F. E. WEIR AND
302
MiiLCOLhI
relation rules shows that the reaction HXs('A') +T\TH('Z-)
+ N2('Zg')
giving products in the ground state is not allowed However, reaction can proceed to the lowest singlet state of NH, thus HNa('A')
-+
"('A)
+
pi2
('E,+)
The l A state is about 28 kcal./mole above the 32state of NH and thus would increase the activation energy for this decomposition to about 37 kcal./ mole. A similar situation is discussed for the N3 radical in a previous section. This activation barrier would render hydrazoic acid fairly stable a t room temperature but would permit rapid decomposition above about 200". 11.e plan to study the kinetics of this reaction in an effort to determine this activation energy more definitely. Consideration of the various tri-nitrogen entities
[CONTRIBUTIOX FROM THE CHEMICAL
1701. 80
DOLE
shows that both N3- and N3+ are quite firmly bonded, but that N3, if it exists a t all, m u s t be veFy unstable. Thus, D(N?-N-) must be approximately equal to the electron affinity of K3 arid D(Nz-N+) is approximately 83 or 59 kcal./mole depending on whether N or N Z retains the charge. On the other hand, apparently D(Nn-N) Q 0. Presumably this is another example of the greater stability of systems having even numbers of electrons. It is striking that either adding or renioving an electron permits formation of a firm bond where none otherwise exists. Acknowledgment.-The authors are indebted to Charles R. Yoltley of the Engine Fuels Section for the purification of the methyl azide by gas chrornatography and to Fred L. Xohler for helpful suggestions and discussion throughout the course of the work. WASHINGTOS, D. C.
LABORATORY O F XORTHTVESTERN
UNIVERSITY]
Diffusion in Sugar Solutions. IV. The Onsager Diffusion Coefficients for Glucose Diffusing in Sucrose Solutions BY F. E. ~ Y E I RAND MALCOLM DOLE RECEIVED OCTOBER1, 1956 Height-area diffusion coefficicnts and reduced second moments have been measured at 25 arid 35" fur aqueous solutions containing on the average 60.5% sucrose, 0.5% glucose and 39% water. From these values the main and cross term diffusion coefficients as defined by Baldwin, Dunlop and Gosting have been calculated using the recently published method of Fujita and Gosting. Equations relating these diffusion coefficlents to those defined by Onsager arc given and the Onsager diffusion coefficients calculated.
I. Introduction This paper represents the fourth in a series on diffusion in concentrated sucrose and glucose solutions and their mixture.s. Having previously studied sucrose and glucoseZ in binary systems with water as solvent and the diffusion of sucrose in the presence of glucosej3in this work the diffusion of glucose was studied in the presence of sucrose. During the course of this work an important paper by Fujita and Gosting4 appeared in which a mathematical technique was developed by which all four diffusion coefficients defined by the relations (first introduced by Baldwin, Dunlop and Gosting5), could be calculated. In eq. 1 the sub-
script 1 refers to sucrose and the subscript 2 to glucose. 9 is the diffusion coefficient in cm.2/sec., x the distance from the initial boundary, c the solute concentration in moles/cc. and t the (1) A. C . English and M. Dole, T H I S J O U R N A L , 72, 3261 (1950). (2) J. K. Gladden and M. Dole, ibid., 75, 3900 (1953). (3) D. M. Clarke and hi. Dole, ibid.,7 6 , 3745 (1854). (4) H. Fujita and L. J . Gosting, ibid., 78, 1099 (1956). (5) R . L. Baldwin, P. J. Dunlop and L. J . Gosting, i b i d . , 7 7 , 5235 (1955). See also P. J . Dunlop and L. J . Gosting. ibid., 77, 5238 (1955) (6) It should be noted t h a t the cross term coefficients a)n and 9 x 1 are different if the concentrations are expressed as grams/liter instead of rnoles/liter a s first recognized b y P . J . Dunlop. (Private
I n this paper measured diffusion coeficients will be indicated by script 3 ' s while coefficients calculated froin the Onsager relations (see below) will be designated by Roman D's. Experiments were concluctecl that made possible the use of Fujita-Gosting method. Experiments similar to those of Clarke and Dole3 were also attempted, but with uncertain succes5. Included i n the present paper are recalculated values of Clarke and Dole who inadvertently failed to add unity to the extrapolated j s-alues to obtain j,. (for definitioii see reference 1). The numerical values of Clarke and Dole were changed slightly by this correction. 11. Reciprocal Relations of Onsager Some years ago, Onsager' generalized Fick's law of diffusion t o systems of more than two components. For unidirectional diffusion in a system of 3 components Jo =
I)oc
sv
C no b-2
ax
D
._
u1
"*ax
(?a)
cornmimication from :,, J . Gosting.) I n this paper the concentrations will al!%-ays be expressed in moles/liter or rndes/cc. (7) L. Onsager, A n n . iV. Y . A r n d . Sci., 4 6 , 2.11 (1815). L . Onsager and K . 11. Fuoss, J . P h y s . C h e ~ ~ f36, . , 2089 (1932). G. J. Hooyrnan Physica, 2 2 , 751 (1956), has also discussed t h e Onsager reciprocal relations a s applied t o diffusion a n d has given this relation for ternary mixtures in terms of f o u r diffusion coeilicicnrs
Jan. 20, 1958
DIFFUSION IN
where J represents the flux in units of moles/cm.2/ sec., and the subscript zero refers to the solvent water. These nine diffusion coefficients can be calculated from the measurable diffusion coefficients, all, BIZ,DZZ, Dtl, by means of the following equations,8 (concentration in moles/cc.) DII
Qi(1
- &ci) - BlCzDiz
Dz2 = Dzz(1 -
UZCZ) - iizclDzi Diz = Diz(1 - OZCZ) - W13ii Dzi = %i(1 Bici) - Bi~zDnz DOO= (Dll 922) (Dll DZZ) CZDIZ) Dio = -iio(~iDii Dzo = - flo(cza)zz ~ 1 3 ~ 1 ) Dol = - &ID11 - ~ D Z I Dol = - 9iDiz -
+
-
-
+ +
+
(3a) (3b) (34 (34 (4a) (4b) (4c) (5a) (5b)
where the 0’s represent partial molal volumes in units of cc./mole, and $1 and 42 represent ijl/b0 and g z / f i o , respectively. 111. Experimental The Gouy interference method, diffusion cell, optical equipment, other apparatus and materials were the same as those used in the previous work.’-3 Considerable difficulty was experienced with water leaks between the glass plates of the diffusion cell and the water-bath, but after it was found that by using heavy grease (“Lubriseal”) dissolved in a small amount of benzene to grease the faces of the stainless steel blocks, no more trouble occurred. The leaks were indicated by a large At (see below) and by an actual shift in the position of the initial boundary during the run. Sucrose solutions approximately 6070 by weight were prepared by dissolving special confectioner’s sucrose in water a t room temperature. Five ml. of 0.1 N KOH was added to the solution per 595 ml. of water in the solution. The refractive index of the solution was then determined with a dipping refractometer in order to find the concentration of the solutions so that we’ghed quantities of “special anhydrous dextrose” or sucrose could be added to bring the solution to the desired concentration by weight. The solutions were of three types: first those made according to the method of Clarke and Dole in which a1 was kept approximately constant; second those made according to the requirements of the Fujita-Gosting method in which a~ was varied from zero to unity while maintaining the average concentration of each component across the boundary approximately constant; and third, a solution of equal moles of sucrose and sucrose plus glucose on the two sides of the boundary which had been equilibrated in the vapor equilibrator used by Clarke and Dole3 to equal vapor pressures of water. The equilibration was followed refractometrically until the refractive index showed no change with further equilibration. The definition of 011 is given by the equation
where RI is the differential refractive index increment of sucrose a t the average concentration of sucrose across the boundary. It was assumed that the total refractive index increment across the boundary, An, could be expressed by the equation A n = RlAcl RZACZ (7) where c represents concentration in moles per liter. The concentration of each component was calculated from the weight percentages and the density. A few density measurements were made, but the density of all the solutions used was calculated taking the partial specific volumes of water, sucrose and glucose to be 0.9908, 0.6373 and 0.6439 cc./g. a t 2500, respectively, and 0.9947, 0.6410 and 0.6472 cc./g. a t 35 Equation 7 can be rearranged to
+
.
(8) M. Dole, J . Chcm. P h y s . , 111, 1082 (1956)
303
SUGAR SOLUTIONS
so that it is easy to solve for R1 and Rp simultaneously by means of the least squares method. Equation 8 requires that An/Acl be a linear function of Acs/Ac~ provided that R1 and RZare constant over the small concentration gradient of interest. The extent to which this was true in the experiments of this research is demonstrated in Fig. 1. Inasmuch as no difference could be detected between the 25 and 35” points on the curve, RI and RI were taken to be 0.04744 and 0.02500 liter/mole (uncertainty possibly several per cent.), respectively, a t both temperatures. 0.24
0.20
’
...2
0.16
2
0.12
0.08
0.04
0
2 4 ACZIACI. Fig. 1.-Test of eq. 8.
8
6
IV. Method of Calculating the Data From the experimental observations of the shift’ in the Gouy fringes an average height-area ratio, DA can be calculated by the methods previously outlined by Gosting and Morrisg and Akeley and Gosting.lo If there is no interaction of flow, the individual diffusion coefficients of the two solutes, D~A and D ~ A are related to DA by the equation published by Gralen, l1 D*-1/1
=
a1[a)1*-’/2
-
32*-I/1]
+
D2.4--’/1
(9)
The use of this equation requires that the average sucrose and glucose concentrations remain constant as al is varied. By plotting DA-’/%as a ~ D?Amay be calculated from function of al, D l and appropriate relations involving the slope and intercept providing that the data follow the required linear relationship. Figure 2 illustrates a plot of D ~ - - l /as l a function of a1 for 25 and 35’. Satisfactory agreement with the required linear relationship is seen to exist. I n the work of Clarke and Dole in which cq was held constant, D ~ A was calculated from eq. 9 assuming DZAto be the same as 32 for the pure glucose-water system (in that work glucose solutions were the solvent). The results of the present work discussed below show that this assumption was reasonable. Another approach t o the interpretation of the a ) values is to follow the method of Fujita and Gosting4 in calculating DU, 3 2 2 , D ~ zand , Dzl, in which the reduced second moment, a)?m,is evaluated by a rather complicated mathematical and graphical method (see the paper of Fujita and Gosting for details), (9) L. J. Gosting and M. S. Morris, THISJ O U R N A L , 71, 1998 (1949). (IO) D.F. Akeley and L. J. Gosting, { b i d . , 711, 5688 (1953). (11) N. Gralen, Kolloid Z.,96, 188 (1941); 0. Quensel, Dissertation, Uppsala, 1942.
~
F. E. WEIR AXD M s L C o L h i DOLE
304
Vol.
so
were 15.T9 X lo-' and 16.44 X lo-', respectively. The values plotted in Fig. 3 and used in the calculations were those calculated by the senior author. Looking at Fig. 3 it can be seen that the 25' value of D2m a t a,, equal to 0.219 is considerably out of line with the other data a t that temperature. The At value given in Table 11 is also inordinately high. In computing the constants of the straight
900
850
2
TABLE I REDUCEDHEIGHT-AREARATIOS AS CALCULATED FROM EQUATION 12 All diffusion coefficientsin cm.'sec. X IO7. Concentration in moles/liter of solution.
800
25'
Exp. no.
750 0.2
0.4
0.6
0.8
Av. Av.
of eq. 12 and 17.
+SZ~~QI
(10)
where Izm and S*m, the intercept and slope, respectively, of the line, are related to D 2 2 , a>,,, D12 by equations given by Fujita and G o ~ t i n g . ~
13.0
18.0
B 12.0
17.0
11.0
16.0
10.0
0
0.2
0.4
0.6
0.8
CI C*
15.0 1.0
2.232 2.218 2.218 0.07153 0.1436 0.1800 - .0373 .0751 ,0946 ,1431 ,2872 ,3600 -1.040 -1.054 -1.062 20 213 341 15.69 14.05 12.20 11.60 11.88 11.92 13.38 12.87 12.05
-
ACi
Figure 3 illustrates a plot of Dzm at 25 and 35' as a function of a, which according to the theory of Fujita and Gosting should result in a straight line inasmuch as Dzm = I z m
35 (Water equilibrated)
1.0
Q2.
Fig, 2.-Test
17
13
11
ACP ffl
At, sec. 9 . 4 331"
%A
-
2.255 0.07144 .lo89 - ,1429 3.384 54 3.504 11.30 25.47
+ -
Fujila-Costing type experiment (see also Table TI) D1.4
%A
25 10.73 17.57 35O 16.39 24.39 a Calculated from data of English and Dole a t the average sucrose concentration shown above.
line for 25' we have omitted this value of DDnm and have substituted for it the value of DA calculated for a, equal to unity, assuming thereby that a t this value of cyl there will be no fringe deviations. The fringe deviations plotted in Fig. 4
0.05
0.04
61.
Fig. 3.-Test
of eq. 16.
0.03
I The value of D U for Z ~ the water vapor equilibrated solution was calculated from eq. 9 making use of the assumption of Clarke and Dole that D ~ A 0.02 is equal to D1in such a system.
V. Experimental Results All of the experimental results are collected in Tables I and 11. It is difficult to estimate the reliability of the data, especially the diffusion coefficients calculated by the Fujita-Gosting method. This is because two graphical integrations have to be carried out and because in the calculations the diffusion coefficients depend upon small differences between large numbers. The critical calculation is that of a s r n . I n one experiment a t 35', one of us obtained 15.95 X lo-' for this value while the other found 16.86 X lo-' (using the same fringe deviation values) ; in another experiment the values
\
\ 0.01
0
0
0.2
0.4
0.6
0.8
1.0
f (r). Fig. 4.-Interference fringe deviation from Gaussian behavior at two values of q.
for two values of a1 suggest that at a1 equal to unity the fringe deviations might be zero. I n the
DIFFUSIONI N
Jan. 20, 1958
305
SUGAR SOLUTIONS
TABLE I1 DATAFOR DIFFUSION COEFFICIENTS CALCULATED BY THE FUJITA-GOSTING METHOD All diffusion coefficients X 101 in cm.l/sec. Concentrations in moles/liter of solutions. 25'
Av. CI Av. cz ACi
Act ai
A f , sec. D A
Pam
350
36
34
24
2.277 0.03231 .0094 .0646 .2186 830 15.63 12.69
2.275 0.03733 .0245 .0746 .3754 68 14.34 11.84
2.278 0.03596 .0621 .0719 ,6175 -41 12.81 11.14
Expt. no.
25' 35O
25
all
%
a 1 1
10.95 17.16
2.37 2.18
5.27 8.32
-
2.280 0.03586 .0532 ,0360 .7363 25 12.07 11.00
THE
TABLEI11 ONSAGER DIFFUSIONCOEFFICIENTS CM.*/SEC. x 107 61% Sucrose-two 25'
DaO Dol DIO
5.34 -64.8 - 0.44 5.36
Dii
5.33 -64.5 -45.2 - 0.45 5.46 2.36 0.025 - 0.34 2.53
Dit
DI~ Dzo Dzi Dzz
IN
components'
350
7.98 -96.7 0.65 7.90
-
8.05 -97.6 -64.0 - 0.71 8.58 3.79 0.09 1.14 2.71
-
I n the introduction i t was stated that Clarke and Dole failed to add unity to their estimates of j,. Table IV contains a summary of Clarke and Dole's data as recalculated using the new values of Im.
VI.
39
40
2.266 0.03569 .0096 .0714 .2026 60 22.27 17.53
2.272 0.03576 ,0498 .06152 .6102 156 19.07 16.86
2.269 0.3550 .0385 .01769 .8172 51 17.60 16.44
2.269 0.03642 .0487 .00517 .9606 20 16.56 16.03
lgfl
Three components 60.5% sucrose, 0.5% gIucose, 39% water av. concn.
Da, Dol DO2 DID
42
-0.64 -2.13
case of the Dzm values a t 35' the straight line extrapolated to a1 equal to unity yields a value of Dzm not much greater than the value of DA calculated for cy1 equal to unity. It is believed that this procedure yielded more reliable values for Izm and SZm a t 25' than would have been obtained if the Dzmvalue a t a1 equal t o 0.219 had been included in the least squares calculation. The straight lines of Figs. 2 and 3 illustrate the values calculated by the least squares equation. Table I11 contains values of the Onsager diffusion coefficients calculated by means of eq. 3a to 5b, inclusive. DATA FOR
-
37
should be less uncertainty in comparing diffusion coefficients for the sucrose which was present a t 60.5 wt. % ' concentration. A t 25' and 61.05% sucrose English and Dole' found 10.71 X lo-' cm.2/sec., while 10.73 X also was calculated from experiments 36, 34, 24 and 25 of this research. TABLEIV CORRECTED REDUCED HEIGHT-AREA RATIOSOF CLARKE AND DOLECOEFFICIENTS IN CM.~/SEC. X 10' Glucose
Av. wt. %
Sucrose
A
25' 5 5.176 3.00 29.82 29.73 29.55 4.00 39.93 50.0 49.70 49.50 49.37 49.23 49.06 60 59.48 59 * 77 58.95 58.29 70.0 70.40 69.99
60.0 59.47 59.51 59.30
0 0.999 0 0.600 0.900 1.500 0 1.501 0 0.601 1.000 1.467 1.540 2.393 0 0.953 1.520 1.998 2.516 0 0.966 1.485 35O 0 1.004 1.017 1.512
47.92 26.44 26.13 25.83 11.56 10.94 11.51 11.18 11.36 11.21 5.192 4.774 5.047 5.099 1.270 1.328
8.567
Discussion of Data 8.254 It is first of all of interest to compare the D D 1 ~ 7.966 and the D2.4 with the D1and D2 diffusion coefficients measured by English and Dole' and Gladden and At 35' the agreement was not as good, 15.83 X Dole2 in the binary systems. Such comparison lo-' found by English and Dole again a t 61.05 7 0 for the component present in small concentration, sucrose, and 16.39 X lo-' from this research. glucose in this work, is questionable because of the These results seem t o be good enough t o justify change in the nature of the solvent and because the assumption made by Clarke and DoleS that a t the there is uncertainty as t o whether the comparison a > , ~of eq. 9 is approximately equal to should be made at constant mole fraction of com- same total weight fraction of sugar if one of the bined sugars or constant weight fraction. There components is present in small percentage.
:EO6
F. E. WEIR AND MALCOLM DOLE
Clarke arid Dole found that D l for ~ the diffusion
Vol. 80
splitting of a fringe into two fringes appeared. In experiments no. 11, 13 and 17 of Table I, Ct greater than Dl if the comparison was made a t the calculated from the larger fringe distances did not same total mole fraction of solute. I n this research change with fringe number as expectetl. For i t was found as expected that QAas calculated fringe numbers above 10, non-Gaussian beha\ ior from eq. 9 a t a1 equal to zero, 17.57 X lop7c ~ n . ~ /existed. The variation of the fringe dcviation 62 sec., was smaller than 32 for glucose, 25.7 X lo-’ with time should be constant, but this was not the calculated from the data of Gladden and Dolez a t case as the peculiar result of zero values of 62 for the same total mole fraction of solute. Thus, su- fringes up to 10 did not appear until diffusion 11ad crose diffuses faster it1 solutions of glucose than in its proceeded for about 5000 seconds. -2 clear exown solutions while glucose diffuses more slowly in planation of these observations is not available a t solutions of sucrose than iii its own solutions. If, the present time. All of these experiments were however, the comparison is made a t the same weight performed before the best technique of assembling rraction, the con7 erse is true, as was also found by the cell was developed. The data are included in Clarke and Assuming no iriteractioii of Table I, nevertheless, because other criteria for solute flows, the diffusion coefficient of glucose, valid experimental results such as not too large A/ D 2 can ~ be calculated using eq. 9 from the three values and fairly good agreement between the DZA types of experiments done in this work, namely, values indicated that the results were possibly from the Fujita-Gostiiig, “opposed gradients” and significant. water equilibrated experiments. The results are, In the mathematical analysis of Gosting and his i n order mentioned at 25’, 17.57, 14.2(value extrap- school, an attempt has been made to develop theoolated t o zero concentration of glucose) and 25.47 retical techniques for determining the extent of x lo-’ cm.2/sec. In the Fujita-Gosting type ex- interaction between different diffusing species. periments the concentration gradients for sucrose If there were no interaction of flows, then the D 1 2 and glucose were parallel, i.e., they had the same and Dzldiffusion coefficients should be zero. From sign. whereas in the Clarke-Dole type the concen- Tables I1 and I11 i t is clear that interaction did octration gradients had the opposite sign. I n the cur. Referring to Table I11 one sees that D o 0 and latter the diffusion of sucrose was in the opposite DI1are practically the same in the three component direction or “opposed” the diffusion of glucose. solution as in the two component as might be exThe a-values were approximately -- 1.0 for the op- pected as the small a m o m t of glucose present posed gradient experiments, +0.3 (average) for should not greatly affect the diffusion of sucrose or the parallel gradient ant1 +3.374 for the water water due to their own concentration gradients. equilibrated experiment Thus there is a very about A surprising result is the low calue of good correlation between the calculated %A \-dues ,me tenth that calculated assuming n o iiiteractioli arid cyl, the larger the sucrose gradient, the larger of flows. From J. physical standpoint such a low is the calculated diffusion coefficient of glucose. result would have to be explained on the basis, perIn fact a plot of a ) . ~for the three types of experi- haps, of complex formation between sucrose and ments as a function of a1 approximates well to a glucose molecules. straight line. The value of DAfor the water equilComparing the Dzl with the Dlz values the foribrated experiment a t el equal to 3.38 does not mer is very small and negative, while D12 is quite “fit” on the extrapolated straight line of Fig. 2, large. In other words the diffusion coefficient nor do the a ) values ~ of the “opposed gradient’’ representing the flow of glucose due to the sucrose typ: agree a t all with the relatioiiship of Fig. 2. concentration gradient is not as great as that for I11 ract, approximately the same XTalues of DA were the flow of sucrose due to the glucose concentration obtained as in the Fujita-Gostitig experlments but gradient a t negative a-values. The authors wish to express their gratitude to At this point it should be noted that many of the E. J. Rrach and Sons, Inc., Chicago, for financial experiments of the Clarke-Dole type which were support and gift of materials and to Professor L. Jttenipted failed to give reliable results. The J. Gosting and Dr. G. J. Hooyman for several fruitfringe photographs were imperfect, the fringes on ful comments and suggestions. the photographic plate showed a variable density from top tn hottoiii ,ir?d sometimes an apparent EVANSTON, 11 LINOI’? o f sucrose i n glucose solutions as solvent was
