This approach was tested by obtaining excellent reproductions of the analytical solutions obtained by Fujita (9, 70) for two specific diffusivity-concentration relationships. This close agreement with Fujita’s exact solutions points out the accuracy of the method in solving nonlinear diffusion problems. The semi-infinite diffusion problem becomes considerably more complex if there is a significant phase volume change, since a moving boundary problem (3, 6, 75) must then be considered.
T = temperature t = time u x
= = cy = 7 = = p = w1 =
x component of mass average velocity space coordinate in direction of transfer and flow
k/PC, x / 2 t”2 p D dwl,’dq
total mass density mass fraction of component 1
SUBSCRIPTS Conclusions
u
= property evaluated a t starting point of asymptotic solu-
T h e developed technique offers a systematic approach to the solution of a \vide range of diffusion problems in infinite media. By the use of one undetermined parameter, solutions including any functional dependence of the diffusivity on concentration as well as any effects of volume change during mixing are readily obtained. This method is particularly attractive, since it has been established that the trial and error selection of initial conditions converges to a correct value after only a few trials. This technique can be extended to the solution of heat conduction problems and to transport problems in semiinfinite media.
e 0
= property evaluated at equilibrium interface = property evaluated a t - m boundary
The authors thank R. H. Foy and J. A. Moffitt, Computations Research Laboratory, T h e Dow Chemical Co., for assistance in computer programing. Nomenclature
A,
= k/& C, = specific heat capacity a t constant pressure = binary diffusion coefficient = constant of integration = thermal conductivity
K k
m
= property evaluated a t
+
m
boundary
Literature Cited
(1) Boltzmann, L., Ann. Phjsik Chem. Wzed. 53, 959 (1894). (2) Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed., Oxford UniLersity Press, London, 1959. (3) Chiang, S. H., Toor, H. L., A.I.Ch.E. J . 10, 398 (1964). (4) Clarke. D. M., J . Chem. Phys. 27, 29 (1957). (5). Crank, J., “The Mathematics of Diffusion,” Oxford University Press, London, 1956. (6) Danckwerts, P. V., Trans. Faraday Soc. 46, 701 (1950). (7) Duda, J. L., Vrentas, J. s., IND.ENG.CHEU.FUNDAMEKTALS 4, 301, (1965). (8) Dullien, F. A. L., Shemilt, L. LV., Can. J . Chem. Eng. 39, 242 11961) \-. - - I -
Acknowledgment
D
tion
(9) Fujita, H., TextiIe Res. J . 22, 757 (1952). (IO) Ibid., p. 823. (11) Gillis, J., Kedem, O., J . PolymerSci. 11, 545 (1953). (12) Gosting, L. J., Fujita, H., J . ‘4m.Chem. SOC.79, 1359 (1957). (13) Hammond, B. R., Stokes, R. H., Trans. Faraday Soc. 49, 890
(1953). (14) Hiidebrand, F. B., “Introduction to Numerical Analysis,” McGraw-Hill, New York, 1956. (15) Knuth, E. L., Phys. Fluids 2, 84 (1959). (16) Philip, J. R.,Trans. Faraday SOC.51, 885 (1955). (17) Stok;!;; R. H., Ibid., 48, 887 (1952). (18) \Vagner, C., J . Metal Trans. 4, 91 (1952). ~
RECEIVED for review November 12, 1964 ACCEPTED July 29, 1965
DIFFUSION COEFFICIENTS FOR FOUR HOMOMORPHIC BINARY LIQUID SYSTEMS R. L. ROBINSON, JR., W. C. E D M I S T E R , AND F. A.
L. D U L L I E N ’
Oklahoma State University, Stillwater, Okla.
N EXPERIMESTAL
A tion
investigation was made to obtain informa-
on two separate facets of the diffusion process:
the effects of polar groups on diffusion rates, and the temperature dependence of the diffusion coefficient. T h e results of this research are presented here. To study the role of polar groups without the complicating influence of changing molecular structure, the “method of homomorphic compounds” was employed for the first time in diffusional work. (Molecules of essentially identical structural arrangement and mass but differing in polarity are sometimes referred to as “homomorphs.” For example, replacing a methyl group in a hydrocarbon by a hydroxyl group results 1
74
Present address, Esso Production Research Co., Houston, Tex. I&EC FUNDAMENTALS
in the “polar homomorph” of the hydrocarbon.) By use of the homomorphic technique, the diffusion coefficients were measured for the following systems :
I. Nonpolar A-nonpolar B (n-octane-methylcyclohexane) 11. Polar homomorph of A-nonpolar B (l-heptanolme thylcyclohexane) 111. Nonpolar A-polar homomorph of B (n-octane-cyclohexanone) 1x7. Polar homomorph of A-polar homomorph of B (1hep tanol-cyclohexanone) These compounds were selected as the result of a search for compounds which would give the above types of interactions and would exist as liquids over most of the temperature range 0’ to 100’ C. T h e above requirements limited the
The diaphragm-cell technique was used to measure mutual diffusion coefficients for the homomorphic systems n-octane-methylcyclohexane, n-octane-cyclohexanone, and 1 -heptanol-methylcyclohexane at 25 C., and for 1 -tieptanol-cyclohexanone at l o " , 25', 55", and 90" C. The complete concentration range was covered for each of these liquid-phase systems. Viscosities and densities were also measured. The four binaries studied have almost identical structural arrangements of the molecules, but the intermolecular forces vary from nonpolar-nonpolar through nonpolar-polar to the polar-polar type. The 80" C. temperature range studied for the 1 -heptanol-cyclohexanone system i s considerably larger than that covered b y previous comparable data. The temperature dependence of the diffusion coefficient i s equally well described b y s8everalmodels including the Arrhenius-type relation.
selection to systems for u hich no thermodynamic activity data, useful in theoretical evaluation of diffusion theories, are a t present available. As the second part of the study, the l-heptanol-cyclohexanone system was studied a t four temperatures in the range 10' to 90' C. These diffusion data span a temperature range double that of any comparable previous data-Le., data covering the complete concentration range-and offer a more demanding test of existing models for the temperature dependence of the diffusion coefficient. Experimental
Chemicals. T h e n-octane and methylcyclohexane were supplied by the Phillips Petroleum Co. and were specified to have a minimum purity of 99 mole yo. T h e cyclohexanone and 1-heptanol were Eastman grade from Eastman Organic Chemicals. .A11 chemicals except heptanol were used as received. T h e heptanol was distilled on a 1-inch-diameter, 30-plate Oldershaw column a t a 10 to 1 reflux ratio. and the first 25 volume yc overhead and final reboiler contents were discarded. Chromatographic analyses showed all four chemicals to have purities in excess of 99.5%. Densities and refractive indices of the chemicals are shown in Table I. Table 1.
Physical Properties of Chemicals
Refractice lndex, 20' C. Exjtl. Lit. 1.3975j 1.59743( 7 )
Chemical n-Octane Methylcyclohexane 1.4231 1.4232(I ) Cyclohexanone 1.45055 1.4505(72) 1-Heptanol Batch 1 1.42425 /I .42351(23) 1.42425 \1.4249(7) Batch 2 a Interpolated calue by authors.
Density, G./Cc., 25' C. Exptl. Lit. 0.70050 0.69849( I ) 0.76524 0.76506( 7) 0.94240 0.94207a(22) 0.81874 0.8188(21) 0.81866
T h e potassium chloride used in the calibration of the diffusion cells was Baker analyzed reagent (J. T. Baker Chemical Co.) and had a stated purity of 99.9 weight Water for these runs was deionized and distilled. Apparatus. A battery of six diaphragm cells was employed in the diffusion (experiments (76). These cells ivere of the Stokes (79) type, \vith diaphragms of F (fine) grade fritted glass. All connections between the compartments a n d surroundings were made through four capillary glass legs, two to each compartment. T h e legs were fitted with valves fashioned from polyethylene tubing (heated and flattened) a n d commercial tubing clips. This arrangement eliminated any glass or other joints in direct contact with the compartments. At the same time, evaporation was negligible through the open capillary adjoining the top of the upper compartment. A compact stirring and support apparatus housed the cells. This apparatus was very similar to that described by Dullien and Shemilt (6). Diffusion runs took place in a n oil bath in which the oil temperature was controlled to ? ~ 0 . 0 5C.~ Viscosities were measlured in standard Ostwald viscometers, and densities were determined in (modified) 20-cc. Sprengel
x.
pycnometers. Calibration of viscometers and pycnometers was with a water reference, and measurements \Yere made in a bath controlled to i0.03O C . Procedure. T h e diffusion cells were calibrated using 0.1 S potassium chloride diffusing into water. T h e cell constants, ?, Were calculated using the data and procedure given by Stokes (20). Kecalibrations were made during the study to allow for wear of the stirrers against the diaphragm. I n preparation for all runs, the lower compartment was filled until the diaphragm was covered Ivith solution. This lower solution was boiled under slight aspirator vacuum to degas the solution and displace any trapped air from the diaphragm. The cell \vas then placed in the bath to attain thermal equilibrium. Next, the upper compartment was rinsed and filled ivith its degassed, thermally equilibrated solution. A period of preliminary diffusion was used to establish a concentration gradient in the diaphragm; duration of preliminary diffusion was estimated as suggested by Gordon ( 7 7). The top cell was then emptied, rinsed, and refilled with fresh solution, and actual diffusion \vas begun. Actual diffusion times varied from 4 to 21 days. Maximum initial concentration differences, consistent with the desired average concentrations, were used in all experiments. Thus, a t least one inirial solution was alivays a pure component. Sampling \vas accomplished by applying compressed air to one leg of a compartment and collecting samples from the other leg. Analyses. Samples from the organic runs xvere analyzed in duplicate pycnometrically. Densities were related to concentrations through experimentally determined density-concentration relations. KC1 samples were analyzed in triplicate by determining the residue \veights from evaporation of 10-cc. aliquots of sample. All lveighings rvere done on a Mettler Gram-atic semimicrobalance, and the necessary buoyancy corrections Lvere applied. T h e average absolute deviation from the mean for 127 pairs of organic densities was 1.5 x 10-j gram per cc., and for 134 KC1 residues from their 46 respective means was 8 x 10-5 gram. Calculation of Diffusion Coefficients. -4s a part of this study a rigorous set of equations \vas derived to allow calculation of the diffusion coefficients from diaphragm cell data regardless of volume changes during diffusion (77). For the present systems the volume changes are negligible and the equations become identical to those of Gordon ( 7 7). Gordon's equations (his Equations 11 through 17) were programmed for a digital computer and used to compute the differential diffusion coefficients from the experimental data. T h e resulting differential diffusion coefficients are s)-mmetrical lvith respect to the t\vo components and are identical to the so-called Chapman-Cowling diffusion coefficient. L-se of the computer program required analytical relations for the diffusion coefficients as functions of concentration. Second- and thirddegree polynomials were used to represent the diffusivity us. concentration relation. T h e n-octane-cyclohexanone system was fitted in sections to three separate 2nd degree polynomials to obtain satisfactory representation ; other systems were fitted by single polynomials. Since the integral and differential diffusion curves were very similar, adequacy of the analytiVOL.
5
NO. 1
FEBRUARY 1966
75
cal expressions was judgcd by their ability ~ C Jfit the integral diffusivity-mean concentration curves. T h e integral diffusivity curve was th n used as a first approximation to the differential diffusivi:. cur\ e to begin the iterative solution to Gordon's equations. Estimates of Errors. ri detailed statistical analysis of errors is available (Id). T h e experimental scatter in the cell constan q. p , from the calibrations \vas approximately 10.1%. T h e orgnnic diffusion data scatter about smooth curves through the data by an average of about & I % . An analysis showed that analytical errors should account for standard deviations of 0.4 and 0.2% in the cell constants and organic diffusivities, respectively. Thus, the excellent agreement of the cell constants (two to four replicate determinations on each cell) may be in part fortuitous. Conversely, some factor or factors other than analytical errors contributed to the scatter in the organic data.
Fvhere 0 is the diffusion time, the single and double primes refer to the respective cell compartments, and subscripts o and f refer to initial and final conditions, respectively. is the average of the four concentrations appearing in Equation 1. Table I11 presents bnoothed values of the differential diffusion coefficient, D, a function of mole fraction, a , for the systems studied. These D values n e r e taken from the solutions to Gordon'. equations, as discussed above. The visccdties, p (cp.), and densities. C (grams per cc.), are given as functions of mass fraction, w , in Tables IV and V, respectively. Viscosity and density measurements were not made on the I-heptanol-cyclohexanone system a t 10' C. because of limitations of the existing apparatus.
cl,
Table 111. Results
Values of the experimental integral diffusion coefficients,
b (sq. cm. per second), and their respective mean concentrations, C i (grams per cc.), are given in Table 11. T h e were calculated from the formula
Table II.
bx
1.738 1.853 1.976 2.041 1.118 2.190 2.231 2.255 2,290 2,249 2.278
0,0588 0.1160 0.2006 0,2728 0,3590 0,4543 0.5428 0,5567 0,6070 0.6088 0,6957
( A ) n-Octane(B) Cyclohexanone, 25 C.
0.708 0.706 0,674 0.661 0.681 0.739 0.759 0.795 0.929 1.139 1.129 1.376 1.686 1.711 1.843
0,0548 0,0552 0.1672 0,2202 0,2683 0,3508 0,3528 0,4341 0.5131 0,5832 0,5845 0,6279 0,6644 0.6691 0,6703
( A ) Methylcyclohexane( B ) 1-Heptanol, 25' C.
0.505 0.528 0,560 0.581 0.601 0.609 0.610 0.616
5 values
Diffusion Data
706 ?A (.A) n-Octane-
(B) Methylcyclohexane, 25 ' C.
76
Smoothed Diffusion Coefficients
Mole Frac-
0.0699 0.1373 0.2550 0.3837 0,4968 0.5931 0,6581 0.7128
I&EC FUNDAMENTALS
bx
106
CA
( A ) 1-Heptanol(B) Cyclohexanone, 10' C. 0.352 0.1025 0 321 0.1803 0,255 0.4077 0,233 0,5431 0.195 0.7226 0.202 0.7306 ( A ) 1-Heptanol(B) Cyclohexanone, 25 C . 0,542 0,0564 0.515 0.1064 0.526 0.1082 0.491 0.1981 0,458 0,2762 0.432 0,3507 0.423 0,3779 0.416 0.4085 0,406 0.4143 0.406 0,4635 0,377 0.5735 0.368 0,6494 0,347 0,7227 0.350 0.7227 ('\) 1-Heptanol-
(B) Cyclohexanone, 55 C. 1.020 0,0702 0.961 0.2145 0,926 0 3129 0 838 0.5130 0.812 0,6141 (.\) 1-Heptanol( B ) Cyclohexanone, 90 C. 1 906 0.0611
tion ... .
StraightChain Component
Smoothed Differential Di'juusion Coe8cient, Sq. Cm,/Sec. X 706 I5 II III IVa IVb IVC 0.0 1.611 0,741 0.618 0.576 0.394 1.051 0.1 1 800 0.680 0.613 0.519 0.351 1.007 0.2 1,940 0.655 0,609 0.475 0.310 0.964 0.3 2.042 0.653 0,604 0.438 0.281 0.929 0.4 2.113 0,686 0.598 0.413 0.255 0.893 0.5 2.166 0.745 0.588 0.395 0.237 0.864 0.6 2.202 0,841 0.575 0.380 0,220 0.837 0.7 2.233 0.980 0,557 0,366 0.209 0.812 0.8 2.258 1.190 0,534 0.355 0,202 0.787 0.9 2.280 1.590 0.505 0.345 0.197 0.765 1.0 2.302 2.200 0,470 0.335 0.194 0.744 a I n-Octane-MCH,' 25' C. II. n-Octane-cyclohexanone, III. I-Heptanol-MCH, 25' C. IV. I-Heptanol-MCH, b I O 0 : c 55": d 90" C. Table IV. w
Viscosity Data
P
(.A) n-Octane(B) Methylcyclohexane, 25" C . 0.0 0,680 0,655 0,0988 0.633 0.1986 0.602 0,3586 0.577 0.5052 0.559 0,6505 0.535 0,8046 0.9035 0,526 1. o 0.517
( A ) Meihylcyclohexane(B) 1-Heptanol, 25 C. n o 5.868 010955 4,608 0.1958 3.546 0,3498 2,371 0,4962 1.643 0,6505 1,192 0.8028 0.890 0.9045 0,757 1. o 0.680
w
P
(.A) n-Octane-
(B) Cyclohexanone, 25' C. 0.0 2,000 0.0956 1,590 0.1955 1.310 0.3515 1.023 0,5061 0.829 0,693 0.6593 0,8072 0,602 0.9071 0.550 1. o 0.517 ( A ) I-Heptanol(B) Cyclohexanone, 25" C. 0.0
0.1178 0.2282 0.3397 0,4441 0.5415 0,6376 0.7315 0,8264 0,9098 1. o
(.4)1-Heptanol(B) Cyclohexanone, 55 C. 1,149 0.0 1.119 0.1122 1.164 0.3013 1.280 0,4993 1.533 0.6991 1.982 0.9011 2,350 1. o
IVd 1.918 1.859 1.812 1.772 1.740 1.714 1.693 1,676 1,663 1.653 1,646 25' C. a 25",
2.000 1.965 2,034 2.160 2.351 2.584 2.914 3.335 3,921 4.663 5,868
( A ) 1-Heptanol(B) Cyclohexanone, 90 C. 0.0 0.670 0.1122 0.658 0.3013 0.659 0,4993 0.689 0.6991 0.770 0.9011 0.891 1. o 0,982
Discussion
Effect of T e m p e r a t u r e on Diffusion Rate. A number of different models for the effect of temperature on the diffusion coefficient have been proposed and used (4, 9, 70, 74, 75, 78, 25). T h e data from this study on the system l-heptanolcyclohcxanone from 10" to 90' C. were used to test several such models. Results for four of the more accurate models are sho\vn in Table \7 along \vith references Lvhere the models are discussed. T h e comparisons in Table \'I, a t mole fractions, s, of 0> 0.5, and 1.0, show the models for temperature dependence to be of comparable accuracies. Average errors are only slightly larger than the average scatter of the data a t
Table
V. Density Data
C
w
C
w
(.I) n-Octane-
(.\) n-Octane-
( B ) Meth)Ic)-cloliexane, 25 C. 0.0 0.76524 0.11358 0,75716 0,22473 0.74940 0.43611 0.73528 0.63572 0,72246 0.91621 0.70534 1. o 0.70050
( B ) Cyclohexanone, 25 C. 0.0 0.94240 0.15174 0.89498 0.30124 0,85296 0.50153 0.80242 0.69907 0.75824 0.84582 0,72868 1 .0 0.70050
(-4) ~leth\-lc)-clohesiIne(B) 1-IHeptanol, 25 C. 0.0 0.81874 0.09910 0.81303 0,29960 0.80150 0.50023 0.79026 0.6241 9 0.-8353 0 .77964 0.69939 1. o 0.76524
(.A) 1-Heptanol(B) Cyclohexanone, 25" C.
0.0 0.22821 0 33966 0.44411 0.54153 0 63764 0.73154 0,82637
0,94240 0.91028 0.89554 0.88226 0 87038 0.85895 0.84814 0.83756 0.82846 0.81874
0.90976
1 .0
(:I) 1-Heptanol(B) Cyclohexanone, 55 C. 0.0 0.91 594 0.11223 0.90050 0.30127 0.87510 0.50138 0,85079 0.69907 0.82854 0.90112 0,80736 1. o 0,79752 Table VI. x,
n-CiOH 0.0
0.5
1 .0
(.I) 1-Heptanol(B) Cyclohexane, 90' C. 0.0 0.88390 0.11223 0,86839 0.30127 0,84440 0.50138 0.82107 0.69907 0.79990 0.90112 0.77978 1. o 0,77044
0 2
2 75
1
1
3 00
3 25
3 53
lo3 Diffusivity-temperature relation for 1 -heptanolRECIPROCAL TEMPERATURE. I/T,
OK-'
x
Figure 1. cyclohexanone system
x = mole fraction 1-heptanol
Comparison of Models for Temperature Dependence of Diffusivities Absolute Error iii Piedicted D" T, C. -xGdul I .tlodel I I .\lode1 III .\lode1 I V 10 0.003 0,013 0,002 0,004 25 0.007 0.008 0.006 0.004 55 0.009 0.010 0,007 0.015 90 0.003 0.005 0.000 0.006 10 0.009 0,021 0 ,008 0,008 25 0.012 0,001 0 ,011 0.000 55 0.028 0,034 0 ,029 0.014 90 0.050 0.013 0 ,049 0,006 10 0.005 0.039 0.006 0.032 0.009 0.013 25 0.011 0.010
55 90
Av. Mas
each isotherm about a smooth curve (about 0.007). O n the basis of these data, no decision as to a "best" model is warranted. At concentrations other than infinite dilution, most theories, such as the absolute rate (70) and statistical mechanical (5) theories, embody an effect of the thermodynamic factor, d In a l d In c, on the diffusion coefficient. I n the comparisons of Table VI, the data at A = 0.5 were not corrected for any effect of the thermodynamic factor. This could possibly explain the increased errors for Models I and I11 a t this intermediate concentration. Model I is the "Arrhenius-type" variation which is most often applied to both diffusion and viscosity. Figures 1 and 2
0.001 0,011 0.012 0.050
0.045 0.015 0,018 0,045
0.002 0.013 0,012 0,049
0.035 0.012 0,012 0,035
Model I In D cs. l / T (10) Model I1 D L ~ S .T / p (25 Model I11 In D L'S. In Model IV D CS. 1 , ' ~ ~ l ~ ~ p( 1 ~ 5~)' " Absolute dzgErencr beticeen experiinental D a r ~ d D calculated from
(Ql
leasl-mean-square,fit of dnta.
2 75
1
3 50
3 25
3 00
R E C I P R 3 C 4 L TEhlPERATUYE, I / T ,
OK:'
I:
IO'
Figure 2. Viscosity-temperature relation for 1 "heptanolcyclohexanone system x = mole fraction 1-heptanol
VOL. 5
NO.
1
FEBRUARY 1966
77
illustrate this model for the data of this study, with straight lines draivn through the data. For the viscosity, the model gives a n excellent representation. Hoivever, the diffusion data, \vhile very linear for the pure cyclohexanone limit, have a slight curvature in the pure 1-heptanol region and more pronounced curvature a t .Y = 0.5. Influence of the thermodynamic factor may be important a t Y = 0.5 but not for the pure 1-heptanol region. Ewe11 a n d Eyring (8) attributed curvature of similar plots of viscosity data in hydrogen-bonded s p t e m s to a breakdown of the hydrogen bonds as temperature increased, ivith subsequent increase in the ease of formation of the "activated complexes" of Eyring's theory (70). Such arguments are in harmony with the results for the present diffusion data, since the alcohol should be hydrogen-bonded. This effect, nevertheless, does not appear in the viscosity data. Defining experimental "activation energies" for diffusion and viscosity by the follou.ing ..irrhenius-type relations
D
= A~ED/RT
(2)
= AI~-E$'/RT
values of EDand E p were evaluated from plots similar to Figures 1 and 2. Results are shoivn in Figure 3. Caldwell a n d Babb (14) found ED Ep to be constant with composition for six ideal systems; other investigators (2, 9) have presented results
-
3 0
0
1
0.1
similar to those in Figure 3 for nonideal systems. Sharp variations or "bumps" in the activation energy curves have been used to infer presence of complex formation (9). O n this basis, the data in Figure 3 would imply an absence of any strong complexes in the 1-heptanol-cyclohexanone. Conversely, from the density data on this system, the maximum excess volume, V', for the system is 0.175, 0.25: and 0.32 cc. per gram mole a t 25', 5 5 O , and 90' C., respectively. Such an increase in V E \vith temperature is sometimes attributed to a decrease in an exothermic complex as temperature rises ( 3 ) . However, other more suitable means must be employed to determine definitely \vhether complex formation is prominent in the system. From statistical mechanical considerations (5) the Dp product for ideal systems has been shoivn to be linear in composition. Since systems tend to approach ideality more closely as temperature increases, the data of this study should shoiv increasing linearity in the Dp product as temperature increases. Figure 4 shoivs this to be the case. T h e factor D p / T is sometimes assumed to be independent of temperature (23): and. indeed, data taken over moderate temperature ranges have been used to substantiate this contention (4: 9, 73). However, for the larger temperature range of this study, the follou.ing tabular data indicate that the group is not, in general, constant.
I
0.2
03
I
0.4
05
M O L E FRACTION
Figure 3.
70
I&EC FUNDAMENTALS
I
08
09
10
Activation energies for 1 -heptanokyclohexanone system
MOLE FRACTION
Figure 4. system
I
06 07 -HEPTANOL
Diffusivity-viscosity
I-HEPTANOL
product for
1 -heptanol-cyclohexanone
Mole Fraction 1-Hebtanol
Temperature, C. 25 55 DI.L/T
10
X
6.82 3.88
0.0 1. o
5.33 3.68
6.62 3.89
90
4.45 3.54
Comparison of Diffusion Rates in Homomorphic Systems. A complete quantitative analysis of these data is not possible a t present, since current theories of diffusion cannot be tested for other than regular solutions and most of these theories require thermodynamic activity data cyhich have not been measured for the systems studied here. Hoirever, in the regions of infinite dilution. where the effect of the thermod l namic factor vanishes, some interesting observations are possible. According to some theories of diffusion (5), for separate binary mixtures in which A and B are a t infinite dilution in a solvent C, the mutual diffusion coefficients are related by the expression
(3)
approached and each solute molecule is surrounded by solvent, essentially insulated from other solute molecules. For our homomorphic systems in this region, QAC and pBC might be expected to become essentially equal and thus the diffusion coefficients would be equal. Table VI1 presents the diffusion coefficients a t infinite dilution in a form to show the effects of changing homomorphic solutes in a given solvent and changing homomorphic solvents for a given solute. As seen in Table T’IIa, the reasoning of the previous paragraph applies qualitatively for each of the four cases except \\.hen meth) Icyclohexane (MCH) is the solvent. T h e possibility exists that a t the lonest concentrations studied, the alcohol is still highly associated in the lMCH a n d extrapolation to infinite dilution is in error. Unfortunately. the diaphragm-cell method is inapplicable a t very lei\ concentrations because of analytical errors. For changes of solvent from nonpolar to polar (Table LTIb) the change in diffusion coefficient is large. This is to be expected, Fince the changes are in a region of high concentration where the polar interactions are important. Here again the I-heptanol-MCH system does not fit into the pattern, perhaps for the reasons given above. Further qualitative discussion of the data. including the intermediate concentration regions, is given elsev here (76).
-
where dz, is the “intermolecular friction coefficient for i j interactions.” I n general, $ t j reflects interactions of the types i - i, i - j , and j - j , b u t when i is infinitely dilute the i - i type interactions vanish. Arguments have sometimes been made that the anomalous behavior of the diffusion coefficient in polar systems is caused by the polar molecules rnoving in pairs or clusters rather than as separate kinetic entities. T h e present data offer a n insight into this argument. If, indeed; the anomalous behavior is d u e to clustering, the anomaly should vanish as infinite dilution is
Table VII.
Comparison of Diffusion Rates at Infinite Dilution
D x 705,
Solute
Soloent a.
Sq. Cm ./See.
Solute Effect
MCH Cyclohexanone
n-Octane
2.2 2.3
n-Octane I-Heptanol
MCH
1.6 0.6
MCH Cyclohexanone
I-Heptanol
n-Octane 1-Heptanol
Cyclohexanone b.
0.5 0.3
0.7 0.6
Solvent Effect
n-Octane
MCH Cyclohexanone
1.6 0.7
MCH
n-Octane 1-Heptanol
2.3 0.4
1-Heptanol
MCH Cyclohexanone
0.6 0.6
Cyclohexanone
n-Octane 1-Heptanol
2.2 0.3
Acknowledgment
T h e authors thank the Sational Science Foundation for financial support to R . L. Robinson, Jr.
Literature Cited
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NO. 1
FEBRUARY 1966
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