4060
J. Phys. Chem. 1902, 86, 4068-4073
Excess Volumes In the Solutions Benzene4 ,/Benzene-d,, Cyclohexane-h,,/Benzene4 , Cyclohexane-h ,,/Benzene-d,, Benzene-h,/Benzene- d,/H,O
and
Malay K. Dutta-Choudhury,f Gerald Dessauges,z and W. Alexander Van Hook* Chemistry Department, University of Tennessee at Knoxviile, Knoxviile, Tennessee 379 16 (Received; February 11, 1982: In Final Form: June 7, 1982)
Excess volumes are reported for the title solutions across the entire concentration range at 25 and 45 "C. For the solution of isotopic isomers, C&/C6&, the excess volume shows a complicated concentration dependence m3/mol near Xch = 0.33 and a maximum of about 2 x m3/mol near with a minimum of about -6 X Xc, = 0.8 (25 "C). The effect is sensitive to trace water content. For benzene/cyclohexane solutions the volumetric isotope effects display simpler concentration dependences. Thus, the difference, pC6Hg/C6Hl2pCeDs,CsH12, within experimental error shows simple minima at Xbenzene = 0.5 of about -7 x (25 "C) and -12 x IO4 (45 "C) m3/moi. After comparing the present results for C6H6/C6H12 solutions with those of previous workers, we discuss isotope effects in terms of the theory of isotope effects on the properties of condensed phases.
Introduction Isotope effects on the thermodynamic properties of solutions, particularly solutions of isotopic isomers, one in the other, are of interest because they constitute a particularly direct measure of the effect of intermolecular forces on the condensed-phase partition function.' Jancso and Van Hook2 have discussed the thermodynamics of solutions of isotopic isomers, one in the other, and the same authors3 have analyzed free energy data (obtained from vapor pressure measurements by Jakli, Tzias, and Van Hook4) for the SOlUtiOnS C&j/C&, C6Hlz/C,jD,z, and various benzene/cyclohexane or deuteriobenzene or cyclohexane solutions using a detailed molecular model. It was shown that the volume dependence of condensedphase internal frequencies makes an important contribution to the partition function in these solutions. In fact, it dominates in determining the excess free energy of solutions of isotopic isomers, one in the other. Thus, molar volume isotope effects were shown to be of critical importance in arriving at an understanding of the excess thermodynamic properties of solution. In the present paper we extend this work to a determination of excess volume effects in mixtures of benzene and deuteriobenzene, and in some cyclohexane/benzene or deuteriobenzene solutions. Experimental Section Densities were measured with a Mettler-Parr DMA-60 vibrating densitometer using techniques previously described by us.&' The hydrocarbons were reagent grade and were further purified by fractional distillation. Deuteriocarbons were reagent grade. Tests for organic impurities in both hydrocarbon and deuteriocarbon samples were made by gas chromatography. The most likely impurity affecting the partial molar volumes is water. The problems which it introduces and our analysis of them are discussed at length in the next section. Hydrocarbons were kept on the vacuum line, over sodium, in closed vessels in the degassed state. In making up solutions, we transferred the degassed liquids via vacuum distillation into the mixing vessel and weighed them (still in the degassed state, i.e.,
TABLE I: Excess Volumes of Benzene-h6/ Cyclohexane-h 12 and Benzene-d /Cyclohexane-h ,2 Solutions XC6H
12
VE( 25 "C)a
VE(45 " c y
C6H6/C6H12
0.1632 0.2136 0.2489 0.3565 0.3891 0.4509 0.4953 0.5460 0.6077 0.6383 0.6600 0.7823 0.84 26
0.3476 0.4397 0.4789 0.5941 0.6444 0.6527 0.6550 0.6259 0.6139 0.5873 0.4560 0.3556
0.3498 0.4445 0.4879 0.6000 0.6301 0.6544 0.6667 0.6359 0.6249 0.6032 0.4819
C6D6/C6H12
0.1777 0.2890 0.3547 0.3763 0.4415 0.5481 0.6622 0.7685 0.8510
0.3792 0.5422 0.5985 0.6154 0.644 1 0.6579 0.5981 0.4834 0.3439
0.3819 0.5448 0.6152 0.6276 0.6622 0.6787 0.6153 0.3514
Units are m3/mol x IO6.
a
under their own pressure, or the vapor pressure of the resulting mixture). Results c6r,/C&lz Solutions. Excess volume data for solutions of protiobenzene in protiocyclohexane are presented in Table I. Handa and Benson8have reviewed the literature (1)Jancso, G.;Van Hook, W. A. Chem. Rev. 1974, 74, 689. (2)Janao, G.;Van Hook, W. A. Physica A (Amsterdam) 1978,91,619. ( 3 ) Jancso, G.;Van Hook, W. A. J . Chem. Phys. 1978,68, 3191. (4)Jakli, G.;Tzias,P.; Van Hook, W. A. J. Chem. Phys. 1978,68,3177. (5) Dessauges, G.;Miljevic, N.; Van Hook, W. A. J. Phys. Chem. 1980, 64, 2587.
(6)Dutta-Choudhury, M. K.;Van Hook, W. A. J . Phys. Chem. 1980,
'Present address: Chemistry Department, P. Anna University, Madras, India. Present address: Ciba-Geigy, CH-1870-Monthey, Switzerland.
*
84., 2735. ~~
~
~~
(7) Dutta-Choudhury, M. K.; Miljevic, N.; Van Hook, W. A. J . Phys. Chem. 1982,86,1711. (8) Handa, Y. P.; Benson, G. C. Fluid Phase Equilib. 1979,3, 185.
0 1982 American Chemical Society
The Journal of pnysical Chemistry, Vol. 86, No. 20, 1982 4069
Excess Volumes in Solutions of Isotoplc Isomers
TABLE 11: Parameters of Fit to Eq 1for Excess Volumes of C6H6/C6HI2Solutions variance
x 10"
ref
10 9 11
t,
d
A0 2.61938 densitometer 2.620 2 25 vibrating 773.598 873.524 2.614 18 densitometer 2.613 13 25 dilution 2.61286
"C method 25 vibrating
12
25
present work
13
25 25
present work
45
23
25
d o C , H I z d d°C,H6
773.87
dilatometer magnetic float 773.87 densitometer pycnometer vibrating 773.94 densitometer vibrating 754.89 densitometer
773.87
present work, 25 vibrating 773.94 densitometer C6D6/C6H12
45 Present fit to eq 1.
754.89
873.63
873.67 873.68 852.26 873.68 943.27' 919.98'
2.614 1
A,
(no. of points)
AI -0.11649 -0.098 0 -0.077 379 -0.10497 -0.101 03
0.064 59 0.043 9 -0.0449 0.033 541
3.6 (10)
0.06444
4 (50)
-0.074 4
0.033 7
A2
note
a b a b 0.02751 b,e
2.4 (31)
b
2.618 0 -0.098 6 -0.063 3 2.615 28 i -0.08288 * 0.023 11 i 0.006 94 0.015 6 0.038 4 2.653 94 i -0.151 54 * 0.14512 i 0.009 99 0.023 6 0.063 9
a a
- (16) 14 (12)
a
27 (11)
2.639 5 f -0.086 75 f 0.062 72 f 0.006 47 0.013 6 0.035 7 2.716 57 i -0.126 32 * -0.059 24 f 0.005 09 0.012 1 0.029 6
a
8.4(9)
a
5.1 (8)
Fit reported by original authors.
' d ° C 6 ~ 6kg/m3. ,
on excess volumes for C6H6/C6H12. It is clear that the present results are in excellent agreement with the best previous data. Comparisons are given in Table 11in terms of our least-squares fits to eq 1, where X is the mole P = X(1 - X)[Ao + Al(1 - 2X) + Az(l - 2m2] (1) fraction of C&12 and P is in units of m3/mol X 10s. At X = 0.5 the average of P for data for three laboratories using the vibrating densitometer (ref 9 and 10 and present work) is 0.6541 f 0.0006. The root mean square (rms) deviation in density in the interlaboratory comparison is equivalent to a precision of a few ppm. Comparison with other techniques is also excellent. Thus, the average of the dilatometric" and magnetic float12results of Benson and co-workers and the pycnometric data of Wood and Austin13 is 0.6538 f O.OOO6, and the calculated densities are in agreement with those determined in the present experiment within 1 ppm or so. It is clear from the intercomparisons of the C6H6/C6H12test system that the algorithm connecting the vibrational period of the Mettler-Paar instrument with density is properly defined for solutions exhibiting a density range typified by the test solution. It is worth noting that the bulk of the literature reports on this systems show excess volumes significantly smaller than the present result or of those cited in Table 11. However, it has been OUT observation in working with thew solutions that careful attention to purity, including removal of water and air, is important in improving precision (and apparently accuracy). The density of pure C6H6as reported in the recent literature ranges from 873.45 to 873.70 kg/m3 and that of cyclohexane from 773.20 to 773.95 kg/m3. The values that we report, 873.68 and 773.94 kg/m3, are at the upper end of these ranges. However, the dry hydrocarbons have higher densities than those containing trace amounts of water (ref 7, vide infra), and we conclude that the present densities are representative of properly dried and degassed materials. The behavior of the excess volumes is similar. For 20 seta of dataa at 25 (9) Kiyohara, 0.; Benson, G. C. Can. J. Chem. 1973,5I,2489. (10)G o a h , J. R.;Ott, J. B.; Moellmer, J. Chem. Thermodyn. 1977, 9,249. DArcy, P.: Benson, G. C. Can. J. Chem. (11)Tanaka, R.;Kiyohara 0.; 1976,53,2262. (12)Weeks, I. A.;Benson, G. C. J. Chem. Thermodyn. 1973,5,107. (13)Wood,S. E.; Austin, A. E. J. Am. Chem. SOC.1945, 67,480.
Units are kg/m3.
e
A, = -0.03163.
X
:02
.25
.5
'75
XC6Hg
Figue 1. EXvokme isotope effect in CeH&& and C6H,dC@, sokrtkns, AVE = (VEw,dw Vw,-) X loe m3/md: (0)25 "C, (X) 45 "C,(-) least-squares fits.
-
"C the average excess volume at X = 0.5 is (0.648 f 0.007) X lo* m3/mol with the entries from Table I1 lying near the top of that distribution, (0.653 f 0.002) X lo4 m3/mol, again,we think, because of more careful attention to proper drying and degassing of the solutions. C@,/c$I,2 Solutions. Excess volumes for solutions of deuteriobenzene in protiocyclohexane at 25 and 45 "C have been reported in Table I and least-squares fits are given in Table 11. The excess volume of C6D6/C6HI2is about 1%more positive than that of C6H6/C6H12at 25 "C and about 2% higher at 45 "C. (The excess volume of C6&/C6HI2is about 1.5% higher at 45 "C than it is at 25 "C.) In comparing the isotope effects, we prefer to calculate the difference A P = P&moothed) - P*(obsd)
(2) where P , (smoothed) represents the three-parameter fit to C6&/C6H12data reported in Table I1 and Pdh (obsd) the individual data points for C6D6/C6H12.This, followed by direct fitting of A P , minimizes uncertainty in the smoothed representation of the isotope effects due to polynomial ripple. Values of A V together with calculated lines representing eq 3 are shown in Figure 1. We find
A P = X(l-X)AAo
(3)
AA0(25 "C) = (-0.0289 f 0.0050) X lo4 m3/mol with a variance of 8.6 X m6/mo12and hA0(45"C) = (-0.0478 f 0.0076) X lo4 m3/mol with a variance of 20 X m6/mo12. X is the mole fracture of cyclohexane. We had hoped to make measurements on C6H6/CsD12mixtures, but the only sample of C&z available in sufficient
4070
The Journal of physical Chemistty, Vol. 86, No. 20, 1982
DuttaGhoudhury et ai.
TABLE 111: Excess Volumes in C,H,/C,D, Mixtures C,D, lot a a a a
b C C C C
C
d d d
XC,H,
0.3419 0.4474 0.5812 0.6409 0.2588 0.1231 0.7819 0.8641 0.8761 0.9373 0.4061 0.4933 0.5906
109vE, m3/moi
temp, "C
-6.5 -4.4 - 1.2 +0.9 -4.0 - 1.5 t 1.7 t 1.2 + 1.5 t 0.2 -7.2 -5.0 -1.1
25 25 25 25 25 25 25 25 25 25 45 45 45
quantity was contaminated with deuterated methylcyclopentane and the results are not amenable to quantitative analysis. Even so, it is useful to remark that the isotope effect on P for cyclohexane deuteration is on the order of -1% at 25 "C and X = 0.5. We are aware of no data in the literature with which to compare the present results for isotope effects on P for benzene/cyclohexane solutions. C&/c,& Solutions. Results for these solutions at 25 and 45 "C are presented in Table 111 and displayed graphically in Figure 2. The data at 25 "C are more nearly complete, extending over a wider concentration range. They clearly demonstrate a complicated concentration dependence for the excess volume of this solution. This is surprising. We had assumed that the deviation from ideality of a solution as simple as this would itself be simple. Least-squares analysis yielded the result given as eq 4, where x symbolizes the mole fraction of C6H6,P 1 0 9 P = X(1 - X)[(-12.939 f 1.331) + (-52.365 f 5 . 0 7 2 ) ~+ (22.904 f 5 . 9 8 3 ) + ~ (67.204 i i2.664)~31 (4)
is in units of m3/mol, and Y = 1 - 2X. The variance of fit is 3.2 X 10-lo. Using the F test we have shown that the fit reported as eq 4 is statistically more significant than the analogous two- or three-parameter fits. The number of parameters is too many, and the correlation between them too large, to attempt a term-by-term physical explanation of the parameters in eq 4. Careful examination of Figure 2 or eq 4 shows that in addition to the experimentally established maximum (around X = 0.75) and minimum (around X = 0.33) there is a second very small maximum at a concentration (about X = 0.05) well below the most dilute data point. This effect is not real. It lies inside the statistical error bounds on the fit and is to be discounted. The data on C6H6/C6D6solutions at 45 "c are restricted to the middle of the concentration range (0.4 < X < 0.6). They are in good agreement with the data taken at the lower temperature but certainly do not eliminate the possiblity that the maximum and the minimum at 45 "C might occur a t higher and lower values of P, respectively, than at 25 "C. It is unfortunate that because of the expense of preparing solutions for these experiments, the concentration dependence at 45 "C could not be explored in more detail. There exists a single data point in the literature with which to compare the present results. Lal and Swinton14 report VE = 4 X m3/mol at 30 "C and a concentration "near equimolar". That data point is shown together with an arbitrary estimate of the uncertainty in P and X in (14) Lal, M.;Swinton, F. L. Physica (Utrecht) 1968,40, 446.
01
.6-0
Flgwe 2. Excess volumes of C,H&,D, solutions: (0)experimental points, 25 OC; (X) experimental pokns, 45 "C; (+) experimentalpoint, ref 14; (-) least-squares fit (eq 4); (a) contribution of effect I (see text), random appoxhnatbn; (b) contribution of effect I, quascchemical approximation, q = 0.2; (c) contribution of effect 11, A = -0.01 X lod cm3/mol; (d) calculated total effect = effect I effect 11.
+
Figure 2. The agreement with the present data is not unreasonable. Following our initial experiments on this system (carried out near the middle of the concentration range) we were quite puzzled by the results and repeated the experiments with extraordinary care using several different C6D6samples purchased at different times and from sources. Cfi,/C&6/H20 Solutions. Recognizing that the most likely complicating impurity in the study of C6H6/C6D6 solutions was H20,we carried out a series of measurements of excess volumea of mixtures of C6H6/Hz0(XH,o= 2.418 X and C6D6/H20(XHZ0 = 1.623 X w3).The experiments were carried out before our recent study7 of thermodynamic properties of water-rich and hydrocarbon-rich solutions of C6H6/Hz0and their isotopic isomers. Under the assumption that the difference VEH = v H 0is small, and concentration and solvent t e . , c6& vs. C ,J independent, the desired information could be obtained straightforwardly at only a modest increase in thermodynamic complexity. As it turns out, however, the excess volume of water in benzene is large. Furthermore, it is quite sensitive to concentration and to isotopic sub~titution.~ For all these reasons the water-dependent excess volume effects overwhelmed those due to C6H6/ C6D6substitution, and the data could not be used as originally intended. Even so,they are of interest in their own right and are presented in Table IV. Several alternate functions describing the departure of these solutions from ideality have been calculated. In columns 6 and 7, respectively, excess volume calculated as P ( 1 ) = V(obsd) and P ( 2 ) = V(obsd) - &Jj(obsd) are entered. In these expressions V(obsd) is the experimental volume as obtained from the measured density and mass of each solution, pi is the volume of the pure component (C&, c,&, or H@), and V,.(obsd) is the volume of a given wet two-component starting solution (C6H6/H20 or C6D6/H O ) , calculated from measured density and mass. Both @(1) and P ( 2 ) are positive at all measured concentrations and increase monotonously with Xc In column 8 we report the apparent molar volume fo%,O,
V"3
The Journal of Physical Chemistry, Vol. 86, No. 20, 1982 4071
Excess Volumes In Solutions of Isotopic Isomers
TABLE IV: Excess Volumes in Some Mixtures of C6H6,C6D6,and H,O at 25 "C soln
densitya
dry C,H6 dry C6D6 wet C,H, wet C6D6
873.680 943.221 873.624 943.133 883.002 896.698 910.820 922.325 930.330
1 2 3 4
5
XC6H6
1 0 0.997 58
0.861 32 0.665 00 0.462 6 2 0.297 70 0.183 22
XH,O
XC6D6
0 1 0 0.998 38 0.136 37 0.332 85 0.535 39 0.700 4 4 0.815 01
0 0 0.002 4 2 0.001 6 2 0.002 31 0.002 15 0.001 99 0.001 86 0.001 77
VE(l)b,c
0.022 0.018 0.015 0.014 0.012
VE(2)b*d @ V ( H , O ) ~ ~ ~
0.0101 0.0068 0.0043 0.0035 0.0015
22.9 24.2 24.8 24.2 26.4 27.6 25.9
a Units are kg/m3. Units are (m3/mol)X lo6. V E ( l )= V(obsd) - XiiVOi. VE(2) = V(obsd) - XjVj, where the V. are the volumes of the wet components. e @V= [ V(obsd) - V'C,H,- V'c,~,]/n,. The error in @V is at least * 1 X m J/ mol.
= [V(obsd) - V",, - V"C ]/nHzO, where ~ H is~ the O number of moles of water. &e precision of the data at the low concentration involved accounts for an uncertainty in 4v of about 5% (fl X lo4 m3/mol). At the C6H6/Hz0 (XHZ0= 2.418 X lob3)limit, 4v = 22.9 X lo4 m3/mol in satisfactory agreement with the value calculated from the relation reported earlier,7 4V(C6H6)= 24.4 -1030XH 0, whereas, at the C6D6/H20(xH= 1.623 X limit, the 6) = -1.3 x 104 m3/moi is difference 4 V ( ~ 6 -~c$~(C~DJ also in good agreement with the earlier value, -1.0 X lo4 m3/mol. At intermediate concentrations the observed values of 4v lie significantly above the line calculated from the simple linear combination of the limiting effects. The limiting effects of solutions of H20 in c(& or H20 in c&D, are described within experimental error by equations of the type' 4v = 4vo(j)+ A;XHZowhere j indicates C6H6or C6D6,and both 4voand A; depend on isotopic substitution in the solvent. The present experiments indicate that, if one chooses to describe the data for three-component mixtures usng empirical equations of the form 4v = q5vo(ij) + AijXH,,, the values of r$vo(ij),or A,, or both involve excess terms, i.e., cannot be described in terms of simple linear combinations of the &,O(j) or the A; for the wet c,& or c&D,components. It is unfortunate that, because of the expense of c&&we were unable to pursue studies which woud have unraveled these interesting effects. A practical conclusion to be drawn from the study is that careful attention to the removal of water as an impurity is required in precise measurements of excess volumes of hydrocarbon solutions in spite of existing statements to the contrary.16
Discussion Excess volumes of C6HlZ/C6H6solutions have been thoroughly discussed and will not be further considered. Instead we focus attention on the volumes of mixing of C6H6/C6D6solutions and on the difference ~ c ~ H ~ ~/c
0'
Solutions. We proceed by discussing the more prominent features of the concentration dependence of P, using these features to develop a model which approximates the observed concentration dependence of the excess volume. The isotope effect on the low-pressure molar volumes of C6H6 and C6D6 is well established (V"C~H~ = 89.432 X 10-6 m3/mol, VO = 89.234 X lo4 m3/mol, both at 25 "C) and is underst03 as a natural consequence of the molecular dynamics of these molecules.'J6 Very briefly, the difference in the molar volumes of the pure separated isotopic isomers is dominated by the contribution of the CH (or CD) stretching motions, in turn described with a c&&/$&6
(15) For references see the review of: Battino, R. Chem. Rev. 1971,7i, 5.
(16)Bartell, L.S.; Roskos,R. R. J. Chem. Phys. 1966, 44, 457.
, c
set of isotope-independent force constants. The mean square amplitude of the CH (CD) stretching motion shows a marked isotope dependence which accounts for most of the observed molar volume isotope effect (MVIE). The contribution of the external overall motions of the molecules to the partition function, and hence to the MVIE, is small and is considerably simplified by the fact that the center of mass and the center of interaction coincide for both molecules.6 Jancso and Van Hook2 and earlier Prigogine17 have discussed aspects of the solutions of isotopes, one in the other, with emphasis on free energies and excess free energies. The point of interest in such solutions is the fact that the properly averaged potential function describing the motions of the condensed-phase molecules is isotope (and hence concentration) independent. However, at one extreme the solution is rich in one isotope, e.g., H, of volume V"H, and at the other is rich in D, of volume VD. We emphasize, as described above, that this MVIE is well understood in terms of the isotope-independent potential function. Even so, to the extent that the partition function and hence thermodynamic properties of a given (e.g., solute) molecule are volume dependent, its partial molar thermodynamic properties will depend on concentration. The implicit assumption has been made that the properties of guest molecule are modified by interactions with the host lattice through the intermolecular potential characteristic of the host. The approach has proved successful in rationalizing free energy effects in a number of solutions of isotopic isomers, one in the ~ther.~BJ'In actual practice one proceeds by expanding the free energy of each component, i, in the solution around its equilibrium volume, Vi. In first approximation the volume is linked to the concentration by ignoring nonlinear (i.e., excess volume) effects. Appropriate differentiation of the free energy ~yields H ~ expressions for the enthalpy, volume, etc., of the solution, but it must be emphasized, especially for the volume effects, that the resulting equations can only be good through first order. The presence of both a minimum and a maximum in the present data for solutions of C6D6 in c6& (Figure 1) makes it clear that the first-order "free energy" approach outlined above, while it may contribute to the effect, is inadequate. The idea that a partial molar property of a solute molecule (at infinite dilution in a solution of isotopic isomers, one in the other) may be defined in terms of that property for the host lattice nicely explains the most prominent features of the present data. Those features are the maximum and the minimum which the excess volume displays as one proceeds from an infinitely dilute (17) See, for example: Prigogine, I.; Bellemans, A.; Mathot, V. "Molecular Theory of Solutions"; North Holland Publishing Co.: Amsterdam, 1957.
4072
The Journal of Physical Chemistry, Vol. 86,
No. 20, 7982
solution of C6D6in C6H6to a dilute solution of C6H6in C6DG.Thus, consider the situation when an infinitesimal quantity of C6D6is added to C6H6. If the partial molar volume of the solute is equal to or approaches that of the cavity defined by the host lattice (Le., of c,& in the given example), then clearly the total volume of the solution will lie above that for an ideal solution (because V D ( S O ~ )VOD = positive = VOH - VOD while VH(soln) ii: VOH). Just as clearly, at the other extreme, the deviation from ideality will be negative. Obviously the properties of the solution do not change discontinuously at some intermediate concentration, so the excess volume must go through a maximum and a minimum, respectively, before joining at some intermediate concentration. We label this effect I and call it a steric effect because of the argument presented in the last few sentences. In a more quantitative treatment, using standard approximations as described below, we show that effect I is expected to be symmetric (i.e., P-(X = 0.75) = -Pmin(X= 0.25) and P ( X = 0.5) = 0; X is the mole fraction of C6H6. However, in Figure 2 we see that the curve is not symmetric, -P- > Pa, and conclude that an additional effect, effect 11, must be operating. The simplest description for effect I1 consistent with the data is of the form P = X H X d . A is a negative parameter which is interpretable in terms of the free energy approach outlined above. Details are presented in a later section. Effect I. This contribution to the excess volume is described in steric terms. Consider the solution as consisting of HH, DD, and HD contacts each with its characteristic contribution to the volume, VHH, VDD, and VHD, and assume V, = (V, + vDD)/2, V, proportional to VOH, and VDD proportional to VOD The number of HH, DD, and HD contacts is proportional to XH2,XD2,and XHXD and it follows that P I = YHVOH + YDVOD + YmVm - XHVOH - XDVOD,where YH = xH2/B, YD = XD2/B, and YHD= XHXD/B with B = x~~ XD’+ XHXD.The approach is that for a regular solution assuming random mixing and follows that of Rice.ls The calculated effect is shown in Figure 2, where it is labeled “random” in the caption. In agreement with experiment it shows a maximum and a minimum in the proper order but the predicted effect is too large, lying at least a factor of 2 above experiment. Further progress can be made either by modifying the combining law or by changing the statistical assumptions. We have elected the latter path and have made further calculations in the quasi-chemical approximation. In this case we have to sufficient precisionl8 YH = XH2/B, YD2/B, and Ym = 2 ( YHYD)1/2eTwhere q is a parameter describing the volumetric part of the interaction. Further, ZH = YH/c, ZD = YD/c, and ZHD= YHD/Cwith c = YH + YD + YHD, SO VE = ZHVOH + ZDVOD + ZHDVHD - X H V O H XDVOD. The excess volume is a function of the parameter q. For q = 0,0.2,0.5,and 1 we calculate maximum excess volumes at XH = 0.75 of 0, 3.6 X lo4, 8.6 X lo4, and 15.4 x lo+ m3/mol, respectively. As in the simpler random calculation a symmetric maximum and minimum are predicted. Calculated values of effect I for q = 0.2 are shown in Figure 2. In the figure the calculated contribution of effect I, = XHXd with A = -0.01 X lo4 m3/mol, is shown as well, as is P(calcd) = effect I + effect 11. It is clear that the calculated excess volume is in good qualitative agreement with experiment. Our attempt to fit the experimental data by least-squares using q and A as parameters failed to converge. This, we think, was due to the fact that the theory is only approximate and higer-order skewing terms will be required to force the cal(18) Rice, 0. K. “Statistical Mechanics, Thermodynamics and Kinetics”; W. H. Freeman: San Francisco, CA, 1967; Chapter 12.
Dutta-Choudhury et ai.
culated line through the data points, particularly at low X H . Even so, it is clear from the analysis above (as depicted in Figure 2 ) that there is very fine qualitative agreement between the calculated (given as the sum, effect I (q = 0.2) + effect I1 (A = -0.01 X 1 0 3 ) and observed effects. In the next section the theoretical orign of effect I1 is discussed and an approximate calculation of A presented. Effect II. The negative sign and the order of magnitude of parameter A, P I I / ( X H X D ) = A, can be qualitatively understood from several different points of view. The excess free energy of C,&/C& solutions is positive: and a straightforward application of regular solution theory leads to the prediction of a positive volume change. However, Hildebrand and Scottlg comment that, in the special case of a mixture of species of approximately the same size but different corresponding states attractive energies, ell and e22, the pure components (1 and 2) will be at different reduced temperature, and consequently at different molar volumes. Furthermore, the curve of V against e/kT (reduced temperature) is concave upwards and for any reasonable combining rule the volume of the mixture will therefore lie below the average volume of unmixed components. Thus,the excess volume is negative even though an analogous approach yields a positive excess free energy. For the C6H6/C6D6system it is clear from an approximate relation presented by Fang and Van Hook,20 Aa/a = Ae/e + 6Aa/a N Ae/e + 2AV/V, and the wellestablished polarizability isotope effect reported by Ravinovitch,21 Aa/a = 0.0054, that there is a significant isotope effect on e. (The presence of an isotope effect on e does not imply the assumption of an isotope-dependent potential function for the interaction. Rather this is a natural consequence of the dynamical properties of the molecules which can be described in terms of an isotopeindependent force constant m a t r i ~ . ~ .The ~ ~ )authorslg caution that this exceptional behavior, evidently exemplified by the present system, occurs only for mixtures of substances sufficiently similar to follow the same law of corresponding state, and which are also very nearly the same in intrinsic size. It is eminently reasonable that these conditions be met by the present solution of isotopic isomers. An alternate approach involves expansion of the free energy in terms of the mixture volume followed by differentiation with respect to pressure to yield an expression for the excess volume of the s~lution.~J’At vanishing external pressure
+
+
where f” = XDfH’’ xHf~” and 7”’ = xHf~”’XDfH”‘. Furthermore, f” = l / @ V where @ is the isothermal compressibility and f the free energy. The primes denote differentiation with respect to volume. To f i t order, then, VE/(XHXD) (VOH - VOD)[1- @HVH/(@DVD)]. It is reasonable to suppose that this effect is negative because, if the H isotope has the greater volume, it should also be the more c ~ m p r e s s i b l e . ~We ~ J ~have applied the method of Jancso and Van Hook2to a harmonic cell model for liquid benzenes and estimated the isotope effect, P H - PD, which is positive as expected but is found to vary between 0.5% (19) Hildebrand, J. H.; Scott, R. L. “Regular Solutions”;PrenticeHall: Englewood Clifta, NJ, 1962; pp 112-13. (20) Fang, A. Y.; Van Hook, W. A. J. Chem. Phys. 1974,60,3513. (21) Rabinovitch, I. B.”Influence of Isotopy on the Physicochemical Properties of Liquids”;Consultants Bureau: New York, 1970; Chapter 4.
J. Phys. Chem. 1882, 86,4073-4077
4073
and 5% depending on the precise values of Gruneisen molar compressibilities of the two benzenes at infinite constants employed to define the volume dependence of dilution in C6H12,APc/(XBXc) = AVO(1 - PmHVmH/ the lattice frequencies, and the relations defining the ~ " D V ~ DThe ) . limiting partial molar compressibility of benzene at infinite dilution in cyclohexane, OmH, is more isotope dependence of the Gruneisen constants. In other than 10% larger than PH(the compressibility of pure words, the excess volume from effect I1 as estimated in eq 5 is consistent in both sign and magnitude with the analysis benzene)22and it follows from model calculations2 that of the data in Figure 2. An experimental value for OH O m ~ / / 3 "is~ larger than, but still of the same order of OD would be of considerable value in delineating the This is a consequence of the larger magnitude as, PH/PD cavity which benzene enjoys at infinite dilution in C6H12 physical origins for the excess volume effect. We are presently engaged in the construction of experimental as compared to benzene itself. The change in the volume apparatus to determine the isotope dependence of 0 in ratio, while of the same sense, is not as marked. The order to further test the thermodynamic framework.24 analysis leads to the conclusion that the excess volume isotope effect should be negative and display a significant Effects in C&12/c& and c&12/c&6Solutions. The temperature dependence in agreement with the present isotope effects for excess volumes on this system, data (Figure 1). More quantitative comparisons await Pmd, - PvdF = A P are shown in Figure 1. The effects are negatwe. o asymmetry due to effect I is found experimental measurements of the partial molar comwithin the experimental precision, which is about *20% pressibility isotope effect which are planned for the near future (vide infra). of the measured effects. However, it must be emphasized O C ~ ~ ~ that, because the differences, vOcd12- VOc and ~ Acknowledgment. This material is based upon work - VOcsD6, are about 100 times larger than - vOcsD6, supported by the National Science Foundation under extremely precise data defining the limiting curvatures of the eXCeSS Volumes Of the C6H12/C& and C ~ H ~ Z / C ~ DGrant ~ CHE 81-12965. solutions at high benzene concentration are required to define the asymmetry. At high C6H12 to the extent that (22) Kiyohara, 0.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodyn. this contribution is defined by the host lattice alone, effect 1978, 10, 721. (23) Zwolinski, B., et. al. "SelectedValues Propertie of Hydrocarbons"; I vanishes. Within the present experimental uncertainty American Petroleum Institute: College Station, TX, 1971. it is therefore useful to ascribe A P as arising only from (24) Note Added in Proof. Preliminary measurements of the comeffect 11, and the A P difference can therefore be appressibility isotope effect give (& - &)/& = 0.010f 0.002 at 15,25, and proximated (eq 5 ) in terms of AVO and the limiting partial 40 OC.
Pc,
Translational Dlffuslon Coeff iclent of a Macropartlculate Probe Species in Salt-Free Poly( acrylic acid)-Water Thy-Hou Lln and George D. J. Phlllles" Department of Chemistry and Micromolecular Research Center, The University of Mlchigen, Ann Arbor, Michigan 48 109 (Received: February 26, 1982; In Final Form: March 9, 1982)
Quasi-elastic light-scattering (QELS) spectroscopy was used to measure the mutual diffusion coefficient D, of 0.038-pm diameter carboxylate-modified polystyrene latex spheres dissolved in poly(acry1ic acid) (mol wt 300000)-water solutions of concentration 0.0-171 g/L. D exhibits a complex dependence on the polymer concentration, viscosity, and temperature. The apparent hydrodynamic radius of the polystyrene spheres, as obtained from the Stokes-Einstein equation, increases markedly with increasing polymer concentration.
I. Introduction In a previous paper, one of us1 reported on the diffusion of polystyrene latex spheres and bovine serum albumin in solutions of water-glycerol and water-sorbitol of various temperatures and compositions, over a range of solvent viscosities q of 0.8-1000 cP. While at low viscosity the electrophoretic mobility A,, and diffusion coefficient D are proportional to q-l, for q 5 10 CPthe q dependence of A,, and D can be more complex. If the solute molecules are small, but comparable in size or larger than the solvent, one2y3finds D q-l. If the solute molecules are smaller than the solvent molecules, D and X arek7 -qQ, for 0.63
-
(1) Phillies, G. D. J. J. Phys. Chem. 1981, 85, 2838. (2) Singh,K. P.; Mullen, J. G. Phys. Rev. A. 1972,6, 2354. (3) Bemer, B.; Kivelson, D. J.Phys. Chem. 1979,83, 1401. (4) Green, W. Heber, J. Chem. SOC.1910,98, 2023. (5) Stokes, Jean M.; Stokes, R. H. J. Phys. Chem. 1966,60, 217. 0022-3654/02/2006-4073$01.25/0
< a < 0.7. For the systems studied in ref 1, it was found that D depends on q and temperature in accordance with the Stokes-Einstein equation regardless of whether q was changed by varying the temperature or the solvent composition. Here ro is the hydrodynamic radius of the probe species. This difference between the results of ref 1-3 and 4-7 suggests that it would be interesting to use light-scattering spectroscopy to study diffusion in highly viscous systems as solute and solvent sizes are gradually made more similar. Making the sizes more similar by using smaller macroparticulate probes would lead to significant experimental (6) Stokes, Jean M.; Stokes, R. H. J. Phys. Chem. 1958, 62, 497. (7) Hiss, T. G.; Cussler, E. L. MChE J. 1973, 14, 698.
0 1902 American Chemical Society
