Sept. 5 , 1964
ST.\TISTIC.\L [CONTRIBUTION FROM T H E
THERMODYN.4MICS O F C H A I N h'IOLECULE
DEPARTMENT O F CHEZIISTRY, STANFORD
LIQUIDS
3515
VXIVERSITY, STANFORD, CALIFORNIA]
Statistical Thermodynamics of Chain Molecule Liquids. 11. Liquid Mixtures of Normal Paraffin Hydrocarbons' BY P. J. FLORY, R. A. ORWOLL,AND A . VRIJ RECEIVED APRIL 2, 1964 T h e partition function and equation of state of the preceding paper are reformulated for mixtures of homologous chain molecules. Thermodynamic properties of mixtures of n-alkanes are interpreted according t o relationships thus derived. T h e excess volumes, always negative for these systems, are readily interpolated from the volumes of the pure components with t h e aid of the reduced equation of state. This scheme is formally equivalent to the Br6nsted principle of congruence a s it applies to t h e volume, T h e theoretical expression for the chemical potential affords an accurate and rational account of experimental results of McGlashan and Williamson for C6-C16 a t 20' and a t 50", and those of van der Waals a n d Hermans for Ci-Czl a t 73' a n d C7-C- a t 73" X single choice in the value of one parameter suffices t o establish agreement between theory and experiment for all of these systems. Previously neglected contributions from characteristics of the pure liquids, a s expressed through parameters determined from their equations of state, are large. T h e contact term-the only one usually considered-is small for these systems. T h e theory correctly predicts a decrease in the excess enthalpy with temperature, its sign ultimately becoming negative a s has been observed. Quantitative comparisons of observed with calculated enthalpies are less satisfactory, owing in p a r t t o the deficiency of equation of state d a t a for t h e n-aikanes. T h e theory accounts in a straightforward way for the appearance of lower critical miscibility among mixtures of lower parafins with higher homologs. L C S T values are calculated for mixtures of several n-alkanes with polyethylene
Mixtures of n-paraffin hydrocarbons have been subjects of some of the most painstaking experiments on solutions of nonpolar liquids. These studies have been concerned with the volume change on mixing,*s3 with the enthalpy of mixing,*s5with the excess chemical and, more recently, with the phenomenon of lower critical miscibility when one component is of low molecular weight and the other very high.Y~'o T h e excess volume of mixin,g is n e g a t i ~ e ~the , ~ ;excess enthalpy is small and positive a t low temperature^,^,^ strongly dependent on t e m p e r a t ~ r eand , ~ recently has been reported t o be negative a t higher temperatures. The excess chemical potentials for mixtures of dissimilar n-paraffin hydrocarbons are negative, but considerably smaller in magnitude than lattice theories would predict.6ss Studies of excess properties of these systems'~s~'2 are monumental demonstrations of the inadequacy of lattice treatments of liquid mixtures. The thermodynamic properties of .mixtures of homologous chain molecules are derived in this paper by extension of the treatment of systems of one component set forth in the preceding paper,I3 hereafter referred to as I. The application of the theory of mixtures of nparaffin hydrocarbons is illustrated through comparisons with representative experimental data. Adaptation of the Theory to Mixtures of Homologs The theory sketched in I concerns a liquid comprising segments of two types: mid-chain segments and ter(1) Presented in p a r t before t h e Division of Physical Chemistry of t h e American Chemical Society, D e n v e r , Colo., J a n . 23, 1064 ( 2 ) A . D e s m y t e r a n d J. H. v a n der X-aals, Rec lrai' chim , 7 7 , 53 (1998). (3) J . D G6mez-IbPRez a n d C -T Liu, J . P h y s . C h r m , 6 7 , 1388 (1963). (4) J . H v a n der Waals a n d J . J . H e r m a n s . R P C LYQV r h i m . , 69, 949 (1950), J . H . v a n der Waals. d i d . , T O , 101 (1951) ( 5 ) 41 L McGlashan a n d K W l l o r c u m , 7'rnns F a r o d o y S o r . , 6 7 , ,581, 007 (1061) J. H van del- IVVaals a n d J J He,-mans. R P C l r a v c h i m , 69, 971 (19.50) J. H vat, der lVaa13, Tivans F a v a d a y S o c , 6 2 , 916 (1956). M L McGlashan a n d A G Williamson, ibrd , 6 1 , ,588 ilR61'. P I Freeman a n d J S liowlinson, P o i y n f r r . 1 , 20 (1960) (IO) A . J D a v e n p o r t a n d J S Howlinson, T i n n s . F a r a d o y S a c , 69, 78 ,196,3), J S I\,,ll
,
\%,
I
-
-.28 \
Excess chemical potentials
are plotted against the mole fraction N? of n-hexadecane in the system n-hexane-n-hexadecane (c6-c16) a t 20' in Fig. 2. The experimental points are from the precise measurements of McGlashan and Williamson, The solid curve has been calculated according to eq. 21 with = 2.4'K., corresponding to b' = 0.40. The agreement is remarkably good. Departures of experimental data from the theoretical curve amount to no more than about 1% in the activity over most of the range in composition. The contribution from the lattice entropy, expressed hy the first bracketed term in eq. 21, is shown b y the dashed curve in Fig. 2 . Other terms in eq. 21, which are the ones contributed by the present theory, are comparable to the lattice excess free energy in magnitude but are opposite in sign. The calculated excess chemical potentials for this system (c6&16) consequently are fairly small. The measure of disagreement with experiment should be judged by the absolute amount of the departure and not by the ratio of observed and calculated values. Experimental results of LIcGlashan and Williamson* for the same system a t 50' are similarly compared with theoretical calculations in Fig. 3 . Also included are data of van der Waals and Hermanse on C,-C32 mixtures a t 7 3 ' . The curves shown have been calculated using t.he same vulues of b' quoted above. The agreement with experiment is comparable to that shown in Fig. 2. Thus, the arbitrary choice of a value Sor only one parameter, b', suffices to fit data for two diverse systems and for t h e same system a t different temperatures. Moreo \'cr, the resulting value of P12 being small, the contrihiition of the contact term governed by it is minor. The volume sensitive term, involving only such param-
r300
0.2
0,6 MOLE FRACTION Nr
04
1 1.0
0.8
Fig. 3.---Excess chemical potentials for the system C 6 - C l na t A O o (McGlashan and \Villiarnson8) and for C,rC1:! a t 73" ( v a n der Waals and Herrnansc) plotted against the mole fraction lir C l f the component of higher molecular weight Solid lines were calculated from e q . 21 x-ith b' = 0.40. T h e dashed line represents t h e lattice contribution for the latter system
eters as are determined by properties of the pure components, is dominant for the systems considered. Conventional interpretations proceeding from the lattice model would prescribe an excess chemical potential consisting of the lattice entropy term in eq. 21 s u p plemented by a contact energy term. Since the latter is small, it alone would raise the dashed curve in Fig. 2 by only a small fraction of the difference between this curve and experiment. If one should choose to supplement the lattice entropy with the observed enthalpy of mixing (see below), the resultant curve for c&16 mixtures at 20' would, to be sure, fall not far below the experimental points. Such agreement as would be achieved is peculiar to this temperature. however. The observed excess enthalpy for this system decreases rapidly with temperature, becoming negative above about G o . Inasn~uch as the excess chemical potential changes little with temperature,B interpretation of the excess chemical potentials in this manner becomes increasingly unsatisfactory for temperatures above 20'. These remarks are by way of illustrating the serious failure of lattice theories alone to account for excess therinodyr~amic properties of these systems. Their deficiencies are abundantly documented. Calculations carried out using the lattice theory i r i the higher approximation of Huggins,IS Orr. Sliller. and (;uggenheixnl: have little effect ( 1 3the agreeiiirrlt attainable, despite the avaiiahilitj- o f a further 1):iraiiieter. namely-. the lattice coordination number 2 .
Sept. 5, 1964
STATISTICAL
THERMODYNAMICS O F CHAIN
The value of Pl2 required to achieve optimum agreement is lowered however. For z = 12, the experimental d a t a require b' S 0.15. van der a'aals and Herman@ also measured activities of n-heptane mixtures with polyethylene a t 108.9'. Their polyethylene sample consisted of a highly branched fraction having a CH3: CH2ratio of about 0.05 and a molecular weight of 26,000. Calculations carried out again on the basis of b' = 0.40 reproduce the experimental d a t a fairly well. T h e significance of this comparison, based on parameters deduced from properties of the n-alkanes, is obscured by the nonlinearity of the polymer component. &;e have not therefore included the analysis of this set of results. The calculated curves shown in Fig. 2 and 3 would not be altered significantly by taking account of the small dependence of b' on the composition of the mixture. T o have performed the calculation in this manner would have required assignment of a numerical value to the ratio se'lsmrwhich can only be estimated crudely from the molecular geometry.
The Excess Enthalpy Ignoring the small difference between enthalpy and energy for a condensed system, we have for t h e excess enthalpy per mole of mixture H~ =
MOLECULE
LIQUIDS
3 5 1!J
Excess enthalpies per mole of mixture calculated according to eq. 25 are compared in Table I with values observed for various systems. All data refer to cornpositions (01 = p2 = Experimental results are from the investigations of van der \\:ads and Hermans4 a r i d McGlashan and Morcum.5 Values of PI* and p2* required for calculation according to eq. 25 were computed from 1-35 wherein the constant a has been assigned the value - 1.5for the express purpose of achieving approximate agreement with the enthalpies of mixing, as previously acknowledged.I3 The required pI2*were computed according to eq. 17 with b' being assigned the same value, 0.30, used for calculating the chemical potentials. Reduced volumes G I and $ 2 v.ere take11from Table I of the previous paperI3; those for 73' were obtained by interpolation from the tabulations for 30 arid 100°. Experimental reduced excess volumes 5 E were used, except for the system CIO--Ca2for which experimental excess volumes are unavailable. 'The required 5" in this instance was calculated in the manner described earlier. TABLE I MOLAREXCESS EKTHALPIES FOR VARIOUS MIXTURES
HY~ROCAKIION
[Eo(mixture) - Eo(1) - Eo(2)I:"
which, in light of eq. 1, can be written H~ =
Zv*(plpl*/fil
+ p2p2*
fi2
-
p*/,ij) (23)
Substitution of eq 15 for p* converts this result to HE =
%Zt*[plpl*(l fi1 - 1/6)
+ p2$'~*(1
52 -
~ l ( C 2 P l S * l it
l/V)
I
+
(2.1)
"
Since 2: is very small, the reciprocal of may be approximated with negligible error by
substitution of which in eq. 24 permits the excess molar enthalpy to be expressed as a sum of three terms as IIE
xo - * -
(p12*i5)p1p*
+
[(El
-
C2),
i'o](p2*,:;2
+ + + - i~z)~'ilI(p2*:'2:* p 1 * / ' i ' l ) ~ l+ ~ 2(cE; z;2j(plpl*+
Pl*
5l)Crl(F2
(p12*. 2:)Pl(C2
[i~E~(5")"1(p1p1*p*p2*)
-
(25)
[(ill
p2p2*) (26')
The first term in eq. 25, or in 2 5 ' , represents the contribution to the excess enthalpy from change in contact pairs upon mixing. This term should be positive, and such is observed to be the case. The second term takes account of the effect on the enthalpy arising from the difference in the volumes available to segments in the two pure components. I t vanishes for $1 = 52, but does not depend on the change in volume on mixing. Depending on values of the parameters, it could conceivably be either positive or negative. For the n-paraffin hydrocarbons it is decidedly positive. The third term accounts for the effect of the volume change on mixing. I t varies directly as 5', and since itE< Ofor the paraffin hydrocarbon mixtures, its contribution is negative. Three such contributions to the excess thermodynamic properties were recognized by McGlashan, Morcum, and M-illiamson.i 2
The relative magnitudes of the three ternis in eq. 25 calculated in this way for the Ce-C16 system :it 20" are illustrative. They are 9.4, 50.5, arid -31.:) cnl. niole~! , respectively. The first is comparatively small, arid the latter two are of opposite sign. The second tefni is sensitive to the d i f e r e n c e s between the reduced \~olunies $ 1 and C 2 and also to the difference between pl* and p>*. Although the former are obtained with rc;tsona!,le accuracy from the thermal expansion coefficients (see Ij, their difference is disturbingly sensitive to s n ~ a l lerrors in the a ' s . The @* are subject to large errors steir:ming from inaccuracies in existing compression tlaca, which we have attempted to circumvent by exercising an arbitrary choice of the parameter (L. lloreover. i c e assume implicitly that the constants Po* and a i n ey 1-35 are independent of temperature. The third term amounts to a t least half of the sum of the first two. Hence, the final result is precariously dependent also on the values of PI* and p?* and V '. The theory correctly predicts a decrease in the excess enthalpy with temperature and it provides a n approximate correlation of the enthalpies of mixing Cor diiferent pairs of n-alkanes. The calculated change with temperature for the C6-CI6system is less than that 01)served. A n error in either 5 or p * for n-hexane a t the higher temperature would account for this most seri,::is discrepancy between calculated and observed result 5 In any case, the theory correctly depicts the dirrctior. :)f changes in the excess enthalpy. I t may be quantit:itively inaccurate in its account of their magnitudes. I t will be observed from figures quoted above t h a t t h y contact term amounts to less than 20Sc o f the vcc:ricl
P.J. FLORY, R. A. ORWOLL, AND A. VRIJ
3520
term for the Ca-Cla mixtures. This term is the only one normally considered in the treatment of the excess enthalpies of solutions. It alone could account for the change of sign of H' only by ad hoc identification of b' as a free energy parameter'? whose entropic contribution is very large, thereby rendering b' strongly dependent upon temperature. According to the present treatment, the excess enthalpy for n-paraffin mixtures arises in a natural way from characteristics of the pure liquids. ,.It depends only to a minor degree on the contact energy between unlike species or segments.
Lower Critical Solution Temperatures (LCST) The series expansion of the chemical potential as given by eq. 21 can be written (111
- lllO)/KT =
-(XI/XZ)(OZ -
(1
