1345
EXCESS PROPERTIES OF SOMEAROMATIC-ALICYCLIC SYSTEMS would only disappear at higher temperatures. Some preliminary measurements on TiC14.2POCl3, which is isomorphous28 with SnCL 2POCl3, seem to indicate that all four expected resonances for chlorine bonded to phosphorus are present.
Acknowledgment. The assistance of Dr. V. Nagarajan and Mr. Michael Buckley in some of the experimental work is gratefully acknowledged. (28) C. Branden, Acta Chem. Scand., 16, 1806 (1962).
Excess Properties of Some Aromatic-Alicyclic Systems, 111. Analysis of HE and VE Data in Terms of the Theory of Flory’ by G. C. Benson and Jaswant Singh2 Division of Pure Chemistry, National Research Council of Canada, Ottawa, Canada
(Received October 23, 1967)
The Flory theory of nonpolar solutions is used to analyze previously published data for the thermodynamic properties of a number of aromatic-alicyclic systems. Values of the excess enthalpy and excess volume can be correlated fairly well by the theory.
Introduction Part I 3 of this series presented the results of measurements of molar excess enthalpies (HE)and molar excess volumes (VE) at 25’ for eight binary aromatic-alicyclic systems formed by mixing either benzene or toluene with cyclopentane, cyclohexane, cycloheptane, and cyclooctane. The data were examined4 from the point of view of the Scatchard-Hildebrand equationls the quasi-lattice theory of Barlier16 and the Prigogine corresponding states average potential model.’ Although each of these approaches provided some basis for analysis of the results, none was particularly satisfactory. Recently Flory8 has developed a statistical theory for mixtures of nonpolar molecules differing in size, which should be quite suitable for describing the properties of aromatic-alicyclic systems. The few calculations for benzenecyclohexane carried out by Abe and Floryg support this view, and the theoretical estimates of the excess volumes of an equimolar mixture are in excellent agreement with the experimental results over a range of temperatures. The present paper describes the analysis of HE and VE data from ref 3 in terms of Flory’s theory. It thus provides a more extensive examination of the applicability of the theory to aromatic-alicyclic systems.
Flory’s Theory of Binary Mixtures This section summarizes the equations of the Flory theory which are needed for the subsequent application. With a few minor exceptions, Flory’s notation has been
followed closely; reference should be made to the original paper8 for details of the derivations. A molecule is considered to be made up of segments (isometric portions), the effective number being r . Each segment has s intermolecular contact sites capable of interacting with neighboring sites. I n the liquid state, the volume per mole of segments is denoted by v, and the corresponding “hard-core” or “characteristic” volume by v*. The molar values of these are indicated by and respectively. ments is
The reduced volume of a mole of segt7 =
v/v* = v/v*
(3)
(1) Issued as Nationa Research Council No. 10070. (2) National Research Council of Canada Postdoctorate Fellow 1965-1967. (3) A. E. P. Watson, I. A. McLure, J. E. Bennett, and G. C. Benson, J . Phys. Chem., 69, 2753 (1965). (4) I. A. McLure, J. E. Bennett, A. E. P. Watson, and G. C. Benson, ibid., 69, 2759 (1965). (5) J. H. Hildebrand and R. L. Scott, “Regular Solutions,” Prentica Hall Inc., Englewood Cliffs, N. J., 1962, Chapter 7. (6) J. A. Barker, J. Chem. Phys., 20, 1526 (1952) (7) I. Prigogine, “The Molecular Theory of Solutions,” North Holland Publishing Co., Amsterdam, 1957, Chapters 10, 11. (8) P. J. Flory, J . Amer. Chem. Soc., 8 7 , 1833 (1965). (9) A. Abe and P. J. Flory, ibid., 87, 1838 (1965); see also P. J. Flory and A. Abe, ibid., 8 6 , 3563 (1964).
Volume 72,Number 4 April 1068
1346
G. C. BENSON AND JAWANT SINCTH
and can be calculated from the coefficient of thermal expansion ( a ) using the expression d =
[(I
+ 4/3a~)/(1 +
m
i
3
(4)
for the excess properties of mixtures only as the ratios and S I / S ~ . Since the same hard core volume is adopted for segments of both species, it follows from eq 2 that r1/1'2
At zero pressure, the reduced volume and reduced temperature ( p ) are related by the equation of state
F
= (5'13
-
(5)
i)pl3
This is the central equation of Flory's theory. The characteristic temperature T* and pressure p* are obtained from the relations
T* = T / P
(6)
p* = yTC2
(7)
Abe and Floryg assumed that the numbers of contact sites on the benzene and cyclohexane molecules were proportional to the surface areas of spheres having the same core volumes, and deduced that
and
where Y =
ff/P
(8)
is the thermal pressure coefficient and P is the isothermal compressibility. It can be shown that the reduced temperature of a mixture of two species of molecuies (indicated by subscripts 1 and 2 ) at mole fraction z1 is given by = ($lP*lFl
+
Table I :a Parameters for the Pure Liquids a t 25'
*
v*,
V,
$2P*2F2)/
($1~*1
+ $2p*2 - 41e2x12) (9)
where the segment fractions $1 and
$2
are defined by
and the site fraction 82 by
The molar excess enthalpy, excess volume, and residual1° entropy of the mixture are
HE
These relations will be used for the eight systems considered here. The values of the properties of the component liquids are summarized in Table I where corresponding values of the characteristic volumes, temperatures, and pressures are also given.
- 5-1) + 22p*2v*z(5z-1- 5-1)
= s1p*,v*1(51-'
+
103&, deg-1
mol-1
C6He C7Hs CsHio CsHn CTHi4
89.43 106.84 94.74 108.85 121.62 134.96
loop,
om3
T*,
atm-1
mol-'
deg K
P*, J cm-8
4720 5040 4531 4720 5231 5260
624.1 542.5 609.0 530.2 479.9 571.5
1.217C'd 98.12e 69.30 84~65 1.071' 95.0' 1.325' 1 3 ~ 5 . 1 ~ 72.28 84.34 1.217' 115.5e 1.00" 98.0h 97.48 0.99' 81.2' 108.34
Data for OL and p are the same as given in ref 4, excepl for correction of printing errors. Based on density values from ref 3. American Petroleum Instit,ute, Research Project 44, Carnegie Press, Pittsburgh, Pa., 19;53, and later revisions. S. E. Wood and A. E. Austin, J . Amer. Chem. Soc., 67, 480 (1945). e G. A. Holder and E. Whalley, Trans. F a m d a y SOC., 58, 2095 (1962). B. Jacobson, Acta C'hern. Scand., 6, 1485 (1952). A. Weissler, J . Amer. Chem. SOC.,71, 419 (1949). Estimated by interpolation of data for the other cycloparaffins. E. Butta, Ric. Sci., 26, 3643 (1956).
'
@
+ ~l~*1ezx125-1(12)
VE = (ICIV*I ~ 2 ~ * 2 ) ( 5 $151 .- $25~)
(13)
and
I n eq 9 and 12, XI2 is a constant characterizing the difference in the energy of interaction between sites on neighboring molecules of species 1 and 2, from the average of the interactions in the pure component liquids. Treatment of Data for Aromatic-Alicyclic Systems The geometric parameters Y and s enter the equations T h e Journal of Physical Chemistry
cma
Evaluation of the excess enthalpy and volume of a binary mixture from eq 12 and 13 requires, in addition to data for the pure component liquids, a knowledge of the interaction energy XIZ. This quantity was treated by Abe and Floryg as an adjustable parameter and its value was chosen to fit HE in an equimolar solution. I n the present investigation, we have calculated Xlz from the individual values of HE and VE over the full concentration range. The results for benzene and toluene systems are shown in Figures 1 and 2, respec(10) It should be noted that Flory employs the term residual to indicate the normal thermodynamic function for mixing exclusive of the combinatorial term. This differs from earlier usage by J. S. Itoivlinson in "Liquids and Liquid Mixtures," Butterworth and Co., Ltd., London, 1959.
1347
EXCESSPROPERTIES OF SOMEAROMATIC-ALICYCLIC SYSTEMS
800
AT
25°C. 700
-
P
6oo
7
i
5
500
U r
z 400 R I
E
Lo
In
35
U W
7
300
2
W.
I
30
-I
zoo
\
BENZENE - ALICYCLIC 100
00
02
04
06
X,, MOLE FRACTION
OF
08
00
10
AT
02 X I , MOLE
BENZENE
Figure 1. Variation of parameter XI2 for benzene systems: solid curves, values determined from HE; broken curves, determined from VE. Curves are labeled with number of carbon atoms in cycloparaffin.
SYSTEMS
25°C
04 FRACTION
06
08
IO
OF BENZENE
Figure 3. Excess enthalpies of benzene systems: solid curves, based on experimental results from ref 3; broken curves, calculated from eq 12 with from second column of Table 11. Curves are labeled with number of carbon atoms in cycloparaffin.
50
b'~~
TOLUENE-ALICYCLIC
45
SYSTEMS
-
B E N Z E N E ALICYCLIC A T 25°C.
SYSTEMS
0.8
40
142
35
e
0.6
c1
5
m
'E
30
u*
0.5
I
7
3
N
$
X
25
20
04
v)
i
w
0.3
W L5 u-
02
15 0.1 10
00
00
02 X I ,MOLE
04
06 08 FRACTION OF TOLUENE
IO
Figure 2. Variation of parameter X12 for toluene systems: solid curves, values determined from H E ; broken curves, from VE. Curves are labeled with number of carbon atoms in cycloparaffin.
tively. It is clear from these that, in general, different values of Xlz are required to fit H E at each concentration. Also, at any one concentration, different values of Xlz are determined from the fit of H E and of VE, values of Xlz obtained from VE generally show a somewhat wider variation with concentration. The relative positions of the curves in Figures 1 and 2 show an in-
00
02
04
06
OB
10
X , , MOLE FRACTION OF BENZENE
Figure 4. Excess volumes of benzene systems: solid curves, based on experimental results from ref 3; broken curves, calculated from eq 13 with XIZfrom second column of Table 11. Curves are labeled with number of carbon atoms in cycloparaffin.
teresting parallel between the variation of XIz for the benzene and toluene systems. The benzene-cyclohexane system conforms most nearly to the behavior required by the Flory theory, but even in this case there is an over-all spread of about 5 J ern4 in the values of XlZ. Volume 72, Number .G April 1968
1348
G. C. BENSONAND JASWANT SINGH
TOLUENE-ALICYCLIC
TOLUENE - A L I C Y C L I C
SYSTEMS
AT 2 5 ° C
700
SYSTEMS
AT 2 5 ° C .
1
c
600
$
-I i
05
J
9 i
4’
04
(0
ln V W
J -
5 400
03
Y.
I L
02
z
W
ln
300
ln W V X W W’
01
200
L
00 00
02
100
02
00 Xi
, MOLE
06
04 FRACTION
OF
08
10
TOLUENE
Figure 5. Excess enthalpies of toluene systems: solid curves, based on experimental results from ref 3; broken curves, from second column of Table calculated from eq 12 with XLZ 11. Curves are labeled with number of carbon atoms in cycloparaffin.
The values of XIZgiving the best (least-squares) fit either of the experimental HE data or of the experimental BE data were determined by minimizing either the integral
S, 1
gH2(X1Z)=
(HEexpti
- HE~~o,y(X~z))2 dzl (17)
or the integral 1
~rv’(Xi,)=
04
X I ,MOLE
(VE,,pti
-
VEFio,,(Xiz))2
dzi
06
08
10
FRACTION OF TOLUENE
Figure 6. Excess volumes of toluene systems: solid curves, based on experimental results from ref 3; broken curves, calculated from eq 13 with XIZfrom second column of Table 11. Curves are labeled with number of carbon atoms in cycloparaffin.
Table I1 : Values of Xl2 Determined from Least-Square Fit of
HE
--Fit System
XlZ, J om-a
and of
VE
of H E data-? UH,
SV,
J mol-1
oms
---Fit XlZ, J
mol-‘
cm-8
33.84 7 . 4 41.02 6 . 9 37.87 3 . 0 38.68 3 . 4 17.69 3 . 1 28.25 1 2 . 7 25.53 13.9 25.68 12.0
0,105 0,005 0.061 0,030 0,076 0.089 0.049 0.108
of VE data----. 0“.
J mol-1
UV,
om8
mol-’
24,76 123.8 0.003 41 25 7 . 6 0.005 32.69 76.7 0.016 40.02 2 0 . 8 0.026 11.80 91.1 0.002 34,94 109.4 0.017 28.79 56.8 0.027 33.89 144.1 0.046
(18)
ideal, the excess Gibbs free energy can be calculated In the case of the fit of HEexpt], the resultingXIz was from the relation used to calculate VEplory,and, alternatively, XI2 determined from VEexptl was used to calculate HE~lory. GE HE TSR (19) These computations are summarized in Table I1 where The broken curves in Figures 7 and 8 were computed XIz, uH, and u v are given for the two ways of fitting in this way. Experimental data for three of the systo the experimental data. I n general the use of HEexpti tems1l-13 are shown as solid curves. In these cases the determine Xlz leads to a better agreement between theoretical values of GE are 1.5 to 2 times larger than theoretical and experimental values. This is not surobserved experimentally. l4 prising since as noted by Abe and F l ~ r yXI2 , ~ is an energy Considering the very simple equation of state on parameter which enters the formula for VE only imwhich the Flory theory is based, the properties of the plicitly through the reduced temperature p of the mixture: thus it appears more logical to determine its (11) G. Scatchard, S. E. Wood, and J. M. Mochel, J . Phys. Chem., value from the enthalpy, which is an explicit function of 43, 119 (1939). XIz, as well as depending on it through p . Figures (12) R. W. Hermsen and J. M. Prausnits, Chem. Eng. Bci,, 18, 486 3-6 compare the experimental values of H E and VE (1963). (solid curves) with those calculated from eq 12 and 13 (13) T. Katayama, E. K. Sung, and E. N. Lightfoot, A.I.Ch.E. J,, 11, 924 (1965). (broken curves) using the XIz values based on the fit of (14) The Barker theory, as applied in ref 4, also led to values of HE (as given in the second column of Table 11). GE which were too large; the discrepancies (not reported previously) Assuming that the combinatorial entropy of mixing is were similar in magnitude t o those noted above.
-
The Journal of Physical Chemistry
1349
EXCESS PROPERTIES OF SOME AROMATIC-ALICYCLIC SYSTEMS
600
400
500
>-
400
$
W >.
5
. P
w
w
300
e IL
a
LL
2
u)
W
E
300
W
200
X
200
Y-
W-
100
100
0
00
02
04
06
08
10
X I , MOLE FRACTION OF BENZINE
0 00
02 X I , MOLE
04
05
08
10
FRACTION OF TOLUENE
Figure 7. Excess Gibbs free energy of benzene s y s t e m : solid curves, based on experimental results from ref 10 and 11; broken curves, calculated from eq 14 and 19 with XISfrom second column of Table 11. Curves are labeled with number of carbon atoms in cycloparaffin.
Figure 8. Excess Gibbs free energy of toluene systems: solid curve, based on experimental results from ref 12; broken curves, calculated from eqs 14 and 19 with X I $from second column of Table 11. Curves are labeled with number of carbon atoms in cycloparaffin.
aromatic-alicyclic systems considered in this paper are correlated fairly well by it. Abe and Floryg have discussed the dependence of the theoretical values on the precision of the data for the pure component liquids. Of the present set of systems, undoubtedly the values of the properties of benzene and cyclohexane are the most
accurate : possibly, some of the discrepancies between theory and experiment may be attributed to inaccuracies of data for the other liquids.
Acknowledgment. The authors wish to thank Mr. A. Potworowski for computational assistance.
Volume 72, A'unzber 4
April 1968
