - exp[ -2rYk,(R)]}
x (1 -$ The Landau-Zener pressed as
exp[-2~y~t;,,(R)]}-‘. parameter
(4)
YA,,( R) may be ex-
(5)
I’, is the electron the Frank-Condon the R dependence
resonance integral and S,_!_!,,,(R) factor. As indicated in eq. (5) of Vc is neglected. For
2,iry,“, > 1
(6)
the transmission coefficient is unity transition). On the other hand, for
(adiabatic
zrry,L;, < 1
(7)
the transmission L XP," =
coefficient
is given by
L
4CY”,,
(8)
(non-adiabatic transitions). The integration in eq. (2) can be avoided. as the integrand displays a sharp maximum at some value Rf;,, (transfer distance). Thus the values of the integrals in eq. (2) may be replaced by the values of the integrands at R!kk,=Z;‘CC x exi[
exp[-(ErL-EFL)/kDT] : U( Rf;,,)/k,T]
x Y?ti(R!in)e7n;,(R~wJThe reaction
(9)
volume IJ~, may be written as
uL mn = 4rr( Rf;,,)?AR.
(10)
It should be noted that eqs. (2), (3) and (9) rest on the assumption of independent transitions be-
tween all possible pairs of vibrational states. This assumption is no longer justified, if the splitting of the potential curves (A&,) near the intersection point and the energy of the proton-stretching vibration are of comparable .magnitkde [3]. According to refs. [2] and [4] A&,, can be approximately calculated as A E,,, = 2V,S~,<‘. For & =45 pm and FH= 3000 cm-’ (S,“)? has the value of 1.07 x lo-‘. With the maximum value V, = 2000 cm- * used in our paper this gives AE,,, = 42.8 cm-‘. Thus we feel that the assumption of independent vibrational transitions can be applied for V, < 2000 cm-‘, see also ref. [4]_ The free energy relations of proton-transfer rate constants (Bransted plots) were studied from the point of view of quantum-statistical mechanical rate theory in ref. [5] for ground state reactions and in ref. [6] fdr excited electronic states. It was shown that this theory can reproduce most of the experimental findings. The adiabaticity effect was not investigated in refs. [5,6]. In ref. [4] the influence of adiabaticity and anharmonicity on the dependence of the kinetic isotope effect (KIE) on the free energy of the reaction was studied assuming transfer distances R,,,, independent of the isotope. It was found that the inclusion of vibration anharmonicity and proton transition adiabaticity has no significant effect on the shape of the free energy relations for the KIE. Recently the intermolecular interaction potential was explicitly taken into account using model potentials of the parabolic and Lennard-Jones [7,8] and of the Morse type [9]. It could be shown that the usually applied assumption of isotope-independent transfer distances is not justified, if realistic interaction potentials are used. However, in refs. [7-91 free energy relations were not studied. Now the question is, do the results of ref. [4] have to be modified if the assumption of equal transfer distances is relaxed. Thus the objective of this paper is to investigate the influence of adiabaticity on the free energy relations of the proton-transfer rate constant (Bransted plot) and of the KIE (kH/kD). It will be shown that the shape of the free energy relations may be qualitatively different for non-adiabatic and adiabatic transitions in the exergonic region.
)
AL
_- -_ _-
_
I_
.
-The- transfer distan’&zs Rk,(ri,,)can be de-termined by the first derivative of- the-integrand in eq. (2) with-respect to -R or i-_ Neglecting the~slight dependence of the transfer distance bn vibrational excitation only Rh or & has to be calculated. The corresponding expression reads (_& = 27r-j&) exp[-&(r)i (2 exp[&(r)] du(r) -k,Tdr
-l}{exp[_&(~)] + -= 2 r+k,
In the non-adiabatic -d&,(r) x&i,
-d&g(;)
dr
dr
-1)
0
(11)
-
limit eq. (11) simplifies to
2 -- dU(r)-= k,Tdr+rtk,
0 .-
(12)
(Rh=&+k Q; k, = rAf, + ~BH; see fig. 1). For a practical application of eqs. (11) and (12) expressions for the r-dependent terms U and S&, have to be introduced. Unfortunately, in most cases it is not possible to arrive at an analytical relation for &_ Only for harmonic Franck-Condon factors, a parabolic interaction potential and a non-adiabatic reaction is an analytical expression obtained, see eq. (9) of ref. [7]. Thus eq. (10) has to be solved numerically_ All the equations given in refs. [7,8] for harmonic and Morse vibrational wavefunctions, and parabolic and Lennard-Jones interaction potentials in the non-adiabatic limit are special cases of eq. (12) and can thus be derived from this equation. In this paper the Franck-Condon factors are calculated in the harmonic approximation llO]_ For modelling the intermolecular interaction a Lennard-Jones potential is applied.
03) In all calculations the Lennard-Jones parameters c = 30 cm-’ and u = 300 pm were used. They should be viewed & effective values as the quantitative description of the interaction of two large molecules in a. polar medium by a Lennard:Jones potential &ur only be of a very approximate-nature. Fortunately, the ~shapes of -the free energy relations are only slightly dependent on the values
~of- the -potential p&ra~meters.:On- the-other- hand; I the-individual -~~_ues.o~:rate-coastants .ork&iic isotope effects-are very sensitive to-the parameters of- the. LentiardJones potential (especially to_:&), .. After having ‘determined the transfer distances r&j or.Rb the -rate:constants can be calculated using eqs. (3)-(5),.(9) and (10). As far as only relative rate constants-and. no..solvent isotope. effects are calculated exp( -- a’) and AR can be assumed to be 1 and 1 pm respectively_: Both terms are assumed to be independent of the isotope, see ref. [2]. This means that they cancel out in the equation for the determination of -the. KIE. All calculations were carried out for T = 300 K and aerr = 10” s-l_ Th e sum of the bond distances <+u and rn.u, see fig. 1. was chosen as kc = 204 pm. Finally, it is assumed in all cases that- the proton-stretching frequencies in the initial and final states are equal. The value of 2, is equal to unity for T= 300 K: 3. Results and discussion In this paper rate-equilibrium relationships of the rate constant k and of the kinetic isotope effect k,,/k, are investigated_ Bronsted and Pedersen proposed the following relation logk=culogK+C
04)
(r is the Bronsted coefficient and C is a constant) [Ill. It states that there is a linear correlation between the logarithms of the rate constants k and equilibrium constants K for a series of closely related proton transfer reactions. However, Bronsted himself has already pointed out that a may vary between 1 and 0 in passing from strongly endergonic to strongly exergonic reactions_ This would result in a curved Brensted plot provided a sufficiently large range of pIC values is- studied. There are a lot of experimental data which confirm this fact. Nevertheless linear correlations over extended pK ranges are also known. A compilation of the corresponding examples isgiven in ref; [12]. Hence this -subject is still being .debated.- More: over, results_on react&series *th a values‘larger than unity- or’ negative were -reported, for a review of -the older literature .and a new example see ref. [13]. .-
IS2
J. SiiJmel.K
Gustao / Free energy reialions of proron -framfer
Westheimer was the first who put forward the idea that the kinetic isotope effect (KIE) should exhibit a maximum for a symmetrical transition state [14]. Using the Hammond postulate and the Leffler principle, both of them interrelating the free energy of activation with the free energy of reaction, it was concluded that the KIE should display a maximum for thermoneutral reactions. There is now experimental evidence for this phenomenon [15]. However. also in this case there are examples of nearly constant X-,/X-, values over larger pK ranges [ 161.
3. I. The Brornted relatiotz We have performed a large number of calculations on free energy relations varying the parameters CH (proton-stretching wavenumber used for the calculation of Z, and Er”. EFL). Es and V,. Typical results are presented in fig. 2. Let us first consider the non-adiabatic limit (Y, = 10 cm-‘). In all cases the expected curvature characterized by a decreasing Bronsted coefficient in passing
Fig. 2. Calculated Bransted plots (solid he: V, = 10 cm-‘. cm-‘. a = 300 pm. 5, = 3OGOcm-‘, P, = 2-‘fli;,).
reactions
from the endergonic to the exergonic -region is. found. With an increasing vakk of the solvent reorganization energy. ES the- B&&d plots_ become more and more flattened_ One should notice here that the range ‘under study (-10000 cm-!
dashed line: V, = 2000 cm-‘;
T=
300 KS rAH + reH = 2~
pm, c = 30
(54.3) 61.7 (53.1) 60.7 i52.3) 60.3 (51.7)
3000 6000 !Noo 3ooo
ICOO 3Ooa 6000 9000
10
1500
1000 3000 6000 9000
3000
1000 3000 6000 9000
_
(0.7296) 0.8032 (0.6874) 0.7839 (0.6631) 0.7577 (0.6616)
46.4 (39.7) 45.4 (39.1) 44.8 (38.6) 44.6
(0.5305) 0.5937 (0.4551) 0.5611 (0.4260) 0.5231
(38.3)
(0.4100)
,:s, 46.4 (41.9) 46.5 (41.9) 46.5 (41.9)
0.6368
~.
(q.9998) l.oooo (0.9993) 0.9999 (0.9987) 0.9998 (0.9984)
(l.ooOo) 1_oocm (1.0000)
-l.o060.‘l.O$Kl
(1.oooo)
(l.ooooj
0.0039 (O.OillS) 0.0022 (0.0010) 0.0015 (0.0007) 0.0012 (0.0006)
0.0186 (0.0103) 0.0108 (0.0059) 0.0075 (0.0042) 0.0061
0.0441 (0.0283) 0.0258 (0.0165) 0.0181 (0.0117) O.Ol.@
(0.0034)
(0.00?6)
37.8 (33.7) 37.8
0.0008 (0.0003) .0.0004
0.0051 (0.0026) 0.0029
0.0163 (0$098) o.oq94
(33.7) 37.8 (33.7) 37.8
(O.ObO2) 0.0003 (0.0001) 0.0002
(O.OOlSj 0.0021 (0.0010) 0.0017
(0.0054) 0.0067 0.0038) 0.0054
(33.7)
(0.0001)
(0.0008)
(0.0031)
ing to proton transfer_ to a certain base family. However, in order to prove this suggestion- the different a! vahres and especially the monotonous decrease in passing- from the endergonic to the exergonic region have to be explained. Finally it has to be,mentioned that the flattening out of the Bronsted plot induced:by diffusion is completely different from the same effect being a result of the Marcus or quantum-statistical rate theory:for the exergonic region. .One can expect to find this latter effect far below the diffusion~limit. .Alf the result_!:presented so far .can also be
-:
(l.OOOOj-
0.99ss (0.9940) 0.9968 (0.9827) 0.9941 (0.9740) 0.9897 (0.9678)
-
-_-
,::E,
..
1.0000 (l.ooooj
(K-E!, 1.0000
-
-_
obtained within the framework of the Marcus theory. On- the other hand, the following results are typical for the quantu&statistical mechanic_$ rate theory: For small values of 4; an @Hating strut- _ ture is found. This effect-is well-known from other_investigatior?s. on electron-transfer . /IS] and proton-transfer reactions [19]. .-q:’ trgsfer distan&es arc. nearly. iridependent of .&see Jable 1. The effekt of_va$ng CH iS not shown in,fig_ 2. For ti, _T,1SOt) cm-’ larger ~transfer distances are :ob: mined as_ compared to jiH-~-3.00Q cm-‘,, table 1.: This leads to ‘a decrease- of -the rate _co$t_%nts by : -.
_
184
J. Siiinel. K Gustav J Free enerm relatiolzs o/proton -transfer reacrions
about two orders of magnitude_ The shape of the Bronsted plot however is not changed. A very interesting effect can be observed if the value of the electron resonance integral is increased up to V, = 2000 cm-‘. Whereas in the endergonic region the shape of the plots for V, = 2000 cm-’ and V, = 10 cm-’ is very similar, the rate constants in the exergonic region reach a maximum at -E, and decrease finally for V, = 2000 cm-‘. An analogous phenomenon is predicted by the Marcus theory for electron-transfer reactions 1201 but not for proton-transfer reactions [21.22]. This result can be understood if one bears in mind that by increasing the electron resonance integral up to V, = 2000 cm-’ most of rhe transitions between the individual vibrational levels will become adiabatic (.X$, = 1). From table 1 it can be seen that all values of the transmission coefficient are between 0.5 and 1. The different shape of the Bronsted plots for V, = 10 cm-’ and for r/,= 2000 cm-’ can be explained by analyzing the a E dependence of the rate expression. eqs. (3) and (9). The last term of eq. (3) gives “subbands“ ‘with maxima at AE=
-E,-(E;‘L-E:)L-)+(En’L-E~L)_
(15)
Provided that. the other terms being dependent on vibrational excitation are for the moment neglected. all these maxima have equal “intensity”. The only terms which can affect the probability of transitions between individual vibrational states and thus determine the shape of the Bronsted plot are. within rhe approximations adopted here. the Bohzmann term. the transmission coefficient and the term exp(-I3;,:2ay,f;)). An analysis of our calculations has shown that in the endergonic region only m ---, 0 vibrational transitions and in the exergonic region only 0 + tl transitions have to be considered_ In the endergonic region the increase of the rate constant in passing from large to small hE values is mainly governed by the increase in the Boltzmann term. The transmission coefficient remains constant in the adiabatic limit or decreases slightly in the non-adiabatic case and the term exp( -Cz:A2ny,,,,) disappears for n = 0.
In the exergonic region we come across a completely different situation. Now the Boltzmann term remains- constant and. the shape ok- the Bronsted plot is detekined by the two other terms. It seems worth noting, that this result should be essential for an analysis of the temperature dependence of free energy relations. The values of the two’ terms mentioned are dependent on adiabaticity. In the non-adiabatic limit only the transmission coefficient has to be taken into account. Its dependence on vibrational excitation is illustrated by the following example. For CH = 3000 cm-’ . r,, = 37.8 pm, Es = 1000 cm-’ and V, = 10 cm-’ it increases from .X$j = 0.0008 to Jib,H = 0.0730 and decreases then the value X& being 0.0079. The transitions m = 0 --, n = 6 and m = 0 + tz = 12 correspond to ?SE values of - 18 000 and - 36000 cm-‘. Thus in the region under study (-10000 cm-’ chE< 10000 cm-‘) the ultimate decrease is not observed_ On the other hand, for V, = 2000 cm-’ first the transmission coefficient increases up to unity and remains then constant in passing to higher vibrational states. But the Landau-Zener parameter increases further. resulting in the reduction of the rate constant by the term exp(-C;1::2?;y,f;,). In this way the decrease in the rate constant in the exergonic region for adiabatic transitions can be explained_ For smaller proton-stretching frequencies this behaviour is more pronounced as a larger number of higher excited vibrational states contributes to the rate constant in the same h E region. As the adiabatic Bronsted plot has its maximum at a E = -Es the decrease of the rate constant occurs for systems with higher Es values in the highly exergonic region. not so easy amenable to experimental studies. In other words. for higher Es values there are to be expected no or only minor differences in the hE region usually under study, see fig. 2. Upon examination of the experimental data it turns out that there is at present no indication of an ultimate decrease in the rate constkt in the highly exergonic region. Because of the difficulties in obtaining reliable data in this region (diffusion control. fast reactions) it. seems however unjustified to use this as an argument against adiabaticity in proton-transfer reactions. Furthermore it should be pointed out that a few of the possible explana-
_. J. SiihneC-K~G~+~tau~/~F~ek&v~
tions suggested.; for.. the. -absence. of-.a- decka+~g electron-tiansfer rate yconstant- in -the Marctis- invetted r&&tiay also apply to proton iliansftif: -_ 3.2.. The free energy dependence of rhe kinetic -. , isorope effect. The results of the calculations are shown in fig. 3. For Y, = 10. cm” the expected behaviour is observed. Small values of ES result in an oscilIatory structure of the free energy plot. The curve is symmetric about the maximum at AE = 0, its curvature being dependent on ~5,. In passing to V, = 2000 cm-’ there are mainly two effects. The KIE at A,?Z= 0 is reduced for the larger V, value and the. whole curve is flattened. The plot of k,/k, versus AE is n9 longer symmetric about AE=O. These effects can be explained in the following way. In the vicinity of AE= 0 mainly the O-O transition contributes to the rate constant. As can be seen from table 1 the ratio *z/X$ is larger for the smaller V, value. On the other hand the term exp{-[U(Rg)U(Rk)]/k,T) is for V,= IO cm-’ smaller than for V, = 2000 cm-’ due to
-
.-
~~~~fi~~‘~~p~~tron,t-~~~~~i~~~ : -‘.
-_ I. 1 .- ;...- _ -. _ .. ^
I
...a
=6ooo A&i9
values the transfer distances had to bei+&ased
to
calculate the correct value of the KIE. This is completely analogous to the result obtained ifi this paper. It appears that the marked asymmetry of the (kH/kD) - AE plot is closely connected with the fact that different transfer distance&for I-I and D were used, see fig. 4_ For V, = 10 cm-’ both for equal and different transfer distances a symmetric plot is calculated_ In the case of V, = 2000 cm-’ a slight asymmetry can be seen for equal transfer distances, but this asymmetry becomes very pronounced for different transfer distances. As equal transfer distances were assumed in ref. [4] no marked differences between the adiabatic and the non-adiabatic case could be n&iced.
As in the case of the Brwsted
plot there is a
scarcity of experimental data in the exergonic region. To the best of our knowledge the most
Es- 3000 mi’
6000 AE@d]
E,.9000cnC~
!w kD
. -8000
‘. 185 -_
order to reproduce a given experimental value ofthe isotope effect. It was found that for larger v,
-6000
-8000
-_-
bh_~1; iThe_ ~t~a~smissi~n.-~~~f~i~~~~~~~~tis,6~ .tJie more dronounced eff&t ieid&g-;o theoverafl & crease of- tl&I&E in thk ad&datic c&e. .-In r&f. [4] the transfer &s&es were fitted in
kD
E;6000 cd
2..
the dmr&&a .;li~~~~~“ce.~of_‘:~~~~~r-.~ddbi~~~,ta;- ;
!T!!’
r(H' kD
.. -.. .-
. . . '?O" AEl$r)~
Fig. 3. Cakulated free energy relations of the kinetic isotope effect (the same-data as in fig. 2 were used).
186
J. Sihei,
V,=lO cm-
K. Gustav / Free energy relations of proton - wansfer readions
-R:,,= !w
-6000
Rk
# Rfa
--- R:,
%I
.- Es_ The rate constant decreases for A E- -z JZtix is very weak in the non-adiabatic &nit but rather pronounced in the adiabatic case. As for the kinetic isotope effect we found evidence contrary to the conclusion drawn in ref. [4] that there should be no marked influence of adiabaticity on-the corresponding free energy relations. According to our results the (kH/kD) - AE plot is no longer syrnmetric about AE = 0 in the adiabatic case. The reason for the different results can be found in the assumption of equal transfer distances for H and D made in ret [4] and relaxed in this paper. In ref[7] it was shown that this assumption is not justified for realistic interaction potentials between the reactants_ The qualitative differences of the shapes of the free energy relations may represent a criterion for the adiabaticity of proton-transfer reactions_ Unfortunately there is still a marked scarcity of experimental data in the exergonic region.
*
6000
A E [cc’]
References Fig_ 4. Calculated free energy relations of the kinetic isotope effect for Rk = Rg and R& f Rk (E, = 6000 cm-‘: other parameters
as in fig. 2).
[II J.R. Murdoch and D.E. tMagnoli. J. Am. Chem. Sot. 104 (19SZ) 3792.
PI E.D. German, pronounced
asymmetry
can
be
Found
for the reaction acetate with bases [23]. However. it present it is not possible to decide if from a symmetric curve is within experimental scatter or not. ....
(kH/kD)
-
AE
plot
in
the
of nitroethylis felt that at this deviation the limits of
4. Conclusions Systematic investigations on the free energy relations of the proton-transfer rate constant (Brcmsted plot) and of the kinetic isotope effect were performed utilizing the quantum-statistical mechanical rate theory. It was found that the Bronsted relation displays a qualitatively different behaviour for small and large values of the electron resonance integral (non-adiabatic or adiabatic transitions) in the exergonic region (AE c -ES). In the adiabatic limit the maximum rate constant occurs at AE,,,, = -Es. whereas in the non-adiabatic case the maximum of the plot is at AE,, <
A.M. Kuznetsov and R.R. Dogonadze. J. Chem. Sot. Faraday Trans. II 76 (1980) 1128. 131 J. Ulstrup, Charge transfer processes in condensed media (Springer. Berlin. 1979). [41 E.D. German and A.M. Kuznetsov. J. Chem. Sot. Faraday Trans. 177(1951)397. R.R. Dogonadze, E.D. German, A.M. 151 V.G. Levich. Kuznetsov and Yu.1. Kharkats. Electrochim. Acta 15 (1970) 3.53. Chem. 324 (1982) 71. 161 J. Sihnel and K. Gustav. J. P&t. [71 J. Siihnel and K. Gustav. Chem. Phys. 70 (1982) 109. 181 J. Siihnel and K. Gustav, Wiss. Z. Friedrich-SchilIer-IJnivJma. Math.-Naturwiss. Reihe 31 (1952) 1113. A E-D. German and A.M. Kuznetsov. J. Chem. Sot. Faraday Trans. II 77 (1981) 2203. 1101 C. Manneback, Physica 27 (1951) 1001. Bronsted and K. Pedersen, Z. ‘Physik. Chem. [l*l J.N. Stoechiom. Verwandtschaftsl. 108 (1924) 185. [121 EC. Bordwell and D.L. Hughes, J. Org. Chem. 47 (1982) 3224. r131 J.R. Murdoch. J-A. Bryson, D-F_ McMillen man, J. Am. Chrm. Sot. 104 (1982) 600.
and J.I. Bmu-
[I41 F-H. Westheimer. Chem. Rev. 61 (1961) 265. reactions..eds. I151 R-4. More O’FerralI. in: Proton-transfer EF. Caldin and V. Gold. (Chapman and HalLLondon. 1975) p. 201. [161 B.C. Challis and EM. Miller. J. ChemI Sot_ Perkin T&sII (1972) X618.
J_ Siihnet, K Gurtm
/ Free energy reiations ofpratm -transfer recu&nu
1171RF. Be& The proton in chemistry,2nd Ed., (Chapman and HalI, London. 1973). flS] J. Wlsnup and J. Jortner.J. Chem. Phys- 63 (1975) 4358. [19] N. Bruniche-OIsen-andJ_ Us&up. J. Chem Sot- Faraday Trans. I 75 (1979) 205. [2Oj R.A. Mrucus and P. Sides. J. Phys:Chem. 86 (1982) 622.
187
l23] R.A. Marcus, 1. Phys, Cbem. 72 (1968) 891. [221 A-0. Cohen and R-k Marcus..J. Phys; Chem. 72 (1968) 4249. f23] F.G. Bordwell and WI Boyle Jr., J. Am Chem. !Zoc_97 (1975) 3447.