PVT RELATIONS IN THE SYSTEM METHANE-TETRAFLUOROMETHANE
3477
Pressure-Volume-Temperature Relations in the System Methane-Tetrafluoromethane.
I.
Gas Densities and the
Principle of Corresponding States1
by D. R. Douslin, R. H. Harrison, and R. T. Moore Contribution N o . 163 from the Thermodynamics Laboratory of the BartlesviUe Petroleum Research Center, Bureau of Mines, U.S. Department of the Interior, Bartlesville, Oklahoma 74003 (Received March 83, 1967)
~
The methane-tetrafluoromethane system was examined for conformity with the principle of corresponding states in terms of the Boyle reference point, the critical reference point, and mixing rules as applied to the reference temperatures and volumes. Experimental values of gas density are given for pure methane, pure tetrafluoromethane, and three mixtures (0.25, 0.50, and 0.75 mole fraction of methane) over a temperature range from 0 to 350°, a pressure range from about 16 to 400 atm, and a density range from 0.75 to 12.5 mole L-'. Values are derived for the second, third, and fourth virial coefficients of the mixtures, B M ,CM, and D M ,respectively, and for the cross-term virial coeficients, BIZ,C112, C122, 0 1 1 1 2 , D1122, and D12z2. Based on the experimental results for BIZ,the geometrical-mean mixing rule was found to be unsatisfactory for a corresponding states correlation. However, in terms of the Boyle temperature, TB,and Boyle volume, VB = ( TdB/dT),T,, the experimental reduced second virial data for the methane-tetrafluoromethane system were found to follow the corresponding states function with high accuracy.
Introduction Gas density measurements were undertaken on the methanetetrafluoromethane system to provide sufficient information, internally consistent over a broad range of temperature, pressure, density, and mole fraction, for a definitive test of the laws of mixing in terms of the principle of corresponding states, the intermolecular potential energy, or the derived thermodynamic properties. The work was carried out as a part of a continuing project of the U. S. Bureau of Mines on the PVT relations of hydrocarbons, fluorocarbons, and mixtures of hydrocarbons and fluorocarbons. Methane and tetrafluoromethane were chosen because they have positions as first members of their respective homologous aliphatic series and because they are structurally similar, nonpolar, quasispherical molecules that are most likely to respond to theoretical treatments developed for central force fields. Thus, they are likely to help bridge the gap in intermolecular potential theory between simple and
complex particles. In addition, the upper temperatures to which these materials are thermally stable are high enough for the repulsive forces to become significant in the second virial coefficient. Values for the second, third, and fourth virial coefficients, B M , CM, and DM, were derived from unsmoothed gas density data on three mixtures. These results were combined with virial coefficients for pure methane and pure tetrafluoromethane, reported previou~ly,~-* to obtain cross-term virial coefficients for (1) The work upon which this research is based was conducted in part under an Interservice Support Agreement between the Air Force Office of Scientific Research, Office of Aerospace Research, U. 5. Air Force, Project No. 9713, Task No. 9713-02, and the Bureau of Mines, U. S. Department of the Interior. (2) D. R. Douslin, R. H. Harrison, R. T. Moore, and J. P. hfcCullough, J. Chem. Eng. Datu, 9, 358 (1964). (3) D. R. Douslin, R. H. Harrison, R. T. Moore, and J. P. McCullough, J . Chem. Phys., 35, 1357 (1961). (4) D. R. Douslin, "Progress in International Research on Thermodynamics and Transport Properties," The American Society of Mechanical Engineers, United Engineering Center, New York, N. Y.,1962, pp 135-146.
Volume 71, Number 11 October 1967
3478
unlike molecule interaction^.^ I n the second and third papers of this series, the mixture phenomena will be examined in terms of the mixed parameters of various potential functions and in terms of the thermodynamic functions, enthalpy, entropy, and Gibbs free energy of the compressed gas mixtures. I n the present paper, examination of the cross-term coefficientswill be limited to the principle of corresponding states and to mixing rules applied to corresponding states parameters. Most of the work reported in the literature on hydrocarbon-perfluorocarbon interactions has concerned phenomena in the liquid phase. A very limited effort has been directed toward a study of interaction effects observed in mixtures of hydrocarbon plus perfluorocarbon gases. In a definitive study, Hamann, Lambert, and Thomas6 reported that they were unable to correlate second virial coefficients for the methane-sulfur hexafluoride system by means of Guggenheim and McGlashan's7 extension of the principle of corresponding states. Garner and McCoubrey* studied two binary hydrocarbon-perfluorocarbon systems: n-pentane-perfluoro-n-pentane and n-pentane-perfluoro-nhexane. They showed that correlations of the second virial coefficients, based on mixing laws for the Lennard-Jones [12, 61 potential parameters, were quite inadequate to explain the experimental results.
Experimental Section Gas densities for three mixtures, 0.749787, 0.500050, and 0.250066 mole fraction of CH4, were determined with the apparatus and by the method used previously on the pure c o m p ~ n e n t s . * ~ These ~ ~ ~ Jmixtures, ~ hereafter referred to in text and tables by the nominal designations 0.75, 0.50, and 0.25 mole fraction of CH,, were prepared from highly purified methane (99.994 mole %) and tetrafluoromethane (99.985 mole %) contained in the weighing bombs shown as part of the mixing manifold (see Figure 1). The gases were condensed from the weighing bombs with liquid hydrogen refrigerant into the highly evacuated sample liner of the compressibility cell, and the filled liner was sealed off a t the Pyrex capillary. The uncondensed sample remaining in the small void space of the capillary below the valve was negligible in terms of total sample or mole fraction. Thus, the final weight of sample in the cell liner was checked directly against the corresponding loss of sample from the weighing bombs. All weighings were made with an accuracy estimated a t 0.1 mg. Since the total sample used was about 0.1 mole, the calculated mole fractions given above were estimated to be accurate to about 0.00003, 0.00006, and 0.00009 mole fraction, respectively. However, the mole fraction values used were retained a t six decimal figures to The Journal of Physical Chemistry
D. R. DOUSLIN, R. H. HARRISON, AND R. T. MOORE
Figure 1. Sample mixing manifold.
provide the maximum possible precision in evaluating the virial coefficients and the effects of mixing. Calculations of moles of sample and mole fractions were based on the following atomic weights: H, 1.0080; C, 12.011; F, 19.000. Adjustments were made in the molal calculations for 0.005 mole % nitrogen in the methane and 0.0034 mole yo oxygen plus 0.012 mole Yo nitrogen in the tetrafluoromethane. In previous work on pure tetrafluoromethane the impurities in the sample were estimated to be 0.04 mole % maximum. Analysis of the present sample, which was somewhat purer, was obtained by a gas-liquid partition chromatographic method known to give more accurate results than those previously obtained from the mass spectrometer. Pressures were measured on a deadweight gauge that was calibrated against the vapor pressure of carbon dioxide a t O", taken to be 26,144.7 mm, the value determined by Bridgeman." A more recent determination by Greig and Dadson12 gave a value approximately 0.025% lower, which appears to be more accurate. Possible small errors in pressure scale can be classed with a number of other small systematic errors that might be introduced through the gas constant, atomic weights, variation in the normal isotopic ratio C12/C13, or weight of sample. Adjustments for these errors, as a group, were made by the method discussed in ref 2. (5) T o preserve internal consistency and possible cancellation of systematic errors in the evaluation of mixture effects, measurements from other laboratories were not included in the present correlations. Comparisons of P V T data for pure methane and pure tetrafluoromethane from various sources were given previously in ref 2 and 3. (6) S. D. Hamann, J. A. Lambert, and R. B. Thomas, Australian J . Chem., 8 , 149 (1955). (7) E. A. Guggenheim and bl. L. McGlashan, Proc. Roy. SOC. (London), A206, 448 (1951). (8) bl. D. G. Garner and J. C. RIcCoubrey, Trans. Faraday Soc., 5 5 , 1524 (1959). (9) J. A. Beattie, Proc. Am. Acad. Arts Sei., 69, 389 (1934). (10) D. R. Douslin, R. T. Moore, J. P. Dawson, and G. Waddington, J . Am. Chem. SOC.,80, 2031 (1958). (11) 0. C. Bridgeman, i b i d . , 49, 1174 (1927). (12) R. P. Greig and R. S. Dadson, Brit. J . AppZ. Phys., 17, 1633 (1966).
PVT RELATIONS IN THE SYSTEM METHANE-TETRAFLUOROMETHANE
Gas density measurements were taken along isotherms a t 25' intcrvals from 0 to 350" and at isometric points spaced a t 0.5 mole 1.-' intervals (except for the first interval) from 0.75 to 12.5 mole 1.-'. The pressure ranged from about 16 to 400 atm. Temperature of the compressibility cell was controlled and measured to 0.001" with platinum resistance thermometers that were calibrated a t the Kational Bureau of Standards in terms of the international temperature scale [T, "K = t, " C (Int., 1948) 273.161. Data previously reported for pure methane and pure tetrafluoromethane were also on this scale. To maintain a systematic basis for the entire system, this temperature definition was retained through the present study. For similar reasons the old definition of the liter (1 1. = 1000.028 cm3) was used in calculating gas densities. However, the cubic centimeter was the volume unit used to express the virial Coefficients. Also, because of the need to refer the present data on the mixtures to previously published data on the pure components, methane and tetrafluoromethane, the recently defined value of the gas constant13 was not used. The following two values of the gas constant were used as appropriate: R = 0.0820544 1. atm deg-l mole-' and R = 82.0567 cm3 atm deg-' mole-'.
+
3479
lated from unsmoothed experimental values of P, 8, and T (Table I), and the values were extrapolated to zero density to obtain B M ,CX,and DII (Table 11) as coefficientsof an infinite series in 1/V. At the lower temperatures the second virial coefficients are most negative for pure tetrafluoromethane (Figure 2), but they follow a trend toward more positive values as concentrations higher in mole fraction of methane are approached. At the upper temperatures a complete reversal in the order has occurred; however, the temperaturemole fraction surface of the plot is smooth and shows no inconsistencies. Also, second virial coefficients exhibit a maximum with respect to mole fraction over a well-defined temperature range extending from about 125 to 335". The third virial coefficients (Figure 3), unlike the second virial coefficients, show regular variation from methane to tetrafluoromethane over the entire experimental temperature range. The fourth virial coefficients are also regular, although the precision and accuracy of the points are considerably lower. The characteristic increase of the fourth virial coefficients with respect to temperature, predicted theoretically by Boys and Shavitt14and by Barker and Monaghan15 for substances that follow
Gas Densities
Unsmoothed values of the measured pressures of the mixtures are given in Table I as functions of density and temperature. Similar data for pure methane and tetrafluoromet hane, also included in the tabulation, may be found in ref 2-4. The number of significant figures carried is consistent with the precision of the measurements. The accuracy, which is roughly a factor of 10 lower than the precision, was discussed in previous publications. 2 , 3 lo
Ot
-20
-
- -40 2 ro '0
E m -6OL
I
4
9
Virial Coefficients
A graphical method3 was used to evaluate the temperature-dependent coefficients, B M ,Cor, and DM, in the virial expansion for mixtures in eq 1. PV
=
RT[1 $- (l?o~/V)
+
+
(CM/V~) ( h ' V 3 )
BM = lim[(PV/RT) l/V-O
Dhf
=
lim { [((RVIRT)
1/v-o
+ ... I
- BM]V
'
0
1
50
1
100
I
150 1.
(1)
- l]V
CM = lim[((PV/RT) - 1)V 1 / v-0
I d
-120
(3)
- l ) V - BMIV- CM]V
(4) At each temperature the terms in eq 2-4 were calcu-
I
1
I
1
200
250
300
350
ec.
Figure 2. Second virial coefficients for the CH4-CF4 system. Insert shows expanded area above 200O: 0, pure CH4; A, 75% CH4-25% CF4; 0, 50% CH4-50% CFa; A, 25% CH4-75% CFa; 0, pure CF4.
(13) F. D. Rossini, Pure AppZ. Chem., 9, 453 (1964). Boys and I. Shavitt, Proc. Roy. SOC.(London), A254, 487
(14) S. F. (1960).
(15) J. A. Barker and J. J. Monaghan, (1962).
J. C h a . Phys., 36, 2564
Volume 7 1 , h'umber 11
October 1067
D. R. DOUSLIN, R. H. HARRISON, AND R. T.MOORE
3480
h
d
x
0
If
U
dY
L9 LS
s 0
v)
N
The Journal of Physical Chemistry
h m
C
s
m $1
P v T RELATIONS I N THE
3481
SYSTEM ~~~ETHANE-TETRAFLUOROMETH.4NE
Volume 71, Sumber I 1
October 1967
3482
The Journal of Physical Chemistry
D. R. DOUSLIN, R. H. HARRISON, AND R. T.MOORE
PV T RELATIONS IN THE SYSTEM METHANE-TETRAFLUOROMETHANE
3483
Volume 71. Number 11
October 1967
D. R. DOULSIN, R. H. HARRISON, AND R. T. MOORE
3484
,
0
7 60
1
1
I
0
50
I
100
I
I
150 t.
200
I
I
250
300
350
'C.
Figure 3. Third virial coefficients: 0, pure CH,; Al 75% cH4-25% CFI; 0 , 50% CHa-50% CF4; Al 25% CH4-75% CF4; 0, pure CF4.
the Lennard-Jones potential in this temperature range, was obtained for the mixtures as well as for the pure materials. Theoretical expressions for the virial coefficients of mixtures in terms of cross-term coefficients can be obtained from the configuration integral for multicomponent systems. The expression for the second virial coefficient of a binary mixture BM = Xi2Bii
~
I
+ 2XiXzBiz + X2'B22
(5)
in which X1 is the mole fraction of methane, X 2 is the mole fraction of tetrafluoromethane, B11 is the second virial coefficient of methane, Biz is the second virial Coefficient for the unlike molecule interaction, and B22 is the second virial coefficient of tetrafluoromethane, is usually referred to as the Lennard-Jones and Cooki6 relationship. Values of Bl2 (Table 111) were calculated from eq 5 for each of the three mixtures by direct substitution of the experimental values of Bof, Xi, X2, B I ~ , andBz2. The variation of B12 with mole fraction reflects the experimental inaccuracy of the data on the two pure substances as well as of the data on the mixtures because the evaluation of B12 by difference collects all of the error in this term. Therefore, the small variational trend of B12 with mole fraction is thought to be systematic in the experimental data rather than indicating that eq 5 is not completely representative of the nature of the system. A simple arithmetical average, BI2,for the three mixtures is in very close agreement (16) J. E. Lennard-Jones and W. R. Cook, Proc. Roy. Soc. (London). A115, 334 (1927).
The Journal of Phusical Chemistrg
PVT RELATIONS IN THE SYSTEM METHANE-TETRAFLUOROMETHANE
Table 111: Cross-Term Second Virial Coefficients of Methane-Tetrafluoromethane as a Function of
Table IV : Mixed Third Virial Coefficients
+
-Nominal OC
0 25 30 50 75 100 125 150 175 200 225 250 275 300 325 350
---
composition, mole fraction of CHd0.50
0.25
-62.30 -48.88 -46.37 -37.64 -28.67 -20.78 -14.44 -5.78 --3.62 +0.45 4.43 7.59 10.69 13.24 13.51 17.56
-62.35 -48.75 -46.35 -37.67 -28.56 -20.63 -14.09 -8.30 -3.26 +0.99 4.90 8.27 11.31 14.07 16.60 18.89
t,
-
0.75
Biz,a cma mole-1
-61.56 -47.81 -45.54 -36.76 -27.70 -19.87 -13.40 -7.90 -2.74 $1.62 5.48 8.98 12.16 14.98 17.55 20.18
-62.07 -48.48 -46.09 -37.36 -28.31 -20.43 -13.98 -8.33 -3.21 $1.02 4.94 8.28 11.39 14.10 16.55 18.88
B I Z cma , mole-'
' Averaged arithmetically.
with the results for the 0.50 mole fraction of CH4 mixture. Expansion of the configuration integral for the third and fourth virial coefficients of binary mixtures
CM =
X13C111
-t3X1'XzC112
+
+
3XiX22C~22
X23C222
(6)
Exptla
Calcdb
0 25 30 50 75 100 125 150 175 200 225 250 275 300 325 350
+34.76 30.26 29.90 28.59 26.65 25.01 23.76 22.87 22.25 21.44 20.91 19.85 18.92 18.32 17.34 16.17
f36.5 32.4 31.7 29.2 26.7 24.7 23.2 22.0 21.2 20.3 19.7 19.1 18.5 18.0 17.7 17.5
a Averaged arithmetically. c222(cllllc222)1/~.
+ ~ X I ~ X ~+D I I ~ ~ 6X1~X2~01122 + ~ X I X Z ~ D+I Z Z ~
X24D2222
(7)
generates two cross-term third virial coefficients and three cross-term fourth virial coefficients. Since the three sets of values that could be derived for Cl12 and C122from the experimental data were in fairly good agreement and showed no significant trends, they were simply averaged arithmetically (Table IV). However, if the averaged values for Cllz and ClZ2were to be resubstituted into eq 6, a small but insignificant error in the calculated values of CMwould necessarily appear. Calculated cross-term coefficients (columns 3 and 5, Table IV), obtained from the empirical combinatory proposed by Connolly, rule C Z j k= (CttzCjjICkkk)l/S are believed to be as accurate as the experimental results. Since there are three cross-term fourth virial coefficients, the data on three mixtures provided only a single numerical solution for each coefficient. The results of this solution are given (Table V) only to
C122
$49.74 45.65 44.28 40.64 37.24 33.96 31.40 29.20 27.39 26.58 23.59 23.54 25.40 25.35 25.70 26.35
+50.9 44.4 43.2 39.6 36.0 33.3 31.3 29.6 28.4 27.3 26.4 25.6 24.9 24.2 23.7 23.3
* CIIZ= C111(C222/Clll)1/a.
CIZZ=
Table V : Experimental Mixed Fourth Virial Coefficients
x lo-', cm9 mole-'
01112
OC
0
X1*D1111
X 10-9,cma mole-2 Exptla CalcdC
Cln X 10-2,cmB mole-2
OC
1,
and
DM =
+ 3X1X22C122 + xzsc222
cx = Xl~Clll 3X12X2ClII
Composition and Temperature
t,
3485
25 30 50 75 100 125 150 175 200 225 250 275 300 325 350
$1.7 5.6 6.5 7.2 4.6 3.2 2.4 2.7 2.3 2.4
3.8 5.1 6.0 5.9 6.8 7.5
01122
x
10 -4,
01222
x
10-4,
ern9 mole-3
cmg mole-3
-1.5 -7.1 -8.1 -8.8 -1.2 $3.8 9.0 9.7 12.0 12.7 12.1 11.4 11.9 13.4 13.9 14.5
$0.7 5.1 6.0 8.3 7.1 7.6 7.7 10.3 11.5 13.1 14.5 15.7 16.0 15.0 14.4 13.8
show order of magnitude and general trend with temperature. Because of the obvious limitations on their accuracies, they are probably of little value for testing combinatory rules, and no attempt was made to use them in this manner. (17) J. F. Connolly, Phys. Ruids, 4, 1494 (1961).
Volume '71, Number I1 October 1967
D. R. DOULSIN, R. H. HARRISON, AND R. T. MOORE
3486
Table VI : Critical and Boyle Constants Experimental data
CH4 CF4 CF4
509.3 518.14
(B12)
467.0 410. 5d
Argon Krypton
54.34 104.13
191.06b 227.6F 194.1 (pseudo)
...
...
...
193.2 (pseudo)
... ... ...
78.75 39. 87d 46.92'
584.4'
99.0
... ...
..
Mixing Rules Vl2B t
TIZB,OK
CHrCF4
v120,
om3 mole-1
513.7 (eq 8) 496.0 (eq 10)
Tis,,
76.56 (eq 9)
...
cm' mole-'
OK
192.6 (eq 11) 185.6 (eq 13)
140.9 (eq 12)
...
'
By definition, VB = ( T d B / d T ) p r , . K. A. Kobe and R. E. Lynn, Jr., Chem. Rev., 52, 117 (1953); American Petroleum Institute Research Project 44 on "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds." D. L. Fiske, Refrig. Eng., 57, 336 (1949). Evaluated from the second virial coefficient data of Michels, et U Z . , ~ ~and Whalley and Schneider.20 e Evaluated from the second virial coefficient data of Beattie, et aZ.21
Application of the Principle of Corresponding States Before an attempt was made to apply the mixing rules to corresponding states parameters,' the experimental second virial coefficients,Bll, BZZ, and B12,were examined for congruity with the principle of corresponding states in terms of the Boyle and critical points. The Boyle-point constants, TB and VB = ( T U / dT)T-TB,18were found to be more tractable parameters than the critical constants T , and V,. The reason is that the reduced Boyle parameters for the unlikemolecule interactions can be evaluated experimentally and tested to determine if they agree with the corresponding states function defined by the pure components. A similar test Rith reduced critical parameters is not possible because the unlike-molecule critical temperature and volume are not experimentally determined quantities. Thus, the examination of a mixture in terms of the Boyle point has the advantage that it allows one t o decide whether the experimental crossterm virial coefficients follow the principle of corresponding states and, if they do, allows the behavior of the mixing laws to be more clearly interpreted. The Boyle-point corresponding states function (Figure 4) is remarkably accurate for the second virial coefficients of the cross-term as we11 as for the second virial coefficients of pure methane and pure tetrafluoromethane. A r g ~ n and ' ~ ~krypton,21 ~ ~ which also agree accurately uith the Boyle function, were included to show that the function does indeed represent true corresponding states behavior. The Boyle constants (Table VI) were derived from accurate empirical representaThe Journal of Physical Chemistry
tions of the second virial data by the Lennard-Jones [n,61 potentials and their first derivatives with respect to temperature. For CH, the L-J [28,'6] potential, 8 = 248.3"K and bo = 54.39 cm3 mole-', was fitted to the data of Table 11; for CF, the L-J [500,6] potential, 6 = 416.6"K and bo = 89.50 cm3 mole-', was fitted to the data of Table 11; for the cross-term Blz the L-J [30, 61 potential, 6 = 234.2"K and bo = 77.98 cm3 mole-', was fitted to the data of Table 11; for argon the L-J [17, 61 potential, 6 = 155.0"K and bo = 44.14 cm3 mole-', was fitted to the data of JIichels, Wijker, and Wijkerlg and Whalley and SchneiderZ0;for krypton the L-J [24, 61 potential, 0 = 266.1"K and bo = 48.20 cm3 mole-', was fitted to the data of Beattie, Brierley, and Barriault.*l Contrary to the general trend of results indicated by Guggenheim and McGlashan7 for a variety of systems following the corresponding states principle, the geometrical mean rule for Boyle temperature yizB=
( ~ i ~ ~ z ~ ) ' / *
(8)
and the Lorenta rule for Boyle volume ~ 1 2 B=
+v
did not furnish values of TlzB and
~ ~ ~ / (9) ~ ) ~ V12,
that corre-
(18) T.Kihara, Rev. Mod. Phys., 25, 831 (1953). (19) A. Michels, Hub.Wijker, and Hk Wijker, Physicu, 15,627 (1949). (20) E. Whalley and W. G. Schneider, J . Chem. Phys., 23, 1644 (1955). (21) J. A. Beattie, J. S. Brierley, and R. J. Barriault, ibid., 20, 1613 (1952).
PVT RELATIONS IN THE SYSTEM METHANE-TETRAFLUOROMETHANE
3487
Results of the examination of the above system in terms of the critical reference points are presented in Figure 5. The critical volume of tetrafluoromethane is not determined. Its experimental critical temperature, reported by FiskeJZ5does not appear to conform to a corresponding states relationship with methane. Therefore, the pseudo-critical temperature and volume (Table VI) were derived by fitting second virial coefficient data for tetrafluoromethane to the corresponding states second virial coefficient function defined by methane. Values for the mixed crit.ica1 or pseudocritical constants, calculated by equations similar to eq 8-10, as follows =
T12.
(TIJ“,)”~
vI2,= 1/8(V1,1/8+
(11)
~ 2 , ” ~ ) ~
(12)
1
failed to provide a satisfactory corresponding states correlation of the cross-term second virial coefficient. The deviations from the corresponding states function were in the same direction and about the same magnitude as found in the Boyle-point correlation.
6 - K r y p t o n , r e f 21
- . 9 y
i
I
05
0.6
I
0.7
I
I
I
1.1
1.2
1.3
I
1
I
0.8
0.9
I .O
c
T/Te Figure 4. Corresponding states correlation in terms of Boyle-point parameters, -; e q 8 a n d 9 for BI1,- - - -; eq 9 and 10 for BIZ, *
- -.
sponded with the experimentally determined values in Table VI. Also, the reduced mixed second virial coefficient values, shown as the dashed line in Figure 4, no longer a g r e d with the corresponding states function. When the mixture rule for Boyle temperature contained a factor for the difference in volumes of the molecules, as suggested by Hudson and McCoubreyZ2
the calculated reduced second virial coefficients moved closer to the corresponding states line, but the agreement was still unsatisfactory. No correction was applied for the ionization potentials because they are not greatly different for methane and tetrafluoromethane. Values for methane vary between 13.1 and 14.5 ev,23 and a recently estimated value24for tetrafluoromethane lies between 14.9 and 15.1 ev. Thus, a correction factor, 2 ( I J 2 ) 1 / 2 / ( 1 1 Iz) = 0.998, based on the greatest differences in the ionization potentials, makes no significant difference in the estimated mixed Boyle temperature.
+
- 0.6
I
I
I .5
2.0
I
2.5 T/Tc
I
3.0
5
Figure 5. Corresponding states correlation in terms of critical point parameters: 0, pure CHI; 0, pure CFI; pseudo-critical constants, Table VI; X, BIZ,eq 11 and 12; 0 , Biz, eq 12 and 13.
Conclusion The geometrical-mean rule for reduced temperature is not an accurate basis for extending the principle of corresponding states to the cross-term second virial (22) G. H. Hudson and J. C. McCoubrey, Trans. Faraday SOC.,56, 761 (1960). (23) T. M. Reed, 111, J . Phys. Chem., 59, 428 (1955); J. D. Morrison and A. J. C. Nicholson, J . Chem. Phys., 20, 1021 (1952); UT.C. Price, Chem. Rev., 41, 257 (1947). (24) R. W. Kiser and D. L. Hobrock, J . Am. Chem. Soc., 87, 922 (1965). (25) D. L. Fiske, Refrig. Eng., 57, 336 (1949).
Volume 71, iVunzber 11 October 1967
J. F. PADDAY
3488
coefficients of the methane-tetrafluoromethane system. This conclusion was reached after examining the system in terms of both the Boyle reference points and the critical reference points. An advantage possessed by the Boyle-point correlation, not widely recognized apparently, is that the cross-term reduced variables can be determined experimentally, thus making a direct test of the corresponding states principle on mixtures possible without involving the mixing rules. Alternatively, the mixing rules can be tested on a corresponding states correlation without having to contend, possibly,
with an unknown factor associated with the exactness of the corresponding states principle itself. It is evident from the values of Table VI that the geometrical-mean Boyle temperature lies between that of methane and tetrafluoromethane and, therefore, is nearly 50°K above the experimental value of the Boyle temperature of the mixture, T12B. If Boyle temperatures are assumed proportional to interaction energy, the obvious conclusion is that the unlike molecule interaction represents a new situation not related to the like molecule interactions by any kind of average.
The Halide Ion Activity of Dyes and Organic Salts in Aqueous Solution
by J. F. Padday Research Laboratories, Kodak Limited, Wealdstone, Harrow, Middlesex, England
(Received N a r c h 27, 1967)
The ionic interaction hypothesis to explain metachromasy of some cationic dyes in aqueous solution has been tested experimentally. Electrometric measurements of the halide counterion activity of these dyes at concentrations where marked metachromasy took place showed no indication of ion-counterion association. Similar measurements of the counterion activity of alkyltrimethylammonium bromides in aqueous solution showed that above the critical micelle concentration, obtained from surface tension measurements, strong ion-counterion association took place. Explanations of metachromasy of the dyes, particularly that in terms of the formation of dimers, are discussed.
The metachromatic behavior of many types of dyes in aqueous solution has been the subject of investigat i o n ~ ' - ~which have been reviewed elsewhere.' The metachromatic behavior of some cyanine, carbocyanine, and thiacyanine dyes in aqueous solution has attracted particular attention because the spectral perturbations that are dependent on dye concentration follow a general pattern. This pattern involves the appearance of a broad shoulder, hypsochromic to the main peak, as the concentration of dye increases until at relatively high concentrations, the shoulder develops into the principal absorption peak. The hypsochromic band has been attributed to an aggregation state of the dye, the dimer, by Scheibe and many other investigator^.^ T h e Journal of .Physical Chemistry
McKay6 found no evidence of aggregation, using a polarographic technique, of one cyanine dye that exhibited metachromasy. Hillson and LIcKayl then attempted to explain metachromasy of cyanine dyes, rhodamine B, methylene blue, and some other dyes in (1) R. B. McKay and P. J. Hillson, Trans. Faraday Soc., 61, 1800 (1965); P.J. Hillson and R. B. McKay, iyature, 210, 297 (1966).
(2) M. J. Blandamer, M. C. R. Symons, and G. S. P. Verma, Chem. Commun., 629 (1965). (3) F. Feichtmayr and J. Schlag. Ber. Bunsenges, Physik. Chem., 68, 95 (1964). (4) W.West and S.Penrce, J . Phys. Chem., 69, 1894 (1965); B. H. Carroll, Phot. Sci. Eng., 5 , 65 (1961); S. E. Sheppard, Rev. Mod. Phys., 14, 303 (1942); G. Scheibe, Kolloid Z.,82, 1 (1938). (5) R. B. McKay, Trans. Faraday SOC.,61, 1787 (1965).
