,July, 196fl
ISOTHERMAL DIFFUSION IN THM SYSTEM Hz0-Na2S04-€Id304
1279
species of t'he aquated cobalt(lI1) ion, presumably between cobalt(I1) and cobalt(II1) could result in a in a high-spin state, aiid that the Franck-CondonZ3 labile form of cobalt(II1) with the rate of electron restriction is minimal for reactions involving the transfer between cobalt (11) and cobalt (111) rate determining. Since the rate of the cobalt(I1)aquated cobalt(II1) ion. It has been suggested that the normal diamag- cobalt(II1) reaction is slower than the iron(I.1)iietic, lo-w-spin state of cobalt(II1) can be fairly cobalt (111) reaction under comparable conditions, easily excited to the paramagnetic high-spin state the possibility of electron transfer between cobaltwhen water, with its relatively weak ligand field (11) and cobalt(II.1) being rate determining is exstrength, is the ligand.21 The rapidity of both the cluded. Data for additional reactions of this charge type water exchange with the aquated cobalt(II1) ionz4 and the cobalt(II)-cobalt(III)8 exchange reactions are included in Table IV for the convenience of other have been explained on the basis that the reactive investigators interested in this type of reaction. species of cobalt(II1) is in the high-spin state, and a TABLE IV similar explanation could be advanced in the case MISCELLANEOUS REACTIONS INVOLVING of the iron(I1)-cobalt (111)reaction. I n view of the preceding discussion, it is possible M ( U ) AND N ( I I I ) IOSS IS 1 M PERCHLORIC ACIDAT 0' transition in the aquated that the ( t 2 & 6 +. (t7g)4(eg)2 M(I1) * N(1II) 'k', see.-' cobalt(II1) ion IS the rate-determining step in the WII) Co(II1) >300a iron(I1)-cobalt (11) reaction. Such a condition Cr(I1) Co(II1) >3OOa would require this reaction to obey zero-order V(I1) Fe(II1) >lob* kinetics with respect to the iron(I1) concentration. Eu( I1 Fe(II1) > I osb The data for this reaction are consistent with firstReaction Tisually observed, initial reactant concenorder kinetics with respect to each reactant, iii- trations were equal and about 5 x 10-3 &f, b Reaction dicating that electron-transfer is rate determining. studied using a flow reactor similar to Gordon and Wah12j and quenching technique used for the iron(I1)-cobaltTherefore, the possibility that the ( t Q + ( t ~ ~ )(111) ~ -the reaction. The reactant concentritions were equal (eg)z transition is rate determining is excluded on and about lo-* d l , the basis that it is not consistent with the observed Acknowledgments.--We wish to acknowledge kinetics. Another possible explanation of the rapidity of the support of this research by the Xational Science the iron(I1)-cobalt(II1) reaction is that it is Foundation. The data found in Table I V were obcatalyzed by the presence of cobalt(II), which tained by L. C. Brown. always is present in these reaction mixtures. In (25) E. A t . Gordon and A. C. Wahl, J . Am. Chem. Soc., 80, 273 this case it might be expected that electron-transfer (1958). M - 1
STUDIES OF ISOTHERR_IAL DIFFUSION A T 25' I N THE SYSTEM WATEB-SODIUbP SULFATE-SULFURIC ACID AND TESTS OF THE ONSAGER RELATION1 B Y R I C H a R D P. T3'E,NDT2 Depurtrrient of Chemistry and the Institute for Enzyme Research, University of Wisconsin, Mudison 6, Wisconsin Recetved December 15, 1961
The isothermal diffusion process at 25" in dilute solutions of the ternary system H~0-NazSO4-II2SO4 has been studied with the Gauy diffusiometer. Values for the four volume-fixed diffusion coefficients which may be used to describe diffusion in a ternary system were obtained at each of four compositions of the system. The cross-term diffusion coefficients responsible for the interacting-flow effect were found to be relatively large and of the same order of magnitude as the mainterm coefficients. Diffusion and density measurements at two compositions of the binary systems HzO-NazSOa and H20HaS04, and partial molal volumes and refractive-index derivatives for thp ternary system, also are reported. Activity data in the literature were analyzed and combined with the diffusion data to test the Onsager reciprocal relation; the relation was found to be satisfied within experimental error at each composition.
Introduction The isot'hermal diffusion process in a threecomponent syst'em can be completely described by two flow equation^^^^ which here are writ'ten for t,he case of diffusion along the x-coordinat,e only 111 these equations the flows (Ji)" are referred to the volume-fixed reference frame and have dimen(1) Portions of this work were submitted as partial requirements for the degree of Doctor of Philosophy a t the University of Wisconsin. sions of moles/(cni.2 see.), the concentrations These studies were presented a t the 140th National Meeting of the n i have units5 of moleslcc., and the four volumeAmerican Chemical Society, Chicago, Illinois, September, 1861. fixed diffusion coefficients ( D i j ) ~have dimensions ( 2 ) Institute for Molecular Physics, University of Maryland, College Park, Maryland. ( 3 ) E. L. Baldwin, P. J. Dunlor,, and L. J. Gosting, J . Am. Chem. Soc., 7 7 , 6235 (1955). (4 G. J. Huoy~nari,I'hysicu, 22, 761 (lY6U).
(5) The concentrations m in eq. 1 and 2 must have units of moLs/cc. if ( J c I v and (DS,Jvare to have the units indicated. However, the concentrations reported for all experiinents in thib work have units of moles/1000 cc.
1280
RICHARD P. WENDT
of cn~.2/sec. The volume-fixed reference frame becomes very nearly identical with the cell-fixed (or apparatus-fixed) reference frame for the small concentration differences in the experiments described I n this paper we report values for the four diffusion coefficients measured with the Gouy diffusiometer a t each of four compositions of the system H20-Xa~S04-H~S04. This system was particularly interesting because the solute R2S04 was expected to behave approximately as it does in dilute binary aqueous solutions,8i.e., it was expected to exhibit the properties of both a strong acid, H2S04 (which dissociates completely into Hfand HS04-), and a weak acid, HSO4- (which only partially dissociates into H+ and s04-). Relatively large values were expected for the cross-term diffusion coefficients (&)v and (Dz1)v because of the appreciable concentrations of very mobile H+ in the solutionss; the measured values for those coefficients a t some compositions in fact were found to be larger than values previously reported for any ternary system. The incomplete dissociation of the bisulfate ion was expected to cause unusual concentration dependence of certain physical properties of the system which could have invalidated assumptions in the procedurelO used to calculate the diffusion coefficients from data obtained with the Gouy diffusiometer. However, for the small concentration differences in these experiments the calculation procedure was found to be applicable. Available thermodynamic activity data for the system, l1-I3 when combined with the diffusion coefficients and density data obtained in this work, enabled tests to be made of the Onsager reciprocal r e l a t i ~ n . ’ ~ JThe ~ Onsager relation was derived by Onsager by using the postulate of microscopic reversibility and is of fundamental importance to the theory of irreversible processes. Our tests of the Onsager relation for this system confirmed its validity, within experimental error, a t each of the four compositions studied. The Onsager relation also has been tested and confirmed from activity and diffusion data obtained a t several compositions of two other systems,10J6J7 HzONaC1-KC1 and H20-glycine-KC1. The tests reported here are the first for a ternary system containing an incompletely-dissociated electrolyte. (6) G. J. Hooyman, et al., Physica, 19, 1095 (1953). (7) J. G. Kirkwood, et at., J. Chem. Phys., 33, 1505 (1960). (8) T. F. Young, et al., in “The Structure of Electrolytic Solutions,” W. J. Hamer, ed., John Wiley and Sons, Inc., New York, N. Y., 1959, pp. 48-59. (9) L. J. Gosting in “Advances in Protein Chemistry,” Vol. XI, Academic Press, Inc., New York, N. Y., 1956, p. 538. Equations 168-172 are only applicable t o systems containing strong electrolytes. but b y assuming the dissociation of HSO4- to be complete, estimates of the diffusion coefficients can be made for the system 1120-NazSOr (10) H. Fujita a n d L. J. Gosting, J. Phg8. Chem., 64, 1256 (1960). (11) H. 8. Harned and It. D. Sturgis, J . Am. Chem. Soc., 47, 945 (1925). (12) G. Akerlbf, ibid., 48, 1160 (1926). (13) M. Randall a n d C. T. Langford, ibid., 49, 1445 (1927). (14) L. Onsager, Phgs. Rev., 37, 405 (1931): 88, 2265 (1931). (15) 6. R. de Groat, “Thermodynamics of Irreversible Processes,” Interscience Publishers, Inc., New York, N. Y., 1958. (16) P. J. Dunlop, J. Phvs. Chem., 6 3 , 612 (1959); see also corrections, dbzd., 63,2089 (1959). (17) L. A. Woolf, D. G. Miller, and L. J. Gosting, J . Am. Chem. Soc., 84, 317 (1962).
Val. 66
Theoretical Definition of Components.-The system H20Na2S04-H2S04is designated a three-component system because the composition at any point in the system can be specified by two independent concentration variables. For this system five constituents, designated by the subscripts1* 0 = HzO, 3 = Ea+, 4 = H+, 5 = SO4-, and G = HS04-, are hypothesized to be present at appreciable concentrations; the dissociation of HzO produces no appreciable concentrations of OH- in the strongly acidic solutions. There are three constraints on the concentrations of these constituents a t each point in the system during the diffusion process. The concentrations Cj (in moles/1000 cc.) and the partial molal volumes (in cc./mole) are related by
c 6
civi
= 1000
i-0 1#1,2
(3)
Another constraint is the condition of local electrical neutrality 6
zici = 0
(4)
i-3
where zi is the valence (including the sign) of constituent i. The condition of local chemical equilibrium for the dissociation of HS04- into H f and SOP- is Q = C&dco (5) where Q is the dissociation quotient for the reaction. Only two of the five concentration variables related by these equations can be independent and the system therefore consists of three components. Although the number of components is definitely fixed by eq. 3-5 the choice of components is arbitrary. Any three constituents could have been chosen as components, but the components H20, Na2S04,and H2S04were chosen because they are neutral and can be separately weighed to prepare solutions for the experiments. Flow Equations.-To describe the diffusion process in this system and to test the Onsager reciprocal relation it is necessary to be certain that the equations for flows of neutral components have the correct form. First the expression for the entropy production will be written to include the forces and flows of the ionic constituents; then that expression will be rewritten in terms of the forces and flows of the two neutral solutes. The forms of the linear laws are obtained by examining the final expression for the entropy production. For the isothermal diffusion process in this system the expression for the entropy production can be written4s6 6
~a =
C
(Ji)Ji
(6)
i=3
Here 2’ is the absolute temperature, r is the local production of entropy per unit volume per second, (JJ0 is the flow of constituent i referred to the sol(18) The subscripts 1 and 2 will be used later to designate NazSOd and I-InSOd, respectively.
ISOTHERMAL DIFFUSION IN
July, 1962
vent-fixed reference frame, and .Xi, the thermodynamic force per mole of i, is defined by
Xi -/- z~FE
.Ti =
(7) (8)
Xi = -3pi/ax I n these equations pi is the chemical potential per mole of constituent i, F is the absolute value of the electrical charge per equivalent of electrons, and E is the local electric field strength. The solvent-fixed reference frame is used because of its mathematical convenien~el~;i.e., by definition (9)
(Jo)o = 0
so the product (J,), 8,does not appear in eq. 6. The required relations between the flows and forces of the neutral components and the ionic constituents now will be written. I n this discussion the subscripts 1 and 2 denote Na2S04 and HzS04,respectively; the subscripts 0,3, . . . , 6 were defined previously. The equations
x1 = 2x3 + xs XZ
f XG 2(Jl)O = ( J d o
2(J2)0
= x 4
=
+
(J4)O
(JG)O
(10) (11) (12) (13)
follow from the definitions of the neutral components 1 and 2. Also XG
=
x4 x6
(14)
because local chemical equilibrium is considered to exist during the diffusion process. From the condition of zero electrical current density during the diffusion process 6 zioini
=0
(15)
i=3
the condition of local electrical neutrality, eq. 4, and the definition of solvent-fixed flows (Jib =
ni(vi
-
110)
(16)
we derive the constraint 6
=
(Ji)oxi
2=3
o
(17)
on the solvent-fixed flows of the ions. In eq. 15 and 16, V i is the velocity of constituent i relative to the cell-fixed reference frame. Equations 7, 8, 10-14, and. 17 are introduced int,o eq. 6 to obtain 2
(Ji)OXi
TfY =
(18)
i=l
Thus eq. 6 for the entropy production can be expressed in terms of flows and forces of the two neutral solutes NazS04 and H2SO4. The linear laws then have the form 2 (Ji)o
(Lij)oXj
=i
(i = 1, 2)
(19)
j=1
according to the thermodynamics of irreversible processes. l5 Furthermore, because the flows (J& and the forces Xj are independent variables, the Onsager reciprocal relation for the solvent-fixed reference f.rame can be written (h2)a
=
(L2l)O
(20)
It has been shown by Rooyman and others4J (19) Ta also can be written for flows referred t o other reference frames; c/. eq. 31, ref. 6,and ea. 8, ref. 17.
THE
SYSTEM H20-NaBO~-HzS04
1281
that equations such as (19) can be transformed to a pair of flow equations for the volume-&xed reference frame. The resulting independent flows (Jl)v and (JZ)v then can be related linearly to the concentration gradients of Na2SO4and HnSOa to obtain eq. 1 and 2, thereby justifying the description of the diffusion process in this system by those two flow equations. Experimental Reference to some earlier descriptions of studies of diffusion in ternary systems1°J7may be helpful in reviewing standard procedures and notation. New procedures used in the present work, and other necessary information, are included here. Materials.-HzO, the solvent used for all experiments, wm prepared by distillation of ordinary distilled H2O from an alkaline solution of KMn04 in a Barnstead Conductance Water Still. The conductance water collected had a specific conductance of 2 X ohm-' cm.-l and was stored in a Pyrex carboy after being saturated with air. Anhydrous Na2S04was prepared by dissolving Baker and Adamson rzagent grade Na2S04.lOHzO in conductance water a t 40 The solution was cooled to 4' and seeded; the precipitated crystals of NaZSO4.10H20 then were collected by centrifugal drainage, heated in a drying oven a t 115' to cpnstant weight of the anhydrous salt, and stored in a desiccator over PzOs. When the NalS04 was transferred into weighing vials t o prepare solutions for the diffusion experiments the desiccator was opened, and the transfer was made in a "drybox" containing- an atmosphere of NZdried over Pz06. Instead of preparing a stock solution of H~O-HZSO~ and then analyzing the solution for H2S04 by gravimetric or acidimetric methods, a stock solution for the diffusion experiments was prepared by weight from conductance water and 100.00% pure H2S04. KunzleP has concluded that the mixture H&S03 having a maximum freezing point of 10.371' is 100.000% H2S04. Two batches of HzSO4 were prepared at different times by mixing solutions of slightly SO,-rich and slightly HzO-rich distilled H z S O in ~ an air-tight 300-ml. freezing-point cell ~imilar to that described by Bunzler. All of the glass apparatus used in the preparation and storage of the acid wa? fabricated of Pyrex brand glass No. 7740, and all ground glass joints were either lubricated with concentrated 11~50~ or fitted with Teflon sleeves. The maximum freezing points of batches 1 and 2 were 10.358 and 10.367', respectively, as determined by an American Instrument Co. platinum resistance thermometer and a Leeds and Northrup Type G-2 Mueller temperature bridge, both of which had been calibrated recently by the National Bureau of Standards. By assuming th?t SO2 was the impurity causing the freezing point depressions of 0.013 and 0.004', and using the cryoscopic constant2' of 6.0 (deg. kg. solvent)/(mole solute) for solutes in the solvent H2S04,the weight per cents of SO2 in batches 1 and 2 were calculated to be 0.021 and 0.007%, respectively. These estimated amounts of impurity were so small that both of the HaO-SOa mixtures were taken to be 100.00% HzS04. For each preparation the batch of HzS04mas transferred from the freezing-point cell to a weighed 2-1. Florence flask through a delivery tube so that no moist air came in contact with the acid. The flask and acid were weighed and conductance water was carefully added to the acid to prepare a stock solution of moderately dilute &SOP. After the flask and stock solution were weighed an automatic buret fitted with a ground-glass joint was inserted into the flask. The buret had been modified so that all air entering the flaskburet assembly first bubbled through a trap containing 30 ml. of the stock solution. From the weights of the stock solution and the 1 0 0 . 0 0 ~ HzSO~ added to the flask, the compositions of batches 1 and 2 of stock solution were calculated to be 44.873 and 48.330% &so4 by weight, respectively. A sodium carbonate analyeis of batch 1 agreed, within the 0.08% relative precision of that analysis, with the calculated weight per cent of HzS04 in the freshly prepared stock solution. Densities
.
(20) J. E. Kunsler, Anal. Chem., 25, 93 (1953). (21) R. J. Gillespie, Rsu. Pure and A p p l . Chsm., 9, 1 (1959).
1282
Vol. gti
of dilute solutions of the stock solution also m r e measured shortly after its preparation. Thereafter, analyses of batches 1 and 2 were made by density measurements. The relative change in the concentration of batch 1 detwmined from these measurements was 0.15% over a period of eight months and the relative change of batch 2 n-as 0.05% over a period of seven months. Because the relative precision of the density analyses was approximately the same as the changes in the stock solution concentrations, no corrections were made and the concentrations originally calculated from the weights of pure acid and stock solution were used for all experiments. The slow changes of the acid concentrations with time are not believed to have influenced the magnitudes of the diffusion coefficients reported here because those quantities showed a relatively small dependence on concentration. Preparation of Solutions.-Two solutions A and B, having eoncentrations ( C I ) ~ (Cz)a, , and ( CI)B, ( CZ)B,respectively, were prepared for each diffusion experiment. Solution B, with a density ( d ) > ~ ( $ ) A , was present in the lower portion of the diffusion cell during the siphoning process for sharpening the initial boundary, and solution A was in the upper portion of the cell. It was desirablejo prepare solutions A and B so that the concentrations C,, where
ments, the coefficients H , and If, in eq. 34 verc sufficiently m-ell-determined so that densities for the remaining two experiments at each composition could be predicted to \Tithin 0.0055%. Densities of the solutions were measured in triplicate in single-stem pycnometers, each holding approximately 30 ml. of solution. The percentage deviation from the average of the three densities measurcd for each solution mas approximately O.OOl%, and over a period of several months the volume of each pycnometer stayed constant m-ithin approximately 0.001 To calibrate the pycnometers the density of mater at 2~ was taken to be 0.997044 g./cc. The observed weights of solutes and solution were corrected to weights in vacuo by using densities of 8.4, 2.66, 1.36, and about 1.12 g./cc. for the brass \?-eights, the anhydrous Sa2S04,the stock solution, and the prepared solutions, respectively. Diffusion Experiments .-The Gouy diffusiometer described previouslyz2 was used for this work without modification. Two different quartz Spinco Rlodel H electrophoresis cells mere used for the diffusion experiments. Cell 1, used for experiment I, had an a distancc of 2.5092 em., and the h distance with this cell in the diffusiometer was 306.67 cm.; cell 2, used for all other experiments, had an a distancc of 2.5075 em., and the b distance corresponding to this cell was 306.64 cm. Exoeriments for the binarv svstems I-120-Sa2S04 and " yere \Tithin approximately 0.05% of the concentrations HzO-H~SQ~ were performed near the compositions c" = C1 and which specified the compositions of the ternary 0.5 and = 1.0 for each system. For the ternary system solutions. The actual concentrations of the solut,ions were I-TzO-SanSQ4-H2S04 four experiments were perfoJmed near known within approximately 0.002%. Deaired values for each of the composition points A ( Ci =_0.5, CZ = 0.5), ACi, where B - ( ~ =I l.?, Ca = 0.5), C = 0.5, C2 = l . O ) , and D ACi = (Ci)B ( C i ) A (22) ( C , = 1.0, C? = 1.0). were calculated according to eq. 39 and 41 of ref. 17 for For experiment I and for the experiments a t composispecified values of the refract,ometric fraction, 011, by re- tions A and B a stainiess steel capillary, electrolytically quiring 40 < J < 80. The number of fringes, J , for the ex- coated with platinum to prevent chemical attack by t'he periment was made as small as possible without seriously acidic solutions, was used to form t,he boundary between the reducing the precision of the measurements, so that the two solutions in t'he diffusion cell. A capillary fabricated effects of concentration dependence of the measured proper- of 80% platinum-20% iridium hard-temper tubing by J. ties of the sy_stem would be minimized. From the values Bishop and Co. xTas used for t8heot8herexperiments. specified for Ci and ACi the desired values for (Ci)h and ( C ~ ) B For all experiments in this work, monochromatic light of were calculated according to eq. 21 and 22. wave length X = 5460.7 A. in air, emitted from a GE HTo determine the relative amounts of each component 100A4 inercury vapor lamp and transmitted by an Eastmm required for t,he solutions, aeight per cents wi tn vacuo Kodak Wratten filter 77A, illuminated the source slit of the corresponding Lo the desired values for Ci, were calculated diffusiometer. All photographs were taken on Eastman from Kodak Kodaline C.T.C. Pan plates xith anti-halation backing. The temperature of the water-bath in which w,d = CiMi/lO (23) the diffusion cell was immersed was within ~ t 0 . 0 0 8of ~ Here d is the solution density (in g.,/cc.), Ci is the con- 25 .OOO" for all experiments and temperature variations were centration of component i (in molesjlOO0 cc.), and Mi is no greater than i:0.005' during any experiment. the molecular weight of i. S o density data for the system The positions of fringe maxima and minima were deH20-NazSOe-HzS04are available in the literature at the concentrations required. Therefore, before performing termined with a newly const,ruct,ed photoelectric null-indithe three-component diffusion experiments a preliminary cator mounted on a Gaertner Model M2001RS toolmaker's solution at each composit~ion was prepared as follows. microscope. The construction of a similar device recently From density data for the binary systems H20-Na2S04 vas reported by Bennett and Koehler.2a Our apparatus and H20-H2SO4(Table V) an estimate was made of the con- contained a rotating glass plate having two parallel light centration dependence of the densities of the binary systems. transmitting surfaces instead of the four-surfaced plate By assuming that this dependence on C1 and CZwas the same used by those authors, and we did not find it necessary to for the ternary systems an estimate was made of the density install a light chopper as they did.24 By using the nullindicator, positions of interference fringes on the photoof the preliminary solution at the composition CI, C1. graphic plates could be reproduced to within approximately Equation 23 then was used to calculate the estimated 5 ,U for fringes having broad maxima and minima and to values for w1 and w2. To prepare the solution a weighing within approximately 1 p for fringes having narrow maxima vial containing KazS04 was emptied into a 300-ml. erlen- and minima. These uncertainties are less by a factor of meyer flask. From the known weight of T\;a&D4and the approximately two than the estimated uncertainties in estimated values for and w ~ the , required x-eights of visual determinations of fringe maxima and minima. For stock solution and HzO mere calculated and added to the each of the 7 or 8 Gouy fringe photographs taken during flask. By using a weight buret it was possible to add, an experiment, the positions of approximately 15 fringe within approximately 4 mg., R predetermined amount of minima throughout the fringe pattern and 5 lower-numbered stock solution to the flask. The mTeights of stock solution fringe maxima were measured. added, and the weights of the SanSOcand of the prepared Horizontal Rayleigh photographs26 were taken during the solutions, actually were known to within about 0.1 mg. siphoning process to find the fractional part of J , the number The density measured for the preliminary solution was used to estimate d( cl, 62)in the density expression (22) P. J. Dunlop and L. J. Gosting, J . Am,. Chem. Sac., 1 7 , 5238 (1955). d = d ( d i , 62) Hi(C1 - 61) H2(C2 - 6 2 ) (24) I.
cz
(e,
-
+
+
for ternary solutions with small values for Ci - Ci. The densities of the solutions for the diffusion experiments then could be estimated to within about. O . O S ~ o . After
the densities of solutions prepared according to the above procedure had been measured for two diffusion experi-
(23) J . 31. Bennett and W, F. Koohler, J . O p t . SOC.Am., 49, 466 (1959). (24) A more detailed description of the null-indicator is given by R.P. Wendt in his Ph.D. thesia, U. of Wisconsin, 1961 (1). C. Card No. l\Iic 61-680). (25) 12. J . Gosting, et al., Rri,. Sei. Insti.., $ 0 , 209 (1948).
1283 of fringes for the experiment, and Rayleigh interferogramsz6 were used t o determine the integral part of J.
Calculations Diffusion Coefficients.-A procedure developed by Fujita, and Gostinglo was used to calculate the four diffusion coefficients, ( D i j ) ~ , from the Gouy diffusiomieter data obtained a t each composition of the system. This procedure requires data from at least two diffusion experiments a t different values of al,the refractometric fraction of component 1. At each composit'ion in this work data were obtained a t four different values of 011 of approximately 0, 0.15, 0.85, and 1.0 to provide checks on the internal consistency of the data and increase t'he accuracy of the results. Values for the quantities required to calculate the diffusion coefficients were obtained from the data shown in Table I. Determinat'iori of the number of fringes, J , was described in the previous section, a,nd values for ACi were calculated according to eq. 22. The reduced height-area ratio, DA,is defined by eq. 35 of ref. 22. It is necessary to find preliminary values2' :oA' for D)~,corresponding to the times t' after the boundary sharpening process is stopped. The measured positions of the Gouy interference fringes were used to calculate DA' for each fringe photograph according to the following procedure. Except a13 indicated, all calculations were performed by a Bentlix G-15 computer using programs written for the Intercom 1000 (double-precision) programming system. Fringe displacements Yj of fringes number j were calculated from measured values for the positions of the fringe minima and maxima and the corrected position of the undeviated slit image. The quan.titiesZsZj and ddj were calculated by using the appropriate equationsz9to find a, and as', the zeros and turning values of the Airy int]egral; f(Sj) then was calculated from eq. 11 and 12 of ref. 28. Values for e-rj' were computed by using a Newton-.Rhapson iteration process. The computer then calculated and printed out values for Y./e-"', 'A graph of Yjle-rj' us. Zj2/r (or Mj2/3) should be linear30 for values of Zj2i8(or Mj2/3)near 0, and the intercept at Zj2/3(or Mj'9 = 0 should equal the maximum displacement of light according to ray optics, Ct, a quantity necessary to calculate DA'. We tried to use the method of least squares to determine Ct by calculating the coefficients in expressioiis for Yjle-rj' as a linear funct,ion, a quadratic! function, and a cubic function of Zjp/8, Fringe displacements Yj for about 15 of the first 30 fringes were used for these calculations, because Zj2l3and Mj'/a become small as j + 0. Values for Ct obtained from these equations were in general significantly different from each other and were (26) J. >I. Creeth, J. Am. Chem. Soc., 77, 6428 (1955). (27) Equation 35, ref. 17. (28) L. J. Gosting and & S.!Morris, I. J . Am. Chem. SOC.,71, 1968 (1949). (29) "The Airy Integral," British Association for the Advanceinsnt of Science Mathematical Tables, Part-Volume B, University Press, Cambridge, 1946, p. B48. The subscripts s in equations there correspond t o fringe numbers j 1 in our notation. (30) See Appendix. ref. 10.
-+
m
I
RICHARD P. WEXDT
1284
Vol. 66
found to differ by 1% or more from values for Ct of ref. 10. To perform the numerical integrations obtained by the graphical method of extrapolation the fringe deviation graphs were divided into 40 to Zj’” = 0. The differences between the evenly-spaced segments from f({j) = 0 to f({j) = 1. graphical and calculated values for Ct were especiRefractive index derivatives R, in Table I1 are a,lly large for experiments with large values for the defined by the equation area of the fringe deviat,ion graph for each experi- 6,) (25) n = 451, 62) d- R ~ C I PI)t ment, Q, probably because the limiting slopes of the graphs of Yj/e-bj* vs. Zj*/a also were large for where n is the refractive index of the solution at these experiments. Because the graphs were composition C1, Cz and % ( e 1 , 6.J is the refractive visually j udged not to behave as lomer-order polj7- index of a solution a t composition The nomials in Zjz/sexcept for small values of j , we derivatives were computed by the method of least decided not to use the method of least squares to squares according to eq. 41 of ref. 17. Values for determine Ct for any experiments in this work. the refractometric fraction, q, were calculated Values for Ct obtained by the graphical method were from eq. 39, ref. 17: the method of least squares entered into the computer along with the cor- then was used to calculate I A and XA in Table I1 responding values for t’, and the method of least from values of DA and cy1 in Table I according to squares was used to calculate the c o e f f i ~ i e n t s ~ ~eq. t ~ ~15, ref. 10. The coefficients Eo, El, and Ez DA and At in the linear equation for D A ’ vs. l/t’. in Table I1 were calculated from eq. 22, ref. 10, by Values for At were less than 20 sec. for all experi- the method of least squares, during the iteration ments except 9, where t = 114.68 sec. This large process used to calculate the diffusion coefficients. At apparently produced no unusual uncertainty in The quantity P which appears in the iteration DA. For every experiment the average deviation process was computed directly from eq. 20, ref. 10. of the observed values for DA’ from the values This equation was found to be as accurate as the calculated from the linear equation containing equation in footnote a of Table I, ref, 10,for finding P DA and At was less than 0.2% of DA. Values ob- when p < 0,9999. The entire computation was tained for DAat the temperature of the experiment stopped when two successive iteration cycles were corrected to 25.000O by using a series form of produced values for each diffusion coefficient that the Stokes-Einstein relation. 33 differed by less than 0.000001 X Chemical Potential Derivatives.-Values of these TABLE IIaVb derivatives, of the ( Q j ) ~ ~and , of the partial molal REFRACTIVE I K D E X DERIVATIVES, PARTIAL MOLAL \‘OLvolumes vi are required to calculate the phenoUMES, AND OTHER PHYSICAL PRQPERTIES OF THE SYSTEM menological coefficients (Llj)o and to test the HzO-NazSOpHzSO, AT 25” Onsager reciprocal relation a t each c o m p ~ s i t i o n . ~ ~ (0 = 13~0,1 = Sa,zS04, 2 = HzSO4) For these computations the derivatives
el, e,.
Coinpositioii
k CZ
A 0,50000
RI X 108 Rz X 108 IA SA
Eo El
E* diCi, ~
Hi I12 170
1‘1 T.2
ml mz mu mlz
mp1 rnzz
22)
0.50000 15.637 7.962 277.71 63,87 4.94074 -2.38804 3.96671 1.08331 0.11164 0,05409 18.0080 30.40 43.97 0.51908 0.61908 1.05455 0,02370 0.01639 1.06186
B 1,00000 0.50000
15.319 5.661 352.37 15.37 2.04391 1.97630 0.21178 1.13829 0.10942 0.04830 17.9314 32.48
-
49,54
1.05674 0.52787 1.09211 0.05548 0.01819
1.08349
C 0.50000 1.00000 13.337 8,961 242.92 102.55 3.90942 3.70018 10.29267 1.11107 0.10464 0.05644 17.9745 37.32 41.54 0.53081 1.06161 1.08269 0,02347 0.04216 1.10854
D 1 00000 I 00000 13.365 7.123 282.34 81.04 6.08065 -I., 35689 6.06297 1.16334 0.10414 0.05161 17.8802 37.62 46.12 1.08318 1.08318 1.12764 0.05453 0.04447 1.13770
Units: concentrations Ci, moles/1000 cc.; refractive index derivatives Ri, 1000 cc./mole; partial molal volumes Vi, cc./mole; molalities ml, moles/1000 g. HzO. Molecular weights used: Mo = 18.0160, .MI = 142.0480, MI = 98.0820.
sure P , were needed to calculate the quantities Aik
(31) L. G. Longsworth, J . Am. Cham. Soo., 69, 2610 (1947). (32) H. Fujita, J. Phys. SOC.Japan, ii, 1018 (1956). (33) Equation 37, ref. 17.
(27)
= (b/-4i/alnk)T,P,mq$k
where ,u, is the chemical potential of component i. Values for the derivatives were found by using eq. lnkJ = (bmk/bCj)T,P,c,+,,o
(k,j
1, 2) (28)
8, ref. 34, and the density data were obtained in conjunction with the diffusion experiments. The chemical potential derivatives pij then were calculated from
E.m.f. measurements for the cell
HZIHiS04(nd, NazSOdrnl)l Hg2S04- Hg
a
The area, &, of the fringe deviation graph for each experiment is a measure of the deviation from gaussian shape of the diffusing boundary; it was calculated by using Simpson’s rule to integrate numerically a smoothed graph of the relative fringe deviation s t j us. f({j), according t’o eq. 9
= [ b log ( y * ) l / b m l ~ , ~ , m q i k
(i, k = 1, 2) (26) of the activity coefficients (yk), with respect to molality mk, at constant temperature T and presrlk
(30)
were used by B a e ~ 3 to ~ calculate values for log(^*)^ from the equation log
(y*)z =
(E” - E)/0.088725
- (1/3)
log (4mz2m)
(31)
+
where Eo = 0.61513 volt and m = nzl m2. He obtained an empirical expression for log(y& as a function of ml and m2 which we have rewritten as (34) P. J. Dunlop and L. J. Gosting, J. Phus. Chem., 63, 85 (1959). (35) C. F. Baes, Jr., J . Am. Cham. Soo., 79, 5611 (1957).
ISOTHERMAL DIFFUSION IN
July, 1962
log (?*I,* - mO.6233 [0.2305 - 0.3355(m2/m) O.lO5O(m~/m)~](32) Here log ~ ( yis~the ) ~value of log (y& for the bi-
log
( y * ) z =:
+
nary syslem H20-H2S04 at a concentration m of H2S04. By differentiating eq. 32 three of the required derivatives rl, are obtaiiied
rzz(kl, k 2 )= rZ2*(h) + fi~-0.3~~~[0.1918 - 0.1264(&~/%)
+
-
rz2*= [a log ( ~ , ) ~ * / a m=l ~ ~ [b log ( ~ * ) z * / a m z lP~ = [blog (r*)z*/3mil~P (36)
which follow from the definition of log (y+!**. Equation 35 is derived from the equations defining the activity coefficients fii
= /I 0
TABLE 111. ACTIVITYCOEFFICIENT DERIVATIVES FOR H20-Na2SO4-HzSO4AT 25’ (1 NazSO,, 2 HzSOa)
+
In these equations we have used the identities
+ RT In [4m3n(Y*)?]
(i
=
1, 2) (37)
where R LS the gas constant in ergs/mole-deg., and from the cross-differentiation ru’e Aiz =
(38)
A21
The derivatives r13are evaluated a t the molalities nil and k2corresponding to the composition-point as calculated from eq. concentrations Cl and 7 of ref. 34. Equation 35 is integrated with respect to m2 and differentiated with respect to ml to find the fourth required derivative
cz,
m2 = h a
(39)
I n this equation
rll”= [a log ( ~ * ) ~ * / a m ~ l ~ , ~(40)
where log (Y*)~* is the value of log (r*)lfor the binary system H20-Na2S04 a t a concentration ml of Na2S04. Equation 34 is inserted into (39), and by using a tab’e of integraW and identity (36) we obtaiii
+
-
rz2*(77a) r22*(h1)rll*(h1) 0.7356m1-0.3’67 i l ~ - O . 3 7 6 7 [1.6314 - 1.7099(ki//7b) 1.0637(~76i/k)~ - 0.2496(fii/77~)~](41)
rl,(;a1,a,)=
+
+
The derivatives rll*and I’22*were obtained by differenthting the Taylor series
1. 2. 3. 4. 5.
Composition
6.
YA. 7B.
adm1 -
=
4
log
(y*)z*
b,(mz
=
r=O 4
log (Ti),* =
- 0.3) 6 ml - L)?
C
D
rZ2*(ljll) 1’11
-0,0860
-0.0886
rii
-0.1996
-0.OY4Z
-0.2329 -0.1059 -0.5586
-0.08S0
-0.0867
-0.OS61
-0.0621 -0.0506
-0.0867 -0.1648
-0.0361
-0.0305
-0.lSS6
1‘12
-0,0830 -0.8517 -0.8655
-0,1619
-0.1456
9A.
T22
-0.2568 -0.2483 -0.oos8
9B.
l-22
-0.0009
-0.1667 -0.1575 0.1080 0.1048
-0,0880 -0,1019 -0.0872
-0,1560 -0.1377 -0.0957 0.0~08 o.1iir 0.0259 0.1098
m2
rl1*m rS2*(&)
8C. I’ll
7 ~ rll . 8A. I’iz ( =
r21)
( = rzi) 8C. ria ( = r2i) 8D. riz ( = T21)
8B.
QC. r12
0.0970
0.0165
0.53081 1.06161 -0.3lOj -0.0068
0.0881
1.08318 1.08318 -0.1631 0.0341 -0.0566 0.0175
0.0947
rn2 0.0041 0.1033 0,0274 0.1054 The letters A and B designate values for rij calculated from eq. 47-49 with n = 3 and n = 4, respectively: C designates values for I’ij calculated from eq. 33, 34, and 41; D designates accepted values for rij. SD.
To confirm 13aes’ eq. 32 and obtain estimates of the accuracy of the activity coefficient derivatives calculated from his equation we performed a separate analysis of the e.m.f. data reported11-13 for the cell in eq. 30. Values calculated for log (y& from eq. 31 were graphically interpolated t.o obtain 168 evenly spaced points on the ml, m,2 plane in the regionbounded by0.2 6 ml 6 1.3andO 6 m2 6 !.3. The coefficients (d& and (era)n in the following four Taylor series then were calculated by the method of least squares a t each of the four compositions &, h2for which diffusion data had been obtained
cc n
log (r*)2 =
n
(drAl(m1
- 770(m2 - f i z p
r=O s = O 0 6 (r+s) n
