CALCULATIQN or’ DQFFUSION
3449
COEFFICIENTS FOR KBr-NBr-M@
from pure vapors, without recourse to any particular model of adsorption,
n‘ = total number of moles adsorbed from mixture per gram of adsorbent np surlace excess of ith component ndwoTbed from liquid pep gram of adsorbent no = total number of moles in bulk liquid per gram of adsorbent (before contact;with adsorbent) P = pressure Pi0 = vapor pressure of sorbate of ith ~ O I X I ~ Q I X Y ~ ~ Po = vapor pressure of bulk liquid R = gasconstauit T = absolute temperature z i = mole fraction of ,ith component in bulk liquid phase xi’ = mole fraction of ith component in adsorbed phase yi = mole Iract,iori oi itb component in vapor phase ziO = mole fraction of itb component’in bulk liquid phase (before contact with adsorbent) y ; = activity coefficient of ith component in bulk liquid phase cpz8 = free energy of immersion of adsorbent in ith liquid7eq 2 =i
~ c ~ ~ , ~ o wfiiriancial l ~ ~ support ~ ? ~ ~ by?the ~ National ~ . Soirnca Foundarior; is ~ r a t e f u ~acknowledged. ~y
=- conJtant ~nLangmmx equation (see Table I ) I: = constant in BET equation (see Table I) K = conatant, eq 16 and 17 m -= number of rioFe6 adsorbed a t monolayer surface coverage G
npO -nz’ =
per gram of otkorhent of ntoles 31 piire Jthcomponent adsorbed per gram of adsorbtnt number of 11101e3 of zth component adsorbed from mixture per gram of arlec1rhent riiindm
ure for Calculating the Four Diffusion Coefficients
Ternary Systems from Gouy Optical Data. Application e System KBr-HBr-H,O at 25”1)2
Deportm,eizt of Chemistry and The Institute for Enzyme Research, University of Wisconsin, Madison, Wisconsin 68706 (Receiaed April 17, 1972)
k new method is presented for calculating the four diflusion coefficients, Dij, for ternary systems from Gouy optical data. I n this approach, the very accurately measurable reduced height-area ratios, DA,are first fit directly to the precisely known values of the solute refractive index fractions, a l , using the method of least squares. Then the experimentally measured reduced fringe deviations, 9, are fit to the corresponding reduced fringe numbers, f ({), by a least-squares procedure using the theoretical expression for i2 us. f ({). This calculation procedure, which is a theoretical improvement over previously used methods, was applied to new data for the system KBr-HBr-H20 a t 25’. The new method gives values for Dij for this part’icular system which are slightly better than those computed using the older “area” method. Gosting’s approximate theory for predicting L ) i j in electrolyte systems gives values for the diffusion coefficients which agree semiquantitatively with the measured Dij, indicating that the interaction between solutes in this system is due mainly Lo the eiectric field created by the rapid diffusion of H+ion relative to Br-- ion. The Onsager reciprocal relation was t’estedand found to hold within experimental error.
~~~~~~~~~i~~ 4
is then possible to derive expressions for each of the Calculation of tbe four diffusion coefficient^,^ D,,, four Dtl in terms of a n y four such “convenient conibinafor ternary liquid systems from Gouy optical data pre(1) This investigation was supported, in parr,, by research Grant sents a very int eresting problem in the treatment of exNo. AM-05177-02 from NIAMD, National Institutes of Health. The author was the recipient of predoctorai fellowships from the perimental results A difficulty arises because it is not Wisconsin Alumni Research Foundation a.nd from the NationaI Infeasible to derive closed-form analytical expressions for stitutes of Health. the D,,in terms of measured quantities. However, (2) *tiom of this work were submitted to the Graduate School of t,he bniversity of Wisconsin in partial fulfillment of the requirements the theoretical equations relating the D,, to experimenfor the Ph.D. degree. tal parameters do contain certain recurring combinat mas (3) Present address: Institute of Molecular Biology, University of of t h e D2,, which can be determined from the data. It Oregon, Eugene, Oregon 97403. The Journal of Physical Chemistru, Vol. 7 8 , N o . 23, 1972
ARNQED REVZIN
34214 tions;” various calculation procedures have been used,6--12u k h the different methods involving diff erent “convenient quantities” from which the Dij are computed. Each of itkese methods was developed to improve on those used previously and even the recent procedure 01 Iluiin and Hatfieldl*appears to give somewhat more Reelirate D,,values than do the older methods developed b y Costing, et al.6-11 The Dunn and H a l fieid method, however, does not use the experimental data in ttie rriost direct manner, so a new computation procedure hair been derived and is presented here. The new approach is theoretically an improvement over previous methods, and so should yield the most accurate D,,values t o date for the types of data currently extracted from Gouy diffusion experiments. Ne117 diflu3ion data are reported for the three-component system MBc-H r-H20, and the D,, values obtained using the new method are compared with those computed by other procedures. The experimental results are discussc:d in ternis of Gosting’s approximate theory for predicting values of D,,in electrolyte systems.I3 7‘hc ternary system studied is one of a series of systems f o p which daLa are now being compiled as part of a program to study the influence of hydrogen ion transport on thcl diffusion of other electrolytes.
rief ~ ~ r n afr ~ na ~ c~ ~ ~aFrocedures ti~n Now in Use A method which has found wide use was developed in 1960 by Fujita and Gosting.Io I n this procedure, two of the four necessary combinations of D,, are determined from data for the reduced height-area ratio,’ TJA. Of the various quantities which can be derived from Gouy f Pingt-1photographs, %A can be determined with the greatest accuracy. If several experiments are performed or the same system, with the mean concentration of eaeh solul e hcld constant from experiment to experimenx but with different solute refractive index fractions, al, then the values of l/dZ are linearly related8to a1
1/dG = IA-k
SAW
(1)
Here, I A and SAare expressions containing the D,,and the (known) refractive index increments, R,. The method involves determination of I A and SAby fitting the l/d%yvalues t o a1 using the method of least squares. ’Ps\o other D2,combinations are obtained from the yringe deviation graphs, l4 consisting of the reduced fringe deviations, Q j , plotted against the corresponding reduced fringe numbers, f (p,). Fujita and , Q is the Gosting’O showed that the ratio Q / d Zwhere area under t le fringe deviation graph, is a quadratic function of a I . Two of the coefficients in this quadratic expression art>used dong with I A and SAin the calculation of the La,, by this “area” method. Dunn and 14atfield12 have pointed out that the “area” meithnd ute3 graphs of reduced fringe deviations which are drawn by hand through the experimental The Journal of Physical Chemistry, Val. YE, No. 88, 1972
points. They observed that it is not very probable that two different workers will draw exactly the same curves, nor is it certain that these curves will be the “best” curves through the data points. To avoid such problems, Dunn and Hatfield have presented a computation method in which they fit the fringe deviation data by the least-squares method [using the theoretical relation between $2 and f({) ] while simultaneously fitting the data for 1/45; through use of the quantity8 rn (where n denotes the particular diffusion experiment), For the system to which their method bas been applied,15 it appears to yield somewhat more accurate D,, values than do other procedures, as judged by comparing the experimentally deduced fringe deviation graphs with those calculated from the theoretical equavalues. However, the Dunn tion using the derived Dz, and Hatfield approach does not, directly fit the l / d s values to a1) but involves an iterative procedure for determining I’n and the Dg,.This means that the fitting of the reduced height-area ratios may be influenced by the fitting of the fringe deviation data. This is undesirable because the l/ds values can be determined with much higher accuracy than can the values of it. Thus, a new calculation procedure has been derived in which the very accurate 1/fiA vs. 011 data are fit independently of the fringe deviation results, after which the Q us. f({) data are fit to their theoretical expression.
New Method For the four diffusion experiments which are ordinarily16 performed a t given mean solute concentrations, p1 and pz (but with different values of aI),there are five curves to fit [one plot of 1 / 4 g vs. a1 plus four Q us. f(p) graphs]. I n this new method, as in most of the previous methods (excepting that of Dunn and Hatfield12),P A and SA are determined using the method of least squares to best fit the data for l/y/zus. a1 according to eq 1. This choice is made because the a ) (4) In this paper, the symbols have the meanings used in related publications (ef., ref 6-11). (5) L. J. Gosting, Advan. Protein Chem., 11, 429 (1956). This review article includes the flow equations in which the D,, appear. The symbol D,, is used to denote the set of four diffusion coefficients, D u , D I Z ,D21, D Z Z . For convenience the reference frame for D,, will be omitted unless it is necessary to specify it. (6) R. L. Baldwin, P. J. Dunlop, and L. J. Gosting, J . Amer. Chem. Soc., 77,5235 (1955). (7) P. J . Dunlop and L. J. Gosting, ibid., 77,5238 (1955). (8) H. Fujita and L. J. Gosting, ibid., 78,1099 (1956). (9) P. J. Dunlop, J . Phys. Chem., 61,994 (1957). (10) H. Fujita and L. J. Gosting, ibid., 64, 1256 (1960). (11) H. Kim, ibid., 70,562 (1966). (12) R. L. Dunn and J. D. Hatfield, ibid., 69,4361 (1965). (13) Seep 535 of ref 5. (14) D. F. Akeley and L. J. Gosting, J , Amer. Chem. Soc., 75, 5685 (1953). (15) 0. W. Edwards, R. L. D u m , J. D. Hatfield, E. 0. Huffman, and K. L. Elmore, J . Phys. Chem., 70,217 (1966). (16) Seep 542 of ref 5.
~
CALCIJLATION OF D~im UBXQN GQEFFICXENT~ FOR KBr-HBr-W20 and rxl values are known to a high degree of accuracy, while the unoert,aiudies in other measured quantities are considerahly higher. There remain two parametersI7-withwt-nich t o f i b the fringe deviation data; these are chosen t,o be p etiid. ed;;, qumtities which are functiom ondy of the .Id, and which are defined by eq 30 and 31 of ref 8,:tlong with. the definition,’o p = 2/Z/ .%/&-~
?‘he the~ret~icei equation for the reduced fringe deviations is found by combining eq 5, A6, A7, and A8 of ref :IO t o give
oj = exp{ - - l j 2 ) ._
1 4-
exp{ -2;/\,yzj
a-.B/,
exp{
P-)%JZ
--’
-1 / 2 y j q
+ r”-dE
_.________ __.
(2)
. . I
““pi
-,.-f,iq ~
{ [ p d F b- (la -I-
wz== I*
cy1
= a?,
=
PO 1- APE
B/
exp( -1/&2y32f p(w’ -
+
a/
