M U T U A L DIFFUSION COEFFICIENTS FROM THERMAL DIFFUSION EXPERIMENTS MICHAEL J. STORY Department of Chemical Engineering, University of Adelaide, Adelaide, South Australia Mutual diffusicin coefficients in concentrated solutions can be obtained to within 1.5% of previously published data, by following the approach to the steady state with a thermal diffusion flow cell.
R E C E X T publication (DiCave and Emery, 1968) showed
A that a vertically mounted thermal diffusion cell with a
sintered-glass membrane gave nonisot'hermal mutual diffusion coefficients higher than the corresponding isothermal mutual diffusion coefficients by 20 to 40%. The present author feels t,hat this work was not sufficiently accurate to form generalized conclusions on this subject. This article establishes thermal diffusion experiments as a means of obtaining reasonably accurate values for mutual diffusion coefficients in conce~it~rated solut,ions. An accurate technique is required, with a c,orrespondingly rigorous phenomenological theory7 f i r calculation of t'he mut'ual diffusion coefficients. Suggestions why the results of Dicave and Emery are in error are: Thermal diffusion cells with sintered-glass membranes have met with little success to date. I t is doubtful whether meaningful results can be obtained with a vertically mountled cell, the volume of each compartment being decreased by sampling during the course of a run. Errors due t o convection currents through the sinteredglass membrane are a t a maximum with a vertically mounted cell, the approach to the steady st'ate (and hence the magnitude of the "apparent" diffusion coefficient) being markedly increased. Experimental Technique
The details of the a p i m a t u s have been given b y Butler and Turner (1966a). The binary liquid feed mixture is passed in laminar flow beti-ieen two flat horizontal plates, the upper plate being heated and the lower cooled. The lengt'h of the cell duct is 15.0 cm., the width 5.0 cm., and the separation of the plates about' 0.03 cm. 1 horizontal knife edge a t t,he exit end of the duct divides the flowing liquid layer into two product streams, which. are drawn off a t constant equal rates, providing a meaiic; of controlling the flow rate t,hrough the cell The two product streams are analyzed for their concentration difference by means; of a Rayleigh interferometer which is calibrated for each system. B y performing a number of run? at different flow rates through the cell, the approach to the steady-state separation can be studied for a given feed composition.
(1966b). The resulting equations are: (TI=----
8
1
Ad1
3 ATP (XI $142
where = Soret coefficient for component 1 (" K.-l), where
u1
u1 is positive for component 1 concentrating at the cold wall Ad1 = volume fraction difference for component 1 between bottom product stream and top product stream = volume fraction of component i in feed stream 6i AT = temperature difference between top plate and bottom plate, O C. F ( X ) = fractional attainment of maximum possible separation
Butler and Turner (1966b) found that
F ( X ) = 1 - 1.045 exp (- 13.66X), for X
> 0.035
(2)
where
X = DLb/2aQ D = mutual diffusion coefficient, sq. em. see? L = length of cell duct, 15.0 cm. b = width of cell duct, 5.0 em. 2a = depth of cell duct, about 0.03 em. Q = volumetric flow rate through cell, cc. see.-'
The calibration of the cell has been given in detail by Butler and Turner (1966a), for which they found
Lb
- = 2130 cm. 2a
Mutual diffusion coefficients may be determined b y the following method. For a fixed feed concentration &, three or four runs are performed at different rates of flow through the cell. Run a is carried out at a very slow flow rate such that the steady state is almost reached. Then by taking any run 6 a t a faster flow rate:
(")
u1 = 8
[ATF(X)l? 6161 B
Thus
Phenomenological Theory
A rigorous phenomenological theory for the calculation of the Soret coefficient has been given by Butler and Turner
if we assume [ F ( X ) l a = 1.000-Le., the attainment of steady state for run a. VOL.
a
NO.
4
NOVEMBER 1 9 6 9
777
1
Results
To determine t.he accuracy of the apparat'us for the det'ermination of Soret coefficients, t,he syst'em carbon tetrachloride f cyclohexane was studied (Turner et d.,1967). Satisfactory agreement. was established between our results and bhose of other workers. Further systems were t,hen st'udied: carbon tetrachloride f benzene and cyclohexane benzene (Story and Turner, 1969). To check the validity of the phenomenological theory for calculat,ioii of mut.ua1 diffusion coefficient,s, dat'a have been tmakenfrom the work on the systems carbon tet'rachloride f cyclohexane and carbon tetrachloride benzene. This technique for determination of mutual diffusion coefficients is of little use in dilute solut,ionand when t,he Soret coefficient approaches zero in the mid-concent,ration range. I n these regions the separat.ions are sufficiently small to cause any inaccuracies in the analysis of t.he separations to be magnified during the calculation of t'he mutual diffusion coefficients. For t'his reason t8hesystem cyclohexane benzene (in which u1 changes sign in the middle of the concentration range) has been excluded from t,he calculations of the mutual diffusion coefficient's. Figures 1 and 2 show plots of mutual diffusion coefficients, calculated from the result's of the t,hernial diffusion experiments, and t'hese are compared wit'h the mutual diffusion coefficients obtained by ot'her workers from isothermal experiments. The mean temperature was 25" C., with AT = 10" C. The standard deviat,ion in D from those of other sq. em. see,-' for the eight point's workers is k0.02 X that have been calculated and plotted. By comparing t'his value with the mean value of 1.49 X 10-5 sq. em. sec.-' for the eight points, it' can be seen that. t'he derived mutual diffusion coefficients are within 1.57, of those previously published.
+
0
.I
.3
.2
.4
.6
.5
.7
.8
.9
1.0
XI
Figure 1 . Mutual diffusion coefficients vs. mole fraction for carbon tetrachloride ( 1 ) cyclohexane (2) a t 25" C.
+
- Data of
Hammond and Stokes ( 1 9 5 6 )
0 This work
+
I .9 I .8 I .7
5
DXlO
I.6 1.5 1.4
0
.I
.2
.3
.4
.5
.G
x,
.7
.8
.9
1.0
Figure 2. Mutual diffusion coefficients vs. mole fraction for carbon tetrachloride ( 1 ) benzene (2) a t 25" C.
+
Discussion
- Data of Caldwell and Babb (1956) 0 This work Now Equation 3 gives
21300
x=-
Q
(5)
and when combined with Equation 2, yields
Thus by obtaining [ F ( X ) ] p from Equation 4, D can be determined by Equation 6. The value of [ F ( X ) ] , must now be checked by use of Equations 5 and 2, using the calculated value of D . This corrected value of [ F ( X ) l Cmust l now be used to correct the value of (A&/&&), used in Equation 4. By this means a corrected value for D may be obtained b y re-use of Equation 6. Runs at other flow rates should give the same value of D for the given feed concentration.
778
l & E C
+
FUNDAMENTALS
Mutual diffusion coefficients can be obtained from nonisothermal experiments if the apparatus and experimental technique are sound, and the phenomenological theory is rigorous and correct. This technique yields accurate values for the Soret coefficient. literature Cited
Butler, B. D., Turner, J. C . R., Trans. Faraday SOC.62, 3114 (1966a). 62, 3121 Butler, B. D., Turner, J. C. R., Trans. Faraday SOC. (1966b). Caldwell, C. S., Babb, .4.L., J . Phys. Chem. 60, 51 (1956). DiCave, S., Emery A . H., IND.ESG.CHEM.FUNDAYEXTALS 7, 95 (1968). Hammond, B. It., Stokes, It. H., Trans. Faraday SOC. 62, 781 (1956). Story, 11. J., Turner, J. C. I{., Trans. Faraday SOC.66, 349 (1969). Turner, J. C. R., Butler, B. D., Story, AI. J., Trans. Faraday SOC.63, 1906 (1967). RECEIVED for review December 9, 1968 J u n e 2, 1969 ACCEPTED
