4124
KJELL-IVARDAHLQVIST AND STURE FORS~N
umes of BiBr3 (1)' and BiCb (1)6 are, significantly smaller than the sum of the molar volumes of their respective elements from which they are formed. On this basis, BiI3 (1) would show a tendency to disproportionate at high pressures whereas BiBr3(l) and BiCL(1) become more stable. This difference in stability with pressure could account for the difference in behavior of the log x us 1/T and 1c us. P curves of BiI3 from those of BiBr3. and BiC13. The activation energies (E,) for BiC13, BiBr3, and BiI3 a t P 5 0.1 kbar were estimated from the slope of
the log x us. 1/T curves for these salts from data given by Grantham and YosimZaa t temperatures immediately above the melting point. These estimated values of E, a t P 5 0.1 kbar are compared with E, at P = 5.4 kbars in Table 111. The activation energies at these two pressures are essentially identical in the cases of BiC4 and BiBr3; for BiI3, E, at 5.4 kbars is almost four times its value at P = 0. This difference in E, between bismuth triiodide and the tribromide and trichloride may be due to a greater effect of pressure upon ionic association in the case of BiIa.
Application of Density Matrix Methods to the Study of Spin Exchange. I. The Barrier to Internal Rotation in N-Acetylpyrrole by Kjell-Ivar Dahlqvist and Sture Forsen Division of Physical Chemistry, The Lund Institute of Technology, Chemical Center, Lund 7, Sweden (Received April 1, 1969)
The density matrix theory of the effects of rate processes on complex nmr spectra, developed by Kaplan, has been applied to a study of the barrier to internal rotation in N-acetylpyrrole. Theoretical spectra of the four-spin system of the ring protons give excellent fits to the experimental lineshapes in the temperature region - 6 5 to $30'. The activation parameters for the rotation of the acetyl group were found to be E, = 12.6 zt 0.5 kcal/ mol, AH = 12.0 f:0.5 kcallmol, A F ~ ~ ~ o=K12.1 * f:0.2 kcal/mol and A 8 = -0.6 f:2.3 eu. The Arrhenius activation energy, E,, is about 7 kcal/mol lower than that for N,N-dimethylacetamide, which indicates that the lone-pair electrons on the nitrogen atom in N-acetylpyrrole are to a large extent localized in the pyrrole ring,
*
*
Introduction Comparatively high values for the barrier to internal rotation about formal single bonds have been observed in a number of molecules. Examples of bonds and compounds in which this effect is manifest are the 3 C-N< bond in amides1w3thioamides4 and vinylogous amides686 the =N-N< bond in nitroso amines7,*, the $ C-C 6 bond in aromatic aldehydes,g and the =N-C bond in aromatic or conjugated nitroso compounds.1° I n all of these compounds, the main contributor to the barrier is assumed to be conjugative interaction between ?r electrons on both sides of the single bond that exhibits restricted rotation. The rotational barrier about the >N-C< bond in amides ((I),X = 0) is among those most extensively investigated, mainly by nmr spectroscopy.
N-C< bond. An interesting observation has been made by Anet and COworkers,”!lz who have shown the ring protons in N-acetylaziridine (11) to give only one sharp nmr signal at temperatures down to - 150”. The barrier in the C-N bond in (11)
0
CH2
1
\
N-c
/
I1 can be estimated to be lower than that in an ordinary N,N-dialkylsubstituted analogue by a t least 10 kcal/mol, which indicates that the “lone pair” electrons on the nitrogen atom in (11) are predominantly localized in the three-membered ring. If the nitrogen atom in an amide also forms part of an aromatic system, a low barrier in the >N-C< bond may be expected. I n order to test the validity of this idea, we have evaluated the activation parameters for rotation of the acetyl group in N-acetylpyrrole (111).
IIIa
IIIb
The evaluation of rate parameters from pmr spectra of N,N-dialkylamides can in most cases be made using the adiabatic Gutowsky-McConnell a p p r ~ a c h . ~ ~I Jn~ order to determine accurately the interconversion rate in N-acetylpyrrole (111) from the four-spin nmr spectrum of the ring protons, it is however, necessary to apply the density matrix theory developed by Kaplan.l5J6 The applicability of this theory to intramolecular exchange in a four-spin system has recently been demonstrated. l7
Experimental Section N-Acetylpyrrole was prepared according to Dennstedt18 and purified by distillation, bp 181-182”. The deuterated methylene chloride (CD2Cl2)used as solvent was of commercial quality (Merck Sharpe & Dohme of Canada, Ltd.) and used without further purification. A 10% solution of N-acetylpyrrole in CDzClz containing 1% tetramethylsilane (TMS) was used in the present investigation. The sample tubes used were standard
4125 5 mm Pyrex nmr tubes, and the sample was degassed and sealed before use. The sample temperatures were obtained from the temperature-dependent shift difference between the signals from the hydroxyl and methylene protons in a 50% solution of ethylene glycol in acidified CD80D. This solution was placed in a sealed capillary which in turn was fixed in the center of the sample tube by means of two teflon plugs. The temperature-dependent shift between the OH and the methylene signals in the solution was separately calibrated against a copper-constantan thermocouple according to a procedure described el~ewhere.’~The temperatures obtained in this way are estimated to be accurate to *0.5”. The nmr spectra were recorded between -60 to f 3 0 ” using a Varian A 60 nmr spectrometer equipped with a variable temperature probe and a V 4343 variable temperature controller. The resolution of the spectrometer was maintained approximately constant by adjusting the homogeneity so as to always obtain the same half width (ca. 0.35 Hz) in the triplet from the traces of CDHClz in the CDzClz solvent. Shifts were measured by means of the ordinary side-band technique using a Hewlett-Packard Model 202 C Audio Oscillator and a Hewlett-Packard Model 5512 Electronic Counter.
The Nmr Spectra of N-Acetylpyrrole The ring protons of N-acetylpyrrole give rise to an ABXY spectrum at -70” and an AA‘XX’ spectrum a t 1-50”, These spectra were analyzed using the LAOCOON I1 computer program developed by Castellano and Bothner-By.20 The ABXY spectrum at -70” consists of two “quartets” in the region 420 Hz to 450 Hz (from TMS) and a “triplet” a t 378 Hz. The two “quartets” in the low temperature spectrum are due to H-2 and H-5 in conformer IIIa. The assignment of H-2 and H-5 is dependent on the model for the diamagnetic anisotropy effects of carbonyl groups. We have tentatively assigned the low field “quartet” (at, 446.3 Hz) to H-2 and the high field “quartet” (at 427.5 Hz) to H-5. The signals from H-3 and H-4 are found as a triplet centered a t 377.8 Hz (from TMS). The chemical shift between H-3 and H-4 obtained from the iterative computer analyses is only (11) F. A. L. Anet and J. M. Osyany, J. Amer. Chem. Soc., 89, 352 (1967). (12) F. A. L. Anet, R. D. Trepka, and D. J. Cram, i b i d . , 89, 367 (1967). (13) H.S. Gutowsky, D. McCall, and C . P. Slichter, J. Chem. Phys., 21, 279 (1953). (14) H. M. McConnell, ibid., 28, 430 (1958). (15) J. Kaplan, ibid., 28,278 (1958). (16) J. Kaplan, ibid., 29,462 (1958). (17) K-I. Dahlqvist and 5. Forsen J . Magnetic Resonance, in press. (18) C. Dennstedt, Bsr., 16,2353 (1883). (19) T. Drakenberg, K-I. Dahlqvist, and S. Forsen, Acta Chem. Scand., in press. (20) S. Castellano and A. A. Bothner-By, J . Chem. Phys., 41, 3863 (1964). Volume 73, Number 1.9 December 1960
4126
KJELL-IVARDAHLQVIST AND STUREFORSSN
=0.8 Hz, which makes it difficult to obtain any assignment of the relative shifts for these protons. At 30” the two “quartets” from H-5 and H-2 have coalesced into a “triplet” which forms the AA’ part of the AA’XX’ spectrum. The methyl protons of the acetyl group give at room temperature a singlet at 150.6 Hz (from TMS). The parameters for the ring protons obtained from the computer analyses are given in Table I.
eq 4. The time dependence of the density matrix element for the transition between the states k and 1 may now be written using the notation of Johnsonz1
Table I: Chemical Shifts from TMS a t 60 MHz and Spin Coupling Constants for the Ring Protons in N-Acetylpyrrole
Temp -70°-Shift, Ha
Proton
H-2 H-3 or H-4 H-4 or H-3 H-5
446.3 378.2 377.4 427.5
h(Pkk
Temp 50°--
I
Spin coupling constants, Proton
Shift, Hz
H-2 H-3 H-4 H-5
435.9 375.6 375.6 435.9
Ha
Jes = 3 . 0 0 = 1.42 525 = 2.06 534 = 3.00 585 = 1.47 J45 = 3.15 J24
Evaluation of the Interconversion Rate The interconversion rate between (IIIa) and (IIIb) has been evaluat,ed from the nmr spectrum of the ring protons. This spectrum is converted from an ABXYtype spectrum on slow rotation of the acetyl group into an AA’XX’-type spectrum on fast rotation. Intramolecular exchange in a spin system can according to Kaplan16J6be represented by an exchange operator R (1) Here J/ and R$ are the wave functions for the spin system before and after the exchange. The density matrix before and after exchange may similarly be represented by *-R*
p =
RpR
(2)
The time dependence of the density matrix in the absence of exchange is given by dp i - = i[P,Hl dt
(3)
where the brackets on the right hand side denote the commutator of the density matrix and the Hamiltonian of the spin system. According to Kaplan the exchange may be accounted for, by adding a “damping term” (RpR p)/r (where r is the mean lifetime of the nuclei in each site) to eq 3
-
- P1Z)c(Iz‘)kl z
(5)
When the spectrum is recorded using a weak radiofrequency field, only transitions between those states for which AM = +1 need be considered. Under [‘slow passage” conditions we may put dpldt = 0 and eq 5 then leads to 56 linear equations in the present four-spin case. However, the corresponding 56 X 56 coefficient matrix decomposes into diagonal blocks, two of dimension 24 X 24 and two of dimension 4 X 4,and these must be inverted for every point in the spectrum. This may be done by means of a digital computer. The absorption intensity of the nmr-spectrum I ( w ) , is given by the expectation value of I , in the rotating framez1
I(w)
=
(Iv}= kIm Tr(pZI+’)
(6)
where k is a proportionality factor, T r denotes the trace, Im stands for the imaginary part and I+’ is the raising operator ( I z il,) for nucleus j . Since eq 6 contains the trace of (pZI+j), I(o) will be independent of the representation used for the basis functions in eq 1-6. For convenience we have chosen simple product functions for this representation. A computer program named DENSMAT has been developedz2 by means of which eq 5 and 6 may be solved numerically for up to four-spin systems. For a given set of r , Tz, chemical shifts, and spin coupling constants, theoretical spectra were calculated using a CDC 3600 computer and plotted on Calcomp plotter Type 565. The rate of rotation for the acetyl group a t each temperature was determined by visual fitting of theoretical spectra to experimental ones. A single point in the theoretical spectrum required about 1.7 sec computation time on a CDC 3600 computer. In general a complete spectrum of the AB part was made up of about 700 points and the X Y part of about 200 points, thus demanding ca 20 min and 6 min of computation time respectively. I n all approximately 150 spectra, some of which however, were cal-
+
(4) The relaxation processes are accounted for by adding a term l/T2 (where T2 is the spin-spin relaxation time) to The Journal of Physical Chemistry
(21) C. 8. Johnson, Jr., “Advances in Magnetic Resonance,” Vol. l , Academic Press, New York, N. Y . ,1905, p 33. (22) K-I. Dahlqvist, 8. ForsBn, and T. Alm, Acta Chem. Scand., in press.
4127
SPINEXCHANGE IN N-ACETYLPYRROLE
I
T.100.0 soc. t m - 7 0 'c
T.0.021 sac, t -32.5 *C
Figure 1. Examples of experimental and theoretical spectra of H-2 and H-5 protons in N-acetylpyrrole.
culated over a small region of the spectrum with less than 700 points, were calculated during the present investigation, and the total computation time for the whole investigation was about 25 to 30 hr. The transverse relaxation time, Tz,was obtained by visual fitting of theoretical curves to the experimental ones obtained at the slow and fast exchange limits where the value of the exchange rate no longer affects the line width. The theoretical spectra at these limits were calculated with the computer program DENSMAT with input parameters obtained in the following way. The chemical shifts and spin-spin coupling constants were obtained by the LAOCOON I1 analyses, and 7 values appropriate for each limit were chosen. The theoretically reproduced line widths are found to depend mainly
on Tz. (See spectra a t -70 and +SO0 in Figure 1.) Since both the high- and low-temperature spectra could be well reproduced with the same set of Tz(0.65 sec) and spin-spin coupling constants (see Table I), these parameters were used for calculation of theoretical spectra a t all intermediate temperatures. The shift difference, AVAB-XY = (VA VB)/2 (VX VY)/~, between the AB part and the XY part of the spectrum showed a small temperature dependence and was 1.2 Hz larger a t +50° than a t -70". As calculated spectra with AVAB-XY f 2 HZshowed no detectable difference in the exchange broadened region the low temperature value of AVAB-XYwas used for all temperatures. The XY part of the N-acetylpyrrole spectrum could
+
+
volume 73,Number 18 December 1060
KJELL-IVARDAHLQVIST AND STUREFORSJ~N I
,
10
"
"
I
"
"
I
"
'
CPS
T=100.0 scc. ts-700 'C
'c- 0.125 sec. t=-L7.9 'C
3.5
4.0
4.5
lJ&4
T
T=0.022sec. t=-32.5 *C
'c- 0.0028 scc. t.-13.7 *C
'c.10'~ s e c t * +50.0 'C
Figure 2. Examples of experimental and theoretical spectra of the H-3 and H-4 protons in N-acetylpyrrole
Figure 3. Arrhenius plot of the kinetic data obtained from the lineshape of the H-2 and H-5 protons in N-acetylpyrrole.
was found, and a mean value of the shifts determined at temperatures below the coalescence point were used for the calculation of spectra at higher temperatures. The evaluation of r is, on the other hand, most accurate a t the coalescence point, but the precision decreases with decreasing exchange broadening both toward the slow and fast exchange limits. The relative error in r a t the coalescence point was about hl-2% and the Arrhenius plot close to the slow and fast exchange limits was terminated when the relative errors amounted to about *4-5%.
Results and Discussion at all temperatures be well reproduced using the same value of BVXY (0.8 Hz). A variation in BVXY of the order of h0.2 Hz had no detectable effect on the lineshape of either the X Y part or the AB part of the spectrum, and the value 0.8 Hz was used for BVXY throughout the investigation. The r value at each temperature was evaluated only from the AB part of the spectrum since the lineshape of the X Y part was very insensitive to variations in T due to the small value of BVXY. I n the temperature region below the coalescence temperature both r and BVAB could be determined independently by curve fitting. Above this temperature the spectrum contains too little information to permit the simultaneous evaluation of ~ Y A and B r. The BVAB values obtained by direct curve fitting are most accurate at the slow exchange limit where the spectrum contains fine structure. The accuracy in the determination of BPAB decreases gradually as the exchange broadening increases and the fine structure disappears. At -70', BVAB could thus be determined within about h 0 . l Hz, at -47.9' within about h0.2 Ha and at the coalescence temperature within about &0.3-0.4 Ha, (cf. Figure 1.) Within these error limits no significant temperature dependence in BVAB The Journal of Physical Chemistry
The ring proton spectrum of N-acetylpyrrole is not a pure ABXY spectrum since N-acetylpyrrole contains seven protons and one nitrogen, and is therefore actually the ABXY part of an eight-spin system. However, the spectrum of the ring protons could be well reproduced at the slow and fast exchange limits by neglecting the spin-spin interactions with the acetyl protons and the nitrogen. Furthermore the absorption lines in the limiting spectra were relatively sharp ( B V ~ , , N 0.5 Hz). It therefore appears that no additional errors should result from a phenomenological treatment of all these couplings as contributing only to the apparent TZvalues. A comparison between experimental and calculated spectra for N-acetylpyrrole (Figures 1 and 2) shows the same excellent agreement for all exchange rates determined between the slow and fast exchange limits. The 7-values obtained by visual fitting of theoretical spectra to experimental ones are plotted as ln(l/T) us. 1/T in Figure 3. The activation parameters for the >N-C Q rotation in N-acetylpyrrole calculated from the theory of absolute reaction rates are given in Table 11. The errors in these activation parameters were calculated assuming only random errors and are probably too low. An investigation of the sources of
SPINEXCHANGE IN N-ACETYLPYRROLE
4129
Table 11: Activation Parameters for the Internal Rotation of the N-Acetyl Group in N-Acetylpyrrole"
*,
AFass'K,
T , OK
Ea, kcal/mol
(111)
298.2
12.55 rt 0.09
11.96
N,N-Dimethylacetamide
298.2
19.0 rt 0.4
18.3 i 0 . 4
Compound
AH
koal/mol
*
koal/mol
=IC0.09
AS
*,
eu
12.14 i 0.10
-0.6 =!= 0 . 6
18.0 =!= 0 . 2
0 . 9 =!= 1 . 3
Ref
present investigation 22
*)
a The errors given for these parameters have been calculated assuming only random errors. More realistic errors are E, and AH =!= 05 kcal/mol; bF*) i 0.2 kcal/mol; AS*, i. 2.3 eu (cf. text). For comparison, corresponding values for the internal rotation of the N-(CHa)n group in N-dimethylacetamide are also given in the table.
error in the evaluation of the 3 C-N< barrier in N,Ndimethyltrichlor~acetamide~~ shows that more realistic values for the errors in E , and in AH* are A0.5 kcal/ mol, in AF f0.2 kcal/mol, and A2.3 eu in AX*. A comparison of the activation parameters for the restricted >N-C 2 rotation in N-acetylpyrrole and N-dimethylacetamide (IV) (see Table 11) shows that Ea, AH *, and AF are about 7 kcal/mol lower in (111) than in (IV). As the steric interactions between the acetyl group and the rest of the molecule should be quite similar in both of these molecules, and since the interaction between the lone pair on the nitrogen and the carbonyl group is believed to give the main contribution to the restricted >N-C Q rotation in amides, we may conclude that the lone pair on the nitrogen in (111) is to a great extent localized in the pyrrole ring.
*
*
The entropy of activation for the >N-CQ rotation in compound (111) was found to be zero within the experimental errors. Careful measurements of the barrier to internal rotation in (IV) and other simple alkylsubstituted amides23have also yielded activation entropies close to zero. Acknowledgments. The authors wish to thank Dr. E. Forslind for his kind interest in this work. Thanks are also due to Dr. R. E. Carter for helpful linguistic criticism and to Dr. R. A. Hoffman for valuable discussions and criticism. The cost of the nmr spectrometer was defrayed by a grant from the Knut and Alice Wallenberg Foundation. (23) T. Drakenberg, K-I. Dahlqvist, and S. Forsb, Acta Chem. Scand., in press.
Volume 78, Number 12 December 1868
