Vibrational spectra of crystalline n-paraffins

JOURNAL

OF MOLECULAR

Vibrational

7, 116-l-14 (1961)

SPECTROSCOPY

Spectra

of Crystalline

n-Paraffins

II. Intermolecular Effects ROBERT G. SNYDER Shell Development

Company,

Emeryville,

California

Intermolecular effects in the infrared spectra of crystalline n-paraffins, n-CzoHn2 through n-CZ0H6? are reported. Absorption bands in the spectra of the triclinic form are singlets. Extensive doubling of bands occurs for the mono clinic and orthorhombic structures in accordance with selection rules based on the factor group symmetry. In particular, the methylene rocking mode bands of the orthorhombic structure display a striking pattern ‘of splitting. Frequency separations of the components can be expressed by a simple function of a single parameter, ka/(m + l), where WLis the number of methylene groups of the n-paraffin and k is an int,eger which characterizes a particular intramolecular mode. From these data several intermolecular force constants have been evaluated. In relating these force constants to specific methylene interactions, methylenes in adjacent planes were found to int,eract more strongly than those in the same plane. By extending Stein’s theory that t,he splitting arises from short range repulsive forces, to include additional interactions, the force constants are explained satisfactorily. However, interactions which involve only dipole-dipole forces are found t,o be inadequate. INTRODUCTION In an earlier paper ( 1) the infrared spectra of a series of crystalline n-paraffins, YL-C&,H~~through n-C&He? , were considered. nlet,hylene wagging and rocking modes were analyzed on the basis of a single isolated molecule and detail consisting for t,he most part of doubling of absorption bands was ignored. In the present paper, attent’ion is focused on these intermolecular effects and their relation to the crystal structures of t’he n-paraffins. Doubling of the strong methylene rocking absorption band near 728 cm-’ has been reported much earlier (2). That the doubling is intermolecular in origin has been well established by t#he following observations: (i) One component (722 cm-‘) of the rocking mode doublet of n-C$~H~o is polarized along the b axis’ of the crystal and the other component (734 cm-‘) along the a axis. Similarly, one component (1475 cm-‘) of the methylene bending mode is polarized along a and the other (1462 cm-‘) along b. These observa-

1For the convention rules.

regarding

crystal

axes, 116

see Fig.

1 and the discussion

of selection

VIBRATIOXAL

SPECTRA

OF CRYSTALLIKE

n-Pi4RAFFINF

117

tions (3) are consistent with the selection rules for this n-paraffin based on its known crystal structure. (ii) Only one rocking mode band (725 em-’ ) appears in t,he spectrum of IZ-C&H~~~for low concentrations of t,his n-paraffin in a matrix of n-Cl,,,,HBo2 .is the concentration of n-C~~Hlno is increased, the doublet appears (3). If the splitting were an intramolecular phenomenon, no dependcnee on crystal cnviroiimcnt should he observed. This is eontraqv to what is found above and contrary to (iii) below. ( iii) W;e observe that the oceurrenct of splitting is dependent on the crystal structure of the n-paraffin. Sptit,ting of absorption bands in the infrared spectra of a considerable numhc~ of crystalline solids has been observed and thi s splitting has been frecmently \ery few systems have IW~II att,rihut.ed to intermolecular effects. Xeverthetess studied in detail partly because very little structural data on vibrationally simple molecular crystals are available and partly because of the limited amount of int,ermoleeulnr coupling data available for any single system. In both these regards a study of the crystalline n-paraffins is attractive since their crystal strmtours are kmnvn (and are comparatively simple 1 and since the homologous series of compounds provides an uncommon number of data. In this paper we describe the splitting of infrared absorption bands, relate this split,ting Tao the c*rystat struc$ure, derive intermolecular force constants from the dat,a where possible, :md fi~~~tty :i.ttempt to re1at.e these force const.:uits to hvdrogell-hydrogetl repi2lsivc fortes. EXPERIhIE1;TBL

Alt,hough experimental procedures have been dcseribcd in t,he earlier paper t 1 ) , they are summarized again below. All infrtlred measurement,s were made on n-paraffins of purity greater than !N “; (4 ) _ .\hsorption curves were recorded by :I Hec~kman IR-7 spectrophot,oneter. Samples of n-paraffins suit,able for infrared apectrophotometSry were prepared in potassium bromide pressed disks. B toluene solution of the n-paraffin was poured upon powdered pot.assium bromide and t,he resulting paste was ground in a mortar unt,il the solvent, had evaporated. The translucent~ disk, obtained by pressing the dry KBr-paraffin powder, was secured in a low temperat22re wll equipped with rocksalt, windows. To determine if the crystal struct.ure of the n-paraffin had been altered and/or if t,here was orientation of cryst,allites as a result, of pressing, t,he spectra of several disks were compared with t,hc speetm of the corresponding n-paraffin, formed by rapid solidificat,ion of a “sandwich” of a melt. Mween two salt windows. Since t,hc two spectra were identical, vve con(+lude t,hat pressing in a KBr matrix had no observable effect. upon the strut*tine or oricntat,ion of the crystallit,es. .ilt sprat ra were recorded with the samples at - I 80°C‘ and all fre~tuencies

SNYDER

118

TABLE I CRYSTALSTRUCTURES OF ~-PARAFFINSIN THE RANGE n-C20H42THROUGHn-C&oH,? Crystal system

Number of carbon atoms

Space group

Molecules per unit cell

Kef.

Orthorhombic Monoclinic Triclinic

odd numbers 21 through 29 even numbers 2 26” even numbers $ 261L

Pbcm P21/a ?

4 2 lb

6 Y 6

9 n-Cz6HS8can have either the triclinic or monoclinic structure (6). The spectra obtained in this research were of samples in the triclinic form. b A. E. Smith (private communiczkion).

recorded in this paper - 180°C. VIBRATIONAL

are, unless

otherwise

indicated,

those

SELECTION RULES FOR CRYSTALLINE

of n-paraffins

at,

n-PARAFFIN8

Vibrational selection rules for the crystalline state are derived from the symmetry of the unit cell toget,her with the number and symmetry of the molecules in t.he unit cell (5). Crystal st,ructures of the n-paraffins have been reported for odd numbered and even-numbered members of the homologous series considered here (6, ‘7). These structures are summarized in Table I. Selection rules for the orthorhombic, monoclinic, and triclinic strurt,ures will be discussed separately. ODD-NUMBERED ~-PSR~~FFIKS HAVING THE ORTHORHOMBIC STRUCTUIIE Figure 1 shows the arrangement, of the four molecules in the orthorhombic unit cell (6). A unit cell is indicated by the dashed lines. Crystal axes, a, b, and c are defined in the figure. Molecules are arranged in ident.ical layers wit,h their long axes perpendicular to t#he ab plane of the layer. The repeating unit’ of a single layer, enclosed by solid lines in the figure, contains two molecules arranged in a manner ana%@‘% t#o that of the two chains traversing t’he unit cell of polyethylene (8). Two layers, parallel but differently disposed cut across the unit cell of the n-paraffins. The factor group symmetry of the unit cell is isomorphic with point group Dn,, , the symmetry elements being E’, C,“(c), C,“(b), Cz’( a), i, (UC). Symmetry elements which leave molecules in their own G( ab), ceCc)(bc), crgCa) layers, E, G’(u), o(ab), 9(a) (UC), form a factor group CfV of the subcell containing two molecules. The only elements not interchanging molecules are E and u(ub). These constitute site group C’, . The point group symmetry of a n-paraffin having an odd number of carbon atoms is CzU. Axes are defined as follows: .L‘and z are in the carbon skeleton with z being parallel to t#he long axis of the molecule; 1~is perpendicular to the skeletal plane. For the molecule in the crystal, z and c are parallel. Selection rules for vibrat.ional modes of an isolated molecule of an odd-num-

VIBRATIONAL

l’ra. 1. Crystal

SPECTRA

structure

OF CRYSTALLINK

of t,he orthorhombic TABLE

INFRARED (IR)

AND

RANAN

I l!,

n-PARAFFINS

form of the odd-numbered

rr-pnrafiw

II

(R) ACTIUTY

OF METHYLESE

~‘IBRATIONS’

.ktiviQ hide C’?,. (odd-numbered

hIethylene blethylrne

n-paratfins,

(;b

Ieven-numbered rz-paratlins)

bending; wagging!

1\Iet,hylenc twisting\ 12Zet.h.vlene rocking j :b !I is ~~erp,rndicrdnr molecule.

to the

plane

of the

nrolccule;

z is pardIe

to the long

:Isis of

I Iw

brrcd N-pwaffin hare been discussed clscwhrrr ( 2 ). Table II summarizw thv of the infrared act,ivity of methylene modes. In 1+3g. 2 ia shown a correlation symmetry speck of the molecular, site, and factor group symmctrics. ‘I’ht site symmet,ry C’,
120

SNYDER Site Group (Molecule)

Point Group (Molecule) Czv

Factor (Complete

Group Unit Cell)

Factor Group (Subcell) Czv

*zh

cs

A’ (XY) -se__

Species Species

FIG. 2. Correlat,ion

which which

are Underlined Have Infrared Activity are Underdashed Have Raman Activity

of molecular,

site, and factor

group symmetries

of odd-numbered

n-paraffins.

modes. Thus those modes belonging to the infrared inactive A, species of point group CPUare now allowed activity. No indication of infrared activity for these modes was found in the spectra, however. The correlation between the factor group and the site group shows t,hat,each molecular mode has four components in the crystal. In so far as infrared activity is concerned, each fundamental vibration of an isolated molecule gives one band for parallel (x) modes and two for perpendicular (2, y) modes. The same selection rule applies also to polyethylene where again one band for parallel and two bands for perpendicular modes are indicated (9). Since we expect greatest interaet,ion t.o occur between molecules in the same ub plane, it is appropriate to choose a unit cell containing only two molecules. As mentioned above, this subcell has factor group symmetry Ctz). The correlation diagram (Fig. 2) shows that the effect of choosing a smaller cell is to remove the distinction between the two possible phase relationships in t,he relative motion of adjacent layers, i.e., between g and u modes of De,& . EVEN-NUMBEREDWPSRAFFINSHAVING THE MONOCLIKICSTRUCTURE The monoclinic structure (7) is very similar to the orthorhombic, differing from it in that the chains are tilted with respect to what would be t,he ab plane of the orthorhombic cell (see Fig. 1). The site symmetry of the molecule is Ci so that its infrared activity is the same as for the isolated molecule. The presence

VIBRATIOSAL

SPECTRA OF CRYSTALLINE

Point Group (Molecule) C2h

N

C2h

Ag

/

Ag\

All (Y)\ -I

_ l“1

Factor Group (Unit Cell)

ci

Ag------l Bg

Site Group (Molecule)

n-PARAFFINS

Bg ~

A, b)

L

B, bc)

&(XYZ)

&l (xz)

FIo. 3. Cor~~el:ttioli of molecular. site, :cud factor group, r?-mrnetriw of t~verl-n~itnl)~retl wp:rmffins

having

the monoclinic

structure.

.\ssigluncnts for mrlst of t.h(h methylcne rocking modes ha\-e been made rarlk ( I ). ‘l’hcsc vihrzations extend from 7%) cm- to somewhere above 1000 cm-‘. 7 A. IX. smith (private communication).

122

SNYDER

/

0

1100

1000

900 Frequency,

800

700

cm-’

FIG. 4. Infrared spectrum of an orthorhomhic m-paraffin, n-Cz:jH,s , at -180°C 100

1000

900 Frequency,

800

700

cm-’

FIG. 5. Infrared spect,rum of a monoclinic ,n-paraffin, n-CzsH,, , at -180°C

The diode which involves most nearly in-phase motion of adjacent, methylene groups is the very strong band near 725 cm-‘. In accordance with the selection rules, this band is a doublet for polyethylene and for n-paraffins having t,he orthorhombic and monoclinic structures, but not for those having t#he kiclinic st,ructure. Figures 4-G show representative spectra for each of the t,hree structures. If the n-paraffins in this molecular weight range are very pure they will upon crystallizing assume one of the three structures according to the rules given in Table I. If they are contaminated by a few per cent of neighboring

VIHRATIONAL

SPECTRA

OF CRYSTALLIXE

1“>I

n-PilRAFFISS

100

I

I

1000

FIN:. 6, Illfrared

spectrllm

/

900 Frequency, of a trivlinic

800

700

cm-’

w1wdFm.

&‘:,H:e

, at -18O’Y

homologs, t’hr preferred structure for all the n-paraffin. q is orthorhomhica ( /i 1. Ikuhling has hccn rtported for the 7%cm ’ hntl of some of the ,~-parafills known t,o havr a t.riclinic st.twt.urt whelr vrry pure ( 20 I ; it is likely thcst spectra :w of imp\wr sumplrs. R~gularitiw in t)ht hand splitting of the rocking modes arc’ evident in l’ig. ‘7 whkh sumnxu-izw t#hc spectra of t,hc odd-numherrtl n-paraffins. In procwdiug from t.hc 725-m-' doublet. toward hightlr frct~uttwics, w find separations of wmponrnts diminishing until in t,hc region near 82.5 cm -’ the two componr~~~ s overlap. 111going to still higher frequcn&s separations increase. Thew are ASI) rrgulzlritiw in the rrlativr peak heights of thr componrnts. With the c>scAept,iotl pairs of lines wry near the 8!)0-(m ’ hnd, nhosr pcrturhiug illfluc1rc.t will tw diwussrd separately, the less int,c‘nw cwmpmwnt is always the OIIP nrarwt in freclucwy to thr posit,ion of minimum separation ( 432.3 c*iim-11. If it is assurucd that. the int.ensitics of t.he two components result from the addition :uld srlt)of the tn-c) t,rwtion of the ~mperturhed dipoles rwultin, u from the vibrations molcculrs of’ the s[~hcrIl, t.hr ratio of the intcnsitics of the tn-o (~ompolwnts is

of scvwal

whrrc 0 is the an& hctwccn tht skdctd planr of :I o-paraffin molrc& and thus (I axis of the lmit ~11 ( I:ig. 13 ). The atlgl~ 0 has hcc~~ dctwmined to hi 12” h .? by x-ray dift’raction st.udies upon a single c*rystal of ~L-<‘~~H~~ at 28°C ( ti ). If thir angle is still 1~s than 45” when t,hr sample is (~~olecl to - 180“(‘, t heI1 I, mlwt tw less t,h:tll I!> . Since hoth components havcl nearly c~lunl halfwidths, thcl n Mompo~wni is thrl hand which has the smallw absorptivity. This assignm~~~t is COII-

124

SNYDER TABLE

III

SUMMARY OF ABSORPTION BAND STRUCTURE FOR CRYSTALLINE n-PAR.mFINs Orthorhombic Methylene rocking No. of components Frequency” (cm-l) Separations (cm-‘) ~L-CLH~~+~measured Methylene bending No. of components Frequency (cm-l) Separation (cm-l) n-C,,H2,+2 measured

(n =)

2 7OG1050 O-12.3 21, 23, 25, 27, 29

(n =)

Monoclinic

2 700-1050 o-12.4

Triclinic

1 7Ol-1050

28, 30

20, 22, 24, 26

2

2

1475, 1462 13.8 23, 29

1475, 1462 12.8 28

1 1473 24

Symmetric methyl bendingb No. of components Frequencyc (cm-‘) Separation (cm-l) n-C,H2.+2 measured (n =)

2 C?)

?

1 (?I

1373, 1378

1367

1368

5 (?) 21, 23, 25, 27, 29

28, 30

24

Terminal C-C stretching No. of components Frequency (em-l) Separation (cm-‘) wC,H~,,+~ measured (n =)

2 893, 890 2.7 21, 23, 25, 27, 29

893, 889 3.2 28

2

1 894 24

a Frequencies are accurate to fl cm-l. Component separations are accurate cm-l. Measurements made with all samples at -180°C. b In all cases these bands were accompanied by weaker bands which obscured structure of the bending mode. c Only the frequency of the dominant band is given.

to ho.2 the true

&tent with t,he a polarization of the 7:3S-cm-’ band measured by Krimm et al. (3). Frequency separations of the components have been carefully measured. These are list,ed in Table IV. At room temperature the majority of the doublets cannot be resolved because the bands overlap badly. When the temperature of the sample is lowered, the bands narrow and the separation between their components increases. Although lowering the temperature causes increased separation between components, each component is affected differently. With the exception of the 725-cm-’ doublet,, decreased temperature results in both components shifting to lower frequencies, with the lower frequency component shifting more. Thus, the mean posit’ion of each doublet shift’s toward lower frequencies while separation of the components increases. The 725-cm-’ doublet, behaves differently. With lowered t,em-

z9-IIIIII z7-l I

II

II

II

II

1,

,I

I

II

I

I I

I

I

I

u

‘c i

25I

c:

II

I

II

II

II

II

1

I

:

2j-I

I

z'-

I

Ii

I

II

II

I

b

I I

II

II

I

I

I

I

I

I

I

I

I

750

780

810

840

870

900

?30

960

990

1 I 720

II

:

u),cm-' FIG.

i.

Met.hylene

rocking

modes for odd-numbered

ri.-paraffins

pwuture

both component#s move t,oward higher frequencies with the higher frr(lucn(‘y component moving more. At, low temperaWes” the half-widths of these bands are very small. At the highest resolution available, -0.4 cm-‘, halfwidths of about 0.5 to 1.0 cm--’ at - 180°C~ were recorded; near O”C, halfwidths are about :i enlCL. hlt~hough the available resolution and band halfwidths are of the same order for the samplrs at - 18O”C, the ratio I,!Ib call be measured a-ith some accurac*y, if thr product of the peak height, and halfwidth is 11sed for band intensity. I;or t,he average of 12 doublets,* I,/Ib is 0.8 f 0.1. For these measurements hands were selected whose components were well resolved and whose frequencies were sufficiently removed from the 890-cm-’ band to precllude significantj interaction. Then by 141. ( 1 ), 0 is calculat#ed to be 4%"+ 3, in good agreement wit,h thr vxlr~ -I-‘” + 5 rcport,rd from the x-ray diffra(*tion data on n-Cz:,H4x at %“C i 6 1.

This band has been examined cnrrflrlly for wC~:~H~~ and IZ-C?~H~O( Fig. 8 1. In this region there are three strong bands which dominate in intensity plus a of equal munber of weak bands. Two ( 1462 and l4i5 cm -“I are approximately 3Recently we have obt,ained tho spectrum of n-C?,H, at -269°C. The spectrum is essentially the $ame as that for the sample -180°C. Halfwidt,hs are slightly smaller and splitting slightly greater. ,* Specificall,v the doublets measured were IL = 21, k = !1, 11, 15; II = 23, k = 11, 25, k = 11, 19; n = 27, k = 21; n = 29, !i = 11, 13, 21. 23.

17; n =

SNYDER

126

TABLE SEPARATION

Iz

21

23

k’

It

1.80 0.77 -2.50 -4.95

7

731

9 11

752 783 830 881

,-9.9 6.1, 2.6~

9 11 13 15 17 19

741 767 804 850 902 952

FOR ORTHORHOMBIC

k 2s __

763 807 863 924

938

IV

ROCKING MODE COMPONENTS %-PARAFFINS AT -180°C

‘;; 3+; 5" x,

9 11 13 15

13 15 17 25

OF ~~ETHYLENE

-,-0.2 -3.34 -5.40 7.2~ 4.28 0.8; -1.911 4.18 -5.1,

7.33 1.24 -1.32 -9.3:, X14.5 9.27 4.08 --0.3 -5.9, -10.1, 10.8, 6.5~ 1.40 -3.2, -7.54 -10.42

27

29

1: 13 15 17 19 21 1 11 13 15 17 19 21 23

755 781 823 870 917 963 727 748 770 802 812 886 930 973

5.58 2.59 -0.0 -2.80 -4.7s -5.99

8.41 4.06 NO.0 -4.81 -8.71 -11.52

12.30 6.6.1 4.05 1.0, -1.22

17.8, 9.9” 6.2, 1.72 -2.05

-3.34 -5.3s -6.42

-5.9, -9.9, -12.49

:t These assignments are given in Ref. 1. Infrared active rocking modes of a given molecule are dintinguished by the odd integer k. The phase difference between the rocking motion of adjwent oscillators (methylene groups) is proportional to k (see Eq. 11).

intensity and are more intense than the third ( l-1-66 cm-’ ). The former are int.erpret,ed as the two eomponems of the met,hylenc bending mode. They are separated by 12.8 cm’. The third hand is probably the asymmetric met,hyl deformation. Terminal

C--C

Stretching Al odes

The relatively strong band near 890 cm-’ which is present in the spectra of all the n-paraffins is commonly attributrd to a methyl out-of-plane rocking mode (1, 3, and 11, for example). However, the results of recent normal coordinate analyses carried out in this laboratory,5 on simple saturat.ed hydrocarbon molecules, indicate that methyl rocking modes occur in the region 1150-1050 cm-’ with t’he in-plane and out-of-plane modes not likely t,o differ in frequency by more than 50 cm-‘. In view of t’hese findings the assignment of the 890-cm-’ 5These calculations were carried out by Dr. J. H. Schachtschneider to whom the author is indebted for suggesting the interpretation of the 890-cm-l band related here.

VIBRATIOSAL

SPECTRA

OF CRYST.4LLI?iE:

r,.-PAR.AFF1N.C

13

”

1480

1410

1460

1480 Frequency,

F‘Ic;.

8.

Mcthylene

beding

modes of the npectrmn

1470

1460

1450

cm-’

of wC,,H,,

. and n&_.Hro 1at -1XOY (“q

h:tntl to :I methyl rocking mode seems untenable. IIowcvcr, these snmr nornx~l voordinatc analyses do point, to the existence of a vihratlion which has :I,frc~lut~~wy IIC1L1!,OO u11-’ :~tltl lvhicah, although involving considerable methyl in-plane rwking motiolr, is appropriately called C----C stretching since stretching of the ttrminal (‘pm (’ hand makes t.he largest, contribution to t.he potent,ial energy of this mode. We deem the assignment, of the 81)O-cm~~’hand t.o this mode mwh mow s:ltisfwtor!than to a methyl rocking mode. The 8N-cn~~’ hand in t,he spectra of the ort,horhomhic wparafins consists of with the higher frequency compouc~rlt t m-o components separated by 2.7 cm--’ lwiiig more intense. This vil)ration ir1teract.s with methylene roeking vibrations when they :IIY~ IKW in freqlwncy. h result of this intertlction is to change the ratio of the ilrtvnsities of the two rocking mode components. A>\ example of such an itlt.cractio~~ is show1 iti I”&. 4 and 9 for wC~,FI~~ . Here t,he n component. of the perturbed rovkiug mode hand is stronger than the 0 compolwnt, a sitlwtion quitr thr IV verse of the lwrmal one. We were, however, ul~ablt to detect vhangrs in the frrquen&~s of the interact’ing bands. Since the 8!)0-cm-’ band is an in-pl:tllcs anode and thr mcthylene rocking modes are out-of-plane, this wry unusual intcrwtion must he between components hclonging to the s:mw spwies in the factor interaction. group, i.e., an intermolecular Jleth$

$!~mvnefric Deformaiion. iIf odes

The st,rurturr of this band is essentially the same for all the odd-numbered v-paraffins awl is represented in Fig. 10 by .t~-(‘?;H~c . The band at 1373 cam ’ has an unresol~-ed companion at 1378 cm-’ which could be a second fundamental f two modes are allowed) or could he an intermolecular component. The rathcl wmplicated structlw of t.his hand observed for the mouoc~linic st,rwtrwc serves to tvmpw speculation.

SNYDER

128

25

0 C

Frequency, FIG.

9. Terminal

C-C

stretching

cm-’

region of the spectrum

of ~-C&HJ~ at -180°C

STRUCTUREOF BANDS FOR THE MONOCLINIC MODIFICATIOS

Methylene

Rocking Modes

Only two n-paraffins (n-Cg8Hgg and n-C30H62) which have this structure were available for study. The doubling pattern for the rocking modes (See IJig. 5 for n-C&H68 and Fig. 11 for n-C30H62) may be more complicated than in the orthorhombic struct’ure. Although the structure of the strong doublet near 725 cm -’ (components separated by -12.4 cm-‘) resembles the corresponding band of the orthorhombic structure, the over-all pattern of splitting for the remaining rocking modes is not the same. The frequency separation of the components is in general less than for n-paraffins of the orthorhombic struct,ure and t,he weaker component is always on the high-frequency side. Xear 780 cm-’ in the spectrum of n-C&H, there is a doublet (components separated by 6.5 cm-‘) which is flanked on each side by unresolved doublet,s whose components are separated by less than 1 cm-‘. Since the rocking modes at the high- and low-frequency ends

VIBRATIONAL

SPECTRA OF CRYSTALLIKE

n-PARAFFINS

129

100

0

I

1 1380 Frequency,

1390

1370 cm-’

1360

FIG. 10. Symmet.ric methyl bending region of the spectrum of t&7Hj6

I

1000

1

900 Frequency,

I

800

at -180°C

I -l 2

700

cm-’

FIG. 11. Infrared spectrum of wC,~~H~?at -180°C

of the spectrum are split by 4 cm-l or more, there appear to he two positions for minimum doublet separation. Unfort,unately, the spcct.rul~~of n-C&H68 (Fig. 5) does little t.o clarify this since its bands are not favorably situated in the regions of interest.

130

SNYDER

100

25 -

1470

1460 Frequency,

FIG. 12. Methylene at -180°C.

Methylene

and asymmetric

methyl

1450

J

1440

cm-’

bending

region of the spectrum

of n-CzaHjs

Bending Modes

The structure in this region C1450-1480 cm-‘) is almost identical to t,hat in the spectrum of the orthorhomhic form as seen in Fig. 12 for n-CzsHss . Again there are two strong bands (1462 and 1475 cm-‘) in between which a medium intensity band occurs (1466 cm-‘). The interpretation of these bands follows that for the orthorhombic case. Terminal

C-C

Stretching hfodes

strucThe splitting of the 890 cm-’ band resembles t#hat for the orthorhombic ture. The int,ensity ratio of t.he high-frequency to the low-frequency component is considerably greater t,han that, in the ort.horhombie ease (Figs. 5 and 11). Methyl Symmetric

Deformation

In the region of the are found for n-C&H,, these occurs near 1367 the surplus of bands, occurs.

Modes

spectrum where t,his band normally occurs, 3 or 4 bands and for n-C30H62 (Fig. 13 for n-C,,H,,). The strongest of cm-’ and is probably the methyl fundamental. Owing to it is not, possible to ascertain if int’ermolecular splitting

STRUCTURE OF BANDY FOR THE TRICLINIC MODIFICATION Selection rules indicate t,hat only one component’ of each band of t,he isolated molecule is expected in the spectrum of the crystal. In all cases where doubling

VIBRATIONAL

SPECTRA OF CRYSTALLINE

Frequency, FIG. 1s. Symmetric

methyl

bending

n-PARAFFINS

cm-’

region of the spectrum

of ~I-C”:~~,H~~ at -IX()“C

was obsrr~td for the orthorhombic and monoclinic st,ructurrs, only single bands arc found for the triclinic strrwtjurr (SW I:ig. 6). The methylenc hcnding wgion of I~-(‘.‘~H~,,show (see Fig. 14) one very strong band ( 1171 cm-‘) which must be the methylene bending mode. One medium intensity hand ( 1-W cm-’ ) again appears and must, he assigned to ont or both of the two methyl asymmc>trit deformation modes. This concludes our clualitative discussion of the relation hrtween thr illtcrmolecular effects and t,he crystal structures of the ,l-paraffins. Thr remuindcr of this paper will be confined to a calculation of the msgnitudc of itlt~ennolec~~l:~~ forws from the data for t,he orthorhomhic strwtrw topet.hrr with a tliscrwsicnl of the possible origins of these forces. (‘.4I,C’~LATION OF INTERPUZULECULAR F( )RCE COXSTBTTS FOR THE ORTHORHOMBIC STRIiCTL~RIC 7’0 make trwtnhlc the problem of calculating intermolrt~ulnr force constaut s from the splitting of methylcnc bending and rocking modes, it in nrwssary to m&r a number of simplifications. First, n-e assume that the methylenc bending antI rocking modrs arc relatively uncoupled from other symmetry related \:ibrations. This i3 indicated t,o be the raw for the rocking modes a~ shown in thv taarlier paper ( z ) and is probably also the cast for bending modes. Swond. we assume that interaction between mcthylenc group, q of two different moleculrs will btl distillguishrd hy the relative separation of thrl ah planes of the methylen~~

132

SNYDER

groups and not by positions relative to the ends of the molecules. This is equivalent to determining interactions on the basis of t’he symmetry for t,he cryst’al which results if the chains become infinitely long. The repeating unit for this crystal is identical to the previously defined subcell of the odd n-paraffins except. that its c dimension is shortened to include only two methylene groups, the reof certain interactions t,hen peating unit of the infinite chain. The equivalence follows from the new symmet,ry elements which result. Figure 15 shows the ab plane of the crystal. The solid circles represent hydrogens in the ith methylene plane and dashed circles hydrogens in the adjacent (i + 1 )th plane. Internal bending coordinat,es ( a and 0) are assigned to two molecules (1 and 2) of t’he unit cell with a superscript and to a particular methylene plane (i) with a subscript. Planes are numbered consecutively beginning at the same end of all molecules in the layer considered. The int,ernal coordinates measure changes in t,he angle between a given CH bond and the plane defined by the carbon atom of the CH bond and its two adjacent carbon atoms in the skeleton. The G elements for t’hese coordinat,es are given in Appendix A. The rocking and bending vibrations belong to separate species for the isolated molecule, but get mixed in the factor group C‘2”of the crystal as is seen below from the symmetry coordinates of the subcell. For a crystal composed of molecules which have m methylene groups, there are m + 1 symmet,ry coordinates in each species. These are S,’ = MN’

&(a)

+ %‘l + W2 + (Y,2)

$3’ = .l,LXPll + Pm1 + 81” + Pm2) s,” = 4/i(a,’

+

wn+1-i

l

+

a2 2 +

$+1-i,

SBi

+

Ptn+l-;

+

Pi’

p%+l-J

s&m+l~‘2

r(m+1)/2 b

Bz(b)

=

fz(d

=

~a L(

_ -

-&D

1 a(m+1)/2

tm+l~12

+

&+1~/2~

+

&t+l,,2

8,’

= fS( aI1 + an1 -

s;

= %(Pl’

+

1

a12 -

(Y,2)

+ L&?&l - P12 -

Pm2)

sai = >i(czi’ + (Y:+1_-i -

(Yi2-

sgi = %(‘(pi’ + PL+1-, - Pi2 fpfl)/2 (I

(m+1)/2 &

=

=

2 4’

-&

1

1 a(m+1)/2

(&+I,,2

-

4m+l~/2~

-

P:m+1,/2).

(2)

&+,_i> &+1-J

(3)

VIBRATIONAL

SPECTRA

OF CRYSTALLI~R

n-PARAFFIN8

133

25 -

01

I 1480

\

/I Frequency.

FIG. 14. Methylene at. -180°C.

I

I

1450

1460

1170

and asymmetric

I

144c

cm-’

methyl bending region of the spectrum of n-(‘rrH:,,

Interactions between internal coordinates arc given in Table I’. The interwtion between planes i and j are assumed to depend only upon t’heir relative separation, 1i - j 1 = 1, 2, 3 . . . are, respectively, referred to by n, b, c . . . It~tramole~ular interactions are dist,inguixhed from iut~crmolecular interaction by a Roman numeral superscript for the former and au Arabic numeral superscript for t,hc latter.. Having wn&ucted !ar determinant

the G and F matrices,

we proceed

to solve t.he uwal

I GF - IA 1 = 0.

SWII-

(4)

It is necessary to separate the bending modes from the rocking modes and then LOseparat,e the intermolecular part from the intramolcc~~lar. The former is readily accomplished wit.h the orthogonal tralwformatioll matrix (of order m + 11, 11-l

0

0

0

0’

0

0

0

0

0 0;).

1

1

0

0

0

l-l

0

0

1

1

0

0

0

0

0

I-1;)

0

0

0

0

I

l;!

i

134

SNYDER

VIBRATIONAL

SPECTRA

OF CRYSTALLINE TABLK

1 3.5

/L-1’AIlhFFI~S

V

INTERACTION BETWEEN INTERNAL CI~ORDIK~TES

h (i.’

The (2 mat,ris has only int8ramoleculsr

clemwts.

hc3

h.,

ha’

tr,.:i

h,,”

II,,1 h ,;I

After we let

H = H" + H'.

H" = 2MGF”JlH’ = t,hr equation

‘,

2,1/GF’Jr-1,

(7)

to he solved is (S)

136

SNYDER

of H”. These are the same for both species B, and B1 . or B (bending).

Below are the elements We use u = R (rocking)

Hqi”” = G,“F,” u, H:,,“,, = Gb”F: u, H:,i”+“2= G,“F: u, 0

uu

H l/z(m+l), 1/2(m-3) = d%X’F,o 0 w1( H l/z(m+l), l/‘J(m-l) = 2/2Gbu~: 0 uu H l/z(m+l). 1/2(m+l) = G,“F: u, H?,&j,

[i

‘,

1, 2, . . .(m

=

-

1)/2],

(!I)

u,

uz(m--1)=

(Ga"+ G,UV':", HqjRB= HTjBR= Q. of H” are zero. The following

All other elements

gi’ + g{‘,

GiH =

F ,“” = far +f:‘,

GiR =

terms used above are defined:

g;r _ g: (i

FzR = fa’ -f:‘.

=

a, 6, c),

(10)

If we let H,” ” = G,“F,” ‘, Hi ” = Gb”F,”‘, H”” = G,%,” “(u = R, B), the solution of the equation 1H” - 2XE 1 = 0 is given closely by 2X:,,“” = H: u” + 2H,” UUcos (pk,?,(f

2H: UUcos 2~k,,~,

lm Pk.m

=

(k = 1,3,5,

___

. . . ,m).

(11)

mfl

methylene groups for the Then (Pk,m is the phase difference bet’ween adjacent kth normal mode of a molecule containing m methylene groups. After the neglect of some small differences in border terms, the following are found for H’ of species Bl(H,’ = H;i, Hbl = &,,+I, etc., Hij = Hji , U, u = R, B). H: yl’ = GaUFLu“+ 2Gb”F; us + 2G:F: “‘, r UL’= G,“F; “’ $ Gb”(F; “” $ FI. “‘) + G,“(F; uL’+ F; u”), Hb

(12)

H:. ‘” = G,“F: liv + G;(Fj) “I’ + F; “) + G,“(F: ‘” + F: “‘), I RB FLRR = (fa’ + fa” - 2.67, FCZ = (fa’ + fa” + 2fa”), F;“’ = (.fbl +.fb2 + 2.67, ’ SB = (j-c’ +je’ + 2f:), FC ’ BB = (fdl + fd2 + 2.fdt), Fd etc.

1’6”

= -(fbl

+fb’) - 2fb3L

F6”” = (fe’ +fc’ FiRR = -(fdl

- 2.0,

+ fB” etc.

2.L?,

(13)

VIRRATIONSL

b-

2-

0

n321

i-J

n=23

d

n = 25

SPECTRA OF CRYSTALLISE

A

On=27 vn=29

0

0

4-

O@

O-

A

-4 -

v

_ 0

-0.8

V

I

I

-0.6

-0.4

-0.2

0 co?.

Fro.

hombic

ov

so

0-

-1.

13

wPARAFFI?;S

I

I

0.2

04

16.

Separation of the components of the methylene rc-paraffins its a function of cos q.

The cross terms solution of

H' xB and H' BR can hc neglecttd ~H’ “‘L -

0.8

0.6

#i

rocking

modes of the orthol

so that, thr

spproximate

2x14;j = 0

is simply

2x :.:%z)

=

H:"'z + 2H', "'cos(PI;,,,? + 2H:flu cos%pk,nr ,

1 0

138

SNYDER TABLE

VI

OBSERVED INTERMOLECULAR FORCE CONSTANTS FOR METHYLENE RCKKING MODES OF ODD-NUMBERED n-PARAFFINS~

i

H!R ,

GLR

6”

-0.310 1.402

5.14 1.55 x 10-14 10-1’

0.056

C

F:Rh (ergs/rad2)

-0.50

x 10-l*

8 The units used are: length, A; mass, y-atom/g; force constant, related to frequency, w in cm-l, by X = 5.889 X lOW* d. the intermolecular b FLR, FL”, and FLR represent, respectively, two methylene groups in the same plane, in adjacent planes, and in further discussion of these constant’s, see Appendices B and C and

Since

F’(a)

= -F’(h),

we have for the separat,ion

AX =

x:,;“(a)

-

X:,;“(b)

=

1.28 2.97 x lo-‘4 10-l* 0.48 x IO-14 ergs/rad2.

Then

X is

interaction between alternate planes. For below.

of components %:,;“(a).

(15)

In Pig. 16 the separations of components of methylene rocking doublets n-paraffins (w.2 - wb2) have been plotted against cos cpfor all the orthorhombic examined here. Assignments of rocking modes are those given in t,he earlier paper (1). All points fall, within experimental error, upon a smooth curve. From this by least squares fit. curve constant,s H: R,Hi R,and Hf R have been obtained Since GaR, GbR, and GCRare known (see Appendix A), the intermolecular force constants have been calculated and are summarized in Table VI. We now turn to t,he problem of relating these force constants to intermolecular forces. INTERPRETATION FOR

OF THE INTERhlOLECULAR FORCE THE ORTHORHOMBIC STRUCTURE

CONSTANTS

Although splitting of vibrational bands has been frequently observed and attributed to intermolecular coupling, we have only an incomplete understanding of the nature of the intermolecular forces involved. It, is generally thought, however, that these forces arise principally from dipole-dipole interactions and short range repulsion, An elegant example of dipole-dipole coupling has been reported by Decius (12) who accounted yuantit,atively for isotopic band struct,ure of the out-of-plane modes of carbonate and nitrat,e ions using a single paramet,er, the dipole derivative with respect to t,he out-of-plane coordinate. ,4n example in which short. range repulsive forces are important has been reported by Dows (13). We now consider consecutively these two t.ypes of int,eraction forces, dipole-dipole and short, range repulsive and determine which of these t,aken by it,self seems to account for the data most satisfact,orily. Only the results of t)he calculations will be discussed in this section. l;or details the reader is referred t,o Appendices B and C.

VIBR.4TIONAL

SPECTRA OF CRYSThLLISE

n-1’hRAFFIXS

I ::9

SHO~T-R_~NC;E REPULSIVE FORCES One earlier effort has been made to explain the rocking mode doubiet. in polyethylene near 735 cm-‘. Stein (I,$) has proposed t,hat, the coupling of the molccwles results from repulsive forces between t,he nearest two hydrogen atoms in I he :iilit cell of the crystal. _Uthough the polarization of the two components ot I hr doublet is prcdictcd correctly by this simple theory, it fails to account for certain 01 her findings. An extension of this calculation to the bendiug mode leads to a.n incorrect8 predict,ion of polarization of its two components (3). The higher frccIuency component ( 1175 cm’ ) for n-ClaHsn is found to be polarized in the CLdirection and the lower component ( 1-W ~11.~‘) in t.hc b direction, whereas the reverse is predirt,ed. l~urthermore, since the pair of hydrogen atoms considered by Stein lie in the same place, the only iiit,ermoleciilar force const,aiit. iiitrrwtic~n which sholtld not be zero is F,‘. We have seen that F{, representing Iwtwrcn mrthylene groups in adjacent, planes, is actttally larger than F,,‘. Wit.h parameters determined from room temperatttre x-ray diffraction clata I li), w i-ild that the pair of hydrogens considered by Stein ( 14 ), which are separat,ed by 2.8 L?i, arc not, the nearest pair. There are two other close pair, l)oth involving hydrogens in adjacent planes. These have separations 2.i :md ‘,‘.!I .I (Table VII ). The next nearest, pair has separation 3.7 A1, and for our pur(*a11he neglected. l~rom poses this pair and all other pairs at grentei scpamtions otir values of Fi”, FL’ and the interactiou constant wlculated from the ohservetl splitting of the hending mode fundamciital, we have calcrilated values of ($l-‘!$,,

‘L, ’

where 1” is the potrnt,ial energy between the pair of hydrogen at,oms i aiid j, separated by distance riJ . The results of these calculations are summarized in Tnhlc VIII. We note t,hat ($I”/&“),., , has a positive sign for all three pairs;, ;md has the largest value for the pair of hydrogens nearest t.ogether. l~urthti support for believing that the intermolecular force constants are dominated l)y repulsive interactions is found in comparing the observed values of ( 13’1” dr’ 1,:, mit,h t,hosc predicted from a theoretically derived expression for hydrogen repulsion energy. Such an expression has been derived by deBoer (16) and used by Dews (13) to compare theoretical and ohserved intermolecular force constants of crystnlliiie CHaCl. The deBoer potential is IT’ = 1.20 x lo-‘” esp( -:2.3’2 x for 1” in in Table observed that, the observed

109 1

i Iti)

ergs. Values of (d2V’/dr?)Vil calculated from this potential are given VIII. Xhhough the numerical agreement in tw cases between the and calculated values is bet.t.er than tve can rightfully expect, it is clwr theoretiral values are of t,he right order of magnitude to produce the splittings of the infrared bands. WC conclude then that, by inclndiiig

140

SNYDER

more interactions than Stein (1.4) did, the magnit’ude of the observed intermolecular coupling and the polarization of the components for the rocking and bending modes can be accounted for on t#he basis of repulsive forces between hydrogen atoms. DIPOLE-DIPOLE

~~ORCES

Calculations were also made to determine if the int,ermolecular force constants could be satisfactorily accounted for by dipole-dipole int’eractions between C-H bonds. Two scalar yuant,ities were calculated from the observed splittings of the rocking modes: pa, the permanent CH bond moment, and J, the dipole derivative, wit,h respect t’o its associated bending coordinate, (Y, defined earlier. The calculation (see Appendix C) leads to p” = f0.15 Debye and p’ = fO.65 Debyelrad as a possible solution. The value of p” is of the order expected for bond m0ment.s and thus calculation of intermolecular potentials should include in addition t,o repulsive forces bond moment’ int8eraction. However, t,here are several disturbing things about, the calculat’ed values of 1’ and ,u’. It is apparent) t#hat, a permanent CH bond moment8 cannot by itself account, for the observed force constants. The introduction of another t#erm, p’, is acceptable but its neccssarily large value is probably not. The like signs of p” and p’ are inconsistent with the infrared absorption band intensity ratio of the methylene bending mode t,o the rocking mode, since to give t,he observed ratio they must have opposite signs. Finally, t,he values of p” and I*’ lead t.o an unsat,isfact,ory value of 1 cm-’ fol the separation of the components of t.he met,hylene bending mode. CONCLUSIOS Intermolecular splitting of infrared absorption bands of the crystalline n-paraffins occurs in accordance with fact#or group selection rules. Relating the splitting of the met.hylene rocking modes of t.he ort,horhombic (odd-numbered) n-paraffins t,o t.hr parameter kr/(m + 1) leads to intermolecular force constants. These force c0nstant.s can be interpreted much more sat.isfactorily in t,erms of hydrogen-hydrogen repulsive forces than in terms of dipole forces. This suggest’s that in the case of the n-paraffins repulsive int,eractions are dominant’, though for a more accurate treatment, dipole interactions will have to be considered since t,hey undoubtedly cont’ribute t#o some extent. APPENDIX A. G ELEMENTS FOH. INTERX~L BEKUISG COORUIN~TE~ For the coordinates

defined

above

VIBRATIONAL

G*R

SPECTRA OF CRYSTALLINE

I

-6Q,(Qx

=

R =

(ICI

%Qk

n-PARAFFINS

141

!+Qr)~c,

-

,

where Qv and QK are the reciprocals

of the (‘-II

ant1 c’-_(’

tlistsnws.

In Table C’II are listed the nearest hydrogen-hydrogen dist)ances found iu cqstallinc IZ-C’~.~H~~ . ?;umbering of hydrogen atoms follows Pig. 15. These interat.omic distalIces are based on the parameters reported for t,his cry&l at 28”(’ ( 6) : a,, = i.45 -4, 60 = 4.96 A, V = 12’; B-P ~SSUI~Pa C -H distanw of 1.09 ;3 and all H(~‘H angle of 109” 28’. Hcsides k-2 ( Fig. 1.3) separated by 2.8 h and considered by Stein ( 1.4) to tw the c*losest pair, there are at least two other pairs which must he taken into ;I(*count, :3-Z and Sfi, separated by 2.7 and 2.9 A, respectively. Both of t.hese lattet pairs involve hydrogenx in adjacent planes. The next pair has a separation of i1.7 A and the hydrogens of this pair arc assumed noninteracting along with all other hydrogens separated by more t.han this dist,anw. Table VII indicates the ~orw constants wrresponding t.o t.he interacting hydrogens. Thus, to ;I first ~pprOxiIll~lti~Jli pi

If z

-&’

F:,

’

-.fb“ .+ 2j.i’

=

FL I’ = 0.

iH.l

I

If we assume that, only t)hese terms are importaut, WC (aan then ut,ilize the il~termolecwlar force const,ants from the rocking modes t,ogether wit,h an intrrmolecxlar forw constant derived from the separation of the component,s of mcthylenc bending mode, to calculate .fh”, .fhB,and ,t’. Thus from Eq. ( 14). wf’ hLVf~”

AX’

since

GgU =

=

GCB =

Gj(F;

0. Then

B

-

ZF;

’

)

=

GaH(

from t,he observed

2,f,ji - 2j”*’ - q; 1 separation:

cH.” 1

AAH = 2.35 X 1W”‘;

(:,rB is O.!)l so that

.fczi:’ - .fh2- 2fi3 = 0.0130 X 1W” c’rgs/rad’. (‘omhining

this result

6 For the coordinates to Cp= 0.

with Eqs.

(B.l)

gives

fa” = -0.0015

X 10-l’ erg, rad’,

.ft = -0.0036

X lo-”

.fz = -0.009-2

X I OK” rrg!rad”.

defined in

erg/rad’,

( B.:< )

Fig. 15, the infrared act.ive bending mode corresponds

142

SNYDER TABLE

VII

INTERMOLECULAR Hi-Hi DISTANCESAND DERIVATIVES OF THIS DISTANCEWITH RESPECTTO INTERNAL BENDING COORDINATESFOR THE ORTHORHOMBK STRUCTURE Interacting hydrogens

Force constant

.fb2

8 All other

(Hi-Hi)

(A)

aoli (A)

3-2’ 3-5 3-6 3-6’

2.79 2.70 2.94 2.94

-0.76 0.24 0.80 -0.98

hydrogen

separations

>3.7

ar

ar

Hydrogen separation”

:; 0.96 -0.90 -0.98 0.80

6% aaiaff, (A) 0.20 0.14 0.62 0.62

A.

TABLE

VIII

VALUES OF (a2V’/ar2)ri, DERIVED FROMTHE OBSERVEUINTERMOLECULAR FORCECONSTANTS.VALUES OF (a*V’/ar2)ri, CALCULATEDFROMTHE DEBOERPOTENTIAL Interacting

Same

pair of hydrogen

plane

atoms

(a2Ti’/W)ri, Observed

Separation rij (A)

fn3 3-2’

2.7,

Adjacent plane

2 x

lo-‘”

12 x 10-14 6 X lo-l4

2.70 2.94

(ergs/AZ) Calculated 8 x 10-14 11 X 10-l” 5 x 10-14

To relate these force const,anbs to interactions between specific hydrogen at,oms we have

where f&j is t,he int’eraction force con&ant between coordinates ciated with hydrogens of Eq. (B.4)

H; and Hj separated

vanishes at equilibrium.

for the closest hydrogen

LY~and aj,

asso-

by a distance rij . The second term

In Table

VII

are list,ed values of &/aai

pairs. The values of ( a21r’,/dr2),.~,calculated

from the

previously determined values of fa3, fb3, and fb2, are listed in Table VIII. It is seen that t,he three values of (a2T”/ar2),,, are all positive in accord with a repulsive potential.

The value of the second derivative

is largest. for the shortest

H;-Hjdistance C. DIPOLE-DIPOLE INTERACTIONS AKD THE INTERMOLECULAR FORCECONSTANTS The total dipole moment for a CH bond is defined ~1 = ec&’

+ w’),

where eCH is a unit vector directed along the CH bond in the direction

(C.1) of the

VIBRATIONAL

SPECTRA OF CRYSTALLIKE

n.-PARAFFINS

1x3

hydrogen, LYis a bending coordinate ((Y or P, as defined earlier) and ,A’ and p’ are cwn~tants. The energy of interaction between the dipoles, p1 and ~2 , of two (‘H bonds is

where @Iis the angIe between p1 and the vector, r12from pL1t,o ~1~, projwted on the a6 plane that. contains pl ; t& is similarly defined; y is the angle hctwectr rlz and its projert~ion on the ah plane that contaiu~ CL, l’hwr for the interaction force cwttstant

+ ; ( po)2+ ( 1 -

3 ~0~ ylr ) ( PI)‘I

COB

:lp’p’ fws

+

er cog ec y12

sin ( e1 + e2 11.

~\'~~lI~ti(J11 of fl; must include contributions from all j's.Individual were enlculated out, to about 12.5 A, after which the remaining wntribtttion estimat.ed by an integral. This leads to

& zt = [i.x

(‘3

terms IWP

+ 3.90 hop’]10--‘4,

tPoj2 + 2.48 ($)’

F; ’ = lo.40 (/.A2 + 0.61 (!.L’)’ + 183

i c.1 J /.&‘]W4

for F in erga/rad and p” and p’ in Dehyes and Dchyesirad, respectively. With the use of the values of F,’ and Fb’ found earlier, these equations then lead to (a)

,u” = f0.82

Debye,

P’ = ho.1 I Debyc/rad,

t’b’l

p” = f0.15

Debye,

J

= +O.c5

( ( ‘.3 1 Debye/rad.

Solubion (a) is immediately rejected heeaune :I very low value of ,.L’ (0.08 I>(~hyc ) is indicated from infrared intensit,y measurements on solid rz-paraffins ( l(i). That solution (11) is also unsatisfactory has already heen indicated itI the section on dipole-dipole forces.

REFERENCES 1. R. G. PIWDER, J. Mol. Spectroscopy 4,411 (1960). .2. H. W. THOMPSON AND P. TORBINGTON, Pvoc.Roy. Sot.A184,3 (IYG). S. S. GRIMM, c. Y. ILANG, AKD G. B. B. h'i. SUTHERLAND, J. Chetrt. Whys.26,539 (1956). 4. A. A. SCHAERER, C. J. Busso, A. E. SMITH, AND T,. IS. SKINNER. 6. .4vl. C’hew. SUC:. 77, 2017 (1955).

144 6. 7. 8. 9. IO. If. 12. IS. I$. 16. 16.

SNYDER

A. E. SMITH, J. Chem. Phys. al,2229 (1953). H. M. SHEARER AND V. VAND, Acta Cryst. 9, 379 (1956). C. W. BUNN, Trans. Faraday Sot. 36,482 (1939). M. C. TOBIN, J. Chem. Phys. 23, 891 (1955). R. S. STEIN AND G. B. B. M. SUTHERLAND, J. Chem. Phys. 22, 1993 (1954). J. K. BROWN, N. SHEPPARD, AND D. M. SIMPSON, Trans. Roy. Sot. A247,35 J. C. DECIUS, J. Chem. Phys. 23,129O (1955). D. A. Dows, J. Chem. Phys. 32, 1342 (1960). R. S. STEIN, J. Chem.. Phys. 23, 734 (1955). J. DEBOER, Physica 2,363 (1942). 0. THEIMER, J. Chem. Phys. 27, 1041 (1957).

(1954)