Jan., 1950
ROTATION OF
LONG-CHAIN
salts exhibit a minimum in their conductance curves and the conductance increases sharply a t higher concentrations. The increase is the more pronounced the lower the conductance a t the minimum. 3. For salts with large anions, such as the picrate ion, the conductance curves are compara[CONTRIBUTION FROM
THE
ALKYL BROMIDES
171
tively simple. With salts of simple ions, the curves are highly inflected. 4. It is suggested that concentrated salt solutions may be looked upon as solution of solvent in fused salt. The properties of concentrated solutions are discussed from this point of view. PROVIDENCE, R. I.
RECEIVED JULY9, 1949
FRICKCHEMICAL LABORATORY, PRINCETON UNIVERSITY ]
The Mechanism of Rotation of Long-chain Alkyl Bromides and Other Molecules in the Solid State' BY JOHN D. HOFFMAN~ AND CHARLES P. SMYTH* I t has been found that the normal long-chain In this work, dielectric evidence of prerotation alcohols rotate about their long molecular axes was sought in ndodecyl, n-hexadecyl, n-octadecyl in the solid state provided the chain contains and n-docosyl bromides down to liquid nitrogen fourteen or more carbon atoms.s Observa- temperatures. n-Hexadecane was measured for tion of orientation polarization due to molecular purposes of comparison. The wide temperature rotation was rendered difscult in the study on range over which prerotation phenomena were alcohols by a direct current conductivity effect observed suggested that Fowler's treatment of which resulted in Maxwell-Wagner polarization rotating diatomic molec~les,~with extensive a t low frequencies. Since i t is to be expected modifications, might predict the dielectric conthat the proton transfer mechanism of conduc- stants and rotational specific heats in the solid tivity postulated for the alcohols could not occur state a t and below the transition point. Formain the bromides the study of molecular rotation tion of liquid due to impurities and premelting* in the latter should not be hampered by conductiv- does not constitute a true prerotation effect, as ity effects. has been discussed in a previous paper on molecMuller4 has shown by X-ray techniques that ular freedom in arninesl9and care has been taken rotation about the long axis in the solid is not to minimize these effects by using pure materials. observed in the normal long-chain hydrocarbons Purification of Materials on warming until the chain contains twenty-two We are indebted to Dr. E. Emmet Reid for the or more carbon atoms. The explanation tentatively advanced by Miiller for the onset of rota- samples of pure n-dodecyl, n-hexadecyl and ntion in the pardins at about twenty-two carbon octadecyl bromides,'O and to Dr. Nathan L. atoms was that the molecules achieved sufficient Drake of the University of Maryland for the chain length to overcome end-group forces. sample of n-docosyl bromide. In addition, nI t has seemed desirable to reconsider this hy- octadecyl bromide from the Matheson Co. pothesis. Data on transition temperatures have (Paragon Div.) was purified by twice crystallizing been collected from the literature for alcohols, from the melt, and then distilling under vacuum three times, each distillate being cut into five iodides and paraffins for purposes of comparison. I t was also found by Miillers that certain fractions and the product designated as I1 in long-chain ketones showed a gradual increase in Table I. This product is compared with that orientation polarization a t 1.5 X lo8 cycles/sec. supplied by Dr. Reid which is designated as I. prior to melting. This increase was marked even The sample of n-docosyl bromide supplied by Dr. 30' below the melting point, and considerably ex- Drake was further purified by high vacuum disceeded the polarization of the correspondingparaf- tillation. n-Hexadecane obtained from Paragon fin, thus indicating molecular freedom in the solid TABLEI state. Tfie term "prerotation" will be used to PHYSICAL PROPERTIES OF THE ALKYLBROMIDE SAMPLES describe the gradual onset of molecular rotational Densities freedom below the transition or melting point.6 Bromide 30' 60D nD M.p.
* Huvard University Ph.D., 1921.
(1) This invertigotion was carried out with the aid of the O5ce of Naval Research. (2) Present address: Research Laboratory, General Electric C o . , Schenectady, N.Y. (3)(n) Hoffman and Smyth, THIS JOURNAL, 71, 431 (1949); (b) Baker and Smyth, ibid., 60, 132 (1938); (c) Ott, 2. ghysik. Chem., 198, 218 (1944). (4) Miiller, Proc. Roy. SOC.(London), 1188, 614 (1932). (6) MUIer, ibid., AlM, 403 (1937). (8) Smyth, Tmns. Forodoy SOC.,49A, 176 (1946).
n-DodWyl 1.029 1.005 1.45747 (25') n-Hexadecyl 0.986 0.962 1.46075 (25') n-Octadecyl(1) ,9760 ,952 1.46145 (30') n-Octadecyl(I1) ,9754 ,952 1.40105 (30') n-Docosyl . . . . . . . 1.46 (est.) (40')
-9.70" 17.33' 27.35' 27 15" 42.7"
(7) Fowler, Proc. ROY.SOC.(London), 81419, 1 (1935). (8) Oldham and Ubbelohde, ibid., A176, 80 (1940). (9) Hoffman and Smyth, THISJOURNAL, 71, 3591 (1949). (10) Meyer and Reid, ibid., 56, 1674 (1933).
172
JOHN
D. HOFFMAN AND CHARLES P. SMYTH
was also purified by vacuum distillation. The liquid densities, the refractive indices for the D sodium line, and the melting points of these compounds are given in Table I. An estimated density of n-docosyl bromide (0.948 a t 40') was obtained by extrapolation of molar volume data calculated from Table I. The good agreement of the physical constants with the available literature values and the sharpness of the cooling curves indicate an adequate degree of purity for these compounds. Experimental Results The details of the method for obtaining dielectric constant-temperature data on an impedance bridge have been described elsewhere.Sa Each compound was run at least three times, and the data presented here represent the most reproducible run on the purest sample. The dielectric measurements were first made at 0.5, 5.0 and 50 kc., but since no dispersion was ever found, only the 5.0 kc. results are recorded in Table I1 and plotted in Fig. 1. The precision was better than
-SO - 40 0 40 Temperature, "C. Fig. 1.-Temperature dependence of dielectric constants of long-chain bromides a t 5.0 kc. : hollow circles, n-dodecyl bromide; half-filled circles, n-hexadecyl bromide; filled circles, n-octadecyl bromide; and squares, n-docosyl bromide. Arrows indicate direction of temperature change.
0.2% but the accuracy was less due to voids. The quantity of n-docosyl bromide available was so small that glass beads had to be used to raise the liquid level in the cell so that the three concentric gold cylinders were filled with dielectric. No beads were between the condenser plates. The material froze just flush with the top of the cylinders so that the dielectric constant values may have been slightly low due to difference in edge effect. It was, therefore, considered inadvisable to measure prerotation effects much below 0". The increase of density in the cell due to freezing calculated from the dielectric change in n-hexadecane is 5%. The presence of some voids causes a smaller density change in the cell than would be obtained under ideal conditions. The average increase in density observed for n-
Vol. 72
TABLEI1 DIELECTRIC CONSTANTS AT 5.0 Kc. t,
oc.
e'
It-Dodecyl bromide Cooling 31.5 4.15 6 . 6 4.38 - 1 . 0 4.46 --4.9 4.50 - 9 . 9 4.55 -11.9 4.57 -10.0 4 . 4 8 -10.6 3 . 8 0 - 1 0 . 7 2.95 -10 8 2 . 5 3 - 1 1 . 0 2.40 --11.2 2 . 3 5 - 1 2 . 3 2.33 -25.0 2 . 2 8 -32.0 2 . 2 7 -45 2.25 -56 2.24 -92 2.22 -134 2.20 Remelted a t
-9.70'
f, o c .
€'
n-Hexadecyl bromide Cooling 37.4 3.66 23.5 3.74 18.7 3.78 1 5 . 7 3.80 1 2 . 0 3.85 16.0 3.47 16.2 3.38 16.2 3 . 2 5 16.1 2.85 16.0 2.71 14.9 2.45 13.9 2.43 12.2 2.40 8.6 2.38 0.0 2.36 -8.0 2.35 -63 2.24 -80 2.22 -102 2.21 -140 2.20 Warming -160 2.20 -60 2.24 9.9 2.39 12.0 2.41 13.0 2.43 14.5 2.48 15.0 2.50 1 5 . 4 2.53 16.3 2.71 1 6 . 5 2.90 16.9 3.48 1 7 . 0 3.73 17.2 3 . 7 8 1 7 . 3 3.83 19.9 3.80
1, OC.
e'
n-Octadecyl bromide
Cooling 58.4 3.40 32.4 3.52 24.0 3.59 2 2 . 5 3.60 25.9 3.32 2 6 . 1 3.24 26.0 3 . 0 8 2 5 . 8 2.60 25.7 2.50 25.4 2.45 24.8 2.40 21.5 2.35 9.4 2.34 -16.0 2.31 -28.0 2.29 - 7 9 . 0 2.24 -120 2.21 -160 2.20 -178 2.20 Warming -53 2.25 -39 2.30 -18.0 2.34 3 . 0 2.37 25.0 2.42 2 5 . 5 2.53 2 6 . 6 2.90 27.3 3.54 30.2 3.53
I,OC. e' n-Docosyl bromide Cooling 55.2 3.12 3 9 . 8 3.20 3 9 . 5 3.10 39.3 3.02 38.5 2.92 36.5 2 . 9 1 34.4 2.90 32.9 2.90 30.9 2 . 8 8 3 0 . 3 2.85 3 0 . 3 2.70 30.0 2.40 29.9 2 . 3 8 27.4 2 . 2 7 23.0 2.26 20.7 2.26 14.0 2.25 Warming 14.0 2.25 2 5 . 3 2.27 29.1 2.30 40.6 2.30 4 1 . 4 2.35 42.1 2.50 42.7 2 . 8 5 42.7 3.20 60.2 3.10
octadecyl bromide on freezing in a pycnometer was 4%. The maximum value found was 7%. Approximate values for the other solids were estimated by increasing the liquid densities at the freezing points by 4%. As was to be expected from the absence of the possibility of proton transfer, no high specific conductivities or Maxwell-Wagner effects were encountered, and the dielectric loss factor, E", may be taken as < 0.03 throughout. Discussion of Results The dielectric behavior of the three shorter bromides is very similar (Fig. 1). All supercool slightly before freezing, and then show a sharp drop in dielectric constant coincident with freezing. There is a slight hysteresis between freezing and melting points as may be seen in Fig. 3, where the dielectric constant-temperature curve of n-hexadecyl bromide is depicted on a large scale. In contrast to the behavior of the three shorter compounds, n-docosyl bromide shows a relatively high dielectric constant below the freezing point (40°),which remains nearly constant at 2.90 until the substance is cooled to 30.3', where the dielectric constant drops sharply. This indicates qualitatively that the molecules of n-
Jan., 1950
ROTATION OF LONG-CHAIN ALKYL BROMIDES
for density change even at -looo, and show a continuous rise with rising temperature. This has been shown in detail in Figs. 6 , 7 and 8. The comparison with n-hexadecane is particularly striking. The fact that the dielectric constant of n-hexadecane varies only 0.02 between 18 and - 1 W indicates that the number of molecules between the plates does not change after the material has solidified. MfiUefl found a similar situation in his study of n-docosane and longchain ketones. It is apparent that the gradual rise in dielectric constant observed in solid bromides may be regarded as an orientation polarization effect.
50
6
40
0
a
k
173
30
8 b
20
e
4 25 50 75 Time in minutes. Fig. 2.-Temperaturetime curves for n-docosyl bromide (upper curves) and n-octadecyl bromide (lower curve). Arrows indicate direction of temperature change.
docosyl bromide rotate about their long molecular axes between the freezing and transition points. The transition in the solid state is characterized by rotational freezing of the dipoles and by a sharp heat of transition (Fig. 2). The transition is monotropic, and does not reappear on warming.
0
0 0
0 0
0
0
,.---0,
A
0 0
0 0
.'\
0 0 6 0 0 . 0 \
I
\
0
0
0
'04'0
0
o o o a o o o 0
0
0
0
0
0
.
4.0
3.5
9 z 8 .Y
-5
3.0
al
E
2.5
10 15 20 Temperature, "C. Fig. 3.-Temperature dependence of dielectric constant of n-hexadecyl bromide a t 5.0 kc. near melting point. Arrows indicate direction of temperature change.
The dielectric constants of the bromides exceed the squares of the refractive indices as corrected
Fig. 4.-Molecula.r structure of long-chain compounds. Upper part represents end view of long-chain molecules in lattice. Black circles represent rotator molecules. E d view of single molecule showing bar shape is depicted on lower left, and a diagrammatic side view of long-chain molecule showing axis of rotation and dipole (arrow) is shown on lower right.
The dipole moment vector, PO, in the bromides is inclined a t an angle, 0, to the axis of rotation (Fig. 4) so that the effective dipole moment, PI, of a molecule rotating about the long axis for ; I polycrystalline material is given bp In =
sin e
the gas moment." S h e PO = $aN/sne/9kT, it is seen that the net result of the dipole vector being inclined a t an angle to the axis of rotation would be a r e d u c t h in orientation polarization and dielectric constant. where
40
is 2.15
D,
(11) Smyth and Mchlpiae, J . C k m . Phys.. 8, 347 (1935).
174
JOHN
D. HOFFMAN AND CHARLES P. SMYTH
If one accepts the simple planar model of a long chain bromide, 0 = 35'. Dipole induction tends to make 0 somewhat smaller, but a slight twisting of the dipole on the chain due to thermal agitation tends to increase 6, so 6 = 35' should be a good approximation. The dielectric constant of the rotator phase in n-docosyl bromide can be calculated at 30' using w = 2.15 D, M = 389, d. = 1.00 (extrapolated from Table I with density correction for solidification), 8 = 35O, n 2 D 2.21 (also corrected for density), T = 303' K.,k = Boltzmann's constant and N = Avogadro's number in the Onsageri2 equation, in which the moment, ~.ca,is replaced by PO sin 0 p 0 = -4-r-N ( p o sin 9kT
-
(E' _ n2) (2c' = Jf _ _
d
f ' (n*
+
+ n*)
2)2
(1)
The calculated dielectric constant is 2.70 which is 6% lower than the observed value of 2.88. Kirkwood13 has modified the Onsager equation by multiplying pi by [l zcoSy] which accounts approximately for the dielectric constant of the medium surrounding the dipole. z is the number of dipoles in the first coordination sphere. y is the angle between dipole moments of a pair of neighboring molecules, and is defined as J f cosy e- WelkTduldwz,where W Ois the potential of average torque acting on a pair of neighboring molecules. I n the present case, following Kirkwood, we assume W O= 0 (free rotation) so that cosy = cos2 B for the six dipoles which are on molecules alongside the one under consideration, For the six dipoles lying in the head-to-head position cosy = - cos2 6, so that the calculated value of [ 1 z Cosy] is unity. The experimental value is 1.2 which indicates Onsager's equation is a fair approximation. The assumption Wo = 0 is not exact, and the small increase in [l z cosy] emphasizes the fact that the rotation is slightly hindered. In either case the dielectric data quantitatively support the picture of molecular rotation about the long axis. The fact that the dielectric constants of the alcoholsaaVb showed no drop on solidification to the rotator state may be attributed to the fact that 0 for the resultant moment lies close to 90°, at which value
+
-
+
+
P1 =
e.
Qualitatively, it may be seen that the prerotation effect is due to the gradual lowering of the potential barrier opposing rotation as a solid is warmed. At any given temperature, thermal fluctuations permit a number of molecules to surmount their barriers and rotate or reorient frequently. A rotating molecule, however, should exert much less orienting influence on its neighbors, thus giving rise to a cooperative effect which increases the fraction of molecules rotating. At some temperature, the cooperative effect may cause all of the molecules to attain rotational freedom. If the lattice must expand to permit (12) Onsager, THISJOURNAL, 68, 1486 (1986). (13) Kirkwood. J. Chem. Phys., 7, 911 (1939).
VOl. 72
the molecules to rotate, the tendency to rotate is reduced. The ideas set forth above may be developed to calculate the fraction of molecules rotating, r, as a function of temperature. This relation between r and T is used to calculate the dielectric constant by replacing N by Nr in equation (1). The molecular model is essentially that given by h/Iiiller,4J who regarded the long rods as bar shaped, much rounded a t the corners. The roughly rectangular cross-section is a result of the fact that carbon atoms in a paraffin chain are arranged in a regular saw-tooth pattern. The essential elements of the model are shown in Fig. 4. The bar shape of the molecule will tend to cause the potential energy to have two minima as one rotates the molecule from 4 = 0 to 2n, but the presence of both end-groups on the same side of the axis of rotation for the even bromides and the bromine atom a t one end of the chain will greatly deepen one of the minima. The analytic form of the angular dependence of the potential energy may be taken as of the form '/zV cos 4/2 abbreviated V(4),or as a step function with one deep minimum of depth V. The barrier against rotation about the long axis for these rods may be treated as composed of two parts*: the first, Ve,is the barrier due to both end groups, and the second, ( n - 2) VCH~,is due to the bar shape of the molecules. VCH~will be constant for any -CH2- group in the even series. The barrier resisting rotation, Vn (exclusive of lattice expansion energy) will be given by vn
v a
f (n
- 2)
VCHr
where n is the number of atoms in the chain. The barrier is seen to increase linearly with increasing chain length. Using the Hamiltonian H for a classical rigid rotator with one degree of freedom and denoting the rotation about the long axis in terms of an angle 4, we may write H = P$ /2I
- V (6)
(2)
where P+ is the conjugate momentum and I is the moment of inertia. The degree of nonrotation, s, may be defined as s = l - - r =
Q ~ Q T
(3)
where QT is the partition function for all states of a molecule and QOis the partition function which satisfies the condition of non-rotation. The condition of non-rotation is given by P; j 2 1
r
v
(4)
which states that a molecule cannot rotate if the kinetic energy available is equal to or less than the prevailing barrier, V. Fowler inserted a factor, /3, in the above expression to represent the fraction of the barrier which the molecule had to overcome in order to rotate, and achieved different kinds of behavior by varying fi arbitrarily. It is difficult to see how /3 could be other than 1 unless
ROTATION OF LONG-CHAIN ALKYLBROMIDES
Jan., 1950
tunnelling were important. The partition function which satisfies (4)is given by
After making the substitutions x 2 = P+z/(21kT) and B = (V/kT)'/zthis becomes
175
Vn/k is a constant in the perature which we denote as B becomes s(T,/T)l/~when for V in (V/IZT)'/~.If we equation (8) becomes s
E
1
- 7 = $E/[$E
+
dimensions of tem-
T, so that the limit
we substitute Vns2 set A E n / V n = a,
e-aB'(rl/z
-
$E)]
(IO)
where
The lattice may have to expand to permit rotation Equation (10) is an implicit equation which deof the molecules. There is ample indication from scribes the variation of r for different values of X-ray4 data that the paraffinic lattice expands T/T,. This relation is found numerically by rather rapidly below the transition point. If we choosing a value of B , evaluating #B and finding let AE equal the difference per molecule in the s from equation (10). The corresponding value lattice energy of the rotator and non-rotator of T / T , is found by putting this value of s in states, the Hamiltonian for the rotator becomes B = S(T,/T)'/~. For a given value of a, r H' (P: /2I) - V(q4) + AE (7) snaps up toward 1 a t a temperature of 1.275Tc, A.& is taken as the difference in lattice energy which we call T,. This is shown in Fig. 5. of the rotator and non-rotator molecules before any cooperative loosening has set in. Since 1.0
we see from equation (3) that s = Qo/QT
-
$B/[$B
+ e-'/kT(d/2
- r c ' ~ J)
(9)
The potential energy, V , is a variable which is dependent on the prevailing degree of rotation. Fowler assumed a cooperative law in the form V = Vns. For a rotator in one degree of freedom we shall adopt a law in the form V = VnS', where V is the ef€ective potential barrier resisting rotation for a given degree of rotation, and Vn is the barrier prior to the operation of the cooperative effect. This cooperative law should probably be regarded as an assumption, although the following considerations based on the nature of the paraffinic lattice make it appear reasonable. If we consider an hexagonal lattice of long-chain molecules (viewed end-wise as depicted in Fig. 4) and let 1/6 of the molecules rotate as indicated by black circles so that s = 5/6, it is seen that five of the nearest nei hbors of a non-rotator, A , possess barriers of 576 of the full value on an average since each has one rotating neighbor where the barrier is assumed to be zero. Thus the average barrier for s 3 5/6 is V = [(0 5 (5/6))/6]1r, = (25/36) Vn about A . Hence V = Vns2is reasonable insofar as this model represents the facts. It is clear that the law V = VnS is an oversimplification in the present case, since it takes account of the nearest neighbors only, the loosening of the nearest neighbors of A by their neighbors being neglected. AE is also a variable depending on the degree of rotation, and will also be assumed to vary with s2, i. e., AE = AEnS2. This may be seen to be reasonable in a manner similar to that given to indicate V = V d .
+
0.8
0.6
r:
0.4
0.2
0.4
0.8
1.2
T/TC. Fig. 5.--Calculated temperature dependence of degree of rotation. Numerals represent values of a.
If V = V,,s had been used as the cooperative law, no snap-up would have developed and r would have risen monotonically to a value near 1 a t T,, then leveled off and become 1 only when T approached infinity. This means the molecule will rotate in a very shallow potential well due to nearest neighbors above T,. V, is obtained from the relation 1.275 V,& = T,, where T , is the observed transition temperature. AE,,may be estimated from the relation a,,-AHt - V,,,where AHt is the observed
176
JOHN
D. HOFFMAN AND CHARLES P. SMYTH
Vol. 72
P
I
-100 -60 -20 20 Temperature, ‘C. Fig. 6.-Temperature dependence of dielectric constant of n-hexadecyl bromide (upper curve) and n-hexadecane (lowercurve) at 6.0 kc. Dotted line represents calculated prerotation effect using equation (1). -140
-100
I
-20 20 Temperature, “C. Fig. 7.-Temperature dependence of dielectric constants of n-dodecyl bromide (half-filled circles), n-octadecyl bromide (filled circles) and n-hexadecane (hollow circles) a t 5.0 kc.
-140
-60
heat of transition. Ubbelohde14 has found that the freezing point. The agreement (Fig. 6) is hexadecene-1 and n-pentadecane show heats of seen to be good. It is necessary to estimate transition of about 1 kcal./mole. In the absence T, for this compound since the transition point is of other data we shall use this figure to obtain an not observable. In the alcoholsaathe appearance approximate a. Thus, for n-docosyl bromide, of a transition point on cooling was attributed to T, = 303’ K., SO V n = 0.47 kcal. and a = A E n / V , an equilibrium between two metastable phases = (1.0-0.47)/0.47 = 1.2. Equation(10)hasbeen cu(rotator) and pl(non-rotator). The 01phase was used to calculate r vs. T/Tc using a = 0, 1, 1.2 supposed to transform to p2, a more stable (tilted) and 1.5, corresponding to an increase in lattice form on standing. DielectricaaPb and X-ray3c expansion energy. The results (Fig. 5) show that evidence was cited which strongly supported the the inclusion of the lattice energy suppresses the existence of these phases. The a phase was somedegree of rotation. At Tr the solution is double- what unstable, and transformed to &. The valued for AEnZO. The solution above Tr failure of a transition to appear in full on rewarmmay be regarded as an “unstable” solution as in ing was ascribed to the fact that the more stable order-disorder theory. The vertical jump of p2 form melted before its transition appeared 7 a t T, obtained when the lattice energy is con(i. e., Vn was higher for 0 2 SO that T,> Tfusion). sidered is a first order transition, and the gradual In the very long hydrocarbons the rotational rise in degree of rotation below T, may be identi- transitions are enanti~tropic~ and in the shorter fied with prerotation. ones, monotropic, indicating that the PI-& If the treatment presented here is essentially transformation is extremely slow for the very long correct, the dielectric constants of the long-chain molecules. The activation energy for the PI-& bromides should be calculable in the region of transformation is probably very high, and nucleaprerotation for an 7 vs. T/Tc curve based on a = tion difficult. Thus, n-docosyl bromide is com1.2. Further, since Vn = V, (n - ~ ) V C H , parable to compounds like n-octadecane and nand T, = 1.275 T.’,Jk it is obvious that the transi- octadecyl alcohol, which show monotropic transition temperatures in the unassociated compounds tions. The shorter bromides, like the alcohols, should increase nearly linearly with increasing when cooled slowly probably crystallize directly chain length. into the stable 02 form. It may be expected The calculation of the dielectric constants will that with proper experimental methods (probably be attempted first, replacing N in equation (1) a quick-freeze technique) the more transient by Nr, the effective number of orienting mole- rotator modifications may be found below the cules, where the values of r are taken from the chain length normally given as the inception point curve in Fig. 5 for a = 1.2, and where the pre- of rotation. Ideally, prerotation effects should be viously used values for density, pa and 0 axe em- calculated for substances where the transition ployed. The calculated dielectric constant of the is enantiotropic and the AHt known. The prerotator state is in good agreement with the application of this statistical mechanical theory observed value as may be seen in Fig. 8. The to these phase transitions in the present case is gradual change in dielectric properties of the pre- legitimate inasmuch as two phases (aand &) are rotator phase and the sharp nature of the transi- in equilibrium at the transition point. tion seem correctly predicted. In Fig. 10 it is seen, for iodides and parafEns,I5 A similar calculation has been made for n- that the transition points of the paraffins inhexadecyl bromide assuming T, = 290’ K., (15) Deffet, “Compos& Organique Polymorphes,” Desoer, Liege,
+
(14) Ubbelohde, Trans Paraday SOL 34, 289 (1938)
1942
ROTATION OF LONG-CHAIN ALKYLBROMIDES
Jan., 1950
I
30
28 32 36 40 Temperature, "C. Fig. S.-Temperature dependence of the dielectric constant of n-docosyl bromide near the freezing and transition point a t 5.0 kc. The dotted lines represent the dielectric constants calculated by equations (1) and (10). The horizontal line a t E' = 2.21 is the value of the index of refraction squared for the liquid corrected for density change. 24
40
c
I I
1
177
I I
I I
-110 -90 -70 crease nearly linearly with increasing chain Temperature, "C. length in accord with prediction. The transition Fig. 9.-Variation of specific heat of t-butyl chloride temperatures of the iodides do not vary quite linearly with chain length, probably because lye with temperature for low temperature solid transition. cannot be regarded as strictly constant with Lower curve observed, upper curve calculated. The temperature since molecular dipole interactions horizontal dashed line represents the assumed vibrational vary as 1/T. The slopes and intercepts of the specific heat and the vertical dashed line represents the lines indicate that VcZ500 cal./mole and V C H ? ~ heat ~ of transition. 5 cal. The total barrier against rotation ranges around 1.0 kcal., before any cooperative lowering state a t a given chain length is due to the fact that of the barrier occurs. This is considerably smaller the B1-pz and a-pz transitions are very rapid for than the potential barrier for internal rotation the shorter compounds owing to lower activation about the carbon-carbon bond, so it is easier for energies and greater ease of nucleation. At a a molecule to rotate as a whole than to permit the somewhat greater chain length the CY form is dipole to turn independently of the remainder of stable and a transition appears. KirkwoodlBhas advanced a rigorous treatment the molecule. This is also borne out by the close correspondence of X-ray3c and dielectric datasa of phase transitions in solids due to hindered for alcohols, which indicate that rotation of the rotation which is able to predict the dielectric paraffin chain and dipole orientation effects constant near the broad lambda point in hydroappear simultaneously. The alcohols show a bromic acid" a t 89' K. with fair accuracy. non-linear variation of T, with chain length as If a two-minima potential between neighboring expected from the temperature sensitive nature molecules had been used, first order transitions of the hydrogen-bond. The unassociated com- could have been obtained. Kirkwood also surpounds show a convergence of the melting point mises that first order transitions could arise from and transition point curves a t high chain lengths the sensitivity of barrier height to lattice paramowing to the flattening off of the melting point eters. Although no explicit case of first order and the approximately linear rise of T, with in- transitions was discussed in detail, it may be profitable t o attempt an interpretation of our creasing chain length. Fowler's original treatment of rotating polar data with Kirkwood's theory, but this is beyond molecules predicted a dielectric Curie point far the scope of the present paper. Frank1* has developed a theory for a rotator below T,. The absence of such a Curie point with one frozen position. Then, employing in the present treatment is a result of using the Onsager rather than the Debye expression to an expression of the form nrot./nfrolen= g e - E / k f relate polarization and dielectric constant. Since and applying the Bragg-WilliamslY cooperative we find no dielectric Curie point experimentally, approximation to the energy, E , Frank obtains vs. T curves for different values of g indicating a serious objection to Fowler's treatment 15 first order transitions similar. to ours. When eliminated. (16) Kirkwood, J . Chcm. Phys., 8, 205 (1940). Muller's concept4 that the sudden appearance (17) Smyth and Hitchcock, THISJOURNAL, I S , 1830 (1933). of a rotator a t a given chain length was due to (18) Frank, Trans. Faraday Soc., A42, 32 (1946). overcoming end-group forces must be modified. (19) Bragg and Williams, Proc. R o y . SOC. (Lowdon), A i l s , 699 In our view the sudden appearance of a rotator (1934).
178
JOHN
D. HOFFMAN AND CHARLES P. SMYTH
Vol. 72
+
I', A E n . The heat capacity due to molecules overcoming the barriers may be obtained by multiplying the energy required for rotation per mole, A€€Ts&, by the rate at which the molecules are assuming the rotational state, dr/dT. Thus dr
Cr S (AE f Vn)s2 p
-
dr/dT may be read from Fig. 5 for the appropriate a. If one plots C,against T for a 0 the specific heat passes through a distinct maximum below T,. If a is about 1/2 or greater, the specific heat rises slowly up to T,, and then the latent heat of transition appears (Fig. 9). No specific d heat data of high precision are available for longchain compounds. Certain solid transitions involving only one degree of freedom of rotation have been studied, however, and the treatment given here should be applicable. The heat capacity has been calculated for t-butyl chloride which shows a transition,21.22 probably involving one degree of freedom a t -90". The motion is presumably around the dipole axis (6 = 0), since only a slight change in dielectric constant is observed. AEIT = 0.41 kcal., T, = 183' K., so Vn = 0.28 klaal., and, therefore, AE, = 0.41 0.28, or 0.13 kcal. and a = 0.13/0.28 or about 1/2. Using a plot of r vs. T / T , calculated for a = 1/2 in conjunction with the expression for C,, the specific heat has been calculated at various temperatures by adding C, to a roughly estimated curve (dashed line, Fig. 9) for the vibrational I I I heat capacity and compared with experiment in Fig. 9. We are indebted to Mr. R. W. CroweZ2 Chain length. for the use of these data. The vibrational conFig. lO.-Melting points and transition points in long-chain tribution to the heat capacity was assumed continuous as the transition took place. Heat compounds. capacity data on such compounds are customarily the statistical weight of the rotators, g, is set obtained with rather rapid warming. This tends equal to 13, the results are nearly identical. The to smear out transitions and give them the assignment of g seems rather arbitrary. appearance of lambda points, e. g., a calorimeter Fr6hlichZo has developed a theory of the warmed a t 1O /min. caused carbon tetrachloride dielectric properties of long-chain compounds to "melt" over a range of go, while warming at based on order-disorder in the dipole array, half this rate narrowed the rnelting range to 5°.22 in which i t is assumed that the interaction energy This may be the cause of the breadth of the is due solely to dipolar forces. As was pointed transitions observed by Ubbe10hde.l~ A small out in the addendum to his paper, the calculations amount of true broadening of transition and suggest a transition of the second order, and do not tnelting points in long-chain compounds may be agree with experiment. Another objection to due to premeltings or chain twisting as treated Frohlich's theory, as he apparently realized, is by F r o h l i ~ h ,but ~ ~ the transition oints and that the transitions in the paraffins must remain melting points observed in bromides, unexplained since they possess no dipole array and seem sharp within the limits of and could therefore have no solid transition. purity, so premelting phenomena are little in A4ctually, the close parallelism between the evidence. I t should be recognized that a-@2 transitions in paraffins, bromides, alcohols and and PI-/32 transitions may slowly take place iodides (Fig. 10) with such varying polar nature during dielectric and calorimetric measurements strongly suggests that the dipole interactions are thus obscuring the true nature of other phase quite weak in comparison to the London forces changes. Heat capacity should be measured with a t the end-group and along the chain. lowering temperature to avoid this difficulty. The energy required to cause a molecule to (2 1) Rushner, Thesis, Princeto8 University, 1948. rotate a t a given temperature is given by AHd/ (22) Crowc, unpublished measurements in this Laboratory N , where N is the Avogadro number and U T 7 = ( 2 3 ) Frohlich, Trans Fluaday Soc., 41, 90 (1945) (211) Frohlich, Pvoc
It",. S O L ( f o i i d o i i ) , 8 1 8 5 , 3011
'1114tj)
1 1 ) 'strer, Pattrrwn and Keyes, TEIS J O U R N A I
, 66, 179 (1944,
Jan., 1950
ROTATION OF LONG-CHAIN ALKYLBROMIDES
179
In any event, the treatment which has been ing temperature in the rotator state is emphadeveloped gives the correct shape and magni- sized by the slight rise in e’ in n-docosyl bromide tude for the specific heat curve in Fig. 9. between the transition and melting point, a.nd The absence of dielectric dispersion in the the flat portion of the C, curve of t-butyl chlonde prerotator state of long-chain ketone3 even a t above T,. Frenkela2,concludes that the thermal frequencies up to 1.5 X 10’ cycles/sec. (2 motions of molecules above T, are of the same meters) has already been mentioned. DanielsZ6 character as below T,, partly from the conhas found that mixtures of long-chain ketones and sideration that the additional drop in specific paraffins show dielectric transitions similar to heat expected for a hindered rotator transforming that described in this paper for n-docosyl bro- to a free rotator (one-half R cal./mole/degree mide. No dispersion was found in the rotator of freedom) is not found experimentally, but is in or prerotator state between los and 3 X los fact replaced by an increase in specific heat. cycles/sec. It may be inferred that the dis- The latter statement is not correct in the cases persion region in certain solid rotator states of methane,33 t-butyl chloride, and other commay occur a t a wave length corresponding to about pounds where drops in specific heat are found, 1.5 X lo9 cycles/sec. (20 cm.) or less. Heston, but this does not revive the concept of free Hennelly, Laquer and SmythZe,27,28 have investi- rotation. Long-range rotational order is degated the wave length, Am, where maximum dielec- stroyed at Tr, but local order remains and tric loss occurs for many liquid long-chain bro- maintains a relatively small hindrance to rotamides. It is interesting to note that Am for liquid tion which prevents the molecules from performn-hexadecyl bromide at 25’ is about 10 cm., which ing free rotation, although this condition may be corresponds to a free energy barrier of AF* = approached32a t high temperatures in certain sym3.5 kcal./mole as calculated from the E ing rate metrical compounds. Insight into the nature of t h e ~ r yusing ~ ~ *the ~ formula Am = (h$) (27~). the change in molecular rotation a t T, may be gained by measuring Am in the rotator and preeA**:/R*, where h = Planck’s constant and t = velocity of light. The orientation process in the rotator states of polar long-chain molecules. The general picture of the rotational transitions solid is evidently very rapid, so it is reasonable to treat the transitions as involving momenta in the bromides is as follows: n-dodecyl, n-hexarather than configurational degrees of freedom decyl and n-octadecyl bromides show a marked but such as are concerned in the glass transformation gradual increase in dielectric constant due to defined by Kauzmannaowhere relaxation times of prerotation. This is due to cooperative loosening several hours or more are often encountered. of the bar-shaped molecules in the lattice, which The theory presented in this paper‘should be permits some of the molecules to rotate. nused only where there is evidence of rapid orienta- Docosyl bromide shows a similar effect as well as an observable rotator phase. The dielectric tion. Fowler’s original theory implicitly involves properties of the rotator phase were shown to be the notion of free molecular rotation above Tr consistent with the picture of the molecules as oirginally suggested by Pauling. 31 Actually, rotating about their long axes, as in the alcohols of course, the nature of rotation is not deter- and paraffins. It was indicated that the three mined only by the moment of inertia, but also shorter bromides did not show transitions beby potential barriers which may be present. cause they crystallized directly into stable The treatment of rotation here developed clearly modifications. The failure of the transition to refers to the breakdown of long-range order, reappear on warming was ascribed to a PI-& as may be seen in the derivation of V = Vasz. transformation in the solid, as was shown for The short-range order corresponding to the law alcohols. Fowler’s theory of rotating moleV = Vn,s, which considers nearest neighbors cules has been modified by (1) solving for moleonly, persists in the rotator state above Tr so cules with one degree of rotational freedom, (2) that free rotation is approached only at very high removing the parameter j3, (3) changing the temperatures. V,,,, the barrier associated with assumed cooperative law from V = Vas to shoit-range order, may be expected to be small V = V,,s2, (4) taking into account the lattice so that e‘ will be reduced by only a small amount expansion energy and its cooperative nature and above Tr. For example, if V,,, were 300 cal./mole, (5) using Onsager’s and Kirkwood’s relations r would be 0.95 a t the transition point and rise to calculate dielectric constant from polarization, gradually to unity as T approached infinity. The results of the calculations indicate that the The gradual loss of short-range order with increas- dielectric constant and specific heat should show (26) Daniels, Nature, 163, 725 (1949). prerotation and then snap up sharply a t a transi(26) Heston, Hennelly and Smyth, THIS JOURNAL, 70, 4093 tion point. Such behavior was shown by n-do(1948). (27) Hennelly, Heston and Smyth, ibid., 70, 4102 (1948). cosy1 bromide on cooling, and by certain very long (28) Laquer and Smyth, ibid., TO, 4097 (1948) hydrocarbons and alcohols on warming and cool(29) Glasstone, Laidler and Eyrinp “Theory of Rate Processes,” McGraw-Hill Book Co.. New York, N. Y., 1941, Chap. IX. (30) Kauzmann, Cham. Rev., 48, 219 (1948). (31) Pauling, Phys. Rcp., 86, 430 (1930).
(32) Frenkel, “ginetic Theory of Liquids.” Oxford University Press, 1946, Chap. 11. (33) Clusius and Palick, 2.physik. Chum., 94, 313 (1924).
1 so
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ing. The theory also is useful in calculating the dielectric constant due to prerotation in the
Summary The dielectric constants and cooling curves of three shorter bromides. The model required that n-hexadecane, n-dodecyl, n-hexadecyl, n-octathe transition temperatures should rise nearly decyl and n-docosyl bromide have been measured linearly with increasing chain length in agreement over a large temperature range. Prerotation was with experiment. Polarity seems to affect the found in the bromides even a t very low temperatransition points only slightly. I n certain cases it is possible to estimate the additional heat capa- tures. On cooling, n-docosyl bromide froze a t city and the dielectric constant due to prerotation 40' and showed a sharp transition at 30°, and a and T,. The theory does not imply suitable analysis showed that the molecules rogiven U T free rotation above the transition point. The the- tated about the long molecular axis between the ory of rotating molecules presented in this paper, freezing and transition point. No unsharp transiin conjunction with the concept of an equilibrium tions were found. A theory of molecular rotation between metastable phases, seems able to give an has been developed and the dielectric properties account of the gross features of the rotational be- of long-chain bromides have .been discussed in terms of the results. Heat capacity also has been havior in long chain molecules. Acknowledgments.-The authors wish to ex- discussed. Comparisons have been made between press their indebtedness to Professor W. J. the properties of long-chain bromides, alcohols, Rauzmann, Dr. A. D. Franklin and Dr. R. A. iodides and paraffins. Oriani for many helpful conversations. PRINCETON, NEWJERSEY RECEIVED JUNE 7, 1949 -_- -___[CONTRIBUTION FROM THE STERLING CHEMISTRY
LABORATORY OF YALE
UNIVERSITY]
Polarographic Studies of Metal Complexes. 11. The Copper(I1) Citrates' BY LOUISMEITES* Other papers in this series have dealt with the polarographic characteristics of copper(I1) in oxalate' and tartrate2 media. The present paper includes the results of a similar study of the various citrate complexes.
the half-wave potentials of, here, the citratoand aquo-copper ions, Kd is the dissociation constant of the compIex, p is the number of complexing groups attached to one copper atom, n is 2 for reduction to the metallic state, and [XI is the concentration of the citrate species involved Experimental in the complex. Laitinen, Onstott, Bailar and The apparatus used will be described e l s e ~ h e r e . ~Swann4 have recently rewritten equation (3) in the related form Data and Discussion 0.0591 Figure 1 shows the variation in the half-wave A E l / $ -log Kd - ----g(pH potential of copper(I1) in 0,5 F potassium citrate PK log [ax11 (4) a9 a function of pH. There are four major sections of interest, bounded, at this citrate where [HX] and K, are the concentration and dissociation constant, respectively, of the acid concentration, by the pH limits 2.5-5.5, 5.5-7.4, corresponding to the complexing anion. They 7.4-11.8,and 11.8-14. In the most acid range, as a first approximation, state that "from this equation it is apparent that the half-wave potential is related to the pH by the the half-wave potential will be a linear function of PH if the dissociation of the acid is small. . . .' ' equations This is only true within the accuracy of the El/, = 0.112 0.0523 PH (0.5 Fcitrate) (1) measurements when PH 6 p K a - 1, and,' as and this is not the case here, equations (1) and ( 2 ) El/,= 0.142 - 0.050 PH (0.05 F citrate) (2) would not be expected to be as accurate as the The working formula for the half-wave potential data themselves. The four regions of the graph can hardly inof a metal complex, neglecting activities, is dicate anything but the formation of complexes with, successively, the dihydrogen and mono(3) hydrogen citrate ions, the citrate ion itself, and, where A&/, is the algebraic difference between at high +H values, with both hydroxyl ion and citrate. In solutions more acid than pH about * Harmrd University Ph.D., 1947. 5, the dihydrogen citrate ion predominates. Its (1) L.Meites, THISJOURNAL, 71, 184 (1949)
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(2) L. Meites, i b i d . , 71, 13269 (1949). (3) L. Meites and T. Meites, to be submitted.
(4) H.A. Laitinen, E. I. Onstott, J C. Bailar, Jr., and S. Swans, Jr , TATS J O U R N A L ,71, 1550 (1949)
