KAROL J. MYSELSAND D. STIGTER
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Vol. 67
A NEW METHOD O F MEASURING DIFFUSION COEFFICIENTS1~2 BY KAROLJ. MYSELSAND D. STIGTER Contribution from the Department of Chemistry, University of Southern California, Los Angeles 7 , Ca1;fornia Received July 88, 1068
The new method uses fritted glass to immobilize the liquid in which diffusion occurs, but differs from the conventional fritted glass diaphragm method in that the whole liquid is immobilized. The need for stirring any part of the liquid is thus obviated. The method is particularly adapted for tracer work and is being used to determine the sdf-diffusiou coefficient of micelles of association colloids tagged by solubilized dyes.
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While many excellent methods of measuring diffusion coefficients are available, there does not seem to be any adapted for precise measurements in macromolecular tracer systems. The need for such a method became apparent in the determination of self-diff usion coefficients of micelles using solubilized water-insoluble dyes for tagging the micelle, as has been suggested earlierS5 The presence of tracers produces no density gradients, hence common methods using wide channels cannot be used because unavoidable convection currents would cause excessive errors. Immobilization of the liquid in small capillaries seems necessary despite the accompanying danger of surface diffusion. The conventional porous diaphragm method depends critically on maintaining uniformity of concentration in the two compartments (which has been shown recently to require vigorous stirring4~6)and' on the complete immobilization of the solution in the diaphragm. A sharp and permanent boundary must exist between these regions. If stirring leads to any mechanical "pumping" of the solution through the diaphragm, it may cause large methodical errors when applied to slowly diffusing macromolecular systems even though it is harmless for rapidly diffusing simple ions. The new method immobilizes the whole solution in two relatively thick (1-2 em.) fritted glass discs. Thus any need for stirring is obviated. At the beginning, one disc (ie., its pores) is filled with solution containing tracer and the other with tracerless solution. The discs are clamped together and submerged in mercury to prevent evaporation while diffusion proceeds. They are then separated and the amount of tracer in each disc determined. The diffusion coefficient is computed from the measured transport by comparison with the behavior of a substance of known diffusion coefficient, D. In our case sodium chloride diffusing between 0.2 and 0.7 N concentrations was chosen, as it has (1) Presented before the twenty-sixth National Colloid Symposiuln which was held under the auspices of the Division of Colloid Chemistry of the American Chemical Society in Los Angeles, California, June 1618. 1952. (2) This work has been conducted as part of Office of Naval Research Project No. NR 054-254. A much more detailed account is
presented under the same title in mimeographed First Technical Report, O N R Project No. NR 054-254, available from the senior author, The Library of the University of Southern California, or the Library of Congress. (3) R. B. Dean and J. B. Vinograd, THISJOURNAL, 46,1091 (1942): H. W. Hoyer and K. J. Mysels. ibid.,64, 966 (1950). (4) R. H. Stokes. J . Am. Cham. Soc., 73, 763 (1950). (5) J. M. Nielsen, A. W. Adamson and J. W. Cobble, ibid., '74, 446 (1952).
an unusually small variation6 of diffusion coefficient with concentration, Calculations , The ratio .R of transport across the plane of separation of the discs a t a time t to the transport at equilibrium is a dimensionless quantity which can be readily shown2 to depend only on t and the diffusion coefficient D for any given system, provided D is independent of concentration and the concentrations are originally uniform on each side of the plane. In particular, $ is independent of the original concentrations and of the dir'ection of diffusion. These relations are true for discs of any arbitrary shape. If the same R is obtained a t times tl and 12 for two substances, then tiDi
= 1,Da
This relation is used to calculate the unknown diffusion coefficient. Since D1for the calibrating substance and tz for the unknown are both known, it requires only the knowledge of the time tl required by sodium chloride to reach the same R. This is interpolated fronl several calibrating experiments This plot is linear up to on a plot of R vs. about R = 0.6 and then gently curves to reach R = 1 a t infinity. One can determine experimentally the volumes Vu and VI of the two discs, the concentration Cn of the solution placed in the lower disc (assume that in the upper = 0) and the amounts A, and Al found in the two discs by analysis a t the end of the experiment. One may then calculate the transport at infinity, as T, = CC,V,V,/(V~ Vu) and an average transport T = (A./2) (C0V,- A1)/2. Of course R is T/T,. If R is thus computed it is very insensitive to small errors in the position of the original boundary. Thus if the boundary is formed at a small volume A below the surface of the lower disc (at which separation occurs), the concentration in this volume A will be substantially equal to C0/2 within a very short time. It is then easy to show2 that T and R are completely independent of A. Experimental
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The discs are cut from fritted glass cylinders of medium porosity (average pore diameter 14 p ) 1" X 2" or 4" whose sides are glazed by fusion immediately after .sintering.? Alternate methods of closing the pores are fusion of the fritted cylinder into a glass tube, which is quite difficult and reduces the porosity near the walls, or imbedding the cylinder in lucite, which gives a less inert coating. Two discs are always cut immediately adjacent to each other and their, (6) R. H. Stokes, ibid., 73, 2243 (1950). (7) Supplied by the Corning Glass Works, Corning, New York.
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A NEWMETHOD OF MEASURING DIFFUSION COEFFICIENTS
,Jan., 1953
I
LUCITE
~1
Upper Disc
H
Lower
9
DISC
12
I
LUCITE
’
I
Fig, 1.-Schematic representation of the diffusion cell showing in an exaggerated manner the spaces I, I1 and I11 occupied by the free liquid and the formation of the boundary below the surface of the disc. original relative position preserved so as to maintain closely matching surfaces. A height of about 13 mm. seems to be optimum. A greater height prolongs the measurement unduly; a lesser one reduces the amount available for analysis. The cell is shown schematically in Fig. 1 and its assembly in a clamp in Fig. 2. The porous discs are mounted with Araldite 101 cements in lucite rings for ease of handling and aligning. The two porous discs are held between two lucite discs in the stainless steel clamp which ensures rigidity and reproducibility of alignment and allows immersion in mercury to prevent evaporation and leakage. Filling of the porous discs with a solution (and its removal) is done by gradual displacement of a miscible liquid. When the porosity of the disc is uniform up to the impervious wall, slow flow under gravitational forces of 1.5 pore volumes is sufficient to replace over 99.9% of the original liquid. When the porosity is not uniform, as many as ten volumes may be necessary. We always use a 50-70% excess above the minimum required. This displacement is rather slow and careful protection from evaporation is necessary. Burets may be used to dispense NaCl solutions but syringes are used with detergent to avoid loss of dye to stopcock grease or elastomers. Analysis .-Sodium chloride concentrations were deter*mined by direct conductimetry after dilution to a known volume, using a Jones-Dike bridge,O properly designed cells and Shedlovsky’sIo equivalent conductivity values. The conductivity of the water used was always determined and subtracted. Dye contents were determined with 10mm. cells in a Beckman UV spectrometer using a wide slit a t the absorption maximum and correcting for any turbidity measured in the region of negligible absorption. Volume Determination .-Because of unavoidable unevenness of the disc surface, there are spaces between the discs, as shown in Fig. 1. Each of them amounts to about 0.02 cc. The liquid in the middle space is always considered a part of the upper disc. The liquid in these free spaces is collected (with a pipet for NaCl solutions and small squares of cleansing tissues for detergent solutions) and added to that contained in the pores of the corresponding disc. The volumes of each disc are determined by filling both discs with the same solution, assembling, disassemblin and determining the solute content of each. Using NaCq solutions the mean deviation is about 0.002 cc. for a volume of approximately 2 cc. Using solubilized dye the mean deviation is about 0.01 cc. due to the lower precision of colorimetric analysis and there is no significant difference between the volumes determined by the two methods. Calibration.-Sodium chloride solutions 0.7 N in the lower and 0.2 N in the upper disc are used for calibration. This avoids the presence of very dilute solutions in which surface diffusion is significant,4~6gives sufficient amounts of solute for precise analysis and utilizes the unusually broad minimuma of the D us. cmcn. curve for this salt. The value for 0.5 N , D = 1.474 X 10-6 cm.Z/sec., is used in further computations. The values of time for various values of R are read from a graph of deviation from linearity with respect to fi. (8) Supplied by CIBA, New York. (9) P. H. Dike, Rev. Sci. Instruments, a, 379 (1931). (10) T. Shedlovsky, J . Am. Chem. Soc., 54, 1405 (1932).
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8 Fig. 2.-The diffusion cell is formed by porous discs U and L (each in a lucite ring) and is closed by lucite discs 3 and 4. I t is held by springs 10, .11 and 12 in a clamp formed by plates 5 and 6 connected by rods 7, 8 and 9. Handle 15 serves to hold the cell under mercury.
The boundary between two different solutions is formed by filling the discs with the two solutions, clamping the lower one in position, drying the outside, submerging its lower edge in mercury and drying its upper surface. The upper porous disc, covered with the upper lucite disc and with a drop of the solution hanging from its lower surface, is lowered into position, the excess liquid squeezed out, and after closing the upper spring and drying the outside, the whole cell is submerged in mercury. In this process there is a small amount of unavoidable mixing of the two solutions, and some of the liquid from the lower disc is pressed out, so that the boundary is effectively formed at a small volume A (Fig. 1) below the surface of the lower disc. The material balance shows that this volume amounts to about 0.01 cc. with an average deviation of about 0.005 cc. As has been shown above, under Calculations, the value of A does not affect the average transport ratio R . That the initial mixing is of no significance may be seen from Table I, which compares results of short time diffusion experiments with values predicted on the basis of long time results. Significant mixing would lead to consistently higher experimental values and require a zero time correction, but this is not the case. TABLE I SHORT TIMEDIFFUSION EXPERIMENTS 0 . 7 m NaCl in lower disc; 0 . 2 rn NaCl in upper disc Time, inin.
A, cc.
+o
.004
+
.012
Calcd.
Transport ratio, % Found Difference
2.1 1.9 -0.2 2.3 2.6 .3 .040 6.0 5.8 .2 5.0% Aerosol MA Orange OT in lower disc; 5 . 0 % Aerosol MA in upper disc 128 $0.004 6.2 6.4 $0.2 164 ,000 7.0 7.1 .I Density gradients which act to st,abilize t.he boundary are present in calibration runs using NaCl. In tracer experiments there are in principle no density gradients and in 2.5 3 9
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F. D. MASUN, M. ALTMAN AND E. R. ABERTH
106
practice there may be even minute destabilizing ones. The method seems, however, to be completely insensitive to such inversions of density because in two runs in which sodium chloride solutions were used in inverted position ( i . e . , 0.7 N in the upper and 0.2 N in the lower disc), the values of R deviated by -0.6% and t;-1.1% from the calibration line for I1 hr. and 17 hr. experiments, respectively. (Dr. R. J. Williams has found that discs of coarse porosity are no longer insensitive to such extreme inversions of density gradient but are still satisfactory for tracer work). Temperature control is provided by an air-bath whose temperature varies by =l=0.05"with a period of approximately 1 minute a t 25 i 0.1". The mercury cups are supported on rubber in an almost vibrationlexs way although strong vibrat.ions seem to have no effect,' Accuracy and Precision.-The above results show that the method seems to eliminate most sources of methodical errors except that of adsorption and surface diffusion in the particular system studied. The presence of these effects is indicated by a discrepanc between volumes determined using tracer and using N a d a n d may be ascertained by using discs of different porosities. The precision depends on the value of R , showing a flat minimum near R = 0.65. The reproducibility of measurements with sodium chloride corresponds to about f 0.5% in the value of D and with dye solution it corresponds to the analytical error. Thus in 5y0 Aerosol MA, Orange OT in three different discs gave four results averaging 1.089 X 10-8 cm.Z/sec. with a mean deviation of 1.2%, while oil blue (which is appreciably water-soluble) gave three results averaging 1.156 X 10-6 cm.*/sec. with a mean deviation of 1.1%.
DISCUSSION 8. C. LIANQ.-wOUld
the dye affect the diffusion of the micelles because of the amount of dye pres'ent in the micelle? K. J. MYsELs.-That is a very good question. However, the amount of dye employed was on the average less than one dye molecule per micelle. Therefore such an effect would be very small. Furthermore the dye was so com-
Vol. 57
pletely water-insoluble and very soluble in oil 'so that it should be located well inside the micelle. Also in electrophoretic measurements we used several different dyes to see if they gave the same results and found that they did. I do not think we are changing things very much but we plan to check this point by comparing different dyes in diffusion.
IRVING REICH.-I would like to just go a stage further on the question just asked. Obviously the molecule or two of dye locked within the micelle will not affect the speed of diffusion very much. But there is so much to be understood about the geometry and the energetics of the micelle. It may be a highly sensitive structure. Might it not be possible for the micelle to let's say, double in size because of the presence of even one molecule of the dye? K. J. MYsELs.-If there was a large effect by one kind of dye, presumably different kinds of dyes would give very different results and we propose to consider this effect a t a later time. MALCOLM DoLE.-Could not dye molecules diffuse from one micelle to another micelle by some kind of chain effect? ' The solubility of the d e in the solvent, even if the dye is very insoluble, may maze this kind of an effect possible. K. J. MYsELs.-If the dye is completely insoluble in water, then in order for them to jump from one micelle to another, the micelles must be very close together, in fact the micelles must touch, which is very unlikely. MALCOLM DoLE.-It is also unlikely that the dye is completely insoluble. K. J. MYsELs.-The solubility of the dye in water is estimated at perhaps 1/10,000of the solubility in the soap solution. Hence for 10 OOO dye molecules which are entrapped inside the micelies there is only one which is in the water. Even if this one were diffusing ten times faster than the micelles it would introduce only an error of 1/10 of 1%.
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PREDICTION 'OF GAS-ADSORBENT EQUILIBRIA* BY F. D. MASLAN,~ M. ALTMAN .4ND E. R. ABERTH Departnmt of Chemicd Engineering, New Yo& University, New York, N . 1.. Received July 22, 1858
It has been found that a modified Polanyi-Dubinin theory can be used to correlate the adsorption of various gases on activated carbon, silica gel and activated alumina with good accuracy both above and below the critical I n this modification the adsorbate is considered as a highly compressed gas, and its volume and fugacity are cagdl%ed at the adsorption temperature. The method has been tested on nine systems and good checks with experimental results are obtained. Binary gas adsorption can be predicted from single gas adsorption data when N12V1e = NIVI N2V2. This method has been tested on the system oxygen-nitrogen-activated carbon wlth success.
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It is desirable ttobe able to predict gas adsorptiou equilibria from a minimum of experimental data. The Polanyi-Dubinin theory offers a route to this goal. Single-Gas Adsorption.-The modification of the Polanyi-Dubinin adsorption theory proposed hy Lewis and his co-workers3 is given by when -
NrV'r = N I ~ V ' I I
(2)
(1) Presented before the twenty-sixth N'ational Colloid Symposinin
which was held under the auspices of the Division of Colloid Chemistry of the American Chemical Society in Los Angeles. California, June 1618, 1952. (2) National Research Corp., Cambridge, Mass. (3) W. K. Lewis, E. R. Oilliland, B. Chertow and W. P. Cadogan, I n d . Eng. Chem., 42, 1826 (1950) ,
where f = fugacity of the gas a t adsorption pressure and temperature, f s = fugacity of saturated liquid at adsorption temperature, N = moles adsorbed per unit weight adsorbent, R = gas law constant, T = adsorption temperature,' degrees absolute, and V' = molal volume of saturated liquid at a vapor pressure equal to adsorption pressure. The subscripts I and I1 refer to different sets of adsorption conditions. I n the above relationships the adsorbate is assumed to be a saturated liquid. Lewis, et aZ., found that good correlation could be obtained using the above equations and definitions below the critical point. Data for many hydrocarbons correlated well. Branching of the generalized correlation curves, NV' vs. (RTIV') In fa/f, were obtained for saturated and unsaturated hydrocarbons. Whereas the substitution of fugacities for pres-
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