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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