Marc D. Porter and Gordon 6. Skinner Wright State University Dayton, Ohio 45431
I II
The Steady-State Approximation in Free-Radical Calculations A numerical example
When an overall chemical reaction consists of a series of elementary steps that involve free radicals, a common approach in making calculations concerning the progress of the reaction is to assume that steady-state concentrations of the free radicals exist throughout the duration of the reaction. It is usual to assume
the reverse reactions, (1) and ( 2 ) are assumed to he secondorder and (3) first-order. Rate constants, approximating the actual values a t 1600 K, were assnmed to be h i = 14 s-' hLl = 1.2 X 10LOmole-' Is-' hp = 1.5 X 109mole-' I SC'
1) The steady-state concentrations of the free radicals are very
mall compared to the concentrations of the reactants and products of the overall reaction. 2) The steady-state coneentrations are expressible, in principle st least, in terms of the concentrations of the reactants and products of the overall reaction. 3) The steady-state concentrations are reached in a time very short compared to the total reaction time.
Under certain conditions, other simplifying assumptions may be made. For example, reverse reactmns may he neglected if only the initial stages of the reaction are being considered. U'e have made three sets of calculations for the free radical decomposition of methane. We have assumed that the steps are
k - 2 = 2.9 X lo7 mole-' Is-'
mole An initial methane concentration, [CHdlo, of 1 X I-' was assumed. We have made three types of calculations to illustrate the possibilities Steady-state calculation of initial rates and concentrations, using only forward reactions. 2) Complete steady-state calculation using bath forward and re^ verse reactions. 3) Accurate numerical integration of both forward and reverse reactions, in which no steady-state assumptions are made. 1)
Methods of Calculation Initial Rates and Concentrations
If only the forward reactions are considered, the steadystate equations are where (4) is, of course, the overall reaction. While it is known that reactions (1)-(3) are not the only steps in the reaction, and that reaction (1) is only first-order a t very high pressures, we have assumed for our illustrative calcnlation that these are the three steps, that reaction (1) is first-order and reactions (2) and (3) are second-order. For
Simultaneous solution of these equations gives
The results of these calculations are shown graphically in the figures. In Figure 1, the steady-state values of [HI and [CH1] are the left-hand ends of the dashed curves which represent the complete steady-state values for these species, while the concentrations of H2 and C2Hs are given by the dashed curve over this limited time span. In Figure 2, the concentrations of Hp and CrH6 are shown as a dotted line. Full Steady-State Calculations
If both forward and reverse rates are considered, then Time, microseconds Figure 1. Product yields calculalad for the free-radical decomposition of methane near the beginning of the reaction. - - - Complete steady-state: -numerical integration. 366 / Journal of Chemical Education
Newton's method to solve ihe cubic equation. Results are shown in the figures. Numerical Integration
This straightforward numerical integration, in which concentrations of all substances but methane are assumed to he zero initially, and which contains no assumptions of steady-states or relative magnitudes of radicals versus molecules, resulted in accurate tables of species concentrations versus time against which to compare the results of the steady-state calculations. To carry out the integration, the total time of interest, 1.4 X 10-' s, is divided up into increments of 10V s. T h e changes in species concentrations occurring in each increment are obtained by multiplying the rates of each of the forward and reverse reactions (calculated from the species concentrations a t the beginning of the increment) by the length of the time increment. For example, a t the heginning of the reaction the decrease in [CHa] in the first time increment due to reaction (1) is
The concentrations of H and CH:I increase, of course, by this same amount. These changes are added (algebraically) to the species concentrations a t the beginning of the time increment, and then the process is repeated for succeeding increments. The calculation may be made as accurate as desired, within the limitations of the computer, by varying the size of the time increment. A computer program for this numerical integration is given in a recent textbook.' The results of the numerical integration are shown as solid lines in the figures. Time, milliseconds Figure 2. Product yields calculated f w the free-radical decomposition of methane at longer times.. Initial steady-state: ---complete steady-state; -numerical integration.
..
Simultaneous solution of these equations can be accomplished by first solving for [HI in terms of [CH4], [CzHfi]. [Hz], and [CHz], and using this relationship and the equations [HPI = [ C Z Hand ~ [CHi] = [CH& - ~ [ C ~ H R ]
(which are correct to the extent that the concentrations of
H and CHg are small compared to the molecules) in the [CHa]equation. The result, after some rearrangement, is
This equation may be solved for [CHz] for any desired value of [C2Hfi],and the rate of formation of product found from reaction (3) as
The time for a small change in [CsHs] to occur may he found by dividing the change by the rate, and therefore a table of concentration of each species versus time may he obtained. These calculations were done by computer, using
Discussion
The three methods of calculation agree much more closely in some details than in others. By coming to a low steady-state concentration in a few microseconds, and remaining within a few percent of that concentration while a significant fraction of the methane reacts, the H atoms follow the first and third assumptions fairly well. Methyl radical concentrations, on the other hand, take longer to reach the initial steady-state value since they have farther to go, and then continue to rise beyond the initial steady-state value, with no pause a t that level. This continued rise can he traced to reaction (3) (reverse), the dissociation of ethane, which becomes appreciable quite early in the reaction. From Figure 1 it can be seen that production of Hz is nearly identical by all three calculations a t the beginning of the reaction, while the actual rate of production of C2Hfi is far less than that predicted by the steady-state methods, due to the slow rise in concentration of CH:]. Over a longer time scale, as shown in Figure 2, the full steady-state calculation gives quite a good account of the course of the reaction, as does the initial steady-state calculation for ahout the first 20% of the reaction. Figure 2 does show that the amount of CH:*that builds up is sufficient to make a difference of 5-10% between the concentrations of HZ and C2H6 throughout most of the reaction. The steady-state method is described in many textbooks, and has its values, but students should be made aware of the assumptions implicit in its use, and also of the possibility of obtaining inaccurate results in some cases in which the assumptions are apparently fulfilled. As students hecome increasingly adept a t computer use, it becomes possible to recommend that free-radical processes may also be approached via the numerical integration route, which is often no more difficult to formulate than the steady-state one, and avoids dependence on its assumptions.
'
Skinner, G . B., "Introduction to Chemical Kinetics," Academic Press, New York, 1974, p. 92.
Volume 53, Number 6, June 1976 / 367
