Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Giant magnetoresistance wikipedia , lookup
Ferromagnetism wikipedia , lookup
Magnetochemistry wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Ising model wikipedia , lookup
Electron paramagnetic resonance wikipedia , lookup
Nuclear magnetic resonance wikipedia , lookup
Nuclear magnetic resonance spectroscopy of proteins wikipedia , lookup
Nuclear magnetic resonance spectroscopy wikipedia , lookup
Two-dimensional nuclear magnetic resonance spectroscopy wikipedia , lookup
A Straightforward Way To Determine Relative Intensities of Spin-Spin Splitting Lines of Equivalent Nuclei in NMR Spectra Ronald H. Orcutt East Tennessee State University, Johnson City, TN 37614 + The well-known n 1 rule predicts the degree of spinspin splitting of proton NMR lines by equivalent protons; the relative intensities of the members of the resulting multiplets are given by the binomial coefficients C(n,r), as defined by where n is the number of equivalent nuclei causing the multiplet splitting, and r, ranging from 0 to n, is the index number of the particular line within the multiplet. A simple way of obtaining a set of relative intensities is by use of Pascal's triangle (see Table 1) rather than calculation of binomial coefficients. Starting with a row of two 1's (corresponding to one nucleus giving rise to the splitting), successive rows can be generated by noting that each entry is the sum of the two entries above and to the left of the one desired. If an element is nonexistent, its value is taken to be zero. Because of the increasingly widespread availability of FTNMR spectrometers with multinuclear capability, it is desirable to have a similar, simple rule for predicting relative intensities to be expected in spin systems other than (-'12, +%I. A quick look at standard NMR reference^',^ does not reveal such a rule for other spin systems. The degree of splitting is given by the formula Table 1. 1 2 3 1 1 1 1 1 1 4 5 6 (S1+ S2 I 3 6 10 15 Pascal's Trlangle 1 4 10 20 1 5 15 1 6 1 + S3P. Expansion yields the result: + + S13+ SZ3+ S33 + 3S12(S2 S3) 3S2'(51 3S3'(Sl+ S2) 6S1S2S3 + + + 53) The term S13 represents the situation when all three equivalent deuterons are in the -1 state; since only one such term is present, the degeneracy of the -3 (-1, -1, -1) spin state is 1. Similarly, the three terms S12S3 represent a spin state of -1 (-1, -1, +I), as do the terms SZ2S1(0,0,-1); thus, the degeneracy of the -1 state is 6. Continuing this process of where I is the nuclear spin of t h e n equivalent nuclei causing the splitting. The possible spin states in a particular case can he generated by taking all possible combinations of the 21 1nuclear spins for each of t h e n equivalent atoms and evaluating relative intensities by counting the number of comhinations that vield the same net suin. One wav- of eeneratine all such possible spin states is t o k i t e the set of nuclear spins as a ~olvnomialand then raise the ~olvnomialto a Dower equai tothe number of equivalent nuclei: Such an expansion can he carried out either by algebraic manipulation or by use of the multinomial expansion. This process generates all the terms; evaluation of the total spin for each term followed by counting the number of terms with the same total spin gives the degeneracy for each total spin state and, hence, a determination of the relative intensity to he expected. Consider, for example, the splitting of the C-13 resonance due to equivalent deuterium atoms (I= 1) in the CD3 moieties in molecules such as CD30H or (CD3)2CO. In these cases seven line multiplets Are expected. If one labels the three equivalent spins S1, S2, and S3 (corresponding to the set of values 1-1,0, +I)), then all the possible spin combinations will be generated as the terms of + ' Becker. E. D. HighResolutionNMR TheoryandChernical Applica tion: Academic: Orlando. FL. 1969. ~ S o h a r ,P. Nuclear Magnetic Resonance Spectroscopy; CRC: Boca Raton, FL, 1983. '%NMR spectra of acetone. d6, and rnethandl.64. Volume 64 Number 9 September 1987 763 Table 2. Relative Intensities of Multiplets for Nuclear Spins Ranging from 1 to 512 collecting like total spins gives the degeneracies as follows: Total Spin State Degeneracy -3 -2 -1 0 1 2 3 1 3 6 7 6 3 1 Thus heptuplets with intensities in the ratios 1:3:6:7:6:3:1 are predicted. This is consistent with the observed spectra as seen in the figure. The above procedure was carried out for systems of three, four, five, and six spins for up to six equivalent nuclei; the results are shown in Table 2. T o the author's astonishment, a rule like that for Pascal's 764 Journal of Chemical Education triangle for generating the resulting relative intensities works for all the cases examined. One can generate Table 2 by first writing down a row of l's, 21 + 1in number; additional rows are then constructed hv s i m ~ l taking v the sum of the 21 + 1entries in the row abo;e and to the Gft of the entry desired (nonexistent entries being assigned values of zero as in Pascal's triangle), until the row number is equal to the number of equivalent nuclei in the moiety under consideration. These results can also he applied to the analysis of the splitting of ESR lines by n equivalent nuclei. The author has not given a general proof of this rule, hut offers it as a convenient way of summarizing the results obtained in all the cases tried. Acknowledgment The author wishes to express his gratitude to his late colleague Boris Franzus, who suggested the problem and enthusiastically greeted this solution.