2176
ELFREDA T. CHANQAND EDGAR F. WESTRUM,JR.
ment between observed and calculated U t , values, the results of the first method indicate a closer coincidence of the main librational directions with the inertia axes of the molecule. The error introduced by putting w12 = W23 = (013 = 0 will therefore be less serious if the origin of w,, is taken a t the center of gravity. This diagonal approximation considerably simplifies
the calculation of corrected bond lengths according to Cruickshank's method.23 The revised bond lengths shown in Figure 8 were obtained by assuming mm symmetry for the molecule and applying the corrections to the molecular parameters of Marsh, et al. As in the case of perylene-PMDA, a value for q 2 = 0.1 was assumed.
The System Tetramethylmethane-Tetrachloromethane. Thermodynamics of Mixing in the Plastically Crystalline Region'
by Elfreda T. Chang and Edgar F. Westrum, Jr.2 Department of Chembtry, University of Michigan, Ann Arbor. Miehigan
(Received November 19, 1964)
Heat capacities of five different compositions (CCl, mole fraction of 0.200, 0.334, 0.501, 0.666, and 0.826) of the system tetramethylmethane-tetrachloromethane were determined by adiabatic calorimetry from 5 to 300°K. Thermodynamic functions calculated by integration of the data, together with similar values for the pure components and mixing data for the liquid solutions, yield the excess enthalpies of mixing in the solid state (ie.,for the plastically crystalline solid solution) and the temperature dependence of this quantity. The excess enthalpy of the equimolal solution is compared with the values predicted by theory.
Introduction During the last two decades a considerable theoretical and experimental endeavor has been devoted to the properties of liquids and liquid mixtures. For example, the cell model, the lattice model, the theory of conformal solutions, the average potential model, etc., have been devised. However, many, if not most, of the quantitative predictions of these theories are not in accord with the existing experimental data. Poor agreement between experiment and theory is due in part to the difficulty of obtaining good experimental mixing data and in part to the imperfection of the present state of development of the theory of fluids. Several of these theories ascribe a more or less regular structure to the liquid state. Since the solid state has many features in common with the liquid state The Journal of Physieal Chemistry
(ie., a large number of first neighbors and local order) and, in addition, a determinable lattice structure, measurement of the excess functions of solid solutions should permit, in many instances, a better test of these existing theories of mixtures than is presently possible in the liquid state. With this end in view, a system was sought that forms a continuous series of solid solutions and for which good enthalpy of mixing data were available. To make the experimental chore (1) Based upon a dissertation submitted to the Horace H. Rackham School of Graduate Studies of the University of Michigan by E. T . Chang in partial fulfillment of the requirements for the degree of Doctor of Philosophy. This work was supported in part by the Division of Research of the U. S. Atomic Energy Commission. Presented at the 146th National Meeting of the American Chemical Society, Denver, Colo., Jan. 1964. (2) To whom correspondence concerning this paper should be sent.
THERMODYNAMICS OF TETRAMETHYLMETHANE-TETRACHLOROMETHANE SYSTEM
Table I : Selected Thermodynamic Properties for the System Tetramethylmethane-Tetrachloromethane"
_---_
I -
Ncc
,--
~
----o.ooob--[C(CHa)*]
230 240 245 250 260 270 280 290 300 273.15 298.15
H o - H'o
29.54
4970.5
29.43 5076.4
29.75 5118.8
5329.1 (61.2) 5489.1 34.93 6604.6 35.61 6972.6 36.28 7351.7 36.88 37.45 38.02 37.70 7091 36.48 37.92
5474.5 6175.9 6528.4 6887.9 7253.8 7625.5 8002,8 7002 7929
34.00 34.26 34.78 35,31 35,83 36.35 36.88 35.47 36.78
33.18 33.39 33.79 34.19 34.60 35.05 35.56 34.32 35.46
31.77 32.23 36.24 37.86 38.45
Units: cal., mole of solution, OK.
Cp
'See ref. 7.
H e - H'Q
---0,501-
H a - Hoc
30.39 4862.9
---~--------l.OOW--
70.334-
-O.ZOO--
C,
C,
T
2177
6041.3 6211.9 6557.2 6907.6 7263.3 7624.2 7990.4 7019 7922
H a - Hoo
C,
6027.3 6193.7 6529.6 6869.5 7213.4 7561.7 7914.7 6977 7849
--0.666--
H" - Hoo
C,
30.55 4991.7 32.29 5859.6 32.57 32.86 33.15 33.43 33.72 34.01 33.24 33.95
[CCIdI
---0.826-----
6183.9 6511.1 6841.1 7174.0 7509.7 7848.4 6946 7786
C,
H" - H'Q
Cp H o
-
Hoc
29.46 4960.6
27.97 4883
31.64 31.75 31.97 32.19 32.41 32.64 32.86 32.26 32.82
29.30 29.75 31.16 31.24 31.33 31.41 31.49 31.27 31.48
5991,5 6150.0 6468.6 6789.4 7112.5 7437.7 7765 2 6891 7704
5312 5460 6372 6684 6997 7310 7625 6783 7567
See ref. 8.
practicable it is essential that equilibrium be achieved a t each stage of the process, that "frozen-in" equilibria be avoided in the solid state, and that the time for equilibration be "finite." For these reasons it was considered desirable to make the initial measurements on systems of globular molecules which have approximately the same diameters and which are spherically symmetrical so that the intermolecular forces may be considered central. Such molecules often form plastically crystalline substances, as defined by TimmerThe calorimetric investigation of the system tetramethylmethane-tetrachloromethane was originally undertaken so that the experimental results, together with existing data on mixing in the liquid phase and heat capacity values for the pure components, would provide an experimental test in the solid state for some of the modern theories of solutions.
Experimental Results The heat capacities of five compositions of the system tetramethylmethane-tetrachloromethane, with mole fractions of tetrachloromethane ( N c c ~ , of ) 0.200, 0.334, 0.501, 0.666, and 0.826, were measured as described elsewhere.4,5 Pertinent thermodynamic functions of the compositions studied obtained by digital computer integration are tabulated over the relevant range of temperature (230 to 300'K.) together with those of the pure components in Table I. For more extensive tabulation the detailed paper5 should be consulted. Enthalpies of mixing for this system a t several temperatures were determined from available enthalpy of mixing data at Oo,6 the heat capacity data of the pure components,7'8and the heat capacity data of this
investigation in the following manner. The enthalpy of mixing H M (or the excess enthalpy, H E )a t teniperature Tomay be expressed as
HE,, Or H M ~ = o Hmix.To - N A H A . T-~ N B H B , T ~ in which H refers to the relative enthalpy a t the temperature To,and N refers to the mole fraction. The subscripts mix, A, and B refer to the mixture and to the components. The excess enthalpy at any teniperature, T , is therefore
+
HET = HETo
*
T
i
To
N J p i dT
+ f
N , ( A H t , or AHm,) z
in which the summations are over the components and the solution in the balanced formation reaction, and AHt and AHm are the enthalpy increments associated with enantiomorphic and melting transitions. Hence
For each composition of this system the derived ( 3 ) J. Timmermans, Phvs. Chem. Solids, 18, 1 (1961). (4) E.T. Chang and E . F. Westrum, Jr., J . Phys. Chem., in press. (5) E. T. Chang, Doctoral Dissertation, University of Michigan, Ann Arbor, Mich., 1962. (6) A. Englert-Chwoles, Doctoral Thesis, Free University of Brussels, Brussels, 1955. (7) J. G. Aston and G. H. Messerly, J . Am. Chem. SOC.,58, 2354 (1936). (8) J. F. G. Hicks, J. G. Hooley, and C. C. Stephenson, ibid., 66, 1064 (1 944).
Volume 69,Number 7
July 1966
2178
ELFREDA T. CHANGAND EDGAR F. WESTRUM, JR.
-Dlrect Experimentol, 273.15”K
Table 11: Temperature Dependence of the Excess Enthalpy and Excess Heat Capacity of the System Tetramethylmethane-Tetrachloromethane“
0 Deri’ved, 230°K
A
Derived, 2 4 5 ° K
-
0 Derived.2 80°K N c c ~ , Property 0 200
/
/
P
/ I
I
f I
\
0 334
\
0 501
\ 1
0 826
I
I I
0
0 666
\
0.2 0.4 0.6 MOLE FRACTION,
I \
-0 47 -0 65 -0 73 -0 65 -0 44
14
$0 47 24 - 0 66 29 - 0 74 28 - 0 66 16 - 0 45
07
Temp-. 260
+ O 39 44
08
+ O 23 65
255b
+ O 49 42
+ O 37 63
16
+ O 10
18
+O 01
+O 08
13
67 - 0 08 46
67 - 0 06 46
74
+O 20 74
245’
230d Solid
- 0 47 177 + O 64 - 0 15 58 24 1 + O 39 +O 58 71 279 + O 22 + 1 77 65 121 - 0 02 + 1 07 71 46
a Units: cal., mole of solution, OK. * Mixing of undercooled liquid C(CHa)awith liquid CClr. Mixing of undercooled liquids C(CH& and CClr. Mixing of solid C(CHa),with solid CCl,.
I 0
\
0.8 CC!,
1.0
Figure I . Temperature dependence of the excess enthalpy for the system tetramethylmethane-ttrachloromethane. The uncertainties of all values (occasioned by propagated uncertainties in the work of Englert-Chwolesa) is f 5 cal. mole-”.
excess enthalpies and the excess heat capacities a t 280, 260, 255, 245, and 230’K. are listed in Table I1 together with the experimental value of EnglertChwoles6 a t 273.15’K. upon which the others are based. The excess enthalpy plotted vs. temperature is also depicted in Figure 1. It is to be noted that the values for 255’K. are the enthalpies of mixing for undercooled liquid tetramethylmethane plus liquid tetrachloromethane forming liquid solution; the values for 245OK. are the enthalpies of mixing for the two undercooled liquid components forming a liquid mixture; and the values for 230’K. are the enthalpies of mixing for the two solid components forming a solid solution. Values a t other temperatures may be derived conveniently by using the relative enthalpy values for the mixtures and the pure components given in Table I.
Discussion Excess thermodynamic properties of nonpolar liquids have often been selected to test the validity of statistical theories of solutions, e.g., as summarized by The Journal of Physical Chemistry
CpE HE CpE HE CpE HE CpE HE CpE HE
280 273.15 ---Liquid--
G ~ g g e n h e i m ,by ~ Hildebrand and Scott,l0 and more recently by Prigoginell and by Rowlinson.12 Owing to the similarity of the molecular shapes and sizes of the tetramethylmethane and tetrachloromethane molecules, the binary system of these substances has been favored for such purposes. Moreover, the tetrahedral symmetry of the molecules sufficiently approaches spherical symmetry to permit the assumption that intermolecular forces are central. The excess functions of this system in the liquid state have been well investigated a t 273’K. Mathot and DesmyteP studied the excess free energy and volume of mixing by measuring the total vapor pressures and densities of several compositions of this system at 273’K. finglert-Chwoles6 determined the enthalpy and entropy of mixing at 273’K. by adiabatic and isothermal calorimetry. The excess compressibility, (dVE/dP)T, was investigated by Jeener,’* who measured the velocity of sound in liquid solutions of tetramethylmethane and tetrachloromethane. As indicated in the reference cited this system is regular a t 273OK., and the respective values of the excess free energy, enthalpy, entropy (expressed as TS”) and volume at 273’K. ate 76, 74, -2 cal./mole, and -0.5 cm. 3, respectively, for the equimolal composition. (9) E. A. Guggenheim, “Mixtures,” Oxfora University Press. Oxford, 1952. (10) J. H. Hildebrand and R. Scott, “Solubility of Non-Electrolytes,” Reinhold Publishing Corp., New York, N. Y.; 1960. (11) I. Prigogine. “The Molecular Theory of Solutions,” NorthHolland Publishing Co., Amsterdam, 1967. (12) J. S. Rowiinson, “Liquids and Liquid Mixtures,” Rutterworths Scientific Publications, London, 1959. (13) V. Mathot and A. Desmyter, J . Chem. Phyu., 21, 782 (1953). (14) J. Jeener, ibid., 25, 584 (1956).
THERMODYNAMICS OF
TETMMETHYLMETHANE-TETRACHLOROMETHANE SYSTEM
The magnitude of the excess volume decreases with increasing pressure. Theoretically calculated excess functions taken from the referen~es~~-*~' cited are compared with the experimental values for this system in Table 111. Quantitative agreement between theory and experiment is generally poor. However, in a few cases ( i e . , the smooth potential cell model and the average potential model) fair agreement prevails. The best accord a t 273'K. (in the liquid phase) is obtained with the average potential model. Lack of accord between the theoretical evaluations are occasioned not only by the differences in the assumptions made concerning the structure of the liquid state and the interactions between species but also on the evaluation of parameters. Since several of the solution theories are predicated upon a more or less regular structure with many features in common with the solid state, the calorimetric investigation of the system tetramethylmethane-tetrachloromethane was made in part to ascertain whether better agreement between theoretical and experimental excess enthalpies is obtainable for solid solutions. Table In : Comparison of Calculated Excess Functions with Experiment for the Equimolal Composition of the System Tetramethylmethane-Tetrachloromethane in the Liquid Phase" HE
TSE
273
55
-5
60 - 0 . 2
293 273
16 71
-98 -9
. . . -3 80 -1.3'8
273 . . . 273 100
...
150 . . , 88 - 0 . 5
T
QE
VE
Theory Cell model, smooth potentiallS Cell model, Lennard-Jones Devonshirele Average potential model" Scatchard-Hildebrand formula18 Random mixture20 Semirandom mixture a t fixed volumez0 Semirandom mixture at fixed pressure"
12
273
93
9
84
-0.3
273
67
5
62
-0.8
76
-0.5
Experiment6 273 a
74
-2
Units: cal., mole of solution, OK., cm.*.
The direct method of measuring the heat of mixing using adiabatic or isothermal calorimetry is that co"only used to obtain satisfactory H E data on liquids. In this method one encouiiters adiabatic calorimetric concerns, ;.e., exchange of heat with surroundings, satisfactory stirring, and rapid and sensitive temperature measurement, plus the further difficulty of ensuring the absence of
The E~~~~~Heal Capac,&,.
2179
any appreciable volume of vapor. In practice, measurements of HE by the direct method may be accurate to about 2% for liquids at room temperature, with lower accuracy at higher temperatures. Measurements of the excess heat capacity, CPE reveal the temperature dependence of HE. Very few experimental investigations of CPE have been made. This neglect has been in part, as Rowlinson12stated, "probably due to the fact that even the best theories of solution are not accurate enough to predict reliably the size of the second derivative of the excess free energy with respect to temperature to permit a test of theory." However, in considering a mixture from the standpoint of its excess functions, it is necessary to know the excess free energy and excess enthalpy a t the same temperature. Usually GE is determined a t a higher temperature than HE, and knowledge of the excess heat capacity CPEof the system is required to correct HE to the temperature of measurement of GE. Calculating HE at one temperature from the experimental value at another demands precise experimental investigation of the heat capacity, for the values of the excess properties are all relatively small, and an error in the heat capacity of the order of 1% would lead to a comparatively large uncertainty in the derived H E values. Therefore, the derived value of the excess enthalpy is limited by the experimental error of the C , nieasurements. Examination of Table I1 reveals that the temperature dependence of the excess enthalpy in the liquid range for the system under investigation is comparatively small (e.g., in comparison with that of the system ben~ene-tetrachloromethane)~~~~ and is slightly negative in the liquid range where both components are liquids and is slightly positive when one or both components are undercooled liquids. The excess enthalpy curve in the liquid solution range tends to be more symmetrical at high temperatures. At lower temperatures within the liquid range the maximum in the curve is shifted slightly to the tetrachloromethane-rich side of the excess enthalpy vs. composition diagram (cf. Table I1 and Figure 1). In the plastically crystalline region the derived excess enthalpy curve a t 230°K. shows a comparatively large and irregular composi(15) 1. PrigOgine and
v. Mathot, J . Chem. Phys., 20, 49 (1952).
21, (16)2169 Z. W. (1g53). Salsburg and
J. G. Kirkwood, ibid., 20, 1538 (1952);
(17) I. Prigogine, A. Bellemans, and A. Englert-Chwoles, &id., 24, 518 (1956). (18) See ref. 11* P. 224. (19) See ref. 11, p. 231. (20) See ref. 12, p. 331. (21) J. R. Goates, R. J. Sullivan, and J. B. Ott. J . Phys. Chem., 6 3 , 689 (1959).
V d u w 69,Number 7 July 1966
2180
ELFREDA T. CHANGAND EDQAR F. WESTRUM, JR.
tion dependence and possibly a discontinuity between the compositions "1, = 0.5 and 0.7. Unfortunately, there are insufficient data in this region to elucidate this situation. The existence of phase separations restricts the plastically crystalline region to a relatively short temperature i n t e r ~ a l so ~ , ~that only in the vicinity of 230'11. does the possibility of a continuous series of solid solutions exist. Hence, even in this system, comparison of values of the excess functions is limited to a rather narrow range of temperature. Comparisons with Theories. The predictions based on the cell model which assumes structures for solutions most closely resembling those of crystals might be expected a priori to be especially appropriate for comparison with the excess enthalpy of soIid solutions. As Prigogine22has suggested, for high densities as in the solid state a harmonic potential approximation for the mean potential may be used. The excess enthalpies for the harmonic oscillator cell model given in Table IV are calculated by the equation15
Table IV : Comparison of Derived Experimental Excess Enthalpies with Calculated Values a t 230°K."3b
--0.200
Ncc I'----0.334
0.666 -__-_
0.501
___-HE
0.826
Harmonic oscillator cell model
43
78
102
101
78
Hildebrand's empirical approach Internal pressure considerations Distortion energy Total
63
88
34 -
49 -
97
137
101 58 159
53 -
92 145
61 36 97
164 172 168
146 154
94 99
279
121
Average potential model
C( CH3)aa8 reference CCh as reference Av.
105 111 108
146 I54 150
150 97
Derived experimental 177
241
a C j , Table ADI-X in ref. 23 for properties used. cal., mole of solution.
71
*Units:
VE
gies of liquid and solid phases is small, the magnitude of the excess enthalpy in a solid solution may be expected to be nearly the same as that in the correspondin which ing liquid phase. However, a distortion energy due to VE differences of size and shape must be added to the ex- = 1 . 4 1 2 5 [ ( k T / A ~-~NA(~T/AAA) ) - N B ( ~ T / A B B ) ]cess enthalpy which is based upon internal pressure V* considerations alone (i.e., the excess enthalpy given Here A is the interaction parameter, k is the Boltzby the Hildebrand-Scatchard seniienipirical formula10). mann constant, N ois Avogadro's number, and N is the The distortion energy is peculiar to the solid phase mole fraction. The exact values of physical properties and is lost on melting; therefore, the part of the excess on which Table IV is based are detailed in a separate enthalpy arising from internal pressure differences should be much the same as it is in the liquid phase. Since the average potential model gives good agreeThe semiempirical excess molal enthalpy formula based ment with experiment for liquid solutions of this syson internal pressure considerations26is tem (cf. Table 111), a comparison is also made at H~ = (NAvA N B v B ) ( ~ A - 6B)'+A+B 230°K. to determine whether the theory is applicable to solid solutions. For the average potential model, where V refers to the molal volume, 6 to the soluHEin Table IV is calculated from the equationz4
(ABB/~T)"']~ -k 4.83 V*
+
HE = NANB
__-
[-HA
+ TCpA][0.25d2 + 9p2] 0.5T2[dC,A/dT - 0.25dCvA/dT]d2
in which the thermodynamic properties refer only to the configurational part of the properties, and d and p are interaction parameters (for which cf. discussion by Prigogine"). Finally, the derived experimental excess enthalpy at 23OoI
