L. J. TICHACHEK, W. S. KMAKAND H. G. DRICKAMER
660
with water.26 If the oxonium ion exists also in small quantities in water, acids should increase the ionic character of the detergents and bases should decrease it. Changes in the ionization of the detergents would be reflected in corresponding changes in CMC values. Hydroxide ions were very effective in decreasing the CMC of NR-15 (decrease in CMC = 69 a t 0.5 N ) . This indicates a marked decrease in the charge on the micelles and supports the postulate that their weak positive charges are caused by oxonium ions. The acid and electrolyte effects nearly compensate each other in dilute acidic solutions. The CMC values of NR-15 in 0.86 N HC1 and HN03 were 120 and 130 pM, respectively. At higher concentrations the acid effect becomes predominant as the electrolyte effect levels off. The CMC values of NR-15 in 3.1 N HC1 and HN03 were 150 and 290 p M , (26)
G.A. Hill and L. Kelley, “Organio Chemistry,” The Blakiston
Co., Phila., Pa., 1943, p. 140.
Vol. 60
respectively. These data show that the relative effectiveness of the chloride and nitrate ions are unchanged but that their actual effectiveness is superimposed on the effect of strong acids. The results shown in Figs. 3 and 4 may be summarized by the two equations
-
log (CMC)o log (CMC) = D (3) A (CMC) = (CMC)o CMC = (270 20N)C., for NR-15
-
-
(4)
where (CMC),, represents the value of CMC when C, = 0, and Cs is the normality of the electrolyte and N is the lyotropic number of the anion. D is a parameter independent of R. By combining these two equations, it may be shown in general A(CMC) = C,(270
- 20N)e0.’366(R-16)
from which the approximate CMC value (of a detergent based on nonylphenol) may be calculated at various values of N , R and C,.
THERMAL DIFFUSION I N LIQUIDS;l THE EFFECT OF NON-IDEALITY AND ASSOCIATION BY L. J. TICHACEK, W. S. KMAKAND H. G. DRICKAMER Department of Chemistry and Chemical Engineering, University of Iilinois, Urbana, Illinois Receivsd November 66. 1066
It is shown that the thermodynamics of irreversible processes, when appropriately applied to a system fixed in the laboratory, predicts that a thermal diffusion separation depends on the difference in the quantities Qi*’/Vi, where Qi*‘is the molal net heat of transport and Pi is the partial molar volume. &i*’ is related to the activation energy for motion by a physically logical ar ument and by a kinetic derivation. The results are applied to a series of binary solutions including one or more associate2 compounds. I t is found that the activation energy for local expansion, AHh is probably the part transported in molecular diffusion. The introduction of the ‘(partial molar activation energy” is shown to predict the correct concentration dependence in thermal diffusion. The role of X (d$/dX) in thermal diffusion is discussed.
*,
In a recent paper2 we have presented a theory of thermal diffusion in liquids which gives very good agreement with experiment for solutions which are ideal or only moderately non-ideal. The theory was, however, based on an abortive and artificial formulation for the “net heat of transport” of a molecule in a solution. It is our purpose here to show that essentially the same result can be obtained from the thermodynamics of irreversible processes by an appropriate transfer of coordinates from the center of mass to the laboratory system with no artificial assumptions necessary, or by a kinetic argument which is a generalization,of that due to Prigogine, ef aZ.3 We shall further show how the theory can be applied to associated and other non-ideal systems. Thermodynamic Theory.-The development of this theory follows in principle that given by de Groot4; the chief differences are: (1) this treatment employs laboratory coordinates instead of center of mass coordinates, and (2) only constant (1) This work was aupported in part by the A.E.C. (2) E. L. Dougherty and H. G. Driokamer, THISJOURNAL, 69, 443 (1055). (3) I. Prigogine, L. de Broukere and R. Amand, Physica, 16, 577, 851 (1950). (4) 8. R. de Grooi, “The Thermodynamios of heversible Processea,” Intersoience Publiahera, New York, N. Y.,1950.
pressure systems are considered so that fluxes of pure heat and of enthalpy are simply related. We define our phenomenological coefficients, L i k , in a laboratory coordinate system by
x=
1
1
4
(1)
LikXb
k=1
-*
where X R is a force associated with component k, +
J, is the flux of component i in laboratory coordinates, and n is the total number of components in the system. For an open system defined as an infinitestimal constant volume fixed in the laboratory coordinate system, the Gibbs equation yields Td(sp) - Whp) at
am
at
k
dt
(2)
where s and h are the total specific entropy and enthalpy, pk is the partial specific free energy of component k, Pk is the mass of component k per unit volume of system, and p is the total density of the system. We now represent the flux of enthalpy in * laboratory coordinates by J h and find that it is composed of two parts for systems at constant pressure + Jh
+
n
+
= J Q 4k=l
hkJCck
(3)
THERMAL DIFFUSION IN LIQUIDS
May, 1956 where
hk
where Cj is the concentration of component j. This relation can be applied to a binary system with the aid of equation 8 to give
is the partial specific enthalpy of compo+
nent k and JQ can be described as the flux of pure heat (conducted energy); moreover, both the nu-
+
+
- LU
JI
merical value of JQ and the validity of equation 3 are found to be independent of the coordinate sys- + +
-*
tem chosen for expression of J h , J Q and J k . We now apply Gauss' theorem to the two time derivatives on the right side of equation 2; with the aid of equation 3 end appropriate rearrangement of terms we find that
+
Xk =
- T g r a d ET
+
J1
(5)
[-grad xi -
= - p ~
-+
Jh
-L
-
JkQk
-t 3
L h z h
+
+ Lhjxi
i
&k
k
where vi is the partial specific volume of component i. We can thus conclude that E
(13)
-
[ZLhjXj 3
+ Lkzh]
(15)
The definition of &k and Onsager's relations can be used to show that the sum of the terms involving
0
+
i
Xj is identically zero; the two terms involving + X h are zero when there is no temperature gradient, so that the left side of (16) is then zero also. Thus
We now pursue a development similar to that of de Groot by defining a Qk to transform coefficients of the enthalpy flux-mass flux interaction n
-+
where P i s the molar volume of the solution ( X I ~ -II x2PJ..We now find the significance of &k* in the follomng manner. We write k
Lik
az1za grs.d T]
The steady-state solution of equation 13 can be compared with equation 12 to find the value of a
(6)
We notice that these forces are the same as those found in a center of mass coordinate system. In the flux equations which are typified by equation l, two types of relations must be found between the coefficients. The first set of relations is known as Onsager's relations; the second set of relations arises from the definition of our system as a fixed volume
Vi
(11)
where PI, V2are the partial molar volumes of components 1 and 2 ; x1 is the mole fraction of component 1; ,u'~is the molar chemical potential of component 1; and Q1*'is now the product of Q1*and MI, the molecular weight of component 1. The definition of a,the thermal diffusion ratio, is commonly taken from the flux equation
4
X n = - -grad T T
-
[$* -grad T + 84 grad Ci1+ -+ L ,5 ~ [$$ 2 T +? !E grad C,] vz ac1
Conversion of specific to molar quantities and the application of the Gibbs-Duhem equation gives for the steady-state condition
where the relation p k = h k - T s k has been used. The first term on the right embodies the entropy flux in the open system; the second and third terms, therefore, give the entropy generation within the system. Since the products of the fluxes and conjugated forces should be equal to the entropy generation, the forces are -*
661
n
The significance of &k and QR* in this laboratory coordinate system will be discussed later. The form of equations 1 and 9 has caused the definition of n different Q's instead of n - 1 Q's employed by de Groot. The information which we can obtain from the thermodynamics of irreversible processes is not altered by the manner of definition of Q; however, if defined as in equation 9 it is simpler to introduce a physical interpretation for Q from outside the thermodynamic theory. Now by a series of steps analogous to those used by de Groot, we find
.
Now &I*' (eq. 14) is M1Q1*, and the significance of &I* can be seen by a study of equation 16 in its various aspects. I n particular, it can be seen that &I*' is the difference between the total enthalpy transported by one mole of moving molecules of type one in the solution and the average enthalpy of one mole of molecules of type one in the same mixture. It is clearly related to the activation energy for motion of type one molecules in this mixture. It is, in fact, that part of the activation energy which is transported with the moving molecule. A simple kinetic argument, given below, serves further to identify these quantities. The same expression for a (equation 14) could be found directly from de Groot's result if one were
L. J. TICHACHEK, W. S. KMAKAND H. G. DRICKAMER
662
able to interpret his heat of transport, Qlm*. For the significanceof QIm*de Groot finds + JQm
n-1
=
* Qkm*Jh
where Vi = partial molar volume of component i
P
= average molar volume of system whose exact form is unim= function of Ti and
0
portant ae the factor involving it will cancel out
(17)
k-1 -c
where Jk, is the flux of component k in his center of
Vol. 60
The other terms have their previous significance (eq. 22). The flux equation is written so that
--c
mass system; a study of the properties of Jqm x i 7 1 +ZVz = 0 (231 shows that it is the flux of enthalpy in that center (24: of mass system. A proper transformation between the two coordinate systems, plus the constraint given in the first quality of equation 8 shows that as it must in a system fixed in the laboratory. One can now write an analogous expressionsfor QIm* = Rl - R2 + -i7i7, i 7 2 - ‘I*’] - (18) the flux of component one in the opposite direction, Vl equate the two for the steady state, and expand the where R1 is the partial molar enthalpy of compo- exponentials as before. Then one obtains nent 1, the &I*’ are used in equations 12 and 14, and &Irn* is the net molar heat of transport in the center of mass system. Substitution into the steady state This is just like equation 14 obtained from the solution of de Groot’s flux equation yields equation thermodynamic theory except that RT replaces 14. X(dp/dX) in the denominator as is understandable Kinetic Theory.-Prigoginea has written the flux since no correction for solution non-ideality is inof component one in a mixture of molecules of the cluded. same size as This kinetic development provides a further confirmation for our intuitively logical identification of the net heat of transport with the activation energy transported in molecular motion. The Activation Energy Transported in Molecular Motion.--The best description of molecular motion of a component in a mixture would be obtained from where measurements of “self-diffusion” of that component in the mixture as a function of temperature (and of XI = mole fraction component i Co = total concentration of molecules pressure) using tagged molecules. Since these p i = “localized” activation energy, Le., that part of the measurements are essentially non-existent, it is activation energy transported with component i necessary to approximate those quantities in some QL = %on-localized” activation energy, Le., that part way. A possible first approximation would be acnot transport with motion TBTb= temperature one molecular jump.apart tivation quantities derived from “self-diffusion” AT = T8 - Tb measurements on the pure components. These are also relatively scarce; however, according to EyThe definition of the q’s used here is somewhat more general than that used by Prigogine, but this ring’s theory6 of molecular motion, the mechanisms does not affect the form of the equation. Prigogine of diffusion and of viscous flow are similar and the wrote a similar expression for the flux of component activation quantities for the two.processesshould be one in the opposite direction. He then equated nearly the same. This has been shown to be true’ those in steady state, expanded the exponentials, for a wide variety of substances, with one or two exceptions. In particular, for CC1, the diffusion acand obtained for a tivation enthalpy is 50% greater than the activation energy for viscous flow. In our discussion we shall use the activation quantities derived from visRutherford and Drickamer6 showed that if the p’s cosity coefficients of the pure components as the are considered as free energies the corresponding first approximation for the activation properties of the molecules in a mixture, except for CC14 where expression for a! is the values from diffusion will be used. (q2 - T Cq?) - (pl - T ?q! According to Eyring the viscosity coefficient can a = bT bT) (21) be written RT
[g
We now propose for the flux in a system of molecules of different size the more general expression
hNo V
where h
NO V AF
R T (5) W. M. Rutherford and H. G . Drickamer, J . Chem. Phys., 43, 1157, 1284 (1954).
AF*
= Planck’s constant = avog&dro’Bnumber = molar volume
* = free energy of activation gas constant = =
absolute temperature
( 6 ) 8. Glasstone, K. L. Lsidler and H. Eyring. ”Theory of Rate Processes,“ McGraw-Hill Book Co.. Inc., New York, N. Y., 1941, (7) E. Fishman, T H IJOURNAL, ~ 89, 469 (1955).
May, 1956
THERMAL DIFFUSIONIN LIQUIDS
Then 0.20 .50 .80
= AH*
-
- -A v *
(29)
RT
These quantities have been discussed in detail elsewhere2BQ although their exact significance is still not clear. AH* is the total activation enthalpy. It seems probable that AHh* is the activation enthalpy associated with a localized expansion to permit motion, and AHj* is the activation enthalpy associated with orientational effects. We are interested in the fraction of the activation enthalpy transported by the molecule when it moves. This could, of course, be AHh*, AHj* or somefraction of each. For most non-associated liquids AHh is 65-75% of the total activation enthalpy a t one atmosphere. For associated liquids AHh* is only 2040% of the total AH*. It has been suggested that AHj* is the measure of thermal diffusion separation, Le., that it is the quantity transported in motion. However, for non-associated systems it is an order of magnitude too small, and frequently fails to predict the correct sign. Dougherty and Drickamer2 predicted the correct sign and magnitude of a for a wide variety of nonassociated mixtures using the total AHo* (the subscript refers to a pure liquid) as the quantity transported. (It is now clear that the factor '/2 used on AHo in that paper is necessary only because a factor of two is introduced spuriously earlier in the derivation, due to the definition used for the net heat of transport.) Since for all these systems AHh* is the major part of AHo*, and about the same fraction in each case, they do not provide an identification of the portion of the activation enthalpy transported in motion. For three systems involving alcohols the use of AHo did not provide correct magnitudes. It will be shown later that if it is assumed that AHh is the enthalpy transported correct magnitudes can be predicted for all systems including associated liquids. This is rather surprising, and it is not a t all clear intuitively why it is so. Now in the above approximation, the concentration depende_nceof a appears to a limited extent in VI, VZand V , and to a much greater extent in XI. ( b m / b X ~ ) .As will be discussed later, this latter quantity predicts trends with concentration correctly, but is not sufficient to give the entire con-
*
*
*
*
TABLE I EXPERIMENTAL A N D CALCULATED VALUESOF Ph
0.20 .50 .80
Benzene (1)-Carbon Tetrachloride (2) at 40" 1.15 1.37 599 1.27 1.21 1.43 585 1.34 1.21 1.48 598 1.34
Cyclohexane (1)-Carbon Tetrachloride (2) a t 40" 1.33 1.25 0.20 60 1 1.50 1.31 1.27 592 1.48 .50 1.24 1.30 .80 605 1.40
- AHh*
T )";:"
663
UBXD
Benzene (1)-Cyclohexane (2) a t 40' 540 0.09 0.17 491 .10 .20 523 .IO .20
( 8 ) A. Bondi, J . Cham. Phya., 14, 591 (1946); Sca., 68, 805 (1951).
ma
Ann.
0.58 .40 .10 N. Y. Acad.
0.20 .50 .80
Benzene (1)-Methanol (2) a t 40" 261 8.6 0.15 127 13.8 .24 .41 61 23.4
-0.80 +O. 15
f1.80
Carbon Tetrachloride (1)-Methanol (2) at, 40' 0.20 259 +6.1 -1.99 -3.0 -6.37 -4.9 63 +19.5 .50 -9.75 -2.8 33 +29.9 .80 n-Butyl Alcohol (1)-Carbon Disulfide (2) a t sod 0.20 Assume -4.63 -0.64 -6.5 .50 RT -4.05 - .56 0 .80 -3.59 - .50 .25
+
Isobutyl alcohol (1)-Carbon Disulfide (2) a t 8"' 0.50 (RT) -5.64 -0.92 -0.93 0.297 ,444 ,615 ,706 .906 ,956
Ethanol (1)-Triethylamine (2) at 50" 446 -7.0 440 -7.8 482 -8.2 528 -8.3 638 -8.2 640 -8.2
-1.08 -1.19 -0.92 - 1.22 -0.89 -0.67
0.430 ,540 .683 ,804 ,909 ,971
Ethanol (1)-Diethylamine (2) a t 50" 745 -3.8 835 -3.6 930 -3.5 955 -3.8 805 -4.9 682 -6.0
-1.11 -0.85 -0.48 -0.59 -0.88 -1.12
0.388 .658 ,794 .846 ,931
Water (1)-Diethylamine (2) a t 49' 400 -10 330 -24 258 -48 180 -65 180 -70
-1.64 -3.24 -2.71 -2.19 - 1.52
0.145 ,266 .582 ,726 ,884
Water (1)-Ethanol (2) at 25" 486 -23.0 1.80 367. -29.5 2.02 239 -26.5 1.52 213 -12.5 0.64 377 -. 8.0 .42
-0.50 -0.93 -1.47 -0.90 $0.29
Water (1)-Methanol (2) a t 40" 0.200 558 -5.0 0.35 -0.46 .360 518 -7.0 .44 -0.93 ,492 500 -9.0 .56 -0.54 .772 442 -9.5 .85 -0.44 ,900 586 -8.4 .71 +0.62 .953 615 -8.0 .70 +1.17 CY is positive when component (1) is enriched a t the hot wall. * OLO calculated using AHo* (activation enthalpy of pure components). c (Yh calculated using AHhO (activation From therenthalpy for expansion of pure components). mal diffusion data of E. 1,. Dougherty, Ph.D. Thesis, University of Illinois, 1955.
*
L. J. TICHACHEK, W. 5. KMAKAND H. G. DRICKAMER
664
c
-0.2l-0.4
Vol. 60
-I
\ \ \
\
-
EXPERIMENTAL E Q U A T I O N I431
0.8
I
I
I
I
0.5 0.8 0.2 MOLE FRACTION BENZENE.
I
Fig: 1.-The concentration de endence of experimental and theoretical values for the tierma1 diffusion ratio of benzene-cyclohexane at 40'.
R2 0.4 0.6 0.8 MOL FRACTION TRIETHYLAMINE.
Fig. 3.-The concentration dependence of experimental and theoretical values for the thermal diffusion ratio of triethylamine-ethanol at 50": -, experimental results; ----, theoretical values based on activation enthalpies of , theoretical values based on partial pure components; molar activation enthalpies.
-----
EQUATIONS 14,JI
W EXPERIMENTAL 2.0
- 1.0
I
I 0.2 0.5 0.8 MOLE FRACTION BENZENE. I
11
AH f m i r
=:
Xi
+A
m X2
(30)
These could then be evaluated from viscosity measurements as a function of temperature and composition. It would have been desirable t o measure values of partial molar activation enthalpies for expansion, AHh but accurate measurements of viscosity as a function of pressure are not obtainable. Results The systems studied include benzene-cyclohexane, benzene-carbon tetrachloride, cyclohexanecarbon tetrachloride, benzene-methanol, carbon tetrachloride-methanol, n-butanol-carbon disul-
*,
0.2
L
0.4L
0.8 L
U 0.8
I
ID
O
MOL FRACTION DIETHYLAMINE.
centration effects in highly non-ideal (particularly associated) solutions. This is not surprising since the activation quantities wanted are those for the components in the mixture, and these could be very concentration dependent. It is clear that in reality the activation enthalpies of molecules in the mixture will not be simply related to those of the pure components, nor to the AH* of the mixture. Nevertheless, it was decided to try, as a next approximation, a partial molar activation enthalpy AHi defined by the equation
*
L
~~
Fig. 2.-The concentration de endence of experimental and theoretical values for the tiermal diffusion ratio of benzene-methanol at 40".
~
-2oa0
Fig. 4.-The concentration de endence of experimental and theoretical values for the tierma1 diffusion ratio of diethylamine-ethanol at 50": -, experimental results; -----, theoretical values based on activation enthalpies of , theoretical values based on partial pure components; molar activation enthalpies.
-----
-1
a' It o W
-- - __---
-1
O
a
U
LK
e
I ~ - 5 0I--
Y
THERMAL DIFFUSION IN LIQUIDS
May, 1956
I
665
I
I
I
I
I
I
I
I
1
is0
i
4
u
O aL
8 I
I-
-50
- 2.00.0
0.2
0.4 0.6 0.8 MOL FRACTION ETHANOL.
’d
-100
I .o
-2.01 0
0.2 0.4 0.0 0.8 MOL FRACTION M E T H A N O L .
I
1.0
Fig. 6.-The concentration dependence of experimental and theoretical values for the thermal diffusion ratio of ethanolwater at 25“: -, experimental results; ---- -, theoretical values based on activation enthalpies of pure components; -----, theoretical values based on partial molar activation enthalpies.
Fig. 7.-The concentration de endence of experimental and theoretical values for the tierma1 diffusion ratio of methanol-water at 40”: -,experimental results; theoretical values based on activation enthal ies of pur; components; theoretical values basef on partial molar activation enthalpies.
fide, isobutyl alcohol-carbon disulfide, triethylamine-ethanol, diethylamine-ethanol, diethylamine water, ethanol-water, methanol-water. These systems were selected because adequate thermodynamic data9910 were available, permitting the evaluation of X ( b p / b X ) . The chemicals, which were reagent grade, were dried and redistilled or recrystallized. The thermal diffusion ratios were obtained in cells similar to those previously described2-ecalibrated as described there. The analyses were performed in a Zeiss interferometer with a 0.5 centimeter cell. The results are shown in Table I. Each point is the average of 2 to 6 runs with a deviation of not over f10% for the extreme values of individual runs.ll In column 3 are listed the values of a calculated using AHo*, the total activation energy for motion of the pure components. For non-associated mixtures the agreement is good, but gets poorer as the liquids become more associated. This corresponds to the results of Dougherty and Drickamer, who used a similar equation and AHo to get good agreement for non-associated mixtures, but poor agreement for systems involving the alcohols. In column 4 are listed the calculated a’s using AHh *, the activation energy of local expansion, of the pure components. These were calculated from the p-v-t data and viscosity data of Bridgman.12 In every case the magnitude is nearly correct, but concentration effects are not very satisfactory. Almost all the concentration dependence in these calculated a’scomes from X(bfi/bX). The effect of this factor is discussed below. Earlier, an approximation for the activation energy of a component in a mixture in terms of a
“partial molar” activation energy was proposed. These were measured for systems not involving CClr since for this compound it was necessary to use -it diffusion activation energies. NOW,evidently would be desirable to use partial molar m h * . Since the necessary pressure measurements on the mixtures could not be made, it is unobtainable. However, for mixtures involving one or no associated compounds, one could assume is a constant fraction of AH*,i.e.
*
(9) G. Scatchard, S. E. W o o d and J. M. Mochel, THISJOURNAL,48, 119 (1939); J . A m . Chem. Soc., 61, 3206 (1939); 63, 712 (1940); 68, 1957, 1960 (1946).
(IO) (a) A. G. Mitahell and W. F. K. Wynne-Jones, Disc. Faraday Soc.. No. 15, 161 (1953); (b) J. L. Copp and D. H. Everett, 4bid., No. 15, 174 (1953). (11) R. L. Saxton, E. L. Dougherty and H. G. Driokamer, J. Chem. Phys., 43, 1166 (1954). (12) P. W. Bridgman, Proc. Amr. Acad. Arts Sci.. 49, 4 (1913); 81, 57 (1926); 88, 185 (1931).
-- -- -
-----
-
(31)
Figures 1 and 2 show the results of this calculation for benzene-cyclohexane and methanol-bensene. The agreement is remarkably good. For mixtures involving water particularly this approximation is certainly not valid. However, it is in- to compare the values calculated from teresting AH* with experiment. This is done in Figs. 3-7. The magnitudes are, of course, incorrect, but the concentration dependence, including a sign change, is predicted very well. The improvement over using AHo for the pure component is marked. Considering the necessity of making rather crude approximations for results obtainable only from selfdiffusion, the agreement with experiment is striking. It i s desirable here to say something about the role of X ( b p / b X ) . As can be seen from the graphs and calculations, this quantity alone does not give adequate concentration dependence. Nevertheless, as can be seen from Table I it differs by as much as a factor of ten from RT in some cases, and almost always corrects the concentration dependence more closely to experiment. Emery and Drick&mer1*have also shown the importance of this term. L. J. Tichacek would like to acknowledge financial assistance from a National Science Foundation Fellowship.
*
(13) A. H. Emery, Jr., and H. G. Drickamer, J . Chem. Phys., 38, 2252 (1955).
