2098
L. John Gagliardi
sumptionslS may lead to an effective V4. The inclusion of V4 is, however, found to be very useful in bringing the calculated frequencies into better agreement with those observed.18 In phenol itself, VI is zero and various values of Vz have been reported varying from 1170 to 1250 cm-1. The values of the V, term found in this study are close to the values published for phenol and some other meta-substituted phenols.19 This confirms the results of an earlier study19 that the twofold term is not affected significantly by various substituents located in the meta position. In the absence or exclusion of V3 and higher order terms, VI represents the energy difference between the two potential minima. Pople and coworkers20 interpreted the theoretically calculated VI values of some aliphatic compounds in terms of dipole-dipole interactions. The more stable form was found to have the dipoles opposed rather than reinforced; such a generalization would suggest the cis form to be more stable in m-fluoro-, mchloro-, and m-trifluoromethylphenol but less stable in mmethyphenol. Our experimental data do not give any definite evidence as to which form is more stable. However, it does suggest that the energy difference between the two rotamers of m-fluorophenol is greater than that in m-trifluoromethylphenol, and this result cannot be explained on the basis of simple dipole-dipole interactions.
Acknowledgments. This work has been supported in part by funds from the National Science Foundation (GP-22943). The authors acknowledge the free computer
time provided by the Mellon Institute nmr facility for Biomedical Research (NIH Grant No. RR00292).
References and Notes (1) Work presented in this paper represents a portion of the thesis to be submitted to Carnegie-Mellon University in partial fulfillment of the requirements for the Ph.D. in the Department of Chemistry. (2) (a) F. A. Miller, in "Molecular Spectroscopy," Hepple, Ed., Elsevier, Amsterdam, 1968; (b) W. G. Fateley and F. A. Miller, 9th European Molecular Spectroscopic Conference, Madrid, 1967. (3) (a) J. H. S. Green, D. J . Harrison, and W. Kynaston, Spectrochim. Acta, Sect. A, 27, 2199 (1971); (b) F. A. Miiler, W. G. Fateiey, and R. E. Witkowski, ibid., 23, 891 (1967). (4) N. L. Owen and R. E. Hester, Spectrochim. Acta, Sect. A, 25, 343 (1969). (5) A. Koll, H. Ratajczak, and L. Sobczyk, Rocz. Chem., 44, (4), 825 (1970). (6) L. Radom, private communication. (7) G. L. Carlson, W. G. Fateley, A. S. Manocha, and F. F. Bentley, J. Phys. Chem., 76, 1553 (1973). (8) G. L. Carlson, W. G. Fatelev, and F. F. Bentlev, Spectrochim. Acta, Sect. A . 28. 177 (1972). (9) J. V. Knopp and C. R . duade, J, Chem. Phys., 48, 3317 (1968). (10) P. Meakin, D. 0. Harris, and E. Hirota, J. Chem. Phys., 51, 3775 (1969). (11) F. Inagaki, 1 . Harada, and T. Shimanouchi, J. Mol. Spectrosc., in press. (12) C. R. Quade, J. Chem. Phys., 48, 5490 (1968). (13) T. Pedersen, N. W. Larsen, and L. Nygaard. J. Mol. Struct., 4, 59 (1969). (14) Part of the graduate project of A. S. Manocha. (15) Subject of afuture publication. (16) H. D. Bist, J. C. D. Brand, and D. R. Williams. J. Mol. Spectrosc., 24, 402 (1967). (17) A weak band was also observed at 274 c m - ' for which no immediate explanation is apparent. (18) A. S. Manocha, W. G. Fateley, and T. Shimanouchi, J. Phys. Chem., 77, 1977 (1973). (19) G. L. Carlson and W. G. Fateley, Proceedings of Macromolecular Symposium in Prague 1972, in press. (20) L. Radom, W. J. Hehre. and J. A . Pople, J. Amer. Chem. SOC.,94, 2371 (1972).
A Possible Mechanism for Protonic Transfer in Aqueous Solutionsi L. John Gagliardi Department ot Physics, Rutgers University, Camden, New Jersey 08102 Revised Manuscript Received February 20, 19731
(Received August 27, 1972,
Pubiicafion costs assisted by Rutgers University
A mathematical model for proton transfer based on a one-dimensional double potential well is described. Unlike previous models, the decision as to whether quantum or classical transfer of the proton dominates is not based upon poorly known experimental information on the shape of the potential barrier. Proton exchange from one well to a n adjacent well is discussed in general quantum mechanical terms, and it is argued that "hard" experimental facts favor a classical transfer process. These experimental facts are the square root of two value for the H+/D+ mobility ratio and the observed Arrhenius temperature dependence of the mobility. The theory is shown to be internally consistent in the sense that it predicts reasonable values for certain microscopic parameters relating to the structure of aqueous solutions of strong acids.
Previous Theories of Protonic Transfer The relatively high mobility of the proton in aqueous acids has been attributed to quantum tunneling of the proton from the species H30+ and (H+.H20).nHnO, n The Journal of Physicai Chemistry, Vol. 77, No. 7 7, 1973
1 1, to suitably oriented, adjacent water Proton transfer has also been treated as a classical process,5-9 notably by Stearn and Eyringg and also Gierer and Wirtz.8 In general, there is not complete agreement among these theories and there are completely conflicting
A
2099
Mechanism for Protonic Transfer in Aqueous Solutions
views of very important aspects of the models. One may cite in this regard the conflict between Conway, et al.,3 who claim that rotation of water molecules is the rate-determining step in the transfer, and Stearn and EyringlO who state that rotation is the faster process. Reviews of previous theories of proton conductance are given by Conway,ll et al., and also by Eigen and De Maeyer.12 The treatmient of Conway, Bockris, and Linton3J1 would appear to be quite complete and comprehensive. However, the temperature dependence of the protonic conductivity does not appear to be satisfactorily derived.11 This failure to derive the Arrhenius temperature dependence weakens the otherwise very exhaustive theory developed by the authors. The treatment by Stearn and Eyring of proi,on transfer as a classical process is based upon the classical rate equation13 which essentially guarantees the correct temperature dependence at the outset. Classical transfer dominates, according to the authors, because the potential barrier is relatively flat and low. They argue that a classical transfer process is favored and evaluate the height of the potential barrier from the observed mobility, assuming a Lorentz local field. Apart from the difficulty in justifying a Lorentz local field14 in aqueous solutiions, it is not clear how the authors arrive a t the conclusioii that classical processes should dominate since the mobility ratio of H to D can be explained within the context of a tunneling model als0.3 34 In addition, their use of an overall transfer distance of approximately 3 A does not appear to be justified.15 It is clear that without detailed information regarding the shape of the potential barrier for proton transfer in aqueous solutions one cannot decide whether classical "barrier hopping" or quantum tunneling will dominate. This decision depends much too critically upon the shape of the barrier to permit a reliable decision. In spite of the fact that such information is not available,4 all of the above-mentioned theories which attempt to explain proton conductance in terms of a microscopic model do decide, on the basis of present experimental information, which process dominates. The present calculation avoids this sort of approach.
The Model We take the potential function along the reaction coordinate to be i2 double potential well as shown in Figure 1. Because of the hydration sheath, this well is not completely symmetric along a reaction coordinate. This relatively small asymmetry is not important for the present calculation of the tunneling frequency which, unlike calculations based on previous models, is not sensitive to the exact shape d the potential function. There is considerable e ~ i d e n c c 3 ~ ~that J l this is qualitatively the correct shape. It is well known that a central barrier causes a splitting, AEn, of the nth energy level of a potential well without the barrier and that one obtains a pair of eigenfunctions associated with each pair of energy levels, with the symmetric combination in each set corresponding to the lower energy of that pair.16 Thus, if In(1)) and ( 4 2 ) ) are the zero-order eigenfunctions in the wells on either side of the barrier, we have for the nth excited state Im) = 2-'/"(ln(l>) - Inca))) (1) E, A E J 2 = E,, (2) and
+
Figure 1. Potential function along a reaction Coordinate
E,
- AE,/2
E,,
(4)
Then
'+
In(x,t>)= 2-1/2(1na)e-'~"~~ Ins)e-LEnqt')
(5)
where In(x,O)) = 2-'/2(Ina)
Since E,,
- E,,
+ Ins))
(6)
= AE,, we have from ( 5 )
~n(x,t))= 2-'/2(1ns) Thus, when t = t o = Th/AEn
+ Inu)e-l~E,l/~)e-lE.Et ( 7 ) li
~n(x, to)) = 2-'/2(1ns) - /na))e-'Ensto'fi (8) and the proton has tunneled through to the other side after a time equal to to =
irfiIAE, = hl2AE, = ( 2 ~ ~ 1 - l
(9) We see that the tunneling frequency, un, from a pair of excited states with energy splitting, AEn, depends on the splitting. For excited states near the top of the barrier, this frequency is approximately equal to the classical frequency of oscillation in either well, since the energy splitting of a pair near the top is of the same order as the difference in energy between successive pairs.17 If we assume parabolic intrawell potential functions, then v c m-1'2, where m is the mass of the tunneling particle. It will be assumed that W,, > uc, where Wf, is the probability per second for collisional excitation from an initial state, i, to a final state, f, and uc is the classical frequency of oscillation in either well (eq 20). Thus hWf, > huc, and the energy splitting is smaller than the line width for intrawell excitations. We see, as a result of this last inequality, that neither Ina) nor Ins) may be excited separately, a result that we have already used in eq 5-8. Assume that the external electric field is as pictured in the figure with a proton in the left well. We calculate now the average time that the proton spends in either well, neglecting for the moment the very small perturbation introduced by the field. The probability per second, Wlr, that the proton tunnels from the left- to the right-hand well from a pair of excited states is given by the probability per second for collisional excitation to that set, Wf,, multiplied by the probability of crossing the barrier, 2un/(2un W,f), where Wlf is the collisional deexcitation rate and 2un the tunneling frequency for a pair of excited states, t o - l . We have
-
+
The Journal of Physical Chemistry, Vol. 77, No. 17, 7973
2100
L. John Gagliardi w 1 1
= 2Wf1Yn/(2V,
+ W,,)
(10)
+
(11)
with a similar expression for Wrl W ~ I 2Wfi~,zl(2~n Wl,> Using
Wf, = W , f exp(-(Ef
- E,)/kT)
(12)
we have
W ~ =I WI, = 2 W f ~ n / ( 2 ~-t n Wfl exp(CEf Consider now the experimental facts In
(T
-
(TH/(TD
- El)/kT)) (13)
-AE/kT =
(14)
($9
2112
where is the dc conductivity and A E is the effective barrier potential, the symbols H and D referring to proton and deuteron, respectively. Equation 13 will be consistent with these two experimental facts if WE1 exp((Ef - E,)/kT) B
U,
(16.)
an'd u,
u,
N
m-112
(17)
Thus (13) becomes
- E,)/kT)
WI, = 2u, exp(-(Ef
(18) Since eq 17 is satisfied only if tunneling occurs for the most part near the top of the barrier, we conclude that most of the collisional excitations put the excited states near the top of the barrier and, hence, we have a predominantly classical process of activation to states near the top of the barrier and the tunneling of protons far below the top of the barrier plays a relatively minor role. This tunneling could occur from excited states slightly below the top, but these will act essentially like classical transfers. Since the next pair of levels lying just below the set that is near the top of the barrier will be considerably lower in energy,ll the tunneling from these and all other lower sets is negligible. In this connection, it will be recognized that the tunneling rate from a pair of states is a very rapidly changing function of the effective barrier height for that pair. Consistently, from the quasi-classical (WKB) approximation for the potential shown in Figure I, we have the following resultle for the splitting between a pair of energy levels
AE,
=
due to excitations near the top of the barrier. Thus, we allow equally for both classical and quantum transfer in this calculation and find that well-established experimental results require predominantly classical transfer with the possibility of quantum tunneling near the top of the barrier. This procedure is to be contrasted with the previously mentioned approaches which decide, on the basis of poorly known information regarding the shape of the potential function, whether classical transfer or quantum tunneling predominates. Now from eq 18 and the Nernst-Einstein expressions
( X w / n ) e ~ p ( - f i ' ' -U~ ~ ~ ldpr l)
(19)
with 0 equal to the angular frequency of the classical motion in a separate well h
2x10 = Z m l d x / p
(20)
Because of the nature of the WKB approximation, these results, unlike those previously mentioned,l7 are not to be taken too seriously in terms of their exact quantitative predictions, but the trends predicted hold up nicely. In particular, we notice that AE, hw/?r for states near the top of the barrier. Thus, for states that tunnel through near the top of the barrier, we again find that W,I = WI, u, exp(-(Ef - E,)/KT) (18') From this discussion, we have seen that the established experimental facts embodied in eq 14 and 15 are sufficient to guarantee, within the framework of the present model, that proton transfer is predominantly a classical process The Journal of Physical Chemistry, Vol. 77, No. 7 7, 1973
I*,lDP = leI/kT
(21)
D, = (r2),/6rP
(22)
and we find cP= nplelpp = (1z,e~(rz)~v~/3hT) exp(-(Ef
- Ei)/kT)
(23) In the above equations, the subscript p designates the contribution from proton transfers. Equation 23 can also be obtained by calculating the net proton flux in the direction of the field by introducing Boltzmann factors for the energy difference a t the two equilibrium positions and averaging over all directions.19 The final expressions agree to within a factor of 2. We may check the internal consistency of the model by using (18) and (23) along with experimental values of Wlr-l and vp to estimate the approximate magnitudes of uc and (Ef - Ej)/kT. From the estimate of Wicke,20 et al., on the average lifetime of H30+ we may set T~ = Wlr-l = 10-13 sec. Clearly, the average lifetime of H30+ which they discuss is essentially Wlr-l, although we do not restrict ourselves here to the species H30+. Using this value of T~ along with experimental values of v for 0.25 N HC1 a t 25" we find that we may take (Ef - E j ) / k T = 4, uc = 4 X 1013/sec and the root-mean-square net proton transfer distance, ((r2))1'2, is approximately 2.5 A. These values are quite reasonable. Similar agreement is obtained for other concentrations and temperatures. References a n d Notes (1) Presented at the 7th Middie Atlantic Regional Meeting of the American Chemical Society, Philadelphia, Pa., Feb 1972. (2) J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1, 515 (1933). (3) B. E. Conway. J. O'M. Bockris, and H. Linton, J. Chem. Phys., 24, 834 (1956). (4) M. Eigen and L. De Maeyer, "The Structure of Electrolytic Solutions," W. J. Hamer, Ed., Wiiey, New York, N. Y., 1959, pp 64-85. (5) E. Darmois, J. Phys. Radium, 2, 577 (1950). (6) E. Huckel, 2. Elektrochem., 34,546 (1928). (7) G. Wannier, Ann. Phys., (5)24, 545, 569 (1935). (8) A. Gierer and K. Wirtz, Ann. Phys., (6)6, 257 (1949). (9) A. E. Stearn and H. Eyring, J. Chem. Phys.,,,5,113 (1937). (IO) S. Glasstone, K. Laidler, and H. Eyring, The Theory of Rate Processes," McGraw-Hill, New York, N. Y., 1941, p 562. (11) B. E. Conway, "Modern Aspects of Electrochemistry," No. 3, J. O'M, Bockris and B. E. Conway Ed., Butterworths, London, 1964, pp 43-148. (12) M. Eigen and L. De Maeyer, Proc. Roy. Soc., Ser. A, 247, 505 (1958). (13) Reference 10, pp 559-567. (14) C. J. F. Bottcher, "Theory of Electric Polarization," Elsevier, Amsterdam, 1952, pp 177-183. (15) L. J. Gagliardi, J. Chem. Phys., 58,,2193 (1973). (16) L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory," Addison Wesley, Reading, Mass., 1958, p 177. (17) J. Brickmann and H. Zimmermann, Ber. Bunsenges. Phys. Chem., 70, 157 (1966). (18) , D. Park, "introduction to the Quantum Theory," McGraw-Hill. New York, N. Y., 1964, pp 100-105. (19) See, for example, ref IO, pp 562-565. (20) E. Wicke, M. Eigen, and T. Ackermann, 2. Phys. Chem. (Frankfurt am Main), 1, 340 (1954).
