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equilibrium approximation, which in some cases is the same as the steady-state approximation. For example, the two are the same in the illustrative example given in ref 1. Libby has shown that the use of this approximation requires some care due to the appearance of indeterminate quantities in the retained rate equations. The neglect of these quantities in no way hinders the numerical calculation, which may produce answers that appear reasonable. Nevertheless, these answers are generally incorrect.
The integration of these equations by standard methods such as Runge-Kutta fails because of the following numerical difficulty. In the square brackets at the right side of eq 2, there are two large terms subtracted from each other. Near chemical equilibrium, these two terms approach each other. This numerical difficulty can easily be overcome by introducing the following set of new variables
(3)
AEROPHYSICS DEPARTMENT GEORQEEYANUEL Equation 3 combined with eq 2 results in AEROSPACE CORPORATION Los ANGELES, CALIFORNIA90045 RECEIVED NOVEMBER 4. 1966
Computer Program for Chemical Kinetics
Sir: Snow’ reported a computer program capable of handling the product distribution of any homogeneous reaction mechanism. This program involves a direct integration of the pertinent equation (2) below as well as an integration based on the assumption of steady state and a switch-over-mechanism going from one type of integration to the other. Besides the validity of the steady-state assumption being questionable for many reactions, the complexity incorporated seems to make the reported methodl inferior to other methods2J which are based on an alteration of the direct integration. In the following, the basic ideas of the latter methods are described. The pert,inent reaction scheme can be represented by a set of elementary reactions of the type aljN1
+
+ . . . + ai5Ni bljNl + + + 615Ni 7
(1)
where ail is the stoichiometric coefficient of the ith component being a reactant for the j t h reaction; bi5 is the stoichiometric coefficient of the ith component being a product for the j t h reaction; k5 is the forward rate constant of the j t h reaction; kl’ is the backward rate constant of the j t h reaction; and N f is the name of ith component. The rate of production of the ith component is given by
where Ct is the concentration of ith component, and t is the reaction time. The J O U Tof~Physical ChmiatTy
REINERKOLLRACK PRATT AND WHITNEY AIRCRAFT DIVISIONOF UNITEDAIRCRAFT EASTHARTFORD, CONNECTICUT RECEIVED DECEMBER 1, 1966
Chemical Kinetics Computer Program for
kj’
bzjN2
(1) R. H. Snow, J . Phys. Chem., 70, 2780 (1966). (2) T. F. Zupnik, E. N. Nilson, and V. J. Sarli, NASA-Report CR-54042 (1964). (3) J. P. Gurney, A I A A J., 3 , 3, 538 (1965).
Reply to Comments on the Paper, “A
ki
A
a2jN2
Equation 4 does not cause any numerical difficulty in respect to its integration by such standard methods as Runge-Kutta. Methods of the type presented do not need any further assumptions like the sometimes questionable assumption of steady state. Due to their simplicity, these methods appear to require much less computer time than the method by Snow.
Homogeneous and Free-Radical Systems of Reactions”
Sir: The main question raised by Emanuel and also by Kollrack is whether it is really necessary to use the special technique given in my paper for avoiding numerical difficulties in solving the differential equations of chemical kinetics. Certainly, for some reactions under limited conditions a satisfactory solution can be obtained by standard numerical methods. Emanuel’s ref 2-6 give further examples of this. However, the methods which they give do not solve the peculiar difficulties encountered in chemical kinetics, and so they are not truly general. Some reaction systems with free-radical chains may
COMMUNICATIONS TO THE EDITOR
be solved by standard integration methods, if the radical concentrations are not too low, as Duff did for the HrOz system (quoted in Emanuel’s ref 5 ) . Nevertheless, this calculation was said to require 5000 iterations. Under conditions such that the free-radical concentrations are lower, the number of iterations will increase rapidly. Emanuel’s ref 6-9 concern mathematical techniques for speeding numerical solution of differential equations. Techniques such as forward difference schemes, instability analysis, etc., can speed such calculations by a factor of perhaps 10 or 100, while there are practical problems in chemical kinetics where the calculation speed of standard methods is too slow by a factor of 106, even with the fastest computers. In these cases, the steady-state hypothesis is the answer. My main contribution was to demonstrate a method of applying it in general for any reacting system where it is needed, and to analyze the situations where it is useful. Emanuel’s remark that my paper gives only the constant-temperature case is irrelevant, since methods of generalizing are known. Emanuel’s final comments concern the validity of the steady-state hypothesis. His ref 10 by Libby concerns possible errors in assuming “a partial equilibrium or steady-state treatment of those reactions with specific rate constants considerably higher than the others involved in the series.” The comment by Kollrack gives essentially the same information as Libby’s publication. Libby essentially says that conservation of matter is satisfied as long as one calculates the extent of each reaction in a mechanism according to the standard rate equations. However, if one abandons some of the rate equations and calculates some components by means of equilibrium expressions, then changes are made in the amounts of certain components without regard to the stoichiometry of the reactions that produce them. He :hen formulates equations to take into account both equilibria and stoichiometry. This argument is answered in my paper: the steadystate assumption is valid only when the components that are to be determined from other-than-stoichiometric equations (the free radicals) are present in such small quantities that neglect of the principle of conservation of matter applied to them will introduce negligible error in the amounts of other components present. Libby’s warning is not important under the conditions for which the steady-state assumption is valid. Incidentally, the standard rate expressions can lead to error in numerical integration. An example is the pyrolysis of ethane, where the forward and reverse chain-propagating steps approach equilibrium, and the
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difference is beyond the number of significant digits carried by the computer. Libby’s method should be of use in this problem. The work of Giddings and Shin on the hydrogenbromine system’ should have been mentioned in my paper. (1) J. C. Giddings and H. K. Shin, J . Phys. Chem., 65, 1164 (1961); J . Chem. Phys., 36, 640 (1962);TTana. Faraday Soc., 57, 468 (1961); J . Chem. Phys., 39, 2937 (1963).
IIT RESEARCH INSTITUTE TECHNOLOQY CENTER ILLINOIS 60616 CHICAGO,
RICHARD H. SNOW
RECEIVED JANUARY 16, 1967
On The Heat Precipitation of Poly-L-proline’
Sir: In a recent paper,2 a study of the change in certain physical-chemical properties that occur during the well-known heat precipitation3v4of a dilute aqueous solution of poly-L-proline (form 11)3,6,6 was reported. The thermodynamic, dimensional, and frictional properties prior to precipitation were interpreted in terms of conventional polymer solution theory appropriate to chains in random-coiled, nonordered conformations.’ The experimental evidences which favors a rigid rodlike structure in dilute solution was ignored. It was concluded in this work2 that the precipitation was a consequence of a liquid-crystal transition rather than a liquid-liquid phase separation. These are the two obvious possibilities to choose from when observing precipitation phenomena from a dilute solution of random-coiled polymer? However, the liquid-crystal transition involving randomly coiled polymers is (1) This work was supported by a contract with the Division of Biology and Medicine, U. 8. Atomic Energy Commission. (2) A. Ciferri and T. A. Orofino, J. Phys. Chem., 70, 3277 (1966). (3) J. Kurtz, A. Berger, and E. Katchalski in “Recent Advances in Gelatin and Glue Research,” G. Stainsby, Ed., Pergamon Press, New York, N. Y., 1958,p 31. (4) W. F. Harrington and M. Sela, Bwchem. Bwphya. Acta., 27, 24 (1958). (5) E. R. Blout and G. D. Fasman, “Recent Advances in Gelatin and Glue Research,” G. Stainsby, Ed., Pergamon Press, New York, N. Y., 1958,p 122. (6) A. Yaron and A. Berger, Bull. Res. Couneil Israel, A10, 46 (1961). (7) P. J. Flory, “Principles of Polymer Chemistry,” Cornel1 University Press, Ithaca, N. Y.,1953. (8) I. Z. Steinberg, W. F. Harrington, A. Berger, M. Sela. and E. Katchalski, J . A m . Chem. Soc., 82,5263 (1960).
Volume 71, Number 4
March 1067
