C. A. EDWARDS and DAVID M. HIMMELBLAU Department of Chemical Engineering, The University of Texas, Austin 12, Tex.
Mass Transfer between Liquid Phases A novel radioactive tracer technique shows that a re-examination of basic design equations is needed
TIE
B A S I C DATA that have been published on liquid-liquid extraction have been concerned mainly with the performance of small-scale units, with emphasis on the effect of flow rates or agitation rates on mass transfer coefficients or HTU. T h e data, therefore, are primarily of value for the equipment and the systems that have been investigated and cannot provide a general understanding of mass transfer. This study similarly was restricted to a particular aspect of liquid-liquid extraction. one that has not been previously examined extensively-namely? the effect of solute concentration on the mass transfer coefficient for a tkvo-phase liquid system. Although it is knoivn that the physical conditions near the interface are quite complicated. \Vhitman's concept ( 79) of a stagnant film on each side of the interface provides a convenient basis for establishing a picture of the diffusional resistance at the interface. This picture. or model of mass transfer, qualitatively states that the turbulence decreases from the main bod) of a fluid phase to a stagnant film, bvith eddy diffusion being most important in the main body and molecular diffusion playing the predominant role just before the interface is reached. T h e assumptions postulated for the two-film theory when applied on the basis of a n over-all driving force (77) lead to the following relationship :
Some investigators have felt that activities rather than concentrations should be used as measures of the driving force. Karr and Scheibel ( 9 ) used activity units in correlating their data on mass transfer betIveen immiscible liquids in a n agitated chamber. It is also believed by some authorities (5) that activity or chemical potential and not concentration is the driving force in diffusion. Nevertheless, since activities are quite inconvenient to use, they were not employed in this study. T h e simple model indicated by Equation 1 appeared
to be quite adequate to correlate the data collected. T h e concept of turbulence at the interface has been the most recent of the proposed mechanisms for explaining the activity taking place a t the boundary between two immiscible liquids containing a common solute. Some investigators (8, 72) have noted the change of mass transfer rate with a change of solute concentrations: they attributed this to a change in the interfacial tension. Sigivart and Nasjenstein (76) have shown: however, by the use of schlieren photographs that many multicomponent systems exhibit eruptive conditions at
the interface and that the intensity of the eruption is mainly dependent upon the presence of surface-active agents and other impurities. Their photographs showed that material transfer across the interface was enhanced by irregularities in the interfacial tension, which \vas i n turn caused by changes in the level of concentration of the common solute. By means of schlieren optics, LeLvis and Pratt (70) also sho\ved the effects of solute concentration upon the interfacial turbulence. I n addition to the factor of higher solute concentrations causing greater interfacial turbulence and hence higher
0SC I LLO M E T E R CELL
TRANSFER VESSEL
-7
IN PL
Cutaway view of mass transfer vessel shows circulating mechanism and interfacial plate. The vessel was constructed of Type 3 16 stainless steel VOL. 53, NO. 3
MARCH 1961
229
rates of mass transfer, it has been shown ( 7 7 ) that, for equal concentration driving forces, the rate of mass transfer of a common solute, such as acetic acid, depends upon whether the solute is going from the organic phase to the water phase or in the opposite direction. Assuming that there is no resistance to transfer a t the interface, as postulated by the two-film theory and the surfacerenewal theory, this phenomenon is presumed to be the result of hydrogen bonding forces more intense when acetic acid is being extracted from water than when it is being extracted from an organic phase ( 7 7, 78). This investigation was designed to side-step many of the interrelated phenomena involved in interphase mass transfer by measuring solute transfer under constant hydrodynamic conditions and in the absence of concentration gradients. To do this, the transfer of acetic acid between phases was made with the system a t equilibrium from the viewpoint of the acetic acid concentration. The mass transfer rate of tracer molecules was detected by means of a radioactive counting device which measured the amount of radioactive (carbon-14-labeled) acetic acid molecules in each of the two immiscible phases as the transfer took place. Because of the large amount of data (2, 3, 6, 8, 9, 72-15) on the system methyl isobutyl ketone-acetic acid-water, this system was chosen so that information obtained from this study could be related to that which already existed. IVith the mass transfer coefficient K defined by the customary equation
the value of K was found to increase with increasing circulation rate, as would be expected. A considerable variation of K with acid concentration was noted for a constant circulation rate with good bulk mixing in each phase. Furthermore, the K us. concentration curves exhibited distinct minimum values. This behavior can probably be ascribed to a combination of solvation effects a t the interface, plus the variation of the acetic acid diffusion coefficient with concentration. I t suggests that mass transfer measurements in nonideal liquid systems can only produce averaged values of K under constant mixing conditions and that further refinement of the transfer model would be advantageous. Equipment
A Type 316 stainless steel tank type vessel (p. 229) about 6 inches in diameter was selected for the transfer to fix a definite volume and interfacial area of each phase. To assure that the transfer
230
vessel was completely level for each run, it was suspended from a pipe frame placed within the constant temperature bath. Three interfacial plates were made providing a 2-inch-diameter interface (20.3 sq. cm.), a 3.5-inch-diameter interface (62.1 sq. cm.), and a 5-inchdiameter interface (126.7 sq. cm.). Since it was anticipated that the interface would be disturbed by ripples or swirls in a nonrandom manner if the two phases were stirred in the transfer vessel, the equipment was designed in such a manner that the fluids were effectively mixed without being stirred in the usual sense of the word. The mixing was accomplished by means of a centrifugal pump arrangement in which the fluid was withdrawn from each phase, directed through the pump corresponding to that phase, and then returned to the transfer vessel. There it was discharged through a spray ring made from 0.25-inch stainless steel tubing. The spray ring was set up in such a manner that none of the individual sprays impinged directly on the interface but rather collectively formed a flow pattern at a slight angle away from it and approximately half an inch from the interface. O n the downstream side of the centrifugal pump for each phase \cas a needle valve, which served to regulate and control the circulation rate in each phase. The circulation rate was measured b>- two interchangeable sets of Fischer and Porter Flowrators which covered a range of 2.8 to approximately 3800 cc. per minute. The volume of each phase within the transfer vessel was 1840 ml., and the volume contained in the circulation system \yas approximately 210 ml., giving a total volume in each phase of about 2050 ml. Initially, it was felt that it \could not only be desirable but also possible to place a radioactivity detecting device within each liquid phase so that the radioactive material in each phase could be monitored continuously. I t was found, however: that \Then the window of the detecting device was placed in direct contact with the radioactive liquid: the radioactive material (acetic acid tracer) not only was absorbed into the window but also was transferred through the \cindow into the chamber of the gas flow counter that was used. LYith such contamination the background count became prohibitively high. For this reason. it was necessary to withdraw liquid samples periodically from each phase and count them in the usual manner under the window of a gas flow counter. After the recirculating fluid in each phase passed through its Flowrator, it passed through a T! where samples of radioactive material or other fluid could be added to or removed from the system by means of a hypodermic needle
INDUSTRIAL AND ENGINEERING CHEMISTRY
and syringe. For operating convenience, the centrifugal pumps, the Flowrators and the T connections were placed outside of the constant temperature bath. For the system, methyl isobutyl ketone-acetic acid-water, equilibrium values taken from the literature (2, 9, 72, 74) show that equilibrium is very insensitive to temperature changes in the vicinity of 75" F. The circulation system was constructed, however, so that prior to re-entering the transfer vessel the recirculating fluid passed through a coil of tubing approximately 48 inches in length, which was within the constant temperature bath. .411 portions of the circulation system were placed below the transfer vessel, which was vented to the atmosphere. In this manner, the centrifugal pumps were primed when the transfer vessel was filled, and after a few minutes of operation the air was completely driven from the lines into the transfer vessel, where it was discharged to the atmosphere. With the exception of short lengths of 0.25-inch polyethylene tubing, which connected the glass Flowrators to the remainder of the system, the tubing in the circulation system was 0.25-inch, Type 316 stainless steel tubing. The centrifugal pumps used were stainless steel Eastern pumps, Model D-6: with mechanical rotary seals. T h e ketone used in this study was a commercial grade purchased from the Carbide and Carbon Chemicals Co. It was analyzed for impurities using a mass spectrometer and was found to compare favorably with the ketone sold as reagent grade. The distilled water used for this investigation had a specific resistance of approximately 1 million ohms, and the acetic acid was Baker Analyzed reagent grade.
Procedure Several days prior to the beginning of an experimental run: approximately equal volumes of distilled \cater and methyl isobutyl ketone were placed together in a glass container and mixed well \yith a measured amount of acetic acid. Just prior to the beginning of a run each phase \cas titrated to a phenolphthalein end point with standard NaOH solution. The calculated equilibrium concentrations could be compared Lvith the known equilibrium distributions to see if equilibrium had been attained. Each experimental run \cas then begun by filling the equipment with the water and methyl isobutyl ketone mixtures which had been previously mutually saturated. After the system had been brought to the desired temperature and circulation rate, 0.03 ml. of radioactive acetic acid was added to one of the two immiscible liquid phases by means of a hypodermic syringe? and the run was begun.
M A S S TRANSFER About 10 minutes after the radioactive tracer had been injected into the system a planchet for one counting tube was filled with approximately 1.4 ml. of distilled water and placed in its tray in the cabinet under the counting window, where a background count was taken for 2 minutes. At the end of this period. the background count was recorded and the planchet dried. A syringe \vas then inserted through a serum cap in the T in one phase, and approximately 1.4 ml. of sample was withdrawn and returned to the system twice to make certain that the syringe was flushed tvith the sample to be taken. A 1.4-ml. sample was then drawn into the syringe and transferred into the planchet. tvhich was then placed under the counting Ivindow where a 10-minute count was taken. Following the counting period, the planchet with its tray was removed from under the counting window, and the sample was drawn into the syringe and returned to the system. Next, the syringe and needle were flushed out prior to removing the succeeding sample. I t was found during this investigation that the sample containers and the bvindoivs of the gas flow counters were readily contaminated even though the windoivs were not in direct contact Ivith the liquid sample. Prior to counting each sample, the ivindow of the counting tube \vas wiped with a piece of wet cotton and dried with a piece of tissue paper. The aluminum planchet for each phase was also scrubbed thoroughly with soap and water and then allowed to soak in a beaker of water for approximately 10 minutes prior to being used with the next radioactive sample. I n this manner the contamination caused by the radioactive acetic acid was held to a tolerable minimum with the background count varying generally between 30 and 80 c.p.m. Calculations and Results Interpreting Equation 1 in terms of the transfer of acetic acid from the water phase to the ketone phase, the following equation is obtained :
where A and V, are constant. For this investigation, there was no acetic acid concentration difference. since the system was a t equilibrium. There was, however, an isotopic concentration difference. which appears in Equation 2 as the difference in the tracer concentrations C, and Ci. It was assumed that the physical and chemical characteristics of the tracer molecules and the nontracer molecules were the same and that the concentration of tracer mole-
cules in each phase at any time was proportional to the counting rate observed a t that time. I t is now desirable to express the concentrations in Equation 2 in terms of the observed counting rates. The term: Cl., in Equation 2 represents the equilibrium concentration of tracer in the water phase corresponding to the concentration of tracer in the ketone phase a t any time. Since the entire transfer operation was conducted a t a single point on the equilibrium distribution curve, it was possible to say for both tracer and nontracer: (3)
The term a is a specific ratio for a given equilibrium concentration level and should not be confused with rn. the slope of the equilibrium distribution line. Since it was possible that a given concentration of tracer molecules in one phase would not produce the same counting rate as the same concentration of tracer molecules in the other phase, a test of the value of a was made by using data from the titration of equilibrium solutions and data from the radiochemical measurements made at isotopic equilibrium. I t was observed that a t isotopic equilibrium, the counting rate in the ketone phase divided by that in the water phase was greater than the total acid concentration in the ketone phase divided by that in the water phase. This is shown in Table I for three acid concentrations.
I t could be caused by the fact that water has a slightly higher vapor pressure than the ketone: which would cause more of the water molecules to evaporate from the surface of the sample being counted, and this might cause a greater resistance to the passage of the beta particles from the liquid sample to the counting lvindow. Also, the vapor concentration of tracer is different than the liquid concentration of tracer, and this phenomenon might affect what the counter “sees.” A more likely explanation is that the absorption cross section in water for the beta particles is higher than that of the ketone. These remarks are merely hypotheses and are not necessarily extended as the true cause of the phenomenon. which was not a subject of this investigation. It )vas necessary, however, to cancel this effect from the observed data to conduct the analysis. Two methods of calculating K were employed. Method 1. It was assumed that the correction factor needed to adjust for the effect shown in Table I was the same at any time during the run even up to the time when the tracer molecules had assumed an equilibrium distribution between the two phases. By combining Equation 2 with Equation 3 and dividing both sides by C r , Equation 4 was obtained :
Since Equation 3 holds true at infinite time-i.e., when the tracer molecules had assumed an equilibrium distribution between the two phases:
c: Table I. Equilibrium Distribution Ratios o f Acetic Acid Determined b y Chemical Analysis W e r e Smaller than Counting Ratios
Counting
13-t. 5; ,kcid i n Water
Rate Ratio,
Phase
(CPX i W 0.602 0.649 0.860
1.137 5.295 26.1
(cP-l1)k
=
c:/o
(3a)
and
Chemical Ratio, Gram Moles Ketone/Ml. Gram Moles Water/AMl. 0.470 0.536 0.787
If it is assumed, as it should be. that the acetic acid tracer molecules are distributed in the same manner as the nontracer molecules, the observations in Table I indicate that the tracer molecules in the water phase are not “seen” as well by the counting device as those in the ketone phase. This could be caused by a difference in the manner in which the acid molecules are arranged in the ketone phase compared with that in the water phase.
PO0
100
k + W l 4
IO00
0
900 0
,
2
3 4 T I M E IN H O U R S
5
6
7
Figure 1. Transient state counting data for both ketone and water phases for Run 9 gave smooth curves VOL. 53, NO. 3
MARCH 1961
231
determining I; might give less subjective results if the observed data could be integrated analytically S O that KA would
09 RUN
NO
9. METHOD 2
V
be the slope of a straight line. For this development it was necessary to make use of a tracer material balance, assuming no hold-up between phases. The following expression for tracer quantities is true.
07
OS
c5
c:v, = C,VW + CkVk = cz V , + CFVk
(9b)
For this investigation, the volume of the water phase and the volume of the ketone phase were very nearly equal, so that:
0 2
01
0 0
1
2
1
4
5
6
7
vu =
T I M E IN H O U R S
T I M E I N HOURS
Figure 2. Graphical integrotion (left) and analytical solution (right) of the rate eauation were the methods used to determine the mass transfer rate constant ooina " " from ketone to water
Substituting Equation 3 into Equation
4: d(C,/C;)
")
K,A _ C ,_ - V , (C: Cy
(6)
The ratios (Cz,C;j and ( C , C;j represent concentration ratios and a t the same time represent ratios of counting rates, because, for example, if C, is proportional to a counting rate. so is
e;,:
intervals, no observed counting rates were available a t identical times. For each experimental run an average line was therefore drawn through the points plotted for the observed counting rates, as illustrated in Figure 1 for Run 9. The average counting rate was then read from these curves at half-hour intervals and these values were the ones used for the purposes of calculation, using Equation 8. First,
C, = P(CP.ZI),.
(7a)
C: = p(CP.tf):
(7b)
c,- -_(_ C P_ M), c: (CP.zii)Z
(7c)
Then
Integrating Equation 6:
c, c: ~~~
1 Ck CY
Vk
(9c)
L-sing Equation 9c, the volume terms can be cancelled in Equations 9a and 9b leaving:
c:
dt
(9a)
+ Ck
(9d)
=:c: +IC:
(9e)
=IC,
From Equations 4 and 9d : d(C,/C::) _ _ _ _ _ - K,'4 dt I'll
c,
(
- (l/a)(C,O- C),
cm,
-)
(10)
B>- rearranging terms and substituting Equation 9e into Equation 10, Equation 11 can be obtained.
was plotted
against (C,,,CG), and the area under this curve was graphically integrated in segments fixed by the half-hour time intervals. The areas for these time intervals were then plotted against time, (as in Figure 2, left) and the slope of the line thus obtained \cas equal to Fw!.
\\here CT Cp is constant. The solution to this equation is:
An
L'W
(8)
Since samples were removed and counted from each phase a t alternate
equation similar to Equation 8 was developed and used in the determination of Kk. Method 2. In Method 1, an average line was visually dra\cn through the observed counling rates. It \vas therefore felt that an alternate method for
=
K,A- t Vu
+ C'
(12)
where C' is the constant of integration. The left side of Equation 12 was plotted against time (Figure 2, right), and the slope of the line, which was calculated
The mass transfer coefficient for acetic acid changed considerably as acetic acid concentration varied going (left) Figure 3. from ketone to water phase ( K k )or (right) from water to ketone phase (K,)
232
INDUSTRIAL AND ENGINEERING CHEMISTRY
M A S S TRANSFER by the method of least squares, was
K A equal to 2.. The term Kk was deterV,. mined in a similar manner. Figure 3 compares K k and Ku for both calculational methods as a function of the concentration of acid that was in the system. When the system contained no acid except for the tracer (approximately 0.03 ml. in approximately 4110 ml.)? K was high. When the acid concentration was increased to approximately 1% (weight) in the water phase a t equilibrium, the value of K \vas measurably decreased. At approximately 57G acid concentration, the transfer rate constant increased from that observed at 1% and then increased again only slightly when the acid concentration was increased to approximately 237,. Although the curves in Figure 3 \vere dra\\m by eye only for the purpose of shoxving the general trend of the data, they are believed to be representative of the phenomena taking place. The relation of the mass transfer coefficient to the circulation rate is demonstrated in Figure 4 for zero concentration of acid. A sharp break occurs when the circulation rate is increased from 400 to 600 ml. per minute. This phenomenon was caused b>-the circulation pattern in the transfer vessel.
Discussion
Mixing Efficiency. During the initial stages of Run 1 (before rhe proportional tubes had sufficient time to become contaminated), a continuous measurement \vas made of the counting rate in each phase. Figure 5 is a plot of the observed counting rates, including background. in the water phase (the phase in \vhich the tracer was injected), for a 0.03ml. slug of tracer for the first 28 minutes of the run. The tracer was mixed into the bulk phase after a period of approximately 15 minutes. This run was conducted at a circulation rate of 100 ml. per minute, Lvhereas most of the remainder of the runs \vere conducted a t 600 ml. per minute. Thus. the bulk mixing for the later runs was considerably better than that shomm in Figure 5. This conclusion was substantiated by visually observing the circulation pattern in the transfer vessel resulting from different circulation rates. .4t the lower rates? the fluid came out of the spray nozzles only near the spray ring adapter where the fluid entered the transfer vessel. The fluid also only trickled out of these nozzles and probably traveled almost directly to the outlet of that phase without going to the opposite side of the vessel.
Figure 5 shokvs that the counting rate initially rose to a high value and then dropped to a lower average value. This can be accounted for by using the above explanation-i.e., the tracer entered the transfer vessel and traveled almost directly to the outlet, from whence it went out to the radioactivity detecting device. At the higher flow rates, the fluid came out of all of the spray nozzles with enough velocity to travel to the opposite side of the transfer vessel. Experimental Runs 2 and 3 were both conducted at three different circulation rates of 100, 200, and 403 ml. per minute. At the conclusion of each of these runs when the circulation rate was 400 ml. per minute, the lid of the transfer vessel was removed, and it \vas observed that the interface was perfectly flat with no turbulence or rippling. .4t higher flow rates, a definite turbulence existed a t the interface with ripples approximately inch high appearing in the center portion of the interface. This, of course, increased the interfacial area slightly, but it is believed that the large increase in the mass transfer coefficient shown in Figure 4 was primarily due to the improved circulation pattern which reduced the region in Jvhich molecular diffusion took place. It was observed during the course of all the runs that once the needle valves had been adjusted to give the desired circulation rate, this circulation rate remained constant throughout the course of the run. The rate of transfer was therefore not affected by a changing circulation rate during any one run or between different runs. Effect of Acid Concentration. When ketone and water with no acid present are placed in contact with each other and shaken together and then allowed to separate, the time required for the two fluids to separate is quite short. The time required for the tlvo fluids to separate at 1% acid concentration is also quite short. At 57,, and especially at 25% acid concentration, the separation time is much longer, as would be expected from the fact that the fluids are much more miscible in each other a t the higher acid concentrations. This indicates that the physical properties of the system, such as the viscosity and interfacial tension: play an important role in the hydrodynamics of the transfer system and could therefore account for the increased rates of transfer observed a t the higher acid concentrations. It seems difficult to believe that these factors could also account for the sharp decrease in the value of K from 0 to 1% acid concentration, as noted in Figure 3. .41so: this study avoided the problem of interfacial turbulence as a
I4
12
IO
$
8
?c' 6
4
2
0 0
200
600
400
CIRCULATION RATE
BOO
-
1200
IO00
ml/m
n
Figure 4. Circulation rate had a strong influence, as expected, on mass transfer coefficient (KJ at constant acetic acid concentration
1000
0
5
10 15 20 T I M E IN MINUTES
25
2E
Figure 5. Rapidity of approaching steady state counting rate in the water phase was used as a measure of mixing efficiency Curve i s for lowest circulation rate
cause of spontaneous mixing at the interface. It is assumed that the K values calculated for the acetic acid tracer molecules are representative of the A- values that would be obtained for the mass transfer of the nontracer molecules if such transfer could be measured a t equilibrium, or almost at equilibrium. between the ketone and water phases. What significance, then. does the observed change of K with acid concentration have? Since K presumably represents a physical transport, the fact that the K values are concentration dependent indicates that other factors inVOL. 53, NO. 3
*
MARCH 1961
233
fluence the transfer besides the concentration driving force. One likely explanation is that solvation effects were taking place. This hypothesis is based upon the fact that a number of investigators have shown that a solute dissolved in a solvent causes the solvent molecules to be assembled in a more ordered pattern. These assemblies can be different for different concentrations, and the rate of and energy required for breaking and making of acetic acid-water assemblies (or possibly acetic acid-ketone assemblies) would be a function of concentration. Thus one might expect the mass transfer coefficient to be affected by concentration. Measurements made on self-diffusion coefficients in nonideal systems (7, 4, 7 ) show variations with concentration similar to that revealed for K in this study. If the transfer measurements made here were diffusion controlled, the mass transfer coefficient would not be expected to be constant with changing concentrations (nor even linear) and might easily have a minimum value. Unfortunately this explanation cannot be verified a t this time because there are no data available for the diffusion coefficients for this system, nor are there any extensive data on the possible structures of unionized acetic acid in water or in ketone. What does the concentration dependence of K mean with regard to the customary mass transfer experiments for this system in which a concentration difference between phases is used to effect the transfer? I n these cases, whatever model is chosen as the rategoverning relation will give some kind of average value of K , which, although reproducible, will change as the concentration differences in the equipment change. I t would seem reasonable to assume that this same phenomenon would hold true for other chemical systems. There: fore, where fundamental mass transfer experiments are being conducted, this phenomenon could be mitigated by conducting the experiment a t as close to equilibrium concentrations as possible. For the chemical system used in this investigation, it would also probably be well to conduct the experiment outside of the range of 0 to 570 acid concentration. Comparison of Methods 1 a n d 2. As shown by the figures presented, the transfer coefficient calculated from Method 1 is somewhat different from that calculated using Method 2. Method 1 utilizes the counting rates obtained from both phases, while Method 2 uses only the counting rate from the phase of high tracer concentration. For the runs in which the data were somewhat scattered, Method 2 offers the advantage of providing a plot of points
234
concentration of solute in one phase corresponding to solute concentration C in other phase, gram moles/ml. = tracer concentration a t time zero when tracer was injected into one of the two phases = tracer concentration a t infinite time when tracer molecules have assumed equilibrium distribution between the two Dhases CPM = counting rate, c.p.m. K = over-all mass transfer coefficient, cm./hr. m = slope of equilibrium distribution curve .V = rate of mass transfer, gram moles/ hr. = time from start of transfer, hr. t = volume of liquid phase, ml. V = concentration of acetic acid in cy ketone phase divided by concentration of acid in water phase p = proportionality factor = equilibrium
Table (I. Material Balance Check for Run 9 Using Equation 1 3 Shows Good Agreement Except for the First Part of the Run
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
1.09 0.9 0.766 0.727 0.675 0.661 0.572 0.507 0.466 0.392 0.330 0.278
-2.90 -1.79 - 1.482 -1.355 -1.205 -1.170 - 1.000 -0.87 -0.78 -0.665 -0.574 -0.489
-0.376 -0.502 -0.516 -0.536 -0.560 -0.565 -0.572 -0.583 -0.598 -0.590 -0.575 -0.568
along what is known to be a straight line, whereas the curves used with Method 1 are visually fitted to the data points. These methods of calculation are based on the same model, and should give the same results, but they do not. At this time, the exact reason for this discrepancy is not known. Material Balance. A material balance of tracer molecules was tested indirectly using Equations 3 and 9, from which the following equation was obtained :
Equation 13 states that if (Cb,tCz) and (C, ’C:) are plotted against time, the slope of the lines so obtained should give a. The slopes and the corresponding values of cy for R u n 9 are shown in Table 11. I n obtaining the slopes, the observed counting rates were divided by the equilibrium counting rate (Ca) and plotted against time, with average lines being drawn through the points. For an acid concentration of 5.295% (Run 9), cy is 0.535. The differences between this value and those in Table I1 can easily be attributed to the process of differentiation itself, since the slopes were taken with a mechanical slope-taker, and a slight error in reading this instrument or in the slope of the line itself could cause a relatively large deviation from cy. This sensitivity would not affect the values of K , because a n integration process was used to obtain these values. The values of K are estimated to be valid within 7%.
*
Nomenclature
A
= interfacial area between phases,
C
= solute
INDUSTRIAL AND ENGINEERING CHEMISTRY
sq. cm. concentration, moles/ ml.
gram
Subscripts 1 = phase 1 2 = phase 2 k = ketone phase w = water phase literature Cited
(1) Anderson, D. K., Hall, J. R., Babb. A. L., J . Phys. Chem. 62,404 (1958). (2) Bak, E., Geankoplis, C. J., Chem. 3 Eng. Data Ser. 3, 256 (1958). (3) Brinsmade, D. S., Bliss, H., Trans. Am. Inst. Chem. Engrs. 39, 679 (1943). (4) Edwards, 0. W., Huffman, E. O., J . Phys. Chem. 63, 1830 (1959). (5) Glasstone, S., Laidler, K. J., Eyrin5: H., “The Theory of Rate Processes, 1st ed., McGraw-Hill, New York, 1941. (6) Handlos, A. E., Baron, T., A.I.CI2.E. Journal 3,127 (1957). (7) Hardt, A. P., Anderson, D. K.. Rathbun, R.! Mar, B. W., Babb, A. L., J . Phys. Chem. 63, 2059 (1959). (8) Johnson, H. F., Bliss, H., Trans. Am. Inst. Chem. Engrs. 42, 331 (1946). (9) Karr, A. E.: Scheibel, E. G., Chem. En,?. Progr. Symposium Ser. 50, No. 10, 7 3 (1954). (10) Lewis, W. B., Pratt, H. R. C., Nature 171, 1155 (1953). (11) Licht, W., Jr., Conway, J. B., ISD. ENG.CHEM.42, 1151 (1950). (12) Oldshue, J. Y., Rushton, J. H., Chem. Eng. Progr. 48, 297 (1952). (13) Othmer, D. F., White, R. E., Treuger, E., IND.END.CHEM.33, 1240 (1941). (14) Scheibel, E. G., Karr, A. E., Ibid., 42, 1048 (1950). (15) Sherwood, T. K., Evans, J. E., Longcor, J. V. A,, Ibid., 31, 1144 (1939) ; Trans. Am. Inst. Chem. Engrs. 35, 597 (1939). (16) Sigwart, K., Nassenstein, H., V D I Zeitschrift 98,453 (1956). (17) Trevbal, R. E., “Liquid Extraction ” ‘ McGraw-Hill, New‘Yor-k, 1951. (18) Weiser, H. B., Colloid Chemistry,” Wiley, New York, 1939. (19) Whitman, W. G., Chem. @ M e t . Eng. 29, 146 (July 23, 1923). RECEIVED for review July 7, 1960 ACCEPTED December 15, 1960 Work supported by U. S. Atomic Energy Commission.
