2211
TRANSLATIONAL FRICTIONAL COEFFICIENTS
Translational Frictional Coefficients of Molecules in Aqueous Solution by Hisashi Uedaira and Hatsuho Uedaira Research Institute for Polymers and Textiles, 4 Sawatari, Kanagawaku, Yokohama, Japan
(Received October 2,1969)
Diffusion coefficients for aqueous fructose and p-cyclodextrin solutions at 25" were measured by the Rayleigh interference method. The translational frictional coefficients of sucrose, raffinose, and p-cyclodextrin were calculated, being based on the limiting diffusion coefficients of fructose and glucose. The frictional coefficients of oligopeptides and dicarboxylic ions were also calculated. If the effect of the local viscosity around a molecule of solute is corrected, the agreement between the calculated and observed frictional coefficients is very good.
Introduction The diffusion coefficients of small molecules in aqueous solutions a t infinite dilution are, in principle, calculable from the Stokes-Einstein relation. The Stokes radius of the solute has been regarded as a measure of hydration and departure from strictly spherical shape. For small molecules of arbitrary structure, it is hardly practical to compare the theoretical translational frictional coefficient with the experimental value. I n a previous report,2 we calculated the frictional coefficient of maltose in aqueous solution by the shell also correcting for the effect of the local viscosity of water around the molecules of maltose. The agreement between the calculated and observed values was good. This method can be applied to a molecule or ion of arbitrary structure. I n the present paper, we report the diffusion coefficients for fructose and p-cyclodextrin in aqueous solutions at 25". The translational frictional coefficients of p-cyclodextrin, oligosaccharides, oligopeptides, and divalent carboxylic ions a t infinite dilution are calculated and compared with the experimental values.
fusion coefficient for the fructose solution are given in Table I. The corresponding data for p-cyclodextrin
Experimental Section Xamples. Fructose (DIFCO Co.) was recrystallized twice from water-ethanol solution. Purified p-cyclodextrin was kindly supplied by Dr. S. Tomita of Government Industrial Research Institute. [His kindness is greatly appreciated. ] All solutions were prepared by weight, using deionized water as the solvent. Diffusion Measurements. The diffusion experiments were performed a t 25 f 0.01" using a Spinco Model H diffusion apparatus as a Rayleigh interferometer. Kodak Type M glass photographic plates (4in. X 5 in.) were used. The concentration difference of the solution on both sides of the diffusion boundary was adjusted so that the total number of Rayleigh fringes became 50 or 60. The procedure was then the same as that previously described.2 Results The mean concentration in each diffusion experiment and the corresponding observed value of the dif-
D X lo6 = 7.002 - 0.0813~ c < 3 wt % (1)
Table I : Diffusion Coefficients of Fructose in Water at 25' 7 -
0,22924 0.94981
106D, cm2/sec
6.981
6.930
c , wt % ---1.51603 2.00038 2.52132
6.880
6 837
6.798
Table I1 : Diffusion Coefficients of p-Cyclodextrin in Water at 25"
---
c, wt % -0.23137 0.50177 1.02443
lOeD, cm2/sec
3.307
3.284
3.239
1
1.52376 1.93773
3.190
3.148
are given in Table 11. To represent the experimental diffusion coefficients the method of least squares was used to obtain the expressions
for the fructose solution, and
D X lo6 = 3.332 - 0.093%
c
< 2 wt %
(2)
for the /I-cyclodextrin solution, with an average deviation of =k0.06%.
Discussion Recently, Bloomfield, Dalton, and Van H ~ l d e , ~ using the shell model, proposed the following expression for the translational frictional coefficient of any macromolecule that can be visualized to be composed of n spherical subunits. (1) H.J. V. Tyrell, "Diffusion and Heat Flow in Liquids," Butterworths and Co. (Publishers) Ltd., London, 1961, Chapter 7. (2) H. Uedaira and H. Uedaira, Bull. Chem. SOC.Jap., 42, 2140 (1969). (3) V. Bloomfield, W. D. Dalton, and K. E. Van Holde, Biopolymers, 5 , 135 (1967). Volume 74, Number 10 May 14, 1970
HISASHI UEDAIRA AND HATSUHO UEDAIRA
2212
f =
n
n
(3)
Here Rlsis the distance between centers of two spherical shells of radius r z and rs, and 90 is the viscosity of the solvent. If all of the subunits are identical, eq 3 reduces to the Kirkwood e q ~ a t i o n . ~ The translational frictional coefficients of homologous molecules in aqueous solution can be calculated from eq 3, if a correction for the local viscosity around a solute molecule is made. I n the following, we calculate the frictional coefficients of molecules made up of identical submolecules and also of those made up of submolecules of different size. (a) Frictional Coejicients of Molecules Composed of Identical Submolecules. The frictional coefficients, fz and f3, of dimer and linear trimer can be calculated from eq 3 to be
fz = 1.33(6qr,) f3
=
1.59(6aqr,)
(4) (5)
where ra is the radius of the submolecule and 7 is the viscosity of water. These relations are obtained from eq 8 and 9, putting k equal to 1. The assumption that the solvent is a continuum in the derivation of the Stokes relation and eq 3 implies that the dimensions of the diffusing particles are large in comparison with those of the solvent molecules. This assumption does not hold in the case of a small molecule, and its frictional coefficient is affected by the structure of solvent. The water structure around a molecule in aqueous solution is different from that of bulk water. For example, it is known from nmr spectroscopyj that the time of reorientation of a water molecule around molecules such as inorganic, organic ions, and alcohols is different from that of a water molecule in pure water. A water molecule close to a solute molecule constantly exchanges its site with other neighboring water molecules. The translational motion of a molecule with tightly bound water molecules in aqueous solution is strongly inhibitedlB,’and the motion is affected by the interaction between water and solute. This mechanism is supported by nmre5 Therefore, a molecule in motion in water encounters a different structure from that of pure water, and so the local viscosity of water around the molecule is different from that of pure water. For small molecules, this effect cannot be neglected. The water structure around a submolecule and the molecule composed of a number of submolecules in aqueous solution is regarded as identical, provided that the submolecules are all the same. Therefore, the ratio of the translational frictional coefficients of the molecule and its submolecule is independent of the The Journal of Physical Chemistry
water structure. Thus, we obtain f2/fl = 1.33 for dimer, and f3/fi = 1.59 for trimer from eq 4 and 5, where f~ = 6nqr,. I n diffusion of a small molecule, fi should be expressed strictly by aagr,,8-10 but this is not important in the calculation of the ratio of the frictional coefficients. The diffusion coefficient is inversely proportional to the frictional Coefficient. Therefore, if the limiting diffusion coefficients of the molecule and submolecule are determined from the experiments, the ratio of the experimental values of the frictional coefficients is obtained. The results for diglycine, triglycine, succinic, and adipinic ions are listed in Table 111. The submolecules which are used in the calculations of the ratios of the frictional coefficients are listed in the second column of Table 111. The limiting diffusion coefficients of the molecules and submolecules are given in the third and fourth columns. For carboxylic anions, the limiting mobilities are given in the corresponding columns. The values of the ratios of the experimental and calculated frictional coefficients are given in the sixth and seventh columns, respectively. For the carboxylic ions, the experimental frictional coefficients were calculated from the limiting mobilities. Agreement between the calculated and observed values is very good. Now, as an example of a more complex molecule, we calculate the frictional coefficient of p-cyclodextrin. As 8-cyclodextrin is a macrocyclic nonreducing Dglucosyl polymer containing seven residues bonded by a! links, we can regard this molecule as a pearl necklace consisting of seven identical spheres. The translational frictional coefficient fcn of this molecule can be calculated from eq 3. As shown in Figure 1, the centers of the glucose units form a regular heptagon. We can calculate each value of R,, in eq 3 by TG and the value of the interior angle of the regular heptagon (128’3’). We obtain the expression ~ C D=
2.33(6avr~)
(6)
where rG is the radius of glucose. We have fCD/fi = 2.33 from eq 6, where fi is the frictional coefficient of (4) J. G. Kirkwood, J . Polym. Sci., 12, 1 (1954). ( 5 ) H. G . Herz and M.D. Zeidler, Ber. Bunsenges. Phys. Chem., 68, 821 (1964). (6) 0. Ya. Samoilov, “Struktura Vodn’kh Rastvorov Elektrolitov i Gidrataziya Ionov,” Akademy Nauk, Mosrow, 1957; Chapter 5 . (7) H. Uedaira and H. Uedaira, Zh. F i r . Khim., 42, 3042 (1968). (8) S. Glasstone, K . J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill Publications, New York, N. Y., 1941, Chapter 9. (9) The deviation from Stokes law also can be caused by failure of the nonslip boundary condition. A slip of a water molecule may occur on the surface of a nonpolar molecule. On the other hand, water molecules around the nonpolar molecule form an ice-like structurelo and become less mobile than in pure water.6.7 Thus the local viscosity of water around the molecule is different from that of pure water. Whether the nonslip boundary condition holds or does not hold, the effect of local viscosity exists in solution for an associating solvent. (10) H . 9. Frank and hf. W. Evans, J. Phys. Chem., 13, 507 (1945).
TRANSLATIONAL FRICTIONAL COEFFICIENTS
2213
Table I11 : Translational Frictional Coefficients of Oligosaccharides, Oligopeptides, and Dicarboxylic Ions at 25’ in Water Moleoules
p-C yclodextrin Sucrose Raffinose Diglycine Trigly cine Glycylalanine (CHzCOO)zz(CzHdC00)z2-
Submolecules
IOSDo, cml/~ec
Glucose Glucose Fructose Glucose Fructose Glycine Glycine Alanine Glycine CHICOOCzHsC00 -
3.332 5 23”
lO’Doi, cml/sec
6.75“ 6 . 7sa 7.002 6.75. 7.002 10.554c 10.5540 9.097” 10.554c 40. 8gd 35. 82d
I
4.359b 7. 90gc 6 . 652c 7. 22lC 60. 65d 52. 5gd
(f/fi)exp t I
k
(f/fl)oalod
2.03 1.29 1.34 1.55 1.60 1.33 1.59 1.26 1.46 1.35 1.36
0.962 1,037 0.962 1.037 0.862 1.160
2.05 1.30 1.36 1.56 1.61 1.33 1.59 1.26 1.45 1.33 1.33
Reference 11. b P. J. Dunlop, J. Phys. Chem., 60,1464 (1956). c T. Shedlovsky, Ed., “Electrochemistry in Biology and Medicine,” John Wiley and Sons, New York, N. Y., 1955, Chapter 12. d The units are cmZ/ohm equiv: J. D’Ans, A. Eucken, G. Joos, and W. A. Roth, Ed., “Landolt-Bornstein Tabellen,” Vol. 2, 6th ed, 7 Part 11, Springer-Verlag, Berlin, 1960. 0.
agreement between this value and the experimental value is satisfactory. The ring model also agrees with the molecular m0de1.l~ ( b ) Frictional Coeficients of Molecules Composed of D i f e r e n t Submolecules. The frictional coefficient of a molecule which consists of two submolecules of radii ra and r b can be calculated from eq 3, and is found to be fz =
Figure 1. Model of p-cyclodextrin.
glucose. The limiting diffusion coefficient of pcyclodextrin is 3.332 X cm2/sec from eq 2. The limiting diffusion coefficient of glucose in aqueous solution at 25” obtained by Gladden and Dole1’ is 6.75 X 10-6 cm2/sec. The ratio of the experimental frictional coefficient of 0-cyclodextrin to that of glucose is fCD/fl
=
DOG/DOCD = 2-03
The calculated value is larger than the experimental value. The molecule of p-cyclodextrin can also be regarded as a ring consisting of seven glucose molecules, instead of the above necklace model (see Figure 1). Tchen12 calculated the translational frictional tensor of the ring molecule, and the following expression for the frictional coefficient was obtained by Zwanaigl3
f
=
36aqL/ll log ( L / r )
(7) where L is the length of the ring. For p-cyclodextrin, the value of L calculated from Figure 1 is equal to 14.47r~. Therefore, we have fCD/fl = 2.05. The
+
+ k)
6‘Vra(l k2)2(1 1 k 2k2 k3
+ +
+ +
k4
(8)
wherek = r b / r s . The frictional coefficient of a molecule which consists of two submolecules of radius rs and a submolecule of radius ?“b (see Figure 2) is given by the following expression 6.rrva(2
f3
= (2
+ k2)’(1 + k)(2 + k)
+ k3)(1 + k ) ( 2 + k) + 2 + 3k + 7k2 + 41~3 (9)
where k = r b / r a . I n the calculation of eq 9, it was assumed that the centers of the three submolecules were on a straight line and that the submolecule of radius rb was located at an end of the molecule. The experimental value of the ratio of the frictional coefficient for sucrose to that of glucose is given in the third line of Table 111. The calculated value of the corresponding ratio is given in the same line. Let ya and Tb be the radii of glucose and fructose, respectively. (11) J. K. Gladden and M. Dole, J . Amer. Chem. SOC.,75, 3900 (1953). (12) Chan-Mou Tchen, J . Appl. Phua., 25, 463 (1954). (13) R. Zwanzig, J . Chem. Phus., 45, 1858 (1966). He pointed out that the Kirkwood e q 4 is not exact, and showed that the correct result obtained from eq 8 was smaller than the incorrect one by the factor 11/ 12. (14) R. L. Whistler and E. F. Paschell, Ed., “Starch: Chemistry and Technology,” Vol. 1, Academic Press, Inc., New York, N. Y., 1965, Chapter 9. Volume 74, Number 10 M a y 14?1970
ALEXANDER APELBLAT
2214
Figui.e 2. Model of raffinose.
As the radius of a molecule is inversely proportional to its diffusion coefficient, the value of k is determined from the ratio of the limiting diffusion coefficients of the submolecules. The limiting diffusion coefficient of fructose in water is 7.002 X cm2/sec from eq 1 and that of glucose is 6.75 X cm2/sec. Hence k = 0.962. Similarly, the ratio of the frictional coefficient for sucrose to that of fructose is obtained. I n this case, the value of k in eq 8 is 1.037 and ramust be substituted by I’b. The experimental and calculated values are given in the fourth line of Table 111.
The values of the ratios of the frictional coefficients for raffinose to that of its submolecules are given in the fifth and sixth lines of Table 111. I n the calculat,ion of the frictional coefficient for raffinose] the radius of galactose was taken to be the same as that of glucose. The values of the ratios of the frictional coefficients for glycylalanine to that of its submolecules are given in the ninth and tenth lines of Table 111. I n all cases, the agreement between the observed and calculated values is very good. Thus, the frictional coefficient of a molecule with any structure can be accurately calculated, relative to the frictional coefficients of its constituent submolecules. The absolute calculation of the frictional coefficient requires a correction for the effect of local viscosity, particularly important for an associating solvent.
Acknowledgment. The authors thank Dr. IC Tsuda in our Institute for helpful discussion.
Thermodynamic Properties of Associated Solutions.
I.
Mixtures of the Type A
+ B + AB,
by Alexander Apelblat Israel Atomic Energu Commiaawn, Nuclear Research Center, Negev, Beer-Sheva, Israel
(Received October 8, 1869)
+ +
The theory of ideal associated solutions of the type A B AB2has been investigated in detail. The chemical equilibrium, the activity coefficients, the excess thermodynamic functions, and the standard reaction parameters were considered. Special attention was directed towards the evaluation of the boiling and condensation temperature curves for these solutions and the conditions necessary for the occurrence of an azeotrope. The theoretical predictions for the A B AB2 model were compared with experimental data for binary methanolpropionic acid systems.
+ +
Introduction Deviations from the ideal behavior in the systems containing components capable to form intermolecular hydrogen bonds are frequently explained by the formation of relatively stable configurations (associates) between like molecules (self-association) or unlike molecules (complexing). The general theory of the ideal associated solutions (effect of different sizes and shapes of Complexes is neglected) is given by PrigOgine and Defay’ and Hildebrand and SCott.2 Kehiaian and coworkers3 treated in detail the mixtures of the type A + B + AB, A + A, + B and the ~ f ~ type of association, A A2 As . . , A, B. The B AB and MCwork of SaroItSa-Mathot4 on A
+ + + + + +
The Journal of Physical Chemistry
+
Glashan and RastogiJ5Rastogi and GirdhaF on A B 4-AB f AB2 type of association are worth noting. A number of theoretical and experimental investiga-
tions’-‘* concern the binary systems containing alcohols. (1) I. Prigogine and R. Defay, “Chemical Thermodynamics,” John Wiley and Sons, New York, N. Y., 1965. (2) J. H.Hildebrand and R. L. Scott, “The Solubility of Nonelectrolytes,” 3rd ed, Dover Publications, Inc., New York, N. Y., 1964. (3) H. Kehiaian and A . Treseczanowice, BUZZ. SOC.Chim. Fr., 1561 (1969). I n this paper there is an extensive list of Kehiaian’s Forks. (4) L. BarolBa-Mathot, Trans. Faraday &e., 49, 8 (1953). (5) M. L. McGlashan and R. P. Rastogi, ibid., 54,496 (1958). ~ k ~ ~ ~ ~ (6) R. P. Rastogi and H . L. Girdhar, Proc. Nat. Instit. Sei., India, A28, 4T0 (1962). (7) J. A. Barker, J . Chem. Phys., 20, 1526 (1952).
~
~
