Thermodynamic Properties of Liquids
1317
three tin tetrahalides by tetrahydrofuran were determined under identical conditions (0.2 M SnX4 0.2-0.3 M T H F / 1000 g of benzene). I t appears from Table I1 that the equilibrium constant increases with decreasing size and increasing electronegativity of the halogen atom. The increasing electronegativity decreases the electron density on the central Sn atom (as shown by the 6 values) and promotes the formation of a donor-acceptor bond between the tin and solvent.
+
References and Notes ( 1 ) U. Mayer and V. Gutmann. Monatsh. Chem., 101. 997 (1970). (2) F. Gaizer and M. T. Beck, J. lnorg. Nucl. Chem., 29, 21 (1967). (3) A. Vertes, S. Nagy. I. NagyCzakb. and E. C&kvti. J. Phys. Chem., 79, 149 (1975). (4) L. G. Sillen, Acta Chem. Scand., 10, 186 (1956). (5) N. Bjerrum, 2:Anorg. Chem., 119, 179 (1921). (6) B. Cshkvary, E. Cshkvty. P. Gomory, and A. Vertes, J. Radioanal. Chem., 25,(2), 275 (1975). (7) V. Gutmann, "Coordination Chemistry in Not?-Aqueous Solutions". Springer Verlag, New York, N.Y.. 1968. (8) The donicity (DN) defined by Gutmann is: the negative reaction heat of solvation of SbCI5 by a donor solvent.
Acknowledgment. The authors wish to express their thanks to Professor M. T. Beck for reviewing the manuscript. Thanks are also due to Dr. S. Nagy who helped in the computer evaluation of the results and to Mrs. Suba for her assistance in the recording of the spectra.
DN = -AH
Thermodynamic Properties of Liquids, Including Solutions. 12. Dependence of Solution Properties on Properties of the Component Molecules' Maurice L. Huggins 135 Northridge Lane, Woodside, California 94062 (Received January 7, 1976)
The fundamental assumptions and equations of the author's recent theory of solution properties are reviewed for specific application to binary mixtures of simple molecules, each treated as a chemically uniform includes three meaningful (monotonic) compound. The theoretical equation for the excess enthalpy (HE) parameters, each assumed to be independent of concentration. For five systems for which accurate experimental data have been published, it is shown t h a t the calculated H Ecurves agree accurately with the experimental ones. The calculated HEparameters for these and ten other systems are listed. The theoretical equation for the excess entropy of mixing (S") involves the ratio of the molal volumes of the components, plus three other meaningful parameters. One of these can be estimated. The other two can then be calculated from the differences between the G E and H Ecurves. T h e magnitudes of the five contributions to the Gibbs energy of mixing are compared.
1. Introduction During the past few year^^-^ I have been trying to relate the thermodynamic properties of solutions, using theoretically reasonable equations, to parameters that measure properties of the component molecules and their interactions. Most of this research has been concerned with polymer solutions. In this paper I treat binary mixtures of simple nonpolymeric molecules. One of my aims is to test the basic equations and assumptions of the theory, using the very accurate data on the concentration dependence of the excess enthalpy and excess Gibbs energy now available for many such mixtures. Another aim is to deduce the parameters in (and magnitudes of) the four contributions to the excess Gibbs energy. 2. Theoretical Approach
2-6
I make the approximation that the change in energy when two liquids are mixed is the change in the sum of the interaction energies between close-neighbor molecules, assuming in effect that the change in the sum of the interaction energies between non-close-neighbors is negligible.
I assume that the actual energy changes are the same as for a hypothetical mixture in which the molecules have surfaces making mutual contacts, with a constant (concentrationindependent) interaction energy per unit area of contact for each type of contact. For such a solution the relative areas of contact of the three types must be those that give the lowest total Gibbs energy. This requirement is equivalent to requiring that the total areas of contact ( u ) of the three types be related to an equilibrium constant by the equation
I tentatively assume that, for each type of molecule, the average molecular surface area that makes contact with other molecular surfaces does not change with concentration. T h e data now used are all for a temperature of 25 "C and a pressure of 1 atm. I assume that the excess enthalpy (HE) and excess energy (E") are equal, since the difference (PV'? between these functions is negligible a t this pressure. The Journal of Physical Chemistry, Vol. 80, No. 72, 1976
Maurice L. Huggins
1318
With these assumptions and approximations, I have deduced2J t h e relation
measures the change of energy, for a contact area (ul0) equal t o the contacting surface area of a type 1 molecule, when like contacts are replaced by unlike contacts. z 1 and 2 2 are “contacting surface fractions”, given, for a mixture of monotonic (chemically uniform) molecules, by the equation (4) 0.2
where r n is the ratio of the contacting surfaces of the two kinds of molecules:
r,, = a2°/u10
(5)
04
0.8
06
1
x2
Figure 1. Theoretical curves for excess molal enthalpies, for c~ = lOOJ:(l)K= l , r n = l ; ( 2 ) K = 5 , r n = 1;(3)K=0.2,r0= 1;(4)K= 1, r,, = 2; (5) K = 1, r, = 0.5.(From ref 5, by courtesy of hFederation
of Societies for Paint Technology.)
g~ is a factor given by t h e equation n
where
K’ = 4
(i
- 1)
(7)
For perfectly random mixing of the molecules, K and g~ both reduce to unity and eq 2 reduces t o
H” thus depends on three meaningful parameters: cA, r,,, and K. Figure 1 shows the theoretical dependence of H Eon these parameters. T h e curves were all drawn for t h e same energy parameter, 6 1 . Changing this parameter would merely alter t h e scale of t h e figure. For curves 1-3 t h e contacting surface ratio was also assumed constant, equal to 1.These curves are symmetrical, but differ in fatness because of differences in the assumed equilibrium constant, K . Curves 1,4, and 5 were all drawn for K = 1, signifying random mixing. They differ in skewness, because of differences in t h e contacting surface ratio, r,,. With the ratio greater than 1,the curve peak is at a n x . value ~ less than 0.5. The three parameters are readily calculated from accurate measurements of H E as a function of the concentration. I customarily use H” a t x 2 = 0.3,0.5, and 0.7, calculated from t h e experimenters’ smoothing equations. Rounded values of t h e parameters for 15 mixtures are listed in Table I, with the standard deviations of t h e experimental points from the experimenters’ smoothing curves and from the theoretical curves. Similar data for 46 binary systems are given in another paper.7 T h e theoretical equation usually gives practically as good agreement with the experimental points as does the smoothing equation, even though the theoretical parameters were not chosen to give the best overall agreement. H E curves for six systems are shown in Figure 2. They are both the experimenters’ smoothing curves and the theoretical curves, since the differences between the two are less than the widths of the lines. The assumptions can also be tested by calculating (from the experimental d a t a ) one of the three parameters (e.g., r,) as The Journal of Physical Chemistry, Vol. 80, No. 12, 1976
Figure 2. Excess enthalpies: (1) benzene -I- cycl~hexane;~ (2) cyclohexane 4-t ~ l u e n e ;(3) ’ ~cyclohexane 4- n - h e ~ a n e ;(4) ’ ~ cyclohexane CCI4;l2( 5 ) benzene CC14;8(6) benzene toluene.1° The component named first is component 1. (From ref 6, by courtesy of Butterworths.)
+
+
+
a function of concentration, assuming the other two parameters t o remain constant. Table I1 shows t h e results of such calculations for six systems. The calculated r n values are very constant, with appreciable deviations only a t t h e extreme concentrations, where the experimental smoothing curves, used for the calculation, are relatively unreliable. Three types of comparison between the experimental measurements and t h e results of calculations using the theoretical equations thus show excellent agreement. T h e theo-
Thermodynarnlc Propertlee of Llqulde
1310
TABLE I: Enthalpy Parameters and Standard Deviations ParametersC Componentsa
K
CA
Benzene t CC4 cyclohexane toluene cc14 t OMCTSb Cyclohexane t CC14 cyclopentane n-hexane toluene Cyclopentane t CC4 cis-decalin t rans-decalin tetrachloroethylene Toluene m-xylene o-xylene p-xylene
+
DeviationsC Smoothing eq
Theor
Ref
476 3137 250 458
0.75 0.87 1-08 0.90
1.09 1.12 1.16 2.68
0.06 0.16 0.28 0.2
0.08 0.26 0.26 0.2
8 9 10 11
708 84 685 2265
0.93 1.15 0.85 1.08
0.91 1.79 1.89 1.18
0.14 0.07 0.09 1.41
0.13 0.08 0.47 0.97
12 12 13 14
343 -546 -1121 931
0.94 1.02 1.03 0.95
0.89 0.89 0.92 1.05
0.14 0.49 1.01 0.33
0.13 0.49 1.01 0.14
12 15 15 16
161 185 67
1.04 0.99 1.02
1.11 1.03 1.28
0.09 0.15 0.10
0.09 0.16 0.11
17 17 17
The component named first is component 1. OMCTS = octamethylcyclotetrasiloxane. are in joules per mole.
The e_\ parameters and standard deviations
TABLE 11: Dependence of rs on Concentration x2
+ +
Benzene CC14 Benzene + cyclohexane Benzene toluene CC14 + cyclohexane CC14 OMCTS Cyclohexane cyclopentane
+
= 0.1
1.084 1.126 1.147 1.106 2.683 1.793
+
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.086 1.127 1.148 1.098 2.685 1.789
1.086 1.126 1.148 1.095 2.685 1.788
1.086 1.126 1.148 1.095 2.686 1.788
1.086 1.126 1.148 1.095 2.685 1.788
1.087 1.127 1.148 1.095 2.685 1.788
1.086 1.126 1.148 1.095 2.685 1.788
1.079 1.123 1.148 1.095 2.648 1.794
1.038 1.105 1.148 1.094 2.543 1.845
retical assumptions are thus amply justified, at least for solutions of the sort considered here.
3. Excess E n t r o p y 4 T h e excess entropy is in many cases even more important than t h e excess enthalpy. Many years ago18-22 I derived a n equation for t h e combinatorial entropy of a solution-that concerned with t h e randomness of placing the molecules in t h e total volume of t h e solution. FloryZ3independently derived an equivalent relation. We obtained:
AS, = -R(xl In 41 t x p In 42)
(9)
where $1 and 4 2 are volume fractions: x1
41 = 1 - 4 2 = XI
+ rvx2
where (10)
Here, rv is the ratio of the molal volumes: rv = VdV1
of flexible chain molecules, but I have since4 shown t h a t the same relation should apply to any mixture of two components having different molal volumes. In our derivations, Flory and I both assumed perfectly random mixing of the molecules. Actually, if the equilibrium constant K is not exactly one, there will be a preference for each molecule to have either like neighbors (if K < 1) or unlike neighbors (if K > 1).(See eq 1.) To allow for this preference, I have derived the equation
(11)
u* = X l U 1 *
+ xpu2*
(14)
and u1* and up* are contacting surface areas per molecule, in a hypothetical solution in which each contact is made independently. Provisionally, I set
For equal molar volumes of t h e components the volume fractions become mole fractions, hence AS, equals A S R ~ , , , , ~ ~ , which leads to Raoult's law. The excess combinatorial entropy is thus the relative volume excess entropy: (see eq 5 ) and estimate the smaller of the two constants ( U I * and up*) to have the value 6. This estimate is based on the fact SrvE = -R[xl In (41/xd t x z In ( ~ z / x z ) ] (12) that, in a liquid consisting of close-packed spherical molecules, Our derivations were specifically for very dilute solutions each molecule contacts 12 others. I guess t h a t t h e entropy The Journal of Physical Chemistry, Vol. 80, No. 12, 1976
Maurice L. Huggins
1320
TABLE 111: Entrow Parameters and Gibbs Energy Deviations Components a
+ +
Benzene CC14c Benzene cyclohexaned CC14 OMCTS Cyclohexane + CC14@ Cyclopentane + CC14f
+
rv
Ol*
km,1
1.09 1.22 3.21 0.89 0.98
6 6 6 6.57 6.73
0.535 0.424 -0.449 0.099 0.131
I G smE - G calcdq av’
kor,2
-0.400 0.453 1.335 0.058 -0.062
0.07 0.06 0.10
0.10 0.01
a The references are to the experimental Gibbs energy data. The excess Gibbs energies are in joules per mole. References 24 and 25. References 26 and 27. e Reference 29. f Reference 30.
TABLE IV: Contributions to the Gibbs Energy of Mixing for Equimolal Mixtures * Components
+ +
Benzene CC14 Benzene cyclohexane CC14 + OMCTS Cyclohexane + CC14 Cyclopentane + CC14 a
HE
-TSrvE
-TSccE
-TSorE
GE
115
-2 -12
11
-42 -475 102 -93 -46
315 -133 70 34
799 163 166 80
-401 -4
3 2 1
0
0
82
- TS~~ouit -1718 -1718 -1718 -1718 -1718
AGM
-1636 -1403 - 1852 - 1648 - 1684
The units, for all columns, are joules per mole.
would be the same as for a hypothetical liquid with half as many independent contacts. For solutions in which t h e departure from perfect randomness is not very large, this estimate is probably sufficiently accurate. In any molecular liquid there is a n entropy contribution related t o t h e randomness of instantaneous location and orientation of each molecule (and each rigid segment of a flexible molecule). This randomness is obviously a function of t h e close-neighbor environment. Using this concept, I have derived, for the orientational, vibrational, and rotational contribution to t h e excess entropy:
See ref 4,eq 22 (with nN:! replaced by x p ) and 24 (with k,‘ replaced by kljr,2). T h e parameter kljr,l (or kor,2) measures the change in orientational, vibrational, and rotational randomness of a type 1 (or type 2) molecule, when its close neighbors change from all of the same type to all of the other type. In dealing with polymer solutions I have, probably justifiably, neglected the , it unimportant relative to term containing k , , , , ~considering that containing hor,2. My recent calculations, however, show that for mixtures of simple molecules both terms should be included. In addition to kljr,land kor,2,eq 16 contains the parameters r,, and K . (See eq 4 , 6 , and 7.) If accurate experimental values of G” and HE for two or more concentrations are available, a t those concentrations can be calculated from t h e relation (17) Then, using eq 16, hor,l and kc,r,2 can be computed. In the calculations here reported, SorEwas calculated by eq 17 for three concentrations ( x 2 = 0.3,0.5,0.7)and t h e two parameters calculated to give accurate agreement a t x 2 = 0.5, with kor,1 having the average of the values giving agreement a t x p = 0.3 and 0.7. Table 111 includes k,lr,l and kC,,,2parameters, obtained in this way, for five systems. Discussion of their magnitudes will be postponed until after they have been computed for more systems. The Journal of Physical Chemistry, Vol. 80,No. 12, 1976
4. Gibbs Energy
Using t h e equation (18) I have calculated G E a t mole fraction intervals of 0.05. Table I11 gives t h e average magnitudes of t h e differences between G E so calculated and G E calculated from the experimenters’ smoothing equations (extrapolated from higher temperature data, in t h e cases of the first, second, and fourth systems listed). The averages are for 18 concentrations, x 2 = 0.5 being excluded, since agreement was assumed at that concentration. Considering the probable errors of the H E and G E smoothing equations, t h e average deviations are as small as could be expected. Table IV lists the (rounded) contributions to t h e Gibbs energy for equimolal mixtures. At other concentrations the relative contributions of the different terms are not, in general, very different. Note especially the large contribution of t h e term (-TSrvE)correcting for the relative volumes of t h e components in the one system in which rv differs greatly from one. Also note that the contributions of t h e orientational, rotational, and vibrational term (-7’Sljr)are of the same order of magnitude as the excess enthalpy terms. If the parameters were all independent of temperature, one could easily calculate J G M as a function of temperature and concentration. Because of the relatively large contribution of the Raoult’s law entropy term, containing no parameters other than the temperature and concentration, G values so calculated would probably not he far from the true ones for these systems. For greater accuracy, one must await the experimental or theoretical determination of the dependence of the various parameters on temperature. Work on this problem is in progress. Experimental measurements (of varying degrees of accuracy) of H E and G” as functions of concentration and temperature have been reported in the literature for many systems. Applying the equations and procedures of this paper to these data will lead to a better knowledge and understanding of the dependence of the Gibbs energy (and properties deducible therefrom) on temperature, concentration, and meaningful molecular and intermolecular properties than has previously been possible. With this knowledge and under-
Generalized Integral Equations of Classical Fluids
standing for many systems, we shall be better able t o predict parameters, and so AG', for other mixtures. 5. Conclusion I have shown t h a t certain theoretical equations, based on a reasonable model for binary solutions, are in quantitative agreement with very accurate experimental excess enthalpy data. I have shown t h a t these equations, with others for t h e excess entropy, also lead to good agreement with experimental Gibbs energies of mixing. The parameters in the equations are all reasonably related to molecular and intermolecular properties. Activities, vapor pressures, solubilities, and other measurable thermodynamic properties of solutions are of course readily deducible by rigous equations from t h e Gibbs energies of mixing.
Acknowledgment. I gratefully acknowledge my indebtedness to the scientists whose fine experimental data I have used and t o those with whom I have discussed some of t h e theoretical aspects of my theory. I also acknowledge t h e fact t h a t some of t h e theoretical concepts have previously been used by others. Some pertinent references to this have been given in earlier papers in this series. Finally, I express my gratitude to the Paint Research Institute for some financial assistance. References and Notes (1) Presented at h169th National Meeting of the American Chemical Society, Philadelphia, Pa., April 1975. See Am. Chem. Soc.. Div. Org. Coat. Plast. Pap., 35, 283 (1975). (2) M. L. Huggins, J. Phys. Chem., 74, 371 (1970).
1321 (3) M. L. Huggins, Polymer, 12, 389 (1971). (4) M. L. Huggins, J. Phys. Chem., 75, 1255 (1971). (5) M. L. Huggins, J. Paint Techno/., 44, 55 (1972). (6) M. L. Huggins in "International Review of Science, Physical Chemistry. Series Two, Volume 8, Macromolecular Science", C. E. H.'Bawn. Ed., Butterworths, London, 1975. (7) Fourth InternationalConference on Chemical Thermodynamics,Montpellier. France, Aug 27. 1975. (8)R. H. Stokes, K. N. Marsh, and R. P. Tomiins, J. Chem. Thermo@m., 1, 21 1 (1969). (9) M.B. Ewing, K. N. Marsh. R. H. Stokes, and C. W. Tuxford, J. Chem. Thermodyn., 2,-751 (1970). (10) J. M. Sturtevant and P. A. Lyons, J. Chem. Thermodyn., I , 201 (1969). (11) K. N. Marsh and R. P. Tomlins, Trans. Faraday SOC.,66, 783 (1970). (12) M. B. Ewing and K. N. Marsh, J. Chem. Thermodyn., 2, 351 (1970). (13) K. N. Marsh and R . H. Stokes, J. Chem. Thermodyn., 1, 223 (1969). (14) A. E. P. Watson, I. A. McLure, J. E. Bennett, and G. C. Benson, J. Phys. Chem.. 69, 2751 (1965). (15) D. E. G. Jones, I. A. Weeks, and G. C. Benson. Can. J. Chem., 49, 2481 (1971). (16) J. Polak, S. Murakami, V. T. Lam, and G. C. Benson, J. Chem. Eng. Data, 15, 323 (1970). (17) S.Murakami, V. T. Lam, and G. C. Benson, J. Chem. Thermodyn., 1,397 (1969). (18) M. L. Huggins. J. Chem. Phys.. 9, 440 (1941). (19) M. L. Huggins, Colloid Symposium Preprint (1941); J. Phys. Chem., 46, 151 (1942). (20) M. L. Huggins, Ann. N.Y. Acad. Sci., 41, 1 (1942). (21) M. L. Huggins, J. Am. Chem. SOC..64, 1712 (1942). (22) M. L. Huggins, "Physical Chemistry of High Polymers", Wiley, New York, N.Y., 1958. (23) P. J. Flory. J. Chem. Phys., 10, 51 (1942). (24) G. Scatchard, S. E. Wood, and J. M. Mochel, J. Am. Chem. SOC.,62, 712 (1940). (25) G. Scatchadd and L. B. Ticknor. J. Am. Chem. SOC., 74,3724 (1952). (26) G. Scatchard, S.E. Wood. and J. M. Mochel, J. Ptys. Chem., 43, 119 (1939). (27) S. E. Wood and A. E. Austin, J. Am. Chem. SOC.,67, 480 (1945). (28) K. N. Marsh, Trans. faraday SOC.,64, 883 (1968). (29) G. Scatchard, S.E. Wood, and J. M. Mochel. J. Am. Chem. Soc., 61, 3206 (1939). , (30) T. Boublik, V. T. Lam, S.Murakami. and G. C. Benson, J. Phys. Chem., 73, 2556 (1969).
Generalized Integral Equations of Classical Fluids H. S. Chung Central Research Division Laboratory, Mobil Research and Development Corporation, Princeton, New Jersey 08540 (Received October 6. 1975) Publication costs assisted by Mobil Research and Development Corporation
Parametrized integral equations of classical fluids have been derived within the framework of functional differentiation. T h e first-order theory is examined in detail for rigid spherical and Gaussian molecules. In the region of lower densities, it appears that t h e lowest order theory is numerically quite similar, in the case of rigid spheres, to the schemes of Rowlinson and of Hurst which are based on different (diagrammatic) arguments. For the Gaussian molecules, the virial coefficients u p to the fourth are given correctly by this method. Extension of this parametrization t o second and higher order theories is indicated.
I. Introduction Functional analysis offers a powerful technique for systematically improving the integral equations of classical fluids. Thus, by retaining the quadratic term in the functional Taylor expansion, Verlet' has indicated how the original PercusYevick and hypernetted chain (hereafter referred to as PY-1 and HNC-1) theories may be generalized t o what are subsequently known as t h e PY-2 and HNC-2 equations. One of
these second generation theories, the PY-2, has been extensively studied and the most important conclusions drawn from this work2 is that ". . . when the PY-1 equation is a decent first approximation, t h a t is for high temperatures or around the critical point, the PY-2 equation significantly improves over those results, and is a useful equation. On the other hand, for dense fluids at low temperatures where the PY-1 and HNC-1 equations are poor, the PY-2 equation is also bad . . ." In view of these assertions, it seems that further progress in the theory The Journal of Physical Chemistry, Vol. 80, No. 12, 1976
