THERMAL DIFFUSION AND
May, 1965
MOLECULAR
MOTIONI N LIQUIDS
443
THERMAL DIFFUSION AND MOLECULAR MOTION IN LIQUIDS1 BY E. L. DOUGHERTY, JR.,AND H. G. DRICKAMER Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois Received December 2,1964
A theory for thermal diffusion in liquids has been developed which permits the prediction of the separation using only molecular weights, molar volumes, activation energies for viscous flow, and excess thermodynamic properties. Measurements have been made on a series of binary mixtures of isomers with CS2 a t atmospheric pressure. These results and previously published results a t high pressure are compared with the theory. There is excellent qualitative agreement, and in many cases the agreement is essentially quantitative.
In a series of recent publications2-6 we have presented data on liquid phase thermal diffusion both a t atmospheric pressure and under pressures to 10,000 atmospheres, and outlined a fairly satisfactory theory qualitatively consistent with our results. I n this paper we present a revised theory, utilizing a sounder basis, which gives considerably more accurate results and which permits prediction of thermal diffusion separations from viscosity data and thermodynamic data on the solutions. The theory is compared with our earlier results and with new results obtained for a series of binary mixtures of isomeric compounds with carbon disulfide. For a binary mixture the thermal diffusion ratio 01 is defined by the equation
where J, = flux of component 1
D = diffusion coefficient p = density XI = mole fraction of component I T = absolute temperature
The solution of equation 1 for the steady state is
where the subscripts H and C refer to hot and cold chambers of a single stage cell. One can see that the sign of o is arbitrary. The convention used in this work is discussed below. The starting point for the revised theory, as for our earlier work, is the thermodynamics of irreversible processes7J which gives for the thermal diffusion ratio
where Mi = molecular weight of component i iP = MiXi MzXz
+
(1) This work was supported in part by the A.E.C. (2) ( a ) W. M. Rutherford and H. G. Drickamer, J . Chem. P h g s . , 89,
1157 (1954); (b) R. L. Saxton, E. L. Dougherty, Jr., and H. G. Drickamer, ibid., 22, 1166 (1954). (3) W. M. Rutherford and H. G. Drickamer, ibid., 24, 1284 (1954). (4) R. L. Saxton and H. G. Drickamer, ibid., 21, 1287 (1954). (5) W. M .Rutherford, E. L. Dougherty, Jr., and H. G. Drickamer, dbid., sa, 1289 (1954). (6) E. L. Dougherty, Jr., and H. G. Driakamer, ibid.. accepted for publication. (7) J. Prigogine, “Etude Thermo. de Processes Irrev.,” Dumond, Paris, 1947. ( 8 ) S. R. deGroot, “Thermodynamics of Irreversible Processes,” Interscience Publishers, New York, N. Y., 1950.
chemical potential of component i
pi
=
&i
= the heat of transport of component i, the net heat
crossing a boundary with one mole of component i in the absence of n temperature gradient
In our earlier work we used an equation, used also by other author^,^^^^ of the form ff=
Q2*
- &I* bPl
(3’)
x1ax, This latter equa,tion is derived under the assumption that the center of mass of the system is stationary. Since, in the actual experiments, the velocity of the center of mass is very small, the numerical difference between the results obtained from (3’) and (3) is negligible. Nevertheless (3) will be used in this work as it is more nearly correct. This fundamental point was first made by Prigogine.7 Our initial formulation for the heat of transport will be similar, but not identical with our previous work2s6and that of Denbigh.“ Consider a large volume of the mixture and in particular consider a region within the mixture with fixed bounding surfaces (these are mathematical, not physical). Assume that the entire volume is in equilibrium with a thermostat thereby assuring the absence of a temperature difference between the region and the remainder of the fluid. Assume that a molecule of component 1 diffuses out of the region. Let W H i be the energy which must be supplied to move one mole of component one across a boundary in a liquid. W H ~is the energy transported when a molecule moves from one equilibrium position t o the next. In general the hole left by the molecule of type 1 will be filled from across the boundary of the region. If the molecules are distributed randomly, it will be filled by either a type one or type two molecule with the relative probability X1/X2. The energy transported in this process will be WL
= XlWHi
+
x‘2wHz
(4)
The net energy transported will be HI - W L = (1 - X,)(WE, - WHz) (5) provided there is a one for one exchange, i e . , if all molecules are the same size. However, in the process described above there has been a net transport out of the region of only (1 - X,) moles of component one. The heat of transport for com(9) K. G.Denbigh, “Thermodynamics of the Steady State,” John Wiley and Sons, New York, N. Y., 1951. (10) S. R. deGroot, “L’Effat Soret,” North Holland Publishing Co., Amsterdam, Netherlands, 1945. (11) K. G. Denbigh, Trana. Faraday Soc., 48, 1 (1952).
E. L. DOUGHERTY, JR.,A N D H. G. DRICKAMER
444
ponent one, which involves the net transport ‘of one mole of this component will then be Similarly
Vol. 59
where
At atmospheric pressure pAV* is negligible and G AU*. This is not true at high pressure. At present it is not possible to analyze the activation energy in terms of fundamental molecular
AH*
&a*
- WHI
= WHI
(7)
Equation 3 then gives for a
Now if molecules of type one and type two are not the same size, on the average +bl molecules will enter a hole left by a type one molecule and &! molecules will enter a hole left by a type two molecule. Then
Now, assuming the volume of the considered region to remain consta.nt, it follows (equating volume of leaving and entering molecules) that
properties, or to say unequivocally what portion must be supplied to the molecule and what portion to the surroundings. Nevertheless certain facts can be pointed out. Bondi13 made important contributions in this field, and much of the following discussion is based on his work. It is not difficult to show that
The term [T(bp/bT),AV*] is A H h * in Bondi’s nomenclature. This is clearly the work t o create the volume AV* against the thermal pressure of the liquid. Bondi has shown that for many liquids AHh * AV * =v
(15)
AEv = energy of vaporization
where V i is the partial molal volume of component i. Then equations 9 and 10 become 91 W H I &1* = WE1
The range of validity of this relationship can be shown as follows. For a van der Waals liqyid
W H ~-
E = -a/V (17) Hildebrand14 has proposed as a generalization of this expression for liquids, E = -a/Vn. Then, a t atmospheric pressure
-
&e*
bE
VI
WH~
(12)
and the expression for a becomes
It remains then to obtain a satisfactory expression for W H i in terms of measurable molecular properties. In previous work w0 considered it to be a fraction of the “cohesive energy” of the liquid, and evaluated this latter quantity from the energy of vaporization. Further consideration, however, indicates that it should be much more closely related to the energy of activation for molecular motion. The energy of activation is the energy supplied to move the molecule from an equilibrium position to the activated state. No further energy need be supplied for the molecule to move to the next equilibrium position. For the systems under study here the activation energy for motion can most conveniently be evaluated from viscosity measurements. According to Eyring12 Then
(12) 8. Glasstone, K. J. Laidler and H. Eyring, “Theory of Rate Prooesses,” MoGraw-Hill Book Co., New York, N . Y.,1941.
a
m=v
(11)
Then AH,,*
=
T
(g~,),
Av+
(16)
AEv
= n __ A V ~
V
Then equation 15 follows whenever n = 1, i.e., for van der Waals liquids. Hildebrand has shown that n G 1 for many simple non-hydrogen bonding liquids. It is not so clear whether this energy is supplied uniformly around the moving molecule or a t a particular point to create a hole. Bondi showed t,hat it was approximately the change in potential energy (calculated from a Lennard-Jones potential) between a molecule and its neighbors when the intermolecular distance is increased from V’/a to (V AV*)’/n. This would indicate it is supplied around the moving molecule more or less uniformly. The activation volume is not the volume of an equilibrium site as it is only l / b to ‘/IO of the molar volume. At low pressures the internal pressure is 3000-5000 atmospheres and A H h * amounts to SO-SO% of AU*. Thus a t atmospheric pressure AHj* amounts to only 10-20% of AU*. bE/bV decreases with increasing pressure and becomes negative for many organic liquids a t
+
(13) A. Bondi, J . Chem. P h w . , 14, 690 (1946). (14) J. H. Hildebrand and R. L. Soott, “Solubility of Non-Electrolytes,” Reinhold Publ. Corp., New York, N . Y., P. 97.
THERMAL DIFFUSION AND MOLECULAR MOTION IN LIQUIDS
May, 1955
7000-9000 atmospheres. Thus a t high pressure AHj is equal to or larger than AU*. Values of AHj evaluated from Bridgrnan’sl6data for several compounds are plotted as a function of pressure in Fig. 1. It can be seen that A H j * increases by an order of magnitude in 10,000 atmospheres. It is unlikely then t,hat it is merely the energy to break cohesive bonds. It is probably associated in part with the necessity for assuming a favorable configuration for motion by rotation or otherwise, and its large increase with pressure is probably caused by a decrease in free volume making the realization of any given configuration more difficult. This viewpoint is quite similar to that adopted by Bondi. A precise understanding of the nature of the activation energy is most important. Experiments are being undertaken in this Laboratory which, we hope, will help clarify our understanding. In the absence of a detailed picture we shall assume that one-half the activation energy is removed by the molecule and one-half is dissipated among the molecules at the old equilibrium site. This gives good agreement with our experiments and is as justifiable as any other assumption at present. Then
445
* *
1
W H ~= -2
0
AUi*
(20)
In principle A ~ I *is the partial molal activation energy where Au*rnixture = X i A u i * 4- XzAoz* (21) Since viscosity data were not available for the mixtures we assume here that
I
t
I
4
PRESSURE
Fig. 1.-The
I 6
ATM.X
I
I
8
IO
I
id3.
energy of motion, A H j , us. pressure for some normal liquids.
A n investigation of thermal diffusion in binary mixtures where the excess thermodynamics quantities are known is being undertaken along with viscosity measurements on both pure components A o i * = (Au*pure component i ) (22) and mixtures. This will provide a still more Then equation 13 becomes quantitative test of the theory. The new experiments reported (Table I) here were performed on cells very nearly identical to those described previously.2b Each point was run several times and on two or more cells as before. calibrating mixture used was 50 mole per indicating the “activation energy density” to be The cent. n-heptane50 mole per cent. n-propyl iodide. the controlling quantity in thermal diffusion.’6 Considerable further work in a variety of two Now chamber cells as well as an open column apparatus indicate that the most reliable value of cy for this system lies in the range 2.25-2.40. The value of (24) 2.30 was used here rather than the value 2.00 Unfortunately, the excess thermodynamic proper- utilized in previous work. All data reported here ties are not known for the mixtures studied here for the first time and also all previously published and present methods for predicting them are data discussed in this paper are corrected to this scarcely quantitative, and totally unsatisfactory figure. Parenthetically, it should be pointed out here for the derivatives. In this work we shall take X l ( b p l / b X I ) = R T , that a limitation on the usefulness of these cells its ideal solution value. This should be fairly has been discovered. As the viscosity of the satisfactory for the mixtures considered here. liquid mixture increases beyond two centipoises It would fail totally near the critical solution the two chambered cell of our design tends to temperature where X ( d p / b X ) approaches zero. give low results. Above about 4 centipoises the values from the two chamber cell may be 50% (15) P. W. Bridgman, Proc. Ant. A d . Arts Sei., 49, 1 (1913); 61, or more below values obtained on an open column 57 (1926); 66, 185 (1931). of liquid. ( 1 6 ) A development leading to rather similar results can be given from a purely kinetic viewpoint"^'* but it appears to us that.the In comparing theory with experiment, it should treatment of rnoleoules of different size and shape arises less naturally be kept in mind that three approximations have in the kinetic picture than it does above. been made which are not inherent in the theory (17) K. Wirtz and J. W. Hiby, Phusik. Z., 44, 221, 369 (1943). but are necessitated by the lack of various thermo(18) I. Prigogine, R . Amand and L. deBroukere, Physica, 16, 577, 851 (1950). dynamic and viscosity data. ’
E. L. DOUGHERTY, JR.,.!~
446
N D H.
G. DRICKAMER
2,2-Dimethylbutane
.5
.8 n-Butyl chloride sec-Butyl chloride Isobutyl chloride t-Butyl chloride n-Butyl alcohol sec-Butyl alcohol Isobutyl alcohol
T = 40 "C.
.2 .5 .8 .2 .5 .8 .2 .5 .8 .2 .5 .8 .5 .5 .5
Vol. 59 6 6 9 6 8 6.5 6 9.5 8 7.5 8 7 6.5 9 7 7 9
+ .34 + .21 + .07 + .ll + .20 + .OO + ,02 + .11 - .13 -
.13 .14 .58 .60 .96 .O -1.11 -0.83
-
-
-
+ .62 + + +
.65 .24 .25 .27 .17 .18 .20 .10 .ll .12 .40 - .42 .45 -4.24 -7.04 -5.83
+ + + + + + -
(2) X ( d p / b X ) has been assumed equal to RT. The molar volumes of the pairs are seldom equal and for many systems a significant heat of mixing would be predicted by the Scatchard-Hildebrand theory. In general, for the systems studied here I I I I the effect should not be larger than 2030%. 2 4 6 8 to The use of the Scatchard-Hildebrand theory to PRESSURE ATt4.x Fig. 2.-Effect of pressure on the energy of activation. predict X ( b p / b X ) ,as was done in previous papers, was considered. However, the theory, although (1) The volumes of the pure components have reasonably satisfactory for p gives values for been used instead of partial molar volumes. The X ( b p / d X ) which deviate widely from experiment effect of this is not large (see Fig. 10). and it was thought best to test the denominator of eq. 23 in a later paper with measured values of the TABLE I thermodynamic parameters. EXPERIMENTAL AND CALCULATED RESULTSFOR ISOMERIC (3) The activation energies of the pure comSYSTEMS ponents were used, rather than activation energies Temp., Isomer Xcsa OC. aexp. ucalc. for the components in the solution. It is not at o-Xy lene 0.2 7 0.0 -0.32 present possible to estimate the error of this - .19 .5 8.5 - .35 approximation. .8 8 - .38 - .39 High Pressure Data.-The activation energies as m Xylene 9 + .17 evaluated from Bridgman's data are shown in Fig. .2 .o p-Xylene
.5 .8 .35 .5
.8 Ethylbenzene cis-Dichloroeth ylene trans-Dichloroethylene 1,l-Dichloroethane 1,2-Dichloroethane n-Hexane 2-Methylpentane 3-Meth ylpentane
.2 .5 .8 .2 .5 .8 .2 .5 .8 .2 .5 .8 .2 .5 .8 .5 .8 .5 .8 .5 .8
2,3-Dimethylbutane
.5 .8
6 8 7 7.5 6 9 9.5 8.5 8 8 7 8 7 8.5 7 6 5.5 6 6 6 5 6 6 7 6 6 G
6
+ .14 - .06 + .14 + .16 + .21 + .12
+
-
-
.19
+ .12 + .08
+ .21 + .16 + .17 + -19 + .ll + .12 + .13
.51 .41 .31 .26 - .15 .18 .75 .84 .66 -1.35 -1.75 -0.96 +1.13 +l.16 +o .90 +1.09 .82 .87 .40 .48
.29 -31 .33 - .22 - .24 - .26 - .33 - .36 - .39 -1.30 -1.40 -1.52 +1.07 $1.10 $1.10 +1.15 +1.03 +1.07 $0.80 -83
-
+ + + +
1.20-
I
1
1
1
1
- EXPERIMENTAL - THEORETICAL
-
0
-\
-
+
4
0
1
:
-0.
-0.8
L9 0
2
4
6
8
10
PRESSURE ATM. x 10-3 Fig. 3.--a! us. P for ?L-C~H~I-CSP (80% CSz, 40").
THERMAL DIFFUSIONA N D MOLECULAR MOTIONIN LIQU~DS
May, 1955 I
-
I
0
-
I
I
447
I 20 Yo c s 2 80 9b cs, 0 0 OBSERVED CALCULATED
EXPERIMENTAL THEORETIC4L
- --__
-0.4
-0.8
d.
1'-
.CALCULATED
-
-0.40-
-
80
CS2
---
-
-
-0.80L
I
I
I
I
0
2
4
6
8
10
loo0
0
2000
PRESSURE, ATM.
Fig. &-Effect
of concentration on I
01
for CS2-CC14 (40'). I
I
ll-BUTYL BROMIDE
- 0.6
n-HEXANE 0.80
7000 ATM.
-
-
ETHYL IODIDE
I-
4
2
0
Fig. 5.-a
us.
4 6 PRESSURE.ATM.
x
8 10-3.
IO
P for n-butyl bromide and ethyl iodide in CSz (50% CSz, 40').
vention used was that a positive a denotes that CSz went to the cold mall. In general the trends are predicted correctly. It is felt that the high pressure thermal diffusion data for CSz-n-butyl bromide may be in error, possibly due to the high viscosity. Besides the general agreement, the following items are worthy of note: (1) a sign change is predicted and observed for the system n-octane-CSz at intermediate pressures (Fig. 4) ; (2) a small pressure effect is predicted and observed for the system ethyl iodide-CSz (Fig. 5 ) ; . (3) theory gives a very nearly quantitative estimate
I
