Chapter 3 Chemical Oscillations

Chapter 3 Chemical OscillationsChapter 3 Chemical Oscillations• Definition:Definition: Resetting of clock reactionsResetting of clock reactions• 3.1 BZ oscillator3.1 BZ oscillatorreactants and catalysisreactants and catalysisBromate malonic acid metal ion or ligand compoundsBromate malonic acid metal ion or ligand compounds• Batch phenomena analysis:Batch phenomena analysis: Br- electrode : Br- Pt electrode: mostly metal ionBr- electrode : Br- Pt electrode: mostly metal ion• Feature: Relaxation oscillations Feature: Relaxation oscillations AB Br- consume slowly slow MetalAB Br- consume slowly slow Metaloxox increase increaseBC: bromide sharp drop quick Metal(ox at state) riseBC: bromide sharp drop quick Metal(ox at state) riseCD: Bromide rise slowly , Mred rise slowlyCD: Bromide rise slowly , Mred rise slowlyDA: Bromide rise suddenly Mred rise rapidly DA: Bromide rise suddenly Mred rise rapidly

• Bacth: inductionBacth: induction→→oscillations oscillations →→ steady state steady state →→equilibriumequilibrium• Bromate and malinic acid decrease stepwiseBromate and malinic acid decrease stepwise

• 3.2 Mechanism for BZ reaction3.2 Mechanism for BZ reaction A Field- Koros -Noyes mechanism FKN(mechanism)A Field- Koros -Noyes mechanism FKN(mechanism) J. Am. Chem. Soc. 1972 94, 8649-64J. Am. Chem. Soc. 1972 94, 8649-64 B B Chemical Oscillatlons, Chaos, and Fluctuations in Flow ReactChemical Oscillatlons, Chaos, and Fluctuations in Flow React

ors(Classification of phenomena)ors(Classification of phenomena) F. W. Schneider' and A. F. MiinsterF. W. Schneider' and A. F. Miinster Institute of Physical Chemistry, University of Wiirzburg, MarcInstitute of Physical Chemistry, University of Wiirzburg, Marc

usstrasse 911 1, usstrasse 911 1, 0-8700 0-8700 Wiirzburg, FRGWiirzburg, FRG J. Phys. Chem. 1991, 95, J. Phys. Chem. 1991, 95, 2130-21382130-2138C Deterministic chaos in the Belousov-Zhabotinskii reaction: exC Deterministic chaos in the Belousov-Zhabotinskii reaction: ex

periments and simulations periments and simulations (mechanism for complex oscillations) (mechanism for complex oscillations) Zhang, Dongmei; Gyorgyi, Laszlo; Peltier, William R. Department Zhang, Dongmei; Gyorgyi, Laszlo; Peltier, William R. Department

of Physics, University of Toronto, Toronto, ON, Can. of Physics, University of Toronto, Toronto, ON, Can. Chaos (1993), 3(4), 723-45Chaos (1993), 3(4), 723-45. .

• Process A Bromide consumptionProcess A Bromide consumptionFKN3 BrOFKN3 BrO33

--+Br+Br--+2H+2H++→ → HBrOHBrO22+HBrO Rate=k3[BrO+HBrO Rate=k3[BrO33--][Br][Br--][H][H++]]22

FKN2 HBrOFKN2 HBrO22+Br+Br- - 2HOBr Rate=k 2HOBr Rate=k22[HBrO[HBrO22][Br][Br--][H][H++]]Process A sum reaction Process A sum reaction BrOBrO33-+2Br-+3H+->3HBrO-+2Br-+3H+->3HBrO

• Process B HBrOProcess B HBrO22 autocatalysis autocatalysis(FKN5) BrO(FKN5) BrO33

--+ HBrO+ HBrO22+H+H++ <->2BrO2·+2H <->2BrO2·+2H++

• Rate=k5[BrORate=k5[BrO33--][HBrO][HBrO22][H][H++]-k]-k-5-5[BrO2·][BrO2·]

(FKN6) BrO(FKN6) BrO22· +M· +Mredred+H+H++ →→ HBrO HBrO22+M+Moxox • Rate=k6[Rate=k6[BrOBrO22· · ][][MMredred][H][H++]]• Process B sum reactionProcess B sum reaction• BrOBrO33

-- +HBrO +HBrO22+2M+2Mredred+3H+3H++ →→2HBrO2HBrO22+2M+2Moxox+H+H22OOProcess B is limited by the self-disproportionation reaction of HBrO2Process B is limited by the self-disproportionation reaction of HBrO2• FKN4 2HBrOFKN4 2HBrO2 2 →→ BrO BrO33

-- +HOBr+H +HOBr+H++ Rate=k4[HBrO Rate=k4[HBrO22]]22

Switch from A to B when consumption of HBrO2 by Bromide to HBrO2 autocatalysisSwitch from A to B when consumption of HBrO2 by Bromide to HBrO2 autocatalysis Rate 2=Rate 5Rate 2=Rate 5 k2[HBrO2][Br-][H+]= k5[BrO3-][ HBrO2][H+]k2[HBrO2][Br-][H+]= k5[BrO3-][ HBrO2][H+] [Br-]cr=(k5/k2)[BrO3-]≈[Br-]cr=(k5/k2)[BrO3-]≈1.4×10-5[BrO3-]1.4×10-5[BrO3-] Bromide ion as inhibitor consume autocatalyst HBrO2Bromide ion as inhibitor consume autocatalyst HBrO2

• Process C Br- reproduction The catalyst return low oxidation statProcess C Br- reproduction The catalyst return low oxidation state resetting the clocke resetting the clock

1Br-→HBrO2+HOBR→2HBrO2+BrMA+2Mox→HOBr+ BrMA+2Mox→2BrMA+2Mox1Br-→HBrO2+HOBR→2HBrO2+BrMA+2Mox→HOBr+ BrMA+2Mox→2BrMA+2Mox HOBr+MA→BrMA+H2OHOBr+MA→BrMA+H2O 2M2Moxox+MA+BrMA→fBr-+2M+MA+BrMA→fBr-+2Mredred+other products Rate=kc[Org][Mox]+other products Rate=kc[Org][Mox]• Special points: 1 Special points: 1 EEMn+1 / MnMn+1 / Mn=0.9=0.9 ～～ 1.5 Mn3+, Ferroin, Ce4+, Ru(bipy)1.5 Mn3+, Ferroin, Ce4+, Ru(bipy)3 3

2+2+

• 2. MA may be replace by other organic compound 2. MA may be replace by other organic compound • which may be brominated by HOBrwhich may be brominated by HOBr• 3. f is a adjusted parameter f>2/3, [Br-] increases after the cycle 3. f is a adjusted parameter f>2/3, [Br-] increases after the cycle

3.3 3.3 From mechanism to oregonator modFrom mechanism to oregonator modelel • A oregonator A oregonator • J Chem Phys. 1974, 60, 1877-84 J Chem Phys. 1974, 60, 1877-84 • O3 A+Y=X+P V3=k3AYO3 A+Y=X+P V3=k3AY• O2 X+Y=2P V2=K2XYO2 X+Y=2P V2=K2XY• O5 A+X=2X+2Z V5=K5AXO5 A+X=2X+2Z V5=K5AX• O4 2X=A+P V4=K4X2O4 2X=A+P V4=K4X2• OC Z+B=0.5fY VC=KcBZOC Z+B=0.5fY VC=KcBZ• A: Bromate, Y: Bromide, P: HBrO hypobramate, X: subbromate HA: Bromate, Y: Bromide, P: HBrO hypobramate, X: subbromate H

BrOBrO22, Z: Mox, B: organic compound, Z: Mox, B: organic compound• DX/dt=v3-V2+V5-2V4DX/dt=v3-V2+V5-2V4• DY/dt=-V3-V2+0.5fVcDY/dt=-V3-V2+0.5fVc• DZ/dt=2V5-VcDZ/dt=2V5-Vc

3.4 BZ 3.4 BZ Complex oscillations and Complex oscillations and ChaosChaos

A Phenomena A Phenomena

In Batch transient complex oscillationsIn Batch transient complex oscillations

In CSTR Hudson Mixed-mode oscillations at High flowrateIn CSTR Hudson Mixed-mode oscillations at High flowrate

Swinney Period-doubling oscillations and Chaos and mixed-mSwinney Period-doubling oscillations and Chaos and mixed-m

ode oscillations at low flowrateode oscillations at low flowrate

Roux Roux Quansiperiodic Oscillations and Chaos Quansiperiodic Oscillations and Chaos 3325-33383325-3338

Hourai Hourai symmetric diagram around P1 mixed-mode oscsymmetric diagram around P1 mixed-mode oscillations at low flowrate JPC 1985,89,1760-1764 Lillations at low flowrate JPC 1985,89,1760-1764 LSS

SS→0→0n n →1→1n n → 1→ 12 2 → chaos → 1→ chaos → 11 1 →chaos → →chaos → 221 1 → 3→ 311→ 1→ 100

• B Mechnism and model B Mechnism and model L Gyorgyi and R J Field Two cycle couplingL Gyorgyi and R J Field Two cycle coupling• Oregonator HBrO2 autocatalysis and consumption and production of Oregonator HBrO2 autocatalysis and consumption and production of

Br- Br- • BrMA cycle production by bromination of MA consumed by oxidation BrMA cycle production by bromination of MA consumed by oxidation

of Catalylystof Catalylyst• 11 variable JPC 1991, 95, 3159-3165 and 6594-660211 variable JPC 1991, 95, 3159-3165 and 6594-6602

9 variables [MA] as constant deleting 9 variables [MA] as constant deleting Br· bromine radical by rate sensitivity analysisBr· bromine radical by rate sensitivity analysis

7 7 variable removing diffusion-controlled reactions (B1 B9), HOBr replaced by Brvariable removing diffusion-controlled reactions (B1 B9), HOBr replaced by BrMAMA

Brmine deletedBrmine deleted

4 variable model 4 variable model delete the Ce(III) by conservation rule delete the Ce(III) by conservation rule remove the MA radical remove the MA radical by QSSA Quasi Steady State Assumptionby QSSA Quasi Steady State Assumption remove the BrO2 radicalremove the BrO2 radical by QSSA or EQA Equilibrium Assumptionby QSSA or EQA Equilibrium Assumption

3 variable model 3 variable model remove the bromide by QSSAremove the bromide by QSSA

Modify the 4 variable model D4 according to mass action law delete D8 N Modify the 4 variable model D4 according to mass action law delete D8 N modelmodel

Nature 1992, 355, 808-810Nature 1992, 355, 808-810

• Questions:Questions: (1)When and how can we get PD (1)When and how can we get PD

Mixed-mode and QP dynamics by two Mixed-mode and QP dynamics by two cycle coupling ? cycle coupling ?

(2)What is Experimental whole (2)What is Experimental whole sequence of BZR in a CSTR? sequence of BZR in a CSTR?

…….3.311 PD 2 PD 211 PD 1 PD 10 0 PD 1 PD 111 PD 1 PD 122 PD 1 PD 133 PD……PD……

(3)Can we use the model to simulate (3)Can we use the model to simulate the spatiotemporal patterns in the the spatiotemporal patterns in the medium of complex and Chaos? medium of complex and Chaos?

3.5 3.5 Dimensionless equation of OregonatorDimensionless equation of Oregonator• Definition: to make the variable dimensionlessDefinition: to make the variable dimensionless• Why: ①variable reduction ② for relaxation oscillatiWhy: ①variable reduction ② for relaxation oscillati

ons and ③f oscillatory range ons and ③f oscillatory range • ④④slow and fast variableslow and fast variable• x=2k4X/k5A y=k2Y/k5A z=kck4BZ/(k5A)x=2k4X/k5A y=k2Y/k5A z=kck4BZ/(k5A)22 τ=k τ=kccBtBt• ε=kcB/k5A=1×10ε=kcB/k5A=1×10-2-2 • ε’=2kck4B/k2k5A=2.5×10ε’=2kck4B/k2k5A=2.5×10-5-5

• q=2k3k4/k2k5=9×10q=2k3k4/k2k5=9×10-5-5

• A=0.06M B=0.02MA=0.06M B=0.02M• X(1-X) quadratic autocatalysisX(1-X) quadratic autocatalysis

• C Steady-state approximation for bromide C Steady-state approximation for bromide ε’ is so small , dy/dt change quickly comparing to dx/dt and dz/dt, Relativε’ is so small , dy/dt change quickly comparing to dx/dt and dz/dt, Relativ

e to x, z, y change quickly to steady state, for whole dynamics, dy/dt≈0e to x, z, y change quickly to steady state, for whole dynamics, dy/dt≈0• y=yy=yssss=fz/(q+x)=fz/(q+x)εdx/dt=x(1-x)-(x-q)fz/(q+x)εdx/dt=x(1-x)-(x-q)fz/(q+x)• dz/dt=x-zdz/dt=x-z• ε<<1 x fast variable z slow variableε<<1 x fast variable z slow variableD. Simulation of kinetics of equation D. Simulation of kinetics of equation f=1/4 high x z stable statef=1/4 high x z stable statef=1 oscillationsf=1 oscillationsf=3 low x z stable statef=3 low x z stable state steady state steady state dx/dt=0 dz/dt=0dx/dt=0 dz/dt=0xss=zss=0.5{1-(f+q)+[(f+q-1)xss=zss=0.5{1-(f+q)+[(f+q-1)22+4q(1+f)]+4q(1+f)]0.50.5}}yss=fzss/(q+xss)yss=fzss/(q+xss)solid line stable statesolid line stable statedash line unstable state 1/2<f<1+2dash line unstable state 1/2<f<1+20.50.5

3.6 3.6 Pictorial explanation of dynamicsPictorial explanation of dynamics Feature: Feature: dx/dt=0 dz/dt=0 dx/dt=0 dz/dt=0 1 f=0.25 intersection at right stable point high x1 f=0.25 intersection at right stable point high x f=1 intersection at the middle branch of nullcline oscillationsf=1 intersection at the middle branch of nullcline oscillations f=3 intersection at left branch of nullcline stable state at low xf=3 intersection at left branch of nullcline stable state at low x2. dx/dt is much bigger than dz/dt x: quick variable z: slow variable2. dx/dt is much bigger than dz/dt x: quick variable z: slow variable3. nullcline dx/dt=0 dz/dt=0 above the lines dx/dt<0 dz/dt<03. nullcline dx/dt=0 dz/dt=0 above the lines dx/dt<0 dz/dt<0 belov the lines dx/dt>0 dz/dt>0belov the lines dx/dt>0 dz/dt>0

4. x nullcline S type Bistable middle line is nonstable4. x nullcline S type Bistable middle line is nonstable 5 State at x nullcline slow movement 5 State at x nullcline slow movement State at other position x quick movementState at other position x quick movement go to x nullcline then slow movement go to x nullcline then slow movement

ExplainExplain the relaxation oscillations the relaxation oscillations The intersection lies at the middle branch of x nullcline , from any point, horizontal The intersection lies at the middle branch of x nullcline , from any point, horizontal

movement and movement along the A ,B branch are quick and slow, respectively, movement and movement along the A ,B branch are quick and slow, respectively, never reach the intersection. The amplitude and period depend on the tempreturnever reach the intersection. The amplitude and period depend on the tempreture and concentrations, not the initial point.e and concentrations, not the initial point. . .

3.7Conditions for oscillations 3.7Conditions for oscillations intersection at middle branch, and at nullclines between minimum and maximum of x intersection at middle branch, and at nullclines between minimum and maximum of x dx/dt=0 εdx/dt=x(1-x)-(x-q)fz/(q+x) dx/dt=0 εdx/dt=x(1-x)-(x-q)fz/(q+x)

dz/dx=0dz/dx=0

minimum point : minimum point :

Maximum point: Maximum point:

• X=ZX=Z• Minimum f=1+1.414Minimum f=1+1.414• Maximum: f=0.5 Maximum: f=0.5 • Oscillations : 0.5< f<1+1.414Oscillations : 0.5< f<1+1.414• Precondition: ε<<1Precondition: ε<<1• When ε rise, the range of f can be decreased. When ε rise, the range of f can be decreased. • ε<<1 ε=kcB/k5A A=[BrO3-]0 B=[MA]0ε<<1 ε=kcB/k5A A=[BrO3-]0 B=[MA]0• [BrO3-]>>(kc/k5)[Org]=0.03[MA][BrO3-]>>(kc/k5)[Org]=0.03[MA]

3.8 3.8 Amplitude and period of oscillationsAmplitude and period of oscillations

Amplitude Amplitude EEBrBr amplitue∞log(y amplitue∞log(yAA/y/yCC) = 6.4 400mv) = 6.4 400mvEEptpt amplitude∞log(z amplitude∞log(zAA/z/zCC) = 2.7 180mv) = 2.7 180mvPeriod: AB slowest Period: AB slowest period≈move time of ABperiod≈move time of ABdz/dt=x-z x is almost constant dz/dt=x-z x is almost constant dz/dt=q-zdz/dt=q-zintergrating the equation from A to B f=1intergrating the equation from A to B f=1

3.9 Excitability3.9 Excitability

For stable steady state For stable steady state small perturbation system return quickly small perturbation system return quickly large perturbation system does not return back immediately. First a single large perturbation system does not return back immediately. First a single

excursion, develop a single oscillation this is excitabilityexcursion, develop a single oscillation this is excitabilityThere is a critical value or threshold of perturbation from small perturbatioThere is a critical value or threshold of perturbation from small perturbatio

n to excitabilityn to excitability

For BZ reactionFor BZ reactionSmall perturbation: from xss to X1 Small perturbation: from xss to X1 ---return xss straightforward ---return xss straightforward Larger perturbation: Larger perturbation: ---from xss to x2 single oscillation ---from xss to x2 single oscillation When system is moving along the x-nullcline , When system is moving along the x-nullcline , it is virtually insensitive tofurther perturbation. it is virtually insensitive tofurther perturbation.

This phenomena is call refractoryThis phenomena is call refractory. .

3. 10 Other oscillatory 3. 10 Other oscillatory systemssystems• A Liquid phase oscillationsA Liquid phase oscillations 1. Bray-Liebhafsky oscillations1. Bray-Liebhafsky oscillations• Iodate-catalysed disproportionation of hydrogen peroxide Iodate-catalysed disproportionation of hydrogen peroxide • 2H2O2=2H2O+O22H2O2=2H2O+O2• JACS 1931, 53,38JACS 1931, 53,38 2. Briggs-Rauscher oscillations 2. Briggs-Rauscher oscillations • BR + Malonic acidBR + Malonic acid• Gold to blue to colourlessGold to blue to colourless• J Chem. Edu 1973, 50,496J Chem. Edu 1973, 50,496• 3. CIMA oscillations3. CIMA oscillations• Chlorite-iodide-malonic acid oscillations Chlorite-iodide-malonic acid oscillations • B. Noszticzius Z., Ouyang Q., McCormick W.D., Swinney, H.L.:B. Noszticzius Z., Ouyang Q., McCormick W.D., Swinney, H.L.:

"Long-lived oscillations in the chlorite-iodide-malonic acid reaction in batch""Long-lived oscillations in the chlorite-iodide-malonic acid reaction in batch"J. Am. Chem. Soc. 114, 4290-4295 (1992)J. Am. Chem. Soc. 114, 4290-4295 (1992)

• Lengyel, I. & Epstein, I. R. [1990] "Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction syLengyel, I. & Epstein, I. R. [1990] "Modeling of Turing structures in the chlorite-iodide-malonic acid-starch reaction system," stem," ScienceScience 251251, 650-652. , 650-652. Lengyel, I., Rábai, I. & Epstein, I. R. [1990a] "Batch oscillation in the reaction of chlorine dioxide with iodine and maloLengyel, I., Rábai, I. & Epstein, I. R. [1990a] "Batch oscillation in the reaction of chlorine dioxide with iodine and malonic acid," nic acid," J. Am. Chem. Soc.J. Am. Chem. Soc. 112112, 4606-4607. , 4606-4607. Lengyel, I., Rábai, I. & Epstein, I. R. [1990b] "Experimental and modeling study of oscillations in the chlorine dioxide-iLengyel, I., Rábai, I. & Epstein, I. R. [1990b] "Experimental and modeling study of oscillations in the chlorine dioxide-iodine-malonic acid reaction," odine-malonic acid reaction," J. Am. Chem. Soc.J. Am. Chem. Soc. 112112, 9104-9110. , 9104-9110. Lengyel, I. & Epstein, I. R. [1992] "A chemical approach to designing Turing patterns in reaction-diffusion systems," Lengyel, I. & Epstein, I. R. [1992] "A chemical approach to designing Turing patterns in reaction-diffusion systems," PProc. Nat. Acad. Sci. (USA)roc. Nat. Acad. Sci. (USA) 8989, 3977-3979. , 3977-3979. Lengyel, I. & Epstein, I. R. [1995] "The chemistry behind the first experimental chemical example of Turing patterns," Lengyel, I. & Epstein, I. R. [1995] "The chemistry behind the first experimental chemical example of Turing patterns," in in Chemical Waves and PatternsChemical Waves and Patterns, eds. Kapral, R. & Showalter, K. (Kluwer, Dordrecht), pp. 297-322., eds. Kapral, R. & Showalter, K. (Kluwer, Dordrecht), pp. 297-322.

• Lengyel, I., Li, J., Kustin, K. & Epstein, I. R. [1996] "Rate constants for reactions between iodine and chlorine species: Lengyel, I., Li, J., Kustin, K. & Epstein, I. R. [1996] "Rate constants for reactions between iodine and chlorine species: A detailed mechanism of the chlorine dioxide-chlorite-iodide reaction," A detailed mechanism of the chlorine dioxide-chlorite-iodide reaction," J. Am. Chem. Soc.J. Am. Chem. Soc. 118118, 3708-3719, 3708-3719

• De Kepper, P., Epstein, I. R., Orbán, M. & Kustin, K. [1982] "Batch oscillations and spatial wave patterns in chlorite osDe Kepper, P., Epstein, I. R., Orbán, M. & Kustin, K. [1982] "Batch oscillations and spatial wave patterns in chlorite oscillating systems," cillating systems," J. Phys. Chem.J. Phys. Chem. 8686, 170-171. , 170-171. De Kepper, P., Boissonade, J. & Epstein, I. R. [1990] "Chlorite-iodide reaction: A versatile system for the study of nonlDe Kepper, P., Boissonade, J. & Epstein, I. R. [1990] "Chlorite-iodide reaction: A versatile system for the study of nonlinear dynamical behavior," inear dynamical behavior," J. Phys. Chem.J. Phys. Chem. 9494, 6525-6536. , 6525-6536. De Kepper, P., Perraud, J. J., Rudovics, B. & Dulos, E. [1994] "Experimental study of stationary Turing patterns and thDe Kepper, P., Perraud, J. J., Rudovics, B. & Dulos, E. [1994] "Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system," eir interaction with traveling waves in a chemical system," Int. J. Bifurcation and ChaosInt. J. Bifurcation and Chaos 44, 1215-1231, 1215-1231

B Gas phase oscillationsB Gas phase oscillations• CO oxidation at Pt single-crystal (110) most studied by ertl group in Max-plank insCO oxidation at Pt single-crystal (110) most studied by ertl group in Max-plank ins

titutetitute• Hydrocarbon oxidation : cool frames Hydrocarbon oxidation : cool frames • J. F. Griffiths Adv. Chem. Phys. 1986, 64, 203-303J. F. Griffiths Adv. Chem. Phys. 1986, 64, 203-303 C Gas evolution oscillationsC Gas evolution oscillations• Decomposition of formic acid by sulphuric acid the morgan reaction Decomposition of formic acid by sulphuric acid the morgan reaction • HCOOH=H2O+CO supersaturation to nucleation then supersaturation againHCOOH=H2O+CO supersaturation to nucleation then supersaturation again• The decomposition of aqueous ammonium nitrite The decomposition of aqueous ammonium nitrite • NH4NO2=N2+H2ONH4NO2=N2+H2O• D. Chemical oscillations by transport of reactants D. Chemical oscillations by transport of reactants • Benzaldehyde oxidation by oxygenBenzaldehyde oxidation by oxygen• Oxidation of p-xylene to terephthalic acid Oxidation of p-xylene to terephthalic acid E. Biochemical oscillationsE. Biochemical oscillations• Glycolysis oscillations ( to alcohol)( in vivi or vitro)Glycolysis oscillations ( to alcohol)( in vivi or vitro)• Michaelis-Menten kinetics Michaelis-Menten kinetics • Activation inhibition Activation inhibition • Allosteric effects Allosteric effects • PO oscillations nicotinamide adenine dinucleotide Hydride NADH oxidation by PO oscillations nicotinamide adenine dinucleotide Hydride NADH oxidation by

oxygen catalyzed by peroxidase oxygen catalyzed by peroxidase • Benno Hess, Benno Hess, Quarterly Reviews of Biophysics Quarterly Reviews of Biophysics 30, 2 (1997), pp. 121±176.30, 2 (1997), pp. 121±176.

3.9 Oscillations in closed system3.9 Oscillations in closed system• Feature of batch:Feature of batch:A.easily set up and operated A.easily set up and operated B. for every oscillations, small reactants are consB. for every oscillations, small reactants are cons

umed, So each oscillations occurs against diffeumed, So each oscillations occurs against different from background concentration of reactarent from background concentration of reactants. Each oscillations is slightly different from ints. Each oscillations is slightly different from its predecessor and subsequent excursion ts predecessor and subsequent excursion

C. when composition move out from range of osC. when composition move out from range of oscillations, then to steady state, to equilibriumcillations, then to steady state, to equilibrium