J. Phys. Chem. 1082, 88, 1302-1305
1302
barrier for electron transfer) fits the experimental data fairly well. Acknowledgment. This work was partially supported by a Scientific Research Grant-in-Aid from the Ministry
of Education of Japan and also by a Scientific Research Grant for Solar Energy from the Institute of Physical and Chemical Research. The authors are grateful to Professor H. E. A. Kramer and Dr. U. Steiner (Universit4tStuttgart) for their interest and useful discussion.
Actlvatlon Energy and Heat Capacity for the Fast Termolecular Two-Proton Transfer (CH3)3NH+ CH30H N(CH3)3In Methanol
+
+
Erne& Grunwald Department of chemleby, Brarhjeis University, Welthem, Messachwetts 02254 (Received: Septembr 30, 1981; I n Final Form: November 24, 1981)
Rate constants (hjfor the title reaction were measured at temperaturea ranging from -95 to 25 O C . The activation energy E& = R P ( d In k2)/dT decreased significantly with increasing T, from 3.8 kcal mol-' at -95 O C to 2.2 kcal mol-' at 25 "C. Assuming that C& (i.e., dEmd/d7')is constant, C,, was -13.2 cal mol-' K-l. Somewhat better fit was obtained with C, = 46.68 X 106)Pcal mol-' K-'. The data suggest a reaction mechanism involving two-proton tunneling in suitable reactive complexes whose hydrogen-bonded structures and solvation shells are more rigid than those of normal hydrogen-bonded complexes. The kinetic measurements also yielded composite kinetic constanta for proton exchange between methoxide ion and methanol in the range -95 to 25 OC.
In hydroxylic solvents the effects of solvation on AH" and TAS" for reaction, and on AH* and T A S for activation, tend individually to be large. But owing to a propensity for compensation, their effecta on the measured free-energy changes AGO = AHo - T A P and AG* = AH* - T A S are often quite small.'-3 Accordingly, there has been a growing interest in supplementing available values of AGO with accurate calorimetric data for AH" and ACp04-6and in obtaining rate constants of high precision over the widest possible temperature range for the evaluation of AH*and ACP*.' This has been particularly true for proton-transfer processes, such as acid-base dissociation, which proceed with direct hydroxylic-solvent participation, and data are now available for many such processes in Among nonaqueous hydroxylic solvents, methanol is especially attractive for study because of its wide liquid range (-97.8to 64.7 "C) and good solvent properties. I have had an abiding interest in fast proton-transfer proceases in methanol, whose rates can be measured accurately by dynamic NMR techniques."J2 (1)Leffler, J. E.J. Org. Chem. 1955,20,1202;J. Chem. Phys. 1955, 23, 2199. (2)Leffler, J. E.;Grunwald, E. 'Rates and Equilibria of Organic Reactions"; Wiley New York, 1963;Chapter 9. (3)Lumry, R.; Rajender, 5.Biopolymers 1970,9,1125. Lumry, R. Probes Struct. Funct. Macromol. Membr., Proc. Colloq. Johnson Res. Found., 5th, l a 9 1971,2,353. (4)Amett, E. M.; McKelvy, D. R. J. Am. Chem. SOC. 1966,88,5031. Amett, E. M.; Small, L. E.; Oaucea, D.; Johnston, D. Zbid. 1976,9!3,7346. Amett, E. M.; Campion, J. J. Zbid. 1970,92,7097. (5)Christensen, J. J.; Izatt, R. M.; Wrathall, D. P.; Hansen, L. D. J. Chem. SOC.A 1969,1212. (6)(a) b u n g , C. S.; Grunwald, E. J.Phys. Chem. 1970,74,687. (b) Ibid. 1970,74,696. (7)Blandamer, M. J.; Robertson, R. E.;Scott, J. M. W.; Vrielink, A. J . Am. Chem. SOC. 1980,102,2585. (8)Bergstrom, S.;Olofaaon, G. J. Chem. Thermodyn. 1977,9,143. (9)Jones, F.M.; Amett, E. M. Prog. Phys. Org. Chem. 1974,1I,263. (10)Grunwald, E. J. Phys. Chem. 1963,67,2208,2211. (11)Grunwald, E.; Jumper, C. F.: Meiboom. S. J. Am. Chem. SOC. 1962,84,4664. 0022-3654/82/2086-1302$01.25/0
This paper reports rate constants for fast proton-transfer processes with solvent participation in buffered solutions of (CH3),NH+ and (CH3)3Nin methanol. In combination with results published earlier,13 the measurements comprise seven temperatures in the range 25 to -95 "C. Precision and accuracy of the rate constants are mostly within 5%. Such accuracy is not spectacular, but, because of the wide temperature range employed, it is sufficient for an initial examination of AH*(T) and AC * ( T ) and for an indication of the degree of flexibility ofhydrogen bonds in activated complexes for proton transfer. Data will be presented for proton transfer between methoxide ion and methanol and for symmetrical termolecular proton transfer involving the buffer components and solvent, according to reaction 1. The kinetic analysis (CH3)Jd"H;
+ CH3OHb + Nb(CH3)3
k2 +
(CHJ3Na + HaOCH3 + HbNb(CH3)3+ (1) will utilize dielectric constants and (to some extent) autoprotolysis constants of methanol in the range 25 to -95 "C, reported by Leung and Grunwald.6b The discussion will deal exclusively with reaction 1. Experimental Section Purification of reagents, preparation and analysis of solutions, and NMR measurements were essentially the same as described previ~usly'~ and need not be repeated here. The temperature of the NMFt probe was maintained constant by allowing a constant flow of thermostated Nz gas to flow into and around it" and was measured with a copper-constantan thermocouple mounted inside an NMR sample tube positioned in the probe so that the (12)(a) Grunwald, E.; Eustace, D. "Proton-Transfer Reactions"; Caldin, E. F., Gold, V., Eds.;Chapman and Hak London, 1975;Chapter 4. (b) Grunwald, E.; Ralph, E. K. "Dynamic NMR Spectroscopy"; Jackman, L. M., Cotton, F. A., Eds.; Academic Press: New York, 1975; Chapter 15. (13)Grunwald, E.J.Phys. Chem. 1967,71,1846.
@ 1982 American Chemical Society
(CH,),NH+
+ CH,OH + N(CH,),
The Journal of Physlcal Chemistry, Vol. 86, No. 8, 1982 1303
in Methanol
TABLE I: Properties of Methanol at Reaction Temperatures CH,-OH 'H NMRC temp, "C
[MeOH]: M
Db
6
J
25.0 -0.7 -19.9 -40.4 -61.1 -79.7 -94.8
24.55 25.3 25.9 26.5 27.1 27.7 28.2
32.62 38.09 42.92 49.06 56.53 64.79 72.93
1.568 1.801 1.966 2.130 2.279 2.403 2.496
32.4 32.1 31.9 31.6 31.4 31.2 31.0
proton TI,d s CH, 8.8 6.3 4.5, 2.9, 1.8, 0.92 0.55,
OH 8.6 5.3 3.4, 2.10 1.24 0.63 0.33
Reference 6b. a Based on densities in "International Critical Tables", Vol. 3, p 27. (1.60 X 10-5)t2;J in rad s-l. Outgassed methanol containing 0.002 M HC1.
6
Kautop (C s a l e ) b
1.21 x 10-17 2.25 X lo-'' 4.5 x 1 0 - 1 9 5.2 X lo-'' 3.4 x 10-2' 1.53 X 10-" 7.1 x 10-24
= 1.796 - (8.89 x lO-")t
-
TABLE 11: Kinetic Summary
T, K
k,,
298.2 272.4 253.3 232.7 KB = 1.02 X
s-l M-I
3.25 X 2.17 X 1.59 X 9.15 x
kMeo-KB, 8 - l M
T,K
k,,
17 60a 1360a 990 615
212.1 193.5 178.3
5.20 x 107 2.48 x 107 1.05 x 107
10' 10' 10' 107
lo-' (298.2 K), (1.14 f 0.37) X
+ (P(T) + R/[MeOH]
(2)
and (TJoH the OH-proton T1relaxation time at the re) a correction whose physical action temperature. v ( ~ is basis still is not well under~tood.'~It becomes significant when R is small and its empirical evaluation in the form V(T) = aT/(1 + bT2) has been described.14 b = 478 s - ~ ;a = 74/(T1)0~ s-'. In the present experiments (P(T) is small , in a few experiments at -95 O C . compared to 1 / ~except Some relevant solvent properties are listed in Table I. Solvent Participation in Reaction 1. There is fair evidence that at 25 and 50 "C reaction 1 involves one solvent molecule, as s h 0 ~ n . lIn ~ addition, the analogue of reaction 1 in water has been shown to involve one water m01ecule.'~ In the present work it was convenient to check the stoichiometry of solvent participation at -79.7 "C. The method used consists of measuring the exchange broadening (EB) of the CH3proton spin-spin doublet, and of the central two lines of the OH-proton spin-spin quadruplet of CH30H under conditions of lifetime broadening and at buffer ratios and concentrations where, according to rate constants to be presented, reaction 1alone is kinetically significant. In this method one exploits statistical factors in exchange broadening due to OH-proton exchange. If reaction 1 is correct as shown, EB(OH)/EB(CHJ = 2. If reaction 1 in fact proceeds with participation by s methanol molecules (s 2 l),EB(OH)/EB(CH3) = [2 + 1.25(s - l)]/s. On the basis of four measurements, the (14) Grunwald, E.; Jumper, C. F.; Puar, M. S. J. Phys. Chem. 1967, 71, 492.
(16) Luz, 2.;Meiboom, S. J . Chem. Phys. 1968, 39,366.
30 3 110 26 t 10
(272.4 K).
thermal junction was at the center of the rf pickup coil. Reported reaction temperatures are believed to be accurate within 0.1 OC at 25 OC and within 0.3 OC at -95 OC. Rates of proton exchange were generally measured in dilute out-gassed solutions at total buffer concentrations of 50.01 M. Buffer ratios (BH+/B) were mostly in the range 1-10, and occasionally as high as 50. Proton-exchange rates were measured for the OH protons of methanol by means of the shape and/or width of the spincoupled CH3 resonance of methanol. Straightforward methods for calculating the mean time T between proton spin inversions of the OH group have been described." As in previous work,14 the desired rate R/(mol L-' s-l) of OH-proton exchange was derived from 1 / according ~ to eq 2, where [MeOH] is the molar concentration of solvent 1 / = ~ (l/Tl)OH
kMeO-KB, S-' M
M-'
actual ratio is 2.11 f 0.15. Thus, the stoichiometry indicated in reaction 1 is essentially correct.
Results The full rate law for proton exchange in methanol (SH) solutions of trimethylammonium ion (BH+) and its conjugate base (B) has been reported around room temperature.13 On that basis, the present experiments were designed so that the rate law for solvent OH-proton exchange is simply eq 3. When one introduces the base dissociation R = k2[BH+][B] + k,&MeO-] (3) KB
= [BH+][OMe-]yi2/[B]
(44
-logy* = (1.825 x 106)P/2(DT)-3/2/[l 251.5P/2(0T)-1/2](4b)
+
R = kZ[BH+][B] + ~M,O-K&',-~[B]/[BH+]
(5)
constant Kb according to eq 4a, eq 3 is transformed into the experimental rate law eq 5. Mean molar activity coefficients ya were estimated from Debye's first approximation, eq 4b, in which I denotes the molar ionic strength of the solution, and a value of 5 A is used for the distance-of-closest-approach parameter. Equation 5 reproduced our data under all conditions. Kinetic analysis at any given temperature involved 12-20 independent buffer concentrations and ratios. Rate constants were calculatedby weighted least squares, statistical weights being assigned in inverse ratio to precision measures of R. Results are listed in Table IL16 When In k2 and In (kMeO-KB) were plotted vs. T',the resulting relationships showed significant curvature. Further analysis will now be reported for In k2. Terminology is defined in eq 6. In eq 6c, n denotes the number E, = R P ( d In k2)/dT (64 ,
tact = dEact/dT ufit =
(c[h(kZ,obsd/k2.c.lcd)12/(n - P)~'/2
(6b) (6c)
of experimental temperatures and p the number of fitting parameters. The use of Arrhenius parameters E& and , C (16) Rate comtanta at 298.2 and 272.4 K are based on probn-exchange rates reported previously.1a At 272.4 K the values in Table I1 differ slightly from those reported beforela because the simple two-parameter rate law, eq 3, was used in the least-squares calculations and buffer ratios >10 were therefore excluded.
1304
The Journal of Physical Chemktfy, Vol. 86,No. 8, 1982
is preferred over that of AH*and AC,' because, in case of a reaction mechanism which involves proton tunneling, the use of concepts based simply on transition theory may not be justified. A simple Arrhenius equation, In k2 = 24.758 - 1506.4/T, where E , = 2993 cal mol-' and C,, = 0, reproduced In k2 poorly, with ufit = 0.106. Deviations were not random. To obtain better fit, two types of three-parameter , was assumed to treatments were tried. In the first, C be constant. This led to eq 7; deviations were random. In k, = 67.514 - 6.63957 In T - 3013.36/T ufit = 0.042
Eect = 5988 - 13.1g3T cal mol-' C,, = -13.2 cal mol-' K-'
(7)
Since the decrease of E,, with T is clearly significant, a second three-parameter treatment was tried in which E , was written in the form E,, = a T " . As n was varied in steps of 0.5, ufitwent through a sharp minimum at n = 1. The corresponding expression for In k2 is eq 8. Deviations In 12, = 21.482 - (1.68154 X 105)T2 ofit
= 0.038
E,, = (6.682 X 105)/T cal mol-' C,, = -(6.682
X
105)T2cal mol-' K-'
(8)
were random. The fit of eq 8, where C lI, decreases with increasing T, is somewhat better than that of eq 7 where lC,I is constant. To probe further, I tried fitting a variety of fourparameter equations, even though the data do not really support four parameters. I found that ufit in no case is clearly smaller than 0.038 and that ufit= 0.038 only in those cases in which ICa,/ decreases with T. The conclusion probably does decrease with T seems justified thatlC,I and that eq 8 is the better representation. However, values predicted from eq 8 for Cact are only semiquantitative.
Discussion Symmetrical proton-transfer reactions such as reaction 1 are relatively insensitive to acid-base strength of the substrate, and this has been taken as evidence that the transfer of the two protons is concerted." By analogy with other fast proton-transfer processes, it is also likely that reaction 18, or most of it, proceeds with proton tunneling.'8 Evidence for a tunneling mechanism is the low E, 2.2-3.8 kcal, whereas the potential barrier for concerted proton transfer is likely to exceed 10 kcal.19 Two distinct proton-tunneling effects have been described. In cases considered especially by Bell'& and by Caldin and Mateo,'" most of the reacting systems pass over the top of the barrier at the upper end of the experimental temperature range, and tunneling becomes substantial only a t rather low temperatures. Here E,,, increases with increasing temperature. (17) Grunwald, E.; Meiboom, S. J. Am. Chem. SOC. 1963,85, 2047. 1965,39, 105. Grunwald, E.; Cocivera, M. Discuss. Faraday SOC. (18)(a) Bell, R. P. "The Tunnel Effect in Chemistry";Chapman and Ha& London, 1980. (b) Grunwald, E. Prog. Phys. Org. Chem. 1965,3, 317. (19) (a) For exothermic one-proton transfers, kinetic parametera which measure the height of the potential barrier range upward from 6 kcal mol-' in nonpolar and dipolar aprotic solvents. I believe that the barrier for thermoneutral two-proton transfer will be at leaet twice this figure. (b) Caldin, E. F.; Mateo, S. J. Chem. Soc., Faraday Tram. 1 1975, 71, 1876, 1894; 1976, 72, 117.
Grunwald
At the other extreme, considered especially by M a r ~ u s , ~ virtually all of the reacting systems penetrate the barrier at all experimental temperatures. I have called this kind of proton transfer ultrafast.'8b It is likely, by analogy with other fast proton-transfer processes involving oxygen and nitrogen acids and bases, that proton transfer in reaction 1 is ultrafast. Note also that E,, for reaction 1 decreases rather than increases with increasing temperature. Marcus has shown in his classic workz0that the freeenergy change for forming a reactive complex capable of tunneling includes a contribution from external solvent reorientation. When the reactive complex itself includes solvent molecules, as it does in reaction 1, there may also be a free-energy contribution due to constraints on solvent motions within the complex.z1 In the case of reaction 1, one may expect both external and internal solvent effects to be significant. External electrostatic effects of the type treated by Marcusmshould be relatively small, however, partly because the reactive complex B,H+.SH.Bb e B,.HS-HBb+ is only a univalent ion,% and also because the interconverting states are dipolar (in addition to being ionic) so that the centroid of positive charge% in tunneling moves a shorter distance than from B, to Bb More important, perhaps, will be effects resulting from constraints on flexibility. Hydrogen-bonded complexes at statistical equilibrium have flexible structures,22Pwhereas two-proton tunneling is fast enough only for a limited range of the accessible molecular geometries. One desideratum is that the potential barrier to be penetrated be narrow for both protons,'& which requires that the motions of the protons be coupled, or at least phased, so as to make it so. Another desideratum is that the hydrogen-bonded geometries of both BH-S and SH.B be linear. External solvent effects can be significant because the flexibility of BH+-SH.B can be reduced, and a suitable hydrogenbonded geometry be imposed, by appropriate construction of the surrounding solvent cage. How will constraints on flexibility affect E,, and C,,? It is well-known that stretching and bending of hydrogen bonds are associated with absorption in the far infrared.% For OH. .N hydrogen-bond stretching, the far-IR absorption band of phenol-pyridine complexes may serve as Wavenumbers assoa model. It occurs at 130 ciated with hydrogen-bond bending should be still smaller, perhaps around 50 cm-'. Vibrations of low wavenumber imply molecular flexibility. To the harmonic oscillator approximation, the width wi of the potential well at vibrational energy ti = (vi + '12)hviis given by eq 9, where
-
-
wi = (2ti/pp/(7rvi)
(9)
pi is the reduced
mass. A representative value for is kBT, where kB = Boltzmann's constant. On substituting T = 300 K, w2 = 5 X lo-%g, and vi/c = 130 cm-', one estimates that wi = 0.3 A. On substituting the same T and p and letting vi/c = 50 cm-*, one estimates that wi = 0.8 A. (20) (a) Marcus, R. A. J. Chem. Phys. 1956,24,966. (b) Marcus, R. A. Discuss. Faraday SOC.1960,29, 21. (c) Reference 20a, p 974. (21) Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. SOC.1980,102,5798. (22) Luck, W. A. P.; Ditter, W. J. Mol. Struct. 1967, 1, 261. (23) Del Bene, J.; Pople, J. A. J. Chem. Phys. 1970, 52, 4859. (24) Rothachild, W. G. "The Hydrogen Bond"; Schuster, P., Zundel, G., Sandorfy, C., Eds.; North-Holland Publishing Co.: Amsterdam, 1976; Vol. 2, p 767. (25) Ghersetti, S.; Giorgianni, S.; Mangini, A.; Spunta, G. Spectrosc. Lett. 1972, 5, 111. Lichtfus, G.; Zeegers-Huyskens, T. J. Mol. Struct. 1971,9,343. Hall, A.; Wood, J. L. Spectrochim. Acta, Part A 1972,28, 2331. (26) I, T.-P.; Grunwald, E. J. Am. Chem. SOC.1974, 96,2387. See especially Figure 3 and related discussion.
J. PhYS. Chem. 1082, 86, 1305-1314
The nonlinear complex B-H-S-HOB has nine vibrational modes, regarding B, S, and B as point masses. Two of them are assignable to B-H and S-H covalent-bond stretching; the remaining seven are assignable to H-bond stretching (two modes) and H-bond bending. Under normal constraints, wavenumbers for H-bond stretching and bending will be small (5130cm-’) so that the mean energy per mole for each mode at statistical equilibrium approaches the classical equipartition value of R T . The corresponding heat capacity per mole for each mode is R . If one makes the trial assumption that, in the reactive complex for two-proton tunneling, all seven H-bond modes ,, become inflexible, then the resulting contribution to C is -7R, or -13.9 cal mol-’ K-’,independent of T. Remarkably, the experimental value of,,C in the fitting treatment (eq 7 ) which assumes that C ,, is independent of T, is -13.2 cal mol-’ K-’,in nearly exact agreement. Having suggestad a quantitative explanation for C, I would let the discussion stop right there, were it not for two considerations. First, the preceding analysis has ignored the participation by the surrounding solvent, whose conformations have been constrained so as to impose on
1305
BH+.SH*Ba structure suitable for two-proton tunneling. The solvent motions thus constrained are likely to be librational or vibrational motions of normally low frequencies. Thus,the total number of quasi-classical vibrational modes whose flexibility becomes reduced is greater than seven. Second, the weight of the available evidence is that ,C in fact is not independent of T. According to the preferred fitting treatment (eq 8),,C varies from -21 cal mol-’ K-’ at 178 K to -7.4 cal mol-’ K-’at 300 K. Temperature dependence of this magnitude can be rationalized in many ways, even within the limited framework of the harmonic oscillator approximation. Nevertheless, it will be instructive to indicate the scope of the phenomenon by means of the following approximate calculation. Suppose that the oscillators which lose flexibility have a uniform wavenumber of 100 cm-’ in the normal BH+* SH-B complex and its solvent shell. The results for C ,, in eq 8 can then be accommodated approximately if an average of 30 such oscillators become constrained in the formation of the reactive complex for two-proton tunneling so that their wavenumber increases to 300 cm-’.
Ab Initio Study of Valence State Potential Energy Curves of N, W. C. Ermkr,’ Depertmcmt of chsmletry and Clwmhl Enghwhg, Stevens Institute of Technobgy, W e n , New Jersey 07030
A. D. McLean, IBM Research Laboratory, Sen &e,
Celifmh 95193
and R. 8. Mulliken Depemnent of m k t r y , Univwsny of @I&@,
Ch&a@, IlUnols 60637 (Recehd: October 7, 1961)
Valence state calculationsare reported for the nitrogen molecule, including the X’Zg+ ground state and A3Z,+, B311g,W3&, B‘?Z;, a’%;, a’%, wl&, C311,, b’II,, and b’lZ,+ states, whose dominant configurationsare single orbital excitations from the ground state, and G3$, H3@u,AISZ,+ and C’”9Iu states, which are predominantly double excitations from the ground state. Configuration interaction (CI) wave functions expanded in a CSF space of all valence configurations plus fmb and second-order confiiations, which use single and double orbital substitutions into the dominant configuration of each state, are shown to well describe these states around Re. Exceptions are C311,, b’II,, and b”Z,+, which need more elaborate treatments that include multiple “dominant”configurations, rydbergization of orbitals, and mixing with Rydberg states. Spectroscopic analyses of the potential curves, along with adiabatic and vertical excitation energies, are presented with experimental comparisons. We predict T,values for A%,+ and C”sIIu approximately 1 eV lower than indirectly derived “experimental”values. Dissociation energies computed with several ground-statewave functions illustrate the necessity for reaching basis set limits in analyzing contributions to an observable.
Introduction The nitrogen molecule poses one of the most difficult challenges for ab initio molecular electronic structure calculations. Since many of the states giving rise to its rich electronic spectrum have been determined experimentally,’ they can calibrate theoretical approaches to the elucidation of spectral data At the same time, there are many featuma of the electronic spectrum that are not well understood and lend themselves to detailed analyses using ab initio procedures.
The complexity of the spectrum and the difficulties in theoretical treatment derive from the large number of valence molecular orbital (MO) configurations that arise when two nitrogen atoms in their three lowest states (*S, 9 , T )approach and form the MOs lug, lu,, 2ug,2uu, 3ug, la,, lag,and 3 ~ , . ~If the atomic 1s electrons doubly occupy the lug and l u u core orbitals, then the remaining ten electrons, distributed in all possible ways among the six valence MO’s, give rise to 96 electronic states of ‘Eg+ symmetry, 100 of 32u+, 164 of 3Hg, etc. These are the
(1) A. Lofthue and P. Krupenie, J. Phys. Chem. Ref. Data, 6, 113 (1977).
(2) R. S. Mulliken in T h e Threshold of Space”, M. Zelikoff, Ed., Pergamon Press, New York, 1967.
0022-3654/82/2086-1305$01.25/0
0 1982 American Chemical Society
