I
K. KEITH INNES, Vanderbilt University, Nashville, Tenn. LYLE F. ALBRIGHT, Purdue University, Lafayette, Ind.
Effect of Temperature and Solute Volume on Liquid Diffusion'Coefficients This method for predicting diffusion coefficients may have wider usage as future data are accumulated R E L I A B L E METHODS for correlating liquid diffusion coefficients over wide ranges of temperature and solute volume are important. The most useful method seems to be that of Wilke (73, 7 4 , which is based on the Stokes-Einstein relationship,
D s / T = K/V113= F (1) where D is the diffusion coefficient; q , viscosity of solvent a t temperature T; V , volume of the diffusing molecule; and K , a constant. Assuming D r / T to be constant, Wilke tested Equation 1 for a given system over only small temperature ranges. The third column of Table I shows the extent of its failure for larger ranges. Unfortunately, it is impossible to say how much of the error in predicted D values comes from multiplication of two large experimental errors in the product Dq. Wilke also found, and it has been confirmed (7), that values of D q / T can be correlated with fair accuracy (10 to 15%) as a function of V , which he estimated from atomic volumes. For rate processes such as viscosity and diffusion in liquids, a large body of experience in chemical kinetics suggests use of a three-constant Arrhenius equation. Except within 20' to 25' C. of the freezing points, viscosity data of liquids can be correlated accurately (5) over large temperature ranges by such an equation. For liquid diffusivities, we expect the similar empirical form, D
= ATne-B/T
(2)
where A, B , and n are the constants. Self-diffusion data are now available which show appreciable deviations from the two-constant Arrhenius equation (3, 72) as well as from Equation 1 (Table I). Equation 2 accurately correlates these data and those of Longsworth ( 9 ) for several solutes in dilute water solutions. An accurate method for predicting diffusion coefficients of various solutes in water is also presented. Effect of Temperature Self-diffusion data for water, heptane, and tin were treated by Equations 1 and 2 (Table I). Both methods represent D values of water within 2% over a temperature range of 55" C. Equation 1, however, gives a poor representation of
heptane data, but within experimental reproducibility shown by Fishman (3), Equation 2 covers the entire range of 175' C. The same is true for the 360' C. range for tin ( 2 ) . From similar results (5) for viscosities of water and heptane, diffusivities calculated by Equation 2 for temperatures of 50' to 75' C. above the ranges of Table I, will be accurate to within 3y0. Also, precise data (9) for diffusion of urea, alanine, and cycloheptaamylose in water a t temperatures from 1' to 37' C. may be accurately fitted by Equation 2 with n = 11, which is the value used for self-diffusion of water. Therefore, n,may be considered as primarily a function of the solvent, while A and B depend on the solute. From the absolute rate theory (4),
-
D = ( V / N ) 2 ' 3( k T / h )
e-AH/RT
(3)
where N , R, k , and h are Avogadro's number, the gas constant, Boltzmann's and Planck's constants, respectively; V is the molar volume of solvent; AH is the heat of activation for diffusion; and A S is the entropy of activation. Since the volume term is only slightly temperature-dependent, comparison with Equa-
tion 2 for n # +I implies some variation of AS and AH with temperature. If it is assumed that the difference in heat capacities, Acp, between nonactivated and activated molecules is independent of temperature, substitution in Equation 3 yields Equation 2 with A
=
( V/N)213( k / h ) e A s o / R e-Acp/R; n = ( A c , / R ) -I- 1 and B = AH,/R (4)
The empirical parameters thus allow calculation of ASo, AH,, Ac,, AS, and AH. I t is particularly interesting to compare A S for substances of different types; for heptane it is negative for all temperatures in Table I. This is a result similar to that found by analyzing the viscosity data of heptane ( 5 ) for a comparable temperature range and indicates that both self-diffusion and viscous flow of heptane are cooperative processes. I n contrast, AS for tin is positive for all temperatures. Thus, for tin, monatomic diffusion is indicated. For reasons already discussed ( 5 ) it is not worthwhile to interpret the empirical equation for liquids near their freezing points. The high value of n found here for water may be taken as symptomatic of a liquid
Table 1.
Temperature Dependence of Diffusivities D ,Calcd. from , Sq. Cm./Sec. Eq. 1 Eq. 2 D20 in HzO"
(D
To,K.
x
106)obsd
273.2 278.2 288.2 298.2 308.2 318.2 328.2 373.2
1.00 1.20 1.61 2.13 2.76 3.45 4.16
194.7 250.3 273.2 305.7 329.7 353.4 369.0 398.0 423.8
0.415 1.52 2.08 3.22 4.21 5.76 6.56
...
1.00 1.19 1.62 2.14 2.75 3.43 4.18 8.01
0.98 1.18 1.63 2.15 2.75 3.43 4.18 8.48
Tritium-Tagged C7Hle in C ~ HE* I
'
... ...
0.314 1.44 2.11 3.44 4.73 6.24 7.39
... ...
0.446 1.43 2.08 3.27 4.36 5.65 6.60 8.62 10.69
Self-Diffusion of SnC 572.2 665.5 673.2 854.2 935.2
3.69 6.55 6.80 11.6 13.5
4.61 6.39 6.56 10.7 12.8
3.79 6.33 6.57 11.8 13.7
Fig. 1
... ... ... ... ... ... ... ...
... ... ... ... ... ... ...
8.74 11.45
... ... ...
...
...
from ( 1 2 ) . I n Equation 1, F = 6.45 X 10-10; in Equation 2, A = 5.4075 X loa', n = -11, and B = 5619.3. * Dobsd. from (8). F = 2.933 X 10-10, A = 2.358 X 10-9, n = 2 , and B = 583.8. Dobsd. from ( e ) ; viscosities from ( 1 1 ) . A = 1.332 X 107, n = -3, B = 5428.4; F = 13.44 X Dobed.
VOL. 49, NO. 10
OCTOBER 1957
1793
l'he fifth column of 'I'able 1 gives diffusivities using approximate values of AHu (70). A comparison with column 4 indicates ielatively good agreement between the two methods. When A H v is known accurately at high temperatures, predictions from Figure 1 might be used to adjust the constants of Equation 2; thus, all predictions are included in a sinqle analytical equation.
400c
300C
2
Effect of 200c
>
I
e
I000
C REDUCED
T E M P E R A T U R E , T/TC
Figure 1. Correlation of heptane selfdiffusion d a t a structure which changes rapidly in the temperature range considered. Extrapolation of D to high temperatures was converted to an interpolation because the ratio AH,,/D where AH, is molar heat of vaporization of the solvent, must be zero at the critical temperature. Extrapolation to Tc for heptane is shown in Figure 1. Assuming that the curve tends smoothly to zero, values of AH,/D a t 124.8" and 150.6' C. can be estimated.
Table II. Substance Methyl Ethyl Propyl Butyl Isoamyl Allyl Resorcinol Glycine
Solute Volume
Volume of the diffusing molecule used by Wilke (13, 74) in his correlation was obtained by the principle of additive volumes; but experimental molal volumes for solutes in water are available, and the diffusion data are the most precise (8, 9). For many other solutes, molal volumes can be calculated from density data. I n all calculations reported here, experimental volumes were used which, because only dilute solutions were considered, are apparent molal volumes. These volumes were constants over the temperature range considered. If the exponents of Equation 3 are combined, free energy of activation may be calculated from observed diffusion coefficients. For solutes not strongly ionized in water as a solvent, free energy is a rather smooth function of the apparent molal volume of the solute, increasing sharply for small volumes (gases), and only gradually for larger volumes. The average error in D values calculated from such a curve is about 10%. The same volumes were plotted against F values (Equation 1) and
1794
DVO.70 = 16.2 X IO-'
DVo.48 = 3.14 X 10-6
Alcohols, 37.7 54.7 70.0 86.0 85 60.6
88
Amino Acids and Sugars, 1' C. 43.5 5.15 58.9 4.50 60.6 4.32 60.8 4.20
12.8 9.8 8.3 7.2 7.2 9.0 7.1
INDUSTRIAL AND ENGINEERING CHEMISTRY
(6)
a t 1' C. (average deviation, 2.5%). Similarly, for the latter class of substances at 25" C., . D V 0 . 4 ~= 61.4 X 10-6 (7I From these results a general relation for a given temperature could be
Dv"
= I('
(8)
where m and K' are determined for each class of substances. For example, if better data were available, a separate equation probably could be determined for diffusivity of gases in water. Equation 8 is recommended for predicting D values when water is the solvent. Data are inadequate for most other solvents since few apparent molal volumes are known and measured L) values are generally not of high accuracy. However, as data for nonaqueous solvents are made available Equation 8 should prove more valuable. Literature
V"
(5)
at 15" C. (average deviation, 3..5%,) while for amino acids and sugars in which hydrogen-bonding is not expected to be so important,
Volume Dependence of Diffusivities
5.14 4.44 dl-a-Alanine 4.38 dl-Serine 4.37 dl-a-Aminon-butyric acid 76.5 3.89 3.92 Diglycine 77.2 3.79 3.90 2-Proline 81.0 4.19 3.81 Hydroxyl-2-proline 84.4 3.93 3.74 &Valine 91.3 3.57 3.60 Z-Histidine 99.3 3.45 3.46 &Leucine 107.5 3.33 3.33 dl-Norleucine 108.4 3.33 3.31 Dextrose 111 3.14 3.28 Glucose 111.9 3.14 3.26 Triglycine 113.5 3.18 3.24 dl-Phenylalanine 121.3 3.24 3.14 Glycyl-&leucine 2.87 139.8 2.93 dl-Leuc ylglycine 143.2 2.83 2.90 Sucrose 209.9 2.41 2.41 Raffinose 300.8 2.01 2.03 a For alcohols, calculated from density data of (6);for amino acids and sugars, from (8). For aIcohoIs from (7); for amino acids and sugars, from (8). dl-P- Alanine
again average deviation of the calculated D values was about 10%. Because O S simplicity, the latter method is preferred. Major deviations in these. two volutnr plots are for alcohols and for gases where available data are least accurate. This suggests that predictions of the preceding paragraph may be improved by considering several broad classes of solutes separately. For alcohols in Table 11, which are taken as defining a class of hydrogen-bonded substances,
Cited
( 1 ) Caldwell, C. S., Babb, A . L., J . Phyr. Chem. 60, 51 (1956). ( 2 ) Careri, G., Paoletti, A.. Nuovo czmento 2, 574 (1955). ( 3 ) Fishman, E., J. Phys. Chem. 59, 469 (1955). ( 4 ) Glasstone, S., Laidler, K. J.: Eyring, H., "Theory of Rate Processes," p. 524, McGraw-Hill, New York, 1941. ( 5 ) Innes, K. K., J . Phjs. Chem. 60, 817 (1956).
( 6 ) International Critical Tables, vol. 3, p. 115, McGraw-Hill, New York, 1929. ( 7 ) Ibid., vol. 5 . p. 63. ( 8 ) Longsworth, L. G., J. Am. Chem. Soc. 74, 4155 (1952); 75, 5706 (1953). ( 9 ) Longsworth, L. G., J . Phys. Chem. 58, 770 (1954). (10) O'Hara, J. B., Fahien, R. W., Am. Doc. Inst. 3327, Library of Congress, Washington, D. C. (11) Stott, V. H., Proc. Phys. SOC.45, 530 (1933). (12) Wang,J. H., J . Am. Chem. SOC.73, 510 (1951 ). (13) Wilke, C. R., Chem. Eng. Progr. 45, 219 (1949). (14) Wilke, C. R., Chang, P., A.1.Ch.E. Journal 1, 264 (1955).
RECEIVED for review April 13, 1956 ACCEPTED March 20, 1957
