1291
DIFFUSIONDATAFOR THE SYSTEM WATER-SUCROSE-MANNITOL
Dihsion and Frictional Coefficients for Four Compositions of the System Water-SucroseMannitol at 25
O.
Tests of the Onsager Reciprocal Relation'
by H. David Ellerton2and Peter J. Dunlop Department of Phy& and Inorganic Chamiatry, Univerrity of Adelaide, Adelaide, South Australicr (Received September 8, 1066)
Ternary diffusion, refractive index, density, and relative viscosity data have been measured for four compositions of the system water-sucrose-mannitol at 25'. Frictional coefficients, Ro,computed from the diffusiondata and the thermodynamic data of Robinson and Stokea, indicate that, within the error of measurement, the Onsager reciprocal relation is valid for this system. Phenomenological coefficients, (L,)v, have also been computed for this system and compared with those predicted by the theory of Lane and Kirkaldy. Diffusion, density, and refractive index data are also reported for the systems watersucrose and water-mannitol.
This paper reports ternary diffusion data which were determined with the Gouy diffusiometer for four compositions of the system water-sucrose-mannitol at 25' in order to test the Onsager reciprocal relation (ORR)a for viscous solutions of nonelectrolytes. The ORR wm tested by using these diffusion data and the thermodynamic data of Robinson and Stokes' to compute frictional coefficients6for each composition of the system. The theory which is necessary to compute these frictional coefficients has been already given in a previous paper.6 Here we use the same notation as in that article. Occasionally, equations in that article will be referred to here and identified by the letter A. Diffusion and refractive index data are also reported for several compositions of the binary systems watersucrose and water-mannitol. Phenomenological coefficients, (L,,)v,Tfor the volume-ked frame of reference have also been computed from the above ternary data and are compared with values computed by the theory of Lane and Kirka1dy.B
Experimental Section Materials. The sucrose and mannitol samples used in all experiments were British Drug Houses microanalytical reagents and were used without further purification. The binary diffusion coefficients and refractive index derivatives obtained with these samples agreed with experimental error (=tO.l%) with the literature values.6sg
All solutions were prepared with doubly distilled water. The molecular weights*O of water, sucrose, and mannitol were taken as 18.015, 342.303, and 182.175, respectively, while the corresponding component densities which were used to reduce all weighings to vacuum were 0.997048, 1.588,and 1.489 g cm-8, respectively. The correct procedure for reducing weights to those under vacuum is described in a Mettler publication.'l All solutions were prepared by weight, and the solute concentrations in moles per 1000 ema, e,, were calculated by means of the solution (1) This investigation was supported in part by research grants from the Colonial Sugar Refining Co. Ltd. of Australia and the University of Adelaide. (2) Department of Chemistry, University of California, Berkeley, Calif. 94720. (3) L. Onsager, Phye. Rev., 37, 405 (1931); 38, 2265 (1931); Ann. N . Y . Acad. Sci., 46, 241 (1945). (4) R. A. Robinson and R. H. Stokes, J. Phys. Chem., 65, 1954
(1961). (5) P.J. Dunlop, ibid., 68, 26 (1964). (6) P. J. Dunlop, ibid., 69, 4276 (1965). (7) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G.Kegeles, J. Chem. Phys., 33, 1505 (1960). (8)J. E. Lane and J. 8.Kirkaldy, Can. J. Chem., 43, 1812 (1965). (9) L. J. Gosting and D. F. Akeley, J. Am. Chem. SOC.,75, 5685 (1953); A. Chatterjee, iba., 86, 793 (1964). (10) Using atomic weights compiled in International Union of Pure and Applied Chemistry, Information Bulletin No. 14b, 1961. (11) Mettler News (1-22)E,1961,p 81. This publication is available from local Mettler agents.
Volume '71,Number 6 April 1067
1292
H. DAVIDELLERTON AND PETER J. DUNLOP
~
Table I :' Diffusion Data for the Binary Systems at 25" System
Water-sucrose Water-sucrose Water-sucrose Water-mannitol Water-mannitol
e
A@
J
0.020502 0.250051 0.500026 0.250107 0.500127
0.041004 0.043497 0.033042 0.078960 0.081446
92.10 97.48 73.94 93.74 96.19
Qx
D X 1V
48.979 48.867 48.797 25.887 25.770
0.5180 0.4588 0.4007 0.6146 0.5668
104
-1.0 0.0 0.0 -1.0 -1.9
Units: concentrations @, moles 1000 cm-*; diffusion coefficients D,cma sec-l; refractive index derivatives (An/Ae), 1000 cms mole-', referred to the velocity of light in air at standard temperature and pressure and for wavelength 5460.7 X 10-8 cm in air.
densities, p, in grams per cubic centimeter, and the solute molecular weights. The solution densities were determined in triplicate or quadruplicate with matched 30-cm8, single-neck Pyrex pycnometers. Apparatus, Experimental Quantities, and Computalions. The Gouy diffusiometer used for all experiments has been adequately described previously.12 All of the experimental quantities used in this paper have been defined previously.1a This paper should be read in conjunction with that article's which also gives a description of the method of performing the diffusion experiments and the calculation procedures for obtaining the final results. The same fused-quart5 cell, QC-1, was used for all experiments: its thickness, a, along the light path was 2.5043 cm and the optical lever arm, b, for this cell was 305.18 cm. All experiments were performed at 25 =F 0.005', and the diffusion coefficients were corrected to 25.000~by means of the Stokes-Einstein relation. The relative viscosity of each solution used in the ternary diffusion experiments was measured with an Ubbelohde viscometer and a small kinetic energy correction applied to each result.
'
Results Binary Data. Several binary diffusion experiments were performed with the systems water-sucrose and water-mannitol. The data obtained are reported in Table I. Each value of 6 is the average of the concentrations of the two solutions which were used in each experiment, while each A e is the corresponding difference between these two concentrations. The total number of fringes, J, obtained in each Gouy experiment together with the corresponding refractive index derivatives, (An/Ae), and diffusion coefficients, D , are also included in the table. The Q values are the areas of the graphs of the relative fringe deviations, a, vs. the reduced fringe numbers f(t).l8 If the Gouy apparatus is optically perfect, the magnitude of Q is a measure of the deviation of the refractive index gradient curve from gaussian shape. The Journal of Physdcal Chemietry
Binary diffusion coefficients for sucrose have been reported for very dilute solutions by several w ~ r k e r s . ~ All of these results agree with the data reported here. At higher sucrose concentrations, results obtained with the diaphragm cell have been reported by Irani and Adamson14 and also by Henrion.15 The former results at e = 0.25 and e = 0.50 are approximately 4 and 875, respectively, lower than our results, while the data of the latter authors are approximately 3.5% higher. It is difficult to explain this disagreement. The binary diffusion data for mannitol agree with data previously reportedla but disagree with other data discussed in that article. Nine density measurements were made with watersucrose solutions. The following equation represents the experimental values with an average deviation of =k0.0006%. p =
0.997048
+ 0.131562e - 0.001622C2 (e Q
0.9)
(1)
Ternary Data. All of the pertinent experimental data for the four compositions of the system watersucrose-mannitol studied in this work are reported in Tables I1 and 111. Throughout this paper, 0 is used to designate the solvent water while 1 and 2 are used to designate the solutes sucrose and mannitol, respectively. Lines 2 and 9 of Table I1 give the concentrations, e,, and the corresponding densities, p, and relative viscosities, of the two solutions A (upper) and B (lower) used in each diffusion experiment, while lines 10 and 11 give the arithmetic averages, of these concentrations for each component. The total number of
e,,
(12) H.D.Ellerton, G. Reinfelds, D. E. Muloahy, and P. J. Dunlop, J . Phys. Chem., 68,403 (1964). (13) L. A. Woolf, D. G. Miller, and L. J. Gosting, J. Am. Chem. SOC.,84, 317 (1962). (14) R. R. Irani and A. W. Adamson, J . Phys. Chem., 62, 1517 (1968). (15) P. N. Henrion, Trans. Furuday SOC.,60,72 (1964).
DIFFUSION DATAFOR
THE
SYSTEM WATER-SUCROSE-MANNITOL
1293
3
m bi 0
8 0 m YYY OffiW
3
a m
0
U
"
hl M o
e
3
3
AOOln a
d0dr;dd
"00r;dO
33
P-
3
P-
m
i?
8
3
3
8
P-
d
33
m
4
ffi
Volume 71, Number 6 April 1067
H. DAVIDELLERTON AND PETER J. DUNLOP
1294
fringes, Jexptl, used for each experiment is listed in the table together with the corresponding values of Jcrrlcd which were computed from the concentration differences, A&, and the two refractive index increments 6il and (Rz which were calculated by the method of O'Donnell and Gosting.'* The values of al,the fraction of the total refractive increment between solutions B and A contributed by component 1, are listed in line 14. The experimental and computed height: area ratios, DA, are listed in lines 15 and 16 while lines 17 and 18 give the experimental and computed areas of the fringe deviation graphs for each experiment. It is believed that each Qexpt1value is known to approximately *2.0 X Lines 19 and 20 report certain intermediate quantities which are obtained in the calculations for the (D& values.
and (5)
Values of p(&, &), XI, XZ,S&I, &), 371, and XZ are reported in Table IV. Also included in the table are the average deviations between the experimental and calculated values for each composition of the system. The X t values were employed to calculate1*the partial molar volumes, vi, listed at the top of the table. Figure 1 gives a graphical representation of PI and V2 as functions of el and ez. The binary data were obtained from eq 1 and the literature.6 Also included
Table IV:" Diffusion and Frictional' Coefficients Table 111:" Partial Molar Volumes and Differeutial Density, Relative Viscosity, and Refractive Index Derivatives for the Ternary System at 25'
el e2
Vo Vl
5 p(&,
et)
X1
XZ
% 9ev~,(ei,ez) 371 372 %dev X 10' 6iz X 10' (R1
0.25 0.25 18.064 212.38 120.25 1.045434 0.13051 0.06226 40.0002 1.4500 1.5892 0.8633 40.09 48.781 25.768
0.25 0.50 18.059 213.04 120.56 1,060951 0.12978 0.06191 40.0012 1.6835 1.7669 1.0667 zk0.07 48.592 25.675
0.50 0.25 18.055 213.37 120.80 1.077889 0.12941 0.06164 40.0007 1.9210 2.3401 1.2590 10.02 48.608 25.602
el e2 (DllIV x 106 0.50 0.50 18.047 213.92 121.07 1.093261 0.12876 0.06138 j=0.0007 2.2740 2.7828 1.6487 f O .17 48.466 25.574
differential density increments Xi, lo00 g mole-'; O Units: differential relative viscosity increments 'Xi, 1000 cm* mole-'; differential refractive index increments (Rj, lo00 cma mole-'.
Stability criteria" which were applied to each ternary diffusion experiment indicated that the diffusion boundaries were gravitationally stable. The method of least squares wm used to express the densities and relative viscosities of the solutions used for each composition of the system by means of the truncated Taylor series P = P(&,
Ed
+ %del- + El)
%(e2
- EZ)
(2)
- E2)
(3)
(DlZIV
x
I@
(DdV
x
lo6
(Da)v x 106
R,~ x 10-17
0.25 0.25 0.4021 'F0.0044 0.0237 FO.0025 0.0441 *0.0103 0.4962 *0.0057
(mro)2 x 10-17
(6Rm)l X lo-" (6Rm)z X lo-'' A% (exptl) A% (calcd)
0.4621 10.0050
0.50' 0.50 0.3027 Fo.0029 0.0381 10.0014 0.0754 '~0.0069 0.3656 70.0036
1.401
1.582
1.851
10.002
10.025 +0.002
3=0.017 3=0.003
3=0.011 10.004
9-46
'F1.11 ( 6 ~ x~10-1' ~ ) ~~ 0 . 1 0 13.12 R81X lO-I7 (8Rz1)t X lo-'' f4.69 ( 6 ~ x ~10-17 ~ ) ~10 12.
R% x 10-17
10.mo
0.50 0.25 0.3499 F0.0032 0.0416 F0.0018 0.0412 10.0077 0.3964 10.0043
1.227
( 6 ~ x~10-17 ~ ) ~f0.019
R~~x 10-17 (6Riz)i X lo-''
0.25 0.50 0.3557 10.0038 0.0206 F0.0022 0.0689
0.927
9.70
12.22
14.29
F1.13 ~0.11
10.55 ~0.12
?cl.lo ~0.13
10.40
17.74
19.92
h2.42 TO. 13
h4.83 TO. 15
h3.64 TO. 14
1.052
70.033 10.023 10.002 10.001 -7.0 -32.4 453.3 137.7
1.141 F0.060 3=0.003 -36.8 3237.7
1.298 10.018 10.004 -32.9 129.3
" Units: concentrations ej, mole 1000 cm-8; diffusion coefficients (Dij)", c m z sec-1; frictional coefficients R j k , ergs cm sec mole-1. The frictional coefficients are independent of the frame of reference used to measure the ( D i j ) ~ .E The errors in the (Dij)~and R i k for this composition of the system were computed by adding 2.0 x 10-4 to the positive Q vdues and -2.0 X 10-4 to the negative Q values. This procedure is just as valid as either adding or subtracting 2.0 X 10-4 to all Q values.
*
and rlr =
s,(&, E2, +
- El) +
and the differential density and relative viscosity derivatives, X t and Xi are defined by the relations The Jour& of Physical Chemistry
(16) I. J. O'Donnell and L. J. Gosting, "The Structure of Electrolytic Solutions," W.J. Hamer, Ed., John Wiley and Sons, Inc., New York, N. Y., 1969. (19~).
(18) P. J. Dunlop and L. J. Gosting, ibid., 63, 86 (1959).
DIFFUSION DATAFOR THE SYSTEM WATER-SUCROSE-MANNITOL
v2
t
f121.0
21L.O
1295
values in Table IV can be represented within the error of measurement by the two equations
( D U )X~ 106 = el(O.0987 - 0.0388%) (D21)y X 106 = a(0.1564 0.0028ei)
f 213.0
.f 212.0
(6)
(7)
which also satisfy the necessary restriction^'^ J g that (D1*)v 0 as el -c 0 and (D& -+ 0 as (% +0. Figure 2 summarizes all the (Drj)vvalues reported in Table 11. Also included in the figure are the results of Mills and Ellertonm for tracer diffusion coefficients of sucrose in aqueous solutions of mannitol and the tracer diffusion coefficients of mannitol in aqueous solutions of sucrose. These points are indicated by solid circles. -+
R1= 103
Rzs
lo3
Testa of the Onsager Reciprocal Relation One way of testing the ORR for isothermal diffusion is to compute the four frictional coefficients Rlo, R12, Ral, and Rmfor each composition of the system studied. If the ORR (microscopic reversibility) is valid, the R I ~
b
v(,
and Merentia1 Figure 1. Partial molar vohmes, refractive increments, CRt, for sucrose. and mannitol in aqueous solution a t 25'. The binary quantities for mannitol were taken from ref 6.
in Table IV are the values of @t1and R2 for each composition. These data are also represented in Figure 1. The binary refractive index data were obtained from Table I and the literature.e As might be expected, the partial molar volumes and differential refractive index derivatives are only slightly concentration dependent. The pertinent data in Tables I1 and I11 were then used to compute the (D& values which appear in Table IV. The method used for obtaining these ternary coefficients has been given in great detail elsewhere.'* All calculations were performed with programs written for electronic computers. The errors Figure 2. Binary, ternary, and tracer diffusion coefficienb in each (D& in Table IV were computed by assuming for the system water-suorose-mannitl at 25'. The open circles are the binary and ternary values; the closed circles in each Q value. As has been errors of ~ 2 . X 0 are the tracer diffusion coefficients of Mills and Ellerton." previously indicated,la the errors in the values of Q cause much greater errors in the (Dtj)Ythan the corresponding estimated errors of *0.1% in the SA. (19) L.-0.Sundelaf, Arkiv Kemi,20, 369 (1963). It is interesting to note that the (D& and (021)~ (20) R. Mills and H. D. Ellerton, J . Phys. Chem., 70, 4089 (1966). ~~~
~
Volums 71, Number 6 April 1067
H. DAVIDELLERTON AND PETER J. DUNLOP
1296
and R21 at each composition should be equal5 within the limits of experimental error. In order to compute the Rtk, values for the (D& and the chemical potential derivatives, pt,, are required.6 These derivatives are defined by pt5 =
(~idbej)T,PlekZO,j
(8)
where pi is the chemical potential of component i and e, is the concentration of component j in moles per cubic centimeter. A detailed account has previously been given for computing p t j from the binary osmotic coefficient data and expressions for the solute activity coefficients in the ternary system.6 The expressions used for the osmotic coefficients and the ternary activity data were those of Robinson and Stokes4 =
1
+ 0.07028m1 + 0.01847m12 -
+
(10)
Cm12mt
(11)
In
72
=
In
y2O
+ Am1 + (B/2)m12 + (C/3)mla + (D/4)m14 + E m m
e2
ml mz
rllo
rzzo rl rI2= rrl rZz A11 x 10-10 (Ai2
p21
p22
(12)
[A = 0.0737; B = 0.04096; C = -0.01425; D = 0.001194; E = 0.01071 where yl0 and yZ0are binary activity coefficients and y1 and y2 are the corresponding ternary values. Using these data the p t j were computed and are listed in Table V together with some intermediate quantities which have been previously defined. Frictional coefficients were then computed from the pertinent data in Tables 111, IV, and V by means of eq 7A and 13A (i.e., eq 7 and 13 of ref 6). The values so obtained are listed in Table IV. In these calculations, the value of the gas constant, R, and the value of the absolute zero of temperature were taken t o be 8.3144 X 10' ergs deg-' mole-' and -273.16", respectively. At the bottom of Table IV, the A% (exptl) values, which were calculated by means of eq 14A, provide a measure of the validity of the test of the ORR for a given composition of the system. The A% (calcd) values in the next line were obtained by eq 34A as follows. The main errors in the Rtk were assumed to be due to the errors in the listed in Table IV and the errors in the I'ln.6 On this basis, it was possible to compute both the errors (6Rit)' caused by the errors in the (DCJvand the errors (6Rcn)zcaused by the estimated errors of *0.002 in r12and r21and *0.003 in
=Ai) X
X Bii X Biz X BZIX Biz X A22
p12
+ 0.0034m2 + 0.0042m22 In y1 = In yl0 + Am2 + Bmlm2 +
+ Dml3mz + (E/2)m2
el
p11
0.004045m1~ 0.000228m14 (9)
42 = 1
Table V Chemical Potential Derivatives, and Intermediate Quantities
x x x x
lo-''
lo-'' 10-8 lo-' lo-' 10-8 10-1s 10-18 10-18 10-18
ptj,
0.25 0.25 0.50 0.25 0.50 0.25 0.27343 0.28271 0.58059 0.27343 0.56543 0.29030 0.15452 0.15496 0.16750 0.01024 0.01392 0.01046 0.16366 0.17372 0.17493 0.08678 0.09022 0.09602 0.01317 0.01339 0.02033 9.47215 9.19934 4.70349 0.21514 0.22365 0.23803 9.09907 4.41753 8.59006 1.15705 1,19880 1.30470 0.03586 0.03845 0.08126 0.06334 0.13589 0.07176 1.12958 1.20775 1.20181 10.9734 11.0585 6.1538 0.5827 0.6238 0.6683 0.8252 0.8684 0.9270 10.2858 5.3439 10.3430
0.5 0.5 0.60167 0.60167 0.16829 0.01438 0.18339 0.09988 0.02082 4.57487 0.24761 4.17185 1.35794 0.08750 0.15460 1.29084 6.2507 0.7199 0.9812 5.4069
a More than the minumum number of significant digits has been retained on the A i j , Bdj, and pij values to minimize the accumulation of errors in calculating the Ria. Ir The pii have units cm* erg mole-a.
rlIo,1'220, rll, and
r22. Values for (6Rtk)1and (6RD)2 are listed in Table IV. The corresponding values of A% (exptl) and A% (calcd) indicate that, within the error of measurement, the ORR are valid21for the four compositions of the system water-sucrose-mannitol studied in this work.
Experimental and Computed Phenomenological CoefEcienta I n recent publications, Lane and Kirkaldy22 and Wendt23 have provided theories for computing phenomenological coefficients for isothermal ternary diffusion from the corresponding binary diffusion data. Wendt formulated his results for systems of electrolytes and hence we shall not discuss his approach here. Lane and Kirkaldy coasidered two models, the first for an exchange mechanism and the second for a vacancy mechanism. Both models assume a lattice structure for the solutions with each molecule of each component occupying one position on the lattice. (21) Because of inescapable experimental errors, it is impossible for any experiments, no matter how carefully they are planned, to prove that a given hypothesis is valid. One can only illustrate, using the best techniques available, that the hypothesis is valid within certain limits. (22) J. E. Lane and J. 9. Kirkaldy, Can. J . Phys., 42, 1643 (1964); 44, 477 (1966).
(23) R. P. Wendt, J. Phy8. Chem., 69, 1227 (1965).
DIFFUSION DATAFOR THE SYSTEM WATER-SUCROSE-MANNITOL
Thus the ternary system of nonelectrolytes studied in this work cannot be expected to conform accurately to either model. Nevertheless, we consider it worthwhile to present the experimental phenomenological coefficients for the volume frame of reference, ( (Lt,)v)exptl,7J8 with the corresponding values calculated by the methods of Lane and Kirkaldy for the vacancy model, ((Li,)v)vac,and also for the exchange ) Exactly the same approximations model, (( L i j V)exch. were used in computing the phenomenological coefficients aa were used by La.ne and Kirkaldy. The relative viscosity data required for the vacancy model is reported in Table I1 and the limiting binary diffusion coefficients are reported at the bottom of Table VI. Table VI :n Experimental and Calculated Phenomenological Coefficients,b(Lfj)", for the Volume Frame of Reference
el e2
[(Lll), x [(Lll)" x [(k)Vx [(Liz)" X [(Ldvx [(L12)v x [(Lzi)v X [(Ldvx [(L21)v x [ ( L z z )X ~ [(LZZ)"x [(L22)V x
10~1exptl 1O~~lv.o 1Oz0]exch 1Ozo]exptl 10~~1vlul 10aO]exch 1OP0]exptl lO~OI.,
10ao]exch 1OPO1exptl 10201vac 1OPo]exch
0.25 0.25 3.47 3.87 4.86 -0.172 0.038 -0.027 -0.108 0.038 -0.027 4-61 4.91 6.19
0.25 0.50 3.04 3.40 4.73 -0.534
0.068 -0.054 -0.314 0.068 -0.054 8.14 8.67 11.98
0.50 0.25 5.01
6.13 9.16 -0.269 0.062 -0.054 -0.148 0.062 -0.054 3.71 3.84 5.86
0.50 0.50 4.31 5.29 8.90 -0.460 0.110 -0.108 -4.276
0.110 -0.108
6.32 6.64 11.31
Units: phenomenological coefficients (L,j)", mole* erg-1 cm-l sec-l; diffusion coefficients D,cmz sec-1. b The following limiting binary diffusion coefficients were used in computing the (L,,)v: DoIuof0Se = 0.5226 X 10-'and D'mannitol = 0.6664 X The self-diffusion coefficient of water was taken to be 2.27 X lo-'.
1297
This table also summarizies the experimental and the corresponding computed values of the phenomenological coefficients for the four compositions of the system water-sucrose-mannitol studied in this work. As one might expect, because of the large difference in the partial molar volumes of the components, the agreement between the experimental and computed coefficients is not particularly good. All the experimental cross-term coefficients, (L&, are negative while those computed for the vacancy model are positive. The cross-term coefficients for the exchange model, on the other hand, are all negative, but differ greatly from the experimental values. It should be pointed out at this stage that, because of the assumption contained in eq 15 of ref 8, the cross-term phenomenological coefficientsmust always be negative. The results in Table VI indicate that neither model appears to predict values for the (Lt,)v which are satisfactory for this system. This may be due to the large difference in the partial molar volumes of the components or to some of the approximations used in utilizing the binary diffusion data to compute the ternary phenomenological coefficients. For instance, the estimates of the jump probabilitiesz2 for the vacancy mechanism have been modified in an arbitrary way by introducing the relative viscosity of the solution. While it is generally recognized that the relative viscosity of the solution plays an important role in the diffusion process, the authors feel and we believe that Lane and Kirkaldy would agree that eventually the relative viscosity must be introduced in a more rigorous fashion.
Acknowledgment. The authors wish to thank Dr. B. J. Steel for helpful discussions during the course of this work.
Volume 71, Number 6 April 1967
