Peter W. Linder, Luigi Fk Nassimbeni, Alfred Polson, and Allen L. Rodaers University of Cape Town Rondebosch. 7700. South Africa
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II
The Diffusion Coefficient of Sucrose in Water A physical chemistry experiment
As part of our undergraduate course in physical chemistry for biochemistry and microbiology majors we have recently introduced a laboratory experiment to determine the diffusion coefficient of sucrose in aqueous solution. Experiments on diffusion in liquids have already been described by Kraus and Tye ( I ) , Watts (2), and de Paz (3). Each requires long or relatively long periods of diffusion. Nishijima and Oster (4) descrihe a shorter experiment hased on a microscopic technique using an interference wedge. In this paper we describe an experiment hased on the free boundary method devised by Polson (5). The entire exercise is readily completed within a 3-hr laboratory period by average students. Theory
The diffusion of a solute in a liquid solution is governed by the same basic laws as gaseous diffusion, namely those due to Fick, except that there is a greater departure from ideality. Fick's first law for one-dimensional diffusion may he written as J=
-D
(E)
d l I
(1)
where the flux, J, is the quantity of suhstance diffusing per unit time through unit area of a plane perpendicular to the ) ~the concentration gradient direction, x , of flow; ( a c l a ~ is of the diffusing suhstance, after a time, t, of diffusion; D is the diffusion coefficient of the diffusing suhstance in the medium concerned. The negative sign indicates that diffusion takes place in the direction of decreasing concentration. Fick's second law (eqn. (2)) is obtained by combining
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eqn. (1) with the law of conservation of mass in a one dimensional diffusing system.
In eqn. (21, (aclat), denotes the time rate of change of the concentration of the diffusing suhstance at a position, x, along the direction of diffusion. In the case of ideal gaseous diffusion, D is independent of the concentration, but this is not generally true for the diffusion of solutes in liquids (6). Thus in the latter type of system it is customary to carry out diffusion measurements a t different concentrations of the solutes and to extrapolate the results to zero concentration. Alternatively, if D is assumed to be independent of the concentration, the value obtained is actually an integral diffusion coefficient corresponding to the concentration interval between zero and the maximum concentration in the particular experimental run concerned (7). Clearly, it is preferable to work with dilute solutions in order to minimize this concentration interval. By making the assumption that D is independent of the concentration, Fick's second law reduces to
Consider the diffusion of a solute from a column of solution to a contieuous column of nure solvent with an initially sharp houniary between the two (Fig, 1).Provided the variation of D with concentration is not unduly marked, eqn. (3) may he assumed to describe the process with reasonable accuracy. In order to obtain the value of D from suitahle experimental measurements the appropriate
very small
Boundary Solution initial conc. = c, e---*
Cross-sectional Area, A Figure 1. Diffusion d a solute across a free boundary
houndary conditions must he applied. If the columns of liquid diagrammed in Figure 1 are effectively of infinite length so that the concentrations of solute a t the top and bottom remain zero and co, respectively, throughout the experimental run, the boundary conditions applicable are as specified in eqns. (4). The origin, x = 0,is taken as the initial position of the houndary between the solution and the solvent
Figure 2. The spreading of an innially sharp boundary at x = 0 , during free diffusion. The wncentration of the solute is shown as a function of the d i k tame, x, ham the Initial position of the boundary, at various times. t (eqn. (5)). Figure 3. The concentration gradient of the solute as a function of the d i s tance. X , from the initial position of the boundary, at various timer. t (eqn. (6)).
att = O e = e o f o r x < O
.Substitution of this expression form into eqn. (8)yields
The solution of eqn. (3) with the houndary conditions of eqns. (4) is (8)
Hence by determining the concentration, c, after diffusion has heen allowed to take place over a time interval, t , the value of D may he obtained by applying eqn. (9).
The term in square brackets is the probability integral and Figure 2 shows a graphical representation of eqn. (5) corresponding to various Galues o f t h e time, t. The derivative of eqn. (5) has the same form as the Gaussian error curve
Equation (6) is represented graphically in Figure 3 for various values of the time, t. At the original position of the houndary x = 0 and hence the concentration gradient a t this position is given by
By combining eqn. (7) with Fick's first law, we obtain the following expression for D in terms of the amount, m, of solute which has diffused across the houndary (area A) in a time t.
In the method described in this paper, the solvent column is totally removed after a diffusion experiment is completed and is stirred to eliminate the concentration eradiknt thereby producing a uniform solution of concent;ation c. If the heieht of the solvent column in the diffusion cell is h, it followsthat
Figure 4. Sketch of the Polson free boundary diffusioncell unit.
Volume 53,Number 5, May 1976 / 331
.. The particular apparatus used in our undergraduate laboratory was designed and constructed many years ago by Polson and has been used by him in several studies on proteins and viruses (Fig. 4). Each diffusion unit consists of four stainless steel circular sections 60 mm in diameter and with thicknesses of 5, 10, 20, and 30 mm, respectively. Six holes 5 mm in diameter are drilled through each of the upper three sections and to a depth of 25 mm into the bottom section. The flat sections of the surfaces are well ground. They turn on a central bolt fixed in a flat stainless steel base. 330 X 250 X 12 mm. In a certain position, the holes koindde to form six cylindrical cavities 6 0 mm deep and 5 mm in diameter. The stainless steel base carries six such cylindrical diffusion units and three adjustable feet to facilitate levelling. The design of the diffusion units may be modified for experimental convenience. In the present application, for exnmnle. ---- r--. onlv two circular sections are necessary, each with thicknesses of 30-35 mm. There would he an advantage for the e x ~ e r i m e nin t drilline the wlindrical cavities to ~ - nreseut ~ a slightly wid; bore in order to increaae the volume of the upper section cavities to just greater than 1cm3. -
A~
~
~
A
~~~
Experimental Procedure
The ground surfaces of the diffusion units are smeared lightly with silicone grease. The circular sections are positioned so that the four members of each set of six cavities per diffusion unit are in coincidence. The prepared solution of the solute under investigation is added to alternate cavities, three in all, so that the level rises above the upper surface of the bottom circular section. In our laboratory we use sucrose solution, about 10 g I-'. The upper three circular sections are then rotated through about 'ha of a revolution so as to isolate the three columns of solution in the bottom circular section. The vacant top cavities are now exactly filled with pure solvent. The upper three circular sections are rotated in the same direction as before through a further ' h z of a revolution thereby hringing the three columns of pure solvent into contact with the three columns of solution, with sharp boundaries farmed between each pair. After s definite period of time, between 1and 5 min, the upper columns of liquid are isolated by rotating the upper three circular sections. These upper columns of liquid are transferred totally to clean dry test tubes and agitated to eliminate the concentration gradients thus produeing solute solutions of uniform eoncentration, e , (eqn. (4)). In the case of our experiment with sucrose, the concentrations, c, are determined according to the method described by Hodge and Hofreiter (9). One milliliter of the sucrose solution is mired with 1 ml of 5% aqueous phenol and 5 ml concentrated sulfuric acid are added in a stream which impinges directly on the Liquid surface thereby producing good mixing with even heat distribution. The last portion of sulfuric acid is used to rinse the inner walls of the container. The resulting solution is agitated, allowed to stand for 10 min, agitated again, and brought to 2H0'C for a further 20 mi". Using distilled water treated similarly with phenol and sulfuric acid, as.a reference blank, the optical absorbance of the sucrose solutions is measured at 490 nm. In order to compro-
332 / Journal of Chemical Education
mise between the precision attainable and the time available for a laboratory session, our studenta normally use only three sucrose standards of known concentration to obtain a Beer's law calihration line. Appropriate concentrations are 1W.50, and 20 rrg ml-'. Results and Discussion Typical results obtained a t 20% for sucrose (10.0 g I-') hy a competent student for four runs are
D = (4.3 i 0.3) X 10-l0 m2s-' No diffusion coefficient values a t 20°C for sucrose are available in the literature. In order to assess the above result therefore, we invoke the value obtained by Gosung and Morr~s( 6 )and ronfirmed bv Chatterlee (10) and Tilley and Mills (11), for 1.0011% sucrose a t 25% namely 5.148 X 10-lo m2 s-l. By applying the Stokes-Einstein equation together with the published values of the viscosity of 1%sucrose a t 20 and 25'C (12), the Gosting and Morris diffusion coefficient when corrected to 20°C yields a value of 4.49 X 10-10 m2 s-'. The agreement of our student's values with the latter is favorable. Although Polson's apparatus can he operated while immersed in a liquid bath thermostat, this facility is not used in our procedure in order to keep manipulation as uncomplicated as possible and also to eliminate the time needed for thermal equilibration. Since very short diffusion periods are used, between one and five minutes, room temperature is generally adequately stable during a run. A disadvantage is that measurements cannot he made a t a preselected temperature. A source of error arises from the fact that eqn. (6)is strictly valid only for two infinitely long columns of liquid, whereas in our experiment relatively short columns are used. Since the time allowed for diffusion is short, however, the concentrations of sucrose a t the extremities of the columns away from the boundary are unlikely to deviate significantly from the values co and zero, respectively. Thus the procedure outlined above can be considered capable of yielding results which are satisfactory for the purpose. Moreover, the experience gained is valuable, educationally, in enriching insight into the phenomenon of diffusion. Literature Cited (11 Kcam, G.. and Tye,R., J. CHEM. EDUC., 26. a 9 (1949). (2) Watt% H., J.CHEM.EDUC.,39,477 (19621. (31 Oe Paz, M., J. CHEM. EOUC., 46,784 (1969).
Nishijime, Y.,andOster,G.,J.CHEM. EDUC.,38,11511961). Poison, A.,Nafure. 154.823 (13441. Gosting. L.Land Morris. M. S.. J. Amor Chem. Sot. 71.1998 (19491. Shoemaker. D. P., Garland, C. W. and Steinfeld. J. I., "Experirnenta in P h y a i d C h e r n i ~ t r y 3rd . ~ Ed., McCraw-Hill, Inc.. New Ymk,1974,~. 205. 18) . . Tanford. C.. "Phnbdl Chernistru of Msnomolecule8..l John Wiley & Sona. New (4) (5) (6) (7)
YO*. iai.p. fsa.
19) Hodee. J. E.. and Hofreiter. B. T.. in ''Methala in Carbohydrate ChemisVy,"Vol. I,
