NOTES
1916
PROTON N.M.R. SPECTROSCOPY. XII. TET KAMETHYLSILANE : SOME: OBSERVATIONS CONCERNING LINE WIDTH AND LINE SHAPE BY GEORGE V A N D Y K E TIERS Contribution X o . 19.9 from the Central Rceearch Dsparlmcnl of ths Minnesolu Mining and Manufacturhg Co., St. Paul 19, Minnesota Rgeioed Mag 16. 1061
It has belcome customary to consider that individual nuclear magnetic resonance (n.m.r.) lines, when not broadened by exchange may be described by the theoretical Lorentzian line shape.’ It is recognized :nevertheless that “in practice, line shapes are not always well represented by the Lorentz-type curve.”’ For mobile liquids such as t,etramethylsilane a t ordinary temperatures the “natural line width” (in cycles per second, c./sec., measured at half-maximal height) is given by the expressiou l/sT2 where Tz E T I as a By direct measurement upon neat tetramethylsilane it has been found that TI 2 16.7 sec8 from which the natural line width would be e s h a t e d as 0.019 c./sec.; this is perhaps tenfold narrower than the best resolution attainable with present spectrometers. Even when air-saturated, a 5% solution of tetramethylsilane in carbon tetrachloride has TI = 2.6 sec., and thus 0.12 c./sec. line width. For this reason the assumption that the observed. line shape can be treated as Lorentzian4 and that consequently all broadenings are additive,4b nmy be incorrect and might result in substantial errors in kinetic measurements. It is therefore of interest to characterize line shape in terms of its approach to Lorentzian behavior, especially when spectrometer resolution has been optimized by careful attention to experimental details. The recently int’roduced Varian A-60 NMR Spectrophotometer has been found particularly suitable for such a study, since it combines inherently excellent resolution, convenie:nce of operation and remarkable reproducibility of signal intensity and of sweep rate.
Vol. 65
both sweep directione. At the best setting of the fieldhomogeneity controls the decay waa roughly exponential, being characterized by a half-life of 0.76 sec.; upon division by 1n 2 one obtains “T,” = 1.1 sec.,6 and from this the Lorentzian line width 0.29 c./sec. The spacings of wiggle maxima and minima were in exact agreement with the. calculated valuea for the sweep rate used.6 The analysls of Jacobsohn and WangsnesseJ gives AV = 0.31 c./sec. and Av’ = 0.79 c./sec., while the observed valuea (14 meaa.) are 0.43 f 0.01 c./sec. and 0.91 & 0.01 c./sec., in fair agreement?; the true resonance position waa taken as the mid-point between first maxima obtained from sweeps in both directions, as these were highly reproducible. The slowest swee rate available, 0.10 c./sec. per second, was used for t h e h e shape studies; since this was about three times the rate below which wiggles are not observed,’ considerable filtering waa introduced (dial setting 0.2) in order to minimize the effects of wiggles. While this procedure is admittedly open to question, the observed AvF 0.18 f 0.01 c./sec. (20 meas.) shows no decrease in agreement with the calculated value, 0.13 cJsec.7 The response rate of the recorder did not ap ear to be limiting, since six of the measurements were m a i e at double the amplitude, without any detectable change in b. The results of twenty line width measurements are given in Table I. Peak heights in fourteen of the runs, made at amplitude setting 0.63, averaged 50.6 mm. ( f l . 7 mm. std. dev., per run) and those from the remaining six rum, at amplitude setting 1.25, averaged 96.9 mm. ( f 1.7, std. dev., per run). This good reproducibility of heights is certainly very reassuring when accurate width measurements at various percentage heights are desired. An additional spectral feature readily observed is the spin-spin coupling between methyl protons and SiB, present in 8.4% abundance. The coupling constant for tetramethylsilane, J(Si*QH)= 6.57 f 0.02 c./sec., was obtained from ten measurements; it may be compared with B previous value of 6.68 f 0.04 c./scc. obtained with a Varian Si4300-240.00 Mc./sec. spectrometer.
Discussion
From the values cited in Table I it is seen that the somewhat broadened line, resulting from moderately rapid conformational isomerization (conformerization) of perfhorocyclohexane,8 does indeed show the virtually Lorentzian shape expected for a line narrow relative to the separation of the exchanging species. However, for the hundredfold narrower line from tetramethylsilane, a significant deviation from the Lorentzian shape is noted. Since the relative line widths, RH, fall about halfway between those for the Lorentzian and Gaussian curves, one might plausibly suggest Experimental an approximately Gaussian distribution of LorentThe equipment used was the Varian A-60 NMR Spectrophotometer, kindly made available for trial and inspection.6 zian lines, resulting perhaps from tiny random in‘rho air-saturated tet,ramethylsilane solution, 5% by volume homogeneities in the magnetic field. Among other in CC,, was Varian Standard Sample No. 902337-05, “experimental” causes of broadening might be and was conta,ined in a 5 mm. 0.d. thin-walled precision (a) slow exchange, (b) unrecognized multiplicity, tube. All spectra were run at maximal expansion of sweep width, 1 c./sw. = 10 mm. on the chart. The minimal and (c) too fast rate of sweep. While in the RF field setting (0.02 milligauss, nominal value) was present case the first two of these can be ruled out chosen in order to minimize saturation. a priori, it will be instructive to examine their For the optimization of “field shape” the slowness of effects along with those of fast sweep. decay of the ringing patt,ern a t sweep rate 0.50 c./sec. per second waa used as the criterion, it being required that exactly the same monotonic decay pattern be observed for
-
(1) J. A. Pople, W. G. Schneider and H. J. Berstain, “ H i g h
Resolution Nuclear Magnetic Resonanae,” McGraw-Hill Hook Co., Inc., New York, N. Y., 1959, p. 37. (2) While Ta i:i never longer than TI, i t has in certsin caseu been found t o be somewhat shorter; see J. G. Powles and D. Cutler, Noturc, 180, 1344 (1957) and 184, 1123 (1959). (3) F. A. Bovoy, private communication. See F&O F. A. Bovey, J . Chem. Phya.. 3 8 , 1877 (lQ60). (4) Ref. 1. (a) p. 45-46; (b) p. 220-224. (5) During the Second Conference on Experimental Aspects of N M R Spectroscopy, Mellon Institute. Pittsburgh. 1961, the e q u i p ment waa put on display by the manufacturer, Varian Asaooisteg
(6) Ref. 1, p. 41: the envelope of wiggle maxima (or minima) is given by the exponential factor in equation 3-83, and from equation 3-84 the spacinga (in c./sec.) of wiggle maxima or minima “far down the tail” ia simply I/A& where At is the time elapsed after traversing the resonance position. (7) R. A. Jacobsohn and R. K. Wangrmesa, Phw. Rev., 73, 942 (1948); the displacement Ar(c./s.) of the first maximum from the true reaOnance position is given by Av = z(dv/dt)’/3/(2r)’/i where z has she value 0.9 to 1.1 for the aweep rates used in the present article. The displacement Au’ of the first minimum is found from the same equation by letting 2 be 2.8. Wiggles are not observed if the sweep rate. du/dt (in c./sec.i) 5 IhrTi. (8) G . V. D. Tien, Proc. Chem. Soc., 389 (1960); dl experimental proasdurea are described there. The h e width s t halfheight, TI/I,WBS 23.93 t 0.08 a./eeo. at 25.5’.
NOTES
Oct., 1961 IiELATlVE LINEINIDTKS,
RH,’
FOR OBSERVED
1917
TABLE I N.M.R. LINES,AND FOR LORENTZIAN AND GAUSSIAN LINESHAPES, AT VARIOUS PERCENTAGE HEIGHTS,if Relative line width, RH‘
RXO
Rio
RIO
RIP
R40
Rio
ROO
RIP
0.500 0.333 1.527 1.225 0.816 0.655 Lorentzian line shape 3.000 2.000 0.52 h 0.33 f 0.82f 0.66f 1.54f 1.24f Obsd.n.m.r.ofcyclO-C6FI1) 3 . 1 3 3 ~ 2 . 0 4 f 0.03 0.01 0.01 0.01 0.01 0.04 0.02 0.01 Obsd. n.m.r. of (CHt),Sic 2 . 4 9 i 1.73 f 1.39 i 0 . 5 3 5c 0.06 0.03 0.01 0.01 Gaussian line shaped 1.823 1.523 1.319 1.150 0.859 0.718 0.567 0.390 The RH values are expressed relative t,o the full line width at half-height, WI/, (occasionally referred to by the incorrect , 0.32s f 0.007 c./sec. d The line width, Li‘i/*, is 2.345 u contraction “half-width”). ‘1 See ref. 8. The line width, W I / ~was where Q is the “standard deviation”; see ref. 9. 0
LINE\vIDTIi
ItATIOS,
TABLE II Lrr, AT VARIOUS PERCENTAGE HEIGHTS, H , FOR SEVERAL I~YPOTIIETICALN.M.R. A LORENTZIAN LINEOF THE SAMEHALF-HEIGHT WIDTH
CASES,
-
Line width ratios. Lx5 LlO
exchange* Wl/* = 0 . 5 WI/! = l,o
Lao
Lzo
L40
LSO
L70
RELATIVE TU LSO
Lea
SOW
Binomial multiplet” Comp. W = 5J Comp. W = 2J Gaustiian line limits Theor. fmt ,sweepd a‘/’Tn := 1 al/’Tz = 2 (CH.,),Si, obscl. dv/dt == 0 . 1 C./SCC.”
0.853 .743
0.878 .790
0.910 .844
0.947 .911
1.075 1.116
1.187 1.287
1.3% 1.580
1.857 2.232
,813 .G84 .GO8
.872 .802 .762
.926 .885 .865
.9GO .952 .938
1.035
1.060 1.087 1.09G
1.086
1.043 1.051
1.095 1.131
i.100 1.143 1.170
.72 .66
.83 .79
.92 .89
.I10
1.04 1.04
1.1 1.07
1.2 1.1
.95
1 .3
!.A
*
.83 f .86 f 0.91 i 1.06 0.02 0.01 0.01 0.02 d v / d t == 0 . 5 c./gec.l’ .69 .80 .89 .95 1.04 1.1 1.1 1.2 0 The La ratios are as described in the text.. * Exact calculation according to ref. I, p. 233, eq. 10-29, where A = Y A - VB. e I’ifteen individual Lorrntzian component lines of half-height width W and spacing J, the intensit,ies being given by thc binomial coefficients. These curves approximate to Gnussinn distribution of Lorontzian lines; see ref. 9 d Taken from ref. 7, Fig. 2; a1/r7’, = ( 2 dv/dl)‘/tT,. ~ See ref. 6. * Calcd. value of a’/tl’z = 0.9. f Experimentd errors a. f0.03. Calcd. v s l w of ail/*Tt = 2.0. W1/* = 0.58 c./sec.
For this purpose it is convenient to take the theoretical Lorentzian curve as the standard, and to define a width ratio, LH, as RH (observed)/i?H (Lorentzian) , where RH signifies the relative line widths based upon the half-height width. If an observed line is Lorentzian, then all LIZare unity; and if not, Lorentzian the curve may be characterized by :t table or plot of LEIus. H . Such a comparisoii is made in Table 11. Slow exchange between two equal sites may be readily recognized experimentally, since it leads to abnormal breadth above the half-height. A single or Leoin principle could permit measurement of a fairly accurate estimate of A, the shielding difference between the sites, if for some experimental reason this could not be observed directly. The effects of multiplicity cannot be characterized as readily. The values listed in Table I1 are only typical of a many-component first-order multiplet which is not a common case. They were included because they provide a close approximation9 to the hypothetical Gaussian distribution of Lorentzian lines. The LH values obtained from the theoretical curves for “fast” sweep? (i.e,> producing “ringing”) are also in-
cluded. It is interesting to note that onc cannot distinguish between the patterns of LH-values for these two cases. To be sure, the fast-sweep curves are not symmetrical, but the skewed Gaussian curve is well known.I0 In order to decide on experimental grounds bctween these two causes of non-Lorentzian behavior it was necessary to examine the effect upon the LEI-values produced by changing thc sweep rate. If a Gaussian distribution of magnetic field inhomogeneities were dominant, little or no change in the L H values should be noted. If, 011 the other hand, one were to assume a fundamentally-Lorentzian line distorted by too-rapid sweep,7 then a predictable change in LH-values mould he observed. The experimental results of Table I1 correspond remarkably well with the numerical values of Ln calculated for “fast sweep.” At sweep rate 0.5 c./sec. per second the a priori calculated value of a1/2T27is approximately 2.0, and the correspondence of the observed LRvalues with the theoretical is excellent. At sweep rate 0.1 c./sec. per second the a priori value of a‘/lT2 is just below 0.9, and it is apparent that slight de-
(9) F. Mosteller. R. E. K. Rourke and G . B. Thomas. Jr., “Probability and Statistioa,” Addison-Wesley, Inc., Reading, Msss., 1901, pp.
(10) J. C. Bartlet and D. M. Smith, Can. J . Chom., 38, 2057 (ISGO), have provided an intereating treatment of asymmetrical gas-chromatographic peaks baaed on the skewed Gauseian.
275-287.
NOTES viations from the theoretical LH values for d / 2 T 2 = 1.0 are in the right direction. There is clea,rly no experimental basis for a claim of non-Lorentzian line shape, since all measurements are quantitatively in accord with theory for a Lorentzian lane of half-height width 0.29 c./sec., slightly distorted and broadened by sweep rates just slightly too rapid. It is interesting that, this is so, since the natural line width is only about 0.12 c./sec. Further and more careful studies of line shape on various n.m.r. spectrometers would perhaps enable a less tentative statement to be made. In this connection it may be noted that the Si29 satellites, for which J(SiZ9H) = 6.63 c./sec., may be used as internal calibration of sweep when the older-type i1.rrt.r. instrument is used; the shortterm instabilities in sweep rate would, however, make necessary a much larger number of measuremcnts than were required in the present work. I am grateful to Varian Associates, Inc., for making available this remarkably fine new ,spec'trometer, and to Mr. Robert Jones of that company for instruction in its use. T H E HEAT CONTENT AND HEAT CAPACITY OF BORON NITRIDE FROM 998 TO 1689OK.l IIY It. A.
MC.L)ONALD A N D
Tlifrmal Lnhornlory, The
11. R. STIJLL
Daw Chemical Company. .Widland,
Yol. 65 TABLE I ENTHALPY OF a-A1203" T.
Ilc
-
Obs.
OK.
Hzoms, cal. mole 1 NBS eq.6
+
VC I)?\ froni N B b
278.8 -353.8 -355.8 2.0 +0 329.6 615.8 619.9 - 4.1 -0 3346.4 448.3 3297.4 - 51.0 -1 539.4 5591 5 - 22 4 5569 1 -0 655.7 8803 2 8712 5 90 7 +1 761.3 11736 11671 6-5 +0 893.8 15455 15506 - 50 -0 1128.2 22436 22500 - 64 -0 $0 1267.0 26758 (26726)' 32 1463.0 $0 32902 (32765)' +137 1619.6 37849 (37630)' 4 219 $0 Molecular weight, 101.Mg. ni01e--~. * Funikawa, d rvf. 5. c Extrapolated.
+ +
+
56 66 55 40 04 56 3% 28 12 42 58 al.,
TABLE I1 OBSERVED ENTHALPY OF 13"' 111. -
7', "K.
IIT I l ? W ~ , rill. I l l O l t . I
HIYIL.I5,
T ,OK.
cal. mole- 1
279.5 279.5 408.3 455.6 656.3 884 3 902.9 Molcnd:tr
-83.2 -83.4 648.1 910 3 2545.1 4728.6 4893.0 weight, 24.83 g.
1120.7 1299.1 1381 . 8 1504 5 1603.0 1682.5 rnole
7183.3 9193.3 10160 11572 12802 13651
-I.
TABLE I11 HEATCAPACITY
Iliichiyair
SMOOTHEL) ~ N T H A L l ' YA N D
Receiced M a y 4 , 1901
This work was undertaken tJo fill in the gap between the low temperature data of Ilworkin, Sasmor and Van ktsdalen2 and the high temperature investigation of Magnus and Danz3 arid to extend tho measurements to higher temperatures. Experimental
Deviation obsd. - calcd.
O h
C" I
1'.
"li.
300 350 400 450 500 550 600 650 700 750 800 850 900 950
Hr Ldl
Ir Ill
