Measurement: of Vibrational Modes of +Paraffins
2261
Longitudinal Acoustic Vibrational Mode of n-Paraffins and Tetraalkylammonium Ions and Estimation of Their Molecular Dimension Hiroyasu Nomura," Shinobu Koda, Fumio Kawaizumi, and Yutaka Mlyahara Department of Chemical Engineering, Synthetic Chemistry, Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya-shi, 464 Japan (Received June 6, 1977)
Using a laser Raman spectrometer, the longitudinal acoustic (LA) modes are measured for n-paraffins in the solid and liquid states, and on five types of tetraalkylammoniumbromide in the solid and dissolved states. A slight deviation from the linear relation between the frequency of LA band and the reciprocal carbon number is observed for n-paraffins having small carbon numbers. This deviation is due to the changes of ( E l p ) ,where E is Young's modulus and p the density. LA bands of tetraalkylammoniumbromides in the dissolved state do not differ much from those in the solid state. For tetramethylammonium and tetrapropylammoniumbromides, splitting of the LA modes into two bands is observed. The ionic dimensions of tetraalkylammoniumcations are calculated from the frequencies of their LA modes. The values of ionic radius thus obtained are compared with those determined by various methods and found to be reasonable.
Introduction It has been reported that the Raman spectrum of solid n-paraffins contains in the low frequency region 26-500 cm-' a peak whose fundamental frequency is inversely proportional to the chain These observations can be interpretated by an elementary theory of longitudinal vibration, in which the carbon chain is treated as a solid rod and the frequency u is given by the equation
where 1 is the length of the rod; E , Young's modulus; p, the density; and m, the vibration order. The values of Young's modulus for a methylene chain given by eq 1are in close agreement with those determined by the different method^.^^^ This fact shows the validity of the vibration theory expressed by eq 1. For chains with less than about 18 carbons, however, the simple relation of eq 1does not hold exactly and an empirical relation between peak frequency and chain length has been proposed.2 Since the vibration under consideration is regarded as a symmetrical accordion-type longitudinal vibration of a string of beads with antinodes at the chain ends, it has been termed as the LA (longitudinal acoustic) modea6 Recently, the length of the straight-chain system in crystallized p~lyethylene'-~and poly(decamethy1ene seb a ~ a t e has ) ~ been determined on the basis of the LA modes. In liquid and in solution, similar LA modes have been observed'O and this fact indicates that a considerable number of molecules in solution exist in the same extended state as in the solid. The purposes of this investigation are twofold; (1) Experimental results of the LA mode are given for nparaffins in the solid and liquid states and also on five types of tetraalkylammonium bromides in the solid and dissolved states. The results are compared to each other. (2) The molecular shapes and ionic dimension of the tetraalkylammonium ions in aqueous solutions thus obtained are discussed.
Experimental Section The apparatus used consisted of an argon ion laser manufactured by Coherent Radiation Go. Ltd. and a JRS-U1 laser Raman spectrometer of Japan Electron
TABLE I: Frequency (cm-' ) of the LA Mode of n-Paraffins in the Solid and Liquid States Solid
cna 7 10 11 12 13 14 16 18 20 a
12.
m = l 311b 231b 194b 150b 133' 114'
Liquid m=l
m=3 315 233 209 197 184 172 152 139 117
311 231 209 196 183 172 150
502 493 450 436 393 358' 327'
Number of carbon atoms in n-paraffins. ' Reference 2.
359 320 Reference
Laboratory Go. Ltd. To obtain the Raman spectra of lower n-paraffins in the solid state, a Harney-Miller variable temperature Raman cell was used. The details of the experimental procedure and data reductions have been previously described.ll The n-paraffins and tetraalkylammonium bromides were of extra pure grade. Aqueous solutions at concentrations of 2 M were used except for tetraamylammonium bromide, for which a saturated solution (concentration less than 2 M) was used.
Results and Discussion (1)n-Paraffins in Solid and Liquid States. The Raman spectra and frequencies of the observed LA bands are summarized in Figure 1 and Table I, respectively. As is seen in Table I, our results are in excellent agreement with those of Mizushima and Shimanouchi.12 For samples with from 7 to 16 carbon atoms, their LA bands were observed in the liquid state. Shaufele13pointed out that there are no longitudinal modes for n-paraffin in the liquid state at frequencies lower than 200 cm-l. The highest Raman peaks in Figure 1 correspond very well to those of (TGT'G'), assigned by Sha~fe1e.l~ However, as is shown in Figure 1,the highest peaks are not due to the LA mode of n-paraffin. The frequency of each LA band in the liquid state was equal to that in the solid state. For n-paraffins with more than 18 carbon atoms, LA bands were hardly detectable, as seen in Figure 1. These experimental results clearly indicate that the conformation of n-paraffin molecules, which is in the all-trans form in the solid state, is partially held even in the liquid state. With an increase The Journal of Physical Chemisfry, Voi. 81, No. 24, 1977
2282
Nomura et al.
TMA
TEA
TPrA
&& ,A. J.;
T BuA
c13
TAmA
1
Figure 1. Low-frequency Raman spectra of n-paraffins in the solid (at -50 'C) and liquid state (at room temperature). The upper lines correspond to liquid state and the lower ones to solid.
TABLE 11: Frequency (cm-l ) of the LA Mode of Tetraalkylammonium Bromides in the Solid and Dissolved State ena
3 5 7 9 11
Solid
Liquid
454,462 428 316,337 265 226
458 418 310,336 253 221
200 400 600 c m-'
200
400 600 cm-'
Figure 2. Low-frequency Raman spectra of tetraalkylammonium bromide in the solid (at -50 "C)and dlssolved state (at room temperature). The solid arrow shows the LA mode of the all-trans form and dotted arrow shows that of the gauche form.
500
400
a Sum of carbon and nitrogen atoms in tetraalkylammonium bromides.
in carbon number, or chain length of n-paraffin, the number of possible conformation of molecules increases owing to internal rotation. This leads to a smaller portion of n-paraffin molecules taking the trans form. Similar results for polymethylene in the liquid state have been ascribed to a kinked conformation of the chains.13 The following equation has been presented for the LA bands of polymethylene chains2
(cm-') = A(rn/n) + B(rn/n)2 t C ( r n / r ~ + )~ D(rn/n)4 + E(rn/n)5 + F(rn/n)6
E
"
200
100
(2)
where A = 2495 f 86 cm-l, B = 45.867 f 2.855) X lo3cm-l, C = (6.253 f 3.537) X lo4 cm-l, D = -(3.485 f 2.058) X lo5 cm-l, E = (7.329 f 5.676) X lo5 cm-l, F = -44.724 f 5.964) X lo5 cm-l and n is the number of chain units. Frequencies calculated using eq 2 are compared with the observed values in Table I. Connolly and Kandalic14have calculated the core width and the length of n-paraffins on the basis of their second virial coefficients. They obtained 1.26 A for the core length of one C-C bond. This value is in excellent agreement with 1.275 A, obtained from eq 1. From these results, it is concluded that the deviation from the linear relation between the frequency of the LA band and the reciprocal carbon number in n-paraffins is mainly due to the changes of ( E / p )in eq 1. The Journal of Physical Chemistry, Vof. 8 1, No. 24, 1977
300
OO
0.1
0.2
Measurement of Vibrational Modes of +Paraffins
2283
TABLE 111: Ionic Radius of the Tetraalkylammonium Ion r(R-S)" r( SPT)b Me, N' Et,N+ (n-Pr),N+ (n-Bu),N' (n-Am),"
3.47 4.00 4.52 4.94
2.51 2.08 3.49 3.81
r(T)c
r( VWId
r( LA mode)
2.70-2.78 3.20-3.37 3.62-3.86 3.94-4.24
2.80 3.37 3.78 4.13 4.43
2.70 2.99 4.17 5.40 6.69
Values determined from comparison with the scaled particle theory and the salting coefficients, ref a Reference 20. Values cited in ref 21. Values calculated from the relationships between the partial molar volume of ions and ionic 21. Effective radius of the van der Waals volume calculated by the present authors using the method of Bondi in ref radius. 22. The following values are used in the calculation, r&N) = 1.55, r&C) = 1.70, rw(H) = 1.20, and ~ C - N= 1.465 A .
relationship between the frequency of the LA band and the reciprocal carbon number is shown in Figure 3 along with the corresponding relationship for n-paraffins. As is seen in Table I1 and Figure 2, the LA band of tetraalkylammonium ions in the dissolved state does not differ much from that in the solid state. The limiting slope of the relationship between the frequency of the LA band of the tetraalkylammonium ion and the reciprocal carbon number is nearly equal to that for n-paraffins. The x-ray crystallographic studies have shown that in tetraethylamm~niuml~ and tetrapropylammonium16 bromides the alkyl chains are in the all-trans form, and the following values are given to their bond lengths and bond angles: C-C, C-N = 1.55 A; LC-N-C = 105' and LC-C-C = 106'. These values for the tetraalkylammonium ions correspond very well to the values of n-paraffins. On the other hand, slight differences exist for the force constant K and for the deformation constant K2: 4.13 and 0.43 mdyn/A for the tetraethylammonium ion,17and 3.2 and 4.0 mdyn/A for n-butane, respectively. As the molecular weight of nitrogen is equivalent to that of a -CH2- unit, the density of the tetraalkylammonium ion should be nearly equal to that of n-paraffins. For this reason, the differences between the frequency of the LA band of the tetraalkyammonium ions and that of nparaffins are attributed mainly to the differences in their Young's modulus. The fact that the frequencies of the LA band of the tetraalkylammonium ions do not differ much from those in the solid state indicates that the molecular dimension of the tetraalkylammonium ions in dissolved state should be comparable to that in the solid state. For tetraalkylammonium ions, the ratio of molecules remaining in the all-trans form in solution was estimated by the following procedure: For each tetraalkylammonium ion, the values of the ratio, r = Ia/11457 were calculated both in the solid and dissolved states, where I, is the intensity of the accordion (LA) band and 11457 is that of the band at 1457 cm-l which has been assigned as the deformation band of -CH3. The ratios r(solution)/r(solid) can be considered as a measure of the population ratio of molecules taking the same molecular structure as in the solid. Figure 4 shows the relation between r(solution)/r(solid) and the number of atoms included a chain (nitrogen atom being considered). In Figure 4, the values of r(so1ution)/r(solid) up to tetrabutylammonium ion are ca. 1,with the exception of the tetrapropylammonium ion. The values of approximately 1 indicate that each tetraalkylammonium ion holds mainly the same molecular dimension in aqueous solutions as in the solid state. The smaller intensity ratio of tetraamylammonium ion is ascribed to the rotational conformation of -CH2- chains in the ion. In Figure 2, the splitting of the LA mode of the Raman spectra into two peaks is seen for tetramethylammonium in the solid state and tetrapropylammonium bromide in the solid and dissolved states. The occurrence of the splitting of the LA mode may be associated with the
2.0'
5 10 number of a t o m s i n c h a i n Flgure 4. Relationship between r(solution)lr(solid)and the number of atoms in the tetraalkyiammonium ion chains.
existence of the terminal methylene groups taking a gauche conformation. The peak at higher frequency is assigned to the LA mode of the gauche conformation. An interpretation of the occurrence of the LA band arising from the molecules taking the gauche form, especially for the tetrapropylammonium ion, is given from the point of its relationship with the phase transition reported for tetraalkylammonium iodides.17 For tetrapropylammonium iodide in the solid state, two phase transitions have been observed. One at lower temperature, -43 "C, is considered to result from a change of crystal structure and the other at higher temperature, 35 OC, from a conformational change in the cations. On the contrary, for tetramethyl-, ethyl-, and butylammonium iodides, the second phase transition has not been observed. The fact that the experiment was carried out at -50 "C and with considerably rapid cooling, 2 'C/min, is strongly related to the appearance of the LA band of the gauche chain in tetrapropylammonium bromide in the solid state. However, as shown in Figure 2, an inversion in the intensity relation of two peaks was observed as to the two LA modes of tetrapropylammonium bromide. In addition the "true" LA band is stronger in the dissolved state. These experimental results indicate that the all-trans form of the methylene chain is more stable than the gauche form in aqueous solutions. The ionic dimension of tetraalkylammonium cations can be estimated from the frequencies of their LA bands; the results are summarized in Table 111. In Table I11 is included other ionic dimensions for tetraalkylammonium ions obtained by various methods. In general, the values of rR4N+(LA mode) are larger than the correspondingvalues of rR,+(SPT). With an increase of the chain length in the tetraalkylammonium ions, the values of rR4N+(LAmode) increase more rapidly than any other values of rR4y+.However, for the first three types of tetraalkylammonium ions, rRa+(LA mode) falls between rR4,+(SPT) and rR4N+(empirical). Each value for the ionic radius shown in Table I11 has each different significance as it should be. The molecular dimension expressed by The Journal of Physical Chemlsti-y, Vol. 81, No. 24, 1977
L. Madsen and L. J. Slutsky
2264
rR4,+(LAmode) is the ionic volume in the extended state, because the LA mode arises from the molecule taking the all-trans form. Therefore r R N+(LAmode) does not correspond to rR4N+(SPT),which represents the hard core volume of ions and it follows the general relation that rR4,+(LA mode) > ~R~N+(SPT). In conclusion, it was confirmed that the molecular structure of the tetraalkylammonium ion in aqueous solution is an extended one, such as the all-trans from in CH2 chains, especially for lower members of n-paraffins. The ionic radii of the tetraalkylammonium ion shown in the first to forth columns of Table 111, correspond to an effective radii in aqueous solution. In addition, n-paraffins containing long CHpchains are, on the average, not in such an extended state and therefore their effective ionic radius should be much smaller than that obtained from the LA band.
References and Notes (1) S . Mizushima and T. Shimanouchi, J . Am. Chem. Soc., 71, 1320 (1949). (2) R. F. Schaufele and T. Shimanouchi, J. Chem. phys., 47, 3605 (1967). (3) R. F. Shaufele, Macromol. Rev., 4, 67 (1970).
(4) I. Sakurada, T. Ito, and K. Nakamai, J. Polym. Sci., C15, 75 (1966). (5) T. Shimanouchi, M. Asahina, and S. Enomoto, J. Polym. Sc/., B9, 93 (1962). (6) W. L. Peticolas, G. W. Hibler, J. L. Lippert, A. Peterlln, and H. G. Olf, Appl. Phys. Lett., 10, 87 (1971). (7) M. J. Folkers, A. Keller, J. Stejny, P. L. Goggin, G. V. Fraser, and P. J. Hendra, Colloid Polym. Sci., 253, 354 (1975). (8) H. 0. Oif, A. Peterlin, and W. L. Peticolas, J . Polym. Sci., 12, 359 (1974). (9) J. L. Koenig and D. L. Tabb, J. h c r o m l . Sci'.,Phys., BO, 141 (1971). (10) H. Okabayashi, M. Okuyama, T. Kitagawa, and T. Miyazawa, Bull. Chem. SOC.Jpn., 47, 1075 (1974). (1 1) H. Nomura and Y. Miyahara, Bull. Chem. Soc.Jpn., 48,2779 (1975). (12) S. Mlzushlma and T. Shimanouchi, Nippon Kagaku Zasshi, 64, 1064 (1943). (13) R. F. Shaufele, J. Chem. Phys., 49,4168 (1968). (14) J. F. Connolly and G. A. Kandalic, Phys. Fluids, 3, 463 (1960). (15) N. C. Stephenson, Acta Crystallogr., Sect. B , 17, 587 (1964). (16) A. Zalkin, Acta Crystallogr., Sect. B , 10, 557 (1957). (17) J. T. Edsail, J. Chem. Phys., 5, 225 (1937). (18) T. Shlmanouchi and S. Mizushima, Nippon Kagaku Zasshi, 64, 1215 (1943). (19) J. Levkov, W. Kohr, and R. A. Mackay, J . Phys. Chem., 75, 2066 (197 1). (20) R. A. Robinson, and R. H. Stokes, "Electrolyte Solutions", 2nd ed, Butterworths, London, 1959. (21) W. L. Masterton, D.Bolocofsky, and T. P. Lee, J. Phys. Chem., 75, 2809 (1971). (22) A. Bondi, J. Phys. Chem., 66, 441 (1964).
Relaxation Times for Acid Ionization and Internal Proton Transfer in Polypeptides in the Neighborhood of the Helix-Coil Transition L. Madsen and L. J. Slutsky" Department of Chemistry, University of Washington, Seattle, Washington 98 I95 (Received January 14, 1977;Revised Manuscript Recelved Ju/y 22, 1977)
Relaxation frequencies and normal coordinates are calculated for the system of coupled reactions formed by the acid ionizations at glutamyl residues in poly-L-glutamic acid. The results suggest that the perturbation of overall and internal ionization equilibria constitute a reasonable alternate interpretation of relaxations which have been attributed to perturbation of the helix-coil equilibrium.
I. Relaxation Times for Intramolecular Proton Transfer The kinetics of the proton-transfer reactions involved in the relaxation of fluctuations in the overall and internal ionization equilibria in solutions of proteins and uniform polypeptides has been the object of some The role of perturbation of proton-transfer equilibria in determining, and perhaps in complicating, the interpretation of relaxation spectra in the neighborhood of a reversible pH-induced conformational transition is also an object of c ~ n c e r n . ~We - ~wish here to discuss the normal modes of the system of acid ionization reactions (reaction 1)of side -AHi
kfi
F=-=
-Ai
+ H'
kbi
chains in the helical (i = 1) and coil (i = 2) regions of a uniform polypeptide coupled to the ionization of an indicator (i = 3), to consider the application of these results to the interpretation of spectra observed in solutions of poly-L-glutamic acid in the neighborhood of the helix-coil The Journal of Physical Chemistry, Vol. 81,No. 24, 1977
io
-
kb~[-A;] kb3[-A3]
W2O-W
kbi [-A;] kbz[-A;]
kb3[-&1
03'-
&i
[-A;]
0
=0
(2)
