J. Phys. Chem. 1992, 96, 1220-1224
1220
being a strongly activating group, leads to a high temperature of explosion; NOz in p-nitroaglinium perchlorate is highly deactivating, and hence the explosion temperature is lowered. However, all of the ortho substituents (o-CH3, 041, o-NH,, and o-CH30) decrease the explosion temperatures (Table IV). This also confirms the contention of Morrison and Boyd that ortho substituent groups generally exert the same effect36irrespective of their electron-demanding or -donating nature. Statistical treatment37v38 of the data shows that R2 values reported in Table V are very high in all cases and fitting of the kinetic data with pK, and the Hammett substituent constant (u) is quite satisfactory. Summarizing, the present results show that the proton-transfer process scems to be the primary step for the thermal decomposition and explosion of these perchlorate salts. Further, weak bonds are broken, and groups of atoms are produced. These charged and uncharged groups of various stabilities have different lifetimes and thus can be considered intermediates possessing enhanced reactivity. Because of this, they propagate solid-phase interactions to chain-like or even branchingchain-like processes having strongly exothermic reactions which may cause an explosion.
Conclusions
the primary (rate-controlling) step for the decomposition and explosion reactions. 2. The Hammett substituent constant (a) showed linear relationship with the kinetic parameters for decomposition and explosion reactions.
Acknowledgment. We thank the Head, Chemistry Department, for laboratory facilities. We also thank DRDO (New Delhi) and TBRL, Chandigarh, for financial assistance. We are grateful to Miss Pratima Srivastava (JRF) for TG data and to Shri Praveen Pratap Singh for discussion of statistical analysis of the data. Appendix coefedentOfMultipleDetormnntioa (R2)?73The calculations of R z (Table V) were made by the equation R:23 = 6 2 + 63 - 2r12r13r23/(l - 43) where r is the correlation coefficient between two variables and the subscript (1.2 3) of R2 is the measure of the joint influence of two independent variables (2,3) on a dependent variable (1). Reaction Constants ( p ) . The slope (p) of the curve log (klk,,) vs u was calculated by a least-squares method.2sv37
P =
1. The kinetic parameters for the thermal decomposition and explosion reactions have been found to be related linearly with the pK, of arylamines. The proton-transfer process seems to be (36)Morrison,R. T.;Boyd, R. N. Organic Chemistry; Prentice-Hall of India: New Delhi, 1989; p 959. (37) Snedecor, G. W.Stofistical Mefhods,4th ed.;The Iowa State College Press: Am=, IA, 1946. (38) Kendall, M. G.;Stuart, A. Aduanced Theory of Statisfics;Griffin: London, 1973; Vol. 11.
CXY/CXZ
It may be noted that values of meta and para perchlorates were considered in all of the calculations. Registry NO. C~HSNH~CIO~, 14796-11-3; o - C H ~ C ~ H ~ N H ~ C ~ O ~ , 41 195-12-4;m-CH3C6H4NH,CI04, 18720-58-6;pCH3C6H4NH3CI04, 14796-13-5;o - N H ~ C ~ ~ N H ~ 137570-00-4; C I O ~ ,pNH2C6H,NH,ClO4, 22755-08-4;o-C1C6H4NH3C1O4, 124454-54-2;m-ClC6H4NH3C104, 72057-68-2;p-C1C6H4NH,C104, 14999-68-9;m-N02C6H4NH3C104, 15873-53-7; pNO2C6H4NH,CIO4,15873-50-4; PCzH@C&NH$IO4, 137570-01-5; o - C H ~ O C ~ H ~ N H ~ C18720-44-0; IO~, pHOOCC6H4NH3CIO4,1 1 1203-38-4.
Simulation of the Effect of Stirring Rate on Bistabiiity in the Br0,--Ce( I I I)-Br- CSTR Reaction L6szli5 Gyiirgyi**+*tand Richard J. Field*” Department of Chemistry, University of Montana, Missoula, Montana 59812, and Institute of Inorganic and Analytical Chemistry, Eatviis Ldrhnd University, P.O.B. 32, Budapest- 1 1 2, H - 1518 Hungary (Received: April 25, 1991; In Final Form: September 24, 1991)
The effect of imperfect mixing on bistability in the BQ--Ce(III)-Br- reaction (minimal BrO; oscillator) in a continuowflow, stirred tank reactor (CSTR)is investigated as a function of flow rate using a phenomenological model of micromixing. There is a shift to higher flowrates and a broadening of the bistable region when mixing is less efficient, just as is found in experiments with nonpremixed feed streams. The simulations identify the source of this phenomenon as the effective initial segregation of incoming Br- from the bulk of the reacting mixture in packets dispersed uniformly throughout the reactor. This Br- segregation destabilizes the high-Br- (flow) state, which does not penetrate to as low flow rates, and stabilizes the low-Br- (thermodynamic) state, which penetrates to higher flow rates. The broadening results because the flow rate range over which the high-Brstate is destablized is smaller than the flow rate range over which the low-Br- state is stabilized.
Introduction Investigation over the last two decades of chemical reactions governed by nonlinear dynamic laws’ has led to the discovery of many exotic phenomena, e.g., oscillations, multistability, deterministic chaos, and spatial pattern formation. The nonlinear nature of the kinetic equations describing these systems also makes them very sensitive2 to the presence of spatial concentration heterogeneities3in apparently well-mixed solutions. This results because average reaction rates cannot be calculated from average To whom correspondence should be addressed. University of Montana. *&tviW M r h d University.
0022-3654/92/2096-1220$03.00/0
concentrations if the dynamic law is nonlinear. Such spatial inhomogeneitiesmay result in a poorly stirred batch reactor from local nucleation of an autocatalytic reaction and in a continuous-flow, stirred tank reactor (CSTR) from noninstantaneous mixing of the feed streams into the bulk. Thus various features of these reactions may depend on the stirring rate in a batch‘ (1) (a) Field, R. J., Burger, M., Eds. Oscillations and Trwcling Waves in Chemical Systems; Wiley-Interscience: New York, 1985. (b) Gray,P.; Scott, S.K. Chemical Oscillations and Instabilities. Non-linear Chemical Kinetics; Clarendon Press: Oxford, U.K., 1990. (2) Eptein, I. R. Nature 1990, 346, 16. (3) (a) Menzinger, M.;Dutt, A. K. J. Phys. Chem. 1990, 94, 4510. (b) Ochiai, E. I.; Menzinger, M. J. Phys. Chem. 1990. 94, 8866.
0 1992 American Chemical Society
Stirring Rate Effect on the BrO,--Ce( 111)-Br- CSTR Reaction The Journal of Physical Chemistry, Vol. 96, No. 3, I992 1221 TABLE I: Mechanism of the Minimal Bromate Oscillator rate constants reaction
Br-
k,
kr
+ BrO< + 2H+ * HOBr +
2.0 M-’s-I
3.2 M-’
+
3.0 x io6 M-I s-I 3.0 x 109 M-I s-I 42.0 M-2 s-’
2.0 x 10-5
8.0 x 104
8.9 x 103 M-I p l 1.0 x 10-8
HBr02 Br- HBrOz + H+ + 2HOBr
+ Br- + Ht + Br2 + (H20) Br0,- + HBr02 + H+ * 2Br02’ + M2OI Ce3+ + Br02’ + H+ == HBr02 + Ce4+ 2HBr02 * BrOC + HOBr + H+ HOBr
M-2 s-I
3.0 x 103
M-I
p l
s-l
M-l s-I
2.0 s-I 4.2 X lo7
M-I
s-I
I
0
reactor and on both stirring rate and feeding mode, i.e., premixed or nonpremixed, in a CSTR.5-8 Three chemical systems have predominated in the study of these effects: the Belousov-Zhabotinsky (BZ) r e a ~ t i o n , ~the . ~ ClO,--I~*~ ~ y s t e m , and ~ , ~ the ~ + ~minimal Br03- oscillator (MBO). BrO,--Ce(III)-Br- s y ~ t e m , the The effectiveness of mixing in chemical reactors is a central problem in chemical engineering as it may affect both the yield and the distribution of products obtained, even if the dynamic law is linear. The mixing process is usually assumedgto have two levels, macromixing and micromixing. Macromixing considers the formation of macroscopic heterogeneities and their breakdown into very small, segregated liquid packets which have not yet mixed on the molecular level. Macromixing may be quite rapid with good stirring. The d a y of this segregation by diffusive processes is called micromixing and may have a characteristic time as long as -1 s. The importance of chaotic mixing has recently been emphasized. lo Imperfect or slow micromixing has a strong effect on the chemical dynamics only if the governing equations of the chemical processes occurring contain nonlinear terms that evolve on a timescale faster than that of micromixing. Both macro-” and (4) (a) Farage, V. J.; Janjic, D. Chimia 1981, 35, 289. (b) NagypHI, I.; Eptein, I. R. J. Phys. Chem. 1986,90,6285. (c) Menzinger, M.; Jankowski, P. J. Phys. Chem. 1986,90, 1217. (d) Sevgik, P.; AdamEikovl, L. J. Chem. Phys. 1989,91, 1012. (e) AdamEikovB, L.; SevEik, P. Chem. Phys. Lett. 1988, 46,419. (f) Menzinger, M.; Jankowski, P. J. Phys. Chem. 1990,94,4123. (g) Lbpez-Tomas, L.; SaguQ, F. J. Phys. Chem. 1991,95,701. (h) Noszticzius, Z.; BodnHr, Z.; Garamszegi, L.; Wittmann, M. J. Phys. Chem., in press. (5) (a) Gradani, K. R.;Hudson, J. L.;Schmitz, R. A. Chem. Eng. J. 1976, 12, 9. (b) De Kepper, P.; Rossi, A,; Pacault, A. Seances Acad. Sci. 1976, 283C, 371. (6) Dutt, A. K.; Menzinger, M. J . Phys. Chem. 1990, 94, 4867. (7) Dutt, A. K.; Menzinger, M. J. Phys. Chem. 1991, 95, 3429. (8) (a) Roux, J.-C.; DeKepper, P.; Boissonade, J. Phys. Letf.A 1983, 97, 168. (b) Luo, Y.; Eptein, I. R. J . Chem. Phys. 1986,85, 5733. (c) Menzinger, M.; Boukalouch, M.; De Kepper, P.; Boissonade, J.; Roux, J.-C.; Saadoui, H. J . Phys. Chem. 1986,90,313. (d) Menzinger, M.; Giraudi, A. J. Phys. Chem. 1987, 91, 4391. (e) Boukalouch, M.; Boissonade, J.; De Kepper, P. J. Chim. Phys. 1987,84, 1353. (9) (a) Nauman, E. B.; Buffham, B. A. Mixing in Continuous Flow Systems; Wiley: New York, 1983. (b) Westerp, K. R.;Van Swaaij, W. P. M.;Beenackers, A. A. C. Chemical Reactor Design and Operation; Wiley: New York,1983. (c) Villermaux, J. ACS Symp. Ser. 1983,226, 135. (d) Villermaux, J. In Encyclopedia of Fluid Mechanics; Chereminisinoff, N. P., Ed.; Gulf Publishing: Houston, 1986; Chapter 27. (10) (a) Ottino, J. M.; Leong, C. W.; Rising, H.; Swanson, P. D. Nature, 1988, 333, 419. (b) Ottino, J. M. Sci. Am. 1989, 260(1), 56. (c) Muzzio, F. J.; Ottino, J. M. Phys. Rev.Left. 1989,63,47; Phys. Rev.A 1990,40,7182; 1990,4Z, 5873. (11) (a) Boissonade, J.; Roux, J. C.; Saadaoui, H.; De Kepper, P. In NonEquilbrium Qvnamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984, p 172. (b) Kumpinski, E.; Epstein, I. R.J . Chem. Phys. 1985.82.53. (c) Bar-Eli, K.; Noyes, R. M. J. Chem. Phys. 1986,85, 3251. (d) Gyargyi, L.; Field, R. J. J . Phys. Chem. 1989, 93, 2865. (e) Gyargyi, L.;Field, R. J. J. Chem. Phys. 1989, 91, 6131.
I
I
I
40
60
80
PUMP SElTlNG (arb. units)
M-2 s-l
OThe terms resulting from the homogeneous kinetics (HOM,) in the Appendix are formed by applying the law of mass action. The rate constants are those defined by Bar-Eli and Field.’* [H20] = 55 M is included in rate constants when (H201appears. The feed-stream compositions in the simulations were as follows: A, [Br03-] = 0.18 M, [H+] = 0.96 M; B, [Br-] = 7.5 X loJ M, [H+] = 0.96 M; C, [Ce(III)] = 9.18 X lo-’ M, [H+] = 0.96 M.
I
20
Figure I. Experimental data from ref 6 showing the effect of different stirring rates on the width and the position of the bistable region in the BrO,’Br%e(IV)
flow system. See text for experimental conditions.
mi~romixing’~J~ models have been used to model CSTR stirring and mixing effects in chemical systems governed by nonlinear dynamic laws. We are concerned here with the effect of different stirring rates on the limits of bistability when the Br03--Ce(III)-Br- reaction (MBO) is run in a CSTR. We have considered only experiments with nonpremixed feed streams in order to emphasize behavior inside the CSTR. Survey of Experimental Results
The MBO is the inorganic part of the BZ reaction. Its machanism is given by Noyes et al.14 (NFT). A slight moditication of the NFT mechanism is shown in Table I. The reaction Ce(IV) + Br02* H 2 0 * Ce(II1) Br03- 2H+ has been omitted as its rate is now known15 to be negligible. The MBO shows bistability16 over a wide range of conditions and oscillations” over a much smaller range. Bar-Eli and Fieldla have recently reviewed the chemistry and updated the rate constants of the NFT mechanism. The MBO-CSTR phenomena depend largely on the competition for HBrO, between Br- in reaction 2 and Br03- in reaction 4. The CSTR flow is the major source of Br-. The system falls into a low-[HBrO,], high-[Br-] steady state (flow branch, FB) if reaction 2 is dominant and into a high-[HBr02], low-[Br-] steady state (thermodynamic branch, TB) if reaction 4 is dominant. Thus, regardless of the initial composition in the CSTR, at a constant CSTR flow rate, Q, reaction 2 eventually will become dominant over reaction 4 at sufficiently high feed-stream [Br-1, [Br-Io, and reaction 4 will become dominant at sufficiently low [Br-1,. The bistable region occus at intermediate values of [Br-Io where the initial [Br-] in the CSTR determined whether the lowor high-[Br-] steady state eventually is reached. Oscillations occur in a small regime where neither steady state is able to achieve dominance. The bistable region also can be mapped out using Q, the CSTR flow rate, at constant [Br-lo because the bifurcation parameter really is QIBr-lO. A detailed experimental study of the effect of stirring rate on the MBO bistability in an isothermal CSTR was performed by Dutt and Men~inger.~.~ The reactor was a 31.5-mL vessel thermostated at 20.0 OC and stirred at 600 or 2820 rpm. The feed-stream compositions were as follows: A, 0.2925 M KBrO,
+
+
+
(12) (a) Horsthemke, W.; Hannon, L. J. Chem. Phys. 1984,81,4363. (b) Boissonade, J.; De Kepper, P. J . Chem. Phys. 1987,87,210. (c) Hannon, L.; Horsthemke, W. J . Chem. Phys. 1987, 86, 140. (d) Puhl, A.; Nicolis, G. Chem. Phys. 1987,87, 107. (13) Villermaux, J. In Spatial Inhomogeneities and TransienfBehaviour in Chemical Kinetics; Gray, P., Nicolis, G.,Baras, F., Borckmans, P., Scott, S. K., Eds.;Manchester University Press: Manchester, 1990; p 119. (14) Noyes, R. M.;Field, R. J.; Thompon, R. C. J. Am. Chem.Soc. 1971, 93, 7315. (15) Field, R. J.; Fijrsterling, H.-D. J. Phys. Chem. 1985, 90, 5400. (16) Geiseler, W.; Bar-Eli, K. J . Phys. Chem. 1981, 85, 908. (17) Orbln, M.; De Kepper, P.;Eptein, I. R. J. Am. Chem. Soc. 1982, 104, 2657. (18) Bar-Eli, K.; Field, R. J. J . Phys. Chem. 1990, 94, 3660.
Gyorgyi and Field
1222 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
'-1 I
i
:rrr
Figure 2. Schematic diagram of the phenomenological micromixing model from ref 13. The mathematical description of the model is shown in Table 11. Only two of the three segregated zones, which actually are dispersed throughout the CSTR, are shown in the figure. See Table I1
I
-1.45 1 1 .OE-4
in 0.75 M H2S04;B, 0.001 M NaBr in 0.75 M H2SO4; and C, O.OOO918 M Ce(NH4)(S04)2in 0.75 M H2S04. The system was monitored by a Pt electrode. It was found that slow stirring destabilizes the high-[Br-] flow branch and stabilizes the low-[Br-] thermodynamic branch (Figure 1). The transition from the flow branch (FB) to the thermodynamic branch (TB) as the flow rate is d d occurs at higher flow rates with the low than with the high stirring rate. The reverse transition from the TB to the FB when the flow rate is increased behaves similarly. Furthermore, the shift in the thermodynamic to flow branch transition is larger than for the reverse transition; thus the region of bistability widens. Figure 1 shows the region of bistability at the two stirring rates. The abscissa shows arbitrary pump settings that can be converted to absolute flow rates by the formula given by Dutt and Menzinger.' The transitions occur at TB 600 rpm:
FB TB
4.0E-4
5.OE-4
Figure 3. Simulated effect of imperfect mixing on the position and width of the MBO bistability. Feed-stream concentrations in the simulation were as follows: A, [BrO,-] = 0.18 M, [H+] = 0.96 M; B, [Br-1 = 7.5 X lo4 M, [H+] = 0.96 M; C, [Ce(III)] = 9.18 X lo4 M, [H+] = 0.96 M. Data for perfect mixing were obtained by solving the mass-action kinetics equations of the mechanism in Table I with CSTR flow terms. Parameters used to simulate imperfect mixing were tM = 2 s, t,, = 100 s, and a = 1. The transition points are FB TB at Q = (2.135 0.005) X 1 0 4 d m 3 s - 1 a n d T B + F B a t Q = ( 4 . 1 1 5 & 0 . 0 0 5 ) X 10-'dm3s-I.
-
*
FB at Q = 3.83 X lo4 dm3 s-l
Unless stated otherwise, a = 1, corresponding to an ideal CSTR, always was used. Numerical integrations were carried out using the Gear method%with the error calculated with respect to the current rather than to the maximum values of the variables and with an estimated Jacobian. An error tolerance value of 1O-Io typically was used. The program automatically mapped the hysteresis region by changing the flow rate and using the final conditions of the previous simulation to start the next one, as is done experimentally. Steady states were automatically detected, and the steady-state concentration values of the variables were recorded.
TB at Q = 4.94 X lom4dm3 s-l
ReSUltS
FB at Q = 7.51
The model shown in Tables I and I1 did not exhibit bistability with the feed concentrations used by Dutt and Menzinger.' The bifurcation analysis program, AUTO,^^ thus was used to find a set of constraints close to the experimental for which bistability a p pears. Such a set of feed concentrations is as follows: A, [ B a r ] = 0.18 M, [H+] = 0.96 M; B, [Br-] = 7.5 X lo" M, [H+] = 0.96 M, C, [Ce(III)] = 9.18 X lo" M, [H'] = 0.96 M. These [BrO
