FLUOROCARBON SOLUTIONS AT Low TEMPERATURES
3853
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Fluorocarbon Solutions at Low Temperatures.
IV.
The Liquid Mixtures
+ CClF,, CH,F, + CClF,, CHF, + CClF,, CF, + CClF,, C,H, + CCIF,, C,H, + CF,, and CHF, + CF, CH,
by Ian M. Croll and Robert L. Scott Contribution N o . 1711 f r o m the Department of Chemistry, University of California, Los Angeles, California OOO84 (Recezved J u l y 86, lS64)
Total vapor pressures have been measured for seven binary liquid mixtures, and frorn these measurements equations for the excess Gibbs free energy have been derived, yielding the following values for GE/cal. a t 5 = 0.5: CH, CClF, a t 105.l0K.,62; CHzFz CClF, a t 178.7"K., 204; CHF3 CClF, a t 178.2"K., 123; CF4 CClF, a t 145.2"K., 26; CClF, a t 178.7"K., 67; CzH6 CF, a t 150.6"K., 169; and CHF, CF, CZH6 CF, has been measa t 145.2"K., 140. The liquid-liquid miscibility diagram for CzH6 ured; the critical solution point is a t T , = 150.loK.,2,: = 0.50. The theoretical significance of these results is discussed.
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Introduction The work reported in this paper is part of a continuing research prlogram on fluorocarbon solutions; previously we have reported1t2vapor pressures and phase diagrams for the liquid binary systems CH, CF4, CH, Kr, CHF, CF,, CHzFz CHF3,Xe CHFa, C2F6, ChH6 CHF3, arid CHF, C2F6. In C2H6 this paper we report siiriilar work on the additional systems CF, CCIFa, CHF, CClF3, CH2F2 CC1F3, CH, CClF,, and C2H6 CC1F3, together with additional studies 011 the systems CHF3 CF, and C2H6 CF,. Moreover, 1 hese vapor pressure data, together with those of thn earlier papers, have all been processed using the comput cr programs recently developed by Myers and Scott.,
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Experimental Materials. Experimental samples of CF, (perBuoromethane), CHF3 (trifluoromethane), CH2F2 (difiuoromethane), and CClFa (chlorotrifluoroniethane) were donated by the Jackson Laboratories of E. I. du Porit de Nemours and Co. The samples of CF, were purified by repeated passage over activated char.. coal held at solid C 0 2 temperatures. The samples of CHF3, CHzFa, ,and CClF, were further purified by partial distillation, the distillate being rejected. Phil-.
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lips research grade methane and ethane were used without further purification. The last traces of air were removed from all samples just prior to use by pumping while alternately freezing and melting Liquids for low temperature baths were laboratory grade materials without further purification. Apparatus and Experimental Procedure. The vacuunl system used was essentially that described by Thorp and Scott.' The system was evacuated by a Welch Duo-seal oil pump in conjunction with a two-stage mercury diffusion pump. The manifold pressure was determined by means of a McLeod gauge. Gas samples were introduced into storage bulbs through a gas purification train. A Toepler pump; equipped with a 0.5-1. cylinder whose voluine was calibrated to a reference mark and a side arm for measuring the pressure of its coiiteiits, was provided to transfer samples of gases froin storage to the experimental vessels and for the measuremeiit of the amount of material transferred. An eight-junction copper-constantan thermopile, referred to a bath of melting ice, was used for tempera(1) N . Thorp and R. L. Scott, J . Phya. Chem., 60,670, 1441 (1956).
(2) I. M. Croll and R. L. Scott, ibid., 62, 954 (1958). (3) D. B. Myers and R. IL. Scott, I n d . Eng. Chem., 5 5 , 43 (19G3)
Volume 68, Number 12 December, 1964
3854
ture nieasureinent, The bulk of the thermoelectric e m f . was balanced by a calibrated variable voltage supply, arid the residual e.m.f. was measured with a 2.5-mv. range Brown recording potentiometer. The temperature scale was calibrated against the triple point of methane and the vapor pressures of oxygen, methane, and ethane. Temperatures could be nieasured to a precision of O . O 2 O , and were considered accurate to 0.1'. Temperature control for the determination of the mutual solubility of the C2He--CF4system was obtained by refluxing mixtures of methane and propane. The refluxing bath was held in a dewar flask whose main features were a relatively long neck (30 cm.), which reduced end eflects and thus allowed clear vision through the vessel by iiiininiizing moisture condensation on the exterior, and a side arm leading to a cold finger filled with liquid nitrogen which acted as a condenser . The bath was magnetically stirred, and sinall reductions in temperature were achieved by reducing the pressure above the bath with an aspirator. Large temperature changes were obtained by varying the composition of the refluxing mixtures. The teniperature control achieved in this manner was adequate for short periods of time, but a slight drift in coinposition due to loss of methane from the system required periodic additions of methane. The method of changing temperature by varying bath composition was time consuming, and temperature control for solubility measurements of subsequent systems was achieved in another manner. A Pyrex inner container with a surrounding jacket which could be evacuated through a stopcock was placed in a flatbottomed dewar flask which had a pair of diametrically opposed unsilvered strips down its sides to allow vision through the apparatus. The dewar flask was filled with liquid nitrogen as a refrigerant. A small miclirome wire heater was placed in the bottom of the inner container which held the CC1F3 (Freon 13) used as the liquid bath, CC1F3 was chosen because of its wide liquid range (92-192'K.), its relatively low viscosity near the freezing point, and its inert chemical nature. To iiialre up the liquid bath, a sinall amount of dry air was allowed into the jacket of the inner container to increase the rate of heat flow to the liquid nitrogen bath. CClF, was then passed into the inner container where it condensed. The jacket was then evacuated. The mechanical action of the inoving carriage of the recording potentionieter was utilized to open and close the primary circuit of an electronic relay at preselected teniperatures in order to activate the heater circuit. By use of this procedure, a preselected bath teniperaThe Journal of Physical Chemistry
IAN11.CROLLAND ROBERTL. SCOTT
ture could be easily obtained and could be maintained indefinitely to + O . O 5 O . Temperature control for vapor pressure iiieasuretiients was obtained by a modification of the refluxing bath method. The pressure of the bath was controlled by a conventional distillation manostat which was connected through a cold trap to a vacuuiii pump. A ballast tank was provided to dainp out minor pressure fluctuations in the system. One-component baths were used so that temperature control would not be affected by loss of material, Methane was used at the lowest temperatures, ethylene at intermediate temperatures, and propane at the highest temperatures. By use of this method the temperature could be controlled to k0.05' for an indefinite period of time, and to 10.02O for shorter periods. The experimental vessel of Thorp and Scott1 was used to investigate mutual solubilities. I t consisted of a capillary stein leading to a sample-holding bulb of about 2 c1n.3 volume. During operation the bulb was immersed in the low temperature bath, and a sample of gas was withdrawn from storage into a Toepler pump where its pressure (always below 20 cm.) was measured at a known volume. The sample was then transferred to the experimental vessel, where it condensed. Small amounts of the second component were added in a similar manner. Mixtures were allowed to evaporate back into the Toepler pump and were recondensed into the experiniental vessel before making observations. The inole fraction composition of the liquid mixture was taken to be the ratio of the gas pressures as measured in the Toepler pump. The volume of the experimental vessel was less than 10% of the volume of the Toepler pump. The precision of the composition was approximately kO.005 mole fraction. Liquid-liquid solubility data were obtained by varying the temperature of the bath until two phases were observed to appear on cooling and disappear on warming. The method of measuring vapor pressure was that described by Thorp and Scott.' The sample holder volume was about 2 cm.S,and it was closed hy a thin, slightly concave, glass membrane. At a reproducible pressure differential, the membrane clicked audibly, and a sudden displacement could be observed visually. The click gauge was calibrated at each working temperature, since the calibration was found to change with temperature (by 2 min. in 100'). The experimental vessel was loaded in the same manner as in the solubility measurements. The pressure on the upper side of the click gauge was varicd untd it operated, and the applied pressure read on a mercury manometer to the nearest millimeter. Several read-
FLUOROCARBON SOLUTIONS AT Low TEMPERATURES
ings were made, with both increasing and decreasing applied pressure. From the calibration of the click gauge the vapor pressure could be calculated with an estimated accuracy of 2 mni. The mixtureR were made up, with a precision of iO.005 mole fraction, in the same manner as for solubility measurements. For systems with large differences in vapor pressure of the pure components, practically all of the vapor was of the lower boiling component. Although the vapor space above the liquid was small, this results in a deviation of the liquid composition from that calculated. This composition error is greatest when the amount of the lower boiling component is smallest. The magnitude of this effect is somewhat reduced by the fact that the vapor pressure above the liquid is generally low in this region. However, when the vapor pressure of mixtures containing small amounts of the lower boiling Component is considerably higher than the vapor pressure of the pure high boiling constituent (as in the C2He CFI, CHF3 CF4,and CHzFz CCIFI systems), this error can be as large as 0.01 mole fraction.
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3855
'"1 t
E
b \
k
110
c F4
x2
'ZH6
Figure 1. Liquid-liquid phase diagram for the system CzHe CF4.
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Results Liquid-Liquid Phase Diagrams. The solubility measu r e m e n t ~on ~ the system C2H6 CFd are shown in
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Fig. 1. The crntical solution temperature is 150.1'K. The phase diagram is s,ymmetrical within experimental error, so the critical mole fraction may be taken as (x& = (x& = 0.50. I n addition, the unmixing temperatures of three other systems were determined near x = 0.5 and are shown in Table I. Table I : Solubility Measurements Near x
=
Mole
I1 I11 VI1
Tunrnixingt
fraction xa
OK.
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0.493 0.504 0.592
178.0 118.0 131.4
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0.2
0.4
0.6
0.8
I
xz CCI Fa Figure 2. Total vapor pressure for the system CHI CCIFt at 105.1"K. The solid curve is calculated from eq. 1 using the parameters given in Table 11.
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0.5
System
C H I F ~ CClFi CHFa CClFa CHF, CF4
0.0 CH4
These unmixing temperatures must be very nearly the same as the critical solution temperatures. The CF4 observed unniixing temperature for CHFa at x2 = 0.59 is 0.6' higher than the unmixing temperature of 130.8'K. at x == 0.60 reported by Thorp and Scott'; we have not ascertained whether this small discrepancy is due to differences in purity of materials or to differences in temperature calibration. Vapor Pressures and Excess Free Energies. Figures 2-8 show tlhe measured total vapor pressures for the
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seven binary systems studied. The circles are the experimental points, while the full curves are those calculated in the data processing computer program. The experimental data were processed by the computer program CH08B developed by Myers and Scott.s This program, a modification of the least-squares procedure proposed by Barker,6 evaluates the set of coefficients a in an equation for the molar excess Gibbs free energy GE
G E / =~ x~ ~ x +~ c y[I ( ~~I -~ x2) + adx1 - x2)2
+ ...]
(1)
(4) A11 the data shown in Pig. 1-8 (as well as the earlier data of ref. 1 and 2) will be found in numerical form in the Ph.D. dissertation of I. M. Croll, University of California, Los Angeles, Calif., 1958. (5) J. A. Barker, Australian J. Chem., 6 , 207 (1953).
Volume 68, Number 1.2 December, 1964
3856
IAN
where x1 and x2 are the mole fractions of components 1 and 2. Differentiation of eq. 1 yields the excess chemical potentials pIE = RT In y1 and pzE = RT In yz (where the y values are the activity coefficient,s). Of special interest are the limiting values In yl* and In y2* evaluated at infinite dilution In TI* = lim (In
rl) =
In
yz* =
lini (In yz)
+ a3 - . . . + az + a3 + . . .
al -
Zl-tO
= 011
xz-0
n1.
CROLL AND
ROBERT L. SCOTT
8001
(2a)
cyz
(2b)
In practice the series in eq. 1 converges slowly, and an arbitrary decision about truncation must be made. In all but two systenis studied, no significant improvement was obtained by including more than three coefficients, so the results in Table I1 are reported uniformly 01
0.0
CF4
0.2
0.4
0.6 X2
0.8
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I
CClFi
Figure 5. Total vapor pressure for the system CF, CClFI at 145.2"K. The solid curve is calculated from eq. 1 using the parameters given in Table 11. 6001
O/ 0.0
I
I
0.2
I
I
I
0.6
0.4
I
1
I
1.0
0.8
xz
C V Z
CCIFS
8
I
300
2
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Figure 3. Total vapor pressure for the system C H ~ F Z CClFa at 178.7"K. The solid curve is calculated from eq. 1 using the parameters given in Table 11.
2ooi Iooi
OI 0.0
, - I 0.2
0.4
CHF,
0.6
xz
Figure 4. Total vapor pressure for the system CHFs CClF, at 178.2"K. The solid curve is calculated from eq. 1 using the parameters given in Table 11.
The Journal of Physical Chemistry
1.0
0.8
CCIF,
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with the three coefficients ao, al,and a?. The results of recalculations of the previously reported measurements of Thorp and Scott are also included in Table 11. Some comments on the computations niay be helpful in interpreting the results shown in Table 11. (1) The Barker niethod and our program CHOSB include corrections for gas imperfections and the effect of applied pressure upon vapor pressure. Table I1 lists the corrections used: Wl = (BI1- V , ) / R T , Wz = (Bzz - V z ) / R T , and 812 = (2B12 - BII - B d / R T , where the B values are second virial coefficients (esti-
FLUOROCARBON SOLUTIONS AT Low TEMPERATURES
3857
I
I2O01
400
200
0.00
0 20
0.40
0.60
I
1.00
0.80
c F4
X2
c2H6
Figure 7 . Total vapor pressure for the system C2He CF, at 150.6"K. The solid curve is calculated from eq. 1 using the parameters given in Table 11.
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I
s"
f
\
I
0.0
0.2
0.4
0.6
0.8
1.0
c f=4
CHF3
Figure 8. Total vapor pressure for the system CHFa -t CF, at 1452°K. The solid curve is calculated from eq. 1 using the parameters given in Table 11.
mated using the Berthelot equation) and the V values are molar volunies of the liquids. However, the effect of these corrections, even at the highest pressures encountered, Waf3 extremely small, the largest being a change of 0.014 in el for C2H, CF,. Since the standard deviations of the coefficients are of this magnitude, the corrections might well have been neglected. (2) The computer program yields the standard deviation CTP for the set of observed pressures as well values. These c as standard deviations for the values range from 0.005 to 0.053 for an, from 0.009 to 0.10 for el, and froin 0.015 to 0.13 for e2; in every case the smallest deviations are for system VIIIB (CH, CF, at 108.5OK.) and the largest for system
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Volum 68, Number 19 December, 1961;
IANM. CROLLAND ROBERT L. SCOTT
3858
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X I (CH2F2 CHFs). However the truncation of the series in eq. 1 to a trinomial (it?., the assumption that the truncated series is “correct” and that all the higher a values are exactly zero) makes these misleading as measures of reliability. The least-squares procedure leads further to standard deviations for the derived quantities ln yl* and In yz* which are unrealistically small (less than 0.1). These limiting activity coefficients are meaningful thermodynamic properties of the mixtures and should not be artifacts of the assumed equation ; yet they can fluctuate wildly as one computes them successively for the 1, 2, 3, 4, or 5-coefficient equations. (Scott and Dunlap6have discussed this problem of series truncation in relation to the evaluation of second virial coefficients.) To avoid this misleading impression of reliability we have suppressed the uvalues except for that of the vapor pressure itself. A measure of the relative precision is the ratio of this standard deviation to the vapor pressure of the equimolar solution ap/P,, (3) The inclusion of a coefficient cy3 for the system CF, reduces the standard deviation up of the CHFs observed vapor pressure from 11.2 to 4.9 mm. The system is not far above the critical solution teniperature and four coefficients (1.920, -0.391, 0.015, -0.307) give a much better fit for the very flat vapor pressure curve.
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Discussion For many mixtures of nonelectrolytes, a reasonable zeroth approximation to the excess free energy per mole is the regular solution-solubility parameter equation7~8
vl
where and 8, are the molar volumes of the two pure liquids; xl and zz, their mole fractions in solution; 41 and their volume fractions; and and bz, their “solubility parameters” (the square roots of the cohesive energy dcrisity or euergy of vaporization per unit volume). In 1958, Scott9 reviewed the published data on fluorocarbon solutions and compared experimental values of the excess free energy with those predicted by eq. 3. For many such solutions, eq. 3 is reasonably adequate, but mixtures of fluorocarbons with aliphatic hydrocarbons are a striking exception, with excess free energies much larger than those corresponding to the factor (61 - 6 J 2 . I t will suffice here to make similar comparisons for the systems reported in this paper. t$z,
The Journal of Physical Chemistry
Table 111 summarizes the physical properties of the pure liquids: the normal boiling point T b , the molar heat of vaporization A R b v at Tb, together with the molar volume 7 and the solubility parameter 6 at the relevant temperatures.
Table 111: Physical Properties of the Pure Liquids &T/
cal.’/2 cm-’/z
Tb/
aribv/
T/
VT/
OK.
kcal.
OK.
cm.8
CH4 CHzFz CHFa
111.7 221.6 189.0
1.96 5.1“ 4.23
CFd
145.1
3.01
CClFa
191.8
3.85
CzHa
184.5
3.52
105 178 i45 178 145 150 105 145 178 150 178
37.0 42 43.5 46.7 54.2 54.9 57.4 62.2 66.5 51.1 54.3
a
6.8 10.6 10.0 9.3 7.1 7.0 8.9 7.9 7.3 8.4 7.6
Estimated.
Table IV compares the observed free energies with those calculated from eq. 3. If we use the rato (Q’obsd - dEBq. 3)/RTas the ineasure of the failure of the simple theory, we see that it is the hydrocarbon fluorocarbon system, CZH6 CF4, which is by far the most anomalous. In striking contrast to this system, the introduction of one chlorine atom into the otherwise coiiipletely fluorinated molecule produces the good agreement (no better can be expected) for iiiixtures of CCIFa with both CHI and CF4; this difference between CF4and CCIFs is hard to explain. The large excess free energy (169 cal. at 151OK. and z = 0.5) for the system CpH6 CF, is confirmed by the high critical solution temperature of 15OoK. This result is completely consistent with the similar ones for all other mixtures of aliphatic hydrocarbons and CHI CF4, n-CsHu n-CeFd, fluorocarbons (q., but, unlike the others, this system consists of two liquids of almost equal molar volumes, a feature which simplifies almost all theories of solutions. Prigogine’o
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(6) R. L. Scott and R. D. Dunlap, J . Phys. Chem., 66, 639 (1962). (7) J. H. Hildebrand and R. L. Scott, “Solubility of Nonelectrolytes,” 3d Ed., Reinhold Publishing Corp., New York, N. Y., 1950. (8) J. H. Hildebrand and R. L. Scott, “Regular Solutions,” PrenticeHall, Inc., Englewood Cliffs, N. J., 1962. (9) R. L. Scott, J. Phw. Chern., 62, 136 (1958). (10) I. Prigogine, “The Molecular Theory of Solutions,” SorthHolland Publishing Co., Amsterdam, 1957, p. 420.
FLUOROCARBON SOLUTIONS AT L O W
TEMPERATURES
3859
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Table IV: Observed and Predicted Excess Free Energies
System
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I CHI CClFa I1 CHzFz CClF3 I11 CHF3 4.. CClFa IV V VI VI1
105,l 178.7 178.2 148.2 178.7 150.6 145.2
-+
+ +
CFa CClFa CpHa CClFs C2He CF4 CHFs CFI
+ +
62 204 123 26 67 169 140
50 I39 55 10 1 26 101
has attributed the large G"" (86 cal. at 110OK. and x = 0.5) for the syskm CR4 CF4to the large difference in molar volumes and has obtained, from his corresponding-states-theory equations, a calculated BE = 55 cal. and for the similar system Kr CF,, a calculated value even larger than that observed. However, this approach founders on this system CzHa CF4. Scott" has shown that a corresponding-states calculation yields only about 9 cal. for BE at x = 0.5, less than 10% of the observed value. It is impossible to escape the conclusion that the difficulty in accounting for the free energies of these mixtiires arises, not from anything in the model of the solution, but rather from the poor approximation that the magnitude of the fluorocarbonhydrocarbon interaction energy is given by the geometric mean of the hydrocarbon-hydrocarbon and fluorocarbon-fluorocarbon energies. Thorp and Scott1 have cited the anomalously large excess free energies for mixtures of CHF3 with CF, and CzF6 as evidence for hydrogen bonding in CHF3. The systems CHZFZ.t CClF3 and CHF3 CCIF3 seem to fit in with this hypothesis. The ratio a l / a o is a measure of the skewness of the excess free energy function and, according to a volume fraction formulation such as eq. 3, should approximate ( I 7 2 - PI)/(P~ Pl) Examination of the a1 values (or the ratios) shows that the seven systems fall rough1.y into three groups: I and IV show positive a1values or a definite skewing of the excess free energy toward the region rich in the component of smaller molar volume, V and VI are nearly symmetrical ( I 7 2 = and only 11, 111, and VI1 are skewed in a direction contrary to prediction. Since these three systems are the only ones which have CH7F2or CHFa as a component their exceptional behavior may be related to hydrogen bonding. The ratio az/cu0is a measure of the flattening of the excess free energy curve. The large positive values of ag/al found" for the systems C6H12 CbH12and C6H14 CeF14 near their critical solution temperatures have been attributed to clustering of the components.
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vl),
+
0.06 0.18 0.19 0.06 0.18 0.48 0.13
T~/OK.
adao
ad a0
0.30 -0.07 -0.10 0.54 -0.04 0.01 -0.21
0.18 0.13 0.18 -0.39 0.11 0.04 0.01
ca. 178 ca. 118
150.1 130.5
Our results here, however, cast doubt on this hypothesis since there is no consistency between the magnitude of a z / a gand the nearness to the critical point. Three of the systems (11, VI, and VII) have very flat regions in the vapor pressure curves, and one might suspect that we had measured the vapor pressure above a two-phase region. Indeed the computed vapor preasure curves (Fig. 3, 7, and 8) do show shallow maxima and minima. I n actuality, however, the measurements were always made at temperatures above the critical solution temperatures, Moreover, for the systems CHzFz CClF, (11) and C2H6 CF4 (VI), where the vapor pressure ineasurements were made very close to the critical temperature, the assumption that the liquid was a one-phase region was checked. For several mixtures in the middle of the composition range where dP/dx is nearly zero, the temperature was lowered after making the vapor pressure measurements, and the onset of phase separation was observed at temperatures about 0.5" below the operating temperatures. One can attempt to calculate critical compositionis and critical solution temperatures from the coefficients a. This requires two assumptions of questionable validity: (a) that the series of eq. 1 may be safely
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Table V : Observed and Calculated Critical Values Tp/
Syetem
I1 I11 VI VI1
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CH2Fz CClFs CHF3 CClFa CzHe CF4 CHFI CFd
OK.
178.7 178.2 150.6 145.2
,---TO/'U.--Cslcd. Obsd.
187 116 170 154 139"
178 118 150 131
--xz~---
Calcd.
Obsd.
0.65 , . . 0.72 ., , 0 . 5 1 0.50 0.61 0.57 0.54"
With four coefficients instead of three.
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(11) R. D. Dunlap, R. G. Bedford, J. C . row, J . Am. Chem. Soc., 131, 2927 (1959).
Woodbrey, and S. D. Fur-
Volume 08,Number 12 December, 1964
BRUCE W. DAVIS
3860
truncated with three coefficients and (b) that the excess free energy GE is temperature independent ( R T a , = g n = constant). With these assumptions one obtains the numbers shown in Table V. I n general, the calculated critical solution temperature is higher than the actual. This is what one would expect from equations which cannot reproduce the very nearly flat coexisterice curve (and free energy isotherms) in the region of the critical point. We see that the introduction of a fourth coefficient 013 imCF,. Better agreeproves the agreement for CHF3 ment is hardly to be expected; indeed, the good
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agreement for CHF3 CClF, is almost certainly fortuitous in view of the great difference between the temperature T p of the vapor pressure measurement and the critical solution temperature T,. Similar conclusions have been reported by Williamson and Scott l2 for the systems perfluoroheptane isooctane and perfluoro-n-hexane n-hexane.
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Acknowledgment. This research was supported by the Atomic Energy Commission. (12) A. G. Williamson and R. L. Scott, J . Phw. Chem., 6 5 , 275 (1961).
On the Theory of Imperfect Gases
by Bruce W. Davis’ Department of Chemistry, University of California, Riverside, California (Received J u l y 8 ~ 7196.4) ~
I t is demonstrated that ;\layer’s theory of imperfect gases can be modified into a free volume theory by assuming a pair potential with a rigid sphere cutoff. Further simplifications and approximations yield a “van der Waals-like” equation of state. The resulting equation is
_ p v - l +
IC T
@/v) [l - 7/24(b/v)]’5/7
+
- -a2
vkT
+
where b = (2r,’3)ua(T), u ( T ) = u(2/(1 dl ~ T / c ) ) ”u2~=, (16n/9)u3eC1(T) exp(8/35e/ k T ) , and Cl(5!’) = [ 3 / 2 ( u / u ( T ) )-3 l/e(u/a(T))g]. Excellent numerical agreement is found between predicted and observed critical constants for argon. Although the liquid state is not formally included, the theory still gives a good estimate for the argon vapor pressure equation.
Introduction Alayer’s treatment of imperfect gases213 has been well developed and gives, within the limitations of siiiiplifying assumptions, a precise and useful expression for the canonical ensemble partition function. The evaluation of the partition function and related thermodynamic quantities is still relatively difficult so that a practical need for further simplification remains. By introducing a hard sphere cutoff in the pairwise interThe Journal of Physical Chemistry
action potential, it will be shown that the partition function may be put in the same form as that for a free volume model of liquids or gases. This modification makes no basic change in the problem of solving ir(1) Petroleum Research Fund Fellow, 1961-1964. (2) J. E. Mayer and M. G. &‘Itlayer,“Statistical Mechanics.” John
Wiley and Sons, Ino., New York, N. Y., 1940, Chapter 13. (3) T. L. Hill, “Statistical Mechanics,” NcGraw-Hill Book Co., Inc., New York, N . Y., 1956, Chapter 5.
