HIROSHI FUJITA AND LOUISJ. GOSTING
1236 exp -
[L]'
.'k = V B co 2 d z i tit 4' erfc [A+ ] ai
Vol. G4
Discussion = 1,J
2dDt
The solution was obtained using the Euler method of numerical integration. Figure 1 shows this chromatographic separation. The solvent front precedes the solute bands. The movement of the solute bands is characterized by the characteristics (constant concentration contours) emanating from the front and back of the initial solute pulse. The chromatographic separation is caused by the difference in the adsorption isotherm constants ai for the t ~ solutes. o
The analysis presented shows that the diffusion model for solvent penetration does represent the physical phenomena and that the paper chromatographic effect can be explained when equilibrium solute adsorption predominates. The secondary effect of solute diffusion has been neglected. Attempts to include this led to equations which preclude the use of the simple method of characteristics. Solvent evaporation from the paper surface also has been neglected since in practice chromatography is accomplished in a saturated atmosphere. This simple analysis gives further insight into the fundamental mechanism of adsorption paper chromatography.
A NEW PROCEDURE FOR CALCULATING THE FOUR DIFFUSION COEFFICIENTS OF THREE-COMPOSEKT SYSTEMS FROPI1 GOUY DIFFUSIOMETER DATA' BY HIROSHI FUJITA~ AND LOUISJ. GOSTIXG Contribution from the Department of Chemistry and the Enzyme Institute, University o j Wisconsin, Madison 6 , Wisconsin Received February 22, 1960
An exact procedure is developed which permits calculation of the four diffusion coefficients at a given composition of a ternary system from suitable free-diffusion experiments performed with the Gouy diffusiometer. This new procedure can be applied regardless of the relative magnitudes of the diffusion coeficients and should yield somewhat more accurate results than do previous methods. The data required are the reduced height-area ratios of the refractive index gradient curves and the areas of graphs of Gouy fringe deviations which summarize deviations of the refrartive index gradient curves from Gaussian shape. Data for the system NaC1-IICl-H20 a t 25" which were reported previously from this Laboratory are reanalyzed to illustrate the use of this proceure. It is found that the new values so obtained for the diffusion coefficients satisfy Onsager's reciprocal relation somewhat better than did the values which were calculated by a former method.
The thermodynamics of irreversible processes requires four diffusion coefficients for complete description of isothermal diffusion in a ternary system. During the past few years several procedures have been published for calculating four such diffusion coefficients, Djj, from experimental data. We review briefly those methods here to show their relation to the new method which is presented in this paper. Baldwin, et aL13presented a method which uses d a b for the reduced second and fourth moments, Szm and Dim,of the refra,ctiveindex gradient,curves obtained from suitable free-diffusion experiments; for given mean concentrations of both solut.es, 1 and 2, a t least t'mo experiments are required with different values of el,the fractional refract,iveindex increment of solute 1 across the sharp initial boundary. Subsequently a procedure was developed by Fujit,a and Gosting4 who used data for t'he 92, and the reduced height-area ratios, 5 ) ~ ,of the refractive index gradient curves for these experiments. The second procedure yields more accurate results than the first because experimental determinations of Dzm a.re quite inaccurate com(1) Presented at t h e 138th meeting of the American Chemicnl Society, Atlantic City, K e w Jersey, September, 1959. (2) On leave from t h e Physical Chemistry Lahoratory, Departm e n t of Fisheries, University of Kyoto, hlaiauru, J a p a n . (3) R. L. Baldwin. P. J. Diinlop and L. J. Gosting. J . A m . Chem. Soc., 7 7 , 5235 (1955). (4) H. Fujita and Id.J. Gosting, ibid.. 78, 1099 (1950).
pared to those of 3 ~ Homver, . results from the second procedure may still contain appreciable errors because the Dz, are somewhat less accurate than the DA (which can be measured to 0.1% or better with interferometric instruments such as the Gouy diffusiometer). Both of these methods are general, being applicable regardless of the values of the diffusion coefficients. A modification by Dunlopj of the second procedure achieves improved accuracy for some systcms by determining experimentally the tn-o values of a1 for which the refractive index gradient curves become Gaussian. These values of cy1, \Then combined with the data for 3.4, permit calculation of aZ, for any value of ai. Unfortunately the applicability of this method is limited in practice because inverted density gradients may be encountered for some systems in determining the required values of ai. A different method4 developed by Fujita and Gosting uses data for the DA and for reduced Gouy fringe deviatioiq6 Q((), corresponding to a particular value of the reduced fringe number, f (f) ( = f ( 4 2 ) = 0.73854) ; experiments for a t least two values of cy1 are required. When applicable thiq method can give more accurate resultq than the first two methods; however it depends on a w i e s expansion which converges satisfactorily only for suitable ( 5 ) P. J. Diinlop, TITIS JOURUAL, 61, 994 (1957). ( G ) D. F. Akeley a n d I . J. Gosting, J . A m . Chem. Soc., 7 5 , 5685 (1953).
Sept., 1960
CALCULBTIKG F O U R
DIFFUSIOK COEFFICIENTS
ratios of the diffusion coefficients (for values of u - / g + not greatly different from unity). Thus it complements Dunlop's procedure5 because these two methods are applicable to systems with different ratios of diffusion coefficients. In the present paper we develop an exact procedure which is applicable regardless of the relative magnitudes of the diffusion coefficients. The data required are the a > and ~ the areas of the Gouy fringe deviation graphs measured from suitable free-diffusion experiments. Because data from all parts of the refractive index curve are used in obtaining each area, this method should yield more accurate values for the diffusion coefficients than does the corresponding previous method4 which uses only one point ( L e . , n ( d 2 ) ) from each fringe deviation graph. A method which uses data from studies of restricted diffusion has been developed by Fujita.' His procedure is applicable to all systems which may be studied with Harned's conductance method.8 A procedure for using the diaphragm cell to determine some of the diffusion coefficients for a ternary system has been announced recently by stoke^.^ The main diffusion Coefficients, Dlland D2*, and the cross-term diffusion coefficients, D12and Dzl, considered in the present paper are defined, as b e f ~ r e , ~by~ ' flow ~ equations for one-dimensional isothermal diffusion of the form
Here bcl/8.?:and bca/dx are the concent,ration gradients of solutes 1 and 2, respectively, with the concentrations c1 and cz expressed as amounts (grams or moles) per unit volume of solution. The solute flows J 1 and Js denote the amount of each solut'e crossing a unit area per second for a volumefixed frame of reference, and therefore the diffusion coefficient's, D i j , should be identified with this reference frame. These coefficients are identical with one set of diffusion coefficients considered by Hooyman"; t'hey differ from, but are related to, diffusion coefficients defined by Larnm'? and by Onsager.lS We restrict our considerations t'o the cust,omary experiment'al condition that coiicentration differences between the two initial solutions in the cell are sufficiently small that the partial specific volume of each component may be considered independent of the conc,entrations. When every partial specific volume is constant the mean volume velocit,y is everywhere zero relative to (7) H.Fujit:t, THISJOL-RYI,, 6 3 , 242 (1959). (8) H. S.Hnrned and D. 11. French, A n n . N. P. A c a d . Sei., 46, 267 (1945); 13. S. t I a r n i 4 and R. L. Nuttall, J . Am. Chem. Soc., 69, 736 (1947). (9) Work by R. H. Stokes and F. J. Kelly reported a t a conference o n the Physicrl Chemistry of Solutions, Perth. Australia: see PTOC. Eo:/. Australian Chem. Inst., 26, KO.10,412 (1959). (IO) P. J. Dunlop and L. J. Gosting, J . A m . Chem. Soc., 77, 5238 (1955). (11) G. J. Hooyman, Physica, 22, 751 (195G). (12) 0. Lanim, Arkiu Kemi, Mineral. Ceol., lSA, No. 2 (1944): lSB, No. 5 (1944); THISJOURNAL,61,948 (1957). (13) L. Onseger, Ann. h'. Y. Acad. Sei., 46, 241 (1945).
OF
THREE-COMPONENT SYSTEMS
1257
the cell,14 and J1 and J 2 in equations 1 and 2 can be identified with the flows of solutes 1 and 2 relative to the cell-fixed frame of reference. Theory In the following development frequent use will be made of symbols and equations appearing in an article by Fujita and G0sting.l To save space we Li-ill refer directly to equation numbers in that article and distinguish such numbers from those in this article by adding the letter F. First me will derive an expression for the area, Q, of Gouy fringe deviation graphs as a function of 011 for experiments with the same value of El and also of C2, the mean solute concentrations defined by equation 7F; cyl (and also a?)has been defined by equations 45F-48F. It is then shown that Q/l/G is a quadratic function of al; this may be compared with equation 56F which indicates l / d %to be a linear function of 011. Exact equations then are derived for the four diffusion coefficients, Dij, in terms of the coefficients relating Q/G and 1/dsto 0 1 ~ . An Expression for the Area, &, of Each Fringe Deviation Graph.-Starting equations for this derivation can be obtained from a paper by Gosting and Onsagerls who gave a general theory of G O L Iinterference ~ patterns. In the Appendix of this article their theory has been specialized to the cace of isothermal free diffusion in ternary systems by using equation 49F for the refractive index distribution. Equations in the Appendix are identified by a letter A before the equation number. The reduced fringe deviation Q,,the reduced cell coordinate yj and the special variable {, for G O U ~ fringe number j mag be treated as continuous functions, EO we omit from these symbols the suhscriptsj. By definiiif; 2
=
467,
(3)
and P =
and remembering that
dzF+
(4
+
r+ r- = I (3 (see equation 52F) equation A-5 may be written f(4 - f(r) = r-[f(z)- f(p4l
(6)
Here the function f([) is defined byI6
Also, by combining equations 3, 4, 5 , A-6 and A-7 with A-8 we obtain
The area Q of a fringe deviation graph is defined as the area enclosed by the curve of n({) versus f (l).
(14) G. J. Hooyman. H. Holtan, Jr., P.Maaur a n d S. R. de Groot, Phgsica, 19, 1095 (1953). (15) L. J. Gosting a n d L. Onsager, J . Am. Chem. Soc., 74, GO66 (1952). (le) s e e equation 12 of G. Regelm and L. J . Gosting, J . Am. Chem. Soc., 69, 2516 (1947).
HIROSHI FUJITA AKD Louis J. GOSTISG
1258
-
Because it can be seen from equations 6 and 7 that 0, and also that { + 03 j- + 0 and x -+ 0 as f({) and z + as f ( l ) 1, equation 9 may, after substitution for IXj-) from equation 8, be put in the form
Differe1itiat)ion of equation 6 with respect to z , rearrangement, and subqtitution of equation 7 gives dfc
