Measurement of Diffusion Coefficients John E. Crooks Pharmacy Department, King's College, London, SW3 6LX, U.K Diffusion is a well-known physico-chemical phenomenon of great theoretical and practical importance, hut it is difficult to find a suitable teaching lahoratory experiment for the measurement of diffusion constants. Diffusion is a slow process, and typical diffusion experiments require many hours. Irinal r e ~ o r t e da s i m ~ l eex~erimentin which diffusion UD from t h e bottom of speckophotometer cuvette was observed for as lone as 40 hours. Clifford and Orchia12 have desrribed a diffus/on cell construrted from a Y-shaped stopcock harrel. A sidearm containing less than I mL of solvent is connected to concentrated soluiion in the bore of the tap barrel. At the end of a time period of 5 to45 minutes, the tap is turned to isolate the sidearm. and the contents are removed for analysis. The concentration of the sample solution rises to onlv a few Der cent of that of the source solution during the diffision time. This method is capable of considerable precision, hut requires some manual dexterity. Several hours are needed to show the progress of diffusion with time, as each run only gives one point on the graph. Moskovits and Derewlany3 have reported an experiment in which a concentrated solution is contained inside a dialvsis hap. The bag is suspended in a large volume of stirred soivent and the increase in solute concentration in that solvent is measured a t a series of times. A whole run takes less than an hour, since only a small fraction of the solute diffusing out is monitored, and the time course of diffusion can he readilv seen. The disadvantage of this experiment is that the measured quantity is not a simple diffusion coefficient, hut rather the permeability coefficient of the dialysis membrane. I t is therefore not possible to compare the results with published data. A major problem in diffusion coefficient measurement is to make sure that solute movement is solelv hv diffusion, i.e.. random molecular motion, and not by bulk liquid move: ment, e.g., by convection. Bulk movement is prevented if the liquid is held in a gel. Agel is a comparatively rigid structure composed of a three-dimensional open network of polymer chains, the voids filled with liquid. Small molecules, of solvent or solute, can move about in a gel as freely as in an ordinarv .liouid. . This can easilv he shown hv measurine the conductivity of an ionir solute in a cooling solutionof agar in wawr. The conductivity falls as the temperature fall% hut
a
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Journal of Chemical Education
there is no discontinuity as the mobile liquid turns into a firm jelly. A convenient experimental system consists of a cylinder of agar gel formed inside a glass tuhe. The gel contains a concentrated solution of the solute under study. The tuhe is held vertical with the lower, open, end immersed in a large volume of stirred solvent. A t the moment of immersion the solvent contains no solute, and the solute concentration from then on increases with time. Diffusion throueh the eel is in one dimension only, down the vertical axis. A mathematical analvsis of this svstem has been eiven hv H a d e ~ a f t . ~ If less than 5%or so of the total amouncof soluie in tbe gel has diffused out, then the fraction, MJM,, diffused out a t time t is
M&
=~
( D ~ I ~ L ~ ) ~ ~11)~
where L is the length of the jelly cylinder. Procedure
The details of the experimental procedure depend on the monitoring technique. The diffusion of a highly colored substance, such as hromoohenol hlue. which can he monitored soeetroohotometrically, g w i a a n rffecti$;evisual display, whereas thb drffuwm of KCI. mmiumd rondurrirnrrrically. giver a value readily compnrnbie with puhliahed dntn. 'The procedure for bromuphenol hlue is described first. The jelly mold is a length of thick-walled glass tubing, about 10 cm by 2 cm diameter, with one end ground flat. The tube has three indentations toward the top to prevent the jelly cylinder from sliding down. The ground end is greased and pressed on to a glass tile to hromomake a seal. A 5-mL samole of a solution of 35 me.. faoorox.) . .. phenol blue and 0.5g wdiurn acetatein 50 mL wnter isdilurrd to250 ml., nnd its abqorbanrr at 5NI n m (A,, is measured. The remaining ~~
' Irina, J. J. Chem. Educ. 1980, 57,676-677.
Clifford,El.; Ochial, E. I. J. Chem. Educ. 1980, 57, 678-679. Moskovits, M.; Derewlany, L. Educ. Chem. 1978, 15, 45-52. Hadgraft, J. Int. J. Pharmacevt 1979,2, 177-194.
~
When the jelly has reached room temperature, which takes about 314 hour, the glass tile is slid away to reveal a plane jelly surface. The tube is clamped vertically, at such a height that the lower end will be approximately 2 cm above the bottom of a 250-mL beaker containing 150 mL of water. The water is vigorously stirred hy a magnetic stirrer. The clampstand is lifted up and set down so that the tube is in the center of the beaker, and a stopwatch is started. Samples are removed so that their ahsarhance, At, at 590 nm can he measured, and then returned tothe beaker. Suitable times are 1,2,5,10,20,30, 40,55,70, and 90 min from the start. Then MtlMo= (At/A,)(150l45)(5/250)
(2)
The procedure for KC1 is similar. About 0.4 g KC1 is dissolved in 50 mL water, and a 5-mL aliquot is diluted to 250 mL. The conductance of this dilute solution is measured. The remaining 45 mL of concentratedsolution is made into a jelly, as described ahove. There is no need to remove samples to monitor the increasing concentration in the stirred water if a dipping conductivity electrode is svailable. At these low concentrations there is no significant error in assuming conductance to be linearly proportional to concentration.
Fraction dlffused as e hmctlon of square root of h e : a. bromophenol blue anlon; b. KCI.
Results and Discusslon Equation 1shows that a plot of MJM,against t1I2 gives a straight line from whose gradient D can be calculated. Typical data are shown in thefieure. T h e deviation a t lone times is probably due to solutionoozing out of the sides of &e jelly and seenine . ..down between the .iellvand . the elass. The leneth of the jelly cylinder (8.7 cm fur the bromophenol l~lueexperiment. 9.3 cm for the KC1 exneriment) is measured with an ordinmy ruler. Hence values-of D a t 25 O C were found to he 1.94 X m2s-I for KC1 (literature value 1.844 X m2 8-1) and 4.4 X 10-10 m2 s-1 for anionic bromophenol blue. The special apparatus required for this experiment is extremely cheap and simple. A diffusion coefficient can be measured in a three-hour lab period. The data give a clear demonstration of the variation of the extent of diffusion with the square root of the time, the most characteristic feature of the diffusion nhenomenon. The exneriment also illustrates amodernapphation of diffusion, slow release of theraneutic drue from an i m ~ l a ninto t the bodv. so that a steady iong-term concentratioh of the drug in dddy fluids can be maintained.
tee
45 mL of concentrated solution is brought to the boil, and 1 g of agar crystals is dissolved in with stirring. The hot solution is poured into the glass tube and left to cool.
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Number 7
July 1989
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