CRITICAL OPALESCENCEOF METHANOL-CYCLOHEXAKE
Oct., 1963
cerning the illterpretation of X-ray data and to the c~~~~~~~~ for permissioll to publish this paper.
L~~~~Brothers
~IsCussIox A. W. Ao,4wsos (University of Southern California) -The peak you attribute to decomposition of the association complex appears to go through a maximum with composition in the case of the MOH-SaLS system (Fig. 4). Do you attribute any sig-
1969
E. D. GoDDaRD.-~~e suspect solid solution phenomena are associated with the temperature shifts in the melt-decomposition peaks; these seem to be more pronounced in the mixed chain length systems. F. M. FOWKES (Sprague Electric Company ).-Is the interaction measured in these nonaqueous mixtures attributable to solvation of the fiodiuni ions by the alcohol groups? E. D. G0DDARD.-This is possible, but how would one establish it?
CRITICAL OPALESCENCE OF METHAKOL-CYCLOHEXANE, TRANSIUISSIOK MEASUREMEYTS BY B. CHU Chenaistry Department, University of Kansas, Lawrence, Kansas Received March 16, 1965 For some binary liquid mixtures in the vicinity of their critical solution temperature, the reciprocal of scattered intensity is a linear function of s2 (s = 2 sin (6/2)) and its slope is connected with the range of molecular forceai. The interaction range ( I ) and the critical solution temperature ( T o ) for the system methanol-cyclohexane have been determined by measurements of the wave length dependence of the total turbidity of t h e solution a t its critical solution concentration as a function of the temperature. Results are in reasonable agreement with values calculated from dissymmetry measurements.
The calculation of intermolecular potential energy functions of one component systems in terms of virial coefficients from P--V-T relations lias been a great tradition. Recently, experimental fourth virial coefficients of 1,etrafluoromethane has been compared with a calculation of Boys and Shavitt‘ of the fourth virial coefficient based on the Lennard-Jones potential. Intermolecular forces also determine most of the properties of liquids, such as solubility in other liquids. The mutual solubility curves for binary liquid mixtures reveal intermolecular interactions peculiar to the individual systems. If optical measurements are carried out on binary liquid mixtures a t small temperature distances above the phase separation temperature, the behavior of concentration fluctuations may be observed from either the angular dependence of scattered intensity or tlie Tmve length dependence of total turbidity. In the vicinity of the critical point, the angular dissymmetry of scattering lias been related to the range of molecular forces ( I ) characteristic for the components of the ~ y s t e m . ~The correlation between concentration fluctuations in neighboring points of binary critical mixtures is characterized by a persistence length ( L or the Debye length), defined as the second moment of a correlation function. It also follows that the square of the Debye length is proportional to the rcciprocal of tlie temperature distance from the critical solution temperature.
turbidity of the system polystyrene-cyclohexane a t its critical solution concentration as a function of the temperature has been investigated.8 We must, however, remember that the approximate theory takes into account only the additional average square of the gradient of the concentration fluctuation. Anomalies have been observed6 and attempts have been made to explain the anomalies, especially on the small angle critical scattering. lo On the other hand, several disagreements arise from experimental difficulties and pitfalls, such as trace impurities and multiple scattering. Therefore, it is desirable to perform our experiments from different approaches. The purpose of this paper is to explore the possibility of estimating the range of molecular forces from the ware length dependence of the total turbidity for biliary critical inixturgs where the interaction range is small (say, I = 5-15 A). The turbidity CY of a biliary mixture a t its critical solution concentration can be calculated by integrating the “critical” scattered intensity over all angles.8 For unpolarized light, we get CY CY
= =
K*3TT K * ~ TJOT r
-
T T // To To 8r2I3 1 +-;sin2+-;sin23 x 3 x
1
e
V
X
2
+ cos2 0 X sin B de 2
1 2
=
12
T/T, - 1
in which l 2 is the second moment of tlie interaction energy. Yarious tyxperiments have verified the theory.*--’ The wave length dependence of the total (1) S. F. Boys and I. Shavitt, Proc. Roy. SOC. (London). A254, 487 (1960). (2) D. R. Douslin, R. H. Harrison, R. T. Moore, a n d J. P. BlcCullough, J . Chem. Phys., 66, 1357 (1961). (3) I?. Debye, ibid., 31, 680 (1959).
(1)
(4) P. Debye, H. COILand D. Woermann, ibid., 33, 1746 (1960). ( 5 ) P. Debye. B. Chu. and D. Woermann, ibid., 36, 1803 (1962).
(6) P. Debye, B. Chu, and H. Kaufmann, ibid., 36, 3373 (1962). (7) P. W. Schmidt and J. Thomas, Physics Department, University of hllssouri, unpublished results on small angle X-ray scattering of argon near the critical point. ( 8 ) P. Debye, D. Woermann, and B. Chu. J . Chem. P h y ~ . ,36, 851 (1962). 19) G. W. Brrsdy and H. L. Frisch. ibid., 35, 2234 (1961). (10) H. L. Frisch and G. W. Brady. ibid., 37, 1514 (1962). (11) W. C. Farrar and H. Brumberger, unpublished results on small angle X-ray scattering of nitrobenzene-n-heptane. (12) D. WcIntyre, A. Wims, and h?. S. Green, J . Chem. Phys., 37,3019 (1962).
B. CHU
197U
T‘ol. 67
,
in which
K*
=
Kh-4
a@) (;c z
TWO DETERMINATIONS
T i = 0 69’5 45.17’C
K is a constant characteristic for the system, c is the critical concentration of the second component expressed in gram per ec., @ is the index of refraction of the solution, T is temperature in absolute units, T , is the critical temperature, - 1 is ( T - T,)/T,, Xis the wave length in the medium, 1 is the average range of forces between component molecules, and 6’ is the scattering angle. Equation 1 holds provided that the reciprocal of scattered intensity against sin2 0/2 gives a straight line. In our case, me have taken methanol as the second component. With s[s = 2 sin 0 / 2 ] as the integration variable, the result is
1 3 r - 1
+ 70
Y
2
In (1
+Y- [
-i
h0-3300d.
0
=4400A
A
I
1
1
40.8
E L
t
0
1
25
30
I
I
I
35
40
Wt, % Methanol. Fig. 1.-Plot of the turbidity a us. concentration for different wave lengths and temperature distances from the maximum phase separation temperature. T,” is the maximum phase separation temperature ( = the critical solution temperature for binary liquid mixtures, such as methanol-cyclohexone, where both components are “monodisperse” liquids), 0.69” is the temperature expressed in relative units.
87r a@=--
(1
1.2
+ y) - 11 +
Y
+
2 In (1 Y ) - 1 Y2 Y
+ 41)
(2b) 0
T-Tc=0.19
i
Experimental Materials.-Methanol (MCB spectro grade) was distilled from magnesium. Cyclohexane (MCB spectro grade) was passed through a column of silica gel and fractionally distilled before use. Method.-The solutions were prepared from weighed amounts of both components which were filtered through a “F” grade sintered disk. The turbidity apparatus was essentially the same as in previous work on the system polystyrene-cyclohexane.* Precautions on multiple scattering and forward scattering have heen taken. Therefore, only details that relate specifically to the methanol-cyclohexane measurements will be added. The Pyrex Beckman cell was fitted with a stopcock using a Teflon plug. Evaporation w-as estimated a t less than 0.5 wt. %. The glass stoppered 0.5-cm. quartz Beckman cell was sealed with epoxy resin cement before immersing it into the thermostated cell holder which was filled with light mineral oil in the light path of the cell block to ensure uniform temperature control. Measurements were carried out with a Cary Model 14 recording spectrophotometer. The absorbancies ( A s = -log 1/10)were determined relative to a cell that contained cyclohexane. Turbidities determined with both Pyrex and quartz cells agree within the accuracy of our measurements. Refractive index measurements near the critical solution temperature were made with a thermostated Abbe refractometer which was equipped with compensating Amici prisms adjustgd so as to yield correct values of refractive index for XO = 5890 A. Refractive indices of both methanol and cyclohexane a t other wave lengths were calculated with the aid of known dispersion curves.10 The index of refraction of the solution was calculated from an empirical mixture rule in which 4 is the volume fraction. The basic prerequisite for the validity of eq. 3 is that the volume should be additive. There is no doubt that this mixture rule 1112
=
# w l
+ +zpz
= Pl
- C#J2(@1
- PZj
(3)
(13) P. Debye, E. Chu, and D. Woermann, J . Polymer Scz., 1, 255 (1963). (14) V. E. Eskin, VysokomoZekuZyarnye Soedzneruya (Soviet High Moleoular Compounds). 2, 1049 (1960) (15) “International Critical Tables,” Vol. 111, McGraw-Hi11 Rook Co New York, N. Y., 1928
.
E
0.4
1-
01
I
I
25
Fig. 2.-Plot
J
I
I
40
30 35 W f . % Methanol.
of the turbidity a a t 3300 and 4400 t,hat a t 3700 A. us. concentration.
A.
relative to
holds when the concentration of one of the two components in the liquid mixture is sufficiently small. The calculated refractive indices of the liquid mixture (or solution), p ~ a ~t dif, ferent wave lengths are essentially being used t o obtain the corresponding wave lengths in the medium, therefore only relative errors are involved in comparing different wave lengths in the medium. A small deviation introduced in this manner has little methanol a t 45.3” significance in the final analysis: 29.8 wt. corresponds t o +to = 0 295. Superscript c denotes critical solution concentration.
r0
Results and Discussion Concentration Fluctuations away from the Critical Solution Concentration.-- The measured absorbalicy As, defined as log Io/I, is related t o the turbidity a a = 2.303As/d
in which d is the thickness of light path. It follows from eq. 1 that the turbidity a is a function of wave length and is essentially proportional to the reciprocal fourth power of the wave length, known as Ragleigh’s law. This strong wave length dependence manifests itself in concentration fluctuations near the temperature of phase separation. Figure 1 shows a plot of ex-
Oct., 1963
1971
CRITICAL OPBLESCEKCE O F &~ETHASOL-CYCLOHEX-4K-E I
I
i/
I
T, = 0.70'= 45.I8 ' C
I
1
7. Fig. 5.-Plot
of @(y ) us. y.
1.0
0.8
0.6
1.2
1.0
1.4
Fig. 3.-Plot of the reciprocal of the turbidity l/a us. temperature for different wave lengths a t the critical solution concentration (29.8 wt. 7oomethanol). The wave length in vucuo varies from 3300 to 6000 A. .70
. .68
r v I4
+
-.
0.9 (Y)
fly) 0.8
I
1
.'\\
T,=070°=45180C
.e
=lid+2A
0.7 04
08
712
16
i/k x I O .
Fig. 6.-Plot of @ ( y ) / @ O ( y vs. ) the reciprocal of the square of corresponds to @(y) at the wave length in the medium. @'(y) Xo = 6000A.
tance over which the fluctuation persists increase strongly as the critical solution temperature is approached. The Debye theory tells us that the reciprocal of the turbidity for large temperature distances from the critical temperature takes 011 the form*,'g Fig. 4.-Plot
of T* us. the reciprocal of the square of the wave length in the medium.
tinction coefficient a us. conceiitration a t small temperature distances from phase separation temperature for two different wave lengths. If we neglect the wave length dependence of p and dp&, then X14CU1(T,C) X24az(T,c) (4) where subscripts 1 and 2 denote measurements St,t two different wave lengths, such as 4400 and 3300 A., respectively. The turbidities in Fig, 2 are relative to a measured a t a wave length of 3700 A. The reciprocal of the turbidity at constant wave length is directly proportional to the temperature over the whole concentration range. This behavior is in good agreement with Smoluchowskil6 and Eiiistein'sl7 fluct uatioii theory. Critical Opalescence. In the vicinity of the critical point, both the magnitude of fluctuation and the dis-
(16) hl. Smoluchowski, Ann. Phvszk, 25, 205 (1908) (17) A. Einstein, zbzd., 38, 1275 (1910)
T/a
=
const. (7' - T*)
T*
=
T o (1 -
8n2
(5)
12
7G)
T* corresponds to the intercept of the temperature axis extrapolated from the linear portion of the " a vs. T curve. For smaller values of I , it is more difficult to determine the interaction range. The slope, - (8?r2/ 6)TCl2,from a plot of T* us. I/X2 is small and only wider range on the wave length dependence of the total turbidity as a function of temperature may partially compensate this inadequacy. For I = 29 8., T* covgrs about 0.2' oveg a wave length range of 36004800 A.; but for I = 11 A., T* covers a range of only about 0.06' for a wave length range of 3300-6000 A. Therefore, temperature control of the sample under observation is very critical aiid the interaction range can only be estimated when I is small. Figure 3 shows a plot of the reciprocal of the turbidity against temperature (in (18) In our temperature range. Tis relatively constant althouxh T - Tois not. 80 eq. 5 can be written as' l / a = oonst [ ( T - To) t(Z*/h*)].
+
B. CHU
1972
relative units). The linear relationship between l / a and T - T , is in agreement with the general prediction of Chows' class I1 niixtures which include the system methanol-cyclohesane.'S We do, however, expect a bend toward the critical solution temperature, To, a t very small temperature distances from T , in the l / c y us. T- T o plot.8 Multiple scattering has preyented us from getting too close to the critical point. I n addition, although a is inversely proportional to (AT 1)' the t which is [(87r2/6)(Z2/X2))T, is actually a function of the wave length. Figure 4 shows a plot of the so determiiied ?'* us. I / X 2 . L, (turbidity) = 11 &. 2 A.; 1 (light scatteringE) = 12.7 A. The critical solution temperature determined by visual observation a t 29.8 wt. yo methanol = 45.17'. T , (extrapolated from eq. 5a) = 45.18'. The absolute magnitude of the critical solution temperature is uncertain since no calibrated resistance thermometer was used. However, the results are reproducible. At a fixed temperature distance from the critical solution temperature, eq. 2 and 2c reduce to
Vol. 67
in which Xo is the wave length in vacuo. The dispersion factor
+
y = Kz12/X2
F4
€-1 4a (nlal € + 2 3
+
( 8 r 2 / 3 ) / ( 7- 1))are coastants. Figure 5 shows a plot of +(y) us. y. When y is small (say
