Oct. , 1962
2015
DIFFUSION IN
the heats of fusion from the terminal slope of a log (atom, % solute) vs. 1/T plot. We estimated the distribution coefficient of Bi between solid and liquid Ag as 0.1 from the data given by Ilanseii and Anderko.2 Solid solubility was neglected in the remaining cases. Table IV shows a comparison between our calculated values and those chosen by Hultgren from heat content measurements.*’ The data agree well, except for bismuth. This may warrant further investigation. TABLE IV H n A m OF FUSION O F CU, Ag, AU, Element
cu
Ag
ASD
Au
IO m i n
I 372’
Bi Bi
2870 3130h 2940 2780 AH,“ calcd. 2955 2600 3120 2855 AH,“ lit. Average of values 3100,3300, a Calories per gram atom. and 3000 from the Cu-Ri, Ag-Bi, and Au-Bi systems, respectively.
Acknowledgments.-We wish to acknowledge the valua,ble help and suggestions afforded us by (17) R. Hultgren, Project Supervisor, “Selected Values, Thermodynamic Prciperties of Metals,” Prepared for the USAEC a t the Minerals Res. Lab., University of California, 1956.
Fig. 4.-Differential
thermogram of AunBi.
the following members of the analytical laboratory. Dr. R. Bystroff, for his help with the spectrophotometric copper determination, Dr. J. Harrar, for his help with the coulometric gold determination, and Mrs. P. Stephas.
DIFFUSION IN AN IDEAL MIXTURE OF THREE COMPLETELY MISCIBLE NOW-ELECTROLYTIC LIQUIDS-TOLUENE, CHLOROBENZENE, BROMOBENZENE BY JOHN K. BURCHARD AND H. L. TQOR Department of Chemical Engineering, Carnegie Institute of Technology, Pittsburgh I S , Pennsylvania Receiaod October 50. 1961
A modification of the diaphragm cell method has been used to investigate isothermal diffusion in the thermodynamically near ideal liquid system, toluene-chlorobenzene-bromobenzene. Measurements of the binary diffusion coefficients were carried out on the borders of the ternary system. The diffusion coefficients of the systems toluene-chlorobenzene and bromobenzene-chlorobenzene agreed with measurements made elsewhere and the diffusion coefficients of all three binary systems are linear in mole fraction. In the ternary system the difference between the binary diffusion coefficients of any two dilute species in a third species is small. The ternary system was studied a t six points covering the whole concentration range. The results are fairly well described by a modification of the diffusion equations for ideal, multicomponent gas mixtures. It i,j shown from the phenomenological equations that in the absence of cross diffusion coefficients the main coefficients are equal. This is approximately true in the system studied, for the cross diffusion coefficients and the differences between the main diffusion coefficien1,s are an order of magnitude less than the main diffusion coefficients. Consequently, one diffusion coefficient will give an adequate description of this ternary system under most conditions. This diffusion coefficient is found to be simply the molar average of the binary diffusion coefficients a t infinite dilution.
Introduction The relationship between the concentration gradients and diffusion fluxes in systems of more than two components is of considerable practical as well as theoretical interest. Although in sufficiently dilute systems Fick’s Law as applied to each species may satisfactorily describe the diffusion process, recent work using the Gouy interferometer has shown that equations which account for interacting flows generally are needed. Such work’-9 (1) D. M. Clarke and M. Dole, J . A m . Chem. Sac., 7 6 , 3745 (1954). (2) R. L. Baldwm, P. J. Dunlop. and L. J. Gostlng, zbzd., 77, 5235 (1956). (3) P. J. Dunlop and L. J. Gosting, ibzd., 77, 5238 (195B). (4) 13. Fujita and L. J. Gosting. zbzd., 78, 1099 (1956). ( 5 ) P. J. Dunlop, J . Phya. Chem., 61, 994 (1957). ( 6 ) P. J. Donlop, ibid., 61, 1619 (1957). (7) P. J. Donlop, ibid., 63, 612 (1959).
has been concerned with aqueous solutions with one or both solutes electrolytes or sugars, and usually one or both of the solutes has been relatively dilute. One very interesting class of multicomponent systems which has not been studied is that which is made up of completely miscible non-electrolytic liquids. In these systems the diffusion may be studied over a composition range which includes all ternary and binary combinations of the components, and the object of the present study is to determine the rates of diffusion in a ternary system of this type as functions of the concentrations and concen( 8 ) I. J. O’Donnell and L. J. Gosting. “The Structure of Electrolytic Solutions,” John Wiley and Sons, New York, N. Y., 19;17, Chapter 11. (9) F. E. Weir and M. Dole, J . Am. Chem. Sac., 80, 302 (1958).
JOHN K. BURCHARD A S D H. L. TOOR
2016
tration gradients over the complete range of compositions. The phenomenological equations for a ternary system may be written in the formZ 2
hirdspecies can be derived from eq. 1 and, if there is no volume change on mixing, it)may he written in the form
Yol. 66
and DZIapproach zero and D,, approaches the binary coefficient for dilute i in 3, DI3O,i = 1,2. Thus, from eq. I and 2 Lim J , c', -+ 0 cz --+ 0
=
-
Lim
=
( D -~ ~ 2 ~3 " ) : ~ vC1 -
c1-
Jn
DIaoOC,, i = 1,2
(5)
fll
213
0 cz + 0
vC3 (6) Clearly eq. 3 and 4 are valid at this limit only if Since each species in turn can be D13O = &O. considered to be species 3, it is concluded that at least at the corners of the concentration field, the necessary and sufficient condition for the absence of cross coefficients and equality of the main coefficients is that13 D23O
ail"
DkJ0=
0; i, j , IC,
=
1,2,3
('7)
1 # ilk
where the g, are the partial molal volumes. Although the above equations contain four diffusion coefficients, the Onsager reciprocal relations indicate that only three are independent.l0 Consider now a ternary system in which the cross diffusion coefficients, DiJqi # j , of all species are assumed to be zero so that
Jl = D,, VC,, i
1,2,3
(3) Since the first bracket in eq. 2 is a cross diffusion coefficient, and since species 3 is arbitrary, it follows from eq. 1 and 2 that the main diffusion coefficientsare all equal =
Although this condition may be strictly valid only in systems in which all species have identical properties, i.e., in an ideal system, a real system can approximate the simple behavior of eq. 3 and 4 only if the left side of eq. 7 is small compared to the DijO. The Onsager reciprocal relationships yield an expression reIating the and for ideal-systems of constant molar density it reduces to the simple form Dn
+ (1 + [ x s ' x ~ ] ) D ~ I + (1 + [~3,'xil)Di2 (8) =
0 2 2
where the z1 are mole fractions. This equation also indicates that the main coefficients are equal when the cross coefficients are zero. Dl1 = Dzz = 0 3 3 F D* (4) Choice of System.-It is logical to choose as the Thus, zero cross coefficients imply equal main first system for study one which approaches the eq. coefficients and vice versa1' It follows from eq. 1 7 condition. The system toluene-chlorobenzeneand 2 that the complement of the above statement bromobeiizene exhibits this behavior. In addition, is also true; non-zero cross coefficients imply un- the binary diffusion coefficients of two of the three equal main coefficients and Lice versa.12 binary pairs have been measured accurately over Although one might expect the above behavior in the complete concentration ranges and found to be ideal systems,13 it is desirable to have some criterion close t o linear with mole fraction.'j (other than "near" ideality) concerning the condiThe materials used were Fisher certified reagents. tions under which eq. 3 and 4 might approximate Analysis by gas chromatography showed negligihlc real systems. impurities. To this end, consider a ternary system in which Experimental cross diffusion coefficients are not assumed to be The diaphragm cell method was used in this study. Prezero, but the concentrations of species 1 and 2 are liminary studies of the binary systems bromobenzeneallowed to approach zero together. The equations chlorobenzene and toluene-chlorobenzene were first carried for these species should then reduce t o the binary out in order to compnre the diffusion coefficients obtained bv form since there should be no interactions between this method to those obtained by Caldwell ?nd BabbI6 by a two sufficiently dilute species, It follows that DE different technique. Following thls, the third blnary paw, (10) L. Onsager, Ann. A' Y. Acad. Scz.. 46, 241 (1945). (11) I n a dilute system (dilute 1 and 2 in 31, Diz and Dzi may approach zero although Dii # 0 2 2 but, as s h o n n later, species 3 uill then have a non-zero cross Coefficient. (12) These results may be generalized to systems of more than three components. (13) Onsager and Fuoss [I,. Onsager and R >I, Fuoss J . Ph?/s. Chem., 86, 2689 (1932)l suggested t h a t crofis diffusion roefficiPntz kould be negligible in ideal syatems,
toluene-bromobenzene, was studied in order to complete the mapping of the diffusional behavior on the borders of the ternary field. The diaphragm cell was patterned after that used by Trevov and Drickamer.16 The diaphragm was Pyrex fritted (14) D . G. Miller, J . P h y e . Chsm., 63, 570 (1959). (15) C. S.Caldwell and A . L.Babb. i b i d . , 60, 51 (1956). (16) D. J. Trevoy and H. G . D r i c h m e r , J . Clism. Phus., S I , 1117
(1949).
Oct., 1962
DIFFUSION IN
glass, porosity A I , with Teflon stopcocks in the capillary tubes. The chamber volumes were about 25 ml. and the difference between the volumes of the chambers in a cell was less than 1.5 nil. The working equation for the diaphragm cell with a binary system is17
AN
IDEAL MIXTURE
2017
stant molar densities (partial molal volumes constant and equal) the concentration ratio in eq. 9 is also a mole fraction ratio.
Binary Results The bromobcnzene-chlorobenzene and toluenechlorobenzene systems mere each investigated at 1 AC," mean concentrations of 5, 50, and 95 mole %, usDI2 = - I n -/?t At?,' ually with concentration differences of approximately 10 mole yo. Since the results were in where agreementzowith the data obtained by Caldwell and Babb15 with a different technique, the cells mere assumed to be reliable in this type of experiment. The toluene-bromobenzene system was investigated a t mean concentrations of 5 , 35, 65, and 95 mole % toluene with initial concentration differences of approximately 10 mole %. Four repThe superscripts refer to initial and final conrentrations and the sltbscripts to left and right hand bulbs. For equal bulb licas were run at each point and the results using and a linear dependence of DIZ on concentration, the more precise peak height measurements are volumes Dlz is the differential coefficient a t the arithmetic average shown in Fig. 1. As in the other binary systems, the conipozition. Since this composition does not change with diffusion coefficient is linear with mole fraction. time, Slz = Sl2. The 95% confidence envelope shows the results to The cell factor, p, was found by calibrating with 0.5 be good to about *a%. The equation of the least HC1 diffusing into water a t 25'. The valueof 3.078 X 10-5 squares line through the data is given on the graph. cm.Z/sec. from Stokes' datal8 was used for 3 1 2 in eq. 9. The The six dilute diffusion coefficients are shown in results of several replicas were: p (cell I ) = 0.2038 =k 0.0013 Table I. The predictions of the Wilke and Chang cm.?; p (cell 2 ) = 0.20i9 5 0.0015 cm.-2. The organic solutions to be used as initial charges to the equationz1are also included for comparison. The two chambers were weighed to the desired compositions. values for chlorobenzene-bromobenzene and toluThe left chalmber was completely filled and rinsed with CLO by introdncing the solution through one tube and taking the ene-chlorobenzene are interpolated from Caldwell overflow through the second tube. The stopcock on the and Babb's'j data and those for the third system overflow tube was then closed and a vacuum applied to the are from the present measurements. It is seen other chamber pulling CLO through the diaphragm. Yigor- that eq. 7 is approximately true. These values of ous flushing of the diaphragm to dislodge bubblesin thepores DIJoare also shown at the corners of Fig. 2. was continued for several minutes. Ternary Equations for a Diaphragm Cell.-As The cell was inverted, and the right chamber drained. The cell was then righted, and the solution CRO was charged in the corresponding binary development,17 it is t o the right chamber, filling through one tube and rinsing by assumed that (1) a quasi-steady state is obtained, overflouing through the second tube. (2) the diffusion can be treated as if it were uniill1 four stopcocks were then closed and the cell was placed in a water bath maintained a t 20.6 =k 0.03" for the pre- directional, (3) the diffusion takes place only in the liminary difl usion period of four hours which is required t o set fluid, (4) the volumetric average velocity within up the quasi-steady state concentration profiles in the dia- the diaphragm is zero, and (5) the contents of the phragm .I8 Bt the end of this period, the cell was removed from f he constant volume bulbs are completely mixed. bath and each chamber was drained separately, rinsed, and With assumptions 1, 2, and 3, eq. 1 reduces to
filled by gravity with fresh charges of the original solutions. The cell W ~ E Ithen returned to the bath for a diffusion run of approximately one day. Several replicas were made of each run. It was found that reproducible results could be obtained only with the diaphragm horizontal au indicated by Stokes19 and with the denser solution on top. Under these conditions the diffusion coefficient was essentially independent of the initial roncentration difference. The final (concentrations of the solulions drained from the cells were determined by gas-liquid chromatography, using a Beckman GC-2 instrument with helium as a carrier gas. -414-it. stainless steel l/q in. diameter column packed with 37, by weight Silicone 560 (Dow Corning Corp.) on firebrick gave good resolution with a short elution time. The recorder was fitted with an integrator so that pezk areas as well as heights could conveniently be used in the analyses. Samples of the original charge solutions as well as a few specially prepared samples were used to calibrate peak height and area ratios as functions of mole fraction. In analyzing an unknown sample, five determinations of height and area ratios were made and the corresponding mole fraction ratios were read from the calibrations and averaged. With this method of analysis, the error in a mole fraction of the wrresponddeterminaticn was usuallv less than ing mole fraction. Since these mixtures have essentially con(17) A. R. Gordon, Ann A' Y . Acad. S e t , 46, 286 (1945). (18) R. H. (stokes. J . A m Chem. Soc.. 72, 2243 (1950). (19) R , Ht tstokes, rbtd., 72, 763 (1960).
Assumptions 1 and 2 also indicate that the J i are independent, of x , so upon integrating from left to right (13)
where ACj = C ~ L C ~ R1,, is the effective diaphragm length, and
From assumption (4),J i is the flux with respect' ( 2 0 ) T h e 95% confidence limits on the individual replicas averaged 13.6% when peak areas were used and & 2 . 3 % when peak heights were used. Since the average difference between the measured diffusion coefficients and values interpolated from Caldwell a n d Babb's d a t a W ~ 3.1% E when peak areas were used and 2.0% when peak heights were used, there is no significant difference between the present measurements and those of Caldwell a n d Babb. (21) C, R. Wilke and Pin Chang, A , I. Ch. E . J., 1, 264 (1055).
2018
'JOTIK K. BTJRCIIARD AND H. IJ. TOOR 2,6
____ d(ACi)
dt
2.2
i
280
r
1,8
0
c
0
l,6
X
rn
'0
2
C Dij ACj, i = 1,2 j=1
-P
(16)
where P is the same cell factor as in eq. 9. All the assumptions except (2) have essentially the same justifications in multicomponent systems as in binary systems. Assumption (1) is valid with a properly designed cell, (3) with large enough pores in the diaphragm, (4) with a horizontal diaphragm and negligible volume changes on mixing, and (5) with mechanical or density stirring.17 The earlier binary experiments indicate that these assumptions are valid h e m z 2 Assumption (2) has been shown to be unnecessary in binary systemsz3 and in a similar manner this assumption can be shown to be unnecessary in a multicomponent system when the diffusion coefficients are independent of concentration. Since the O i j can be made to remain essentially constant with time, eq. 16 may be treated as a set of two simultaneous linear equations and solved by elementary methods. The solution for ACi a t time t is
2,4
\
Vol. 66
1*4 I a2
ACi
=
+
HieXat Kiexbt,i,j = 1,2
(17)
i ZJ
I
where
to
0,O X, Fig. 1.-Binary
0,2 0,4 0,6 0,8 I,O MOLE FRACTION TOLUENE.
H i
=
diffusion coefficients toluene-bromobenzene.
/,
2.5
Ki
1
- 2, [(Ab
=
1 - [(A,
20
+
PDii)ACio
+ PDijAc,"I
+ pDii)ACio + P
A, = - 2
(Dll
pDijACjoI
+ DB) + a
These equations are transcendental in the diffusion coefficients, but simple approximations which are valid in the present experiments are obtained as follows. From eq. 16 1 ACiO pt1n =
ac,f
8
Fig. 2.-Main
diffusion coefficients vs. concentration of toluene, chlorobenzene, bromobenzene.
to a coordinate fixed in the cell: so using assumption ( 5 )
where V is the bulb volume and A , the effective area of the diaphragm. Combining eq. 13 and 1:
If at is small,24exp ( d ) = 1 then gives
+ at.
Equation 17
(22) Experiments with gases 1.J. B. Duncan and H. L. Toor, A . I Ch. E. J.,S, 38 (1962) ] indicate that assumption (1) will be valid in a ternary experiment if it is valid in a binary experiment. (23) H. L. Toor, J. Phys. Chem.. 64, 1580 (1960). (24) T h e time scale of a n experiment is set b y the requirement t h a t In (ACio/ACi') in eq, 18 be large enough for accurate measurement. This requires Bt t.o be of the order 10' sec./cm.z. Hence for rrt t o be small. the cram diffusion coefficients and differences between the main diffusion cot.fficient,s must be finin11 cornnnred to Ihe ifinin diffusion cocfficientq.
AC - -j ACi
AC jo ACiO
i
= 1,2 i#j
(19) Xow if A C j o / A C i o is not large the variation of A C j / A C i with time as given by eq. 19 is approximately linear, so if AClo is not small compared to ACsoeq. 18 and 19 yield
where subscript a refers to the arithmetic average of the initial and final concentration differences. Similarly, if ACS" is not small compared to AGIO
which must be used in the two experiments will also depend upon the type of concentration dependence chosen. These problems are avoided if the initial concentration differences are made small enough to keep the variation of the Dij across the diffusion path small, for then the integral values may be assumed to be the point values at the arithmetic average composition.26 Under these conditions the same "mean" concentration is obtained in two experiments with different concentration differences by making the initial arithmetic averages of the concentrations in the two bulbs the same in both experiments. The arithmetic mean concentrations mill not change with time if the bulb volumes are equal and they change only a negligible amount if, as in these experiments, the bulb volumes differ by a small amount. I n the system under study the partial molal volumes are all essentially equal so eq. 20c simplifies to
When the differences between the main diffusion coefficients and the cross coefficients are small compared to the main coefficients, eq. 20 approximates eq. 17 very closely if all the ACT? are of the same magnitude or if A C 3 0 is small. Equation 20a is invalid if AClois very smallz5but since ACa0 cannot be small when AClo is small, replacing subscript 1 by 3 in eq. 18 and 19 leads to a valid equation which can be reduced to
and all concentration ratios in eq. 20 may be considered to be mole fraction ratios. In studying an unknown system the errors caused by approximating eq. 17 by eq. 20 are unknown, so the diffusion coefficients obtained from eq. 20 must be treated as first approximations. With these values the concentration changes predicted by eq. 17 may then be compared with the measured changes or with those predicted by eq. 20. In this manner it was found that in the present study the errors introduced by the use of eq. 20 in place of eq. 17 were negligible. If ACzois small, eq. 20b is invalid but replacing Ternary Experiments and Results.-The same subscript 2 by 3 in eq. 18 and 19 also leads to eq. equipment was used in the ternary and binary experi20c. Consequently, for any arrangement of the ments except for the replacement of one of the cells. initial concentration differences there are two le- The technique used in the ternary work differed from gitimate independent equations which contain the that used in the binary study only in the analysis of four diffusion coefficients to be determined-any composition. In the ternary work an extra set of two of eci. 20 if all the initial concentration dif- calibration curves had to be constructed and inferences are of the same magnitude, eq. 20b and c if stead of calculating separate mole fractions from AClo is small, eq. 20a and c if ACzO is small and the measurements of peak height and area ratios, eq. 20a and b if AC30 is small. the mole fraction ratios determined by both of It is apparent that two experiments with dif- these methods were immediately averaged and the ferent concentration differences a t the same "mean" average values were then used to calculate mole concentration level will be needed to determine the fractions. four integral diffusion coefficients. The*extraction I n order to cover the concentration range inside of the point coefficients from the integral values the ternary concentration field, data were taken a t may be carried out in general by assuming a par- six mean concentrations. At each of these mean ticular concentration dependence of the Dij over the concentrations two or three experiments were carrange of the experiment and numerically solving ried out with different initial concentration difeq. 12. However, the proper mean concentrations ferences as shown in Table 11. Subscript T refers. ( 2 6 ) If ACi0 is identically zero a n d rrt is small, eq. 17 gives apto toluene, C to chlorobenzene, and B to bromoproximately benzene. Each experiment was repeated a t least, once. Init, this equation is not very useful for t h e determination of diffusion coefficients. I f b21 = ~ I -I & = 0 the equation is exaot, b u t if eq. 8 is valid. Diz = 0,and the r e w l t is trivial.
(26) If the cross diffusion coefficients are zero and t h e main C O P E cients are linear witti composition, i t can be shown t h a t the meastired integral corificienta are still the point vrdiirs itt fbr arirhnirtic average comi)osition.
JOHN K. BCRCHARD AND H. L. TOOR
2020 TABLE I
EXPERIMENTAL AND cALCLJL.4TEDz' DILUTE BINARY DIFFUSIONCOEFFICIENTS, CM~./SEC. x Temperature f 29 6' System (first component dilote)
9,p
9,,0
(expl.)
(ralrd.)
Chlorobenzene in bromobrnzenr Toluene in bromobenzene Bromobenzene in cblorobenzenc Toluene in chlorobenzene Chlorobenzene in toluene Bromobenxene in toluene
1 36 1 41 1 76 1 80 2 36 2 27
1 58 1 58 2 09 1 89 2 60 2.58
TABLEI1 LfEAN COSCENTRATIOXS AXD NOMISAL I X I T f A L CONCENTRATION DIPPEREWES Expt 1T ?c fg A?T~ Ax(4 AZB~ 1-1 O 25 0 50 0 25 0 0 10 -0 10 1-11 25 50 25 -0 10 0 10 1-111 25 50 25 06 - 14 08 2-1 .26 03 71 08 - 06 14 2-11 26 03 71 - 14 06 08 3-1 70 15 15 15 - 05 10 8-11 70 15 15 10 05 - 15 4-I 15 70 15 16 - 05 IO 441 15 70 15 10 05 15 5-1 45 26 38 - 06 16 - 10 5-11 45 25 30 05 10 - 15 6-1 18 28 54 05 15 - 10 6-11 18 28 54 .05 .10 - .15
-
-
-
Vol. 66
differences close to zero. From the earlier discussion and Table 11, it follows that eq. 20b and c apply to the first experiment, and eq. 20a and c to the second. As a test of the over-all procedure the four diffusion coefficients obtained from these equations were then used to predict the final concentration differences in the third experiment a t point 1 which was carried out with all the initial concentration differences of the same magnitude. These calculated values are compared to the measured values in Table V. The check is satisfactory . The uncertainty in a cross diffusion coefficient depends primarily upon the choice of the initial concentration differences and when the uncertainty of one cross coefficient is decreased tjhe other is increased. At points 2 to 4 inclusive DcT was determined yith minimum uncertainty and a t points 5 and 6 DCT was determined with minimum uncertaint y The confidence ranges given for the D,j were obtained from an analysis of variances contributed by each experimental value used in the calculations. It can be seen from Table IV that the confidence limits on the main diffusion coefficients run about &A%, whereas the percentage uncertainty of the cross diffusion coefficients is much greater and more variable.
.
Discussion
The differences between the main diffusion coSince the diffusion coefficients are assumed to efficients average 5% while the cross diffusion codepend only upon the mean concentration, two of efficients average 2Oj, of the main coefficients so this these experiments yield four equations for the four system approximates the behavior of eq. 3 and 4. I t is not statistically certain that the two main unknown diffusion coefficients at the mean concendiffusion coefficients a t a point actually differ, tration used in the experiments. The diffusion coefficients obtained at the six since the confidence limits overlap in all but one points are given in Table IV together vith the case. However, most of the cross diffusion coestimated 95% confidence limits. The main dif- efficients which were determined with minimum uncertainty are significantly different from zero, infusion coefficients are also shown in Fig. 2. At each of the points 2 to 6 all three values of dicating that the main diffusion coefficients probAC,O are the same magnitude so eq. 20a and b are ably do differ. As shown in Fig. 2, both of the valid.27 Rewriting eq. 20a in the present nomen- main diffusion coefficients lie close to the plane whose borders are described by the binary difclature and summing over repeated experiments fusion coefficients. In Table YI the left and right hand sides of eq. 8 are compared. It is seen that the equation is satisfied a t all points within experimental error. However, when the cross diffusion coefficients DTC and (21) DCT are zero the comparison is almost trivial since where I and I1 refer to the two sets of experiments the equation merely asserts the equality of DTTand with the same mean concentration, but with dif- DCC, an equality which holds irrespective of the ferent concentration differences. and the caps refer reciprocal relationships. Thus, in the present ext o the average values. Substituting the measured periments, the satisfactory check indicates that, the values of the capped terms into eq. 21 gives iwo magnitude of the difference between the main difsimultaneous equations from which DTT and DTC fusion coefficients is consistent with the magnitude may be calculated. Similarly eq. 20b yieJds two of the cross diffusion coefficients. more independent equations from which DCCand Kinetic Theory of Gases.-The kinetic theory of DCTmay be obtained. describes the diffusion in a multicomponent The data and calculations a t point 4 are shown in gases system in terms of the binary diffusion coefficients Table 111. Complete data are given elsewhere.28 of the system it is of interest to consider the Each of the first two experiments at point 1 were application of and these equations to liquids. Thc carried out with one of the initial concentration Curtiss-Hirschfelder equations29for a mixture of (27) Sub-crwt 1 is taken as toluene, 2 as chlorobenzene, and 3 as three ideal gases yielcf2feq. 22a-22e. bromobenzene. (28) J. K. Burchard, Ph.D. Theeis. CnrnPaie Institute of Technoloay, Pittzhurph, Pennsylvania, 1861.
(29) C . F. Curtisa and .I. 0. Hirsolif~ldar.J. Chem. Phus. 17, 550 (1949).
DJFFCSION IK
Oct., 1962
AX
202 1
IDEAL RIIXTCRE
TABLEI11 n.4TA4 AND CALCUL.4TIONS-POINT
4-1 ( a )
I 0)) IT (:I) IT I I ) )
-0.0502 - .0502 .0484 .0484
0.1,503 .I503 .I009 ,1009
0,14713 1,8611
. 14643 18418
0.1146 . to60 .0760 .0713
-0.0373 - ,0341 ,0876 .0356
4
-o.:125,
.4869 From
From eq. 20a
-3,04R
ey.
20b
2 0481 = Dcc - 3 0 4 9 D c ~ 1 8368 = DTT - 0 3 2 8 1 0 ~ ~ 2.0540~~ 1,7019 = Dcc 1 8764 = DTT 0.4869D~c Hence" DTT = 1.853, DTC= 0.049, Dcc = 1.841. DCT= -0.068 Units of l / P t and diffusion coeffieiente are crn.z/aec. X 10-j.
+
+
a
2.054
6Wt.
DTT
1
1.848 f 0.066 1.570 f ,088 2.132 i ,098 1.853 i .lo8 2.006 i ,108 1.774 i . 1 O i
2 3 4 5 6
ficc 1.797 f 0.076 1.606 f. .I09 2.062 i .I08 1.841 & ,108 1.890 f. ,108 1.518 i . l o 7
TABLET COMPARISOS OF CAI,CITIATED AND MEASVRED COKCENTRATIONS, EXPT.1-111
DCT
DTC
-0.063 f 0.109 - ,077 f . l o 0 ,051 i ,163 ,049 f ,162 - ,020 f .026 - .037 i ,026
3 1. - --Z j 3 x1
+ xJ
DijO
-0,052 r- ,012 - .071 - ,068 - .198 .0026
f 0.093 f. .033 f ,026 f ,026 f .163 i ,162
+ xi +
2,
Xi
a>,,0
~
(23)
The multicomponent diffusion coefficients were calculated from these equations using the values of Di,ofrom Table I and are compared to the measured diffusion coefficients in Table VI1 and in Fig. 2. It can be seen that the experimental main difTABLEVI fusion coefficients are, in most cases, well within exCOMPARIW\ OF ~ , E F TA W RIGHT SIDES O F EQ 8 perimental error of the predictions of eq. 22 and 23. Expt LHS RHS It is of interest to note that the equations properly 1 1 '77 f 0 16 1 67 f 0 22 predict the largest and smallest Dii except for expt. 2 I 2'7 f .82 1 32 f. 39 2. Usually the experimental values lie between the 2 12 f 23 3 1 99 f 11 predicted values. 1 94 i 34 4 1 77 =I= .11 For the cross terms the experimental values are 1 86 f. 12 5 1 57 f 38 usually smaller in magnitude than predicted, and 1 37 f. 15 6 1 78 f 49 where the confidence ranges are relatively small 1,IIS = i?TT (1 SB/zC)DCT this difference is statistically significant in most RTF = D < r + ( 1 + SR/ST)DTC cases. Thus, eq. 22 and 23 predict the ternary diffusion = D13[(1 - x1)D12 -t-ZlD231/11, (224 reasonably well from binary diffusion data. The predictions are better for the main terms, but a t D12 = 2 1 a ) 2 3 ( 5 ) 1 1 - 5)1*)/+ (22b) least the order of magnitude of the cross terms is handled correctly. I)?? = D2l[(l - zz)a)21 ZPDl?]' i ) (22c) Use of a Single Diffusion Coefficient.-Because D?, = Zz D,?(Do21 - D21) 'I// (224 of the small values of the cross diffusion coefficients, the cross terms in eq. 1 will generally make only a 1c. = 2 1 9 2 1 2 2 5 3 1 1 r3D12 (22e) small contribution to the flux. Consequently, a simple, usually adequate approximation to the and D , ~= Q, diffusion in this system is obtained by neglecting These equations are ambiguous for liquids since. the cross terms altogether, for then only one ternary unlike ideal gases, the binary diffusion coefficients diffusion coefficient, D",is needed to describe the are composition dependent. In order to allow for diffusion. The D* can then be calculated by the third component, one might evaluate the averaging all the log terms, for all the replicas in a DIJto be used in eq. 22 at a species i mole fraction given experiment, i.e., by assuming that differences in a binary mixture of species i and j of x,/(xl between the initial conditions I and I1 give indisx,),where x, and x, are the mole fractions in the tinguishable _results. However, simply averaging ternary mixture. Since the binary diffusion co- the DTT and DCCalready calculated gives essentially efficients are linear in mole fraction in the present the same results and these values are shown in t o be used in eq. 22 are given by study, the 91j column 1 of Table VIII, Arc
4XTf
Ercpt,. C:1!cd.
0,0483 0,0320
-0.1017 - 0 . 101'7
+ +
a,
+
+
+
+
JOFIYIC. BCRCIIARD ASD 11. I,. Toori
2022
Vol. 06
TABLE VI1 COMPARISONOF EXPERIMENTAL DIFFUSION COEFFICIENTS~ WITH VALUESCALCULATED FROM EQ. 22 c------D'pp----Expt.
1 2 3
4 5 6 Units
Expt.
Calcd.
1.85 1.57 2.13 1.85 2.01 1.77 are cm.2/?ec.
1.92 1.66 2.14 1.88 1.99 1.72 10-5.
x
D* ( I ) 1.82 1.59 2.10 I 85 1 95 1 65
1 2 3 4 5 6
Expt.
Calcd.
Expt.
Calcd.
1.70 1.54 2.00 1.72 1.78 1.57
-0.06 - .08 .05 $- .05 - .02 - .04
-0.03 - .12 - .09 - .01 - ,09 - .06
-0.05 - .01 - .07 - .07 - .20 .00
-0.18 - .02 - .13 - .15 - .16 - .13
1.80 1.61 2.06 1.84 1.89 1.52
D * (2)
D* (3)
1.84 1.59 2.12 1 84 1.91 1.64
1.81 1.63 2 09 1.80 1 90 1 66
D* = 2 . 3 6 ~+~1 . 8 4 + ~ ~1 . 3 0 ~ (24) ~ The multiple correlation coefficient for the equation is 0.995. Values of D* calculated from this equation are given in column 2 of Table VI11 and may be compared with the measured values in column 1. This linear behavior is a natural generalization of the linear behavior of the binary diffusion coefficients. The binary diffusion coefficients in a system obeying eq. 7 form the borders of the plane given by 3
=
i =1
X~D~,O, j#
i
DcT.-
I--
+
identically equal to a)kjo. If these differences are neglected and the arithmet'ic averages of the t'wo dilute coefficients a t each corner of Fig. 2 (which are given in Table I) are used in eq. 2.5 one obtains
D*
From a regression analysis of these D* a t the six concentration points, it is found that they form a plane in D*-concentration space3n
D"
23
,------DTC--
TABLE VI11 TERNARY DIFFUSION COEFFICIEST, cJI.p/s~:c. x 105 Expt.
AND
,------Dee-----. Expt. Calod.
(25)
In the system used here a)i,n is close to but (30) Since the main diffus~oncoefficients ddfer from D* by only B small amount, tlie main diffusion coefficients a l e also rssentiallj Iinrar w i t h inolr fraction.
+
= 2 . 3 1 ~ ~1 . 7 8 4~ ~1 . 3 8 ~ (26) ~
and this is the equation of the plane shown in Fig. " 2. The very close similarity between eq. 24 and 26 is obvious. The values of D" calculated from eq. 26 are given in column 3, Table T'III. It is apparent that the two planes coincide within experimental error, indicating that the D" given by the ternary measurements goes to the proper binary limits. It follows that for many purposes liquid diffusion in the system toluene-chlorobenzene-bromobenzene can be adequately described by eq. 3 and 4 with D" given by eq. 21; or 26. This is a pleasantly concise result-one diffusion coefficient, easily predicted from dilute binary data, serves for a ternary system. If this behavior is a general characteristic of multicomponent systems which obey eq. 7, then an approximate description of the diffusion in such systems can be obtained without the necessity of any experimental measurements, for the dilute binary diffusion coefficients can be obtained with a t least iair accuracy from available correlations. Acknowledgment.-The authors are grateful to the Natiollal Science Foundatloll for3'' fi11ancial support of this work.
