R. W. HASKELL, H. B. HOLLINGER, AND H. C. VAN NESS
4534 to irradiation than are the aromatic hydrocarbons. The most important product, by amount, is the residue. Dealkylation of methylpyridines is a relatively unimportant reaction, suggesting that production of pyridine from methylpyridines by y irradiation alone would not be of significant value in enhancing the value of shale oil products.
Acknowledgments. The work upon which this report is based was done under a cooperative agreement between the Bureau of Mines, U. S. Department of the Interior, and the University of Wyoming. Reference to specific brand names is made for identification only and does not imply endorsement by the Bureau of Mines.
The Chemical Model as Applied to Associated Liquid Solutions. The Ethanol-Heptane System by R. W. Haskell," H. B. Hollinger, and H. C. Van Nesslb Chemistry Department, Walker Laboratories, Rensselaer Polytechnic Institute, Troy, New York 12181 (Received June 10, 1968)
The chemical model is developed for solutions where one component is associated. In essence, the idea is to consider the associated component (A) to consist of many species, AI, Az, . . ., A,, which are in equilibrium according to reactions of the form AI Aj-l = Aj. Based on this assumption, it is shown that for all of the presently used models the chemical contribution to the free energy of mixing is derivable from the ideal entropy of mixing ( i e . , in terms of mole fractions) for a multicomponent system or from Flory's expression for the entropy of a heterogeneous polymer and a monomeric solvent. In this development, it is found that the thermodynamic functions (GE, H E , XE) are indicators of the degree of association expressed as a chain size and also of the bond energies. Because of this connection, the success of a chemical model is shown to be sensitive to the size of r assumed in the above sequence. In particular, the case when r + and the equilibrium constants determine the concentrations of the various species is shown to be the best approach. A two-equilibrium constant (dimer constant and another which represents constants beyond the dimer) mole fraction model is found to give a good fit to the observed thermodynamic properties of the ethanol-heptane system up to the equimolar composition. Enthalpies and entropies of formation for the dimer were found to be about -8.5 kcal/mol and -22.0 cal/deg mol, respectively, Corresponding values for the linear species were found to be about -5.6 kcal/mol and -11 cal/deg mol, Also investigated is a comparison of the mole fraction and volume fraction formulations.
+
I. Introduction A model which postulates chemical association, the chemical model, has been used extensively in the molecular interpretation of properties of certain classes of liquid solutions. We are especially interested in solutions containing one associating component A and one nonassociating component B, e.g., an alcohol in a nonpolar The associating component A is regarded as a collection of species AI, Az, . , ,, A, in Al-l % A,. equilibrium according to the reaction AI By recognizing these chemical species we change our position from one of trying to describe strong interactions, which produce nonrandomness in the spatial distribution of the individual A molecules, to one of trying to describe weak interactions, which are ineffective in disturbing randomness in the spatial distribution of the assumed species.
+
The Journal of Physical Chemistry
The repulsive forces which one attributes to the interaction between a polymeric Aj molecule and other molecules may be represented to some extent by attributing a volume to the A, molecule. This volume varies with the number j of A units involved. Since the advent of polymer theories based on volume fraction statistics, it has been natural to assume that volume fraction statistics are required for the statistical (1) (a) Address all correspondence to the University of Utah, Salt Lake City, Utah; (b) Union Carbide Professor of Chemical Engineering, Rennselaer Polytechnic Institute. (2) (a) 0. Redlich and A. T. Kister, J . Chem. Phys., 15,849 (1947) ; (b) C. B. Kretschmer and R. Wiebe, ibid., 22, 1697 (1964). (3) A. C. P. Hwa and W. T. Ziegler, J. Phys. Chem., 70, 2672 (1966). (4) H. Renon and J. M. Prausnitz, Chem. Eng. Sci., 22, 299 (1967). (5) I. A. Wiehe and E. B. Bagley, Ind. Eng. Chem., Fundamentals, 6, 209 (1967).
THECHEMICAL MODELAS APPLIEDTO ASSOCIATED LIQUIDSOLUTIONS treatment of associated liquids and their solutions. Flory's equations for heterogeneous polymer solutionse have been carried over to the case of associated solutions. These treatments have been fairly successful in the description of vapor-liquid equilibria, not so successful in the description of enthalpy and entropy changes upon mixing, and unsuccessful in explanation of the temperature dependence of the thermodynamic functions.*J It is possible that the volume fraction statistics are not completely suited to the treatment of associated solutions. The purpose of this paper is to develop the chemical model from an expression for the entropy of mixing based on mole fraction statistics as applied to an ideal multicomponent system. The ethanol-heptane system is examined in detail because extensive thermodynamic data have been supplied by Van Ness, et al.7-g Dielectric10-12and spectroscopic (ir18-1'jand nmr17-19) evidence concerning the structure of alcohols and their solutions will also be examined. Finally a comparison will be made of the volume fraction and mole fraction models.
4535
k, = X,/XlJ for the reactions described by jAl % A,. of (4)into (3) yields AS" =
In x B f
-R[nBO
Rx(n,'
+
In (x:/xli)]
~ A O
+ SO)' =
g,O/T = -R In le, = (h:/T) - (sjo - ja0) (6) where h,O is the enthalpy change for the formation reaction j A % A,. Putting (6) into (5), we find AS" =
-R[~Bo
In x B f
+
nA0
In
In XBf
En: In x,']
+
+ Cn,'sj0 +
(I) where sp is the molar entropy of pure j-mer. The superscipt f denotes the final or equilibrium state of the mixture, and the superscript 0 indicates a pure species at the temperature and pressure of the mixture. Before mixing the components A and B, the entropy of the system is
9=
(As"
~BOSBO
-t so)' = -RCn,' In x j
+ Cnjs? +
~BOSBO
(2)
where the superscript i denotes the initial or unmixed state containing the same species as the mixture at the same temperature and pressure. Thus the entropy of mixing for the binary system, ASm = S' - Si,is
+
(X~'/X:)
1+ - 4)h?/T
(7) The first term on the right side of (7) is AG"/T, which can be verified by using the integrated Gibbs equation (G = Zn,pj) to form AG" = Gf - @ =
C(n,:,' -
- pBi)
&ji)
+
= nAo(pif - pii) ~B'(PB'
- R [ ~ B ~
AS" = - ~ [ n ~Inoxgf In xlf - n j In xi) ]
+ C(nlf - nji)s;
(3)
Equations 1 and 2 represent mixing processes in which there is no degradation of the polymer species. The chemical reaction which takes place in the actual mixing process is accounted for in (3) by the different mole numbers and mole fractions of the j-mers in the initial and final states. Equation 3 can be simplified by introducing the equilibrium constants (see Appendix 11)
-
- n,') In k j + E(%,'- n,')s,O
11. Model l~evelopment
S' = (AS"
Substitution
(5) where use has been made of the relation 2jnj = nA0. The second and third terms on the right side of ( 5 ) can be simplified further by using
nB0(pBf
We take as our starting point the entropy of mixing for an ideal multicomponent system. The entropy of the mixture relative to a standard state in which the species are separated is (see Appendix I)
(4)
+ -p
~ ~(8))
and then inserting the appropriate formulas for the chemical potentials together with the equilibrium condition, jp1 = f i j . Thus the second term on the right side of (7) represents the enthalpy change of mixing divided by the temperature. An analysis similar to the one given above can be developed using Flory's expression for the entropy of mixing for a disoriented standard state ASm = -R[Cn, In 4,
+ nB In
(9) or the more general equation based on the quasicrystalline standard stateae I n either case, it can be shown that the free energy of mixing is (6) P. J. Flory, J. Chem. Phys., 12, 425 (1944). (7) H. C. Van Ness, C. A. Soczek, G. L. Peloquin, and R. L. Machado, J. Chem. Eng. Data, 12, 217 (1967). (8) H. C. Van Ness, C. A. Soczek, and N. K. Kochar, ibid., 12, 346 (1967). (9) F. Pardo and H. C. Van Ness, ibid., 10, 163 (1965). (10) G.Oster, J. Am. Chem. SOC.,68, 2036 (1946). (11) P. I. Gold and R. L. Perrine, J. Chem. Eng. Data, 12, 4 (1967). (12) G. Oster and J. Kirkwood, J. Chem. Phys., 11, 175 (1943). (13) H. C. Van Ness, J. V. Van Winkle, H. H. Richtol, and H. B. Hollinger, J. Phys. Chem., 71, 1483 (1967). (14) N. D. Coggeshall and E. L. Saier, J. Am. Chem. Soc., 73, 5414 (1951). (15) U. Liddel and E. D. Becker, Spectrochim. Acta, 10, 70 (1957). (16) M. Van Thiel, E. D. Becker, and G. C . Pimentel, J . Chem. Phys., 27, 95 (1957). (17) G. C. Pimentel and A. L. McClellan, "The Hydrogen Bond," W. H. Freeman and Co., San Francisco, Calif., 1960. (18) J. C. David, Jr., K. 8. Pitzer, and C. N. R. Rao, J. Phys. Chem., 64, 1744 (1960). (19) C. M. Huggins, G. C. Pimentel, and J. N. Shooling, ibid., 60, 1311 (1956). Volume 79,Number 13 December 1968
R. W. HASKELL, H. B. HOLLINGER, AND H. C. VANNESS
4536
AGm/RT =
[nBoIn $Bf
+
+
In (+lf/41i)I C(n,' - nil (10) and the enthalpy of mixing is the second term on the right side of (7) multiplied by T. The chemical model as developed above has some interesting general features. For example, thermodynamic properties can be expressed in terms of an average chain length J for the associated component nA0
0.2
0.0
If J' and Jf are the average chain lengths before and after mixing, then the erlthalpy of mixing per stoichiometric mole of the mixture can be expressed as
0.4 0.6 MOLE FRACTION ALCOHOL
I.o
0.8
Figure 1. Theoretical (2K model) and experimental values of A.Hrn/z~O~~O for ethanol-heptane solutions a t 333°K.
where H is the energy of 1 mol of associative links, h,O = ( j - l)H. It is assumed here that H is independent of j . Since Jfapproaches unity in the limit of small X A ~ , the limiting value of AH"/zAOXBOis given by
(-)AH" XA'ZB'
=($-1)H
(13)
tao=o
In a highly associated liquid, such as ethanol, J' may be so large that the right side of (13) is approximately -H. It should be noted that the left-hand side of (13) is given by the left intercept in Figure 1. Other physically significant intercepts which can be derived from (7) are In
( G ~ / R T ~ A ~ ~ B O ) ~= ~ ~ , Oy A 0
(GE/RTx~o~Bo)Zao,l = In
yBo
= In
= In
J'
(14)
- nl) = voxA"[(l/Jf) - (l/Ji)] (15)
where v, is the cell volume. Equation 15 has the same form as formula 12 for the heat of mixing. This similarity between the two properties is observed for the alcohol-normal hydrocarbon systems investigated by Van Ness and coworkers. (See Figure 2.) The JOUTTUL~ of Physical Chemietry
x
x
x
x
x X
4
0.2 I
'
04 0.6 " " ' MOLE FRACTION ALCOHOL
DENOTES
1.0 "
0.8 I
Figure 2. Theoretical (2K model) and experimental values of A.Vm/z~oz,o for ethanol-heptane solutions at 333'K.
XI'
where GE = AG" - G"(idea1) is the excess free energy of mixing. Another interesting feature of the mole fraction model is its prediction of a volume change on mixing. Unlike the volume fraction model, the mole fraction model does predict such a change, and the prediction agrees qualitatively with observed changes. The volume change can be interpreted as follows. Expressions 1 and 2 for the entropy can be regarded as expressions of S = k In W , where W is the number of ways to assign j-mers and solvent molecules to cells or lattice sites. Since the number of j-mers changes upon mixing, there must be a corresponding change in the number of available cells. Thus the volume change can be formulated as A v " = v,X(nff
x
111. Extent of Dissociation From the discussion above we see that the changes of thermodynamic functions with mixing are completely determined for the mole fraction model by the dependence of x:, xlf, and xBf on the stoichiometric composition XAO and the equilibrium constants k,. The equations to be solved are
ex; + xgf = 1 Cjx; k,
=
= X1'/(Xlf)i,
(16)
(xA0/xBo)xB
j = 1, 2, 3,
...
The equilibrium constants are to be determined from experimental data. It is very difficult and not very profitable to determine all of the k, values by curve fitting. On the other hand, it may be profitable to test various assumptions concerning the relative values of k,. For example, if it is assumed that the free energy change for the addition of one A unit, A1 Ai-1 +Is A,, is independent of j, then it follows that k, is Kj-I, where K is kz, and there is only one adjustable parameter to be determined from the data. This case has been studied by Redlich and Kister.2a They let j go to in-
+
THECHEMICAL MODELAS APPLIEDTO ASSOCIATED LIQUIDSOLUTIONS
4537
ETHANOL
-0.200
- HEPTANE
-
EXCESS ENTROPY OF MIXING
-0.400
L l l l l l l l l t ’ 1.0 0.2 0.4 0.6 0.8
0
MOLE
FRACTION
ETHANOL
Figure 3. Theoretical (2K model) and experimental values of excess functions for ethanol-heptane solutions. 0.600
r
0
I
0.2 MOLE
0.4 FRACTION
0.6 ETHANOL
0.8
1.0
Figure 4. Theoretical (2K model) and experimental values of excess functions for ethanol-heptane solutions.
finity in order to allow for association to an indefinite extent and to obtain what is called the continuous-
-0.400 0
0.2 MOLE
0.4 FRACTION
0.6 ETHANOL
0.8
1.0
Figure 5 . Theoretical (2K model) and experimental values of excess functions for ethanol-heptane solutions.
association model. As pointed out by Scatchard,20 the continuous-association model of Redlich and Kister predicts thermodynamic functions which are symmetric about the equimolar composition. This kind of symmetry is not observed experimentally (see Figures 3-5) and we are led to ask whether or not there is something wrong with continuous-association models. Let us consider models for which the extent of association is finite. Suppose r is the largest number of monomer units associated in a single chain. Then the largest j-mer is A,, and r is the upper limit for the sums in (16). One of the interesting predictions of this model is the maximum value of GE, which is sensitive to r. I n the limit of infinite IC,, where GE is maximized for fixed r, it is easy to solve eq 16 for GE as a function of the stoichiometric composition (see Appendix 111). If we maximize again, this time with respect to composition, we obtain the values listed in Table I. The experimental values of (GE/RT),,, for the ethanol-nheptane system range from 0.5 to 0.6 (see Figures 3-5), and these values are typical for low-molecular-weight alcohols in inert solvents. Looking at Table I we see that the predicted maxima for GE reach up to experimental maxima only if the chain length r is of the order of 10 or greater. Clearly we cannot expect to describe the alcohol-n-hydrocarbon systems on the basis of a mole fraction chemical model with just dimers and (20) G . Soatchard, Chem. Rev., 44, 7 (1949).
Volume 7.9,Number 13 December 1068
R. W. HASKELL, H. B. HOLLINGER, AND H. C. VANNESS
4538 ~
x1
Table I: Estimation of Chain Length from the Maximum of the GE Curve Chain length, r (GE/RT)m,,
2 4 6 8 10
0.223 0.387 0.459 0.501 0.529
Chain length,
0.400 0.427 0.442 0.450 0.456
15 21 50 100 m
(ZA? st ( G ~ / R T ) ~ max ~ ~
0.571 0.594 0.647 0.668 0.692
0.466 0.472 0.485 0.493 0.500
+
IV. The Two-Constant Model We have applied the two-constant model to the data of Van Ness and coworkers by determining the two equilibrium constlants from GE data. The fitting procedure is based on iteration equations for solving two simultaneous nonlinear equations in two unknowns.21 The procedure can be described as follows. We start with eq 16 which can be combined to form a cubic equation for the monomer mole fraction y = Xlf
+ [(K3' + (1
K2)XA'
+ 2(K3
+ 2K3X~')y -
K2)]y2
XA' =
0
+
(17)
The analogous equation for the initial monomer composition is a quadratic equation which has one physically significant root The Journal of Physical Chemistry
+ K3 - [(1.0 + K3)2 - 4(K3 - K2)]'/2 2.O(K3
Compn
trimers and other small polymers. Furthermore, if the k, values are not infinite, as assumed for Table I, then the maximum chain length r must be even greater than indicated in order to bring GE up to the level of experimental values. Thus it appears that we may as In view of this result we are well let r go to infinity. led to reconsider the continuous-association model. The symmetry of Redlich and Ester's model can be destroyed by altering the relation among the equilibrium constants k,. Redlich and Kister used only one equilibrium constant as an adjustable parameter, and the introduction of additional parameters could certainly improve the model. On the other hand, this approach is fruitless unless there is a drastic improvement for a modest increase in the number of parameters. We have found that the continuous association model with two independent equilibrium constants, k2 and k3, is very successful in accounting for some of the properties of the ethanol-heptane system. It is convenient here to deal with the constants K , for the reactions A1 A,-1 k s A,. We treat Kz = kz and K3 = k3/k2 as fitting parameters and assume that K , = Ks, j > 3. There are good physical grounds for separating Kz and K3 even if the other K , terms are not independent. The evidence will be discussed following the development of the two-constant model.
K3(K3 - K2)y8
=
1.0
Compn ( Z A ~a t max
r
i
- K2)
(18)
The mole fraction xgf can be determined from xlf, K2, and K3. Thus the formula for GE
GE(x~O,Kz,K3) = XAO
+
In 7 X1f $1 $Ao
XBO
""1
In -
(19)
XB'
can be used to calculate GE at a particular composition from K2 and Ka. Alternatively, experimental data for GE at two different compositions provides a determination of K Z and Ka. This is accomplished, following Singer, by introducing qa(K2, K3) = ( E T ) - [GE(&, K3, ZAP)
- GEil
(20) where the index i refers to a particular composition XAO, and GEi is the corresponding experimental value for GE. The equations to be solved are g1 = 0 and g2 = 0. The iteration formulas are
Kz' = K Z - D-'[g23g, =
K3
g13g~I
+ D-l[gzzgi - gizgzl
(21) where g2, is the derivative of gr with respect to K,, and D is the determinant of the g, matrix. More detailed formulas and the iteration program are given elsewhere.22 It was found that the iteration procedure outlined above converges only for solutions containing less than 50 mol % alcohol. Values obtained for K 3 were not sensitive to the composition pairs, but values for K2 were fairly sensitive. The following criteria were imposed in order to assess the reliability of the K , values. (1) The values of K2 and K3 inserted into (19) must generate the GE data. (2) The temperature dependence of the K j values must satisfy the van't Hoff relation. (3) The K, values together with the formation enthalpies, derived from the temperature dependence of the K g ,must provide a good fit of the AHmdata. It was possible to find K, values that satisfied all of the requirements mentioned above, as illustrated in Figures 3-5. The following equations give the adjustable parameters K3l
H,(T)
=
H,(To)
+ aj(T - T o )
(22)
&(To) = 53.88, HZ(T0) = -8850 cal, and a2 = 7.50 cal deg-l; &(To) = 76.80, Ha(T0) = -5500 cal, and a3 = (21) J. Singer, "Elements of Numerical Analysis," Academic Press, New York, N. Y . , 1964. (22) R. W. Haskell, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, N. Y., 1967; available from University Microfilms, Ann Arbor, Mioh.
THECHEMICAL MODELAS APPLIEDTO ASSOCIATED LIQUIDSOLUTIONS -3.83 cal deg-'. TO is 288.15"K. H,is the enthalpy change for the reaction AI A+1 = A,. Enthalpies and equilibrium constants are listed in Table I1 for
+
Table I1 : Two-Constant Model Fitting Parameters HI,
T,
kcal/mol
OK
Ka
Ka
348.15 338.15 328.15 318.15 308 15 298.15 288.15
4.00 5.74 8.44 12.77 19.92 32.15 53.88
14.20 18.13 23.44 30.76 40.99 55.58 76.80
I
- 8400
-8475 -8550 -8625 8700 - 8775 8850
-
Ha, kcal/mol
- 5730
-5691
- 5653 - 5615 - 5577
- 5538 -5500
several temperatures. The temperature dependence of the enthalpies is quite small, but recognizing this temperature dependence leads to substantial improvements in fitting the model to the data. The value obtained for Ha suggests that one hydrogen bond is formed in the reaction. The value for the temperature coefficient a3 is reasonable if it is interpreted as a loss of heat capacity upon the formation of a bond. I n the case of Hz, the value obtained suggests that the dimer is predominantly double bonded (see Figure 6a). This interpretation is supported by the increase of H2 with temperature (positive az), which can be attributed to an increasing contribution from single-bonded dimers which would be expected at higher temperatures. The predominance of double bonds in dimers is also indicated by the values for the entropy changes S2 and 53 associated with the formation of dimers and the extension of higher polymers. The entropy changes can be calculated from
-RT In K j = Hj
- TSj
(23)
We find the respective values for T , SZ (eu), and Sa (eu): 288, -22.7, -10.5; 348, -21.3, -11.2. These
R-0.-
,"% .'O-R
% .
(0)
.-H
/
4539
values are reasonable when interpreted as entropy changes for the formation of single and double bonds.
V. The Alcohol Structure We have found that a two-constant model can account for the thermodynamic properties of ethanol-nheptane mixtures over a substantial range of compositions and temperatures. The temperature dependence of the equilibrium constants suggests the existence of double-bonded dimers and higher polymers held together primarily by single bonds. I n this section we will briefly review evidence from other sources concerning the structure of the alcohols. There are considerable infrared data13-'6 for dilute alcohol solutions which show the existence of three absorption peaks corresponding to the hydroxyl stretching frequency of three apparently different hydroxyl environments or spectral species. These spectral species and their spectral characteristics are listed in Table I11 for ethanol. The important point in the present
Table I11 : Ir Spectral Species Observed in Dilute Alcohol Solutions
Spectral species
Wave number, cm -1
Monomer Dimer Polymer
3630 3500 3350
discussion is the appearance of a dimer peak. It probably represents a cyclic structure (see Figure 6a) as argued by Van Ness, et uZ.,'~ since if it were an openchain structure, it would presumably contribute to the spectral monomer and polymer absorption peaks. Further infrared evidence indicating the existence of small cyclic structures comes from the matrix isolation studies of Van Thiel, Becker, and Pimentel.lO In this work a gas mixture of N2 and methanol was condensed onto a silver chloride window a t 20OK. Several spectral species were observed as shown in Table IV. The designation of the various peaks involves interpretations, but it seems clear that these peaks represent
CYCLIC DIMER Table IV : Ir Spectral Species Observed in Matrix-Isolation Studies
(b) LINEAR POLYMER Figure 6.
Structures of some alcohol species.
Spectral species
Wave number, cm -1
Monomer Dimer Trimer Tetramer Polymer
3660 3490 3445 3290 3250 ~~~
~~~
Volume 72,Number 19 December 1968
R. W. HASKELL, H. B. HOLLINGER, AND H. C. VANNESS
4540 short-chain structures in which the hydroxyl environment is different from that of a hydroxyl group participating in a linear-chain structure. I n agreement with the above conclusion is the evidence available from nuclear magnetic resonance spectra.l7-l9 Here it has been possible to determine the dimer bond energyz8 by finding (~/XAOX~O)~.~=~ a t infinite dilution of the alcohol for several temperatures. For methanol and ethanol in CCl,, the Hz values were found to be -9.4 and -7.6 kcal/mol, respectively.’* Some of the best evidence at present concerning the structure of the alcohols comes from dielectric constant measurements. Such measurements allow determination of Kirkwood’s correlation parameter,24g. For the alcohols, we may represent g approximately by the formula g = 1
T2
4.01
sol.
’Ti
T,
Figure 7. The correlation parameter for alcohol-inert solvent solutions.
20.0 -1
//
+ 2C(cOs r)j
(24) Here (cos r)r represents the average cosine of the angle between a given dipole and its j t h shell of neighbors. For the pure alcohols experimentally determined g values are in the range 2.8-3.2. Kirkwood found a g value of 2.6 for an infinite chain. I n his calculation, rotation is allowed about the bond but bending is not. Figure 6b shows the type of structure assumed for the linear polymer upon which Kirkwood’s calculation was based.26 Experimental results10-12 for solutions of an alcohol in an inert solvent give the typical g vs. xA0 curve shown in Figure 7. The value g = 1.0 a t infinite dilution means that there is no correlation between the dipole moments of the alcohol molecules; Le., the solution consists of monomers so greatly separated that the individual dipoles are randomly oriented. However, the g value quickly becomes less than unity with increasing alcohol concentration. According to (24), this can happen only when there is a tendency for the dipoles to be in an antiparallel orientation. From a chemical point of view this indicates the presence of cyclic structures. As the alcohol mole fraction approaches 0.2, g approaches unity. This rise in g must result from an increase in the ratio of open-chain to closed-chain structures. At XAO E 0.2, g is again unity, and there must be a rough equivalence between the number of open- and closed-chain structures. I t is interesting to notice that the chemical models predict average chain lengths of 2-6 as the temperature varies from 348 to 288°K at this composition. (See Figure 8.) These numbers are probably indicative of the size of cyclic structures to be found in the pure alcohol. As the alcohol concentration exceeds 0.2, the correlation parameter exceeds unity, indicating a predominance of open-chain structures. This predominance apparently reaches its maximum in the pure alcohol. It is interesting to note that lowering the temperature causes g to decrease in the low-concentration region and inThe Journal of Physical Chemistry
Figure 8. Theoretical chain lengths based on the 2K model for ethanol-heptane solutions.
crease in the high-concentration region. Apparently both open- and closed-chain structures become more stable. The thermodynamic, spectroscopic, and dielectric evidence leads us to the following description of the alcohol structure. Near the boiling point it probably consists mostly of double-bonded dimers and open chains of, say, 10 segments on the average. At lower temperatures the average chain length increases and the cyclic size increases. At 288” the former may be 20 or 30, and the largest cyclic structures may be hexamers. We do not mean to imply by this that there are no small linear structures (2-6 segments) or no large cyclic structures (greater than 6 segments). However it would appear that the values of Hz and S2, together with the correlation parameter values at low concentrations and also the above-mentioned spectroscopic evidence, strongly favor a large predominance of double-bonded dimers over the linear variety. As the number of segments becomes larger, entropy considerations make the existence of higher cyclics less likely. Average chain length considerations together with the composition value a t which the correlation coefficient begins to exceed 1 would seem to indicate that cyclics beyond 6 segments are of insufficient concentration to affect the (23) The slope of the chemical shift 8 (Le,, the shift in the proton frequency measured from some standard such as a proton resonance in the methyl group) 9s. composition at infinite dilution is obtained by extrapolating the measured shifts to infinite dilution. (24) J. Kirkwood, J . Chem. Phys., 7 , 911 (1939). (25) L. Pauling, “Nature of the Chemical Bond,” Cornel1 University Press, Ithaca, N. Y.,1945.
THECHEMICAL MODELAS APPLIEDTO ASSOCIATED LIQUIDSOLUTIONS
MOLE FRACTION
VOLUME FRACTION
Figure 9. Two-dimensional representations of lattices for mole fraction and volume fraction statistics.
alcohol properties measurably in the temperature range studied. Indeed, the thermal properties of ethanol seem to be well represented by just allowing the dimer equilibrium constant to be different.26 Thus it seems proper to conclude that the double-bonded dimer is the major cyclic at all temperatures but that dielectric constant values would require the inclusion of a few higher cyclics in the low-temperature range.
VI. Mole Fractions and Volume Fractions In this section we will compare the descriptions of the chemical models which are based on mole fraction statistics with those based on volume fraction statistics. The basic difference between the two types of descriptions can be illustrated by the cell diagrams in Figure 9. In both diagrams there are solvent molecules B, one monomer unit of alcohol AI, one dimer A2,and one tetramer Ad. Notice that the mole fraction description attributes one cell to anyj-mer whereas the volume fraction description attributes one cell to each formula molecule. If we imagine these species to be in chemical equilibrium, associating and disassociating, we can make the following comparisons. At infinite dilution of component A, only monomers are present; hence the two descriptions will be nearly equivalent. (If the molar volumes of A and B are equal, the descriptions are identical.) At infinite dilution of component B the two descriptions will be greatly different. An example of the difference between the descriptions appears in the determination from the model of the limiting value of A H m / x ~ O x ~at0 xAO = 1. This is the right-hand intercept in Figure 1. The value is not so easily found as in the case of the left-hand intercept, which is given by (13). The value of the right-hand intercept depends on the details of the model and the relations assumed for the k j values. On the other hand, the cell diagrams provide a means of approximating the intercept value. We assume that the intercept represents the enthalpy change for addition of a solvent molecule to a lattice containing only component A. I n the case of the mole fraction description, this addition would nearly always prevent an associative reaction from occurring. I n the case of the volume fraction description, this addition would only prevent about 1 out of J associative reactions from occurring since, roughly speaking, the solvent molecule will only be in the neighborhood of a reacting site ( L e . , end of chain neighbor) this fraction of the time. Thus
4541
we expect the right-hand intercept to be about the same as the bond energy, - H o , in the mole fraction description, and only a fraction of that value (of order -H o / J ) in the volume fraction description. Exact derivations lead to the formulas listed in Table V. These expressions are all based on the one-constant model, k, = P - I , for the evaluation of the sums in (11). It is interesting to note that the volume fraction relations explain at least qualitatively the appreciable increase of this intercept with temperature as noted by Van Ness and coworkers. Thus, as the temperature increases, the average chain length decreases and the intercept increases. Table V: Right-Hand Intercept Expressions for Heat of Mixing Description
-
(AH/ZAOZB~)~AO1.0
[(l/Ji) - 11HO
Mole fraction Volume fraction Disoriented std state
PJi
Quasi-crystalline std state
(K
+ 1) In ( K + 1) 1
Comparing the predictions shown in Table V with the experimental data, Figure 1, we find that the one-constant mole fraction model greatly overestimates the intercept. The two-constant model is better, as indicated in Figure 1, but it leaves much to be desired. It seems clear that for high alcohol concentrations the mole fraction description is bound to fail because it takes no account of the spatial extent of the associated species. I n this range, the volume fraction description is undoubtedly a more accurate description. However, there are several difficulties with the volume fraction description; these are outlined below. As noted by Wiehe and Bagley,5 even when one allows for infinite association (k,s --+ m), one still finds difficulty in reaching the magnitude of GE found experimentally.27~2sMoreover, when the association constants are large, AHm is small) so that it is not possible (using the same parameters) to fit both AHm and AGE data. This difficulty is resolved by allowing the ratio of the molar volumes to become a fitting parameter or by including a physical interaction term of the Van Laar-Hidebrand-Scatchard form. The latter alternative leads to the models of Kretschmer and Wiebe2b and Renon and Prausnitz.* We have not (26) A. V. Tobolsky and R. E. Thach, J. Colloid Sci., 17, 419 (1962). (27) It is possible that part of the reason for the failure of the volume
fraction model to achieve the proper magnitude is the fact that Flory’s approximation of W in S = k In W overcounts. (28) H. Tompa, “Polymer Solutions,” Academic Press, New York, N. Y., 1956.
Volume 72, Number 13 December 1968
R. W. HASKELL, H. B. HOLLINGER, AND H. C. VANNESS
4542 included a physical interaction term in the two-constant mole fraction model. Both Redlich and Kister2a and Stokes29found that the term does not account for more than a few calories in the mole fraction description. Furthermore, in practice, the term assumes the role of a curve-fitting parameter which takes up the slack between the chemical model predictions and observed data and thus tends to make all models appear reasonable. Another difficulty with the volume fraction description enters because the model is a constant-volume model whereas most data are for constant pressure. Thus in applying the model one should transform the constant ( T , P ) data into constant ( T , V ) data. Only Kretschmer and Wiebe have considered this correction, which is significant with regard to AH" and AS" even though it can be neglected in the case of AGm, Of the more recent volume fraction models we should also mention that of Hwa and Ziegler.a They considered a two-constant volume fraction model with a physical interaction term, but they did not make the correction due to a volume change, as mentioned above. They assumed that the dimer and higher polymer formation enthalpies were the same, and they did not try to satisfy the van't Hoff relation. These restrictions may partly explain the strong temperature dependence of their calculated bond energies for the ethanolmethylcyclohexane system (Le., - 3590 kcal at 308"K, -1779 kcal at 273"K, and -942 kcal at 248°K). In summary, we have used an ideal-solution model of a multicomponent system to describe an alcohol-inert solvent solution. In particular we have determined two equilibrium constants (one for double-bonded dimers and the other for the linear-chain polymers) from the GE data. We have used the van't Hoff relation to describe their temperature dependence and thereby determine the formation enthalpies and the heat of mixing. We have then tried to show that these formation enthalpies are consistent with data from other sources. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
Appendix I Glossary of Symbols for Composition Variables ngo
nBf = ngi
=
n3f,nli
+
nf = nB En,* nA0 = Zjn,f = Zjn,i xAo
=
nm
xjf
=
njf/nf
%A0
The Journal of Physical Chemistru
Moles of solvent (not associated) Moles of j-mer in solution and associated component, respectively Total moles of solution Stoichiometric moles of associated component Stoichiometric mole fraction j-Mer mole fraction in sohtion
xji = n j i / W i = x , (~s ~ o= 1)
Xj =
vj/vB
df =
+ Z~jnji = df (zAo = 1) X,n.'/df 3 1
+Mer mole fraction in pure alcohol Ratio of j-mer to solvent molar volumes
= jP
nBf
di = ZXjnif $if = $ii
+Mer volume fraction in solution j-Mer volume fraction in the associated component Solvent volume fraction
= X.n.i/di 1 3
+ B ~=
na'/df
Appendix Equation 4 follows from (1) and (2). Since (1) represents an athermal mixing process, we may write
(AG")' = -T(AS")'
=
RT(nBoIn xBf
+ Enl' In x,')
Taking the partial derivatives of both sides with respect to nJfgives pif
- p:
=
R T In xlf
Assuming chemical equilibrium, we have jp, = p j which gives (4). This procedure applied to (2) gives
- 1.110 = R T In x;
p:
and similarly for the solvent using (1) we have f
pB
- pB0 = RT In xgf
Appendix I11 We consider the case where the associated liquid contains the species A,, A2, ..., A,, but K + a,so that the following approximations are possible
xBf
+
XXJ' =
xBf f
Xlf =
1
(AI)
These equations can be solved to obtain
xlf = XAO
xAo rxBo
+
(-44)
Now the condition of equilibrium gives p1 = ( l / ~ ) p ,so that G' and G' may be formulated in terms of the r-mer chemical potentials instead of the monomer chemical potentials. Theref ore
so that for 1 stoichiometric mol of solution
(29)
R. H. Stokes, unpublished data,
IONIC POLYMERIZATION UNDER A N ELECTRIC FIELD Now K --t means energy is given by Q)
2 :
+
1 and thus the excess free
4543
+
where c = r / ( r - 1) and'q = 1 [(r - l)/rc]. Equations A3 and A10 allow us to find the solvent activity coeffoient as a function of the chain length. This relation is
At the maximum value of GE the activity coefficients of A and B are equal, so we have This gives US for (GE/RT)m,x the equation Equations A3 and A4 may be combined with eq A9 to give the composition at the maximum. This calculation gives Table I gives (GE/RT)max and XAO for the various values of r.
Ionic Polymerization under an Electric Field. XII.
Living Anionic
Polymerization of Styrene in the Binary Mixtures of Benzene and Tetrahydrofuran by Norio Ise, Hideo Hirohara, Tetsuo Makino, and Ichiro Sakurada Department of Polyner Chemistry, Kyoto University, Kyoto, Japan (Received June 11, 1068)
The anionic polymerization of the lithium salt of living polystyrene was investigated in the binary mixtures of benzene and tetrahydrofuran (THF) a t 25' in the presence and absence of an electric field. The apparent propagation rate constant increased with increasing field strength and dielectric constant. The electric field did not affect the ion-pair propagation rate constant] whereas it increased the k,"K'IP term; k," is the free-ion propagation rate constant and K is the dissociation constant of the ion pair. The observed field effect was larger than that calculated by the Onsager theory on the second Wien effect. This discrepancy is discussed in terms of k,", which was found to increase with increasing field intensity. At a T H F content of 60 vol %, the k," reached a limiting value of -lo5 M-lsec-l when the field was intensified over 3 kV/cm. Evidence is presented which shows that the observed acceleration effect was not caused by the electroinitiation mechanism,
In previous papers, the influence of an electric field on polymerization reactions has been studied. The results have shown that cationic polymerizations can generally be accelerated, and the degree of polymerization of the polymers produced can be increased in the presence of a high-intensity electric field, --B whereas free-radical polymerizations are n o t influenced. Most recently it has been demonstrated that the monomer reactivity ratio in copolymerizationscan also be affected by the field.? These field influences have been shown to coincide with the changes caused by an increase in the dielectric constant of the solvent. This agreement
supports our interpretation that the field effects observed are due to the field-facilitated dissociation8of the (1) I. Sakurada, N. Ise, and T. Ashida, Makromol. Chem., 8 2 , 284 (1964); 95, l ( l 9 6 6 ) . (2) I. Sakurada, N. Ise, Y. Tanaka, and Y. Hayashi, J . Polyn. Sci., Part A-1, 4, 2801 (1966). (3) I. Sakurada, N. Ise, and S. Hori, Kobunshi Kaoaku, 24, 145 (1967). (4) I. Sakurada, N. Ise, and Y. Hayashi, J. Macromol. Sci. Chem., A I , 1039 (1967). (6) I. Sakurada, N. Ise, and Y. Tanaka, Polymer, 8 , 626 (1967). (6) I. Sakurada, N. Ise, Y. Hayashi, and M. Nakao, Kobunshi Kagaku, 25,41 (1968).
Volume 72, Number 13 December 1968
