J. Phys. Chem. 1984,88, 6075-6080 thesis of V.A.W. The polar tensors obtained from each of the two pyridines are available as supplementary material.
This investigation was by Grant CHE-83-18955 from the National Science Foundation. The FT-IR spectrometer and the computer used in the calculations were obtained in part via department instrument grants from the National Science Foundation. We thank Professor Boggs for informing us of their work on pyridine15 prior to publication.
6075
Valerie Walters thanks the Hey1 Foundation for a fellowship. Registry No. Pyridine, 110-86-1;pyridine-d5,7291-22-7;pyridine-4-d, 10259-15-1; pyridine-2,6-d2,17265-96-2;pyridine-3,5-d2,50535-44-9; pyridine-3,4,5-d3,7 1150-39-5;pyridine-2,4,6-d3,71 150-38-4;pyridine2,3,5,6-d,, 71 150-40-8.
Supplementary Material Available: Experimental atomic polar tensors for pyridine and pyridine-d5 (1 page). Ordering information is available on any current masthead page.
A Carbon-I 3 Nuclear Magnetic Resonance Spin-Lattice Relaxation Study of Hydrogen Bonding in 2-Methyl-2-propanol. Relaxation Mechanisms Linda M. Sweeting* Department of Chemistry, Towson State University, Baltimore, Maryland 21 204
and Edwin D. Becker National Institutes of Health. Bethesda, Maryland 20205 (Received: December 8, 1983; In Final Form: July 23, 1984) The NMR spin-lattice relaxation times of the central carbon atom of 2-methyl-2-propanolwere measured for concentrations from 0.01 to 1.06 M in hexadecane-d3.+The dipolar relaxation rates were calculated from the nuclear Overhauser enhancement, and the data compared with published models for the association equilibrium by using the microviscosity correction to a Stokes law model. Excellent fits were obtained to models of the association incorporating monomer, trimer, and hexamer or monomer, trimer, and infinite sets of associated species. The residual relaxation of the quaternary carbon appears to result from spin-rotation interaction with the methyl or tert-butyl groups. Introduction Much of the literature on hydrogen bonding is concerned with the self-association of alcohols. Studies have been carried out with classical physicochemical methods, such as vapor pressure, PVT and calorimetric methods, and spectroscopic techniques, principally infrared and proton magnetic resonance. Data have been interpreted with a variety of models including some postulating only one or two associated species and others based on a large number of species related by regularly varying equilibrium constants. Virtually all studies of alcohol hydrogen bonding have used only a single technique, and the failure of these studies to arrive at a consistent model can be ascribed largely to the lack of correlation of more than one type of data for the same alcoholsolvent mixture over a common range of concentration and temperature. The study of the association of 2-methyl-2-propanol (tert-butyl alcohol) in hexadecane by Tucker and Beckerl used three complementary methods-vapor pressure, infrared spectroscopy, and proton N M R chemical shift measurements. They concluded that the smallest significant associated species is the trimer, rather than the dimer; larger associated species were present, but the data did not allow an unambiguous determination of their size. The dipolar N M R spin-lattice relaxation time of a nucleus is sensitive to the size of the tumbling molecule and thus should be a sensitive probe of the degree of aggregation. Previous studies of alcohol association by this technique*J were unable to examine the aggregation a t concentrations where the monomer predominates and thus were unable to make conclusive remarks about the nature of the aggregation. W e have undertaken a study of the association of 2-methyl2-propanol at low concentrations by examining the T I of the 90% 13C-labeled central carbon. Measurement of the nuclear Overhauser effect (NOE) as a function of concentration revealed two relaxation mechanisms-dipolar and spin rotation. Comparison of the dipolar relaxation rate with the Debye-Stokes model for (1) Tucker, E.E.; Becker, E. D. J . Phys. Chem. 1973, 77, 1783-95. (2) Tucker, E. E.; Clem, T. R.; Seeman, J. I.; Becker, E. D. J. Phys. Chem. 1975. 79. 1005-8. ( 3 ) Dais, P.; Gibb, V.; Kenney-Wallace, G. A.; Reynolds, W. F. Chem. Phys. 1980, 47, 405-15.
molecular tumbling and dipolar relaxation provides useful insight into the limitations of the model. A microviscosity correction to this model applied to our data provides support for the conclusion of Tucker and Becker' that trimer is the smallest associated species, but it does not identify unambiguously the higher aggregates. Experimental Section To ensure that the relaxation is predominantly dipolar and the tumbling as isotropic as possible, we have chosen to study the central carbon of 2-methyl-2-propano1, labeled with 13Cto enhance the signal. The methyl carbons would be expected to relax significantly by spin-rotation interactions because of their internal rotation, and hydrogen relaxation is complicated by intermolecular interactions. In an earlier study,2 Tucker et al. determined that the central carbon had an N O E of 1.60 for a 1.O M solution in hexadecane, and they assumed that the dipolar mechanism dominates throughout the concentration range. In order to maintain a viscosity that is invariant with alcohol concentration, we have chosen as solvent he~adecane-d~~,'J whose viscosity (3.08 c P ) ~is almost identical with that of 2-methyl-2propanol (3.31 C P ) . ~ 13C-labeled2-methyl-2-propanol was prepared from 90% 13Clabeled acetone (Merck) by reaction with methyllithium in diethyl ether; the product was purified by distillation and gas chromatography and kept dry with calcium hydride. Unlabeled 2methyl-2-propanol (Fisher) was dried over calcium hydride. Hexadecane-d3, (Merck, Lot F-589) was not purified prior to use. Solutions of labeled alcohol less than 0.45 M were prepared by weighing the h e ~ a d e c a n e - dsolvent ~ ~ into a 10-mm cylindrical microcell (Wilmad) and adding the 2-methyl-2-propanol to the frozen solvent via a calibrated vacuum line at 28 OC. A calibrated Mensor quartz manometer and the ideal gas law were used to determine the amount of alcohol (maximum pressure 2 torr). It was assumed that the volume of the solvent was unchanged by the addition of the alcohol.' Samples of higher concentration were prepared by weighing the unlabeled alcohol into calibrated 1-mL volumetric flasks. Aliquots of these solutions were transferred
~
0022-365418412088-6075$01.50/0
(4) Ancian, B.; Tiffon, B.; Dubois, J.-E. J. Chem. Phys. 1981, 74, 5857-62.
0 1984 American Chemical Society
Sweeting and Becker
6076 The Journal of Physical Chemistry, Vol. 88, No. 24, 1984
TABLE I: Deoendence on Concentration of the Relaxation Rates" for the Central Carbon of 2-Methvl-2-orosanol in Hexadecane
~~
~
0.01 16 0.01 72 0.0264 0.0564 0.0673 0.0867 0.145 0.451 0.658 1.017' 1.056
76 (8) 83 (6) 79 ( 5 ) 59 (2) 57 (7) 49.5 (0.7) 39 (2) 32 (4) 23 (3) 24 (1) 24 (4)
0.0132 0.0120 0.0127 0.0169 0.0175 0.0202 0.0256 0.0313 0.0435 0.0417 0.0417
RI (0.0014) (0.0007) (0.0008) (0.0006) (0.0021) (0.0003) (0.0013) (0.0039) (0.0057) (0.0017) (0.0070)
NOE 0.41 (0.04)b 0.75 (0.03) 0.56 (0.04) 0.88 (0.03)b 0.97 (0.06) 1.13 (0.02) 1.34 (0.05) 1.66 (0.20)b 1.58 (0.19)b 1.60 (0.30) 1.56 (0.19)
"All relaxation rates are in s-I; standard deviations are reported in parentheses. of the TI and N O E values. OFrom ref 3.
to the microcells. All samples were degassed by at least five freeze-pump-thaw cycles before sealing. The concentrations are estimated to be accurate to 0.5%. The sample tubes were soaked in a NazEDTA solution, rinsed at least five times in deionized water, soaked overnight in reagent grade dilute hydrochloric acid (Fisher, less than 0.00001% Fe), rinsed at least five times with deionized water, and dried for several days at 120 "C. Access to the cylindrical bulb was accomplished with a teflon and glass syringe assembly; the syringe and volumetric flasks were subjected to the same cleaning procedure as the cells. The microcell, filled a few millimeters up the narrow stem (ca. 1 mm i,d,), was used to minimize the surface area and the gas volume above the sample and thus minimize the gas-phase spin-rotation relaxation mechanism. The microcells were inserted into a standard 10-mm tube containing carbon tetrachloride; this ensured that the rf coil was filled with material of approximately the same magnetic susceptibility and thus that the Bohomogeneity was optimized for that probe. Spectra for the carbon spin-lattice relaxation time (TI) measurements were obtained at 68 MHz on the N I H spectrometer which consists of a Bruker magnet and 10-mm carbon probe with a T & T Controls solid-state proportional counter to control the temperature, a Nicolet 1180 data system and 301 Butterworth filters, a General Radio Co. 1164-A frequency synthesizer, and a Amplifier Research power amplifier. The spectrometer was operated with an 8000-Hz bandwidth, with the signal of interest offset from resonance 600 Hz;ABo/Bl = 0.01. The 180' pulse width was determined before each run with an ethylene glycol sample of similar geometry; the 90° pulse width could not be determined as accurately and was set at half of the 180" width. The relaxation times were measured at 28 f 1 OC with a 180-T-90 sequence: usually the fast inversion-recovery sequence6 with a preparation time7 of approximately 3TI. The variable delays were arranged "randomly" so as to minimize the systematic errors resulting from drift of component characteristics with time. From 7 to 25 variable delays were used per sample; the spacing of the delay times was approximately exponential. The total time for each measurement varied from half a day to a week, with the number of spectra accumulated for each variable delay between 16 and 400. The peak areas for both TI and nuclear Overhauser effect (NOE) were measured by the NTCFT integral routine and are the average of at least three integrations of the peak. The Tl values were determined by an unweighted nonlinear least-squares fit to M ( t ) = Mo[l - (1 - k [ l - exp(-W/T,)]) exp(-t/T1)l where W is the preparation time' in the fast inversion-recovery sequence and k is an adjustable parameter to account for offset and imperfect rf pulses. The fit of Mo, k , and T, to the recovery of the magnetization with time was accomplished with the modified Marquardt-Levenberg method incorporated into MLAB.8 The standard deviations of the TI values were used as ( 5 ) Levy, G . C.; Peat, I. R. J . Ma n. Reson. 1975, 18, 500-21. ( 6 ) Canet, D.; Levy, G. C.; Peat, I. J. Magn. Reson. 197518, 199-204. (7) Sweeting, L. M. J . Magn. Reson. 1982, 48, 311-3.
ff.
R,DD 0.0027 (0.0004) 0.0045 (0.0004) 0.0036 (0.0003) 0.0074 (0.0004) 0.0085 (0.001 1) 0.01 15 (0.0002) 0.0174 (0.0010) 0.0261 (0.0047) 0.0345 (0.0062) 0.0335 (0.0064) 0.0326 (0.0078)
R,(other) 0.0105 (0.0015) 0.0075 (0.0008) 0.0092 (0.0009) 0.0094 (0.0007) 0.0090 (0.0023) 0.0087 (0.0004) 0.0082 (0.0016) 0.0052 (0.0061) 0.0090 (0.0084) 0.0082 (0.0068) 0.0091 (0.0104)
Estimated from the correlation between the standard deviations
-
3 ,
1
0
- 2 E
m i I
c o
4
I
-1
-- -2 Y
-3
-7 -5 -4
0
-E -6 Y
C -7
d
+ -8
-
L
0
Time, seconds
Figure 1. Plot of n + In [ ( M , - M,)/BMO]vs. time (seconds), where B = 1 - k(1 - exp(-W/T,)): (A)n = 0, 0.145 M 2-methyl-2-propanol in C16D34; (0) n = 1, 0.0867 M 2-methyl-2-propanol in C,,D,4; ( 0 ) n = 2, 0.0172 M 2-methyl-2-propanol in CI6D3,+
estimates of the probable error of the determinations and are reported with the values of TI in Table I; the value of k was about -0.85 in each case. Typical data are shown in Figure 1, plotted on a logarithmic scale; it is clear from the fit of the data to a straight line that, within the time the magnetization was examined, the relaxation can be described by a single exponential. The NOE values were measured after a preparation time of at least five times the carbon Tl since both the proton and carbon populations should be completely recovered in that time.g All those NOE values measured in successive experiments with the decoupler on and off were repeated when the alternating acquisition technique became available on the spectrometer; standard deviations reported in Table I are for two-four measurements. Time prohibited replication of all the NOE measurements; since the percent error in the NOE was observed to be proportional to the percent error in the TI,the errors in the other NOE values were estimated from this correlation. The large estimated error in Rl(other) at high concentrations is a consequence of obtaining the value by subtracting two very similar numbers. The dipolar relaxation rates (Table I) were determined from the observed rates and NOE with the equation RIDD= Rl(NOE/1.998)
The residual rate (Table I) &(other) = R I - RIDD.The estimated errors in RIDD and &(other) were calculated from the standard deviations in the T I and N O E values. The comparison of the experimental dipolar relaxation rates with the rates predicted by the association models was accom(8) Knott, G. D. Comput. Programs Biomed. 1979, 10, 271-80. ( 9 ) Canet, D. J . Magn. Reson. 1976, 23, 361-4.
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 6077
N M R Study of 2-Methyl-2-propanol plished by numerically determining the root of the equation describing the stoichiometric concentration C a s a function of monomer concentration m (eq 4, 6, and 8) using the equilibrium constants of Tucker and Beckerl and MLABa8 The relaxation rate of the monomer RIDD(m)was adjusted to fit the observed relaxation rate RIDDand the appropriate monomer concentration m and equilibrium constants by applying the modified Marquardt-Levenberg method of MLAB8 to eq 5 , 7, and 9. Those models incorporating an infinite series were approximated by polynomials; terms with powers of m higher than 8 contribute less than 3% to the total concentration and were omitted.
Results and Discussion Table I gives the experimental TI and N O E values over the concentration range 0.01-1.1 M for 2-methyl-2-propanol in hexadecane-d3.+ As expected, TI varies significantly (by a factor of three) over this concentration range. Unexpectedly, the NOE also changes by a factor of at least two. Although there are two relaxation mechanisms, the recovery of the magnetization follows a single exponential (Figure 1) and thus the N O E can be used reliably to extract the dipolar relaxation rate. Calculation yields a dipolar relaxation rate that varies by about a factor of ten, and a “residual” relaxation rate that is approximately constant over the concentration range (Table I). We discuss first the interpretation of the dipolar relaxation, then return to a consideration of the residual effect. Dipolar Relaxation. In the extreme narrowing limit characteristic of small molecules, the dipolar relaxation rate, RIDD,of a carbon nucleus resulting from interaction with a single proton is given by RIDD= l/TiDD = yC2yH2h7C/r6 (1) where yc and yH are the magnetogyric ratios of the two nuclei, r is the distance between them, and 7c is the rotational correlation time. If there are several protons, the relaxation rate will be the sum of the individual rates; if there is a single correlation time, the relaxation will be exponential, as observed. With the reasonable assumption that the intramolecular geometry of a single alcohol molecule does not vary significantly as the monomers associate via hydrogen bonding, all terms in eq 1 are constant except for rc, whose variation provides a probe of the molecular association. The simple Debye-Stokes model for rotation of a spherical molecule of diameter a in a continuous fluid of viscosity 9 predicts that the rotational correlation time rc = 4?ra3q/3kT
(2)
where k is the Boltzmann constant and T is the absolute temperature. Combining eq 1 and 2 reveals that the dipolar relaxation rate is proportional to the volume of the molecule and the viscosity and inversely proportional to the temperature, assuming that the number of protons causing the dipolar relaxation is constant. As the concentration is increased and the alcohol molecules aggregate, the effective volume of the molecule will increase. If the aggregates are spherical and the viscosity and temperature are unchanged, there will be a proportionate change in RIDD.For an aggregate of i alcohol molecules, the relaxation rate will be RIDD(i)= iRIDD(m) (3) where RIDD(m)is the relaxation rate of the monomer. The measured value of RIDDis then the weighted average of the RIDD(i),provided that the rate of exchange between hydrogenbonded species is fast compared with the relaxation rate and slow compared with l/Tc. The estimated exchange rates for alcohol hydrogen bonds (from ultrasonic measurements) are in the range of lo6 to lo8 which is well above the relaxation rates of s-I and below the typical tumbling rates for small molecules of 1010 to 1012 s-I. (10) See, for example: Musa, R. S.;Eisner, M. J . Chem. Phys. 1959,30, 227-33. Yasunaga, T.; Tatsumoto, N.; Inoue, H.; Miura, M. J. Phys. Chem. 1969, 73, 477-82.
27
9
!f
Omo4
i 8 l
I
0.03
n c, c,
(g 0.02 C
.-0
c,
$
0.01
a
d
& 0
I
‘
01
I
O‘.5 1‘
0‘.1
2
Concentration, moles/litre Figure 2. Simple Debye-Stokes approximation to (a) - 1-3-6 model (eq 5), K, = 34.1, K6 = 6593, RIDD(m)= 0.0078 0.0004 s-I, rms error = 0.00346; (b) --- 1-3- chain model (eq 7), K3 = 24.6, K, = 3.72, RIDD(m)= 0.0080 0.0004 s-’, rms error = 0.00368; (c) 1-3cyclic model (eq 9), K3 = 25.5, K, = 4.55, RIDD(m)= 0.0088 f 0.0006 s-’, rms error = 0.00464. Note that the logarithmic concentration scale is chosen for ease of visual evaluation. Equilibrium constants are from
*
*
-e--
Tucker and Becker.’
In principle, the relaxation data alone can be interpreted in terms of the various aggregates present, but the precision of the data is inadequate for a priori determination of the nature and extent of the aggregation. Thus we have chosen to use models and equilibrium constants derived by Tucker and Becker’ as a basis for our data analysis. In their extensive, careful studies in the same solvent, they found a negligible concentration of dimer but identified the presence of trimer. They found their datavapor pressure (several hundred points), infrared, and proton N M R chemical shift-could be adequately represented with only one higher species, the hexamer (the “1-3-6” model). Their data were equally compatible with infinite subsequent aggregation onto the trimer which could be described by a single equilibrium constant for the n-mer to ( n 1)-mer equilibrium (the “1-3-m” model). The 1-3- model has two forms, one for chain polymers and the other for cyclic or cooperative polymers. For the 1-3-6 model, the analytical concentration C is related to the monomer concentration m by
+
C=m
+ 3K3m3 + 6K6m6
(4)
The average dipolar relaxation rate is thus given by RIDD(obsd)= (RIDD(m)/C)(m + 9K3m3
+ 36K6m6) ( 5 )
For the 1-3-m models, chain aggregation is represented by
C=m
+ 3K3m3+ 4K3K,m4 + ... iK3K,i-3mi
RIDD(obsd)= (RIDD(m)/C)(m + 9K3m3+ 16K3K,m4
(6)
+ ...) (7)
and cyclic aggregation by C=m
+ 3K3m3(1 + K,m + Km2m2+ ...)
RIDD(obsd)= (RIDD(m)/C)(m
+ 3K3m3(3 + 4K-m + 5K,2m2...))
(8) (9)
The fits of eq 5 , 7, and 9 to the measured relaxation rates, RIDD, adjusting only the relaxation rate of the monomer, RIDD(m),are given in Figure 2, along with the equilibrium constants. It is clear that all thrde models underestimate the magnitude of the change in the dipolar relaxation rate with concentration and overestimate the relaxation rate of the monomer. Treating the equilibrium
6078
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984
Sweeting and Becker
TABLE II: Dependence on Solvent and Solute Shape of the Microviscosity 1-3-6 Aggregation Model" solvent spherical aggregate linear aggregate radiusb RIDDW rms error RIDD(m) 7 0.00350 (0.0001 1) 0.00224 0.00178 (0.00009) 5 0.00352 (0.00012) 0.00222 0.00184 (0.00009) 3 0.00360 (0.00012) 0.00218 0.00202 (0.00009) "Relaxation rates are in s-I. conformation.
rms error 0.00339 0.00331 0.00308
Standard deviations are reported in parentheses. b u n i t s are CH2 diameters; a radius of 7 represents a completely anti
constants as adjustable parameters does not improve the fit to the model significantly. The poor fit must result from deficiencies in the Debye-Stokes model which assumes the surface of the molecule is rough enough that stick (not slip or intermediate) boundary conditions apply and that solvent molecules have zero volume.I1 The nonzero solvent volume must certainly contribute to the failure of the model to predict the dipolar relaxation behavior of this system where the solvent molecules are larger than those of the solute. The Debye-Stokes model treats the solvent as a continuous fluid in which the solute sphere rotates under viscous drag. One approach to treating systems where the solvent molecules are too large to constitute a continuous fluid is the microviscosity theory of Gierer and Wirtz.lz They calculate that the correlation time for a solute i in a solvent s is dependent on their radii ai and a,. .r,-(microvisc) = (6a,/ai (1 a,/ai)-3)-1~C (Stokes) (10)
+ +
E
0.03
II)
V c,
8
0.02
C
.-0
c,
;0.01
a Although this and other models of the liquid state have been ~ r i t i c i z e d , they ~ ~ J ~have also been used with some s u c c e s ~ . ~ ~ ~ ~ ~ ~ ~ ~ W e can distinguish two limiting cases of alcohol I 0 aggregation-spherical, in which the molecules are assumed to d.01 0'.1 0'.5 1' : .2 be close-packed, and linear, in which the hydrogen-bonded chains Concentrat ion, m o l e s / l itre are extended. From simple geometric considerations, we can calculate an approximate radius for each aggregate. It is not Figure 3. Microviscosity model for 1-3-6 association, assuming (a) spherical alcohol aggregates, RIDD(m)= 0.00351 0.00012 s-l, rms obvious what radius would be appropriate for the solvent C16D34; error = 0.00223 (eq 11); (b) --- linear alcohol aggregates, RIDD(m)= hence a series of possible radii from about two CH2 units for a 0.00180 =k 0.00008 s-I, rms error = 0.00335. Solvent is assumed to be close-packed, almost spherical solvent to seven CH2 units for an almost anti, with a radius of six CH2 units. Equilibrium constants are all anti conformation was tested. Equations 5, 7, and 9 were in Figure 2a. rewritten for spherical and linear alcohol aggregation and for three to five values of the solvent radius by correcting the coefficients of each hydrogen-bonded species in the relaxation equation by the appropriate microviscosity factor (six to ten equations for each 0.04 model). For example, for spherical alcohol aggregates and a solvent with an effective radius of six CH2units, the microviscosity adjusted eq 5 is RIDD= (A/C)(O.O417m 0.711K3m3 4.03K6m6) (11) 0.03 The relaxation rate of the monomer, RIDD(m),is given by the coefficient of monomer concentration, multiplied by the adjustable parameter A . The results of typical fits, given in Table 11, illustrate that the model is insensitive to the effective radius of the solvent 0.02 molecules and slightly sensitive to the assumed shape of the aggregate; thus we have used a reasonable solvent diameter of six CH2, mostly anti. All three association models give a significantly better fit to the experimental data than the Debye-Stokes model, 0.01 as illustrated in Figures 3 and 4, with spherical alcohol aggregates giving the best fit in each case. All three spherical association models give a value for the relaxation rate of the monomer which is indistinguishable from the mean observed rate for the three I 0 lowest concentrations (0.0036 f 0.0007 s-l), where the monomer 0'.01 0'. I OI.5 i' .2 comprises greater than 95% of the species present. The relaxation rate of the monomer and the carbon-hydrogen Concentrat ion, moles/l itre distance (r = 2.2 X lo-* cm) can be substituted into eq 1, 2, and Figure 4. Microviscosity model for 1-3-m chain association, using a 10 to calculate the effective radius of the tumbling monomer. If modified version of eq 7 and assuming a solvent with a radius of six CH2 we assume that all ten hydrogens contribute to the dipolar reunits and spherical aggregates; equilibrium constants are in Figure 2b. d
cr"
I
I
*
+
+
I
*
(11) Dote, J. L.; Kivelson, D.; Schwartz, R. N. J. Phys. Chem. 1981,85, 2169-80. (12) Gierer, A.; Wirtz, K. Z . Naturforsch., A 1953, 8A, 532-8. (13) Green, D. K.; Powles, J. G. Proc. Phys. SOC.,London 1965, 85, 87-102. (14) Prichard, A. M.; Richards, R. E. Trans. Faraday SOC.1966, 62, 1388-99. (15) Gillen, K. T.; Noggle, J. H. J . Chem. Phys. 1970, 53, 801-9.
RIDD(m)= 0.00375 0.00012 s-l, rms error = 0.00209. The similarly modified version of eq 9 for cyclic association, whose equilibrium constants are in Figure 2c, gives almost identical results: RIDD(m)= 0.00377 f 0.00013 s-l, rms error = 0.00214.
laxation, the Gierer-Wirtz microviscosity model gives a value for the radius a of (2.1 f 1.1) X lo-* cm, within experimental error the same as that obtained from Dreiding models (3.0 X lo-* cm).
The Journal of Physical Chemistry, Vol. 88, No. 24, 1984 6079
N M R Study of 2-Methyl-2-propanol TABLE III: Relaxation Rates" as a Function of Temperature temp, K R1
301 333
NOE
RIDD
0.0202 (0.0002)
1.13 (0.02)
0.01 15 (0.0002)
0.0132 (0.0017)
0.34 (0.04)
0.0023 (0.0004) 0.0023 (0.0006)b
333
SR rate 0.0087 (0.0004) 0.01 10 (0.0017) 0.0096 (0.0005)c
a Concentration,0.086 M. All relaxation rates are in units of s-I; standard deviations are reported in parentheses. Calculated from microviscosity 1-3-6 model based on eq 1, 2, and 10, using literature values6 for the viscosity of hexadecane and extrapolating from the 300-320 K equilibrium constants.' cCalculated by assuming the rate is proportional to temperature.
*
The simple Debye-Stokes model gives a = (0.72 0.36) X cm, much too small. The dipolar relaxation rate should decrease as the temperature is increased, from the effect of temperature and viscosity (eq 1) and from the change in the equilibrium constants (eq 5, 7, and 9). Using the 1-3-6 microviscosity model (eq 1l), the viscosity data of A n ~ i a nand , ~ extrapolating Tucker and Becker's 30-50 OC equilibrium constant data' to 60 OC, one calculates that the relaxation rate should be 0.0023 s-' at 60 OC, exactly what is observed (Table 111). Internal methyl rotation may well contribute to the dipolar relaxation but this motion should provide only a constant, small background in the relaxation rate. Because the spin-rotation relaxation rate of the central carbon is independent of concentration (see Residual Relaxation), the rotation rate of the methyl and/or tert-butyl groups and the dipolar relaxation it causes must be constant in these solutions. Such hindered methyls with sixfold barriers have been shown to be free rotorsI6 and thus to have a dipolar relaxation rate for the methyl carbon 1/9 that of a rigid methyl carbon in an isotropic tumbler. A similar effect at the quaternary carbon would be no greater than the experimental error of the measurements and thus too small to be detected. It is unlikely that anisotropic motion of the aggregates complicates the interpretation of the relaxation rates if, as Table I1 indicates, the aggregates are spherical. Even molecules with high anisotropy such as methylbenz~furans'~ exhibit at most a 30% difference in the dipolar relaxation rate between carbons on and remote from the preferred axis. Without detailed knowledge of the structure of the aggregate, it is impossible to estimate the effect of such a n i s o t r ~ p y . ' ~ J ~ The value of the relaxation rate for the central carbon of the monomer is comparable to that obtained by Szeverenyi, Vold, and VoldIg for cyanoacetylene. Its central carbon undergoes dipolar relaxation due to the terminal proton at a rate of 0.0006 s-'. If the correlation time of this thin rod were the same as that of the spherical 2-methyl-2-propano1, ten protons similarly situated (as they are in the alcohol) would cause a relaxation rate of about 0.006 s-'. Given the differences in distance and correlation time (resulting from the differences in molecular shape), the dipolar relaxation rate of 2-methyl-2-propanol monomer (0.0036 s-') is surprisingly close to this estimated value. The methylated quaternary carbons in methylbenzofurans exhibit dipolar relaxation rates of 0.0075 to 0.013 s-',17 Since the smaller spherical 2methyl-2-propanol must undergo much faster tumbling, its quaternary carbon should relax more slowly, as observed. The paucity of T I data on quaternary carbons makes further comparisons difficult. The dipolar relaxation rate of the quaternary carbon gives a satisfactory fit to the equilibrium models for alcohol association previously determined, provided the Gierer-Wirtz microviscosity model12is used to calculate the effect of changing molecular size on the correlation time. The model correctly predicts the surprisingly large temperature dependence of the dipolar relaxation rate, and the relaxation rate of the monomer agrees well with that observed for other molecules of similar size. However, the model is unable to determine unambiguously the size distribution of
aggregates larger than trimer. If the shapes of the aggregates were known, it would be possible to allow for the effect of free volume;" such a model predicts experimental observations better than that of Gierer and WirtzI2 in other systems where small solute molecules are dissolved by large solvent molecules. Residual Relaxation. The relaxation of the quaternary carbon occurs by a second mechanism (Table I) which is independent of concentration. (A linear least-squares analysis of the rates as a function of concentration yields a mean of 0.0086 f 0.0017 s-l, a slope of O.OOO05, and a correlation coefficient of -0.166.) Scalar relaxation of the carbon from chemical exchange of the OH can be ruled out as a relaxation mechanism, and relaxation by paramagnetic impurities from contamination of the glassware is unlikely because the residual relaxation rate would be expected to be highly erratic, rather than almost constant. Chemical shift anisotropy (CSA) is unlikely as a relaxation mechanism for this sp3 carbon in an isotropic environment (an sp carbon relaxes at only 0.005 s-' by CSA'9,20). The relaxation time of the quaternary carbon in 0.0867 M alcohol is 51 f 2 s at 15 MHz and 49.5 f 0.7 s at 68 MHz; if all of the residual relaxation were caused by CSA, a T1 at 15 MHz of 51 s would correspond to a T, at 68 M H z of 30 s. Thus chemical shift anisotropy does not contribute significantly to the relaxation. The only remaining known relaxation mechanism is spin-rotation. Interruptions in the motion of fast rotors create magnetic fields fluctuating at the Larmor frequency and thus provide a relaxation mechanism; the magnitude of such field fluctuat i o n ~ ' ~ determines * ~ ~ * ~ ~the* spin-rotation ~~ relaxation rate. This mechanism may be detected by the fact that the relaxation rate increases with t e m p e r a t ~ r e , exactly ~ ~ ~ * ~the reverse of the temperature dependence of other relaxation mechanisms in mobile liquids. The relaxation time and NOE of the 0.0867 M solution were measured at 60 "C and are reported in Table 111. Within experimental error, the calculated rate for spin-rotation from simple proportionality to temperature is just what is observed. We have taken care to minimize the possibility that such relaxation occurs by rotation of the molecules in the gas phase with the relaxed molecules being reabsorbed into the solution. If gas-phase relaxation were occurring, the high vapor pressure of the alcohol at 60 OC would result in a much greater increase in relaxation rate with temperature than that predicted by theory. It is not obvious whether the spin-rotation relaxation is caused by molecular rotation or by internal rotation of the methyl groups. The spin-rotation mechanism has been observed in the liquid phase for molecules of the size of the monomeric alcohol in cases where other relaxation mechanisms are barely competitive, and the rates are similar to ours.17,19,20However, the fraction of monomer decreases rapidly as the concentration increases (from 99% at 0.01 16 M to 14% at 1.056 M); the overall spin-rotation relaxation rate would be expected to decrease with monomer concentration. Although rapid exchange of individual molecules between monomer and aggregate would permit relaxation of the carbon by spin-rotation while it is in the monomer, the rate should decrease with concentration because of the higher fraction of time that the individual molecule spends as part of a large molecule which relaxes at a negligible rate by this mechanism.
(16) Alger, T.D.; Grant, D. M.; Harris, R. K. J . Phys. Chem. 1972, 76,
(20) Vold, R. R.; Vold, R. L.; Szeverenyi, N. M. J. Chem. Phys. 1979, 70, 5213-7. (21) Flygare, W. H. J . Chem. Phys. 1964, 41, 793-800. (22) Dubin, A. S.; Chan, S. I. Chem. Phys. 1967, 46, 4533-5. (23) Burke, T. E.; Chan, S.I. J. Mugn. Reson. 1970, 2, 120-40. (24) Hubbard, P. S. Phys. Rev. 1963, 131, 1155-65.
'
281-3.
(17) Platzer, N. Org. Magn. Reson. 1978, 11, 350-6. (18) Woessner, D. E. J . Chem. Phys. 1965, 42, 1855-9. (19) Szeverenyi, N. M.; Vold, R. R.; Vold, R. L. Chem. Phys. 1976, 18,
23-30.
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The Journal of Physical Chemistry, Vol. 88, No. 24, 1984
Most previous studies, both theoretical and experimental, of the internal spin-rotation relaxation of methyl groups have looked only at the H14,23,24 and C25-27 of the methyl group itself and thus give little insight into the relaxation of the quaternary carbon. Spin-rotation relaxation from internal molecular rotation can be distinguished from that from overall rotation by theoretical calculations and determination of spectral densities; this has been done for simpler molecule^.'^-^^^^^,^^ Such calculations require more detailed knowledge of the structure than is currently available for the hydrogen-bonded aggregate. Margalit29 reports for 100% 2-methyl-2-propanol a reasonablez5activation energy for methyl rotation which is about 1/3 that for molecular rotation at 300 K other model compounds16suggest the barrier may be less, but both indicate that methyl rotation is fast enough to dominate the spin-rotation interaction. The theoretical work of Flygare*' and H ~ b b a r show d ~ ~ that the spin-rotation relaxation resulting from rotation of whole molecules is dependent on the angular momentum of the molecule and thus on the viscosity of the medium, suggesting that the relaxation rate from molecular rotation in liquids should be more sensitive to temperature than simple proportionality. Indeed, Burke and ChanZ3have calculated the temperature dependence of the internal and overall spin-rotation
Additions and Corrections rates for the trifluoromethyl carbon of benzotrifluoride and found that the internal spin-rotation rate was approximately linear with temperature but the overall spin-rotation rate changed much more dramatically, the curve approximating a fifth power dependence. Since the viscosity of the alcohol solutions changes by a factor of two from 28 to 60 O C , " the spin-rotation rate for 2-methyl2-propanol should be more sensitive to temperature than is observed (Table 111) if it results from rotation of the whole molecule. We conclude that the lack of a strong temperature and concentration dependence of this relaxation rate argues against rotation of the whole molecule causing the observed spin-rotation relaxation and in favor of internal rotation. It is possible that the geometry of the aggregates permits rotation of the tert-butyl group about the CO bond axis, and this rotation might well have a correlation time similar to that of the free molecule. The proposed cyclic structure for the trimer30would permit this kind of motion. We are exploring the possibility of distinguishing spin-rotation relaxation caused by tert-butyl group rotation from that caused by distant methyl group rotation with model compounds. Acknowledgment. Part of this work was done while L.M.S. was on sabbatical from Towson State University in the Laboratory of Chemical Phvsics. NIADDK. National Institutes of Health. We thank G. D.'Knott for helpfui discussions of the data analysis and the reviewers for several valuable suggestions,
(25) Zens, A. P.; Ellis, P.D.J . Am. Chem. SOC.1975, 97, 5685-8. (26) Tancredo, A.; Pizani, P. S.; Mendoca, C.;Farach, H. A.; Poole, Jr., C. P.; Ellis, P. D.; Byrd, R. A. J . Magn. Reson. 1978, 32, 227-31. (27) Hopkins, H. P.; Ali, S.Z . J . Phys. Chem. 1980, 84, 203-6. (28) Vold, R. R.; Vold, R. L. J . Chem. Phys. 1979, 71, 1508-9. Vold, R. R.; Vold, R. L.; Szeverenyi, N. M. J . Phys. Chem. 1981, 85, 1934-43. (29) Margalit, V. J. Chem. Phys. 1971, 55, 3072-8.
Registry No. CH3C(OH)(CH3),, 75-65-0. (30) Saunders, M.; Hyne, J. B. J. Chem. Phys. 1958, 29, 1319-23.
ADDITIONS AND CORRECTIONS 1980, Volume 84
Henry S.Judeikis: Mass Transport and Chemical Reaction in Cylindrical and Annular Flow Tubes. Page 2482. Equation 6 should appear as follows: 2 VW p=-----P T
~
4
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