762
J . Phys. Chem. 1984,88, 762-766
impurities must have the same effect on the kinetic rigidity of the amorphous phase as is observed, through glass transition temperature vs. composition studies, in the better known “network + modifier” systems, Si02+ sodium oxide, and ZnCI, modifier chloides.26-2* In each case the disruption of the network leads to a rapid decrease in Tgup to a limit where Tgbecomes at least temporarily independent of composition. In the region of steeply decreasing glass transition temperatures, major fragments of the initial network still exist. In the case of the aqueous system it is these fragments which produce the easy nucleation of ice. It seems reasonable, then, that by the point in composition where glass formation can be observed in normally cooled aqueous solutions the glass transition temperature should have reached its composition-independent regime. It should be possible to confirm this interpretation by thermal studies of vapor-deposited binary amorphous solids formed by, e.g., deposition from H202+ H 2 0 vapors in equilibrium with H 2 0 2 + H 2 0 solutions of known concentrations. We are at the moment unable to account for the observations of McMillan and Los,’ and Sugisaki et al.’ The heat released on crystallization in our study was 1.5 0.1 kJ mol-’, based on quantitative study of three samples. This is in reasonable agreement with the maximum value, 1.64 kJ mol-’, reported by Sugisaki et al. Ghormley, whose DTA traces also show no glass transition, reported a slightly larger heat of crystallization, 1.8 kJ mol-‘.6 An important consequence of this discussion is the new light it throws on the problem raised by Johari.14 Johari argued, on the basis of the increase of heat capacity at 130 K reported by Sugisaki et al.,’ that the rate of increase of entropy in the supercooled liquid was too great to be compatible with the known entropy of water at the limit of C, measurement at -38 0C.29 Johari concluded that the amorphous solid was thermodynamically
unconnected to the normal pressure liquid state, and suggested that it might bear a closer relation to the high-pressure polymorphs of ice. But if, as we have concluded above, the amorphous state cannot commence producing entropy at a liquidlike rate at temperatures less than 160 K (or even higher, as would be the case if one were to heat the sample at a rate sufficient to avoid the crystallization and connect directly to the liquid), then the problem raised by Johari simply goes away. This leaves the ground free to establish the hoped-for connections between the vapor-deposited amorphous solid and the snap-quenched liquid, which are apparently now being tentatively established experimentally through the identification of an amorphous solid from rapidly quenched liquid in the work of Brugeller and Mayerz1and Dubochet et aLZ2 Although there are some unexplained differences between jet-quenched and vapor-deposited ASW X-ray patterns in the reports of Brugeller Mayer,’ Dubochet et aL30have recently given good electron diffraction and kinetic evidence for indistinguishability of the liquid-route and vapor-route forms. Their characterizations were performed under identical conditions to minimize ambiguities. The warm-up behavior of an initially vitreous sample when heated as rapidly as-or more rapidly than-was necessary to vitrify it remains a matter for speculation. It now seems it would look more like the suggestion of Rice et aLL8than that of Angell and T ~ c k e r ;however, ’~ if supercooled water is characterized by fast “normal” configurational relaxations as well as by slow “anomalous” configuration relaxation modes, then the warm-up curve should still show two relaxation steps rather than the single step considered in ref 18.
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+
*
~~~
Acknowledgment. This work was supported by the Office of Naval Research under agreement No. N00014-78-C-0035. Registry No. Water, 7732-18-5
~-
(26) Green, R. L. J . Am. Ceram. S O ~1942, . 25, 83. (27) Easteal, A. J.; Angell, C. A. J . Phys. Chem. 1970, 74, 3987. (28) Paik, J. S.; Perepezko, J. H., J . Non-Cryst. Solids 1983, 56, 405.
(29) Angell, C. A.; Sichina, W. J.; Oguni, M. J . Phys. Chem. 1982, 86, 998. (30) Dubochet, J.; Adrian, M.; Vogel, R. H. Cryolrtters 1983, 4, 233.
Asymptotic Solutions of a Reduced Oregonator Model of the Belousov-Zhabotinsky Reaction Michael F. Crowley and Richard J. Field* Department of Chemistry, University of Montana, Missoula, Montana 5981 2 (Received: January 7, 1983; In Final Form: May 9, 1983)
Singular perturbation techniques have been used to find relatively simple, closed-form solutions to a reduced, two-variable version (due to Tyson) of the Oregonator model of the oscillatory Belousov-Zhabotinsky (BZ) reaction. The reduction is accomplished by assuming that the value of Y in the Oregonator very rapidly follows that of X . Thus, dY/dt is set equal to zero and an algebraic relation between X and Y obtained.
Introduction The Belousov-Zhabotinsky (BZ) reaction’ is the most thoroughly studied and understood oscillating chemical reaction.2 In its classic form it is the metal-ion-catalyzed oxidation and bromination of malonic acid by bromate ion in a strongly acid medium. It shows a striking variety of interesting and important behaviors in both batch and CSTR (continuous-flow, stirred tank (,l) Field, R. J. In “Theoretical Chemistry: Advances and Perspectives”;
Eyring, H., Henderson, D., Eds.; Academic Press: New York, 1978; Vol. 4, Chapter 2. (2) Field, R. J.; KBros, E.; Noyes, R. M.J . Am. Chem. SOC.1972, 94, 8649.
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reactor) experiments. These behaviors involve the concentrations of the oxidized and reduced forms of the metal-ion catalyst as well as those of various intermediate species and include the following: periodic2 and aperiodic3 temporal oscillations, bistability; hystere~is,~ and traveling waves.6 These phenomena are interesting and important not only because of the complex chemistry occurring but also because they are relatively easily (3) Turner, J. S.; Roux, J.-C.; McCormick, W. D.; Swinney, H. L. Phys. Lett. A 1981, H A , 9. (4) DeKepper, P.; Boissonade, J. J . Chem. Phys. 1981, 75, 189. ( 5 ) (a) Marek, M.; Svobodova, E. Biophys. Chem. 1975,3,263. (b) Janz, R. D.; Vanecek, D. J.; Field, R. J. J . Chem. Phys. 1980, 73, 3132. (6) Field, R. J.; Noyes, R. M. J . Am. Chem. SOC.1974, 96, 2001.
0 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 4, 1984 763
Reduced Oregonator Model of the BZ Reaction studied examples of types of complex dynamic phenomena that occur in other areas as diverse as biology7 and engineeringe8 The connection between oscillating chemical reactions and similar phenomena in other areas is through the class of nonlinear differential equations that describe them alL9 The differential equations which describe the BZ reaction are based upon a chemical mechanism proposed by Field, Koros, and Noyes (FKN)., This mechanism is complex and leads to a set of differential equations that is so large and difficult that it can be handled practically only by numerical techniques.1° Thus, Field and Noyes” used chemical arguments to simplify the mechanism to a point such that it can be described by a set of differential equations involving only three variables. This reduced model is called the “Oregonator”: k3
A+Y-X+P kl
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X+Y-P+P A
+ X -% 2X + Z ki
X+X-A+P
(A)
(B) (C)
(D)
The relationship of this model to the BZ mechanism is through the following identities: X = HBr02, Y = Br-, Z = Ce(IV), A = Br03-, and P = HOBr. The rate constants for the above reactions are numbered according to the FKN scheme, by analogy to which numerical values were assigned to k3, k2, k5a,and k4 by Field and Noyes.” However, k5 and the stoichiometric factor, J are usually treated as expendable parameters whose values are of the order of 1. WolfeI2 has shown that the full FKN mechanism is a singular pert~rbation’~ of the Oregonator.” Tyson14has shown that the Oregonator is only one of a number of reduced models that can be derived from the full FKN mechanism by means of perturbation techniques13but concluded that the Oregonator is the best of these in terms of reproducing the experimentally observed behavior of the BZ reaction. Substantial numerical and qualitative mathematical analysis has been done on the Oregonator. The existence of limit cycle oscillations corresponding to the experimentally observed BZ oscillations has been proved,15 and the form of these oscillations has been calculated numerically.L1Stanshine and Howardt6used singular perturbation techniques to find a zero-order, asymptotic solution13of the Oregonator equations and have given a formula for the period of the oscillatons. Their formulas, however, are rather cumbersome to be of great practical use. The dual problems of quantitative reproduction of the BZ oscillations and the need to do mathematical analysis can be ameliorated by use of an important modification of the Oregonator due to Tyson.I4 He showed that good quantitative agreement between the Oregonator and the BZ reaction can be obtained by suitable adjustment of the Oregonator rate constants away from the values suggested by the FKN mechanism. Furthermore, he showed by a rescaling of the original Oregonator differential (7) Hoppensteadt, F. C., Ed. ”Nonlinear Oscillations in Biology”; American Mathematical Society: Providence, RI, 1979. (8) Slin’ko, M. G.;Slin’ko, M. M. Caral. Rev.-Sci. Eng. 1978, 17, 119. (9) Hirsch, M. W.; Smale, S. “Differential Equations, Dynamical Systems, and Linear Algebra”; Academic Press: New York, 1974. (10) Edelson, D.; Noyes, R. M.; Field, R. I. Int. J. Chem. Kiner. 1979, 1 1 , 155. (11) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. (12) Wolfe, R. J. Arch. Ration. Mech. Anal. 1978, 67, 225. (1 3) OMalley, Robert E. “Introduction to Singular Perturbations”;Academic Press: New York, 1974. (14) (a) Tyson, J. J. J. Phys. Chem. 1982, 86, 3006. (b) Tyson, J. J. In “Nonlinear Phenomena in Chemical Dynamics”; Vidal, C., Pacault, A,, Eds.; Springer-Verlag: New York, 1981; p 222. (15) Hastings, S. P.; Murray, J. SIAM J . Appl. Math. 1975, 28, 678. (16) Stanshine, J. A.; Howard, L. N . Stud. Appl. Math. 1976, 55, 129.
equations that the three-variable set can be reduced to a twovariable set of a much more tractable form than a two-variable set derived by Field and Noyes.” We report here analytical solutions to Tyson’s two-variable set of equations. These have been obtained by a singular perturbation methodI3 and are of sufficiently simple form that they have been of significant use to us in the analysis of electrically coupled BZ oscillator^.^^ Other people may find them useful both for mathematical analysis work and for decreasing the computation time required for numerical work.
Solutions Tyson14expanded and improved the Oregonator by breaking reaction E into reactions F and G. He also recognized explicitly 6Ce(IV)
-+ + +
+ CH2(COOH)2+ 2 H 2 0
kp
+ HCOOH kio 4Ce(IV) + BrCH(COOH)2 + 2 H 2 0 4Ce(III) + HCOOH Br6Ce(III)
2 C 0 2 + 6H+ (F) 2C0,
+ 5H+ (G)
the conservation of metal-ion catalyst by defining C = [Ce(III)] + [Ce(IV)] and using [Ce(III)] = C - [Ce(IV)] in the rate terms arising from reaction C in the original Oregonator. The differential equations resulting from this modified Oregonator are given in Tyson’s scaling by eq 1. p,(dx/dt)
=
X(
1 - X) - Y ( X - 4 )
(1.1)
P,(dY/dt) = - A x + 4 ) + bgz
(1.2)
& / d t = 2~ - bz
(1.3)
In eq 1
=X
[HBrO,]
= ((k5,HA)/(2k4)lx = Xox
y = {(k5aA)/(k2)b =
[Br-l
[Ce(IV)] [Br03-]
A
[Ce(III)]
time (s)
ug
Z = Cz
+ [Ce(IV)] = C
[H+] 5 H
T = {(2k4c)/(k5,HA)2) t = Tor
To z 800 s g = k10/(6k9 + 4k10) 4 = (2k3k4)/(k2ksa) N 4.0
x
+ 4kl0)(2k40/(k5,HA)~Z p x = x o / c= 1.2 x 10-4
b = (6k9
10
pv = Y , / C 2 5 x 10-6
The parameters b and g are related to the f and k5, of the Oregonator” (process C of the FKN mechanism2) and are treated as expendable parameters. The subscripts of the rate constants are according to the conventions of FKN” and the values for them used are those of Tyson.14 This system can be reduced even further by considering it to be a singular perturbationt3of a smaller set of equations with pv as the perturbation parameter. The argument is essentially that, because pv < p x O below the SM and is > q. The solution for IIa is 7
- (t*I/€)
= In ( b o - (Y)/(XIIO - a)) + [(q + a)(xo - x110)l/~(x0- a)(xIIo - .)I
4.0
‘-5.0
-1.0
-2.0
-3.0
1.0
0.0
LOCIX)
Figure 3. Superposition of the numerical (dashed line) and the asymptotic (solid line) phase-plane solutions of eq 2 for g = 0.35.
(IW)
where xlIois the beginning of region IIa, a = (To - q)/2, and t*I is the value of t at the end of region I. The boundary between regions IIa and IIb is arbitrarily chosen at x = 0.01. The solution for IIb is x0(7) = [0.5 - a
C=
+ Ce20r(a+ O.S)]/(Cezor+ 1)
[ ( a - 0.5
(II(ii))
+ O.Ol)/(a + 0.5 - 0.01)]e-20r*~~~
where a = 0.25 - loand T * is ~the ~value ~ of T a t the end of IIa. This trajectory continues through IIb until it is within a small interval of the SM at which p i n t it jumps onto it and into region 111. This jump eliminates creating another region or going to higher order solutions.
Region 111 In this region, x E [1.0,0.5], and q is negligible compared to x. The motion is slow as in region I and the overhead dot signifies d/dt. The differential equations are tx = x(l - x) - { (10.1)
f
= 2bgx
- b{
+ 72) + ((72 - 2g)/(g- 1/21] In (2g(lo + 1 / 2 ) / ( E O + 2g - 1/21)]
(EO
g - l*IIb = -(1/b)[2 In
((0
+ 72 +
g=
+ (O.5
-h)/(1/2
+ 50)1
+ 72
(III(iii))
k
- loZ
(HI(iv) )
=
1/4
(=ET)
I
1
1.6
2.0
identical with those of region I1 except that f takes on the value + 6 where 6 is some small number to move the trajectory off SM. The solutions in this region are region IVbl,
at the end of region IIb and
Region IV When {gets larger than 1/4, the trajectory moves off the SM and the time scale is defined again as T = t / t and the overhead dot signifies d / d s = t(d/dt). The differential equations are
xE xo =
xo =
x
region IVa, 7
fiva(x0)
0 4
[1/2, 0.471
- .rO)(.
-
t*111/c)
+ 1/2
(IV(i))
x E [OM, 3.4 x 10-31
region IVb2,
(III(ii))
Y2
112
TIME ISEC./BOOl Figure 4. Superposition of the time evolution of the variable x for the numerical (dashed line) and asymptotic (solid line) solutions of eq 2 for g = 0.35.
(IWi))
xo = l o
where, t*Ilb is the value of t to 0.
logoes from
1/21
ole
olu
(10.2)
The zereorder solution takes on two forms depending on the value of g. These are
- f*lIb = -(1/b)[2 In
’
1/2
- 0.03/(0.03(~- T * I v ~+~ )1)
E L3.4 x
q (1
(IV(ii))
+ 2q/(o)]
- T*IVb2 =frva(xo) -frva(3.4 X
(IV(iii))
= (1/2) In (xO2 - {oxo + loq) + - xo)/(r - ff + xo)) (IV(iV))
((lo + 2d/(4Y) In (7+ CY
= f0/2 = 0.1265 for
lo= 0.253
y = [{0(0(56/4 - q)]’/’ = 0.1261
This brings the trajectory back to the beginning of region I.
J. Phys. Chem. 1984, 88, 766-769
766
Parametric equations have thus been obtained for the time behavior of this reduced model. Figure 3 shows an overlay in the x,< plane of the asymptotic solutions over those calculated directly from eq 2 by numerical integration.
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Period of Oscillations The period of oscillations is the sum of the time spent in all parts of the limit cycle. However, the time spent in regions I1 and IV is negligible and does not need to be taken into account. The period is, then, the sum of the time spent in regions I and 111. To put the formulas for period into chemically more familiar terms, we change variables back to those of the Oregonator of ref 11. The parameters g, b, q, and T used here are related to f, A , and the rate constants of ref 11 by g = f/2, b = (2kM5kM4C)/(kM3A)2, = (2kMlkM4)/(kM2kM3), and TO = (2kM4o/(kM3A)'. The formulas for period then become AT = AT(1) + AT(II1)
+
ATcf#1) = -(l/kMS)[hl [ 8 ( k ~ 4 k ~ 1 ) ( 3 2(2'/') n/(kM&h13)1 - (1/(f- 1)) In (0.5 (2f/(2f - 1))"-')1 (11.1) ATcf=l) = -(l/kM5)[ln (16(1 + 2'")kMlk~4/(kM2kM3)) - (2 In 2 - 1)1 (11.2) These equations give the period of oscillations to within 3% of that calculated by numerical integration of eq 2 for g < cf
< l), and they are identical with those reported by Tyson in ref 19, Table 2. Two interesting features of eq 11 are the appearance of the multiplying term, l/kM5,and the absence of the dependence of the period on A , the bromate ion concentration. The l/kM5 term is confirmed by numerical integration of the three-variable Oregonator equations. However, the periods calculated from the three-variable Oregonator do have a 1/A dependence which has apparently been lost in the reduction to two variables. From the FKN mechanism and the formula for g by Tyson, g cannot assume a value larger than 0.25 for any real concentration of malonic acid or bromomalonic acid. In the oscillations of eq 2, the contribution of region I11 to the total period (the second term on the right-hand side of eq 11 exceeds 15% of the total period for g < 0.35 and is an important correction for g < 0.6. The first term on the right-hand side of eq 11, AT(I), is a good approximation to the period for larger values of g where the contribution of region I11 is as large as the error in eq 11. Figure 4 shows an overlay of the x vs. t solution calculated from eq 11 and by direct numerical integration of eq 2. Acknowledgment. We thank Professor John Barr of the University of Montana Computer Science Department for technical advice in setting up the LSI 11/23 computer system used for the calculations described here. (19) Tyson, J. J. Ann. N . Y. Acad. Sci. 1979, 316, 279.
Fourier Transform Infrared Study of the Gas-Phase Reaction of ''0, with trans-CHCI=CHCI in ''O,-Rich Mixtures. Branching Ratio for 0-Atom Production via Dissociation of the Primary Criegee Intermediate H. Niki,* P. D. Maker, C. M. Savage, L. P. Breitenbacb, Research, Ford Motor Company, Dearborn, Michigan 481 21
and R. I. Martinez Center for Chemical Physics, Chemical Kinetics Division, National Bureau of Standards, Washington, D.C. 20234 (Received: March 31, 1983; In Final Form: June 23, 1983)
Using the FTIR spectroscopicmethod, we identified 1603 among the products formed in the gas-phase reaction of 1 8 0 3 with trans-CHCl=CHCl in '602-richmixtures. The primary yield of 1 6 0 3 was determined to be 17 f 3% of the reactants consumed in the presence of a C1-atom scavenger such as C2H6 or n-C4HIo.This finding can be explained by the formation of atomic oxygen in the unimolecular dissociation of the Criegee intermediateH(Cl)C00, Le., lSO3+ truns-CHCI=CHCl- H(Cl)CI80+ H(CI)C=180 (1); H(Cl)C1801s0 1sO(3P)+ H(C1)C=180 (2'); I8O + 1602 l 6 0 + 160180 ( 5 ) ; and I6O+ I6O2(+M) 160(+M) 3 (3). Hence, we conclude that the branching ratio of 0-atom production via eq 2' is ca. 20% relative to all the decomposition channels available to the primary Criegee intermediate H(C1)COO.. For H2CO0. produced via the 03-C2H4 reaction, the corresponding branching ratio for 0-atom production appears to be zero. The implication of these findings is discussed with reference to the photooxidation of formaldehyde, 0 + oxoalkane reactions, the reactions of carbenes with 02,and the reaction of CI atoms with H(Cl)C=O.
-
-
-
Introduction Although there is now abundant evidene'v2 that the O3-a1kene reactions in the qas phase proceed via the formation of the Criegee intermediate >COO. as in the solution phase3s4 03
t
' /c=c,
/ -0"o
-' c,=o
0
!--A ,b-L
\t /c-0-0.
(1)
I I'
quantitative kinetic data on numerous thermochemically accessible reaction channels for the >coo.and its isomers are Still largely lacking. On possible fate of the >COO- is its unimolecular dissociation to atomic oxygen and a carbonyl product via reaction 2.1b Re-
-
>coo.
>C=O
+ o(~P)
(2)
action 2, if it occurs, is particularly interesting, since it provides the reaction path linking the O(3P) >C=O reaction system
+
(1) See, for example: (a) J. T.Herron, R. I. Martinez, and R. E. Huie, Int. J . Chem. Kine!., 14, 201 (1982); (b) ibid., 14, 225 (1982). (2) R. I. Martinez, R. E. Huie, and J. T.Herron, J . Am. Chem. Soc., 103, 3807 (1981).
0022-3654/84/2088-0766$01.50/0
(3) R. Criegee, Angew. Chem., Inr. Ed. Engl., 14, 745 (1975). (4) P. S. Bailey, "Ozonation in Organic Chemistry", Vol. 1, Academic Press, New York, 1978.
0 1984 American Chemical Society
