Temperature Dependence of Oxygen Diffusion in H2O and D2O - The

Studies of oxygen diffusion at room temperature have been carried out with numerous methods, including a diaphragm cell,7 light scattering,8 electron ...
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J. Phys. Chem. 1996, 100, 5597-5602

5597

Temperature Dependence of Oxygen Diffusion in H2O and D2O† Ping Han and David M. Bartels* Chemistry DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: October 3, 1995; In Final Form: January 11, 1996X

The diffusion coefficient of O2 in H2O and D2O has been determined as a function of temperature from -0.5 to 95 °C, using the Taylor dispersion technique with optical absorbance detection at 200 nm. Over this temperature range, significant deviation from both Arrhenius and Stokes-Einstein behavior is found. A practical interpolation formula for the H2O solvent is (T in Kelvin) log10[D/cm2 s-1] ) -4.410 + 773.8/T (506.4/T)2 and, for the D2O solvent, log10[D/cm2 s-1] ) -4.706 + 903.6/T - (526.6/T)2. As a test of the apparatus, the diffusion coefficient of nitrobenzene in water was carefully measured and found to be 1.04 × 10-5 cm2 s-1 at 26 °C and 2.14 × 10-5 cm2 s-1 at 60.1 °C.

I. Introduction In a recent study of the diffusion-limited reactions of H and D atoms with O2 in water,1,2 we desired to find in the literature a high-precision, critically reviewed set of diffusion coefficients for the oxygen molecule in water. We were astonished to find that such data did not exist. Since it is clear that this information is needed in many scientific and engineering venues, we set about performing the measurements ourselves, using the Taylor dispersion technique3-6 with a modified HPLC instrument. This manuscript, we believe, describes the first set of data worthy of inclusion in the critical review article that we desired. There are of course a number of experimental studies7-16 and reviews17,18 of oxygen diffusion in liquids, including water in particular. Studies of oxygen diffusion at room temperature have been carried out with numerous methods, including a diaphragm cell,7 light scattering,8 electron paramagnetic resonance,9 time response of an oxygen electrode,10-12 and the Taylor dispersion method used in our study.13,14 All of the results tend to cluster around 2 × 10-5 cm2/s, but with a great deal of scatter.13,17 Temperature-dependent data are sparse. Some of the earliest data as a function of temperature are derived from observations of the dissolution rate of small oxygen bubbles.15 This method is difficult to implement without introducing systematic errors and actually requires knowledge of the rate of oxygen transfer across the gas-water interface for a correct interpretation. The study of Tham et al.16 measured the O2 diffusion using a stagnant microelectrode over a wide temperature range in concentrated KOH solutions. The data are probably reliable for electrolyte solutions, but extrapolation to the dilute solution limit may be little more accurate than the bubble dissolution method (and certainly gives a different result). Ferrell and Himmelblau14 used the Taylor dispersion method in the temperature range 10-55 °C, but their mass spectrometer detection method lacked high precision and might have introduced some distortion into the concentration derivatives. Much of the “best” data were summarized in Figure 1 of ref 13, which makes it clear that scatter in the literature data is at least 50%. More recently, Wong and Hayduk18 proposed an empirical correlation of many selected sets of dissolved gas diffusion data in a simple Arrhenius form with a molecular size-dependent preexponential and universal activation energy of 18.9 kJ/mol. † Work performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Science, US-DOE under contract number W-31-109-ENG-38. X Abstract published in AdVance ACS Abstracts, March 1, 1996.

0022-3654/96/20100-5597$12.00/0

This correlation gave an average error of 10% relative to the measured values. The easiest method available for diffusion coefficient measurement, that is capable of moderate precision and accuracy (better than 2%, typically5), is the Taylor dispersion, or capillary broadening method.4-6 An apparatus can quickly be constructed from commercial HPLC equipment, assuming a suitable detector for the solute of interest is available. Very simply, given laminar flow of a fluid through a straight cylindrical tube, the shape of a pulse of injected solute will become Gaussian, with variance given by3-6

σ2 )

r2tmax 24D

(1)

where tmax is the time of elution of the peak maximum, r is the radius of the tubing, and D is the diffusion coefficient of the solute through the fluid. The width of the peak is controlled by the rate of radial diffusion coupled to the laminar fluid velocity distribution; this produces the counterintuitive result that faster diffusion is correlated with narrower peaks. The conditions for validity of this simple expression have been reviewed in detail by Alizadeh et al.,4 who also recommend correction terms for practical measurements. In section II we describe our apparatus, while in section III the results are set forth in detail. Comparison with earlier results and a brief discussion of implications is included in section IV. II. Experimental Section The apparatus used for this study is indicated in Figure 1. A commercial HPLC instrument (SpectraSystem from Spectra Physics Analytical) was modified by removal of the column and insertion of a length of 316 stainless steel tubing with nominal internal diameter of 0.02 in., jacketed with 0.25 in. plastic tubing for temperature control. A flow restrictor was introduced between the pump (model P2000) and autosampler (model AS3000) to maintain the proper pressure range at the pump for stable flow. In most experiments, pressure in the flow tube was near atmospheric. For temperatures above 60 °C, a flow restriction was added after the detector to maintain several atmospheres pressure to keep the oxygen in solution. In order to avoid problems with secondary flow (see below), the tubing was coiled in a relatively large 85 cm diameter loop. Temperature-controlled ethylene glycol/water solution was flowed at a high rate through the plastic jacket to maintain © 1996 American Chemical Society

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Han and Bartels the tubing volume is given simply by Vt ) πr2L, we find the average internal radius is r ) 0.0267 cm. Alizadeh et al.4 have shown that the experimental method is insensitive to imperfections in the tubing cross section, and that the average radius as calculated above is the appropriate quantity to use in eq 1. The calibration uncertainty of less than 0.5% is smaller than the other errors in the experiment. III. Results

Figure 1. Block diagram of the modified HPLC apparatus used for the O2 diffusion measurements.

temperature stability to within (0.1 °C. Temperature of the cooling solution was measured by thermocouples insterted into the flow at the inlet and outlet of the jacket. As indicated in Tables 1 and 2, the difference in inlet and outlet temperature for measurements above 90 °C was as much as 1.4 °C. The average of inlet and outlet temperature is listed in the tables for the purpose of analysis, but it should be realized that the diffusional activation is ∼2% per degree near room temperature. The thermocouple readout used (Omega Engineering model HH23) has specified accuracy of ((0.1% of reading + 0.6) °C. Thus temperature measurement and control is among the largest sources of error in our experiment. The main problem to be overcome in this experiment is detection of the oxygen. Typical detectors used for Taylor dispersion experiments are “universal” refractive index devices,4 but we estimated insufficient sensitivity for oxygen due to its limited solubility in water. We found that with injection of 100 µL of oxygen-bubbled water into the degassed solvent, there is just sufficient signal-to-noise ratio to use optical absorption in the range 200-210 nm with the standard 1 cm pathlength cell of the SpectraFOCUS scanning detector. Detection of oxygen was very linear in this wavelength region, as determined from peak heights and integrated areas of various injected sample volumes. Typical baseline noise was 10-5 OD units, but baseline drift or shifts due to refractive index changes could sometimes be 10 times this large, forcing rejection of the run. Since almost any impurity also absorbs in this wavelength region, care was taken to use well-purified water. In filling the sample vials with oxygenated water, a 0.10 µm nylon filter (Corning Glass Works) was used to exclude dust. Both H2O and D2O were purified in a Barnstead Nanopure water purification system before use. The D2O for all of the measurements was recirculated, and solvent samples for O2 saturation and injection were taken directly from the reservoir in case a background impurity absorption should build up. Calibration of the tubing radius was accomplished by weighing it both dry and filled with water to establish the total volume of 3.52 ( 0.01 mL. This agreed very well with the volume calculated from the ratio of pump flow rate over observed retention time. Approximate 40 cm sections were cut from both ends of the tubing loop, carefully weighed, and measured with a ruler to within 0.5 mm. From the weights per unit length, we found no apparent tapering of the tube, and calculated a total length of L ) 1567 ( 3 cm. Assuming that

Some typical data collected in this experiment are illustrated in Figures 2 and 3. In Figure 2A, a pure Gaussian fit is superimposed on the oxygen signal detected at 200 nm in D2O at 36.6 °C following injection of 100 µL of O2-saturated solution. The fit is very good, but plot of the residual of the fit shows it is not perfect. The same oscillatory pattern was present in the residual of all data sets. We experimented with fitting these peaks to an Edgeworth-Cramer series19 to calculate the moments of the signal distribution about the mean and found that the peaks typically have negligible skewness (third moment), but nonnegligible excess (fourth moment). Fitting to the Edgeworth-Cramer series gave much smaller residuals, but exactly the same variance (second moment), and consequently the same diffusion coefficients. Therefore most data were just fit with the simple Gaussian form. In Figure 2B, a slightly asymmetric oxygen peak is displayed together with its residual plot. Although the full scale fit still looks good to the eye, the slight tailing and baseline shift visible upon expansion tend to give diffusion coefficients on the order of 1-2% low relative to symmetric peaks. For reasons never fully understood, the asymmetric peaks appeared frequently, but were rejected in the final data selection process. As mentioned in section II, impurities are a particular concern because of the sensitivity of optical absorption at 200 nm. Data was always collected at 205 nm and 210 nm as well to be certain the peak shapes were the same, and not affected by an impurity. Figure 3 illustrates a non-Gaussian peak with a very broad base that was obtained when some slower diffusing impurity contaminated the sample vial filling syringe. The peak can easily be fit to the sum of two Gaussian peaks with common centers but different widths, as shown in Figure 3B. Moreover, the ratio of the two Gaussians was different for detection at 200, 205, and 210 nm, reflecting the different absorption spectra. For the purpose of calibration of the apparatus and confirmation of our data analysis we attempted to measure the diffusion coefficient of several well-studied standards at room temperature.19 Samples of urea and sucrose that were available were obviously contaminated by impurities absorbing at 200 nm. Extensive data was collected on very pure glycine from two different sources (Aldrich 99+% and recrystallized reagent grade), but we ultimately concluded that signal from some slower diffusing impurity molecule contaminated the glycine peaks at about the 5% level. The peaks detected at 205 and 210 nm consistently gave lower apparent diffusion coefficients than 200 nm, by about 1.5% and 3%, respectively. The most likely impurity is the dimer glycylglycine, which we discovered has a 55 times larger extinction coefficient than the glycine monomer, and roughly 30% slower diffusion at room temperature. The ∆G° for the dimerization reaction is 9.2 kJ/mol,20 so that in a 10-1 molar aqueous stock solution, one can expect a contamination of 0.25% dimer. This contamination would not be observed by a standard refractive index detector, but because of the large glycyl glycine extinction coefficient, our experiment is sensitive to its presence. In order to obtain reliable diffusion coefficients from the data, correction of eq 1 for several experimental imperfections is

Temperature Dependence of Oxygen Diffusion

J. Phys. Chem., Vol. 100, No. 13, 1996 5599

TABLE 1: Oxygen Diffusion in H2O T (°C)

Tin - Tout (°C)

no. of samples

D(O2) (10-5 cm2 s-1)

95% confidence (10-7 cm2 s-1)

-0.5 2.7 3.8 9.2 9.5 12.0 14.7 20.6 21.0 21.0 24.0 24.0 25.3 26.2 26.2 30.2 35.1 40.2 40.8

0.8 0.8 1.0 0.8 1.0 0.8 0.5 0.4 1.0 1.0 0.5 0.5 0.2 0.1 0.1 0.3 0.3 0.3 0.2

4 8 6 3 2 20 7 2 6 7 5 3 3 5 5 6 6 4 4

0.951 1.08 1.09 1.29 1.24 1.45 1.55 1.80 1.78 1.77 2.00 1.98 1.96 2.10 2.08 2.26 2.52 2.78 2.91

1.5 3.4 2.1 4.1 5.4 11 4.6 9.4 7.2 6.1 6.0 15 6.3 3.9 5.3 8.3 6.3 12 6.2

T (°C)

Tin - Tout (°C)

no. of samples

D(O2) (10-5 cm2 s-1)

95% confidence (10-7 cm2 s-1)

45.1 50.0 50.3 55.0 60.5 64.8 65.7 70.3 70.5 75.0 75.4 80.1 80.3 86.3 86.3 89.5 90.3 95.0

0.2 0.4 0.6 0.6 0.4 0.6 0.8 0.8 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.2 1.0 1.4

6 3 12 7 13 4 12 6 7 3 16 4 13 3 12 8 2 3

3.05 3.31 3.45 3.64 4.03 4.25 4.39 4.63 4.70 4.91 5.05 5.31 5.42 5.74 5.64 6.08 6.13 6.80

9.1 6.2 12 9.0 9.0 15 14 5.8 18 14 9.9 11 17 21 31 24 60 52

TABLE 2: Oxygen Diffusion in D2O T (°C)

Tin - Tout (°C)

no. of samples

D(O2) (10-5 cm2 s-1)

95% confidence (10-7 cm2 s-1)

-0.7 3.7 10.2 15.8 20.9 27.0 31.8 36.6 41.8 46.6 51.6 56.4 61.3 66.6 71.0 76.3 85.0 89.9 96.6

1.0 0.8 0.6 0.4 0.3 0.2 0.3 0.2 0.2 0.4 0.4 0.4 0.4 0.8 0.6 0.8 1.0 1.2 1.4

6 7 6 5 4 3 6 7 8 4 8 4 5 4 6 4 5 4 3

0.746 0.883 1.03 1.25 1.41 1.73 1.93 2.14 2.35 2.49 2.73 2.90 3.30 3.41 3.69 3.90 4.63 4.85 5.10

2.3 2.3 5.8 7.1 4.0 15 12 8.8 2.3 11 18 11 8.5 15 20 41 16 26 44

needed. Following Alizadeh et al.,4 we correct our results for finite injection and detector volumes with eqs 2a,b:

(

to ) tmax 1 σ ) to 2

2

{

) ( ) ( )}

Vi Vd 2Vt 2Vt

Vd r2 1 Vi 2 + +a 24Dto 12 Vt Vt

(2a)

2

(2b)

tmax and σ2 are the observed peak maximum and variance, respectively. Vi and Vd are the injection and detector volumes, while Vt is the tubing volume. The constant “a” is to be determined below. In simplifying the equations of Alizadeh et al. we have ignored correction terms which are less than 0.5%, since other errors are larger. Note that in the bracketed sum of eq 2b, the small correction terms involving Vi and Vd are constants, but the dominant diffusion term becomes smaller with time. As a result the corrections become more important for long retention times (low flow rates). The corrections reflected in eqs 2a,b are illustrated with the nitrobenzene diffusion data in Figure 4. Nitrobenzene was chosen for this purpose because consistent results were obtained and the solubility and absorbance are large enough to allow small injection volumes. Apparent diffusion coefficients are plotted vs the time of elution tmax. Uncorrected data, using the

Figure 2. Typical data obtained in the study of O2 diffusion. On the left, a symmetric peak is found following 100 µL injection of O2-bubbled D2O at 36.6 °C. A Gaussian fit is superimposed and the residuals are plotted below. The peak on the right, observed in D2O at 66.6 °C, shows some tailing and a baseline shift and was rejected in the data analysis.

measured peak widths and times in eq 1 directly, are shown for 100 (asterisks), 50 (triangles), and 10 µL (crosses) injections of the nitrobenzene stock solution, respectively. The effect due to injection volume is not unexpected, given that 100 µL approaches 3% of the total system volume. Application of the injection volume correction in eq 2 brings the 100 and 50 µL injections into good agreement with the 10 µL injection data, but a slight tailing off of the diffusion coefficient at low flow rate still persists, as shown in the inset of Figure 4 for the 10 µL data. We ascribe the residual tailing off at low flow rates to the effect of the detector volume, which is 15 µL according to the manufacturer. (The small diameter tubing connecting the diffusion tube to the absorbance cell contributes negligible additional volume or flow time.) The third term of eq 2b reflects this correction, but includes an undetermined coefficient “a”. Alizadeh et al. suggest two limiting models by which to estimate the coefficient.4 If the detector integrates signal over a finite volume ) πr2Ld of the diffusion tube, with the laminar flow undisturbed, then the constant “a” should be set to 1/12, and the

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Figure 3. Oxygen sample at 16.1 °C in D2O contaminated with some impurity that absorbs at 200 nm. On the left, a single Gaussian fit is superimposed. On the right, two Gaussians with common center are used to obtain an excellent fit.

Figure 4. Apparent diffusion of nitrobenzene in water at 60.1 °C vs retention time, calculated with and without corrections for injection and detector volumes. Injection volumes are 100 µL (asterisks), 50 µL (triangles), and 10 µL (crosses). Fully corrected 10µl injection data is indicated by the solid circles. Dashed line corresponds to a ) 1, solid line to a ) 0.85, and dotted line to a ) 0.7 in eq 2b. The solid line fit for small tmax is calculated for tubing coiling radius of 54 cm.

correction is just the same as for finite injection volume. If the detector integrates over a volume of solution which is perfectly mixed in the detector after exiting the diffusion tube, the constant “a” is set to unity.4 We assume our detector falls somewhere between these limits. The lines in the inset of Figure 4 illustrate predicted results with “a” set to 0.7, 0.85, and 1. We choose a ) 0.85 as the best value with which to correct our data to get consistent results for long retention times or low flow rates. The nitrobenzene diffusion coefficient at 60.1 °C is thus found to be 2.14 × 10-5 cm2 s-1. A similar experiment at 26.0 °C yielded 1.04 × 10-5 cm2 s-1. At these two temperatures, the group Dη/T is constant to within 1%. Thus nitrobenzene diffusion appears to follow Stokes-Einstein (hydrodynamic) behavior very well, a result which has also been found for other benzene derivatives in water.21 Thus far we have ignored the short retention time or high flow rate data in Figure 4, which gives anomalously narrow

Han and Bartels

Figure 5. Diffusion coefficients for O2 in H2O vs retention time. Corrections are applied for finite injection and detector volumes. Apparent larger values at retention times less than 12 min result from tube coiling.

peaks or high diffusion coefficients. A finite time is required for the laminar flow to convert the sample concentration profile from rectangular to Gaussian. However, from eqs 14-21 of ref 4 we can estimate that the transient terms of the concentration profile should decay well within 1 min in our apparatus. Another reason for incorrect results at high flow rates is secondary flow caused by coiling of the tubing. An approximate correction factor depending on the solvent viscosity and density, the average flow velocity, tubing radius, coiling radius, and solute diffusion coefficient is proposed by Alizadeh et al.4 on the basis of the results of Nunge et al.22 By using the nitrobenzene diffusion coefficient determined above, the solid line fitting the short retention time data of Figure 4 can be calculated for an effective coiling radius of 54.4 cm from eqs 66-71 of ref 4. This is somewhat larger than the actual physical radius of ca. 43 cm, but the qualitative agreement demonstrates the source of the anomalously high results. Corrected data for oxygen diffusion in H2O over a wide temperature range are illustrated in Figure 5. The same corrections used for the nitrobenzene data appear to work well for oxygen over the entire temperature and flow rate regime. Because the 100 µL injection volume was required to obtain sufficient signalto-noise ratio, the corrections needed in eq 2b were on the order of 5-30% of the measured peak variance. The high diffusion result due to the tube coiling is visible in the shortest retention time data of Figure 5 as well. In averaging the final results, only retention times greater than 12 min were used. A summary of all the data collected for H2O is included in Table 1, and data for D2O are compiled in Table 2. The average of jacket inlet and outlet temperatures is listed together with the difference so that the temperature uncertainty can be evaluated. The average at each temperature is calculated from the number of different experiments indicated. Confidence intervals of 95% are then calculated from the standard deviation of the mean and the Student’s t distribution. All results for both H2O and D2O are plotted in Arrhenius form in Figure 6, with error bars indicating the 95% confidence intervals. A practical interpolation formula for the H2O result weighted according to the confidence intervals is (T in Kelvin)

log10[D/cm2 s-1] ) -4.410 + 773.8/T - (506.4/T)2

(3)

and for the D2O

log10[D/cm2 s-1] ) -4.706 + 903.6/T - (526.6/T)2

(4)

Temperature Dependence of Oxygen Diffusion

Figure 6. Arrhenius plot of oxygen diffusion coefficient in H2O and D2O. Error bars represent 95% confidence intervals.

Figure 7. Comparison of our oxygen diffusion results (solid line) with data of previous workers: Ferrell and Himmelblau,14 Wise and Houghton,15 Vivian and King,7 and O’Brien and Hyslop.8 Dashed line represents our results multiplied by 1.3, which interpolates the Wise and Houghton data.

IV. Discussion The results of this study (solid line) are compared with some selected earlier work in Figure 7. The present work finds consistently lower diffusion coefficients than the other studies presented in the figure, although many lower values at room temperature can certainly be found in the literature.17 It is interesting to note that multiplication of our results by 1.2 provides an excellent interpolation for the results (10-55 °C) of Ferrell and Himmelblau,14 whereas multiplication by 1.3 (dashed line) nicely interpolates the bubble dissolution results (10-60 °C) of Wise and Houghton.15 Thus all three studies agree on the temperature derivative of the oxygen diffusion, and only the scale factor is at issue. The microelectrode study of Tham et al.16 apparently found a higher activation energy than these three, and is not included in the figure. At room temperature, we should note the early study of Vivian and King,7 whose diaphragm cell experiment reportedly produced a result of 2.41 × 10-5 cm2 s-1 with 3% precision. This method is generally cited as one of the most reliable for diffusion coefficient measurements,5,16 but the result is 20% higher than our own, and we can provide no explanation. Also of interest is the transient interferometry study of O’Brien and Hyslop.8 Their result of 2.00 × 10-5 cm2 s-1 at 22 °C, with uncertainty of ∼2.5%, is slightly higher than our own (interpolated) result of 1.85 × 10-5 cm2 s-1, although an honest accounting of errors, particularly in the temperature, might find the experiments in good agreement. Wise and Houghton’s

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Figure 8. The group D(O2)*η/T plotted against η (viscosity in cP) for both H2O and D2O solvents. For pure Stokes-Einstein (hydrodynamic) friction, D(O2)*η/T should be constant.

bubble dissolution method15 was criticized by Ferrell and Himmelblau14 as relying on a complex mathematical model which probably omits certain important corrections. Comparison of the Wise and Houghton results for other tracer gases to the very precise recent diaphragm cell measurements of Ja¨hne et al.23 seems to confirm this assessment: the Wise and Houghton results are scattered above and below the Ja¨hne et al. results, although with generally the correct temperature dependence. Explanation of our disagreement with Ferrell and Himmelblau14 is more difficult. These workers used the Taylor dispersion technique with a very straight six meter tube, and mass spectrometric detection of the dissolved gases. A concentration jump method was adopted rather than the “delta pulse” injection used in our study. The reproducibility of the measurements were on the order of 6-12% standard deviation, partly owing to the difficulty of separating the gases from water for analysis. Given the relatively poor precision, and the single edge detected per experiment rather than two as in our work, we suspect that a systematic distortion error on the order of 20% might have been very difficult to discern. The Ferrell and Himmelblau results on He and H2 diffusion24 are about 15% lower than those of Ja¨hne et al.23 However, their result for CO2 is in excellent agreement with many other determinations14,23 at room temperature. Turning now to the temperature dependence of the O2 diffusion coefficient, we can note immediately from the curvature in Figure 6 that a simple Arrhenius law as proposed by Wong and Hayduk18 does not describe the data well. A number of workers have found that diffusion coefficients of tracer molecules can be fit over a very wide temperature range with a formula of the form D ) DoTηb, where η is the viscosity of the fluid.25-27 For b ) -1, one recovers the form of the Stokes-Einstein hydrodynamic formula. In Figure 8 we plot the ratio Dη/T vs η in linear log format for both the H2O and D2O solvents. One can see this formula does a reasonable job of correlating the data, with b ) -0.898 for H2O and b ) -0.836 for D2O. Deviation from pure hydrodynamic (StokesEinstein) behavior is substantial. However, no particular meaning has been assigned to the parameters of this formula. To our knowledge, no rigorous theoretical treatment of diffusion has successfully been applied to small hydrophobic tracer molecules in water. An analytical theory for tracer diffusion and viscosity in hard sphere fluids has been formulated, and found to reproduce the results of molecular dynamic simulations as a function of density.28 In order to treat real

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Han and Bartels

molecules, the “rough hard sphere” approximation is invoked, which suggests that the diffusion of the (quasi-spherical) tracer is directly proportional to, but slower than, the corresponding smooth hard sphere.6,29 Experimental data for tracer diffusion in hydrocarbon liquids has been shown to follow this behavior very well over a large range of density, viscosity, temperature, size, and mass.25,30-32 However, the approximation is not expected to hold in associated liquids, and particularly in water,29 because short-range attractive forces are present in addition to the short-range repulsions that dominate most liquid transport properties. Nevertheless, a modification of the smooth hard sphere equations was found to correlate tracer diffusion in associated liquids (simple alcohols) by invoking an effective association number β(T) for the solvent molecules:33

D12(βM2)1/3M11/6V11/3 2/3

1/2

(βV2) (RT)

) 2.49 × 10-9

[

]

V -1 (βV2)

(5)

In this equation, D12 is the diffusion coefficient of tracer 1 in solvent 2, Mi is the molar mass of 1 and 2, Vi is the molar volume of close-packed 1 and 2, and V is the actual molar volume of the fluid. (The equation is based on a fit of hard sphere molecular dynamics results valid for V1/V2 from 0.5 to 1.6, M1/M2 from 0.5 to 4.0, and V/V2 from 1.5 to 3.0.27) This “effective association number” ansatz was also found to correlate the viscosity η2 of the associated liquids using the same parameters:33

(βM2RT)1/2 η2(βV2)2/3

) 1.50 × 108

[

]

V -1 (βV2)

(6)

One could possibly apply the same equation to correlate tracer diffusion and viscosity in water. Elimination of [V/(βV2) - 1] from eqs 5 and 6 implies that the the group D12η2/T should always obey the same temperature dependence in a given solvent. Indeed, benzene diffusion and water self-diffusion deviate from the smooth hard sphere equations in exactly the same way as a function of temperature (closely following Stokes-Einstein behavior),21 implying a common β(T) function. However, our results for O2, and the results of Ja¨hne et al.,23 indicate that this equation cannot also explain the diffusion coefficients of small hydrophobic gases around room temper-

ature, as the activation energy for diffusion in these studies is dependent on the size of the solute. We conclude that a “sticky hard sphere” theory is probably required to treat transport properties in water. References and Notes (1) Han, P.; Bartels, D. M. In Ultrafast Reaction Dynamics and SolVent Effects; Gauduel, Y.; Rossky, P. J., Eds.; AIP Conference Proceedings 298; American Institute of Physics: New York, 1994. (2) Han, P.; Bartels, D. M. J. Phys. Chem., to be submitted. (3) Taylor, G. I. Proc. R. Soc. 1953, A219, 186. (4) Alizadeh, A.; Nieto de Castro, C. A.; Wakeham, W. A. Int. J. Thermophys. 1980, 1, 243. (5) Cussler, E. L. Diffusion. Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, 1984. (6) Tyrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworths: London, 1984. (7) Vivian, J. E.; King, C. J. AIChE J. 1964, 10, 220. (8) Obrien, R. N.; Hyslop, W. F. Can. J. Chem. 1977, 55, 1415. (9) Sridhar, T.; Potter, O. E. Can J. Chem. Eng. 1978, 56, 399. (10) Ju, L.-K.; Ho, C. S. Biotech. Bioeng. 1985, 27, 1495. (11) Ju, L.-K.; Ho, C. S.; Baddour, R. F.; Wang, D. I. C. Chem. Eng. Sci. 1988, 43, 3093. (12) Linek, V.; Vacek, V. Biotech. Bioeng. 1988, 31, 1010. (13) Sridhar, T.; Potter, O. E. Chem. Eng. Commun. 1983, 21, 47. (14) Ferrell, R. T.; Himmelblau, D. M. J. Chem. Eng. Data 1967, 12, 111. (15) Wise, G. L.; Houghton, G. Chem. Eng. Sci. 1966, 21, 999. (16) Tham, M. K.; Walker, R. D.; Gubbins, K. E. J. Phys. Chem. 1970, 74, 1747. (17) St. Denis, C. E.; Fell, C. J. D. Can J. Chem. Eng. 1971, 49, 885. (18) Wong, C. F.; Hayduk, W. Can. J. Chem. Eng. 1990, 68, 849. (19) Harris, K. R. J. Solution Chem. 1991, 20, 595. (20) Lehninger, A. L. Biochemistry; Worth Publishers: New York, 1975. (21) Tominaga, T.; Matsumoto, S.; Ishii, T. J. Phys. Chem. 1986, 90, 139. (22) Nunge, R. J.; Lin, T. S.; Gill, W. N. J. Fluid Mech. 1972, 51, 363. (23) Jahne, B.; Heinz, G.; Dietrich, W. J. Geophys. Res. 1987, 92, 10767. (24) Ferrell, R. T.; Himmelblau, D. M. AIChE J. 1967, 13, 702. (25) Chen, S.-H.; Davis, H. T.; Evans, D. F. J. Chem. Phys. 1982, 77, 2540. (26) Chen, S.-H.; Evans, D. F.; Davis, H. T. AIChE J. 1983, 29, 640. (27) Sun, C. K. J. ; Chen, S.-H. AIChE J. 1985, 31, 1510. (28) Sung, W; Stell, G. J. Chem. Phys. 1984, 80, 3350; 1984, 80, 3367. (29) Chandler, D. J. Chem. Phys. 1974, 60, 3500; 1975, 62, 1358. (30) Evans, D. F.; Tominaga, T.; Davis, H. T. J. Chem. Phys. 1981, 74, 1298. (31) Chen, S.-H.; Davis, H. T.; Evans, D. F. J. Chem. Phys. 1981, 75, 1422. (32) Chen, B. H. C.; Sun, C. K. J.; Chen, S.-H. J. Chem. Phys. 1985, 82, 2052. (33) Sun, C. K. J. ; Chen, S.-H. Ind. Eng. Chem. Res. 1987, 26, 815.

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