lo00
J. Phys. Chem. 1992,96, 1000-1005
an insignificant part of important studies which we can foresee. We encourage others to undertake such studies which they find to be of interest. We would appreciate it if they notified us of their intentions so that extensive duplication of effort can be avoided.
Acknowledgment. This work was supported in part by Grant No. CHE8717791 from the National Science Foundation. Professor Peter Bowers of Simmons College was also present during part of the time and is developing some theoretical interpretations of some of the observations.
An Alternative Model for Transport of Molecules between Gas and Solution Richard M. Noyes,* Mordecai B. Rubin,**+and Peter C. Bowers*i* Department of Chemistry, University of Oregon, Eugene, Oregon 97403 (Received: July 19, 1991; In Final Form: September 24, 1991)
For almost 70 years, transport of molecules like Nz between gas and solution has been assumed to take place by diffusion through a surface. film. We believe that gases and liquids have such different viscosities that such a film would be unstable to even modest stirring of the liquid. We propose instead that transport of a molecule between gas and surface takes place in a single translational step and that transport between surface and bulk solution is accomplished by physical mixing without significant contribution from molecular diffusion. The two models applied to data collected 50 years apart lead to identical limiting rates of transport at high stirring rates. Our model also predicts that limiting rate of transport is independent of apparatus, and it permits direct calculation of the accommodation coefficient, y, or probability that a molecule striking the surface will become dissolved in the bulk liquid. Because models of transport employed previously assumed diffusion through a surface film of unknown thickness, rates of transport could not be used to measure accommodation coefficients. If our model is valid, we find that 7 is in the neighborhood of lo4 for the hydrogen-bonded solvents water, aqueous glycerol, and concentrated sulfuric acid. It rises to lo-’ for the nonpolar solvent heptane. These values of y are much less than those around 10-’ which have been reported for solutions like HCl or SOzin water where gas molecules ionize almost instantaneously upon dissolving. If our proposed model is indeed valid, measurement of rates of transport of gases to sufficiently well mixed solutions should provide data of direct applicability to many problems of interphase transport. We believe that for decades chemical engineers have been depriving themselves of techniques which could have been useful for their work.
I. The Cooventiooal Model for Transport Devebpmed of the Accepted ModeL The transport of molecules between gas and solution is important to many processes of concem to physical chemists and chemical engineers. As nearly as we can tell, attempts in the past and current literature to describe such a process have postulated that transport takes place by Fick’s law diffusion through a stagnant film which separates the bulk solution from the phase in contact with it. The first reference to such a film which we have identified was due to Noyes and Whitneyl in 1897 when physical chemistry was scarcely identified as a specific discipline. Those authors prepared previously fused cylinders of benzoic acid and of lead chloride and measured their rates of solution in water during rotation. Nernst2also considered solution of solids and in 1905 specifically postulated a surface layer whose thickness depended upon the rate at which the solution was stirred. The first reference we have found which extended the model to solutions of gases was by Lewis and Whitman’ in 1924. We have not been able to determine whether their development was independent of Nemst and whether there were intermediate papers. The discussion below will show that the extension of the stagnant film model from solutions of solids to solutions of gases may not have been valid. The model of Lewis and Whitman can be illustrated for the transport of nitrogen by the process of eq 1. N2(gas) it NJgaseous surface film] e Nz[liquid surface film] it N,(bulk) (1) Each film in eq 1 is considered to be in equilibrium with an adjoining region at the interface where they are in contact, and transport between gas and liquid is by diffusion through the films. ‘Permanent address: Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel. :Permanent address: Department of Chemistry, Simmons College, 300 the Fenway, Boston, MA 021 1 5 .
Subsequent applications of the model have generally treated the surface films to be stagnant while diffusion takes place through them, although Higbie4 and DanckwertsS have considered the possibility that material in the liquid surface film might be interchanged with the bulk solution as a result of stirring. The use of this mode4 for various situationshas been presented in a textbook by A s t a r i d and by many other authors. Because diffusion coefficients in gases at 1 atm or less are at least 3 orders of magnitude greater than those in liquid solutions, applications of the model often assume that the gaseous surface film offers negligible resistance to transport. Equation 1 can then be simplified to eq 2. N2(gas) e N2(surface layer) e N,(bulk) (2)
A Previous Experimeatrl Application of the Model A typical application of the model is a study in 1937 by Hutchinson and Sherwood’ (HS) which used eq 3. ( V / A ) dC/dt = kL(Ci - C‘) (3) In this equation, Vis the volume of solution of concentration C, A is the area of solution in contact with gas, and C, is the concentration of the volatile solute at the gas-liquid interface of the liquid surface film. The rate constant kL has the dimensions cm s-I. The quantity on either side of eq 3 is the flux in mol cmw2 s-I through the surface between solution and gas. HS established that the value of kL increased with rate of stirring, and Figure 1 taken from their paper shows a log-log plot ( 1 ) Noyes, A. A.; Whitney, W. R. 2.Phys. Chem. 1897, 23, 689-692.
(2) Nernst, W.Z. Phys. Chem. 1904, 47, 150-158. (3) Lewis,W. K.;Whitman, W. G. Ind. Eng. Chem. 1924,16,1215-1220. (4) Higbie, R. Trans. AIChE 1935, 31, 365-389. (5) Danchverts, P. V. Ind. Eng. Chem. 1M1,43, 1460-1467. ( 6 ) Astarita, G. Mass Transfer with Chemical Reaction; Elsevier: Amsterdam, 1967. (7) Hutchinson, M. H.; Sherwood, T. K. Ind. Eng. Chem. 1937, 29, 836-840.
0022-3654/92/2096- 1000$03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 1001
Transport of Molecules between Gas and Solution
4 1.0
0.6
0.4
‘‘TO
2
4
4 8
w)
20 40 bobom STIRRING SPffO. R.PU.
200
4 0 0 b00
loo0
Figure 1. Rates of transport of four different gases to water at 25 OC as a function of stirrer speed. Experiments used 50 cm3of water in a cylindrical absorption flask. Reproduced from ref 7.
of their data for four different gases in water at 25 OC. The rate of transport is quite small at very slow stirring speeds but then rises rapidly. The slope in the right side of Figure 1 is less than unity, indicating that in this region relative increases in rates of transport are less than those in stirring rates. HS treated the data on the right side of Figure 1 by means of eq 4. I/kL = x / D + RE (4)
In this equation, x is the thickness (unmeasured and not really known) of the surface layer and will presumably approach zero at indefinitely large rates of stirring, D is a parameter having dimensions cm2 t-I, and RE is called ”resistance of eddy layer” with dimensions cm-l t . The numerical data reported for N2 in water can be summarized by the equations kL(171 rpm) = 2.6 cm h-’ = 7.2 X
kL(x=O) = I / &
cm
(5)
= 4.6 cm h-’ = 1.28 X lo-’ cm s-’ (6)
In these equations, kL (171 rpm) is the value at the indicated stirring speed, while kL(x=O) is the extrapolated value at infinite stirring speed when the surface film will have disappeared. HS take pains to claim that the numerical values in eq 5 and 6 are for their specific apparatus and are not of general applicability. Equations 4 and 6 will be discussed further in section 111. A Question about tbe Validity of tbe Conventional Model. The experimental results reported in the present paper have made us concerned about the validity of the conventional model because of problems which we have not encountered in anything we have read about the subject. The model was originally developed by NernstZto treat the flow of a stirred liquid over a solid surface. If stirring is mild enough that flow is laminar at the interface between liquid and solid, relative motion of the two phases will be zero a t that interface, and the Nernst concept of a stagnant film in the liquid would seem to be valid. However, if the stirred liquid solution is in contact with a gas as proposed by Lewis and Whitman,3 the viscosity of the gas will be orders of magnitude less than that of the liquid. Under such circumstances, we wonder whether the viscous drag by the gas on the surface of the liquid would be sufficient to maintain laminar flow with zero surface velocity for a liquid subject to even very modest stirring. We are not authorities in hydrodynamics and are not in a position to answer the question we have just raised. We are, however, surprised that we do not recall seeing any effort by real authorities to answer that question. If the question has not been answered over 65 years after the model was proposed, we very much hope that a competent authority will examine it. 11. Our Alternative Model for Transport Descriptionof the Alternative Model. We shall now propose
a model alternative to (some critics may say a reductio ad ab-
surdum of) a model first presented by Kaushik and Noyes.* In the following, we shall use the word “surface” only as a noun and never as an adjective and shall not use the words “layer” or “film” a t all. In a typical physical system, a stirred liquid solution and a gas at about I-atm pressure will differ in density by about 3 orders of magnitude and will be separated by an interface which is discontinuous at all distance scales greater than a few times lo-* cm. Virtually all molecules can be cleanly assigned either to bulk liquid or to gas phase. However, an infinitesimal fraction of molecules will each be in contact with several molecules of liquid and will also be capable of entering the gas phase by means of a single (not a diffusive) translational motion of the order of the gas-phase mean free path of cm. Those molecules and only those molecules which satisfy both of these criteria simultaneously are said to be in the surface. No molecule persists in the surface. A molecule in the gas which impinges upon the surface will most often undergo a quasi-elastic collision and return to gas. Sometimes it will temporarily become a surface molecule. In the simplified model presented here, a molecule in the surface will never enter the liquid by a diffusive interchange with a molecule which was immediately previously in contact with it. Rather, molecules in the surface will enter bulk liquid and vice versa only because of the physical flow of material generated by stirring. Definitions of Composition. We shall need a dimensionally consistent set of definitions to define the compositions of solution, gas, and surface. The composition of a liquid solution can be defined unambiguously by means of concentrations expressed in units of mol ~ m - ~ . At least for gases which obey Dalton’s law of combining volumes, we can define partial pressures for a mixture of gases by means of total pressure multiplied by mole fractions of components. To the extent that Henry’s law is a good approximation, we can then speak of “concentration” of a chemical species in a gas as identical with the concentration of that species in a liquid in equilibrium with a gas having the same pressure as the partial pressure of that species in the gaseous system of interest. Because a surface is two-dimensional on the molecular level, we cannot describe its composition in terms of concentrations as defined above. However, we can define the mole fraction of any species in a surface. In the following, “concentration” of a specific surface species will mean the concentration which that species would have in a solution whose mole fractions were the same as those in the surface. Our Eq”MApplication of the Model. We used this model to treat our measurements of rates of interphase transport. Our experimental procedure was virtually identical with that employed by Kaushik and Noyes: and the apparatus for measurements was that used in our paper on nucleation? A stirred sample of solution was evacuated with a water aspirator until gas bubbles were no longer evolved (15-60 min depending on solvent). Then the system was filled with air or nitrogen to about 1 atm and closed off. The solution was stirred magnetically at a rate measured with an electronic tachometer, and the pressure, P, was followed with the transducer and chart recorder described in the other paper. The change of P could be described with excellent precision by the simple first-order kinetics of eq 7. -dP/dt = k,(P - P-)
(7)
Figure 2 shows the fit of eq 7 for a single run at constant stirring rate with air and water at 25 “C.The ordinate is the logarithm of the infinite time value minus the reading in millimeters (mm) on a chart recorder for which readings were a linear function of pressure. The line through the individual points shows excellent fit to first-order decay toward equilibrium for over 3.5 half-lives. The equation of the line in that figure is 5.288-0.06192rlmin. The root-mean-square deviation between calculated and observed (8) Kaushik, S. M.; Noyes, R. M. J . Phys. Chem. 1985.89, 2027-2031. (9) Rubin, M.B.; Noyes, R. M. J . Phys. Chem., preceding paper in this issue.
1002 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992
Noyes et al. 10 I
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300
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Figure 2. Test of eq 1 at constant stirring speed for air in water at 25 O C . Experiment involved 62.15 cm3of water in a flask whose spherical part had a volume of 106.6 cm3with a radius of 2.94 cm. Rate of stirring was 275 rpm, and 207 was reading in millimeters on chart recorder after equilibrium pressure had been attained.
values of mm for each of the 40 points is 1.3%of (207 - mm) for that point, and the maximum is 3.6% in the region where equilibrium is being approached and the deviation should be largest. Put another way, the maximum deviations between the point read visually from the chart recarder and the value calculated for that time is 2.4 mm at short times and 1.2 mm at long times. Scatter of individual deviations suggests that points on the chart could be read to a very few tenths of a millimeter. There were consistent deviations which placed the points in Figure 2 above the line at both ends and below in the middle. However, these may have been caused by nonlinearities in the response of the transducer or transducer indicator or to differences in the rates of transport of O2and N2 molecules in the gas mixture (air) which we were using. Certainly, we have shown that individual pressure changes could be fitted to within a few tenths of a percent of total change by means of a single exponential for over 95% of the way to equilibrium. Only rarely does a chemical kineticist see such excellent fit of data by means of a single parameter. Henry’s law is obeyed well at these low pressures, and as molecules are transported between gas and solution the changes in pressure and in concentration are directly proportional to each other. Then we can use exactly the same k, from eq 7 to write eq 8. dC/dt = k,(C, - C) In order to describe the flux between solution and gas, we can define kflby eq 9. Note that kn is identical with the kL used by HS and defined in eq 3. (9)
A single filling of the flask could be used to measure dP/dt at several different stirring rates before a single dermination of P, permitted calculation of the various values of k, for the different stirring rates. Figure 3 shows several values of k, in eq 7 obtained for air in water also at 25 OC. The excellent linearity in Figure 2 permitted us to evaluate all of the points in Figure 3 from a single filling of the flask with use of a single value of P, at the end of the run. Some replicate runs supported the validity of such a treatment. Figure 3 shows that k, increases monotonically with increased stirring rate and rises to a plateau in the middle of the figure. We use kpto designate the value of k, on the plateau. According to our model, the increased flux with increased stirring rate arises because the bulk solution and the surface are being more and more efficiently mixed. Ultimately, that mixing becomes so rapid that the composition of the surface has become virtually identical with the composition of the bulk. Then the flux to gas phase will be
‘Pm
Figure 3. First-order rate constants for solution of air in water at 25 OC as a function of stirrer speed. Experiment involved 63.38 cm3of water in a flask and side arm having a total volume of 147.1 cm3. The area of the air-water interface was 26.7 cm2,and the ordinate is the value of k, defined by eq 1. The value of k, on the horizontal plateau is called k,.
determined by the rate of transport between the surface and the gas. If k,, is the rate constant for that flux of transport, we can write eq 10.
ktr = M p / A
(10)
The data in Figure 3 lead to the value in eq 11. Note that this value is almost identical with that in eq 6 based on the data of HS.
k,, = 1.32 X cm s-I (11) The right side of Figure 3 shows a further increase of transport flux with stirring rate which was not observed by Hutchinson and Sherwood in Figure 1. At or slightly below the stirring rates at which the rate of transport in Figure 3 began to increase, we noticed a dimple in the surface which began to increase its area. At still greater stirring rates, the dimple became a vortex which pulled air into the solution and obviously would be expected to increase rate of transport between phases. This difference in behavior of our equipment and of that of HS will be discussed further in section 111. The flux between solution and gas results from two successive processes. One is the mixing between bulk and surface as a result of stirring described by kmx,The other is the transport between surface and gas described by k,,. The reciprocal rate constants can be combined just as electrical resistances in series to generate eq 12. 1/kfl = 1 /kmx + 1/ktr (12) The form of eq 12 is identical with that of eq 4, and the methods for treatment of data according to our model are very similar to those employed by HS over half a century ago. Furthermore, numerical data which can be compared are in quantitative agreement. However, there are also differences in behavior between the two sets of observations. Comparisons and differences are discussed further in section 111.
111. Comparison of Experimental Observations of Transport Rates Differences between Sections I and II. Questions have been raised about apparent differences between our data on the right side of Figure 3 and the data of Hutchinson and Sherwood in Figure 1. Both figures describe rates of interphase transport as stirring rates increase to large values. Rates of transport measured by HS at the fastest stirring they studied were still approaching a limit which had not been attained. We used very comparable
Transport of Molecules between Gas and Solution rates of stirring but by the middle of Figure 3 had already attained as a plateau the limiting transport rate for a smooth surface while HS never attained that rate but could extrapolate their data to a value identical with our plateau. At still faster stirring, we observed increased transport rates which we could explain qualitatively by vortex formation and entrainment of gas. It thus appears that our stirring methods were much more efficient than those available 50 years previously. HS used a propeller on a rotating shaft driven by a motor. We used a plasticcovered magnet driven by a rotating magnet of a type quite common to modern laboratory technique. We believe that another factor may also have contributed to our more efficient stirring. The diagram of apparatus in their paper’ suggests that HS absorbed their gases in a cylindrical reactor. We used a spherical flask which was approximately half filled. Our surface-to-volume ratios were much larger than theirs, and we could more easily attain large values of k,, at which eq 12 shows that the flux between phases was limited by k,,. It is easy to see why engineers would prefer a cylindrical reactor where rates of rotation of liquid would be more nearly comparable at all depths. Because we had not studied the prior literature, we more or less accidentally stumbled onto a design of apparatus which was more effetive for what we wanted to accomplish. Comparative Measurements of k,,in Apparatus of Laboratory scale. The numbers in eqs 6 and 11 differ by only about 3%. This excellent agreement is certainly fortuitous. Hutchinson and Shenvood apparently based their estimate of RE(which they called resistance eddy layer) on an average of at least five different gases. Our value of k,, (=l/RE) was derived from the k, in Figure 3 which was obtained with the gas mixture air. Our subsequent measurement with pure nitrogen reported in the table differed by about 20% from the value based on Figure 3. We maintain that all three sets of measurements were evaluating essentially the same quantity, but we do not claim the sort of accuracy implied by the agreement of eqs 6 and 11. It is also of interest to compare the results by HS and by us of k,, for Nz in water with some other measurements with other solvents and gases. Kaushik and Noyes* developed the method employed again by us here and studied the transport of N2 and of CO to and from 96% sulfuric acid. They used volumes of solution about 5 times as large as we did and at 25 OC obtained cm s-l. k,, = 2.6 X Extension to Very Small Surface Areas. The data of Kaushik and Noyes were then used to model the oscillatory evolution of CO during he dehydration of formic acid in concentrated sulfuric acid. The experimental behavior was characterized by Kaushik, Rich, and Noyes,’O and the computations were carried out by Yuan, Ruoff, and Noyes.” The computations required one disposable parameter which increased the assumed surface area of the foaming solution by about a factor of 3 over the area of the still liquid. By introducing that parameter, the computations reproduced simultaneously the frequency and amplitude of the observed oscillations. In order to perform the computations, transport between solution and bubbles used essentially the model of section I1 of this paper and assumed the value of kflwas equal to k,, measured by Kaushik and Noyes with a surface area of a few hundred cmz. The bubble nuclei were computed to have diameters of 1.6 X 10-6 cm, corresponding to surface areas of the order of lo-” cm2 differing by a factor of about lOI3 from the areas used for the experimental measurements of rates of transport. The computations were only possible because there was no need to try to estimate the thickness of a “liquid surface film” around a bubble with a diameter much less than a micrometer. Hutchinson and Sherwood took considerable pains to point out that they thought their experimental results applied only to their specific apparatus. We believe there is more than a glimmer of (10) Kaushik, S. M.; Rich, R. L.; Noyes, R. M. J . Phys. Chem. 1985.89, 5122-5125. (11) Yuan, 2.;Ruoff, P.; Noyes, R. M. J . Phys. Chem. 1985, 89, 5126-5132.
The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 1003 suggetion that they were measuring parameters of much more fundamental value than they realized. If mixing is sufficiently efficient to make the composition of the surface almost identical with that of the bulk solution, the rate of interphase transport will be the same independent of scale of system or of further improvements of mixing efficiency. A Possible Test by Measurements of Much Larger Scale. One of us (RMN) was invited to attend a UNESCO sponsored CHEMRAWN Conferenceizon the Oceans and their interactions with the atmosphere. Informal discussions at that conference indicated that predictions from our data in ref 8 for transport of CO to concentrated sulfuric acid were consistent with the estimam of oceanographers and meteorologists for rates of transport of COz between the Oceans and the atmosphere. The comparison was made without need to try to estimate the thickness of surface layers on a wind-swept Ocean subject to a wide range of climatic conditions. If that comparison were supported by more accurate measurements by our method of the rate of transport of C 0 2 to seawater, it would further validate our procedures. Although the necessary accuracy probably would not be attainable, rates of production of argon by *K decay in the Oceans and rates of transport measured by our method might be combined to permit an improved estimate for the rate of turnover between abyssal and surface waters. If our measurements could indeed be extended to oceanic surface areas of the order of lOI4 m2,then the applications of the previous section would indicate that our procedures were valid for surface areas extending over a range a few orders of magnitude greater than Avogadro’s number!
IV. Applications to Evaluations of Accommodation Coefficients In sections I1 and I11 above we have shown why we think that for solutions of any gas in any solvent we can in principle evaluate k,,, the limiting rate constant for the flux between the gas phase and the surface of a solution stirred so efficiently that the mole fraction composition of the surface is the same at that of the bulk. If we also know the solubility of the gas in the solvent, we can calculate the accommodation coefficient, 7,or probability that a gas molecule striking the surface will enter the solution. The necessary equation is (13). Both sides of eq 13 have the dimensions mol cm-z s-l and describe a flux of molecules per unit time through unit surface area. The left side is rate of escape from the solution with C, the equilibrium solubility at 1 atm as determined in the other paper.9 On the right side, v i s the molar volume in the gas, and the square root is the average speed at temperature T of a molecule of molecular weight M. The derivation of the expression multiplying 7 is presented in uliy textbook of physical chemistry. Because previous investigators thought that they were looking at diffusive transport through a film of thickness dependent on stirring, they did not realize that they could get limiting rates of transport which provided direct measurements of accommodation coefficients. We measured transport rates which could be combined with eq 13 to determine k,, for nitrogen in water and in all of the solvents for which solubhties are reported in Table IX in the other paper? For the quantities of solution we used, we did not always obtain as clearly defined plateaus at different stirring rates as are seen in Figure 1 of ref 2 and Figure 3 of this paper, but we did obtain near constancy of rate of transport over at least three successive stirring speeds. Figure 3 encourages us to think we could have found combinations of stirring rate and surface area which showed even better constancy of transport rates than we found in some of these preliminary experiments. The result for (12) CHEMRAWN (Chemical Research Applied to World Needs) IV, Modern Chemistry and Chemical Technology Applied to the Ocean and its Resources, Keystone, CO, Oct 5-9, 1987.
1004 The Journal of Physical Chemistry, Vol. 96, No. 2, 1992 TABLE I: Transport Rates, Solubilities of N2,and A c c o m d t i o n Coefficients for Several Solvents at 25 OC k,,, cm s-I soiv&t ,c, mol Y 6.03 x 10-7 1 x 10-9 9 x 10-4 water 6 X lo4 3.77 X lob7 4X 15% aqueous
glycerol ethanol
acetonitrile dioxane
heptane acetic acid DMSO ethyl acetate
3 7 3 1 4 1 6
x x x x x x X
10-3 10-3 10-3 10-2
10-3 10-3
4.64 x 3.20 x 3.12 x 6.63 x 4.34 x 1.07 x 5.06 X
10-6
io” 104 104 io” 104 10”
3x 4x 2x
10-8 10-8
Noyes et al. 0 ,
-4
10-8
each solvent is presented in Table I along with the value of the solubilityfrom the other paper’ and the accommodation coefficient y calculated with the use of eq 3. The y for water is within about a factor of 5 of that reported previously8for concentrated sulfuric acid. We only report one significant figure for the transport rates and accommodation coefficients in Table I. However, the entries show definite patterns. Thus, y for the hydrogen-bonded liquids water and aqueous glycerol are less than for any of the organic liquids. Values for those organics range by a factor of 50 from a high for heptane to a low for DMSO. Water and aqueous glycerol are structured liquids with strong hydrogen bonds which resist penetration by light nonpolar nitrogen molecules. Heptane contains nonpolar molecules between which nitrogen molecules can be inserted relatively easily without having to Overcome strong polar attractions between solvent molecules. DMSO molecules contain the heavy central sulfur atom which cannot easily be displaced during collisions with the light nitrogen molecules. Thus, even the approximate results in Table I can be rationalized in terms of known molecular properties, and the major conclusion from that table is that some very intersting information could be obtained by more careful measurements. Especially because of the small effects of surface tension on the threshold concentration for nucleation of bubbles,’ we are surprised that the observations in Table I indicate that dynamic parameters for transport between liquid and gas phases may range by over an order of magnitude for different solvents. It is obvious there is a great deal to learn about the factors which are important in these processes. We have found few other efforts to determine the accommodation coefficient, y, for solution of gases in liquids. Kaushik and Noyes8 by the same method used here estimated a value in concentrated sulfuric acid about 4 times the value in water in Table I but less than the values for most of the organic solvents. Other reported values of which we are aware were obtained by very different methods and are several orders of magnitude larger than those reported in Table I. Thus, Worsnop et al.I3-l5 studied solution of the gases SO2,H202,HN03, N205,and HCl present at low concentrations in N2gas. They dissolved these gases in water droplets with diameters of a few hundred micrometers and determined rates of solution by depletion of the gas composition. They defined a quantity y identically to our use of the same symbol and obtained values never less than and usually more than 10-I while our values in Table I for N2 in different Even this large difsolvents range between lo-’ and 4 X ference need not reflect serious failings of either method. Worsnop et al. were looking at low gas concentrations of polar molecules which could form hydrogen bonds and most of which could also ionize easily when dissolved in small droplets of water. We were working with bulk solutions having surface areas of several cm2 and which were dissolving nonpolar molecules which made up (13) Worsnop, D. R.; Zahniser, M. S.;Kolb, C. E.; Gardner, J. A,; Watson, L. R.; Van Doren, J. M.; Jayne, J. T.; Davidovits, P. J . Phys. Chem.
1989, 93, 1159-1172. (14) Van Doren, J. M.; Watson, L. R.; Davidovits, P.; Worsnop, D. R.; Zahniser, M. S.;Kolb, C. E. J . Phys. Chem. 1990, 94, 3265-3269. (15) Jayne, J. T.; Davidovits, P.; Worsnop, D. R.; Zahniser, M. S.;Kolb, C. E.J . Phys. Chem. 1990, 94, 6041-6048.
I
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Figure 4. Logarithmicplot of y against C., Circles are data from Table I for N, in various solvents. Triangle is the value for SO2in water from data of Worsnop et al.”
almost all of the gas phase. If our method could be extended to polar gases such as those studied by Worsnop et al., a direct comparison of the two methods would be possible. However, we might well find that the rate of solution of HCl in water was too fast for us to be able to follow it. Figure 4 compares logarithms of accommodation coefficients and solubilities for measurements with different gases in different solvents by different methods. The circles plot our data from Table I for N2 in various solvents. All fall close to a straight line with slope of about 2. Such behavior implies that y over a range of more than 100-fold is approximately proportional to Cq2. We cannot devise any theoretical reason to expect such a correlation, but we find this empirical observation to be interesting. The single triangle in Figure 4 represents the observations of Worsnop et al.13 for SO2 in water. It comes surprisingly close to falling on the extrapolation over several orders of magnitude of our data for N2 in several solvents. Worsnop et al. also estimated accommodation coefficients for polar gases such as HCl and H202. We cannot meaningfully include those data in Figure 4. Thus, aqueous HCl is strongly ionized even at its saturation concentration of about 0.012 mol and we have no assurance that the mole fraction of HCl molecules in the surface of the saturated solution is the same as in the bulk. Furthermore, liquid H202is miscible with water in all proportions, and the observed accommodation coefficient for this substance cannot be used to obtain a meaningful point in Figure 4. Because the entries in Table 1 represent the application of different procedures to different gases in different solvents and cover ranges of several orders of magnitude, any inferences must be guarded. It is of interest that the results of Worsnop et al. are at least somewhat consistent with a great extrapolation of our own. Solubility data for many of these combinations of gas and solvent have been available for many decades if not a full century. Accommodation coefficients have been virtually unmeasurable during that period except for special situations. This impossibility arises because if transport between gas and liquid phases takes place by the conventional model discussed in section I, transport by diffusion through a film of unknown thickness will not permit evaluation of the data necessary to determine accommodation coefficients. However, if transport takes place by our alternative model described in section 11, transport rates in a plateau like that in Figure 3 permit accommodation coefficients to be calculated by straightforward procedures. If the model presented in section I1 is assumed to be valid, the way is now open to prepare analogues of Figure 4 for a single gas in many different solvents and for several gases in a single solvent. We believe that such correlations would be very useful to obtain. If measurements of the type proposed generated internally con-
J. Phys. Chem. 1992,96, 1005-1009 sistent values for many different solvents and solute gases, those values would provide justification for deciding the relative validity of the two models.
V. Some Concldig Comments This paper is concerned with transport of molecules between a gas and a stirred salution. All of the various studies have been in unanimous agreement that the rate of transport increases monotonically with the rate of stirring. A model which has been regularly invoked for over 60 years asserts that the transport takes place by diffusion through a virtually stagnant surface layer whose thickness becomes less as the rate of stirring increases. The alternative model which we propose is that molecules in the surface and molecules in the gas pass between one phase and the other in a single step; the increase of rate with stirring results from more efficient mixing between the surface and the bulk solution. Both models predict that there is a limiting rate of transport at which either the thickness of the surface layer has become zero or the mole fraction composition of the surface has become identical with that of the bulk. The mathematics for extrapolating experimental data to that limiting rate are identical for the two models (eqs 4 and 12). Furthermore, measurements for transport of nitrogen to water made over 50 years apart' and interpreted by the different models extrapolate to virtually identical values (eqs 6 and 11) for the limiting rate.
100s
If the objective is to treat a particular set of experimental measurements, there is little reason to choose one model or the other for making computations. However, the conventional model requires that each experimental situation must be studied empirically to determine the necessary parameters. The model which we now propose claims that it is possible once and for all to measure the limiting rate of transport for a specific gas to a specific solvent at a specific temperature. Experiments exhibiting slower rates are indications of less than ideal efficiency of mixing. If our model is the one which is more valid, it offers the bonus that it will be possible to measure accommodationcoefficients for solutions of different in different solvents. We believe that for decades chemical engineers have been depriving themselves of techniques for obtaining information of value to many different problems of design.
Acknowledgment. This work was supported in part by Grant CHE8717791 from the National Science Foundation. This paper is No. 95 in the series 'Chemical Oscillations and Instabilities". No. 94 is ref 9. We belatedly express our appreciation to the anonymous reviewers of previous versions of this manuscript submitted beginning in June 1990. If it had not been for their intransigent refusal to allow publication, we would not have come to recognize the true significance of our own observations. R e t r y NO.DMSO, 67-68-5; N2,7727-37-9;H20,7732-18-5; glycerol, 56-81-5;ethanol, 64-17-5; acetonitrile,75-05-8;dioxane, 123-91-1; heptane, 142-82-5;acetic acid, 64-19-7;ethyl acetate, 141-78-6.
Decomposition of Ozone in Aqueous Acetic Acid Solutions (pH 0-4) K. Sehested,* H.Corfitzen, J. Holcman, and Edwin J. Hartt Environmental Science and Technology Department, Rise National Laboratory, DK 4000 Roskilde, Denmark (Received: July I , 1991; In Final Form: October 14, 1991)
The decomposition of acidic O3solutions containing acetic acid has been studied with the purpose of determining the rate of the initiation reaction for the radical-induced chain decomposition. The effects of acetic acid, ozone, oxygen, and hydrogen peroxide have been studied as a function of temperature over the pH range 0-4. Our interpretation of the mechanism in the 03-aceticacid system gives a rate constant at 31 OC of k = (3.0 f 0.5) X lod s-I for the precursor of radical formation. As expected this rate is constant in the acid pH range. The activation energy EA = 82.5 f 8.0 kJ mol-' as determined over the temperature range 15-50 OC is identical to that found in acidic solutions without acetic acid. It is found that O3reacts with undissociated hydrogen peroxide, with a rate constant k = 0.065 dm3 mol-' s-l at 31 OC and an activation energy of EA = 73.5 f 8.0 kJ mol-' measured in the temperature range 15-45 O C . We tentatively suggest an electronic excited state of ozone, formed in the dissociation/recombinationreaction of 03,to be the precursor for the radical formation by its reaction with water.
Introduction Acetic acid/acetate is a well-known stabilizer of aqueous O3 solutions, but the mechanism is not fully understood, although scavenging of the OH radical, which is the chain-propagating radical in O3 decomposition, has been e~tablished.l-~ In the oxygenated acetic acid solution, the scavenging leads to formation of H 2 0 2via the reaction sequence (1)-(3).4 OH
+ CH3COOH
'CHZCOOH
+0 2
2'00CH2COOH
'CH2COOH -1. H2O
(1)
'OOCHZCOOH
(2)
0.7H202+ products
(3)
-
+
-
The acetic acid radical formed in reaction 1 is scavenged by oxygen to form the peroxy acetic acid radical which disproportionates (reaction 3) to give ca. 70% H 2 0 2and other products such as formaldehyde, glyoxylic acid, glycolic acid, and organic peroxides! 'Permanent address: Port Angela, WA 98362.
Acetate has been used in competition kinetic studies with the aim to determine the rate constant of the reaction of OH radicals with 0 , but due to the complexity of the reaction system, it failed to give a reliable because of a chain reaction between the acetate radical and ozone. Acetic acid was used earlier6 to determine the rate constant of a reaction of ozone with H 2 0 2at 0 OC in 0.018 M HC104, k = 2.3 X dm3 mol-I s-I. In a recent paper,' we proposed the ozone dissociation/re"bination reaction O3 0 + O2 (49-4) (1) Hoign6, J.; Bader, H. Water Res. 1976,IO, 377r386. (2) Forni, L.;Bahnemann, D.; Hart, E. J. J. Phys. Chem. 1982,86, 255-259. ( 3 ) Sehestcd, K.;Holcman, J.; Bjergbakke, E.; Hart, E.J. J. Phys. Chem. 1987,91,2359-2361. (4) Schuchmann, M. N.; Zegota, H.; Sonntag von, C. Z . Narurforsch. 1985,40B,215-221. (5) Bahnemann, D.;Hart, E.J. J. Phys. Chem. 1982,86, 252-255. (6) Taube. H.; Bray, W. C. J. Am. Chem. Soc. 1940,62, 3357-3373. (7)Sehested, K.;Corfitzen, H.; Holcman, J.; Fischer, Chr.-H.; Hart, E. J. Erwiron. Sci. Technol. 1991, 25, 1589-1596.
0022-3654/92/2096- 1005$03.00/0 0 1992 American Chemical Society
