Transition State Theory for Diffusion Coefficients in Multicomponent Liquids In application of transition state theory to diffusion in a ternary system with a molar volume nearly independent of composition, Mortimer and Clark identified experimental diffusion coefficients with those on a Kirkendall reference frame. This i s incorrect, and it i s shown here that the data used, together with other data reported in the literature, are consistent with a mechanism of simple movement of a particle into a vacancy.
I n the M C paper, it is stated t h a t the volume- or numberfixed reference frame is to be identified with the Kirkendall frame, so that J i N = J i K and LikN = LsK. Thence, since it is found that LikNcross-coefficients have substantial values, i t was concluded that the vacancy mechanism is not simple movement into a vacancy, but involves a considerable degree of concerted movement of two particles. This reasoning has no justification. The relationship between L z k Kand LzkNcoefficients is comples, since the reference frames involved are not identical. Thus, if the crosscoefficients in a ternary system are zero in the Kirkendall frame, Ld" is not zero, but is given by
I n a recent paper of the above title, Mortimer and Clark
(MC) (1971) have adopted the viewpoint that diffusion in at least most ordinary liquids is most appropriately represented by a model based on vacancy mechanisms. I n such a model, each species of particle for a given composition and a given set of concentration or chemical potential gradients possesses an independent flus,for which the appropriate reference frame is the Darken frame or, as M C prefer to name it, the Kirkendall frame. Using JtKfor the flus of i referred to this frame, and L k z Kfor the corresponding phenomenological coefficients, the phenomenological diffusion equation is
LizN = N I N z L ~ Nz(1 ~ ~ k
when there are components. When the mechanism is simple movement of a particle into a vacancy, the "cross-term coefficients," LzlK ( k # i), are zero, but if a concerted movement of two particles is involved, this is not the case. The coefficients have a real physical meaning, and expression of this in terms of Eyring transition state theory is given in the RIC paper. The only point to be noted here is that, where cross terms are not zero, transition state theory shows that they should be symmetrical, Le. LfkK =
L kiK
The authors apply the theory to (i) a ternary system of argon isotopes which, as it is fictitious, will not be further discussed here and (ii) t o data of Burchard and Toor (1962) for the system toluene (component l), chlorobenzene (component 2), and bromobenzene (component 3). The present note is concerned with conclusions on this system. Now, actual diffusion measurements are usually based on a volume-fixed reference frame, for which the fluxes are not independent but are related by n
Jivpa
=
0
z
in which pi is the partial molar volume. In the particular system, molar volumes of the components are almost identical and volume changes on mixing are small enough to be neglected, so that we can take the molar volume as a constant, Ti, within the limits of error. As a consequence, the volume-fixed reference frame is essentially the same as the number-fixed reference frame, i.e. n
Ji"
=
0
i
On this reference frame J i v 'v J i N
=
-
2Likh'Vpk k
484
Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
- Ni)LzzK - Ni(1 - Nz)L2zK
where N t is the mole fraction of i, i.e., N i = c Z Bif ci is ex, pressed in moles per unit volume. We have L I Z N= L Z I Nbut this follows necessarily from the transformation between reference frames, and the cross-coefficients are purely formal, without any physical significance. The experimental data of Burchard and Toor are given as diffusion coefficients Ji' = I
JZ"
=
-
3a)ik"ck
in which k and i have only the values 1 and 2, since 3
3
civ
Nz
=
i
=
1
i
and thus there are only two independent values of v c k . The left-hand superscript in 3 a ) i k "indicates that bromobenzene (component 3) has been arbitrarily chosen as reference component, and the script form is to denote that 3 a ) i k 0coefficients are different in kind from the single Dlz" coefficient in a binary system. The significance of this is that, from the same experimental data, a set of, say, 2 a ) z k wcoefficients could becalculated, with component 2 as reference component, which would differ in value from 3 a ) t k u coefficients, though they are naturally related to them. For this particular system, Burchard and Toor pointed out that all 3Dtk' cross-coefficients are small, and that 3 a ) ~ 1 0 N 3a)z2u for any given composition. Consequently, data could be approximately expressed by the simple equation J L ' 'V
D'VC~
in which D' is the same for all components and choice of a reference component is immaterial. I n a previous discussion (Carman, 1968), the writer pointed that this corresponds t o all components having the same mobility for a given composition. I n the following, this highly simplified version will not be used, but the simplest assumption will be made, namely, that the mechanism consists of movement of a single particle into
Table 1. Calculation of Mobilities from Diffusion Data of Burchard and Toor Experiment
N1 N2 AT3
x x axw x
3%u
aa)220
x
lo5 105 105 lo5
1
2
3
0.25 0.50 0.25 1.848 1.797 -0.063 -0.052
0.26 0.03 0.71 1.570 1.606 -0.077 -0.012 1.27 1.32 1.30 1.86 1.62
0.70 0.15 0.15 2.132 2.062 0.051 -0.071 1.99 2.12 __ 2.06 2.31 2.06
q3RT X lo5
jg
Average qlRT X lo5 q2RT X lo5
1.72 1.88 1.87
a vacancy. Then LikK cross-coefficients are zero, and LiiK coefficients can be represented as mobilities, i.e. J , K = c.21.K = a t
- L a.i K V pa.
+
=
RT{ (
3D12'
=
RTXi{qo -
~
+
3 ~ 2 ) q l
N 2
Nlq3)
42)
3Xh1"= RTN2{q3- ql} 3 a Q U
=
RT( ( N ,
+ AT^)^, + ~ ~ p , }
From this, we obtain
The equality of the two values for RTq, is a n expression of that fact t h a t L k t Nmust equal LitN as a formal result of transformation t o a number-fixed reference frame. Burchard and Toor showed t h a t the two values for RTq3 agreed reasonably within the rather wide limits of esperimental error, and while this does not prove anything in itself, it means that a n average of the two gives a closer approximation to the true value. Using this, it follows t h a t
RTqi = -{3Dii' - NiRTq3) 1 - Ni
(i
=
1,2)
The values obtained for the ternary compositions are given in Table I. For comparisoii, the limiting values of the binary diffusion coefficients give RTqi directly, where i is a trace component, and these are quoted in Table 11. The bracketed values are estimates of self-diffusion coefficients in the pure components at 30°C. T h a t for broniobenzene is obtained by applying a viscosity correction to a value a t 20°C (Carman, 1968). For chlorobenzene, a viscosity correction is applied to the value obtained by Harris, et al. (1970), using a diaphragm cell. -4 larger value is quoted by O'Reilly and Peterson (1971), using
5
6
0.45 0.25 0.30 2.006 1.890 -0.020 -0.198 1.57 1.86 1.72 2.24 1.95
0.18 0.28 0.54 1.774 1.518 -0.037 0.0026 1.78 1.37 __ 1.58 1.82 1.49
Table 11. limiting Mobilities in Binary Systems N1
- - c . a 4 a.VP i
where p i is the mobility or the diffusive velocity when the driving force, --Cbt, is unity. I n the solid state, Vignes and Sabatier (1969) have studied a closely related ternary system, Fe-Co-Ni, where i t can also be assumed t h a t the system is ideal and has a molar volume independent of composition. As shown by Kirkaldy and Lane (1966), and used by Vignes and Sabatier, very simple relatioilships between 3 DikU and mobilities can be derived 3 ~ 1 1 '
4
0.15 0.70 0.15 1,853 1.841 0.049 -0.068 1.77 1.94 1.86 1.85 1.81
N 3
RTq1 X lo5 RTq, X lo5 RTq3 X lo6
0 0 1 1.41 1.36 (1 .32)
0 1 0 1.80 (1,86) 1.76
1 0 0 (2.53) 2.36 2.27
a n iimr method, but their equation permits a rather large scatter. The only value for toluene is that of O'Reilly and Peterson (1972), also by nmr. Under the assumption t h a t only a simple mobility is involved, RTqi should be equal to Di*, the self-diffusion coefficients of component i for any given composition, and a critical test would be measurement of such values experimentally. These are not available in this system, but it is to be noted t h a t values of ql, 92, and 43 are reasonably self-consistent. The values of ql, 42, and q3 in Table I are roughly equal, and insofar as they deviate from this, (i) q1 > q2 > q3, with a few sporadic except'ions, which is in accord with the limiting values in Table 11; (ii) the largest values occur in toluenerich solutions and the smallest in bromobenzene-rich solutions, which is in accord with expectation if solution viscosity is a dominant factor. The paper of Dubrovskii and Afonina (1966) on viscosities for the system show that ternary viscosities can be expected to lie between those of the pure components, and qi values show similar behavior. The rather large range of esperimental error i n the data does not permit any greater degree of consistency in the foregoing statements. There is thus no reason to assume a mechanism more complex than movement of a single particle into a vacancy. It was shown b y the writer (Carman, 1968) that the same conclusions should be valid for the system methanol-l-propanolisobutanol, studied by Shuck and Toor (1963). This was also assumed thermodynamically ideal, but calculations mere more complex because molar volumes differed widely. The limits of experimental error were much narrower, so that the sporadic deviations from a systematic trend observed in Table I did not appear. Here, again, however, no self-diff usion measurements are available. The only case in which self-diffusion measurements as well as mutual diffusion measurements have been made is by Kett and A4nderson(1969) for a single composition of hesane-dodecane-hesadecane,also a system with widely different molar volumes. These authors found that interpretation of the self-diffusion coefficients as simple "friction coefficients," i.e., inverse mobilities, was consistent with their mutual diffusion data. Ind. Eng. Chem. Fundam., Vol. 12, No.
4, 1973 485
Literature Cited
Burchard, J. K., Toor, H. L., J . Phys. Chem. 66,2016 (1962). Carnian, P. C., J . Phys. Chem. 72, 1707 (1968). Dubrovskii, S. M., Afonina, K. V., Zh. Obsch. Khim. 36, 1869
O’Reilly, D. E., Peterson, E. M., J. Chem. Phys. 56, 2262 (1972). Shuck, F. O., Toor, H. L., J . Phys. Chem. 67,540 (1963). Vignes, A., Sabatier, J. P., Trans. Met. SOC.A I M E 245, 1795 (1969).
P. C.CARMAN
(1966).
Harris, K. R., Pua, C. K. N., Dunlop, P. J., J . Phys. Chem. 74, 3518 (1970).
Kett, T. K., Anderson, D. K., J . Phys. Chem. 73, 1268 (1969). Kirkaldy, J. S.,Lane, J. E., Can. J . Phys. 44, 2059 (1966). Mortimer, R. G., Clark, N. H., IND. ENG.CHEM.,FUNDAX 10,
National Chemical Research Laboratory Pretoria, Republic of South Africa
604 (1971).
RECEIVED for review November 27, 1972 ACCEPTED June 22, 1973
O’Reilly, I). E., Peterson, E. M., J . Chem. Phys. 55, 2155 (1971).
A Curious Anomaly in Parametric Pumping Experimental evidence i s presented to illustrate a curious reversed separation effect in a closed, direct thermal mode parametric pump operating at high frequency. A comparison with a theory that predicts such reversals at high frequency shows that simple velocity lag effects cannot explain the current phenomenon.
High-frequenc y operation of liquid parametric pumps has not been exploited to date. X recent theory (Rice, 1973) suggests the possibility that larger separations may occur at higher frequencies owing to resonance effects. Indeed, this theory suggests that a reversed separation may occur. Some recent experiments in our laboratory have produced this curious “reversed separation’J effect in the enrichment of oxalic acid from aqueous solutions on activiated carbon using the closed, direct thermal mode parametric pump. When operating this type of parametric pump in a single solutesolvent system, one usually synchronizes fluid pulsation and heat addition so that the phase difference between these essential driving forces is 0’ or 180’. For example, when fluid motion is upward in a vertical packed column, heat is removed from the bed (for example, by passing cooling water through a jacket surrounding the bed) and this cooling causes solute to be adsorbed and hence retarded. When fluid motion reverses and moves downward, heat is added to the bed, allowing the solute to desorb and be swept downward with the applied fluid motion. It appears obvious that this “bucketbrigade” should cause solute eventually to accumulate in the well-stirred lower reservoir, while solute should be depleted from the upper reservoir. On the other hand, if we heat the bed on upflow and cool it on downflow, one naturally expects the upper reservoir to be eventually enriched in solute, while the lower reservoir is depleted. Quite the opposite occurred in our experiments. I n the former case (cooling during upflow) we observed the upper reservoir vias enriched; in the latter case (heating on upflow) we observed the lower reservoir was enriched. One of the experiments demonstrating this rather odd reversal is presented in Table I. Initially, we thought this curious reversal was caused by velocity lags in the pore structure of the bed, that is the socalled “annular effect” (Richardson, 1929) which occurs when a fluid is pulsed in a tube a t high frequency. This interesting hydrodynamic phenomenon is discussed by Schlicting (1960) where it is demonstrated that the flow along the axis of the tube lags behind that in the layers near the wall. A recent theory to predict ultiniate separations in the parametric pump (Rice, 1973) exploits this interesting velocity effect and 486 Ind.
Eng. Chem. Fundam., Vol. 12, No. 4, 1973
indeed shows that when u R 2 / 6 ) > 650, a reverse separation should occur. We tested this theoretical prediction in the following way. High-frequency experiments using aqueous oxalic acid were performed in a vertical bed packed with activated carbon (cylindrical in shape with average particle dimensions of 4mm length X 1.5-mm diameter). The jacketed bed was in. i.d. and contained 40 in. of packed length. Well-stirred reservoirs having capacities of 9.5 (upper) and 20 (lower) i n a awere attached to the column ends. Fluid pulsing was accomplished using a lever arm-cam arrangement to produce very nearly (=k20/,) sinusoidal fluid motion. Cooling water a t approximately 25OC was circulated in the jacket surrounding the bed during upflow, while hot water (56OC) was circulated during downflow. Under such conditions, it was expected that a high concentration would prevail in the lower reservoir and a low concentration in the upper reservoir. Operation in this manner was deemed sensible, since if the reverse were true (heating upflow, etc.) a high concentration should prevail in the upper reservoir and density driven flow would be a problem. On observing the reversed separation, we then applied heating on upflow, cooling downflow, and found the lower reservoir was enriched (Table I) showing a t least the curious reversal was consistent, but still unexplained. Details of the experimental work are presented elsewhere (Mackenzie, 1972). We next propose to test the previously mentioned theory. Taking the Schmidt number as 355 and a computed pore hydraulic radius (based on a measured flow void fraction of 0.5) as 0.315 mm, the theory predicts that reverse separation should occur when w > 18 radians/sec or for cycle times less than 0.35 see. This prediction is a factor of 100 off the mark in predicting the current results, where cycle times as long as 210 see produced the curious reversed separation. The possibility that natural convection could have caused the anomaly was tested by the simple expedient of reversing the thermal flux (Le., heating upflow, cooling downflow as in Table I) and still the reversed separation persisted. We concluded some effect other than velocity lag and natural convection caused the reversed separation, such as: (1) abnormal equilibrium isotherm; (2) interparticle thermal and mass diffusion in the radial direction or thermal lag; (3) inter-
