J. R. ZIMMERMANAND M. R. FOSTER
282
It would appear that the age-old question concerning whether serum is undersaturated, saturated or supersaturated can be resolved only by reference to the presence or absence of solid phase. In the absence of solid phase, serum is undersaturated. It is below the point of spontaneous precipitation. In the presence of solid phase, it appears to be supersaturated since dissolution experiments at pH over 7 have never attained an activity product a c a + + X &HPO,- equalling that observed in normal serum. l 4 Space does not permit an adequate discussion of the many literature reports in relation to this interpretation but it is the authors’ opinion that there exists virtuaIly no disagreement. The conclusion that serum is normally supersaturated (since solid phase is always present) with
Vol. 61
respect to [Ca++] and [HPOd-] has important physiological implications. I n the first place, some cellular mechanism must be postulated by which such a supersaturated condition can be maintained in vivo. Secondly, the driving force of the mineralization process in calcification is easily. . seen to be present in serum itself. Given a “seed” crystal the supersaturated serum will spontaneously carry it to full mineralization. Thirdly, some active mechanism must be present in the gut wall to permit the absorption of calcium against an iongradient. The first of these three corollaries already has been established.a6 The second and third corollaries are under study. (35) W. F. Neuman, H. E. Firschein, P. 8. Chen, and V. DiStefano, J . Am. Cham. Soc., 78,3803 (1956).
E. J. Mulryan
STANDARDIZATION OF N.M.R. HIGH RESOLUTION SPECTRA BYJ. R. ZIMMERMANAND M. R. FOSTER Magnolia Petroleum Company, Field Research Laboratories, Dallas, Texas Rscaivsd Auguai 18, 1868
An accurate method for referencing high resolution nuclear magnetic resonance s ectra with res ect to an external standard is described. Mathematical expressions are developed for determining chemicarshift errors an$ effective line broadenin arising from imperfections in a glass coaxial spinnin system. With a properly designed precision system! errors in chemic3 shifts arising from tube imperfections can be reduce3 to less than 1 part in lo8. The magnetic field at the immediate vicinity of a nucleus is discussed in terms of the applied external field, susceptibilit corrections, shape of sample and intermolecular association effects. Several experiments are discussed which illustrate mo?kular association in alcohol-water and in normal hexane-benzene mixtures and the corresponding necessity of exercising extreme caution in the use of internal reference standards.
Introduction High resolution spectroscopy involves the study of narrow line resonances-the fine structure phenomena of nuclear resonance arising from the electronic environment of nuclei. These narrow line resonances are usually described in terms of such and Jmeasurable quantities as chemical coupling splittings.4~~Because of the small separations of multiple narrow line resonances, high resolution spectroscopy is more often restricted to low viscosity liquids or gaseous samples, rather than solids. Chemical shifts of resonance lines are field dependent and arise because the magnetic field at the nucleus is in general different from the applied external magnetic field. Any such difference can arise from either intermolecular or intramolecular effects or both. In the intramolecular sense line separations or shifts are evident because of differences in the effective electron distribution about the nuclei being studied. Intermolecularly, these differences are derived from several causes, such as (a) intermolecular interactions, (b) exchange rate phenomena between different chemical environments and (0) bulk sample susceptibilities. J-Coupling splitting of resonance lines is an intra(1) W. D. Knight, Phus. Rsv., 76, 1258 (1848). (2) W. D. Dickineon, ibid., 77, 736 (1850). (3) W. G. Proctar and F. C. Yu, ibid., 77, 717 (1950). (4) H. Gutownky, D. AfaCsll, C. Sliohter and E. MoNeil, ibid., Ea, 748 (1951). (5) E. L. Hahn and D. E. Maxwell, ibid., 84, 1246 (1951); 88, 1070 (1852).
molecular, field independent, effect arising from an indirect interaction between the nuclear moments. The magnitude and multiplicity of this splitting depends on the number of nuclei involved and the types of chemical bonds separating the nuclei. The general reasons previously advanced in order for chemical shift and J-coupling phenomena to occur clearly suggest why high resolution spectroscopy has become an important tool for structural analysis and characterization of chemical systems. As the number of chemical applications has increased, there has been an ever increasing refinement in observations which has pointed out the needs for better instrumentation, for higher resolution instruments, for proper corrections for small effects not necessarily related directly to the specific problem, and for more accurate measurements of resonance line positions. If an accurate quantitative anaIysis of eIectron distributions about nuclei in different molecular species is desired then appropriate procedures for standard line resonance comparisons must be adhered to. It is this problem of standardization of nuclear magnetic resonance high resolution spectra which forms the basis of this paper. At the present time several laboratories, both academic and industrial, are already reporting intricate structural resonance lines separated by approximately one part in 100 million of the total applied magnetic field. One ultimate objective in n.m.r. chemical analysis is, of course, to be able to
,
STANDARDIZATION OF N.M.R. HIGHRESOLUTION SPECTRA
March, 1957
st>andardizesuch small differenccs from one molecular species to another. Both internal and external standardization procedures have been widely used for several ycars. However, now with the increased resolut,ion and slability of the spectrometer systems, differences in measurements from separate research laboratmies are often common and sometimes even contradictory in qualitative deductions. I n some instances diff erences of referenced chemical shift measurcrnents exceed by more than ten times the reported deviations of the measurements. The importance of a proper referenced resonance line procedure for n.m.r. high resolution applications to chemical analysis has rapidly become apparent. Chemical Shifts.-The precessional frequency of a nucleus when subjected to a magnetic field is given by the relation w = rH,, wherc w is the angular precessional frequency, y is called the gyromagnetic ratio and is a property of the particular nuclear species, and where H, is the magnetic field a t the nucleus. The magnetic field applied externally to a sample compound is generally not the field H n at the immediate vicinity of the nucleus. Likewise the local field at the vicinity of a particular molecule is generally not equal to the field Hn. It is the small differences in magnetic field, of course, that occur at the nuclei within the molecule that give rise to basic chemical shifts. I n a molecular system, the local field a t the vicinity of a particular molecule is made up of the -+ macroscopic field H in the bulk sample plus con+ tributions arising from the total magnetization M of the system. If one assumes that the contribution of the molecules outside a physically infinitesimal sphere surrounding the molecule in question is the total contribution arising from any magneti+ zation M of the systcm, then by the method of Lorentz, the local field a,t the vicinity of the par+ + titular molecule is given by H 47rM/3. This assumption, of course, neglects any contribution to the local field of the molecule from molecules inside the infinitesimal sphere. In instances where intermolecular interactions involve the electron distribution about the nuclei being investigated, this assumption no longer holds. I n principle one would like to study the chemical shifts of nuclei within an isolated molecule. With a -+ macroscopic field H existing in the molecular + sample, the magnetic field Hn at a particular nucleus might for convenience of discussion be written in the form
283
+ + hz) is while for a cylindrical sample (hl + hz) = -+
spherical samples, the contribution (hl
4
3
zero; ( - 2 s / 3 ) ~Ho,where K is the volume susceptibility of + the sample. h8is an intermolecular interaction effect, -+ and hp is the intramolecular chemical shift in an isolated molecule. In instances where a strong intermolecular association does occur between certain chemical groups of a chemical species but -.b
-+
does not occur with other groups, the quantity hr can become an important factor. I n a monomolecular species sample the question may arise regarding the relative importance of measuring shifts either when the molecule is isolated or when it is associated with its neighbors. But in a mixture of molecular species, erroneous conclusions can result regarding chemical shift measurements if + proper allowance for variations of either the ha -+ -+ factor or combination of ha and hr factors is not made. Internal Standardization.-An internal standardization procedure amounts to dissolving a small amount of the known reference standard into the sample. There are two specific advantages for using an internal reference standard : (a) simultaneous sweeping of the field or frequency spectrum for the unknown sample and reference standard and (b) the susceptibility corrections hl and hz in expression (1) are presumably common to both the reference and sample molecular systems. I n the measurement of chemical shift differences between nuclei in a liquid system of a single molecular species, only the quantities h3 and hd are factors for consideration. If a small amount of the reference standard can be dissolved into the system without affecting the h3 and hd components of either the reference or unknown species, then one could conclude that an internal referencing procedure for comparing other molecular species would be satisfactory. However, this is not generally a valid assumption; and molecular association effects can, in many instances, contribute to effective chemical shift variations in magnetic field by as much as several parts in ten million. A common example of this effect is observed when using water as an internal standard in ethyl alcohol. I n Fig. 1 is shown a plot of the position of the internal water line with respect to the hydrogens of the methyl group as a function of the concentration of water. This chemical shift of the water line in comparison with pure water can vary as + - + - + + + + much as 11 sec.-l in 40 X los see.-' (-2.6 milliH n (Ho 4- hi -k h8 4- ha 3- hr) (1) gauss) ,6 + where HO is the applied external magnetic field, It might be argued from Fig. 1 that so long as one 4 9 --+ small concentrations ( a v = [I - ' / r ( p a - ~L)IHo (8)
STANDARDIZATION OF N.M.R. HIQHRESOLUTION SPECTRA
March, 1957
287
A comparison of this result with the average bulk
T o further emphasize this point, suppose a normalized variable t; is defined by field in the first region av HI= 11 - '/Z(CCI - P5)IHO (9) Uk-'(Pk+I - P k ) P3k 00s 6 3 k ) shows that the average bulk magnetic field experi- t i = (Vl/Ho) enced by a sample in region 3 is the same field that f Ut-' ( P k + l - Pk)P8k sin +3k (15) the sample would experience if it were placed in k-i region 1 in the ideal case. Effects of Imperfect Tubing.-In order to in- The number of molecules experiencing a value of the vestigate the effect of imperfections in the glass normalized variable ti between t and t dt is tubing, standard perturbation techniques may be n(t) dt. This field distribution function is shown applied. The radius of the outer boundary of the in Fig. 7 for the interior (first) region. For the ith region can be written as q(0) = ai ei(O), third region the distribution depends upon (an/aa) where €,(e) is assumed to be so small that [ei(e>]' is and resembles the distributions graphed in Fig. 6. negligible. The potential function is The first-order effect of imperfections on the average magnetic fields experienced by the nuclei W , e ) = [Air ( B i / r ) ] cos e 4- V i (r,e) (10) where the first term on the right is the potential in regions 1 and 3 under rapid spinning conditions function for the unperturbed problem, and where may be summarized as follows. I n region 1 there an effective broadening of the resonance line Vi(r,0 ) is the contribution due to the imperfec- is width with a single maximum (see Fig. 7). In tions and is of the same order of magnitude as ei(0). ei(e) can be represented by a trigonometric region 3 there is a narrow spectral distribution of Larmor frequencies similar in shape to the disseries tribution shown in Fig. 6 .
{ [gi
1 1')-"'
[g
+
+
+
W
ei(e) =
+C
'/zpoi
mi
cos ( k e
- @ki)
(11)
k=l
and Vi(r,O) can be written as a series of solutions of Laplace's equation in the form m
Vi(r,e)
(bnir"
n-1
+
COB
dnir-")
n~
(12)
m
+ n = l (c.1~" -I-
enirn)sin
I
n~
I
\
The procedure is to determine the coefficients of the vi(r,e) series in terms of the coefficients Pki and $ki subject to the requirement that t,bi(r,e) satisfy the boundary conditions to first order. The result of this calculation is that the additional contribution qi(r,e) to the field strength is Vl(r,e) =
-
'/Z
Ho k=i
w { 4 Po+I,k
Cos [(n - 1)e
(Pk+l
- Pk)?%T"-'
Uk-" ( P k + 1
n=l
- Pk)Pn-Igk
i-1
+ cos [(n + 110 -
- @n+l,k]
Uk" ( % T - " - ' )
k=l
#'n-l,k]
(13)
Effect of Rotation.-If the tube system is rotated at an angular frequency il then expression 13 may be used to determine the additional field strength at apoint rotating with the tubing. If the coordinates of this point at time t = 0 are (r,&),then the coordinates at time t will be (r,& Ot). Moreover, since the tubing itself also rotates with the ilt. angular frequency fl, $n,k(t) = & , k ( O ) If these values are inserted in expression 13 and averaged over time, the following result is obtained
+
+
4
Vi(T$O)
=
-HO
ak-'(Pk+l
- Pk)P3,k
r
COS
[eo
- @a,k(O)]
k-i
(14)
Therefore, not even in a rotating system do all the particles experience the same average field. The reason is that the inhomogeneities in field strength arising from imperfections of the tubing are not statio if the system rotates.
-I
-8 -6-4
-2 0
I!
4
.6 .B
I
t NORMALIZED F I E L D VARIABLE.
Fig. 7.-Particle density us. average field strength (spinning system).
It should be observed that it is only imperfections exterior to a given region which affect that region. Moreover, the effect of imperfections in the ith boundary decreases as the inverse square of the radius of that boundary. It follows that the inner surfaces of both tubes in the coaxial system must be particularly precise. Effect of Tilting the Coaxial System.-If a perfect tube system makes an angle 7r/2 with the magnetic field lines, then the average field strength in the ith region can be shown to be Hi = [1 - '/z(Pi - 11.dHa 4- '/z sin2P(pi - !&)Ha (16) where p is the complement of the angle between the field lines and the axis of the tubing. Hence, there is a uniform shift of the average field by an amount
+
J. R. ZIMMERMANAND M.
288
R.FOSTER
Vol. 61
I MOL RATIO (PROPANOL-WATER)
EXTERNAL REFERENCE STANDARD:
225
I
0
1
I
I
I
1
1
1
I
IO
20
30
40
50
60
70
80
I
90
I
100
VOLUME PERCENT WATER. Fig. 8.4usceptibility shift of water-propanol system. '/P sinWpi - Ir6)HO = '/d%.u - P ~ H O(17) One might attempt to deduce this result from the perturbation theory by assuming that the effect of tilting is to distort the circular cross-sections into ellipses. If p is assumed small, this approach results in a field shift of
'/4['/z@*(pi
- dHol
which depends in the same way on the parameters as does the correct result. It, therefore, seems reasonable that the effect of tilting only one of the tubes can be roughly calculated in the same way. This results in the conclusion that the average fields in the first and third regions are shifted with respect to one another by roughly an amount PI d H o (18) The condition that this displacement be negligible together with the condition that the broadening effects previously considered be unimportant imposes rather stringent requirements on the precision of the glass tubing to be used in such a coaxial system of standardization. The required precision can be estimated from the preceding mathematical expressions.
-
Conclusion The use of internal reference standards does (a) permit simultaneous sweeping of the field or frequency spectrum for the reference standard and unknown sample and (b) make susceptibility corrections presumably common to both the reference and sample molecular systems. However, errors as large as several parts in lo7in chemical shift values for intramolecular hydrogen nuclei can arise from intermolecular effects. Hence, internal standardization methods should be used only with extreme caution. Chemical shift measurements, without quantitative corrections for molecular interaction effects, can lead to erroneous interpretations of data. External reference standard measurements of pure monomolecular systems can be made (a) without polluting the pure system and (b) without introducing intermolecular effects arising from the presence of the reference standard compound. Association effects of a monomolecular species are not separable from "isolated" molecular chemical shifts except by an independent experiment but this is also true in internal reference standard
procedures. I n external referencing for the case of cylindrically shaped samples, it is necessary to correct for magnetic susceptibilities of the reference and sample compounds. It should be emphasized that in the case of mixtures proper corrections for susceptibilities can become troublesome. The validity of simple extrapolation procedures often employed to correct for susceptibilities of mixtures is questionable. A plot of the position of the methyl group in n-propanol-water mixtures with respect to an external standard is shown in Fig. 8 to illustrate this point. The primary problem in the use of an external referencing procedure is to be certain that both reference and unknown systems are subjected to a common effective external magnetic field. An accurate spinning coaxial system for external referencing has been developed a t our laboratories which limits the errors arising from tube imperfections to less than 1 part in lo8. Any sharp line reference standard may be used in this system a t the mere convenience of the observer. Side-band modulation techniques for frequency comparisons are utilized. Ultimate limitations of the accuracy of this system are determined by the width of the resonance lines, the field sweep rate, and the accuracy of the modulating audio source. A typical spectral display of this external referencing procedure is shown in Fig. 9. Here the external reference standard is water; the sample spectrum is the water line in a pyridine-water mixture. Line broadening in the reference and sample compounds due to glass tubing imperfections has been considered under both static and spinning conditions. Certain line broadening phenomena are predicted in theory and observed experimentally which cannot be erased by rapid spinning of the coaxial system. This is due to the fact that the inhomogeneities in field strength due to imperfections of the tubing are not static if the system is spinning. The effects of tube tilting have been considered to show that the average fields in the reference and sample compound regions are shifted with respect to one another. Experimental measurements show that even in selected standard glass stock for the coaxial system, field shifts are often as large as 1 part in 107. The condition that the effects, line broadening and average field shifts, be negligible impose rather stringent requirements on the precision of the glass
March, 1057
Low 'I'EMPEI~ATURE CALORIMETRY OF SEVEN 1-OLEFINS hr
t
5 C.P.S.
-I
WATER LINE
EXTERNAL WATER LINE
Fig. 9.-Comparison
280
IN SOLUTION
No Correction For Susceptibilities of external reference water resonance with water resonance in pyridine solution.
tubing to be used in such a coaxial system of stand- permitting reliable data exchange between research ardization. However. such a Drecise coaxial svstem laboratories. is now to spectrosco~ists*'2 (12) The Wilmad Glass Company, Landisville, New Jersey, has reThis method for oently marketed a precision coaxial glass system which fulfills the reresonance spectra does give a practical means Of quirements necessary for accurate chemical shift measurements.
LOW TEMPERATURE CALORIMETRIC STUDIES OF SEVEN I-OLEFINS: EFFECT OF ORIENTATIONAL DISORDER IN THE SOLID STATE BY J. P. MCCULLOUGH, H. L. FINKE,M. E. GROSS,J. F. MESSERLY AND GUYWADDINGTON Contribution No. 60 from the Thermodynamics Laboratory, Petroleum Experiment Station, Bureau of Mines, U.S. Department of the Interior, Bartlesville, Oklahoma Raceiued Awunt W ,1066
From low temperature calorimetric mcasurements, values of the heat capacities in the solid and liquid states and of the heats and temperatures of phase changes were obtained for the following corn ounds: 1-hexene, 1-heptene, 1-octene, 1decene, 1-undecene, 1-dodccene and 1-hexadecene. Values of the entro y and3 other thermodynamic properties of each compound were calculated a t selected temperatures in the range 10 to 360°!k. The thermal behavior of this series of,l-olefin hydrocarbons is more complex than that of the corresponding series of n- arafFins. B comparing the entropy values of 1-olefins with those of the n-paraffins, i t was found that 1-undecene, 1-dofecene and 1-iexadecene crystallize in a state of orientational disorder. In 1-hexadccene crystals, virtually complete end-for-end disorder occurs. Because 1-undecene and higher l-olefins do not form perfect crystals, the Third Law cannot be used to calculate absolute entropy values from the experimental data for these three compounds. However, the results reported here and elsewhere for the Cg to CMhydrocarbons, inclusive, show that the entropy of these l-olefins in the liquid state at 298.16'K. is 0.19 f 0.02 cal. deg.-l mole-' less than that of the corresponding n-paraffins. Thus, reliable values of the entropy of the higher 1-olefins may be calculated by subtracting 0.19 cal. deg.-' mole-'. from the entropy value of the corresponding n-paraffin. The experimental and calculated entropy values of the l-olefin hydrocarbons from 1-octene to 1-hexadecene are represented within 0.04% by the 24.349 7.726 N cal. deg.-l mole-1, where N is the number of carbon atoms in the equation, &.a. (liq., 298.16"K.) molecule. =I
+
Entropy data for selected members of homologous series of compounds found in petroleum are determined from low temperature calorimetric measurements as part of the program of this Laboratory. A recent study of the nine normal paraffin hydrocarbons from n-octane to n-hexadecane' showed, for the liquid state, that the increment of entropy per CH?group is constant above n-octane (7.725 f 0.04 cal. deg.-'mole-Iat 298.16"K.). The results of this investigation provided essential data (1) H. L. Finke, M. E. Gross. Guy Waddington and €1. M. Huffman, J . Am. Cham. SOC.,7 6 , 883 (1964).
for calculation and compila.tion of the thermodynamic properties of the normal paraffins through n-tetrac~ntane.~'~ Low bemperature calorimetric studies of selected 1-olefin hydrocarbons have been made to obtain data needed for similar calculations of the thermodynamic properties of the homologous series of 1-olefins. Results obtained for the following compounds are presented in this paper: l-hex(2) W. B. Person and G. C. Pimentel, ibid.. 78, 532 (1953).
(3) American Petroleum Institute Research Project 44, "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds," Carnegie Preaa, Carnegie Institute of Technology, Pittsburgh, Pennsylvania. 1963.
