CERTAIN PROPERTIES OF LONG-CHAIN COMPOUNDS AS FUNCTIONS OF CHAIN LENGTH"' MAURICE L. HUGGINS
Kodak Research Laboratories, Rochester, New York Received August 7 , 1959
In this paper the dependence on chain length of the heat content, heat capacity, entropy, and free energy of typical long-chain compounds in the solid, liquid, and gaseous states will be considered theoretically and the relationships deduced will be applied to changes of state. Comparisons will be made with experimental data for normal paraffins. Extension to other cases can readily be made. HEAT CONTENT AND HEAT CAPACITY
The heat content per mole of a long-chain compound is given by the following equations, for the solid, liquid, and gaseous states, respectively:
Hs
= E,
+
Evib
Erot
- Ebond
- EvdW
PV
(1)
E, is the energy that a mole of the substance would have at the absolute zero, if the molecules were completely dissociated into widely separated atoms. E v i b , E:ib, and E:ib are the vibrational energies per mole. For the solid and liquid they should be nearly equal at the same temperature. In the gas the vibrational energy is all intramolecular. For the liquid or solid state there is also an energy of vibration of each molecule &s a whole relative to its neighbors. This intermolecular vibrational energy should approximately equal the translational energy (3/2 RT) of the molecules in the gaseous state:
+
E v i b = E:ib = Etib EYL (4) Erot,the rotational energy in the solid state, arises almost entirely from rotation of atoms or small groups of atoms, rather than from rotation of 1 Presented at the Sixteenth Colloid Symposium, held at Stanford University, California, July 6-8, 1939. * Communication No. 733 from the Kodak Research Laboratories.
1083
1084
MAURICE L. HUOQINS
the molecule as a whole. Except for a small correction for the chain ends, therefore, Erotshould be proportional to the number of repeating units :
+
Erot= a0 an (5) In the liquid there is a larger amount of rotation of this sort, and for short chains there is also some rotation of the whole molecule:
+ a’n
EiOt= ai
a’ > a (7) (Strictly, ai is not independent of n. It should decrease, however, aa n increases, approaching a small, constant value.) For the gaseous state, similarly,
Eot= a:
+ ann
a’’ = a’
(8)
(9)
Ehnd,Eknd,and EtOnd, the sums of the bond energies, may be considered equal for the three states:
(For a long-chain molecule, obviously, this sum is proportional to the chain length.) &dW, E&, and ECdw, the van der Waals energy terms, result from interactions (chiefly attractions) between atoms not directly bonded together. In the solid and liquid states these are both intermolecular and intramolecular; in a gas they are exclusively intramolecular, except insofar as there are deviations from the perfect gas laws. Crystalline normal paraffins (12) and some other long-chain compounds in the solid state (e.g., polymethylene oxide (8) and polyesters (5, 6)) contain parallel, extended, zigzag chains. Many others (e.g., many vinyl polymers) consist of chains which are kinked in a more or less random manner. Such kinked molecules are doubtless the rule in the liquid state. For both extended and kinked models, the van der Waals energy contribution of each unit in the chain would be expected to be independent of the chain length, except for a small end correction: Evdw = bo E& = b:
+ bn
+ b’n
In general,
b
> b’
(12) (13)
PROPERTIES OF LONG-CHAIN COMPOUNDS
1085
while bo and bi are of the order of magnitude of b and b' but probably smaller. In the liquid there are, on the average, fewer interatomic "contacts" than in the solid. Also, the average interatomic distance is greater and so the average attraction energy per contact is smaller. I n the gaseous state the van der Waals attractions between atoms not directly bonded together tend to make a flexible long-chain molecule coil up into a ball (11, 1). An atom in the interior of such a ball molecule would have practically the same van der Waals energy as if the molecules were in the liquid state a t the same temperature. An atom in the surface of the ball, on the other hand, is not completely surrounded by other atoms. Hence its van der Waals energy is less than it would be in the liquid. A large sphere composed of n closely packed atoms (each with twelve close neighbors) would have a surface area of 2 1 ~ a 3 2 / 3 d ~ a d ~do n2/3, being the distance between the centers of adjacent atoms. For the lowest energy (greatest stability), each surface atom would have six other surface atoms and three interior atoms adjacent to it. It would occupy a surface or area of 2-131/2di. The number of surface atoms is then 24/331/6d/3n2/3 4 . 4 3 7 ~ ~ ' Assuming ~. approximate close-packing of atoms in both liquid and gaseous states, one deduces, therefore, as a limiting law for large n,
E& = b r
+ b"n - cNnm
(15)
where
m z -2 3
(18)
Also, b; and bb' are probably not very different. The last terms in equations 1 and 2 may ordinarily be neglected, since the molal volumes, V and V', are relatively small and nearly equal :
PV = PV'
0
(19) The PV" term in equation 3 is important at high temperatures or pressures, P. As a first approximation, =;i
PV" = RT (20) Substitution into equations 1, 2, and 3 and subtpction gives the varik tion with n to be expected for the heats of fusion and vaporization, on the assumption that all are measured a t the same temperature: (AH/)T = H ,
(AH,)= =H G
+ bo - b:) + (u' - u + b - b')n (21) - Ht = (+RT+ b: - b'd) + l . l l b ' d ' a (n) - H s = (4 -
1086
a U R I C E L. HUGGINB
Experimentally, heats of fusion of different compounds are not measured at the same temperature, but at the melting points, hence equation 21 is directly applicable only when n is so large that all the compounds being compared have nearly the same melting point, T,,m=. To make the relation between the heats of fusion a t the melting points applicable to all values of n, one must make a correction for the difference in heat capacities :
Equation 22 is compared with the experimental heats of vaporization of paraffins at 298°K. in figure 1. The values plotted are those chosen
FIG.1. Heats of vaporization of normal paraffin hydrocarbons a t 298°K.and a t the boiling point. 0, Rossini, a t 298°K.;X, Sage, Lacey, and Schaafsma, a t Tb; -I-,Schultz, a t Tb. The straight line represents the equation ( A H , ) 2 ~=~ 2.56nf1S. The other curve represents (AH,)T,, calculated on the assumption that Rossini’s ( A H . ) ~ values u~ are correct.
as “best” by Rossini (17) after a critical survey of the literature. The experimental curve approaches proportionality with n2/aas n increases. To compare heats of vaporization at the normal boiling points, it is again necessary to correct for the heat capacity differences:
Since
the relationships given above for the terms in the expressions for the heat content lead to the conclusions that the heat capacities in the solid, liquid,
1087
PROPERTIES OF L O N W H A I N COMPOUNDS
and gaseous states, respectively, should depend on the chain length in the following manner:
Here lo, f, g", etc. are functions of temperature but not of n. The last term in equation 28 would be expected to be small, the change in van der Waals energy with temperature being small relative to the change with
L
FIQ.2. Heat capacities of crystalline normal paraffin hydrocarbons. Variation of C,(s) with n at various temperatures. Data are those of Parks and coworkers.
temperature of the oscillational and rotational energy. Hence, approximately,
Cp(d = Jd
+ f"n
(29)
Experimental data for solid and liquid normal paraffins (9,13,14,15,lS) agree well with equations 26 and 27, as shown in figures 2 and 3. Over the temperature ranges studied, one can put fo = 6 cal.
(regardlees of the crystal form), and
p: = 0
(30)
,.
A.
0 2 4 8
6 IO I2 I4 I6 Ib 20 22 14 26 20 90 52 Y
FIQ.3. Heat capacities of liquid normal paraffin hydrocarbons. Variation of C,(1) with n at various temperatures. Data are those of Parks and coworkers.
FIG.4. Variation of the heat capacity functions for normal paraftin hydrocarbons h’P -n*.
with the temperature. C,(s) = 6 ffn; Cp(Z) = j’n; c,(g) = ft ff”n
+
T’
PROPERTIES OF LONG-CHAIN COMPOUND8
1089
while f and f depend on temperature in the manner indicated in figure 4. It may be noted that the function for the liquid agrees well with the relation f = 6.57 0.033 10-'Ta (32)
+
between 140°K. and 380°K. As Edmister (3) has pointed out, the data available for the heat capacity of paraffins (having 3) in the gaseous state between 250°K. and 600°K. are in fair agreement with the relation
nr
C,
= (2.56
+ 0.51%)+ (0.0042 + 0.0130n)T
(33)
This can obviously be rearranged t o
C,
= (2.56
+ 0.00422') + (0.51 + 0.01302')n
(34)
in agreement with equation 29. Equations for the temperature dependence which are probably more accurate have been obtained by Bennewitz and Rossner (2), using Einstein functions. Reexpressing their results iq terms of power series in T, Fugassi and Rudy have arrived a t equations, which, for normal paraffins, reduce to
C, = $+ j " n where
+ hT" BP n2
(35)
fi = 4.43 + 0.324.10-8T + 6.543 10-'T2
and
f" = -2.784
+ 25.98. 10-aT - 10.647.10-'T2
The n* term is a correction, usually negligible, for deviations from the perfect gas law. The quantity h" is a function of n which seems to be approaching a constant value of about 0.5.lo8 as n increases (figure 5). The difference between equations 34 and 35 is illustrated, for two values of n, in figure 6. Calculations show that the heat capacity correction term in equation 23 for the heats of fusion is always relatively small and decreases as the length of the chain increases. The simple equation AH, = A
+ Bn
(36)
should, therefore, be applicable to heats of fusion measured a t the melting points, except when n is quite small. The experimental results are in agreement with this conclusion (figure 7).
1090
MAURICE L. HUGGINS
A
FIG.5. Variation with n of the function ha in the gas imperfection correction term (h”Pn*/T*)for the heat capacities of gases.
n.o-
-
m m-
u.o-
n-
u-
FIQ.6. Comparison of Edmister’s equation with that of Fugassi and Rudy for the heat capacities of normal paraffins. Heat capacities of gaseous normal butane and normal decane. Experimental points are from Sage and Lacey.
-8yz BO-
1.2
-
I8 -
20
IC 14
-
12-
IO
-
0
2
4
6
6
IO I 2
14 I6 I6 10 22 14 ‘26 28
FIG.7. Heats of fusion of normal paraffins. Van Bibber, and King.
M
32 34 36
+, Parks and coworkers; 0, Garner,
PROPERTIES OF LONQ-CHAIN COMrOUNDB
1091
The heat capacity correction in equation 24 for the heats of vaporization, on the other hand, is not negligible. Assuming Rossini’s values for AH, a t 298°K. to be correct and making use of equations 27, 31, 32, and 35 and also figure 5, the lower curve in figure 1, representing AH, a t the boiling point, has been obtained. The agreement with the experimental points (19, 20) is not bad. ENTROPY
Entropy is a measure of randomness. For convenience, the entropy per mole of a long-chain compound in the solid or liquid state may be divided into four parts, according to the source of the randomness: (1) randomness resulting from the temperature vibrations of each molecule as a whole; ( 2 ) randomness of position and orientation of each molecule as a whole; ( 3 ) randomness resulting from alternative orientations of the chain bonds relative to each other and from more or less free rotation about the bonds; and ( 4 ) randomness due to vibrations of the atoms in each molecule relative to their neighbors.
+ s’ = s:, + S
= Sev
Sp&o
+ + Sir
Siv
+ sl, + Si“
(37) (38)
For the gaseous state, similarly,
S” =
s;: + srr+ si: + s’i:
(39)
the first two terms measuring the randomness resulting from the translational and the rotational motion, respectively, of the molecule as a whole. The entropy of external vibrations, S,, and &, is a function of the temperature, but (for non-rigid molecules) may be assumed to be approximately independent of chain length a t any given (not too low) temperature, both in the solid and in the liquid: sev
= ffo
(40)
s:,
= a;
(41)
As the immediate environment of each atom or group is about the same in the liquid as in the solid,
= a; (42) (Since, for long-chain molecules, this term is relatively small, considerable inaccuracy in this assumption does not matter.) is also temThe entropy of position and orientation, Sp.&o and perature-dependent, but for long molecules practically independent of chain length: Spa0 = Bo (43) ffo
S L =
(44)
1092
MAURICE L. HUGOINS
If the solid is crystalline, B equals zero; otherwise, it is finite. The value for the liquid, as a rule and perhaps always, is greater than that for the solid:
dl > Bo
(45)
For large values of n, both the first and the sacond entropy terms in equations 37 and 38 can be neglected. The entropy of internal randomness, Sir,Sll, and Si:,is a function of both temperature and chain length. At a given temperature it should be proportional to n (except for a small end correction), the addition of one unit to the chain increasing t h o entropy the same amount, regardless of the length of the chain: Sir
= yo
+ yn
(46)
Each C - C bond in a paraffin chain has three alternative equilibrium orientations relative to the preceding bonds in the chain. If these were equally probable, the antropy of internal randomness of a mole of rigid paraffin molecules would be R(ln 3 ) ( n - 3 ) or, with 7t large, R(ln 3)n. Actually, especially in the solid and liquid states, the three orientations are not equally probable. On the other hand, since the potential energy minima are not sharp, oscillations about the equilibrium orientations and (especially at high temperatures) rotations over the energy humps between them occur. This increases the entropy of internal randomness, but leaves it still proportional to n. Semi-quantitative calculations (10) have led to the value 8.9 for the proportionality constant, for a normal paraffin chain at 25"C., on the assumption of free rotation and have shown that restricting potentials of 3000 cal. per mole or less would not alter this constant by much. It is probably correct to write y
< y' < y" < 8.9
(49)
The entropy of internal vibrations, Si, and Si,, should also be proportional to n, for long chains: Si, = 60
+ 6n
(50)
For normal paraffins at 25'C., using the frequency assignments of Pitzer (le), the proportionality constant has been calculated (10) to be about
1093
PROPERTIES O F LONG-CHAIN COMPOUNDS
1.8. This factor should be only slightly dependent on the state, except in cases where intermolecular attractions are strong. Hence, 6 = 6’ 6t’ t: 1.8 (53) The translational entropy, Si:, has been shown (lo), using well-known, generally accepted equations, to be given, at a definite temperature and pressure, by an equation of the form
~=l :&‘
+ e” In n
(54)
where
and, for normal paraffins at 298°K. and 1 atmosphere, (56) The entropy, Syr,due to rotation of the molecule as a whole, obeys (10) an equation of similar form: e:
=
srr= {T
33.8
+
(57)
In n
Hypothetically, assuming the molecular chaim to be so coiled as to give each molecule a spherical shape, with an internal density equal to that in the liquid, one can calculate, again for a paraffin a t 298”K.,
ST = 20.7
(58)
and
If one assumes, instead, complete randomness of kinking (giving nonspherical, less dense molecules), constants which are slightly different, but not markedly so, are computed (10). Substituting into equations 37, 38, and 39, one now obtains
S” =
(7;
+ 6; + + $) + (7’’ + 6”)n + (c + f ) l n 6;
n
(62)
As shown in figures 8 and 9, the entropies calculated from the experimental data on liquid and gaseous paraffins are in agreement with equations of the form of equations 61 and 62. From equations 60 and 61 it follows that the entropy of fusion at the same temperature for all values of n should be a linear function of n. Since
1094
MAURICE L. HUQQINS
+-
FIG.8. Entropies of liquid normal paraffins (data from Parks and coworkers)
IO0
-
meo70-
*.*-
Lwm*LOU.
sow-
m
40-
ea
I
R
1 FIG.10. Entropies
IO
to
. O n
of fusion of normal paraffin hydrocarbons. a-form. Data
are those of Garner, Van Bibber, and King.
PROPERTIES OF LONG-CHAIN COMPOUNDS
1095
the correction for temperature ditrerences is not large and since it is reasonable to suppose that the change in entropy of fusion with temperature is %ear with respect to n, one should expect the entropy of fusion at the melting point to obey a similar law:
A& = C
+ Dn
(63)
This is in line with the experimental results of Garner, Van Bibber, and King (7) (figure 10). For the entropy of vaporization at a uniform temperature, a relation of the form ASv = K Ln M l n n (64)
+ +
is similarly deduced. At the boiling point, according to Trouton’s rule, the entropy of vaporization varies but little with n. FREE ENERGY, MELTING POINT, BOILING POINT, VAPOR PRESSURE
From the relation
AFzAH-TAS
(65)
and equations 21, 22, 63, and 64, one obtains for the differences in free energy between solid and liquid and between liquid and gas, at any given temperature,
FL - FS = P + Qn
and
Fa - FL = U
(66)
+ Vn2’a- W n - X In n
P,Q,U , V , W , and X are functions of T but not of
(67)
n.
At the melting point, the free energy change on fusion is zero. From equations 65, 36, and 63,
Garner and coworkers (7) have shown that an equation of this form fits the experimental dota very well for hydrocarbons (also acids and esters). For large n, l/T,, minus a constant, is proportional to l/n (figure 11). If the heats of vaporization at the boiling points were proportional to n213,like those a t a uniform temperature (equation 22), then Trouton’s rule of constant entropy of vaporization at the boiling point would imply that the boiling point also is proportional to n213. Since the heat of vaporization at Tb increases less rapidly than n2/81however (figure l), so also does the boiling point (figure 12). By means of the specific heat
1096
MAURICE L. HUGGINS
curves for the liquid and gas or the equations therefor, boiling point values in good agreement with experiment can readily be calculated, but only
,00460-
.oosoo
.oo*oo
-
.00250 .oofoo -
e 0
.maw .00300
*
0
-e e o &40°
or'
*
.m1w .00100
-
-
. m 5 0
I
0
I
DtO
I
-0
I
I
.oCO
.OS0
-n
I
I
I
.IO
12
.I4
I
FIG.11. Melting point data for normal paraffins
nS; FIG.12. Boiling points of normal paraffin hydrocarbons
over a limited range, because of the limited amount of data now available on the specific heats of the liquids.
PROPEk+'IES
6$ L6NQ-CHAW
1097
COMPOUND6
To compare vapor pressures of a series of long-chain compounds at a uniform temperature, one may use the relation AF, = AHg -
T U ,=
-RTlnp
(69)
Substituting equation 07, one obtains l n p = -u RT
--
RT
n*/*
X Inn + .E n + RT
RT
The experimental data for the paraffins (from International Critical Tables and Landolt-Bornstein Tabellen) approach agreement, as n increases, with an equation of this form, in which the constants are derived from those in the equations in figures 1, 8, and 9 (see figure 13). The deviations for small n are due almost entirely to the deviations of AH,,from proportionality with n2I3(figure 1).
"e 0
I
2
4
5
0
1
a B
IO
II
It I8
I4
IC
U
I1
I8 1 9 R
FIG.13. Vapor pressures of liquid normal paraffin hydrocarbons at 298°K. SUMMARY
From considerations which are largely qualitative in nature certain conclusions have been reached regarding the variation with chain length of various properties,-heat content, entropy, heat of fusion, heat of vaporization, vapor pressure,-of long-chain compounds at constant temperature. With the aid of experimental heat capacity data, expressions have been derived for the dependence on chain length of the melting point, heat of fusion and entropy of fusion a t the melting point, and heat of vaporization a t the boiling point. The variation of boiling point with chain length has also been briefly discussed. The conclusions reached have been tested, where possible, with experimental data for normal paraffin hydrocarbons. The writer is glad to express his thanks to Dr. F. D. Rossini of the National Bureau of Standards for permission to use his unpublished heat of vaporization values, and also to Miss Dorothy Owen for much help with the calculations leading to the results reported here.
t098
MAURICE L. HUQQINS
REFERENCES (1) ATEN,A. H. W.: J. Chem. Phys. 5, 264 (1937). (2) BENNEWITZ, K.,AND ROSSNER,W.: 21. physik. Chem. B39, 126 (1938). (3) EDMISTER, W. C.: Ind. Eng. Chem. 30, 352 (1938). (4) FUGASSI, P., AND RUDY,C. E., JR.: Ind. Eng. Chem. 30, 1029 (1938). (5) FULLER, C. S.,AND ERICKSON, C. L.: J. Am. Chem. Sac. 69, 344 (1937). (6) FULLER, C. S.,AND FROSCH, C. J.: J. Phys. Chem. 43, 323 (1939). (7) GARNER,W. E., VANBIBBER,K., AND KING,A. M.: J . Chem. Soc. 1931, 1533. (8) HENGSTENBERG, J . : Ann. Physik 84,245 (1927). (gl HUFFMAN,H. M., PARKS, G. S., AND BARMORE, M.: J . Am. Chem. Sac. 63, 3876 (1931). (10) HUGGINS, M.L. : Unpublished calculations, reported a t the Ninety-sixth Meeting of the American Chemical Society, held in Milwaukee, Wisconsin, September, 1938. (11) LANGMUIR, I.: In Colloid Chemistry, edited by Jerome Alexander, Val. I, p. 525. The Chemical Catalog Co., Inc., New York (1926). (12) M ~ L L E RA.: , Proc. Roy. Sac. (London) Al20, 437 (1938). (13) PARKS, G. S.,AND HUFFMAN, H. M.: J. Am. Chem. SOC.62,4381 (1930). (14) PARKS,G. S.,HUFFMAN, H. M., AND THOMAS, S. B.: J. Am. Chem. Sac. 62, 1032 (1930). G. S.,AND LIQHT,D. W.: J. Am. Chem. Sac. 66,1511 (1934). (15) PARKS, (16) PITZER, K . S.:J. Chem. Phys. 6,473 (1937). (17) ROSSINI,F. D.:Private communication. (18) SAGE,B. H.,AND LACEY,W. N.: Ind. Eng. Chem. 27, 1484 (1935). (19) SAOE,B. H., LACEY,W. N., A N D SCHAAFSMA, J. G.: Ind. Eng. Chem. 27, 48 (1935). (20) SCHULTZ, J. W.: Ind. Eng. Chem. 21, 557 (1929). (21) SPAQHT, M. E.,THOMAS, S. B., AND PARKS, G. 5.: J. Phys. Chem. 36, 882 (1932).
