J . Phys. Chem. 1987, 91, 4193-4198 actions between the guest and water vibrations have also been vindicated from the substantially smaller ratio in the frequency shift of the translational vibration at 230 cm-' in the infrared absorption spectra between the deuteriated and nondeuteriated ethylene oxide hydrate from the value deduced from simple harmonic oscillator
Conclusion The unit cell parameters for the structure I clathrate of ethylene oxide have been measured by the X-ray powder diffraction method from 18 to 260 K. We confirmed previous findings that the linear expansion coefficient of the hydrate is greater than that in hexagonal ice and so is the anharmonicity. The essence of the experimental and theoretical results for ethylene oxide hydrate are summarized in Figure 5 . The relative volume change over the temperature range 70 to 270 K is about 2% for hexagonal ice and 3% for the structure I hydrate of ethylene oxide. The corresponding cakculated volume change is 4% and 7%, respectively. Even though the absolute magnitudes are overestimated by the calculations, the trend is duly reproduced. The small calculated relative volume expansion between the hypothetical empty structure I hydrate and hexagonal ice suggest that the structural (63) Bertie, J. E.; Othen, D. A. Can. J . Chem. 1972,50,3443. Discussed
in ref 3.
4193
difference between the two compounds only contribute a minor part in the thermal expansion. The interactions between the host lattice and the guest molecules enhance the anharmonic terms in the total crystal potential of the hydrate and is responsible for the larger thermal expansitivity. These interactions also cause the large thermal expansion reported for a number of structure I1 hydrate.56,58 We have demonstrated the usefulness of constant pressure molecular dynamics simulations (NMD) in helping to rationalize the mechanisms of thermal expansion in molecular solids. Even though the simple pairwise additive intermolecular potential models employed in this study are very crude, the major difference between the thermal expansion of hexagonal ice and hydrate is correctly reproduced. Moreover, our experience41has shown that the transverse acoustic and optic modes of hexagonal ice are predicted to be too soft by the SPC model. This failure is due to a basic deficiency inherent with the model which treats the water molecule as rigid and nonpolari~able.~~ This effect is likely more important when the simulation is performed at high temperature.
Acknowledgment. The authors express their gratitude to M. L. Klein for his encouragement and to R. W. Impey for technical advice in using his constant pressure M D program. Thanks are extended to J. R. Dahn for his assistance with the X-ray experiments at the early stage of this study. We also thank the referees for many helpful suggestions.
Measurements of Critical Supersaturation for Homogeneous Nucleation of Bubbled Mordecai B. Rubin*$ and Richard M. Noyes* Department of Chemistry, University of Oregon, Eugene, Oregon 97403 (Received: December 9, 1986)
If a solution of a gas in a liquid becomes sufficiently supersaturated, it attains a threshold for homogeneous nucleation at which bubbles form spontaneously. Most previous efforts to measure this threshold have required cumbersome saturation at high pressures. We have worked instead at ambient temperature and pressure with use of chemical reactions to produce supersaturated solutions. The degree of supersaturation is then measured as the amount of gas released when rapid stirring or sonication is suddenly initiated. The threshold for nucleation is the limit beyond which it is impossible to push the level of supersaturation. Thresholds for the diatomic gases Hz, N2, Oz, CO, and NO in aqueous solutions all lie between 0.012 and 0.07 M; that for C02is at least 0.4 M. Continued sonication appeared to accelerate the release of hydrogen from sodium borohydride but not the release of carbon dioxide from acetonedicarboxylic acid. Measured thresholds vary surprisingly little with temperature, indicating that the barrier to nucleation is primarily entropic rather than energetic. The threshold seems to decrease with increasing ionic strength. The effects of changing various parameters such as external hydrostatic pressure have not yet been examined.
Introduction Chemists understand well the equilibrium partition of a volatile solute X between liquid and gas phases. A unique function relates the partial pressure Px in the gas phase to the concentration Cx in the liquid. If the solution is sufficiently dilute, Px will be directly proportional to C x . Even if the proportionality breaks down at greater concentrations, Px will increase monotonically with increasing Cx. Because transport processes in gases are much faster than in liquids, an excess of X in the gas phase will be reduced to the equilibrium value much more rapidly than will an excess of X in a comparable volume of liquid. In other words, if both gas and liquid phases are present, then supersaturated solutions will persist much longer than will supersaturated vapors. *Permanent address: Department of Chemistry, Technion-Israel Institute of Technology, Haifa, Israel. 'This paper is No. 77 in the Series 'Chemical Oscillations and Instabilities". No. 76 is: Ruoff, P.; Hansen, E. W.; Noyes, R. M.J . Phys. Chem., in press.
Because of this persistence of supersaturation, it is possible in principle to observe another function relating Px and Cx which we shall call the threshoZd for homogeneous nucleation. If Cx is increased to this threshold, many bubble nuclei will be created virtually discontinuously. Those bubbles will then grow rapidly and deplete the supersaturation while the equilibrium state is approached. The situation can be illustrated in Figure 1 representing a closed isothermal system containing an inert solvent and an inert gas. If no X is present, the total pressure including vapor pressure of solvent is Po. Now let molecules of X be generated in the liquid phase by some means. If sufficient time is allowed for the system to attain equilibrium, the state can be described by a point on the curve designated "equilibrium" in Figure 1. That point is a unique function of the amount of X in the system. Now let agitation be sufficiently rapid that the compositions of the gas and liquid phases are each virtually uniform. However, let the time scale of observations be short enough that the supersaturated solution can be studied far from equilibrium with
0022-3654/87/2091-4193$OlSO/O 0 1987 American Chemical Society
4194
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Rubin and Noyes
Figure 2. Schematic system to measure degree of supersaturation. Flask A containing reacting solution is immersed in thermostat B, and pressure is measured with transducer C which is connected to a recorder. Magnetic stirrer D or sonicator E can be activated on command, and the extent of supersaturation is calculated from the resulting increase in pressure. ‘0
cx Figure 1. Phase diagram for closed system containing inert solvent and gas at constant temperature. Po is pressure in absence of solute X, and P is total pressure of gas in contact with solution having composition C,. Slanted lines represent trajectories along which a supersaturatedsolution would approach equilibrium. If any addition of X is made only to the liquid phase, no system can exist outside the region of these trajectories.
the gas phase. Then the state of the system will be described by a point to the right of and below the equilibrium curve in Figure 1. The slanted lines represent trajectories describing how a supersaturated solution would approach equilibrium if the total amount of X remained constant in the closed system. The slanted-line trajectories in Figure 1 occupy only a limited region of the phase plane. Thus, if systems are created by introducing X to the liquid phase, the state of the system will never correspond to a point above and to the left of the equilibrium curve. Similarly, the state can never correspond to a pressure less than the Po observed in the absence of any X. Finally, the experimentally accessible states are limited by the curve on the right marked “threshold of nucleation”. That curve defines the limit of supersaturation which can be exceeded only transiently in any experimental system subjected to a specific external pressure. Thresholds of nucleation are understood much less well than are equilibrium saturations. All studies refer to the monumental treatise of Volmer.’ However, Strey, Wagner, and Schmeling2 recently commented, “Even though more than 50 years old, the classical nucleation theory has neither been uniquely proved to be correct or wrong by experiments.” Most studies of phase nucleation have used the cloud chamber principle. Rapid adiabatic expansion or compression of a fluid changes the temperature until another phase forms spontaneously by homogeneous nucleation. Such systems are extremely sensitive to adventitious impurities, and a major use of the technique has been to study tra’cks of ionizing particles under conditions such that the threshold for truly homogeneous nucleation was not quite attained. We have discovered very few studies of nucleation of bubbles in liquids. Most have been concerned with bubble formation in pure liquids. Those studies of solutions of which we are aware were carried out by saturating at a pressure of many atmospheres and then suddenly reducing the external pressure by a known amount. A paper by Forest and Ward3 illustrates the complexity ( 1 ) Volmer, M. Kinetic der Phasen Bildung; Steinkopf: Dresden, 1939. ( 2 ) Strey, R.; Wagner, P. E.; Schmeling, T. J . Chem. Phys. 1986, 84, 2325-2335. ( 3 ) Forest, T. W.; Ward, C. A. J . Chem. Phys. 1978, 69, 2221-2230.
of apparatus and the precautions necessary in order to obtain significant data. During a series of studies of gas-evolution oscillator^,^-'^ we have developed much simpler procedures for measuring the threshold for nucleation by appropriate gases. Instead of saturating the solution with gas at a high pressure and rapidly decompressing, we can maintain the solution at a specific temperature and pressure while supersaturation is created by generating the solute molecule by molecule with a chemical reaction. The extent of supersaturation can then be measured as the amount of dissolved gas which is evolved after initiation of rapid stirring or sonication. We find well-defined reproducible limiting supersaturations which seem to represent the desired thresholds for nucleation. The present paper describes a number of chemical reactions which can be used to measure thresholds of nucleation. Such a reaction should involve a solute species which is soluble to at least several times 0.1 mol dm-3 and which produces the desired gas with an effective first-order rate constant of the order of s-I. We have found ways to meet these conditions for several wellknown simple gases. The work reported here has been primarily exploratory. The threshold of nucleation is obviously influenced by many factors including the solute gas, the solvent, temperature, external pressure, surface tension, ionic strength, etc. The procedures described here demonstrate that these influences could be studied over significant ranges of all of these factors. Such studies should contribute greatly to our understanding of homogeneous nucleation of one phase in another.
Experimental Procedure The necessary equipment has been described previouslyI6 and (4) Showalter, K.; Noyes, R. M. J . Am. Chem. Soc. 1978,/00,1042-1049.
( 5 ) Smith, K . W. Ph.D. Thesis, University of Oregon, Eugene, OR, 1981. ( 6 ) Ganapathisubramanian, N.; Noyes, R. M. J . Phys. Chem. 1981,85, 1103-1 105. ( 7 ) Bowers, P. G.;Noyes, R. M. J. Am. Chem. SOC.1983,105,2572-2574. ( 8 ) Smith, K. W.; Noyes, R. M.; Bowers, P. G. J . Phys. Chem. 1983,87, 15 14-15 19. (9) Smith, K. W.; Noyes, R. M. J . Phys. Chem. 1983, 87, 1520-1524. (IO) Noyes, R. M. J . Phys. Chem. 1984, 88, 2827-2833. (1 1 ) Bowers, P. G.;Noyes, R. M. In Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley: New York, 1985; pp 473-492. ( 1 2 ) Kaushik, S. M.; Noyes, R. M. J . Phys. Chem. 1985,89,2027-2031. ( 1 3 ) Kaushik, S. M.; Rich, R. L.; Noyes, R. M. J . Phys. Chem. 1985,89, 5722-5725. ( 1 4 ) Yuan, 2.; Ruoff, P.; Noyes, R. M. J . Phys. Chem. 1985, 89, 5726-5732. ( 1 5 ) Kaushik, S. M.; Yuan, 2.;Noyes, R. M. J . Chem. Educ. 1986,63, 76-79. ( 1 6 ) Rubin, M. B.; Noyes, R. M.; Smith, K. W. J . Phys. Chem. 1987, 91, 1618-1622.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4195
Homogeneous Nucleation of Bubbles is illustrated schematically in Figure 2. Chemical reaction takes place in the liquid phase in a closed thermostated flask of known volume, and the pressure in the gas phase is measured with a transducer whose signal is fed to a recorder. The reaction proceeds continuously in the solution, which is at first unstirred. At a selected time, the dissolved gas is released either by sudden initiation of rapid stirring or by turning on a sonicator of the type used to promote cleaning of dirty glassware. The decrease in concentration of the solution, Cs,can be calculated from the pressure increase and the known volumes of gas and liquid phases. As pointed out previously,16the relevant relationship is
The procedure measures the extent of supersaturation of a particular solution at a specific time. We previously proposedI6 that if such measurements are to define thresholds of nucleation, then the observed supersaturations should satisfy the following criteria: (a) The rate of pressure increase immediately before stirring was initiated equals the limiting rate with continuous stirring. (b) Any particular solution composition produces a reproducible upper limit of supersaturation. (c) This limiting supersaturation is independent of changes in composition which may influence the rate of chemical reaction but do not seriously impact the physical properties of the bulk medium or its surface. The validity of the method is supported by our ability to obtain reproducible supersaturations which satisfy these criteria for thresholds of nucleation. The presentation of results is organized in terms of the gases whose supersaturations were being measured.
Y
YI
2 2 x l o 4 moles gas
I
Figure 3. Variation of pressure in a closed system during evolution of nitrogen by reaction 3. Experimental conditions were [NH&l]o = 1.6 M, [NaN0210 = 2.5 M, [H'] (from H2S04)= 0.1 M, V,,, = 15 cm3, temperature = 32 OC (courtesy of Professor Peter G. Bowers of Simmons College).
J 0.1 Volt
Results Carbon Monoxide. The first suggestion that critical supersaturation by any gas could be measured in this way came from studies by Smith5**of the dehydration of formic acid in concentrated sulfuric acid. The essential reaction is HC02H
-
CO(g)
+ HzO
(2)
Those studies indicated that solutions could be prepared with concentrations up to but not significantly beyond 0.07 M in CO. This limit seemed to be rather little affected by changes in temperature or in composition of the sulfuric acid. The equilibrium solubility of CO under these conditions is only about 0.0009 M, so the threshold for homogeneous nucleation a t 1-atm external pressure corresponds to a solution which would be saturated with gas at about 80 atm. Nitrogen. This gas can be generated in principle from any amine which forms an unstable diazonium salt. The most convenient reaction is probably
NH4'
+ NO2-
-
N2(g)
+ 2H20
(3)
Figure 2 and Table I11 in ref 16 present evidence obtained by rapid initiation of stirring of a solution a t an ionic strength about 6 M with about half of that ionic strength due to sodium perchlorate. The threshold of nucleation in this solvent at ambient temperatures is about 0.012 M. Very much the same threshold was indicated if the supersaturation was released by sonication instead of by rapid stirring. The equilibrium solubility of N 2 in water is 0.00064 M at 1 atm and 25 O C . These data suggest that the critical supersaturation of N2 in water is about 19 times the equilibrium solubility at 1 atm or only about a quarter of the ratio reported above for the isoelectronic CO in sulfuric acid. Much of the difference may be associated with a salt effect. If we cut the ionic strength to about 3 M by leaving out the sodium perchlorate, the critical supersaturation went up to about 0.02 M or about 30 times the solubility in pure water at 1 atm. The same threshold of nucleation a t about the same ionic strength was obtained by Bowers" by the method of Figure 3. That method involves running the reaction as an oscillator in a closed system and calculating the supersaturation released in each
7 5 sec
I
-
I min /
Figure 4. Release of N 2 by initiation of sonication. Reactant solution consisted of 5 cm3 of solution B (ref 16), 2.5 cm3 of solution A, and 2.5 cm3 of water. Flask volume was 160 cm3, and temperature was 25 OC. Pressure change was 0.16 atm VI.Note overshoot when sonication was initiated.
step. The Bowers procedure does not guarantee that supersaturation is entirely released in each step, but the consistency with our results is gratifying. We also made a few measurements in which the ionic strength was reduced to about 1.5 or 1 M and pushed the threshold of nucleation to about 40 times the reported equilibrium solubility in pure water. It is obvious that ionic strength effects should be measured both on the equilibrium solubility and on the threshold for nucleation. Both our sonication experiments and the observations of Bowers in Figure 3 suggest an effect for which we have no satisfactory explanation. Figure 4 shows a typical trace to measure the pressure increase when the sonicator was turned on. We ignored the obvious overshoot. The rise of pressure took only a second or two, and we thought we were looking at an artifact due to inertia of the recorder pen. Bowers also noted the overshoot in Figure 3 on a somewhat longer time scale and thought the pulse was also releasing N O which subsequently reacted with the air in the flask. Our own experiments were done under 1 atm of argon in order to avoid this potential complication. The overshoots in Figures 3 and 4 are certainly real and may or may not be due to the same cause. We prefer for the moment to report the observations without attempting explanations. Oxygen. Decomposition of hydrogen peroxide is an obvious way to generate supersaturated solutions of oxygen by means of reaction 4. (4) Ganapathisubramaniad used iron-salt catalysis of this reaction to generate rather erratic oscillations in gas evolution.
4196 The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
We made a number of measurements with [H20,] in the range 4-7 M, with [Fe3+] about 0.1 M and with [H2S0,] up to about 0.6 M. At 35 O C and with supersaturation released by stirring, the apparent critical supersaturations were in the range 0.04-0.05 M. When supersaturation was released by sonication, the range was more like 0.05-0.06 M. If the apparent difference was real, we are not in a position to say whether it involved more efficient release of supersaturation or was caused by sonication-induced decomposition of hydrogen peroxide. At 25 O C , the critical supersaturation seemed to be somewhat higher and was pushed to 0.068 M in one set of three consistent measurements. This concentration is about 50 times the equilibrium solubility of 0.001 26 M. These observations suggest that the threshold of nucleation for O2 is appreciably greater than that for N2, but this conclusion is clouded by the possibility the difference is an ionic strength effect. More studies are obviously needed. Hydrogen. An obvious reaction to generate hydrogen is the decomposition of aqueous sodium borohydride according to reaction 5 . NaBH,
+ 4H20
-
4H2(g) + H$O,
+ NaOH
(5)
Bowers7 was unsuccessful in an effort to generate oscillatory gas evolution by this reaction. We did manage to obtain erratic pulses of varying height separated by 10s or more, but the trace looked more like random noise than like anything which could fairly be called oscillations. Because of its large diffusion coefficient, hydrogen escapes from solution more rapidly than the other gases we studied. Thus, when the stirring of a solution was stopped, a slow rise of pressure began almost immediately. With the other reactions, there was a period of several seconds of almost static pressure if stirring was suddenly stopped. We did not try to measure the rate constant for transport through a surface by the method of Kaushik and Noyes.I2 The kinetics of reaction 5 have been studied by several workers such as Davis, Bromels, and Kibby." The rate is first-order in borohydride and is very dependent on pH. Our initial rates at pH 10 agreed with theirs but then slowed down somewhat as our weakly buffered solutions became more alkaline. In the first measurements of critical supersaturation, up to 100 mg of NaBH, was weighed into the reaction flask. Following addition of a measured amount of borate buffer at pH 10 with capacity 0.1 M, the flask was closed and measurements were made at 25 O C . With 15 cm3 of solution in a 160-cm3 flask, the supersaturation could not be pushed above about 0.03 M. The volume of liquid was reduced in subsequent experiments, and fairly reproducible results indicated critical supersaturations up to 0.04 M with 4 cm3 of solution and 0.06-0.07 M with only 1 cm3. Some of the differences in these first measurements with small volumes of solution may have been associated with changes in geometry such as extent of surface. A complication of the work with hydrogen is that a large volume of gas is produced from a reasonable weight of reactant. In our subsequent measurements, we added a buffer flask to make the gas volume about 300 cm3, increased the weight of borohydride to 200-400 mg, and used 10 cm3 of borate buffer. These measurements were appreciably more reproducible and indicated a threshold for nucleation of 0.060-0.064 M. We found no significant difference whether the supersaturation was released with rapid stirring or with sonication. A few measurements at 16 "C suggested the critical supersaturation at this temperature was only about 0.045 M. The equilibrium solubility of H2 in water at 25 O C and 1 atm is 0.000 78 M, and the threshold for nucleation is about 80 times this value. This ratio is larger than that for any of the other gases in aqueous solution which we report here and is about the same as the value obtaineds for carbon monoxide in concentrated sulfuric acid. (17) Davis, R.E.;Bromels, E.; Kibby, C . L. J . Am. Chem. SOC.1962,84, 885-892.
Rubin and Noyes
0.1 Volt
i
1
Figure 5. Effect of sonication on rate of reaction of sodium borohydride. Solid line (including small overshoot) is pressure before and after initiation of sonication. Dashed lines are extrapolations to show that rate of gas evolution was greater during continued sonication than it had previously been in the unstirred solution. Solution volume of 2 cm3 was initially 0.34 M in NaBH,, 0.025 M in borax, and 3 M in NaCI. Flask volume was 139 cm3 and temperature was 25 O C .
Even though stirring and sonication appeared to be about equally effective at releasing dissolved hydrogen, they had significantly different effects on the rate of formation of that gas. An important criterion for a trace like Figure 2 in ref 16 is that except for transients the rate of pressure increase before release of supersaturation should be the same as the rate afterward. This criterion was satisfied in our hydrogen experiments if supersaturation was released by sitrring. Figure 5 shows a trace in which the rate with continued sonication was appreciably more than what it had been immediately before sonication was initiated. We concluded that the system had indeed been in a steady state before sonication was initiated and that the sonication increased the rate of the reaction itself. This postulate was tested with a solution made from 21.6 mg of NaBH4 (0.022 M) in 25 cm3 of pH 10 buffer. The solution was allowed to react with vigorous magnetic stirring for several minutes and then transferred to an ultrasonic bath for a comparable period while the pressure was monitored continuously. Except for the disturbances caused by transfers from one bath to the other, reasonably good first-order kinetics were obtained. For four periods of stirring, the rate constant was (1.26 f 0.06) X s-l; for four periods of sonication, it was (2.73 f 0.43) X lo4 s-I. There was some unavoidable warming of the thermostat by up to 2 OC during prolonged sonication, but the factor of 2.2 in rate constants appears to be a real effect of sonication. I t is well-known that sonication can affect rates of chemical reactions, and a recent discussion is by Fischer, Hart, and Henglein.I8 Nitric Oxide. Data in the literature suggested a feasible study of NO could be made by means of reaction 6. 3HN0,
-
2NO(g)
+ HNO, + H,O
We made a few measurements with a reaction mixture prepared from 7.5 cm3 of 3.5 M N a N 0 2 and 2.5 cm3 of 3, M H2S04.The solution was purged with argon and run as for the other systems. The behavior was not very reproducible, but up to about 0.02 M of supersaturation could be released with stirring and up to 0.03 M with sonication. This value is somewhat lower than for most of the other gases studied and is only about 16 times the equilibrium solubility at 1 atm. We had little time to study this system, and the threshold for nucleation may be somewhat greater than the preliminary value reported here. Moreover, the system is somewhat unpleasant, and brown fumes and blue color in the solution indicate that the NO2 intermediate attains considerable concentrations at times. Our data encourage us to think it should nevertheless be possible to measure the threshold for nucleation in this system. (18) Fischer, C.-H.; Hart, E. J.; Henglein, A. J . Phys. Chem. 1986, 90, 1954-1956.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4197
Homogeneous Nucleation of Bubbles TABLE I: Measurements of Supersaturation in Solutions of Acetonedicarboxylic Acid
tpc
[NaClO,]/M
supersaturation/M
40 45
0.1 0 0 0 0.1
0.30 0.25, 0.35 0.28, 0.36 0.38, 0.40 0.24 0.28
3 .O 50
0.27, 0.37 0.37, 0.41
3.0 0 3.0
0.32, 0.33
Carbon Dioxide. The beverage industry has long been interested in supersaturated aqueous solutons of carbon dioxide. Biochemical research has led to many kinetic studies of decarboxylation of dicarboxylic and P-ketocarboxylic acids. We selected reaction 7 involving acetonedicarboxylic acid. H02CCH2COCH2C02H 2C02(g) + CH3COCH3 (7) +
Preliminary studies confirmed the work of Hay and LeongI9 with this substrate. We also considered oxaloacetic acid, which was studied by Gelles and Salama,20 but we abandoned these efforts because of its lower solubility. The critical supersaturation of carbon dioxide is quite large, and we had to work with small volumes of concentrated solutions. A reasonable rate required going to at least 40 OC, and our method based on stirring could not be pushed to such temperatures. Therefore, all of our efforts to measure supersaturation quantitatively were based on release by sonication. A known amount of solid acetonedicarboxylic acid (obtained from Aldrich Chemical Co. and assumed to be 90% pure) was dissolved with warming in about 3 cm3 of water or NaC104 solution in a flask of total volume 139 cm3. The flask was connected to the apparatus, and extents of supersaturation were measured one or two times with each sample. Pressures before sonication never exceeded 0.05 atm, and those after sonication never exceeded 0.25 atm greater than atmospheric. The results are presented in Table I. Measurements from 40 to 50 O C and for NaC104 concentrations up to 3 M indicate that the threshold for nucleation of bubbles of carbon dioxide is 0.3-0.4 M and is little dependent on temperature or on ionic strength. This threshold is only about 10 times the solubility of 0.034 M; such a factor is much smaller than those observed for any of the diatomic gases. Unlike the measurements with NaBH4 reported above, the rates of gas evolution seemed to be about the same for solutions which had been unstirred long enough to attain a presumed steady state or for the same solutions during sonication continued after the burst of gas evolution.
Discussion The work presented here is preliminary. Table I in this paper and Table I11 in ref 16 illustrate the kinds of data obtained with the procedures we have developed. We did not believe it useful to tabulate here the results of over 100 measurements we made of supersaturations for individual solutions of other gases. The data encourage us to believe we could considerably improve the accuracy with which thresholds for nucleation could be measured. The thresholds for different diatomic gases show rather little variation. The minimum is 0.012 M for N2 at an ionic strength of 6 M as reported in ref 16. At no time with any diatomic gas did we observe a supersaturation much over 0.07 M. However, supersaturations with C 0 2 were several times as large as that figure. Of course, those measurements included concentrations of H 2 C 0 3and of HC03- if the appropriate equilibria were established rapidly enough during release of supersaturation. The results have been presented as the difference between the concentration at the threshold for nucleation (Cnuc)and that for saturation at 1 atm (C,,,). This quantity is directly calculable (19)Hay, R. W.;Leong, K. N. Chem. Commun. 1967, 800-801. (20) Gelles, E.;Salama, A. J . Chem. SOC.1958, 3689-3693.
from our measurements. A reviewer has suggested it would be more appropriate to report the supersaturation, S, as the ratio of those concentrations. Then S = Cnuc/Csat. We have presented handbook values for C,,,and rough estimates of S for all of the systems studied. We agree that S is probably a useful parameter for discussing the nucleation of a solid or of another liquid phase, but there does not seem to be any reason why solubility at 760 Torr should be a better parameter than that at any other pressure. For the diatomic gases we looked a t in aqueous solution, values and C,,, each ranged over a factor of about 3 but showed of CnUc rather little correlation with each other so that S ranged over a factor of about 5 . It is still early to say what sort of parameter will be best suited for discussing nucleation of bubbles. The same reviewer also asked how we could be sure we were not looking at nucleation on the walls or on colloidal particles. When the sonicator is turned on, the entire solution immediately turns milky. We are obviously looking at a supersaturation which has developed throughout the entire system. A major advantage of this procedure is that we can repeatedly nucleate thousands of bubbles in the same solution. Those bubbles should then sweep out any dust particles and colloids initially present. We are satisfied that the limiting thresholds we measure are for truly homogeneous nucleation. Several of these or similar reactions could be studied in nonaqueous solvents. We have made little effort to explore such possibilities. Increasing ionic strength seems to depress rather strongly the critical supersaturation of N2 but to have little effect on that of H2 or COz. We are surprised by the small effects of temperature. The very limited data suggest that critical supersaturation decreases with increasing temperature for N2, 0 2 , and C 0 2 but increases for H2. All such effects would correspond to activation energies much less than 10 kcal mol-'. The data strongly indicate that the rate of nucletion of bubbles is dominated by a very negative entropy of activation. Our apparatus did not permit us to explore the effect of external hydrostatic pressure on the threshold for nucleation of bubbles. We are convinced such an effect must exist but have no idea as to its magnitude. It should not be difficult to design apparatus for measuring that effect. Surface tension is a key factor in determining the internal pressure of a bubble of a particular size. That fact suggests that surfactants should greatly influence thresholds for nucleation of bubbles. However, it has been pointed out to us that surfactants concentrate in the macroscopic surface of a solution and may have rather little influence on the rate of homogeneous nucleation. We previouslyI5 observed rather little effect of surfactants on the dynamic behavior of a gas-evolution oscillator. We continue to be amazed that almost nobody seems previously to have thought of using a chemical reaction in order to generate very supersaturated solutions. Koch and Haaf2' did use the dehydration of formic acid to obtain supersaturated solutions of CO and carried out carbonylation reactions which otherwise would have required high-pressure apparatus. LaMer and Dinegar22used the decomposition of thiosulfuric acid to prepare exceptionally monodisperse sulfur sols where all particles had been nucleated homogeneously at almost the same time. We are not aware of other efforts to use chemistry for the quantitative study of phase nucleation. Of course, this chemical constraint can limit the gases for which thresholds of nucleation might be measured. For instance, divers use helium instead of nitrogen as a carrier gas for oxygen in order to reduce the danger of "bends" due to bubble nucleation. We see no prospect of using a chemical reaction to prepare supersaturated solutions of helium for study. Perhaps supersaturated solutions at ambient external pressures could be prepared if a gas at high pressure were allowed to diffuse through a membrane to (21) Koch, H.;Haaf, W. Ann. Chem. 1958, 618, 251-266. (22) LaMer, V. K.;Dinegar, R. H. J . A m . Chem. SOC.1950, 72,
4847-4854.
4198
J . Phys. Chem. 1987, 91, 4198-4203
enter solution as molecules rather than as microbubbles. We do not know whether membranes could be made which were simultaneously strong enough and permeable enough to function as desired. The above recital makes it clear we believe that this paper opens up the possibility of a major research program which could significantly impact the whole field of nucleation of one phase in another. Although we intend to pursue at least some of the openings suggested, we are in no position to explore all of the possibilities. In any field, it is desirable that there be some duplication in different laboratories to ensure that conclusions can be trusted. It is also true that extensive competitive duplication
is wasteful of scientific manpower. If our work suggests studies which others might wish to undertake, we would appreciate it if we were informed as to their plans. We would be happy to reciprocate.
Acknowledgment. This work was supported in part by Grant No. CHE-8405518 from the National Science Foundation. We are indebted to Professor Peter G. Bowers of Simmons College for making the trace of Figure 3 available to us. Professors Arthur W. Adamson of the University of Southern California and Howard Reiss of UCLA helped to introduce us to a literature with which we were unfamiliar; we accept responsibility for any oversights.
Vapor Pressure Isotope Effects in Liquid Methylene Difluoride Arundhati Kanungo, Takao Oi,+ Anthony Popowicz,*and Takanobu Ishida* Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 1 1 794 (Received: November 7 , 1986)
The vapor pressures of the isotopic methylene difluorides '*H2F2, I3CH2F2,and '%D2F2 have been measured at temperatures between 149.36and 244.82 K by differential manometric techniques in a precision cryostat. Throughout the whole temperature range of the measurement, P(13CHzF2)> P(I2CH2F2)> P("CD2F2). The data are best represented by T In cfc/fg)= -(31.64 f 1.97)/T- (0.4069 f 0.0107) for the I2C/l3C effect and by Tln cf,/f,) = (632.26 f 97.62)/T- (19.175 f 1.016) - (0.0532 f 0.0025)T for the H/D effect. The vapor pressure of the natural abundance methylene difluoride is given by log P(Torr) = 7.1990 - 842.31/[t ("C) + 246.811. The normal H/D vapor pressure isotope effect (VPIE) in liquid methylene difluoride is due to the zero-point energy shift upon condensation of the deuterio species being greater than that of the protio species, a fact which is also found for the H/D VPIEs in liquid methyl fluoride and fluoroform. The H/D effect in methylene difluoride is unusually higher than those found in methyl fluoride and fluoroform due to an enhanced perturbation of the C-H stretching modes by a relatively large intermolecular force in liquid methylene difluoride. Results of the normal-coordinate analyses of the cell model are presented. Temperature dependency of the external-internal interaction force constants is a necessary requirement for the satisfactory reproduction of the observed H/D and l2C/l3CVPIEs and spectroscopic data on the liquid.
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Introduction Studies of vapor pressure isotope effects (VPIE) have in the past led to elucidations of intermolecular forces and perturbations of internal vibrations in the condensed phases in a wide variety of The rather surprising success of the simple cell model which the analyses of these VPIE data have been based on is mainly due to the fact that all configurational factors of the condensed-phase partition function are independent of isotopic substitutions under the Born-Oppenheimer approximation and, thus, cancel 0ut2324when the isotopic ratio of the reduced partition function (RPFR) is taken. Following an isotope effect convention we define RPFR for an isotopic pair of a chemical species as In (s/s')J and s and s'are the symmetry numbers of the heavier and lighter isotopic species of the pair, respectively, and f is the ratio of partition function of the lighter to that of the heavier divided by the classical limit of the ratio. All quantities with a prime refer to the lighter species and the unprimed counterpart to the heavier one. The method of differential manometry using a cryostat such as the BBJR cryostat,25which is capable of tempetature stability and uniformity better than 0.001 K, provides precise (50.1%)data on the logarithm of the isotopic vapor pressure ratio, In (P'IP). The latter is directly related' to the ratio of the RPFR's of the condensed phase, f,,and the gas phase, fg In WfJ = (1 + P[Bo- (V,/RTIj In ( P ' / P ) (1) where Bo is the second virial coefficient of the gas (PVIRT = 1 t Present address: Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, Japan 'Present address: Rockefeller University, 1230 York Avenue, New York, NY 10021
0022-3654/87/2091-4198$01.50/0
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B f l ...) and V, the molal volume of the condensed phase. Usually, the internal forces of the gaseous molecules are well-
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0 1987 American Chemical Society
