11636
J. Phys. Chem. 1996, 100, 11636-11642
Kinetic Model for the Sonochemical Degradation of Monocyclic Aromatic Compounds in Aqueous Solution Alex De Visscher,* Peter Van Eenoo, David Drijvers, and Herman Van Langenhove Department of Organic Chemistry, Faculty of Agricultural & Applied Biological Sciences, UniVersity of Ghent, Coupure Links 653, B-9000 Gent, Belgium ReceiVed: December 13, 1995; In Final Form: April 3, 1996X
The breakdown of benzene, ethylbenzene, styrene, and o-chlorotoluene in aqueous solution by 520 kHz ultrasonic waves was studied at various initial concentrations in the millimolar range. First-order reaction rates depend upon both initial concentration and sonication time. These variations can be explained by a model that combines some physical and chemical aspects of sonochemistry. The basic assumptions of the model are first-order pyrolysis in the cavitations yielding both reactive/volatile and inert/nonvolatile products, and lowering of the maximum cavitation temperature due to the presence of the organic compounds in the bubble phase. Despite the necessary assumptions and approximations in order to limit the number of adjustable parameters, the lack of fit standard deviation after regression was as low as 4-9.2%.
Introduction It is well established that the violent collapse of cavitation bubbles in an ultrasonic field causes extremely high local temperatures, although estimated values vary widely.1-6 However, the present knowledge about the mechanism of sonochemical breakdown of organic compounds is limited. Two reaction mechanisms have mainly been proposed to explain the chemical and kinetic data. The first mechanism is pyrolysis in the cavitation bubbles. The second mechanism is the generation of OH radicals in the cavitation bubbles, which subsequently oxidize the organic compounds. Pyrolysis is expected to be the main reaction path for the degradation of apolar compounds. Lamy et al.7 found that the sonication of benzene and chlorobenzene yields acetylene and no hydroxylated compounds. Kinetic data of the sonochemical degradation of apolar compounds is scarce. Bhatnagar and Cheung8 studied the destruction of some chlorinated hydrocarbons by 20 kHz ultrasonic waves and fitted first-order kinetics to their data. The influence of the initial concentration was not reported to be checked. The reaction rates of degradation were found not to depend upon the presence of other reacting compounds, but the scatter of the kinetic data is up to 35%, too large to rely on this conclusion. Cheung and Kurup9 studied the sonochemical degradation of two CFC’s in aqueous solution and found that the reaction rate decreased during the course of the experiment. The pH decrease during the course of the experiment was assumed to cause this phenomenon. The effect was modeled kinetically by introducing a “baseline concentration" of CFC that is not degraded by the ultrasonic waves. Hart et al.6 investigated the sonolysis of methane and ethane in aqueous solution by 300 kHz ultrasound and found a considerable decrease of the reaction rate at high concentrations. Most hydrocarbons ranging from C1 to C4 could be found as reaction products. By comparing the yield of ethane to the sum of the yields of ethylene and acetylene, it was shown that the temperature of cavitation decreased with increasing methane concentration. X
Abstract published in AdVance ACS Abstracts, June 1, 1996.
S0022-3654(95)03688-4 CCC: $12.00
The production of OH radicals in an ultrasonic field can be detected by ESR.10 OH radicals are not expected to influence the breakdown of apolar compounds considerably, as CCl4 is efficiently broken down by ultrasonic waves,8 although the breakdown of CCl4 by OH radicals is negligible, compared to other chlorinated hydrocarbons.11 Polar compounds exposed to ultrasonic waves are expected to break down by OH radicals.7 Pe´trier et al.12 investigated the sonolysis of phenol in aqueous solution and found hydroquinone, catechol, and benzoquinone as reaction products. These reaction products are attributed to the attack of the hydroxyl radical on phenol. Further evidence is found in the fact that the degradation rate of phenol is closely related to the production rate of hydrogen peroxide. The zero-order degradation rate of phenol increases with increasing initial concentration and reaches a plateau at concentrations between 5 and 10 mM. The production rate of H2O2 decreases with increasing initial phenol concentration, reaching a plateau in the same concentration range. The recombination of OH radicals, yielding H2O2, appears to compete with the oxidation reactions. Still, pyrolysis also occurs when polar compounds are sonicated. Currell et al.13 found that the sonication of phenol yields acetylene as well. The formation could be inhibited almost completely by increasing the pH, thus ionizing the phenol molecule. Increasing the pH also decreased the degradation of phenol significantly, which shows that phenol degrades at least partially by pyrolysis in the bubble phase. Kotronarou et al.14 studied the sonolytical degradation of p-nitrophenol in aqueous solution and interpreted the results using an elaborate model involving both hydroxyl attack and pyrolysis reactions. The model was incorporated in a reaction scheme for the sonolytical degradation of parathion, a compound that yields p-nitrophenol as reaction product.15 Hua et al.5 studied the hydrolysis of p-nitrophenol and p-nitrophenyl acetate in the presence of ultrasound, with various cavitating gases. The reaction rate appeared to depend on the cavitating gas considerably. In both cases the reaction rates were lowest in the case of He and highest in the case of Kr. The use of Ar as cavitating gas yielded rates in between. Murali Krishna et al.16 demonstrated that the composition of alcohol-water mixtures influences the amount of pyrolysis in the cavitation bubbles. This is due to a change in the © 1996 American Chemical Society
Sonochemical Breakdown of Aromatic Compounds
Figure 1. Concentration vs time profiles of benzene in a semilogarithmic diagram.
J. Phys. Chem., Vol. 100, No. 28, 1996 11637
Figure 2. Concentration vs time profiles of ethylbenzene in a semilogarithmic diagram.
physicochemical properties of the solution. Fitzgerald et al.17 established that the nature of the gas dissolved into the liquid influences the rate of the sonolysis reactions. Griffing18 noted that volatile compounds such as diethyl ether lower the sonolysis reactions due to a decrease of the specific heat ratio in the cavitation phase. Gutie´rrez and Henglein19 assumed that alcohols inhibit the sonolysis of tetranitromethane the same way. No attempt has so far been made to incorporate these changes into a kinetic model of sonochemical degradation reactions. Materials and Methods Sonication experiments were performed with an Undatim Ortho Reactor, producing 520 ( 1 kHz ultrasonic waves. The high-frequency generator was equipped with an extra volt meter to allow more accurate power calibration. Power was controlled manually. Power transferred to the liquid (150 mL) was 14.6 W, measured calorimetrically. The reactor was a 200 mL glass vessel with a cooling jacket. The cooling water was kept constant at 25 °C, using a water bath with thermostat. Due to the transferred power, the steady state reaction temperature was 29 ( 1 °C. The reactor was sealed with a screw cap. A syringe needle was pierced through the septum of the screw cap for sampling. Samples (2 mL) were taken at various time intervals during sonication, with a glass syringe. These samples were transferred to small bottles and sealed with Mininert stoppers. A 1 vol % 2-hexanone solution (50 µL) was added as an internal standard. A 1 µL sample of the mixture was analyzed by a Varian 3700 GC, equipped with an FID. A 15 m DB-5 capillary column of 0.53 mm i.d. with a 1.5 µm stationary phase was used for separation. Helium was used as carrier gas, with a flow rate of 3.7 mL/min. Benzene was purchased from Merck, and 2-hexanone, from Janssen Chimica. All other chemicals were purchased from Fluka. Purity of all chemicals was 99+%. Experimental Results Benzene was sonicated at initial concentrations of 3.38, 1.69, 0.9, and 0.45 mM. Figure 1 shows the concentration vs irradiation time profiles of these experiments in a semilogarithmic plot. For first-order reactions, this representation yields straight lines. As can be seen from Figure 1, simple first-order kinetics are not applicable to the sonolysis of benzene. The first-order reaction rate appears to depend upon both concentration and sonication time. The experiments with initial concentrations 3.38 and 0.45 mM allow linear regression with good fit. The first-order reaction rates are 0.001 71 ( 0.000 09 min-1 (n ) 13) and 0.023 08 ( 0.000 96 min-1 (n ) 8), respectively. r exceeds 0.99 in both cases. The increase of the first-order degradation rate is more than 13-fold, although pyrolysis reactions are expected to be first order.
Figure 3. Concentration vs time profiles of o-chlorotoluene in a semilogarithmic diagram.
TABLE 1: First-Order Reaction Rate Coefficients of the Sonochemical Degradation of Ethylbenzene initial concn (mM)
sonication time (min)
k (min-1)
r
1 0.5 0.33
140 100 70
0.01667 ( 0.00191 0.03564 ( 0.00379 0.04333 ( 0.00243
0.993 (n ) 8) 0.997 (n ) 6) 0.998 (n ) 8)
The dimensionless Henry coefficient of benzene at 29 °C is 0.229.20 If it is assumed that headspace-water partitioning of benzene after prolonged sonication is governed by Henry’s law, a loss of 7% due to volatilization is expected. This would lead to an overestimation of the degradation rate of 0.0012 min-1 in a 1 h experiment. The overestimation is inversely proportional to the sonication time. This means an overestimation of the rates above of 11.7% and 2.2%, respectively. Volatilization beyond the headspace is not expected, as the solution becomes yellow upon sonication, even when the reaction rate is extremely slow. A broad range of reaction products is formed, including polar and nonpolar volatiles, semivolatiles, and polymers. Work is currently being done to identify these products. Ethylbenzene sonolysis experiments were performed at initial concentrations of 1, 0.5, and 0.33 mM. Higher concentrations were not used because the solubility of ethylbenzene in water is about 1.5 mM. The concentration vs time profiles are presented in Figure 2, in a semilogarithmic plot. The deviation from first-order kinetics is much less pronounced, but the reaction rate depends on the initial concentration. Table 1 shows the reaction rates. The increase is almost 3-fold upon decreasing the initial concentration by a factor of 3. Although the sonication time does not have a pronounced influence on the reaction rate, the influence of the initial concentration is comparable to the case of benzene. Sonication of o-chlorotoluene was performed at initial concentrations of 0.68, 0.34, and 0.17 mM. Figure 3 shows the concentration vs time profiles in a semilogarithmic plot. The profiles are very linear. The benzene experiment within this concentration range yielded a linear plot as well.
11638 J. Phys. Chem., Vol. 100, No. 28, 1996
De Visscher et al.
Figure 4. Concentration vs time profiles of styrene in a semilogarithmic diagram.
TABLE 2: First-Order Reaction Rate Coefficients of the Sonochemical Degradation of o-Chlorotoluene initial concn (mM)
sonication time (min)
k (min-1)
r
0.68 0.34 0.17
105 75 40
0.02637 ( 0.00138 0.02923 ( 0.00225 0.04145 ( 0.00223
0.999 (n ) 8) 0.998 (n ) 6) 0.999 (n ) 5)
TABLE 3: First-Order Reaction Rate Coefficients of the Sonochemical Degradation of Styrene initial concn (mM)
sonication time (min)
k (min-1)
r
0.97 0.49 0.25
80 80 40
0.01262 ( 0.00064 0.02409 ( 0.00233 0.03292 ( 0.00303
0.999 (n ) 9) 0.995 (n ) 9) 0.998 (n ) 5)
Table 2 shows the degradation rates for o-chlorotoluene. The influence of the initial concentration is small, but this might be due to the low concentrations used. Concentrations were kept low because exact solubility data of o-chlorotoluene was not available. The solubility was estimated using the estimated value of the Henry coefficient of o-chlorotoluene21 and the vapor pressure.22 The result was 1.5 mM. As the uncertainty upon an estimated Henry coefficient is very large, a broad margin was taken. Styrene was sonicated at initial concentrations of 0.97, 0.49, and 0.25 mM. The concentration vs time profiles are shown in Figure 4 in a semilogarithmic plot. The first-order plots deviate slightly from linearity and are not parallel. Table 3 shows the first-order reaction rates obtained in these experiments. The Model Pyrolysis reactions are expected to follow simple first-order kinetics. This means that parallel straight lines are expected in a semilogarithmic plot of the concentration vs time profiles. This is clearly not the case. Reaction rates are higher at low concentrations. Therefore, it can be assumed that the reaction conditions change as the concentration of the organic compound decreases. As a cavitation bubble collapses, the gas inside is compressed almost adiabatically. The temperature rise depends, among other factors, upon the specific heat of the gas mixture. The vapor specific heat cp of aromatic compounds is of the order of 120 J/(mol K),23 much higher than the specific heat of air (30 J/(mol K)) or water vapor (36 J/(mol K)).24 Therefore, the adiabatic temperature rise upon compression of a cavitation is much lower if an aromatic vapor is present. Excessive amounts of aromatic compounds lower the temperature of a collapsing cavitation bubble, which lowers the rate of degradation. If incorporation of the reactant concentration in the rate constant suffices to model the reaction kinetics, then the slopes of the concentration profiles for a given compound at a given
concentration should be the same, irrespective of the initial concentration. As can be seen from Figures 1-4, this is not the case. Concentration profiles are always less steep when the initial concentration is high. This can be explained by considering the presence of reaction products that lower the temperature rise upon cavitation as well. Therefore an adequate model of sonochemical degradation should assume both reactant and reaction products to influence the temperature upon cavitation collapse. If the number of cavitation bubbles is constant throughout the experiment, the reaction rate observed is proportional to the reaction rate in the cavitations during collapse. Second, the concentration of the organics in the bulk liquid phase is assumed proportional to the concentration in the bubble phase. This could be due to Henry’s law, if diffusion of the compounds into the bubble is sufficiently rapid, or Fick’s law, if the transfer is diffusion-controlled. As a result of these assumptions, the reactions can be described with respect to the liquid phase, i.e.
r ) kCl
(1)
with r the reaction rate related to the liquid phase (mM min-1), k the rate constant (min-1), and Cl the concentration of the compound in the liquid (mM). k obeys Arrhenius’ law:
k ) A exp(-E/RTc)
(2)
with A the Arrhenius proportionality factor (min-1), E the activation energy (J/mol), R the universal gas constant (8.314 J/(mol K)), and Tc the temperature of the cavitation during collapse (K). Tc is attained by adiabatic compression:25
Tc ) T(γ - 1)Pmax/Pmin
(3)
with Pmax the maximum pressure in the liquid phase, Pmin the minimum pressure in the vapor phase, and T the minimum temperature of the cavitation (K), assumed equal to the liquid temperature. γ is the specific heat ratio cp/cv. This ratio depends on the composition of the gas mixture in the cavitation. If the mole fraction of the organic compounds in the cavitation is relatively small, then a linear relationship between the specific heat ratio of the mixture γ and the concentration Cl in the liquid phase can be assumed.
γ ) γ0 - KCl
(4)
K is a proportionality constant (mM-1). This expression is derived in Appendix A. The assumption of linearity will be discussed later on. Combining eqs 2-4 yields a relationship between the reaction rate constant and the liquid concentration:
(
k ) A exp -
)
E RT(Pmax/Pmin)(γ0 - KC1 - 1)
(5)
The rate constant at infinite dilution is
(
k0 ) A exp -
)
E RT(Pmax/Pmin)(γ0 - 1)
(6)
Combining eqs 5 and 6 yields
(
k ) k0 exp -
EPminKC1 RTPmax(γ0 - 1)(γ0 - 1 - KC1)
)
(7)
If KCl is small compared to γ - 1, then KCl is negligible in the
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J. Phys. Chem., Vol. 100, No. 28, 1996 11639
SCHEME 1: Assumed Reaction Scheme (S, reactant; P, volatile products)
denominator. Thus it follows that
(
k ) k0 exp -
EPminKC1 RTPmax(γ0 - 1)2
)
(8) Modeling Results and Discussion
This assumption will be discussed later on. The only factor that is not a constant in the exponent is Cl. Therefore, the equation can be written as follows:
k ) k0 exp(-aCl)
with a fourth-order Runge-Kutta routine.26 The time increment was 0.5 min. The model is fitted to a series of experimental concentration profiles by nonlinear regression, using one set of adjustable parameters. These parameters are k1,0, a, f, and the initial concentration of every experiment. Regression was performed on the logarithm of the concentrations, in order to obtain best fit assuming constant relative uncertainty of the experimental results. A Marquardt-Levenberg routine27 was used for regression.
(9)
Assuming that the concentration of the organic compounds is sufficiently small leads to a very simple expression for the reaction rate. As mentioned before, the occurrence of reaction products has to be accounted for. The reaction scheme assumed is shown in Scheme 1. The reactant S is transformed to reactive products P and inert products. In this context, a reaction product is assumed reactive if it is volatile. Heavy products precipitating in the liquid can be considered inert. Several assumptions are made at this point. First, the degradation rates of the reactant S and the reactive products P into inert molecules are assumed equal. This reduces the number of adjustable parameters by one, and it allows consideration of the group S + P as one pseudospecies with concentration Cl,2. This pseudospecies degrades with rate constant k2 ) ka. The reactant degrades with reaction rate k1 ) ka + kb. Second, it is assumed that the concentration Cl in eq 4 can simply be replaced by the concentration of the pseudospecies Cl,2. This involves the assumption that the proportionality factor between the concentration in the liquid phase and the concentration in the cavitations is the same for S and P. It also involves the assumption that the specific heats of S and P are the same. Third, it is assumed that the activation energies of all reactions are the same. This allows the use of eq 9 with the same parameter a for all reactions. In view of all these assumptions and approximations, this model should be considered largely empirical. Eventually, the model consists of two reaction rates:
r1 ) k1,0 exp(-aCl,2)Cl,1
(10)
r2 ) fk1,0 exp(-aCl,2)Cl,2
(11)
r1 is the reaction rate of the reactant S, k1,0 is the rate constant of the degradation of S at infinite dilution, and Cl,1 is the concentration of S in the liquid phase. r2 and Cl,2 are the reaction rate and the concentration of the pseudospecies S + P. f is the fraction of S that is converted directly to inert reaction products, and due to the aforementioned assumptions, f is also the ratio between the rates of reactions 10 and 11. Concentration vs time profiles are obtained by numerical integration of the following differential equations:
dCl,1/dt ) r1
(12)
dCl,2/dt ) r2
(13)
The model results are represented in Figures 1-4 by the solid lines. Agreement is obtained with no systematic error. The parameters k1,0, a, and f are shown in Table 4, together with the lack of fit standard deviation and the correlation coefficient. All parameters are significantly different from zero, except the values of f in the case of ethylbenzene and o-chlorotoluene. This parameter influences the curvature of the concentration vs time profiles. The insignificant values of f in the case of ethylbenzene and o-chlorotoluene reflect the straightness of the concentration profiles in a semilogarithmic plot. From a mechanistic point of view, f equal to zero means that no significant amount of inert products is formed. This is unlikely, since the solution becomes turbid upon sonication. It is more likely to assume that the reactive products formed have a stronger influence on the temperature of the collapsing bubbles, because of a larger partition coefficient between the liquid and the bubble, a more rapid diffusion into the bubble phase, or a higher specific heat, in conjunction with the stoichiometric factor of the reaction product. More research is necessary to elucidate this aspect of the mechanism. Therefore, no attempt has been made to simplify the model in the case of ethylbenzene and o-chlorotoluene by assuming that f equals zero. This would provide too optimistic an estimation of the confidence intervals of the parameters k1,0 and a. All values of f differ significantly from 1, which indicates that there are products that influence the cavitation indeed. Work is currently being done to identify these products. All values of a are significantly positive. This proves that the presented model is superior to simple first-order kinetics. The values of k1,0 and a do not vary substantially. k1,0 reflects the reactivity of the compounds upon sonication. The partition coefficient of the compound between the liquid phase and the bubble phase will clearly influence k1,0. As mentioned before, this partition coefficient can depend on the diffusion rate of the organic compounds toward the gas bubbles and on the Henry coefficient. In Figure 5, k1,0 is plotted versus the dimensionless Henry coefficient (i.e., concentration in the gas phase (mol/L), divided by the concentration in the water phase (mol/L), at equilibrium) at 25 °C. Any relationship, if existing, is not very clear. It may be argued that styrene is more reactive than expected, due to polymerization. Benzene might be less reactive than expected because it has no substituent on the aromatic ring. Another way to determine the factor governing the partitioning between liquid and cavitations is by studying the parameter a. This parameter indicates how efficient the compounds lower the cavitation temperature during collapse. However, the physicochemical properties of the reactive reaction products also influence a, which complicates comparison. Figure 6 shows parameter a vs the dimensionless Henry coefficient. Again, there is no clear relationship. Contrarily, the only straight line that falls within all confidence intervals, is a descending one. These results suggest that diffusion limitations govern the partitioning of the aromatic compounds between the liquid and
11640 J. Phys. Chem., Vol. 100, No. 28, 1996
De Visscher et al.
TABLE 4: Parameter Values after Regression (with 95% Confidence Intervals), Lack of Fit Standard Deviations, and Correlation Coefficients compd
k1,0 (min-1)
a (mM-1)
f
s (%)
r
benzene ethylbenzene o-chlorotoluene styrene
0.0271 ( 0.0016 0.0622 ( 0.0107 0.0431 ( 0.0072 0.0446 ( 0.0057
0.996 ( 0.101 1.352 ( 0.458 0.828 ( 0.552 1.569 ( 0.28
0.657 ( 0.175 0.065 ( 0.142 0.069 ( 0.286 0.309 ( 0.221
6.1 9.2 7.3 4.0
0.999 (n ) 40) 0.998 (n ) 22) 0.998 (n ) 19) 0.999 (n ) 23)
of the frequency of the ultrasonic waves on the sonolysis process. The frequency has an influence on the resonance radius of the cavitation. Bubbles larger than resonance size are not assumed to collapse violently.28 Pe´trier et al.12 assume that the bubble collapse is most violent at resonance size. The resonance frequency of a vibrating gas bubble is28
νres )
Figure 5. k1,0 vs dimensionless Henry coefficient at 25 °C (H benzene and ethylbenzene from De Wulf et al.;20 H o-chlorotoluene estimated from Schwarzenbach et al.;21 H styrene calculated from Willink32).
1 2πR0
x
3γ(P0 + (2σ/R0)) F
(14)
with R0 the bubble radius (m) at hydrostatic pressure P0 (Pa), γ the specific heat ratio, σ the surface tension (N/m), and F the density of the liquid (kg/m3). γ can be set equal to 1.33 for an air-water vapor mixture,28 and σ equals 0.07 197 N/m for water at 25 °C.22 Solving eq 14 for νres equal to 520 000 Hz yields a resonant bubble radius of 6.77 µm; if νres equals 20 000 Hz, the resonant bubble radius is 161 µm, 23.7 times as large. The non-steady-state diffusion theory in spherical symmetry29 shows that the equilibration time of a partitioning process is proportional to the square of the sphere radius. Equation 17 in the next paragraph confirms this. In the case of 20 kHz ultrasonic waves, equilibration time for a resonant bubble will be more than 560 times longer than in the case of 520 kHz ultrasonic waves. Despite the longer cycle time, equilibrium between the liquid phase and the bubble phase is harder to obtain with lowfrequency ultrasonic waves. This might explain why Bhatnagar and Cheung8 did not find any deviations from first-order kinetics. This might also explain why sonochemical degradation is more efficient at high frequency.7 Estimation of the Organic Vapor Content of the Cavitation Bubbles During growth of a cavitation bubble, the organic molecules diffuse into it. Since the growth time is very short, on the order of 1 µs, the diffusion is restricted to a very thin liquid layer surrounding the bubble. In this case planar geometry can be assumed. The concentration profile of a compound near an interface of zero concentration is29
C ) Cl erf(z/(4Dlt)1/2) Figure 6. a vs dimensionless Henry coefficient at 25 °C (H benzene and ethylbenzene from De Wulf et al.;20 H o-chlorotoluene estimated from Schwarzenbach et al.;21 H styrene calculated from Willink32).
the cavitation bubbles. This is confirmed by calculations made in the next paragraph. Comparison with Other Studies The finding that the degradation rate of the compounds decreases with increasing concentration is consistent with the findings of Hart et al.,6 but such a behavior is not clear from Bhatnagar and Cheung’s study.8 To compare the results presented above with the work of Bhatnagar and Cheung,8 it is necessary to consider the influence
(15)
with C the concentration in the liquid boundary layer (mM ) mol/m3), Cl the bulk liquid concentration, z the distance from the bubble interface (m), Dl the diffusion coefficient (m2/s), and t the time (s). The flux F (mmol/(m2 s)) at the interface is
F ) SCl(Dl/πt)1/2
(16)
with S the interface surface (m2), which equals 4πR2, R being the radius of the bubble. Integrating the flux and dividing by the bubble volume yields the concentration in the bubble as a function of time:
Cg ) (6/R)(Dlt/π)1/2Cl
(17)
Taking Dl ) 1.2 × 10-9 m2/s for benzene at 29 °C,21 R ) 6.77
Sonochemical Breakdown of Aromatic Compounds
J. Phys. Chem., Vol. 100, No. 28, 1996 11641
× 10-6 m (resonant size), t ) 0.96 × 10-6 s (half cycle30), and a liquid concentration of 3.38 mol/m3, which is the highest concentration used in the regression, a gas concentration of 5.74 × 10-2 mol/m3 is obtained. The saturated water vapor concentration22 at 29 °C is 1.594 mol/m3, assuming ideal gas behavior. The mole fraction of benzene in the bubble is 0.0347. According to eq 15, the concentration reaches 99% of the bulk concentration at a distance of 1.821(4Dlt)1/2, which equals 1.2 ×10-7 m, less than 2% of the bubble radius. The use of planar geometry is justified. Taking 0.229 for the dimensionless Henry coefficient of benzene at 29 °C,20 the liquid concentration in equilibrium with 5.74 ×10-2 mol/m3 gas is 0.25 mM, about 7.4% of the bulk concentration of benzene. The assumption of a zero interface concentration is justified. The specific heat ratio of benzene is 1.1.24 The term (γ1 1)x0 in eq A6 of Appendix A equals 0.09653; (γ0 - 1)x1 equals 0.01145, using 1.33 for the specific heat ratio of water. The slope of γ vs x1 will change only slightly over 10% in the concentration range used. This means that the linearization used in eq A7 is justified. In eq 8, another approximation is made, based on the low value of KCl. Using the above values, and comparing eq A6 with eq 4, a value of 0.0244 is obtained at a liquid concentration of 3.38 mM, about 7.4% of the term γ0 - 1. The approximation is justified. With the assumptions as above, the activation energy of the reaction can be estimated using the definition of parameter a.
E)
aRTPmax(γ0 - 1)2 KPmin
(18)
If Pmax is chosen equal to the hydrostatic pressure and Pmin equal to the vapor pressure of water,28 this equation yields an activation energy of 954 kJ/mol for benzene, about twice the hydrogen-carbon bond strength in benzene (465 kJ/mol31). According to eqs 3 and 4, the parameters used suggest a temperature of 2521 K for the imploding cavitations in pure water and a temperature of 2335 K in the presence of 3.38 mM benzene. This is within the range obtained by Misˇ´ık et al.4 and by Hart et al.6 Therefore, it is assumed that the calculations are reasonably accurate. According to these calculations, all approximations made in the derivation of the reaction rate as a function of temperature appear to be justified. Conclusions The sonochemical degradation of monocyclic aromatic compounds can be described by a model, assuming that the presence of the organic compounds lowers the temperature of the collapsing cavitation bubble, by lowering the specific heat ratio. It was necessary to account for the existence of reaction products that also influence this temperature. A simple triangular reaction scheme was used for this purpose. Despite the linearization performed in order to simplify the model and the assumptions made to limit the number of adjustable parameters, a good fit was obtained. Lack of fit standard deviations ranged from 4 to 9.2%, and regression correlation coefficients were well over 0.99 in all cases. Correlations between the parameters and the Henry coefficient are not clear, which suggests that partitioning of the organic compounds between the liquid phase and the bubble phase is mainly governed by the diffusion rate. Some brief calculations support this assumption. The calculations also confirm that the approximations made are justified. It can be explained why deviations from first-order kinetics are less clear in the case of 20 kHz ultrasound.
List of Symbols exponential coefficient (mM-1) Arrhenius proportionality factor (min-1) concentration of organic compound in the vicinity of an interface (mM) concentration of S in the bubble phase (mol/m3 ) mM) Cg concentration in the liquid phase (mM) Cl concentration of S in the liquid phase (mM) Cl,1 concentration of pseudospecies S + P in the liquid phase (mM) Cl,2 specific heat at constant pressure (J/(mol K)) cp specific heat of the air-water vapor mixture in the cavitation cp,0 bubbles, at constant pressure (J/(mol K)) specific heat of the organic compound vapor at constant pressure cp,1 (J/(mol K)) specific heat at constant volume (J/(mol K)) cv specific heat of the air-water vapor mixture in the cavitation cv,0 bubbles, at constant volume (J/(mol K)) specific heat of the organic compound vapor at constant volume cv,1 (J/(mol K)) diffusion coefficient in the liquid phase (m2/s) Dl E activation energy (J) k first-order rate constant (min-1) first-order rate constant at infinite dilution (min-1) k0 first-order rate constant of reactant S (min-1) k1 first-order rate constant of reactant S at infinite dilution (min-1) k1,0 first-order rate constant of pseudospecies S + P (min-1) k2 K proportionality factor (mM-1) hydrostatic pressure (Pa) P0 solvent vapor pressure (Pa) Ps Pmax maximum pressure in the liquid during cavitation collapse (Pa) Pmin minimum pressure in the bubble during rarefaction (Pa) r reaction rate (mM min-1) reaction rate of reactant S (mM min-1) r1 reaction rate of pseudospecies S + P (mM min-1) r2 R universal gas constant (8.314 J/(mol K)) or bubble radius (m) bubble radius at equilibrium with hydrostatic pressure (m) R0 S surface of bubble interface (m2) t time (min or s) T bulk liquid temperature (K) temperature of the cavitation during collapse (K) Tc mole fraction of water vapor and air in the cavitation bubble x0 mole fraction of the organic compound in the cavitation bubble x1 z distance from the bubble interface (m) Greek Symbols γ specific heat ratio of the gas-water-organic vapor mixture in the cavitation bubble specific heat ratio of the gas-water vapor mixture in the γ0 cavitation bubbles specific heat ratio of the organic vapor in the cavitation bubbles γ1 resonance frequency of a cavitation bubble (s-1) νres F liquid density (kg/m3) σ surface tension (N/m) a A C
Acknowledgment. A.D.V. wishes to thank the Vlaams Impulsprogramma Milieutechnologie (VLIM) for financial support. D.D. wishes to thank the Vlaams Instituut voor de Bevordering van het Wetenschappelijk-Technologisch Onderzoek in de Industrie (IWT) for financial support. Appendix A. Calculation of the Specific Heat Ratio of Binary Gas Mixtures The specific heat ratio γ is defined as
11642 J. Phys. Chem., Vol. 100, No. 28, 1996
γ)
cp cv
De Visscher et al.
(A1)
with cp the specific heat at constant pressure (J/(mol K)) and cv the specific heat at constant volume (J/(mol K)). For ideal gases, the relation between cp and cv is
c p ) cv + R
(A2)
with R the universal gas constant (8.314 J/(mol K)). The specific heats of a gas mixture are
cp ) cp,0x0 + cp,1x1
(A3)
cv ) cv,0x0 + cv,1x1
(A4)
Subscript 0 refers to the air-vapor mixture in the cavitation of pure water and is considered as a pseudospecies with mole fraction x0. Subscript 1 refers to the organic compound, with mole fraction x1. Substitution of eqs A3 and A4 in eq A1, multiplying both numerator and denominator by R/(cv,0cv,1), and substituting eqs A1 and A2, applied to both species, yields
γ)
γ0(γ1 - 1)x0 + γ1(γ0 - 1)x1 (γ1 - 1)x0 + (γ0 - 1)x1
(A5)
γ can be written as a deviation from γ0:
γ ) γ0 +
-γ0(γ0 - 1)x1 + γ1(γ0 - 1)x1 (γ1 - 1)x0 + (γ0 - 1)x1
(A6)
If x1 is sufficiently small, then x0 is almost equal to unity. In that case, (γ1 - 1)x0 is much larger than (γ0 - 1)x1. Equation A6 becomes
γ ) γ0 + x1
(γ0 - 1)(γ1 - γ0) (γ1 - 1)
(A7)
The factor [(γ0 - 1)(γ1 - γ0)/(γ1 - 1)] is a constant that will be negative in the present case since the specific heat ratio of aromatic compounds is smaller than the specific heat ratio of water or air. If x1 is small, this factor will be proportional to the liquid concentration of the organic compound. Therefore, eq A7 can be written as
γ ) γ0 - KCl
(4)
References and Notes (1) Suslick, K. S.; Hammerton, D. A.; Cline, R. E., Jr. J. Am. Chem. Soc. 1986, 108, 5641.
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