770
L. G. LONGSWORTH
TEMPERATURE DEPENDENCE OF DIFFUSION
VOl. 58
m AQUEOUS
SOLTJTIONS
BYL. G. LONGSWORTH Laboratories of the Rockefeller Institute for Medical Research, New York, N . Y . Received March 11, 196.4
The diffusion coefficients, in dilute aqueous solution, of some representative materials whose molecular weights range from 19 to 68,000 have been determined at several temperatures. The results indicate that the temperature coefficient for diffusion increases only slightly with increasing particle size and is less than that predicted by the Stokes-Einstein relation (assuming the diffusion radius to be independent of the temperature) only for solute molecules so small that the solvent, water, cannot be considered a continuum.
Within the last few years methods have been the same as in previous work by the author and has been described d e ~ e l o p e d l - ~that permit the determination of adequately Measurements have been made at 1, 13, 25 and 37". diffusion in liquids with a precision of about 0.1%. A t 37" two sources of difficulty were encountered only one Since the older data were often uncertain by several of which has been overcome. Solutions prepared at room per cent. or more, this development has not only temperature are frequently supersaturated at 37" with reto dissolved air and this is likely to appear as bubbles made possible the study of many interesting prob- spect on the walls of the diffusion channel. The use of freshly lems such as the concentration dependence of the boiled water in preparing the solutions and storing them diffusion coefficient and the variation in the diffu- under reduced pressure effectively prevented bubble formasion behavior of isomers but has also necessitated a tion. The more 8erious difficulty arises from the relatively temperatufje coefficient, dnldt, of the refractive index re-examination of some of the prevalent ideas based highwater at 37 . In the case of a thermostat regulating on the older results. One of these is that the of near room temperature little energy enters or leaves the temperature coefficient of diffusion increases with system. This is not true, however, at other thermostat the size of the diffusing particle. This conclusion settings and appreciable thermal gradients are then present the bath water. At settings near 0" dn/dt is quite small7 is based on a table published by Oholm4 in 1913 in and the thermal gradients are not accompanied by schlieren. and cited by Taylor.6 After conversion of Oholm's At 37" on the other hand dn/dt is relatively large and the data to cm.2/sec. they are as follows resulting schlieren in the path of the light through the bath 2-1 Dm X 106 28 23-21 18-16 14-13 9-8 5-3 Dzo/Dio 1.18 1.20 1.22 1.25 1.29 1.35 1.40 ED 2730 3010 3280 3680 4200 4950 5550
and predict that if a substance has a diffusion coefficient a t 20" in the range given in the first row the ratio of this to the value at 10" will be approximately as given in the second row. Reference to the activation energies, ED, in the third row will be made later in this report. Since there are reasons to think that the dimensions of a particle are nearly independent of the temperature the variation of the ratio Dzo/Dlo with D represents a contradiction of the StokesEinstein relation and will be discussed below. The purpose of this report is to present the results of diffusion measurements of selected solutes a t a sufficient number of temperatures to establish not only the value of the temperature coefficient but also the manner in which this changes with the temperature. The materials selected have a molecular weight range of 19 to 68,000 and include those that previous work6 has shown to have extreme values for the temperature coeffcient. Experimental.-The method employed in the present research was that of free diffusion from an initially sharp boundary between solution and solvent, the spreading of the boundary with time being followed optically with the aid of Rayleigh interference fringes. The Tiselius electrophoresis cell, modified to facilitate the formation of the Rayeigh fringes, has served as the diffusion cell. Except for the use of a platinum resistance thermometer to measure the thermostat temperature the experimental procedure was (1) G . Xegeles and L. J. Gosting, J . A m . Chem. Soc., 69, 2516 (1947). (2) L. J. Goating and M. S. Morris, ibid., 71, 1998 (1949). (3) L. G. Longsworth, {bid.. 74,4155 (1952). (4) L. W. Oholm, Medd. Vetenskapsakad. Nobelinot., 8, 1 (1913). (5) H.9. Taylor. J . Chem. Phys., 6 , 331 (1938). (6) L. G. Longsworth, J . A m . Chem. Soc., 7 6 , 5705 (1953).
produce "quivering" of the Rayleigh fringes at the photographic plate. Although this fringe movement is largely averaged out over the 3- to 4-second interval required for an exposure it impairs the fringe definition. At 37" the diffusion coefficients computed from successive exposures show larger deviations from the mean value for the experiment than at the lower temperatures. Partial immobilization, with the aid of a metal trough immersed in the bath, of the water in the light path appeared to reduce the schlieren somewhat. The thermoregulator used in the present research was a Rudolph Graves instrument (Stockholm, Sweden) with a sensitivity of 0.01 to 0.02'. In future work it is hoped that the use of a more sensitive regulator, together with immobilization of the bath water in the light path, will minimize the schlieren. It should be noted that at 1" and 37" gradients of as much as 0.03" existed between the hottest and coldest regions of the bath. More vigorous circulation is not contemplated, however, owing to the possibility of disturbing the diffusion process thereby. Materials.-The deuterium oxide used in this work was the material recovered from a previous investigation.B Dr. Dexter French generously supplied the cycloheptaamylose.g The bovine plasma albumin was a crystalline product, .4rmour lot number G4502, and was found, on electrophoretic analysis by Dr. Gertrude Perlmann of these Laboratories, to be free of globulins. A second crystalline sample, Armour lot number 128-175, was also examined a t 13' and gave a diffusion coefficient of 0.4672 X a value 1% higher than that of the first sample. These results, together with those of Akeley and Gostinglo for this protein, suggest that a recrystallization immediately prior to use may be essential if the values obtained arc to be independent of the sample. However, the results for the three samples are in sufficiently close agreement to warrant the conclusion that the effect of temperature on the diffusion of a "pure" sample would not differ significantly from the values reported below. All other materials were of reagent grade and were used without further purification. I n the diffusion of D20high concentrations were required to give a reasonable number of fringes and the concentration dependence of D made it desirable to extrapolate to infinite dilution. The extrapolations are shown in Fig, 1. Except (7) L.W. Tilton and J. K.Taylor, J . Research Natl. Bur. Standards, ao, 419 ~1938). ( 8 ) L. G. Longsworth, J . A n . Chem. Soc., 69, 1483 (1937). (9) D. French and R. E. Rundle, ibfd., 6 4 , 1651 (1942).
(IO) D.F. Akeley and L. J. Goating, abid., 76, 5685 (1953).
TEMPERATURE DEPENDENCE OF DIFFUSION IN AQUEOUSSOLUTIONS
Sept., 1954
77 1
TABLE I DIFFUSION COEFFICIENTS, D X 108, AND THE STOKES RADII,r X lo8,OF SOMESOLUTES IN AQUEOUS SOLUTION AT DIFFERENT TEMPERATURES 1
2
Solute
wt.C, %
3 DI
orn.dec.
4
5
6
7
8
9
10
11
D ~ S
Dgs
Dar
ri
7.18
ras
787
rv
(kT/67r7) X loL3 1.1512 1.7255 2,4230 HDO 0.00 11.28 22.61 1.021 1.072 Urea 0.38 6.885 10.043 13.781 18.08 1.672 1.718 1.758 Glycine .30 5.151 7.606 10,554 13.97 2.235 2.269 2.296 Alanine .32 4.317 6.484 9.097 12.17 2.667 2.661 2.664 Dextrose .38 3.137 4.736 6.728 9.088 3.670 3.643 3.601 Cycloheptaamylose .38 1.492 2.274 3.224 4.362 7.716 7.588 7.516 Bovine plasma albumin .25 .3066 ,4626 .6577 37.55 37.30 36.84 for the protein all other solutes were used at such a concentration as to give about fifty fringes. The values of D for the albumin refer to a 44 fringe pattern and a solvent.of pH 4.6 that was 0.01 molal in both sodium acetate and acetic acid and 0.15 molal in sodium chloride. Prior to the diffusion measurement the protein solution was dialyzed overnight against the buffer at 4-6" and then a t the temperature of the experiment for 1-2 hours. The dialyzate was used to form the boundary with the protein solution.
3.2573 1.802 2.332 2.676 3.584 7.467
1.93 2.59 2.58 2.88 3.53 6.56 27.0
only Perrin's extension of the Stokes-Einstein relation but also Simha's for viscous flow. Since the assumption of the fluid as a continuum in the derivation of the Stokes relation implies that the dimensions of the diffusing particles are large in comparison with those of the solvent molecules it is not surprising that r < rv for small molecules. The reaction rate theory'' suggests that for such solutes the numerical factor in the denominator of the Stokes relation is less than 6a. In the case of the most rapidly diffusing material of Table I, HDO. a factor of 37r would still allow for some asymmetry and interaction with the solvent.
1.06 1.04 0 0.02 0.04 0.06 0.08 0.100.12 0.140.16
Mole fraction D20. Fig. 1.-Variation of the diffusion coefficient of HDO with the mean concentration in the boundary.
Results.-In Table I the solute is listed in the first column and then the mean concentration to which the diffusion coefficients correspond. Subsequent columns of the table contain the observed values of these coefficients a t the temperature denoted by the subscript. These are followed by values of the radius r a t each temperature as computed from the Stokes-Einstein .relation r = kT/GnqD where k is the gas constant, T the absolute temperature and q the viscosity of water. The values for kTI67rq used in the computation of r a t each temperature are given a t the head of the appropriate column. I n the final column of Table I are values of the radius rv computed from the molal volume V of the solute a t zero concentration, Le., V = 4/3?rNrv3 where N is Avogadro's number. I n Fig. 2 the Stokes radius rt for a given solute a t the temperature t has been divided by the value a t 1" and plotted as ordinate against the temperature as abscissa. Discussion.-From the data of Table I it will be seen that the variation of the Stokes radius with temperature is generally small in comparison with the discrepancy between this radius and that computed from the molal volume. In the case of the larger molecules r > rV and the difference is adequately accounted for by hydration, or asymmetry, or such a combination of the two as to satisfy not
$1.02
1.00
Alanine
0.98 0.96
Cyclohepta. amylose
1
13
25 tz,
37
"C.
Fig. 2.-Th,e variation of the Stokes' radius, relative to its value a t 1 , with the temperature for different solutea.
In the case of large molecules the decreasing Stokes radius with increasing temperature is usually ascribed to decreasing hydration, a reasonable explanation since most solutions become more nearly ideal a t high temperatures. An explanation for the fact that the radius of small molecules increases with the temperature must await further development of the theory of the liquid state. However, the data of Table I suggest the generalization that if the solute is large enough for the solvent to be considered a continuum the temperature coefficient of its diffusion will not be less than that predicted by the Stokes-Einstein relation. Most of the twofold increase in D on raising the temperature from 1 to 25" is due to a decrease in the viscosity of the solvent, less than 10% being due to the increase in the kinetic energy, kT,of the diffusing particles. Although the Stokes-Einstein (11) S. Glasstone, K. J. Laidler and H. Eyring, "The Theory of Rate Processes," McGraw-Hill Book Co., Inc., New York, N. Y.,1941, p. 521.
L. G. LONGSWORTH
772
relation predicts a relation between T and the Dq product it does not indicate the temperature dependence of either factor. The reaction rate theory is more useful in this regard, as may be illustrated with the aid of the data of Table I for alanine. I n Table I1 the coefficients given in the second column TABLE I1
TEMPERATURE COEFFICIENTS FOR THE DIFFUSION OF ALANINE 2
1 6
1 13 25 37
Dt
x
105
4.317 6.484 9.097 12.171
3 E D , oal.
4 IO0 01
52% 4783 4457
3.34 2.80 2.41
mobility of the hydrogen ion is one of the arguments for its “non-Stokesian” conductance.12 At infinite dilution the diffusion and electric mobilities of ions are identical, the relation being D = RTX/F2for univalent ions, where X is limiting ionic conductance, F faraday equivalent and D ion diffusion coefficient as measured, for example, with the aid of tracers.la-ls Thus from the extensive conductance and transference data on electrolytes inferences may be drawn as to the diffusion behavior of ions. Some typical data are given in Table IV. The ion diffusion coefficients in TABLEIV ACTIVATION ENERGIES FOR IONDIFFUSION Ion
,
are those for alanine at the temperatures of column 1. Subsequent columns of Table I1 contain derived quantities that are useful in considering the temperature dependence of the diffusion coeffcient. Thus the activation energy, ED-column 3, defined by the relation
H OH
c1
K Nos Na Acetate
Li
varies by 17% over the 36” interval of the table whereas the conventional temperature coefficient 2 Dz - DI (ye-------D2 DI tz - ti changes by some 33%;_column 4. Since E D is more nearly indeFendent of t than a it provides a better index of the temperature dependence of the diffusion coefficient for the comparison of data obtained over different temperature intervals. Values of the activation energy for diffusion have been computed from the data of Table I for the 13-25” interval and are given in Table 111.
Vol. 58
Dzs
x
10‘
93.11 52.7 20.32 19.57 19.01 13.34 10.89 10.30
E D , cal. 18’
3230 3610 4220 4230 4030 4690 4590 5040
column 2 were computed from the conductances a t 2501’ and are thus comparable with the values of Table 111for non-electrolytes. This is also true of the activation energies since those of Table IV were computed from the values of a = (l/X) (dX/dt) at 18” as given in the I.C.T.18 The relation is ED = RT(l UT). If the hydrogen and hydroxyl ions be excluded from consideration, owing to their abnormal transport in water, it is clear that the activation energies for ionic diffusion are similar to those for non-electrolytes and also tend t o increase slightly, though irregularly, with increasing size until the particles are large enough to conform to Stokes relation. I n no instance, TABLE 111 however, are activation energies encountered as ACTIVATIONENERGIES FOR DIFFUSION OVER THE INTERVALlow as those derivable from Oholm’s data. FROM 13 TO 25” Since the variation in the radius r for a given En temperature interval is less than that in ED the Solute DO6 x cal. io. Stokes-Einstein relation is a better interpolation 13.78 4470 Urea formula than equation 1. However, in using this Glycine 10.55 4627 relation, for example, to correct the diffusion Alanine 9.097 4783 coefficient of a protein to the temperature a t which Dextrose 6.728 4959 its sedimentation constant has been determined, Cycloheptaamylose 3.224 493 1 in the evaluation of its molecular weight, account Bovine plasma albumin 0.6577 4969 should be taken of the variation in r. Acknowledgment.-As mentioned above Dr. Oholm appears t o have been correct in his conclusion that the temperature coefficient tends t o Dexter French of Iowa State College supplied the increase with the size of the diffusing particle. cycloheptaamylose used in this research. I am I n order to compare his results with those of Table grateful to him for this interesting material and to 111activation energies have been computed with the Dr. D. A. MacInnes of these Laboratories for his aid of equation 1 and are given in the third row of criticism of this manuscript. the table in the opening paragraph. Although (12) J. D. Bernal and R. H. Fowler J . Cham. Phye., 1, 515 Table I11 covers much the same range of particle (1933). (13) A. W. Adamson. ibid., I S , 760 (1947). size as Oholm’s the new results show a variation in (14) J. H. Wang and J. W. Kennedy, J . Am. Chem. Soc., 72, 2080 ED of only some 11% whereas in the case of the (1950). older data the variation is about 100%. Appar(15) J. H. Wang. ibid., 78, 510 (1951). (16) R. Mills and J. W:, Kennedy, i b i d . , 75, 5696 (1953). ently HCI was the most rapidly diffusing solute (17) D. A. Maclnnes, The Principles of Electrochemistry,” Reinthat Oholm studied. This accounts, in part, for hold Publ. Corp., New York, N . Y.,1039. p. 342. the low value of ED derived from his results since (18) “International Critical Tables,’’ McGraw-Hill Book Co., Inc., the low temperature coefficient for the electric New York, N . Y..1926, Vol. VI, p. 230.
+
+
Sept., 1954
TEMPERATURE DEPENDENCE OF DIFFUSION IN AQUEOUS SOLUTIONS DISCUSSION
D. A. MACINNES (Rockefeller Institute).-How does the hydronium ion fit into your system? L. G. LoNwwoRTH.-If one prepares a plot of the limiting diffusion coefficient as abscissa with the temperature coefficient, e.g., the activation energy for diffusion, as ordinate, the points are scattered about a line with a negative slope. With such a plot there is no clear segregation of the points for the few non-electrolytes for which the data are available from those for ions, Le., a rapidly diffusing particle tends to have a low activation energy whether it is charged or uncharged. Although isolated from any neighbors, owing to its very high mobility, the point for the hydronium ion is not markedly out of line with those for other solutes. Thus it is only in conjunction with other properties of the hydronium ion that its high mobilityiand low temperature coefficient can
773
be taken as evidence for the abnormal transport mechanism of this species. Cox and Wolfenden (Proc. Roy. Soc: (London), A145, 475 (1934)) have discussed the correlation between the ion mobility, its variation with temperature, and the B coefficient in the Debye-Falkenhagen viscosity equation. Small and negative B values are generally associated with high mobilities and low temperature coefficients. It would be of interest to see whether sufficiently mobile non-electrolytes would reduce the viscosity of water, ie., have negative B values. R. M. Fuoss (Yale Universit ) The correlation between the constant B of the Debye-kknhagen viscofiity equation and the diffusion constant suggests that the diffusion coefficient of bi-bolaform electrolytes would be quite small, because their B-values are very much larger than those of simple electrolytes (P. Goldberg and R. M. FUOSS, Proc. Natl. Acad. Sn'. U.S., 38,758 (1952)).
