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COMMUN~CATIONS TO THE EDITOR
and the measuring light did not influence the reaction to any measurable degree. Since the oscillation was also found when Na202or distilled H202 was used, the inhibitor added to commercial hydrogen peroxide can be disregarded in connection with the mechanism of the oscillation. The very high reproducibility of the oscillation also rules out any critical influence of dust contamination. l1 The additional finding that the oscillation does not depend on stirring would appear to rule out heterogeneous reactions. If the reaction were heterogeneous, there would have to be concentration gradients of reactants a t the surface where the reaction P. E. BLATZ takes place. Such gradients would be strongly inDEPARTMENT OF CHEMISTRY UNIVERSITY OF WYOMING N. BAUMGARTNERfluenced by stirring, and the time course of the reaction LARAMI~, Mi YOMING 82070 S. DEWHURBTwould depend on stirring. It may also be noted that stirring does not affect the oscillatory kinetics of the RECEIVED APRIL19, 1968 glycolytic intermediates in heart muscle extract. Shaw and Pritchard’s theoretical comments are misleading, to say the least. I n his treatise, Bak12demonstrated that open or closed systems composed of only first-order reactions have real eigenvalues due to the The Existence of Homogeneous condition of microscopic reversibility ; such systems can exhibit only overshoot and undershoot phenomena in Oscillating Reactions the approach to the stationary or equilibrium state. Bak also demonstrated that systems composed of only Sir: The communication entitled, “Homogeneous pseudo-first-order reactions could have complex eigenPeriodic Reactions,” by Shaw and Pritchard’ contains values leading to a damped oscillation in the approach a number of errors which we feel need correction. It is to the stationary state. For general types of reaction stated in the communication that two reactions which schemes ( i e . , involving second-order reactions), Prigooscillate in homogeneous phase are known. It apgine13 has demonstrated that the equilibrium state for pears that the oscillating reaction of ceric ions, bromate, a closed system has real eigenvalues and consequently and malonic acid2-4 and the oscillating reaction in the oscillations cannot occur in the neighborhood of the glycolytic pathways in cell free extracts of yeast6 or equilibrium state. However, this result cannot be heart muscle6 have escaped the attention of Shaw and extrapolated to the behavior of the system in the large, Pritchard. These authors also give a secondary referthat is when the initial concentrations are far from the ence to the explanation of the oscillations in cool flame equilibrium concentrations. It is only within the past combustion of hydrocarbons which was originally proyear that a satisfactory proof has been given that inposed by Day and Pease.’ In their restatement of this definiteIy sustained oscillations cannot occur in a closed explanation they miss the important point that the reacsystem. 14r16 Nevertheless, both experiment and theory tion is a branched-chain reaction, i.e., an autocatalytic show clearly that closed homogeneous systems may reaction. The negative temperature coefficient alone cannot lead to oscillations. It should also be mentioned that analogous oscillations are found in the gas-phase (1) D. H. Shaw and H. 0. Pritchard, J . Phys. Chem., 7 2 , 1403 (1968). oxidation of hydrogen sulfide.s (2) B. P. Belousov, Sb. ref. radiats. med. za 1968, Medgiz, Moscow, 1959. Shaw and Pritchard also report some experiments on (3) A. ,M.Zhabotinskij, Biofizika, 9, 306 (1964). the oscillating reaction of hydrogen peroxide and iodate. (4) H. Degn, Nature, 213, 689 (1967). We assume that they have not used periodate as they (6) K. Pye and B. Chance, h o c . NatZ. Acad. Sci. U.S.,55,888(1966). write; a t least Brayg did not. They claim in italics (6) R.Frenkel, Arch. Biochem. Biophys., 115, 112 (1966). that the exclusion of daylight suppressed the oscilla(7) R.A. Day and R. N. Pease, J. Amer. Chem. Soc., 62, 2234 (1940). tion. It is possible that such an effect can be observed (8) H. W. Thompson, Nature, 127,629 (1931). under their conditions. However, that the oscillations (9) W. C. Bray, J. Amer. Chem. Soc., 43, 1262 (1921). can readily occur in the dark has been demonstrated by (10) H. Degn, Acta Chen. Scund., 21, 1057 (1967). (11) F. 0. Rice and M. Reiff, J . Phys. Chem., 31, 1362 (1927). one of us,l0 who studied this oscillating reaction spec(12) T. A. Bak, “Contributions to the Theory of Chemical Kinetics,” trophotometrically, the only light present being the W. A, Benjamin, Inc., New York, N. Y., 1963. weak measuring beam. Though not reported, it was (13) I. Prigogine, “Thermodynamics of Irreversible Processes," found that turning off the measuring light for a period John Wiley and Sons, Inc., New York, N. Y., 1966. of time did not produce any phase shift of the oscilla(14) D. Shear, J . Theoret. Biol., 16,212 (1967). (16) J. Higgins, ibid., in press. tion. Apparently, the oscillation went on in darkness conjugated double bond sequence of the parent polyene. For a carotene with 11 double bonds, the A, of the complex is predicted to be 1100 mp. For anhydroof the complex retinol with six double bonds, the A,, is calculated to be 600 mp. Consequently, the experimentally determined A,, value of the iodine complex of anhydroretinol is in good agreement with a similar species obtained from protonation of anhydroretinol and also a calculated value from FEMO considerations. This evidence supports the suggestion of Ebrey that the new band might be explained in terms of a shift in absorption of the polyene band.
The Journal of Physical Chemistry
COMMUN~CATIONS TO THE EDITOR
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exhibit prolonged series of oscillations, before approaching the final equilibrium state. This is due to the fact that a complex system will typically pass through a sequence of nearly stationary states in the approach to the equilibrium state. l6 As recognized by Prigogine,ls indefinitely sustained oscillations can occur in open systems. For example, if hydrogen peroxide were supplied at a suitable constant rate, the oscillation would persist indefinitely in the hydrogen peroxide-iodate system. As long ago as 1920, Lotka” proposed a reaction mechanism based on the law of mass action which could exhibit oscillations. Recent theoretical developments have dealt with the general requirements for oscillatory behavior in a chemical reaction scheme. 18-20 The simplicity of these requirements has led to the development of a wide variety of reaction schemes, based on known types of reactions, which can give rise to oscillatory kinetics without introducing any heterogeneous effects. I n computer studies, qualitative reproduction of experimental results for the glycolytic system18 and the hydrogen peroxide-iodate2’ system have been obtained. While the mechanism of the hydrogen peroxideiodate reaction may be complex, we believe that the heterogeneous and photochemical effects are so subtle that they can be ignored with regard to the mechanism of the oscillation. I n our view, that reaction and the others mentioned are excellent examples of homogeneous reactions which can exhibit prolonged oscillations in closed homogeneous systems. (16) J. Higgins in “Control of Energy Metabolism,” B. Chance, R. W. Estabrook and J. R. Williamson, Ed., Academic Press, New York, N. Y., 1966. (17) A. J. Lotka, J. Amer. Chem. Soc., 42, 1696 (1920). (18) J. Higgins, Ind. Eng, Chem., 59, 18 (1967), (19) E. E. Sel’kov in “Symposium on Oscillating Processes in Biological and Chemical Systems,” Nauka, Moscow, 1967. (20) R.A. Spangler and F. M. Snell, J. Theoret. Biol., 16,381 (1967). (21) P. Lindblad and H. Degn, Actu Chem. Scand., 21, 791 (1967).
JOHNSON FOUNDATION UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PENNSYLVANIA 19104
H. DECN J,
HIWINS
RECEIVED APRIL29, 1968
Homogeneous Periodic Reactions
Sir: Degn and Higginsl are indeed correct that we2 accidentally ascribed periodic behavior to the reaction between periodate and peroxide; in fact, we used iodate as described by Bray. We too studied the reaction in a spectrophotometer, but in our instrument the cell space was too small for effective mechanical stirring.
Instead, we used streams of 02 or NZand found that a t low flow rates, the oscillatory behavior persisted; however, using rapid flows, we appeared to suppress the oscillations. We concluded that there was probably a periodic buildup in opacity due to bubble formation followed by sudden effervescence, masking any genuine variations in iodine concentration. Unfortunately, the work of Degns escaped our attention, since it was published after we had completed these experiments. illany theoretical reaction schemes4 have been put forward in the past to account for periodic behavior. It is well known that some of these do not conserve particles (e.g., scheme I of ref 4 where the over-all reacC out of nothing); others tion is the creation of B conserve particles, but include creation (A -t A A) or annihilation (A B -+ B) reactions. We have difficulty in envisaging how such processes can be microscopically reversible when they take place at the molecular level in a homogeneous phase and in a closed system (but accept that they may have symbolic relevance to heterogeneous and/or open systems). From our own work6 on the Master Equation, we have learned that unless meticulous attention is paid to conservation of particles and of energy and to microscopic reversibility, spurious oscillatory or divergent behavior usually results in either the concentrations, or the total entropy, or both. Since it is always possible to symmetrize the Master Equation for a closed system, even when nonlinearities are present,6 the eigenvalues are necessarily real throughout the process; these eigenvalues will be time dependent, but this effect cannot lead to sustained oscillation once any initial transients have died out. Finally, it is clear that the evolution of a gas (either 0 2 in the iodate-peroxide reaction, or C02 in the cericbromate-malonate systems) cannot in any case be r e garded as homogeneous when considered from the point of view of microscopic reversibility-the reverse process must involve at least one heterogeneous step ; basically, if one wants to regard the system as homogeneous, it must also be considered open-otherwise one has to regard it as a closed system ( i e . , liquid plus vapor), in which case it is heterogeneous.
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(1) H. Degn and J. Higgins, J. Phya. Chem., 72,2692 (1968). (2) D. H. Shaw and H. 0. Pritchard, ibid., 72, 1403 (1968). (3) H.Degn, Actu Chem. Scund., 21, 1067 (1967). (4) P. Linbald and H. Degn, ibid., 21, 791 (1967). (6) V. A. LoDato and H. 0.Pritchard, “The Master Equation for the Dissociation of a Dilute Diatomic Gas,” in preparation, (6) H. Degn, Nature, 213,689 (1967).
C.R.E.S.S.
D. H. SHAW H. 0. PRITCHARD
YORK UNIVERSITY
TORONTO, CANADA RECEIVED MAY27, 1968 Volume 76, NumbeT 7 July 1068
