4970
J. Phys. Chem. 1986, 90. 4910-4975
NMR Conformation Studies of Rotational Energetics about the C-C Bond in Trisubstituted Ethanes of the Types CH,X-CHX, and CH2X-CHY, Jenn-Shing Chen, Randall B. Shirts,* Department af Chemistry, Unicersity of Utah, Salt Lake City, Utah 841 12
and Wei-Chuwan Lin Department of Chemistry, National Taiwan University, Taipei, Taiwan, ROC (Received: January 8, 1986)
We have derived, based on rotational averaging and the Karplus equation, an equation governing the temperature dependence of the H-H vicinal coupling constant, J, for substituted ethanes of the form CH2XCHX2or CH2XCHY2. This model contrasts with the conventional rotational isomeric state model in that all rotational conformers are included instead of only the discrete stable forms. The model depends on an energy parameter, V I ,which is a measure of the maximum interaction energy between two C-X bonds or between C-X and C-Y bonds as a function of rotational configuration about the C-C axis. From the temperature variation of J in an inert solvent (e.g., cyclohexane), the characteristic constant A in the Karplus equation is determined. The value of A thus obtained is assumed to be constant for that particular compound regardless of temperature and solvent conditions and is employed to obtain the energy difference between gauche and trans conformers, AE, in various solvents at various temperatures. The value of AE in the gas phase can also be estimated by using solvent theory. Temperature and/or solvent dependence of J for CHzCICHClzand CH,BrCHBr, are taken as examples to illustrate the utility and accuracy of this new approach.
Introduction N M R spectroscopy has been widely employed to investigate conformational equilibria as well as conformational dynamics of substituted ethanes. Such work has led to quantitative information concerning internal rotational energetics and N M R parameters characteristic of each gauche or trans stable conformer.’ In some substituted ethanes, the energy barrier to internal rotation about the C-C bond is sufficiently high to display “frozen-out’’ spectra which allow one to determine the relative populations of the corresponding conformers as well as N M R parameters directly. In intermediate cases or at intermediate temperatures, “coalescence” spectra are seen which allow one to determine the rate of internal rotation (dynamic NMR).* However, most substituted ethanes have barriers to rotation which are sufficiently low to allow rapid hindered internal rotation on the NMR time scale; consequently, observed chemical shifts and coupling constants are weighted averages of the parameters for individual conformers. Determination of physical parameters from these rotationally averaged spectra is difficult, and such parameters are seldom as accurate as from frozen-out or coalescence spectra. Most approaches to the problem of low-energy barriers have followed the best-fit method pioneered by Gutowsky et aL3which involved searching a set of parameters characteristic of each stable conformer to find the best fit to the experimental data. These methods also tacitly assume that only stable conformers are important and that a rapid equilibrium between them is established (conventional rotational isomeric state model). However, Govil and B e r ~ t e i nand , ~ Heatley and Allen,5 have pointed out that this method can only give accurate results with difficulty or in some special cases. By averaging over rotation angles, Lin has developed the theory of observed H-H vicinal coupling constants as a function of temperature and energy barrier to internal rotation about the C-C bond in substituted ethanes of the type CH2XCH2Y,6 CH2XCHXY,’ and CH2XCHX2.8 The purpose of this paper (1) For a review, see: Internal Rotation in Molecules; Orville-Thomas, W. J., Ed.; Wiley: London, 1974. (2) Jackman, L. M.; Cotton, F. A. Dynamic NMR Spectroscopy: Wiley: New York, 1975. (3) Gutowsky, H. S.; Belford, G. G.; McMahon, P. E. J . Chem. Phys. 1962, 36, 3353. (4) Govil, G.; Berstein, H. J. J . Chem. Phys. 1967, 47. 2818. ( 5 ) Heatley, F.; Allen, G. Mol. Phys. 1969, 16, 77. (6) Lin, W. C. J . Chem. Phys. 1970, 52, 2805. (7) Lin, W. C. J . Chem. Phys. 1973, 58, 4971.
is to futher develop Link method and extend this theory to the case of ethanes of the type CH’XCHY,. Theory The dependence of the H-H vicinal coupling constant on the dihedral angle, 4, in substituted ethanes was investigated theoretically by K a r p l ~ s . ~Further work has resulted in a general form of the vicinal coupling constant as reviewed by Barfield and Grant.lo In the case of 1,1,2-trisubstituted ethanes, the vicinal coupling constants between the three distinct protons (see Figure l a ) are
+ B COS 4 + C (4 + 7 3 ~+) B COS (4 + 73~)+ C
Jl,(4) = A J13(4) = A
COS’
COS’
4
(la) (Ib)
where A , B, and C are constants characteristic of a given compound: ’/g is the constant phase difference between CH, and C H 3 moieties (see Figure la). Following the empirical assignment” B = -O.lA, and C = 0, eq 1 can be reduced to
Jl2(4) = A cos2 4 - 0.1A J13(4) = A
COS’
(4
+ 7 3 ~ -) 0.1A
COS
4
COS(^
(2a)
+
%T)
(2b)
The observed H-H vicinal coupling constants J12 and J 1 3 assume values in which eq 2a and 2b are averaged over a cycle of internal rotation in which each conformation of dihedral angle, 4, is assigned a weighting factor of exp(-E(+)/RT), where E ( 4 ) is the potential function for internal rotation, Tis absolute temperature, and R is universal gas constant; i.e. J12
=A
(COS’ J,3
= A ( C O S(4 ~
= (J12(4)) C#J) -
O.lA(cos 6 )
(3a)
= (Jd4))
+ 73~))- O.lA(cos (4 + 7 3 ~ ) )
(3b)
The notation (f(q5)) is defined by
so’”rc4) exp(-E(@)/RT) d@ (f(4)) =
(4)
(8) Lin, W. C.; Chen, J. S.; Chen, L. S.; LiR, S. J . Chinese Chem. SOC. 1976, 23, 117.
(9) Karplus, M. J . Chem. Phys. 1959, 30, 11. (10) Barfield, M.; Grant, D. M. Adu. Magn. Reson. 1965, 1, 149.
0022-3654/86/2090-4970.$01.50/00 1986 American Chemical Society
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The Journal of Physical Chemistry, Vol. 90, NO. 21, 1986 4971
Rotational Energetics in Trisubstituted Ethanes a
X
For most purposes, this energy difference is well approximated by 3 / 4 V 1 .It should be noted that eq 5 is only an approximation to the true potential for internal rotation. More Fourier coefficients might be added if additional experimental information were available. Usually, however, not even AE is known; this is especially true in solution. It is the purpose of this paper to describe a method to determine V , , and thus AE for a potential approximated by eq 5. One further notational simplification of eq 5 will now be introduced. Let OL = V3/2RT and @ = V l / 2 R T , and we express E(4) in units of R T
X
X
E ( 4 ) / R T = OL(COS 34 - 1)
+ COS 4 +
COS
(4
+ y 3 ~ -) 2 ) (8)
l b
To p r d further following Cin,+' we express the exponential terms within exp(-E($)/RT) in terms of hyperbolic Bessel functions, I,( z), through the identity14 m
exp(z cos e) = Io(z) + 2EI,(z) cos (le) I= I
(9)
we then have exp(-cu(cos 34 - 1)) = exp(a) exp(-a cos 34) m
+ 2CIl(-a) cos (314)) I= I exp(a){Zo(a) - 2Z1(a)cos 34 + 2Z2(a) cos 6 4 + . .I
(10)
Likewise exp(-P(cos 4 - 1)) N exp(P) X [Io(@)- 2Z,(P) cos 4 + 2Z,(P) cos 24 - 21,(P) cos 34
+ ..*I
= exp(a)(Zo(-a)
-5 0.00
1.57 3.14 4.71 ROTATION ANGLE (radians)
6.28
Figure 1. Details of modified Gwinn-Pitzer potential (eq 5): (a) Newman projection showing how conformational angle $I is defined for eq 5; (b) plot of eq 5 vs. 4. Dashed curve is the V, contribution (first term in eq 5); dotted curves are VI contributions (second and third terms in eq 5); solid line is the total. Trans conformer is at $I = 5a/3,gauche conformers are near 4 = a/3 and $I = a.
Since the potential function is periodic in the angle 4, E ( 4 ) can rigorously be expressed as a Fourier series. For our purposes, the Fourier series may be truncated and expressed as a modified Gwinn-Pitzer" potential:
(1 1,)
Here we have applied the parity property of hyperbolic Bessel function" I2A-z) =
I2&)
(13a)
E(4) =
v3
-(cos 2
34 - 1)
VI + -(cos 2
4 - 1)
VI + -(cos 2
(4
+ Y37r) - 1)
(5)
In eq 5, V3 is the amplitude of the threefold barrier potential common to rotation about C-C bonds'* and is given the value 3.0 kcal/m01.'~ The parameter V I is the amplitude of the onefold barrier due to the interaction of bond moments of substituents; it controls the energy difference between conformers. The trans conformer (4 = 5 ~ / 3 is) more stable when VI < 0. There are two gauche conformers a t angles 4 = a / 3 q and = 7~ - q, where q is a small angular correction shown by elementary manipulation to be approximated by
+
S S2 7S3 +--19S4 32805 l - - 18 + - - - 81 2916
17s' 118098
+
...)
where S = Vl/V3. The gauche conformer is more stable when VI > 0. The potential difference between the two conformers may similarly be evaluated to be AE =
Etrans
3v1 4
(
- Epauche =
vi2 48V3
s
s2 s3+ 7s4 + 864 31104
I--+--18 144
...)
for n = 0, 1, 2, .... The Bessel function expansion may be truncated by neglecting series terms of sufficiently high order. This truncation can be done because, for the systems investigated, VI I 1 kcal/rnol. Thus, within the experimental temperature range, I1, the Bessel functions of high order become negligibly small compared with the first few.I5 Substituting eq 9-11 into eq 3 and 4 and making use of well-known trigonometric identities and the orthogonality of trigonometric functions, we arrive at the following expressionss (COS' (COS
4) = (cos2 (4
+ Y3r))= K 2 / K l
4 ) = (cos (4
+ y37r)) = K3/KI
(14) (15)
where K l , K2, and K 3 are defined below, and
As expected, for substituted ethanes of the type CH2XCHX2 or CH2XCHY2with low barriers to internal rotation, the two protons in the CH2X moiety are magnetically equivalent.16 K l , K2. and K3 assume the following values:
(7)
(11) Gwinn, W. D.; Pitzer, K. S . J . Chem. Phys. 1948, 16, 303. (12) Wilson, E. B.,Jr. Adu. Chem. Phys. 1959, 2, 307. (13) Other researchers have adopted values for V, ranging from 2.7 to 3.3 kcal/mol. None of the results of this work are sensitive to changes of V, within this range.
(14) Abramowitz, M., Stegun, I. A,, Eds. Handbook of Mathematical Functions; Dover: New York, 1965; p 376. (15) Watson, G. N. Theory of Bessel Function, 2nd ed.; Cambridge University: Cambridge, U.K.,1958. (16) Pople, J. A. et al. High Resolution NMR; McGraw-Hill: New York, 1959.
4972
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
Chen et al.
K2 = Io(a)[ro2(P) + XIo(P) 1203) - 3/21i2(P) - 122(P)3 + I d d [ Y d O ( P ) I,(@)- 3 I l ( P ) M P ) + 4 W ) 13(P)1 + )/212(4[-122(P) + Il(P) I 3 ( P ) + [,(PI W)l (18) K3 = I o ( a ) [ - I o ( P ) I , ( @ )+ 2 I , ( P ) Ia(P)I + II(4VI2(P)- I o ( P ) I2(P) + - fl(P) I,(@) - Io(P) 14(P)1 (19) Terms in (17-19) that have sums of the orders of the Bessel functions of P greater than 4 have been neglected. This approximation is good to four decimal places for /3 near 0.5." For computational convenience, hyperbolic Bessel function can either be calculated by available software'* or may be approxiseries representation20 mated either by p ~ l y n o m i a or l~~
I&)
= ')/*z'";o
-
0. It is also interesting to note that J / A = 0.5 independent of temperature if VI = 0. These facts allow a quick determination of which conformer is more stable if VI is independent of temperature. In such a case, a coupling constant which increases with temperature indicates the trans conformer is more stable, or a coupling constant which decreases with temperature, indicates the gauche conformer is more stable. The value of VI, the maximum interaction between two vicinal bond moments for a given compound in a given solvent, is expected to vary with temperature due to the variation of the dielectric constant of the medium with temperature. This effect, however, may be minimized by choosing a reference solvent whose dielectric constant exhibits a very slight temperature dependence. We have chosen cyclohexane as a reference solvent because the temperature dependence of its dielectric constant is only 1.3 X lo4 per degree and, in addition, cyclohexane is relatively inert in forming solvent-specific interactions with solutes. We will assume, therefore, that the VI value of a given compound in cyclohexane is constant as temperature varies. We can also rewrite eq 16 as follows
A =
JK, K , - 0.1K3 = f(V1)
where T and V, are fixed. The combination of eq 21 and the physically plausible assumption of temperature independence of VI in cyclohexane will facilitate the determination of A through the following proposed scheme: For a given system (e.g., CH2XCHX2in cyclohexane), we plot A vs. VI calculated from eq 21 for a series of temperature values using experimental values of J at these temperatures. If both VI and A are independent of ( 1 7) Additional terms have been included in eq 17-19 over those found in ref 8. These terms do not make a significant difference in the result. The last term in eq 17 is about a 1% correction. The rest of the additional terms are even less significant. (18) For example, IMSL Library. (19) Reference 12, p 378. (20) Reference 12, p 375. (21) Abraham, R. J.; Bretscheider, E. Reference 1, Chapter 13.
t
4
'=.
0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 -
uu
-
I
Figure 2. Equal value contours of VI plotted for J / A vs. temperature as given by eq 16.
1
11.0
-
10.8
-
10.6 10.4
-
0.0
i 0.2
0.4
06
0.8
V, (kcal/rnole)
1.0
1.2
1.4
Figure 3. Determination of A value for CH,CICHCI, in cyclohexane. Each curve represents the calculation of A from eq 21 by using the measured value of J a t each of four different temperatures (see Table I) for varying values of VI. The intersection of the four curves gives the correct value of A and V , .
temperature in cyclohexane as assumed, contours corresponding to different temperatures intersect at a unique point having coordinates A and VI which are the desired values. The determined value of A can then be used to determine VI for other solvents by interpolation in Figure 2. This scheme is analogous to the one used by Rose and Drago2, to determine the association constant for a molecular complex. Results and Discussion Although the proposed method and the foregoing analysis is applicable to compounds of the type CH2XCHX, and CHzXCHY2,we have chosen only compounds of the first type, specifically CH2C1CHC12and CH2BrCHBr2,to demonstrate the accuracy and utility of this method. These two solutes have been intensively studied by using NMR spectroscopy, and some conformation information is available for CH2C1CHCI2. NMR spectra of CH2ClCHCIzand CH2BrCHBr2were taken in various solvents. Samples were prepared with 10% solute mol (22) Rose, N. J.; Drago, R. S. J . Am. Chem. SOC.1959, 81, 6138.
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 4973
Rotational Energetics in Trisubstituted Ethanes
TABLE I: Experimental NMR and Physical Parameters for CH2CICHCI, in Various Solvents A = 11.10 Hz solvent T, K J, Hz kcal/mol AE? kcal/mol cyclohexane 273 6.25 0.70 0.53 12.5
neat liquid
N
5 U
12.3 12.1
chloroform
11.9 11.7 11.5 11.3
acetone
1
. ...
0.0
0.2
0.4
06 0.8 V, (kcal/rnole)
1.0
1.2
1.4
Figure 4. Determination of A value for CHzBrCHBr2in cyclohexane. Each curve represents the calculation of A from eq 21 by using the measured value of J at each of four different temperatures (see Table 11) for varying values of VI. The intersection of the four curves gives the correct values of A and V , . 12.6
I
12.4
L\
I
I
I
I
I
I
I
I
\
12.0 -
neat liquid
11.4
t-
11.2
-
acetone
11.0
-
'
I
1
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
V, (kcal/rnole)
0.7
0.6
Figure 5. Demonstration of the failure of the proposed method to yield simultaneous intersection points when VI varies with temperature. This plot is similar to Figure 4, but in acetone. %. A Varian T-60 spectrometer equipped with a T-6055 permalock unit and T-6080 variable-temperature accessory was used. Measured temperatures are believed to be accurate to within f l "C. The standard sideband technique16 was used to calibrate the frequency to an accuracy better than 0.05 Hz. The vicinal coupling constant for CH2C1CHCI, and CH2BrCHBr2was obtained by the analysis of AB, spectra.16 Each coupling constant was determined by averaging four separate measurements. Our results were within experimental error (0.02 Hz) compared to the previously reported values taken under similar c o n d i t i o n ~ . ~ Figures 3 and 4 demonstrate the determination of the characteristic constant A of CH2CICHC12 and CH2BrCHBr2, respectively, through the proposed scheme. The coincidence of the intersection points in both cases strongly indicates that both V , and A for the two solutes in cyclohexane are substantially correct. The determined values are A = 11.10 H z for CH2ClCHC12and A = 12.15 Hz for CH,BrCHBr,. Estimated uncertainty is f0.03 Hz. In contrast to Figures 3 and 4,Figure 5 of CH2BrCHBr2 in acetone displays a scattering of the intersection points. This
0.71 0.71 0.70 0.44 0.43 0.42 0.40 0.48 0.54 0.55 0.59 0.63 0.0
0.0 0.0 0.0 0.0
0.53 0.53 0.53 0.33 0.32 0.31 0.30 0.36 0.40 0.41 0.44 0.48
0.0 0.0 0.0 0.0 0.0
TABLE II: Experimental NMR and Physical Parameters for BrCH2CHBr2in Various Solvents A = 12.15 Hz solvent T, K J , Hz VI,' kcal/mol AE? kcal/mol cyclohexane 273 6.78 0.63 0.47
chloroform
10.8 0.0
6.17 6.11 6.05 6.03 5.96 5.91 5.86 6.15 6.i2 6.06 6.03 6.02 5.52 5.49 5.50 5.51 5.50
'Determined by using A given above and interpolating on Figure 2. *Determinedfrom eq 7.
12.2
';i 1 1 . 8 I v
308 338 368 273 308 338 368 238 273 308 338 368 238 273 308 338 358
308 338 363 238 213 308 338 363 238 273 308 338 368 238 269 308 338 358
6.69 6.63 6.58 6.69 6.64 6.58 6.55 6.50 6.81 6.67 6.65 6.63 6.58 6.30 6.29 6.24 6.19 6.18
0.62 0.63 0.62 0.44 0.48 0.49 0.52 0.51 0.55 0.51 0.57 0.63 0.63 0.21 0.16 0.14 0.1 1 0.11
0.47 0.47 0.47 0.33 0.36 0.37 0.39 0.38 0.41 0.38 0.43 0.47 0.47 0.16 0.12 0.1 1 0.08 0.08
'Determined by using A given above and interpolating on Figure 2. *Determined from eq 7. failure might be attributed to variation of the dielectric constant or solvent-specific interactions during the course of temperature variation. The measured values of the vicinal coupling constants as well as related conformational energetic data VI and AE, for CH,ClCHCl, and CH2BrCHBr, at various temperatures and in various solvents as determined from eq 16 (or Figure 2) are listed in Tables I and 11. We found that temperature affected VI for the two solutes differently depending on the solvent. For example, for CH2C1CHCl, in acetone, the gauche and trans conformers are equally stable within the range of temperature variation. All the other cases in Tables I and I1 show the gauche form as the stable conformer. For CHzBrCHBr2 in acetone, the gauche conformation is more stable with hE decreasing with temperature. In chloroform, AE for both solutes increased slightly with increasing temperature. In contrast, AE decreased with increasing temperature in neat liquid CH2ClCHCl, while the reverse is true in neat CH2BrCHBr2. The proposed method was also applied to the same solutes in other solvents by using published coupling constants.21 Table I11 presents the values for AE for CH,ClCHCI, in various solvents at 30 "C determined by using the proposed method as well as the
4974
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
Chen et al.
TABLE III: Proton Coupling Constants and Calculated Conformer Energy Differences for 1,1,2-Trichloroethane in Various Solventso
solvent n-hexaneb pentane n-hexane decalin carbon tetrachloride benzene carbon disulfide trichloroethylene isopropyl ether ethyl ether chloroform n-heptyl bromide neat liquid methylene chloride
1,2-dichloroethane mesityl oxide butanone acetone acetonitrile nitromethane acetonitrileb dimethylformamide N,N-dimethylacetamide sulfolane dimethyl sulfoxide
1.95 2.07 2.10 2.32 2.45 2.60 2.95 3.55 4.10 4.50 4.90 5.38 7.15 8.25 9.15 15.0 17.3 19.2
32.0 32.6 34.4 35.0 35.8 42.2 44.9
6.24 6.25 6.21 6.17 6.17 5.94 6.13 6.13 6.04 6.05 6.03 5.99 5.92 5.93 5.87 5.64 5.53 5.52 5.52 5.51 5.52 5.34 5.28 5.31 5.17
0.6 1 0.62 0.57 0.53 0.53 0.30 0.48 0.48 0.39 0.40 0.39 0.35 0.30 0.30 0.25 0.08 0.00 -0.01 -0.01 -0.01 -0.0 1 -0. I 3 -0.16 -0.15 -0.24
1.28 1.25 1.24 1.17 1.14 0.63 1.03 0.92 0.84 0.79 0.75 0.71 0.58 0.52 0.46 0.32 0.27 0.24 0.08 0.07 0.05 0.05 0.04 -0.02 -0.04
"10% solutions. b5% solutions.
