Quantum-mechanical Tunnelling and the Dimensions of Energy-barriers in Pro ton-transfer Reactions in SolutionBY E. F. CALDIN AND (MISS) M. ISASPAWAN"Physical Chemistry Dept., The University, Leeds 2Received 15th January, 1965Quantum-mechanical tunnelling in proton-transfer reactions should lead to non-linear Arrheniusplots and to anomalous isotope effects; both these phenomena have been observed. By fittingtheoretical equations to the data, it is possible to derive values for the dimensions of the energybarriers in these reactions. The results to date are compared. The width of the energy barrierappears to depend on the atoms between which the proton is transferred, the charges on the atoms,the structures of the reacting molecules, and the solvent. An instance of a non-linear Arrheniusplot is reported, for the reaction between hydrofluoric acid and the 2,4,6-trinitrobenzyl anion inethanol over the range -90 to +25".It is well known1 that if quantum-mechanical tunnelling occurs in a proton-transfer reaction, it will lead to deviations from linearity in the Arrhenius plot atsufficiently low temperatures, and also to anomalous isotope' effects, in particularto an A-factor for H+-transfer less than that for D+-transfer.The magnitude ofthese effects will depend markedly on the dimensions of the energy-barrier, andespecially on its width. Both effects have been observed, first in Bell's laboratory 2, 3and later elsewhere. By assuming the energy-barrier to have a particular shape(a parabola is the simplest to handle), it is possible to deduce its width and height.As very little is known about the widths of energy-barriers, the results of such in-vestigations are of great interest.The largest effects observed in the work of Bell, Fendley and Hulett 2, 3 were fora proton-transfer between carbon and fluorine (CH .. . F). A reaction in whichthe rate-determining step is a proton-transfer from fluorine to carbon (C . . . HF)might be expected to have an energy-barrier of similar shape. Such a reaction isthat between hydrofluoric acid and the 2,4,6-trinitrobenzyl anion [C6H2(N02)3CH2]-,derived from 2,4,6-trinitrotoluene :The rate of this reaction was studied aver the temperature range -90 to - 50" in anearlier investigation ; 4 the Arrhenius plot proved to be linear within experimentalerror.We have now extended the temperature-range upwards to +25" by usinga stopped-flow apparatus.5 A curvature of the Arrhenius plot is now apparent,and we interpret this in terms of quantum-mechanical tunnelling. The resultscan be compared with those already obtained for the reaction of acetic acid withthe same anion.6EXPERIMENTALThe stopped-flow apparatus and its operation have already been described? The changesof optical density with time are recorded by photographing the trace produced on an oscillo-* present address : Department of Chemistry, American University of Beirut, Beirut, Lebanon.226 TUNNELLING I N PROTON-TRANSFERscope using a time-base calibrated against 50-cycle a.c. Several traces (3-6) were photo-graphed in each run; sometimes two different time-bases were used.The times for half-change varied from 0.04 to about 0.4 sec. An Ilford 623 filter was used ; this has a maxi-mum transmission at 495m,u, corresponding to the absorption maximum of the 2,4,6-trinitrobenzyl anion at about 500 mp. Temperatures were measured by platinum resistancethermometry to within fO.05 deg.The materials were purified as in previous work.4 A stock solution of T.N.T. was pre-pared fresh each day. The solvent was the same as in Caldin and Jackson's work,4 viz.,ethanol containing 0.4 % by volume of toluene (not 0.8 %, as erroneously stated in theirpaper). The ionic strength was made up to 0.002 M by addition of lithium iodide.Kinetic runs were carried out at 0, 7 and 25°C. At each temperature, a series of runsat nearly constant buffer ratio (around 16) was performed.As HF was always in large excess(not less than 30-fold), linear first-order plots were obtained.RESULTSThe results of the kinetic runs are summarized in table 1, along with the resultsat -50 to -90" from earlier work.4 The symbols used are the same as in the pre-vious work, and are as follows :-d = conc. of 2,4,6-trinitrotoluene (T.N.T.) ;b = formal conc. of ethoxide = conc. of F- in reaction mixture ;c = formal initial conc. of HF ;r = (c - b)/b = buffer ratio ;c- b = initial. conc. of HF in reaction mixture ;s" = slope of first-order plot (decadic logarithms), sec-1.TABLE 1 .--RESULTS OF KINETIC EXPERIMENTS ON HF+ C~H~(NOZ)~CH~concentrations in mole 1.-1; s" in sec-1 (decadic logs)s = slower time-base ; f = faster time-basetemp.O C0.00f0-057-00f0.0525-00i 0.05no oftrices lWd6(f) 1-544 (s) 1.545 1-544(s) 1.544(f) 1-543 1.545 1-244 (s) 1.243 (f) 1.244(f) 1-244(s) 1.245 1.244 1.244(s) 1.245 ( f ) 1.24104b5.685-6811-0016-3016-302.978.3014.7014.705-085.083.797.352.0 12-0110%0.9600.9601.8902.8302.8300.4711.4202.5602.5600.8540-8540.6251 -2503.1303.1301 O*(C - b)0-9030.9031.7802-6672.6670.4411.3372.4132-41 30.8030.8035.8711-772.932.93r15.915.916.216.416.414-916.116-416.415-815-S15.516.014.614.6s* (obs.)1.53 f0.151.44 f0-032.45 f0.063-73 f0.053.57 50.230.84 50.042-78 f0.234.96 f0.065.21 10-651.93 50.111-76 f0.144.26 f0.308-21 50.512.16 f0.072.42 A0.06s" (corr.)1.54 f0.151.54 rt0-032.42 f0.063.68 50.053.52 f0-230.99 f0-092.77 ~k0.234-90 50.075.15 f0-651-96f0.111.79 f0.144-33 f0.308.21 310.512.35 50.1 12.61 f0.11sa (Calc.)1 *491 -492.523-553.550.932-885.005.001-831.834.378.202.472.47All concentrations are in mole 1.-1, corrected for the change in volume onchanging the temperature. Where two time-bases have been used, they are desig-nated as f and s, meaning faster and slower.The values of s" derived from eachset of oscillograms have been averaged and the mean is denoted by s"(obs.); theerror indicated is the standard deviation from the mean.So that all results shallrefer to a buffer ratio of 16, small corrections based on the results of earlier worE . F . CALDIN AND M. RASPARIAN 27at different buffer ratios4 have been applied to the first-order constants when thebuffer ratio differs froni 16 ; these corrections are usually within the limits of experi-mental error (see table 1). The corrected values are denoted by s” (corr.). Thevalues of s” calculated from the “ best ” values of the rate constants (below) areshown as s” (calc.) for comparison.Rate constants for the reactions of HF and H-t with the anion C~H~(NOZ)~CH;were determined as before4 from plots of the first-order constants s” against theconcentration of HF, which is ( c - b ) .Taking account of the reactions of solvent,HF and H+, the equation for s’’ isHere kY refers to reaction with solvent ; k2 is the second-order constant for HF ;and k; is a first-order rate constant for hydrogen ions at the buffer ratio Y. Theslope of a plot of sff against (c- b) gives k2 ; the intercept (since kYl is known fromearlier work) gives ki. Straight lines were fitted to the plots by least-squares. Thebest values of k2 and kj, with the standard deviations, are shown in table 2.2.303s” = kE 1 + k2(~ - b) + kkr.temp. “C+ 25.00 f0.05 + 7.00k0.050.00 k 0.05- 49.94 f0.03- 59.92 f0.03- 69.88 f0.03- 79.86 f0.03- 89.82 f0-03TABLE 2.-RATE CONSTANTS FOR HF+ C6H2(NO2)3CHzk2 in 1. mole-1 sec-1; k j in sec-1ki k2 log k2 (obs.) log k2 (Arrh.) -10-2 (7-8 f0.07)10-2 (6.2 f0.7)10-4 (4.1 3t0.5)lO-S(3-1f1.2)lO-5(1-4f0-3)10-6 (4.8 f3-2)10-2 (3.4 50.4)10-5 (7.5 *o.s)103 (1.49 f0.04) + 3.173 f0.012102 (4.54 f0.21) + 2,657 *0.020102 (2.70f0.09) +2-431f0-0153-25 f0.18 +0.512 f0-0241.11 kO.10 +0.045&0-038- 0.445 h0.026- 1.005 f0.020- 1.565 f0.05910-1 (3.59 k0.21)10-2 (9.88 f0.44)10-2 (2.72 f0.35)+3*173 + 2.653 + 2432 + 0.448- 0.06 1- 0.61 5- 1.230- 1.910log kz (obs.)-log k2 (Arrh.)o*ooo + 0.004-0.001 + 0.064$0.106+0.170 + 0.225 + 0.345In the earlier work covering the range from -50 to -go”, the temperaturesin the reaction cell differ slightly from the measured temperature of the thermostat,the maximum difference being 0.18”.The correction has been found to dependon the liquid in the cell. Caldin and Jackson used values determined with petroleumether instead of ethanol ; we have therefore corrected their values, using the correc-tions reported by Caldin and Harbron 6 for ethanol. The uncertainty in these tem-peratures is about 0.03 deg.6REACTION WITH HYDROGEN IONSThe Arrhenius plot for the reaction with hydrogen ions (logk; against l/T),which is subject to fairly large experimental errors, shows no clear deviation fromlinearity. The best straight line through the points gives the Arrhenius parametersas E; = 9.96+0-51 kcal mole-1, log A j = 6.35+0-50. The rate constant k;, how-ever, is composite. If the rate constant for the reaction of ethoxoniurn ions EtOHzis k 3 ~ , and that for hydroxonium ions (due to traces of water in the solvent) is k 3 ~ ,then k; is given by 4wherek i = k3,Ki + ~ ~ H K H [ H ~ O ]Kg = [EtOH:][F-]/[HF], and KH = [H30’][F-]/[HF][H20].To obtain approximate values for the Arrhenius parameters for the reaction ofEtOHi, we assume, as in the earlier work,4 (a) that Kg has at all temperatures th28 TUNNELLING I N PROTON-TRANSFERsame value as at 25", which may be estimated by assuming that the pK of hydro-fluoric acid increases by 5.56 on passing from water to ethanol, like that of carboxylicacids; and (b) that the term k 3 ~ K ~ [ H 2 0 ] contributes about 10 % of kj at alltemperatures. We then obtain E3E = 9.96 kcal mole-1, loglo A3E" 15.1 (A1. mole-1 sec-I), for the reaction EtOH: + C6H2(N02)3CH;.REACTION WITH HF MOLECULESThe Arrhenius plot for the reaction with HF molecules (log k2 against l/T),shown in fig.1, is not linear over the whole temperature range now available. It isof the same general shape as that for the reaction with HOAc, but the deviationfrom linearity begins at a higher temperature. The points from 25 to 0°C lie on1 0 3 / ~FIG. 1.-Arrhenius plot for reaction of 2,4,6-trinitrobenzyl anion with HF: loglo k2 against lO3/T.a straight line within 5 1 %; the best line calculated by least squares yields theArrhenius parameters E2 = 11.1+0.1 kcal mole-1, logloA2 = 11*3+0-1 (A in1. mole-1 sec-1). From this line are calculated the values of log k2 (Arrh.) in table 2.The deviations from this line at lower temperatures are shown in fig.2. At -80and -go", the observed rate constants are about 70 :(, and 120 % faster than thecalculated rate constants. These deviations are far outside the standard deviationsin the rate constants, which are 5 % and 14 % respectively. The temperature errorrequired to account for them, if the Arrhenius plot were linear, would be more than4", which is many times the actual uncertainty of around 0.03".Bell's equations for quantum-mechanical tunnelling through a symmetricalparabolic energy barrier 7 have been fitted to the data. Only the first term in Bell'seqn. (12) was used, i.e., Q = (ncr/P)/sin (nn/P). The calculated values (kqmt) arevery sensitive to the assumed width of the barrier (2a) and depend also on its height(Eqmt).From fig. 2 it may be seen that the calculated values reproduce the observedvalues to within about 10 % over the whole range of temperature when Eqmt =11.91 kcal mole-1 and a = 0-73 A. The results when Eqmt = 12.00 kcal mole-1and a = 0-72 A also represent the observed values within about 11 o/o, the deviationsat each temperature being slightly larger than before. The best value for the half-width of the barrier is probably 0-73 A within 0.01 A. With Eqmt = 12-20 kcal mole-1and a = 0.73 A, the calculated points lie well outside the standard deviations.The values for Eqmt and a may be too high because we have assumed that thE. F. CALDIN AND M. KASPARIAN 29energy of activation is determined by the motion of the proton alone. Contribu-tions from changes in solvation, and from the changes in the bond-lengths C-CH3,G-N and N-0 due to electronic reorganization, have been ignored. Thesecontributions could be quite considerable; in that case the energy of activationassociated with the motion of the proton would be less than the overall energy ofactivation derived from the Arrhenius plot, and to fit the results we should needsmaller values of Eqmt and of the barrier-width.Some evidence that these contribu-tions are important comes from the fact that when various acids are used in place3.7 4') 4.5 4.9 5 - 3 . 5-71 @/TFIG. 2.-Deviations from the Arrhenius line : plots of [loglo k2-loglo k2 (Arrh.)] against lO3/T.Vertical lines : experimental values of k2, with standard deviations. -.-.-.- calc.curve withEqmt = 11.91 kcal mole-1, a = 0.73 8, ; - - - - talc. curve with Eqmt = 12-00 kcal mole-1,of HF the values of EA lie between 8.5 and 10 kcal/mole,4 whereas if they werewholly attributable to the motion of the proton they should, on the simplest assump-tions, cover a range of about 5 kcal/mole. This constancy suggests that the activa-tion energy may be in large part associated with motions in the substrate molecule,other than those of the proton transferred.Alternative explanations for the curvature of the Arrhenius plot 8b have beeninvestigated, with negative results, in the same way as for the reaction betweenthe trinitrobenzyl anion and acetic acid.6 (a) A change in the mechanism of thereaction a.t low temperatures is unlikely, because the absorption spectra give noindication of any new species at low temperatures,6 and the original T.N.T.isquantitatively regenerated when HF is added to a solution of the anion, as maybe shown by determining the optical density of such a solution with a Unicamspectrophotometer, adding HF and then ethoxide at low temperature, and redeter-mining the optical density. (b) An equation derived from the assumption ofvariations in AH* due to a constant AC:, such as might arise from changes ofa = 0.72 8, ; . . . . calc. curve with Eqmt = 12.20 kcal mole-1, a = 0.73 A30 TUNNELLING IN PROTON-TRANSFERsolvation on forming the transition state, was tested but could not be fitted to thedata. (c) The results cannot be attributed to changes in the structure of the solvent,since deviations occur at widely different temperatures for hydrofluoric, acetic 6and monochloroacetic acids.6z I n2RzIR2 n.. . . . . - . . . . E. F. CALDIN AND M. KASPARIAN 31OTHER REACTIONSIn table 3 are collected the results so far available on the dimensions of energy-barriers in proton-transfer reactions. They have all been obtained by applyingBell's 1959 equations for tunnelling through symmetrical parabolic barriers 1, 7 tovarious reactions which show either (a) a non-linear Arrhenius plot or (b) an isotopeeffect in which AD/& is significantly greater than unity. These two criteria aredesignated in the column headed " method " by " Arrh." and " D-H " respec-tively (or " T-D-H " in one instance where tritium exchange has also been studied).The width of the parabolic energy barrier at the base is 2a (column 6), and the heightfor the proton is E,,t. Under the heading " type " are indicated the atoms betweenwhich the proton is transferred and its initial position, e.g., " CH .. . 0 " indicatesthat the transfer is from C to 0.Reactions (l), (2) and (3) in table 3 are those studied by Bell, Fendley andHulett ; 2 the values of a were recalculated by Hulett 3a using Bell's 1959 equations 7in the first approximation, i.e., only the first term in Bell's eqn. (12) was used. Forreaction (4) we fitted the equations to Hulett's results,3a and obtained 2a = 1-42 A,the calculated and observed rate constants agreeing within 10 "/o over the wholetemperature range ; on including the second term of Bell's eqn.(12), Hulett 3b ob-tained nearly the same value, 1.40 A. Hulett 8, 3b has also found that for reaction( 5 ) the inclusion of the second term does not appreciably alter the best value of a,though it improves the fit of the equations to the experimental data.Reaction (9) has been found 12 to give a linear Arrhenius plot over the rangefrom + 19 to -78" ; taking the maximum curvature compatible with the experi-mental accuracy, we have computed that 2a must be at least 1.92A. Reaction(1 1) gives a linear Arrhenius plot from +20" to - 32" ; 13 here we find the minimumvalue of 2a comes out as 1-22 A. Reaction (12) shows clear evidence of tunnelling 11but the value of 2a has not been published.DISCUSSIONGENERAL CONSIDERATIONS.-The values collected in table 3 for the barrier-width2a are of the same order as those expected from the bond lengths and van derWaals distances, viz., 1.35 A for type CH .. . 0 or C . . . HO, and 1.34 A for typeCH . . . F or C . . . HF. No great emphasis can be placed on the absolute valueof 2a, however. On the one hand, the actual energy-barrier is probably bell-shaped,rather than parabolic, and the effective width at the base will be greater than 2a.On the other hand, the use of a two-dimensional model has been criticized, and itappears that Bell's equations may over-estimate the tunnelling correction at lowtemperatures; 14 the value of a obtained will then be larger than the true value.The relation between the value of 2a and the true barrier width is thus not knownwith precision.The values of 2a derived from experiment are themselves subject to some un-certainties.It has already been mentioned that the values will be too large ifsolvation changes, or electronic reorganization and consequent changes of con-figuration, contribute appreciably to the energy of activation. Moreover, in allthe calculations of a it has been assumed that the energy-barrier is symmetrical,so that the reaction has zero AH. If AH is not zero, the effect of tunnelling onthe rate is smaller ; 15 consequently, if the rate measurements are analyzed by meansof the equations for symmetrical barriers, the value obtained for a will be largerthan if the correct equations had been used.Unfortunately, the values of A32 TUNNELLING I N PROTON-TRANSFERare not known for most of the reactions listed in table 3. For reaction (lo), AHis + 1.4 kcal/mole ; 10 for reaction (9), it is +3.6 kcal/mole.l2 For reactions (l),(2), (3) and (4) the known pK together with estimated values of AS" give AH asaround 17 kcal/mole ; use of the correct equations would therefore give values ofa smaller than those in table 3, but no calculations have been made. For reactions(7) and (8), which go effectively to completion, and would be expected to have AS"around -20 cal deg.-1 mole-1, AH is probably more negative than -9 kcal/mole.Values of AH could be experimentally determined and the calculations extendedto take account of them.It seems worth while, however, in spite of these uncertainties, to attempt somecomparisons between reactions (1)-( 1 l), on the provisional assumption that the rela-tive values of 2a reflect the variations in barrier width, though the results may haveto be revised when values of AH and more accurate computations become avail-able.Valid comparisons might be expected particularly for closely-related pairsof reactions such as (2) and (3), which differ only in the base (types CH . . . 0 andCH . . . F); or (7) and (8), which differ only in the acid (types C . . . HO andC . . . HF) ; or (3) and (4), which differ only in the medium and method of analysis.The factors that might be expected to be important are (i) the atoms between whichthe proton is transferred ; (ii) the charges on these atoms ; (iii) the groups attachedto these atoms, i.e., the structures of the reactant molecules; and (iv) the medium.We consider these in turn.THE ATOMS BETWEEN WHICH THE PROTON IS TRANSFERRED.-We might expect,on a simple potential-energy picture of proton-transfer, that these atoms wouldbe the most important influence on the barrier-width, since their repulsions willdetermine the minimum van der Waals distance of approach in a non-reactivecollision (AH.. . B) and the bond lengths A-H and H-B+ will then fix thethe distance that the proton has to travel in the reaction. Examination of table 3suggests, however, that other factors are also involved. The following comparisonsare relevant.(i) The barrier width 2a varies considerably for a given type ofreaction. Thus, for reactions of type CH . . . 0 and C . . . HO (reactions (l), (2),(5), (6), (7), (lo), (9) and (11)) the values are 1.26, 1.17, 1-13, 1.59, 1.66, 1.64, <1.92,and +1.22k For reactions of type CH . . . F and C . . . HF (reactions (3), (4)and (8)), they are 1-17, 1-40 and 1.46A. (ii) The barrier width is not the same forreaction (7) as for reaction (9) which is closely related to the reverse of reaction (7).The difference is at least 0.26A. This would probably not be reduced by using theequations for an unsymmetrical barrier, since AH is numerically larger for reaction(7) (cf. above), so that on revision the value of 2a would probably be reduced morethan for reaction (9).(iii) The difference between barrier widths for reactions oftypes CH . . . 0 and CH . . . F varies considerably, even when the groups attachedto the carbon atom are the same and the conditions similar; thus for reactions(2) and (3) the two values of 2a are nearly the same, but for reactions (7) and (8)they differ by 0.2A. The values calculated from bond lengths and van der Waalsdistances agree closely (1 -35 and 1.34 A) ; apparently some additional factor isinvolved.THE CHARGES ON THE ATOMS.-A comparison of reactions (1) and (2) may indicatethe effect of the charge on the oxygen atom when the other conditions are as similaras possible. The value of 2a is smaller by 0.09 A for reaction (2) than for reaction(l), possibly because of the extra attraction of the negatively-charged oxygen of theanion for the protonE .F . CALDIN AND M. KASPARIAN 33THE STRUCTURES OF THE REACTANT MomcuLw-The groups attached to theatoms between which the proton is transferred might affect the barrier width in severalways. (i) As donors or acceptors of electrons, they will decrease or increase thestrength of any hydrogen bond that may be formed before the proton-transfer occurs,and will also affect the repulsion between the two reactants. (ii) If conjugationcan occur, it will alter the distribution of charge and the effective charge on thecarbon atom, and so affect the repulsive forces. However, if we compare the pairof reactions (2) and (7) (type CH . . . 0) with the relatedpair(3) and(8) (typeCH .. . F),the predicted effects are the reverse of those observed; in reactions (7) and (8)the carbon atom is conjugated with C ~ H ~ C H ~ ( N O ~ Z , whereas in reactions (2)and (3) it is conjugated with C 4 whose effect should be weaker, yet the values of2a are markedly shorter (by 0.5 and 0.3 A) for reactions (2) and (3).THE SIZES OF THE SUBSTITUENT GROUPS may be important; they may lead todifferent degrees of steric hindrance and thus affect the barrier width. Examinationof models does not, however, suggest that steric hindrance will be much greaterfor reactions (7) and (8) than for (2) and (3). Lewis has recently suggested 11 thatsteric hindrance is a major factor for reaction (1) ; the values of k ~ / k ~ increasemarkedly when B is changed from pyridine (9.84) through 2-picoline and 2,6-lutidine to 2,4,6-collidine (24.2).Lewis points out that in a sterically-hinderedtransition state much of the energy results not from the stretching or bending ofbonds, but from compression (repulsion) and therefore depends on a high powerof the distance ; consequently the barrier is high and thin, and tunnelling is favoured.Lewis' suggestion is clearly an important one, and should be followed up.THE mDIuM.-Table 3 shows that there is a group of reactions ((l), (2), (3), (5))with values of 2a around 1.1-1.2A; another group ((6), (7), (10)) with markedlyhigher values around 1.6 A ; one reaction (9) with the exceptionally high value1.92 A ; and two reactions ((4), (8)) with intermediate values.The high and lowvalues do not correlate with the type of reaction (CH . . . 0 or CH . . . F), with themethod of analysis (isotope effect or Arrhenius plot), or with temperature of observation(for example, reaction (6) was studied at 25-65", reactions (7) and (10) down tobelow -looo). They are correlated, however, with a difference of solvent; thelower values refer to reactions in water and the higher to reactions in ethanolicsolutions. It is also clear from a comparison of reactions (3) and (4) that themedium can be of considerable importance ; the value of 2a is higher by 0.23 Afor reaction (4), carried out in 5.2 M aqueous sodium bromide solution, than forreaction (3), which was identical except that the solvent was D20 containing 0.2 MKBr.It is therefore important to consider how the solvent might influence thebarrier width (apart from any consequences due, as mentioned earlier, to the effectof a change of solvation on the height of the energy barrier).The anions concerned in these reactions are no doubt solvated by one or moresolvent molecules, with the hydroxylic H (or D) atoms of the solvent adjacent tothe anion. The initial state in a reaction of the general type CH . . . X- in a solventROH or ROD (R = H or Et) may thus be represented asR/0IH\ I-C-H . . . X-34 TUNNELLING I N PROTON-TRANSFERA solvating ethanol molecule would be expected to exert a stronger repulsion onthe carbon atom, and so give rise to a wider energy barrier, than a solvating HzOor D20 molecule, both because it is more bulky and because the oxygen atom inethanol is more negative than the oxygen in water.This effect may explain thehigher values for reactions (6), (7), (8) and (10) compared with those for reactionsThe difference between the barrier widths for C . . . HO and C . . . HFreactions in ethanol ((7) and (8)) may be due to the hydrogen-bonding propertiesof fluorine. There is no direct evidence for hydrogen bonds involving carbon andfluorine, but HF is known to form strong hydrogen bonds, and CH-0 and CH-Nhydrogen bonds are well established, in chloroform solutions and HCN respectively.16Hydrogen bonding of -CH to the anions may thus occur in ethanolic soluticn,and more strongly with fluorine than with oxygen, with a consequent shorteningof the CH-F distance relative to the CH-0 distance. In water, with its higherdielectric constant, this hydrogen bonding would be expected to be weaker and itseffect on the barrier width less marked.17).It is larger by 0 - 2 3 A than for reaction (3), from which reaction (4) differs only inthe high concentration of sodium bromide in the medium; the ratio [Na+]/[H20]is about 0.1.This implies that a F- ion will generally be near a Na+ ion, which willreduce the attraction of F- for the partial charge on the proton in -CH(6+).This will lead to a greater minimum distance of non-reactive collision, and so toa wider energy-barrier. (On this interpretation the approximate agreement betweenthe values of 2a for reactions (4) and (8) is largely fortuitous.)(21, (3) and (5).\/The barrier-width for reaction (4) finally requires comment (cf.Hulett\/SUMMARY OF THE PRESENT POSITIONQuantum-mechanical tunnelling in proton-transfer reactions is now past thestage of discovery and entering that of systematic study. If we take the values of2a in table 3 as representing the barrier widths, we can see clear instances wherethis width is affected, as we should expect, by the atoms between which the protonis transferred (compare reaction (7) with (8)), by the charges on the atoms (compare(1) with (2), by the groups attached to them (compare (7) and (8) with (2) and (3)),and-to a surprising extent-by the medium (compare (3) with (4)). There are,however, some considerable difficulties in determining barrier widths, and in theinterpretation of the values; the differences between the highest and lowest valuesof 2a do not seem to be satisfactorily explained. Systematic experimental workis needed, especially on closely-related series of reactions ; and the computationsshould be extended to take into account the effect of non-zero heats of reaction.We are grateful to Mr. R. P. Bell and Dr. J. Hulett for helpful discussions.One of us (M. K.) acknowledges a British Council maintenance grant.1 Bell, The Proton in Chemistry (Methuen, London, 1959), chap. 11.2 Bell, Fendley and Hulett, Proc. Roy. Soc. A, 1956,235,453.3 Hulett, (a) Proc. Roy. Soc. A , 1959, 251, 274 ; (b) personal communication.4 Caldin and Jackson, J. Chem. Soc., 1960, 2413E. F. CALDIN AND M. KASPARIAN 355 Allen, Brook and Caldin, Trans. Faraaky SOC., 1960,56, 789.6 Caldin and Harbron, J. Chem. SOC., 1962,3454.7 Bell, Trans. Faraday SOC., 1959, 55, 1.8 Hulett, (a) J. Chem. SOC., 1965,430 ; (b) Quart. Reo., 1964, 18,227.9 Shiner and Martin, Pure Appl. Chem., 1964, 8, 371.10 Caldin and Kasparian, in preparation.11 Funderburk and Lewis, J. Amer. Chem. SOC., 1964,86, 2531.12 Caldin and Long, Proc. Roy. SOC. A, 1955, 228, 263.13 Bell and Norris, J. Chem. SOC., 1941, 854.14 Johnston and Rapp, J. Amer. Chem. SOC., 1961, 83,l. Sharp and Johnston, J. Chem. Physics,15 Bell, Proc. Roy. SOC. A, 1936, 154,423.16 Pimentel and McClellan, The Hydrogen Bond (Reinhold, New York, 1960), chap. 6.17 Hulett, Trans. Faraday SOC., 1963, 59, 1815.1962,37,1541