EXPERIMENTAL TECHNIQUES
Diffusion in Liquid Systems. I. A Simple and Fast Method of Measuring Diffusion Constants A. C. Ouano IBlV Research Laboratory, San Jose, Calif. 95114
A simple and fast method of measuring diffusion constants based on the hydrodynamic continuity equation for laminar flow in small diameter tubing rather than the classical Fick's second law is prcposed. The apparatus for measuring the diffusion constants consists mainly of a constant volume pump, a sample injection port, a 168 m long X '/18 in. 0.d. X 0.020 i.d. stainless steel tube coiled into a 1.5-ft diameter coil, uv and differential refractometer detectors, a volume counter, and a recorder. The diffusion constant D was calculated from the variance of the residence time distribution of the solute in the tubing. The values of D obtained from the intercept (Q/DL = 0)of a plot of D vs. Q/DL (Q being the flow rate and L the tubing length) gave limiting values of D which agree reasonably well with diffusion constants measured by the free diffusion cells method reported in literature.
I t is readily apparent from a literature survey of physicochemical constants of solutions that diffusion data even for binary systems are relatively rare and difficult to obtain compared to other solution properties. Even though diffusion const'ants have been measured as early as 1850 by Graham (1850), it is only recently that satisfactorily accurate data have been obtained. Most of these reliable data have been obtained through the use of highly sophisticated but painfully slox experimental techniques, usually involving the use of free or porous diaphragm diffusion cells. One of the difficulties encountered in the free diffusion cell t,echnique is in establishing an initially sharp boundary bet,ween the pure solvent and t,he solution. Maintaining an undisturbed medium (minimizing vibration) throughout the experiment has also been a major problem. These problems are associated with diffusion in a static medium, which could be minimized by conducting diffusion esperiments in a flowing medium, such as diffusion in Poiseuille flow or the porous diaphragm method. Alt,hough the porous diaphragm diffusion cell method minimizes most of the problems associated with the free diffusion cell technique, it has some drawbacks of its own. Since the technique requires a porous membrane, with some resistance to the transport of the solute, the technique does not yield diffusioii coefficients directly and needs to be calibrated with a knowii standard. Other problems related to the use of the porous diaphragm such as adsorption of solute on the surface of t'he porous membrane, bulk streaming, and semipermeability due t o improper selection of diaphragm porosity can also occur. The porous diaphragm diffusion technique does not lend itself easily to a continuous method of observation and requires a relatively long period of time (t\vo or more days) to complete a diffusion measurement. The flowing medium diffusion cell method described in this paper mininiizes many of the problems associated with both the free and the porous diaphragm diffusion cell methods. In addition to being a relatively fast method (a few hours t o 268
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
complete the measurement), it also requires only microgram sized samples instead of the gram amomits required by the other tlvo methods. Additionally, the apparatus and the computational techniques required to obtain diffusion constants are relatively simple and straightforward. It is felt that these advantages make this new method an improvement to the earlier techniques of measuring diffusion coefficients in liquids. Theory
Diffusivity in liquids is usually measured in terms of the diffusion coefficient, D defined by Fick's classical equation for diffusion through a single plane
F
=
-D-bC
bX
in which bcl'bx is the concentration gradieut and P' is the mass flus. The difficulty of measuring I;, however, limits the usefulness of eq 1. Thus, D is usually obtained using Fick's second law of diffusion
For simple boundary and initial co~iditions,i.e., linear diffusion from an originally sharp boundary between a pure solvent and a solution in a vertical column of effectively infinite length, eq 2 can be integrated to yield
(3) in which Co and C are the iiiitial concentrations of the solution and the concentration at level I of the diffusion cell, respectively. Therefore, by measuriiip the colicelitration at a different level x a t any fised time t , the diffusivity D can be calculated from eq 3. D can also be calculated from concentration gradient measurements at any fised time t after the start of the experiment from the equation.
dC -
ax
--co
W,Dt
exp- [x2/4Dt]
(4)
Since the relationship between &/ax and x is Gaussian, D can be calculated from t,he variance of the dcldx us. x plot. The experimental measurement of diffusion constants in liquids is normally carried out using free diffusion type cells of various designs and sophistication. An extensive description of the different free diffusion t,ype cells used is given by Geddes and Pontius (1949). I n the free diffusion cell method, the solution and pure solvent are loaded into the lower and upper compartments, respectively, of the cell, which are separated by a thin and retract,able partit,ion. At the start of the esperiment,, t'he partit,ion is rehacted, allowing the solute t o diffuse into t h e pure solvent compartment'. After a sufficient t'ime has elapsed, t,he concentration along the length of the diffusion cell is measured by either chemical, optical, or electrical methods. Under ideal conditions, t,he free diffusion cell method should yield results of high accuracy and precision. However, due t o problems associated with t'he retractable partition (leakage, grease deposit,.. on the cell window, disturbance in the solvent and solutiou interface) and vibration, accurate diffusion data from free diffusion cell measurements are very difficult t o obtain. Quo.ting Harned (1947), "There are few domains of physical science in which so much experimental effort over nearly a century has yielded so litt,le accurate data as the field of diffusion in liquid systems." The problems mentioned in the foregoing paragraph, which are associated with the boundary and initial couditions necessary for the solution of Fick's second law, could be avoided by using the hydrodynamic equation of continuity in cylindrical coordinates for fluids iii laminar parabolic flow (Poiseuille flow)
in lieu of eq 2. Although the solution to eq 5 may a t first glance look formidable, it is, in fact, analogous to the solution of Fick's second law equation when certain Conditions are met. The advantage of using the solution to eq 5 is the ease in which the boundary and initial conditions necessary for the calculation of the diffusion constant D can be met esperimentally with little, if any, loss in the accuracy of the results obtained. Taylor (1953), in his pioneering work 011 the dispersion of solutes in liquid systems undergoing Poiseuille flow through small diameter tubing, showed that the dispersion equation ( 5 ) can be simplified to
if the condition
R2
L
D
G
- 'DI,,is simply the inverse of the asial Peclet number (Pe = 7-I, !D 0. I n order for &,'DL to approach zero, the ratio of Q to 1, must become extremely sinall in value since TI -10-j cm2 ' s w in liquids. Xs in t'he det'erinination of molecular w i g h t Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
269
-ELECTRICAL LiauiD LINES
*-ACETONE .-BENZENE A-ETHANOL *-ETHYLBENZENE 0-DI ETHYLBENZENE
LINES
I
0.20
,
I
0.60
l
,
1.00
,
l
1.40
,
/
I
1.60
1
-
1
2.00
QiOL x 10' iOlMENSlONLESSi A -SOLVENT RESERVOIR F -DIFFUSION CELL 1168 METERS B -SOLVENT DEGASER O F COILEDSTAINLESSSTEEL C -CONSTANT VOLUME PUMP COIL 1% FT. D I A . ) D REFERENCE FLOW CONTROL G -CONSTANTTEMP. BATH '-VALVE H -UV DETECTOR D SAMPLE FLOW CONTROL I -DIFFUSION REFRACTOMETER '-VALVE DETECTOR E -SAMPLE INJECTION VALVE J -SYPHON K -RECORDER
Figure 3. A plot of the inverse of the axial Peclet number vs. the diffusivity D. The intercept represents the limiting diffusion constant of the solute
Figure 1 . Schematic diagram of the flowing medium diffusion cell apparatus .-ACETONE A T 19'C .-BENZENE A T 20°C A-ETHANOL A T 15°C
>
1.00
c
z Y
n
4'
CONC. I V O L . PERCENT)
SAMPLE INJECTION
U
I
Figure 4. A plot of diffusion constants vs. concentration. These values were obtained from Timmermons ( 1 959) I
I - V .
I N ' 0
1
I
1
I '
'
"
2 3 4 5 6 7 8 9 RETENTION VOLUME COUNTS 15 ZOMLICOUNT)
I
10
Figure 2. Typical retention volume distribution of solute in the diffusion cell which illustrates the graphical method of measuring W and V 1
averages from the colligative properties of dilute solutions (Le., osmotic pressure extrapolated to zero concentration), D can be obtained from a series of measurements of D conducted for different values of &. Limiting D is obtained by plotting D vs. &,'DL and extrapolating &/DL t o zero. Thus, in this manner, diffusion constants can be obtained within practical limits of Q and L values. Experimental Procedure
The flow diagram and description of the equipment used in this work are shown in Figure 1. The diffusion cell (F) is made of 168 m of continuous length 0.020-in. i.d. X VI6-in.304 stainless steel tubing coiled into a 1.5-ft diameter coil. The tubing was made by the cold drawing process and manufactured by the Superior Tube Co. of Korristown, Pa. The tubing is finished t o 100 pin. smoothness. The sample injection valve used was the Rater's Associates six-port valve which allows a pulse type injection of solution into the diffusion cell (Ouano and Biesenberger, 1970). The retention volume distribution of the solute was monitored by a two detector system (uv a t 254 nm and differential refractometer) for the concentration measurement, and a syphon collector as a retention volume counter. The experimental technique required to operate the equipment and obtain diffusion constants D includes only the ability to prepare solutions of known concentration and the ability t o inject the sample solution into the solvent stream by opening and closing an injection valve. In this work, diffusion constants of five different solutes (benzene, acetone, ethylbenzene, diethylbenzene, and ethanol) 270 Ind.
Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
in chloroform were measured a t room t'emperature (23 i l°C). About 2 ml of 0.1% solution was loaded into the sample injection loop and at a n appropriate time (when the syphon counter dumps,, which is recorded by an event marker in the recorder), a pulse of solution containing 125 pg of solute was injected into t'he solvent stream. Since the injection time, ti, was very short relat,ive to the average residence time t, (ti/ t.
