Steady-state approximation validity

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

One way to examine the validity of the steady-state approximation is to compare concentration—time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , 2 = 2 s . The period during which the concentration of the intermediate builds up from its initial value of zero to the quasi-steady-state when dcfjdt is vei small is called the pre-steady-state or transient stage in Fig. 3-10 this lasts for about 2 s. For the remainder of the reaction (over 500 s) the steady-state and exact solutions are in excellent agreement. Because the concen-... 

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

If a reaction system consists of more than one elementary reversible reaction, there will be more than one relaxation time in general, the number of relaxation times is equal to the number of states of the system minus one. (However, even for multistep reactions, only a single relaxation time will be observed if all intermediates are present at vanishingly low concentrations, that is, if the steady-state approximation is valid.) The relaxation times are coupled, in that each relaxation time includes contributions from all of the system rate constants. A system of more than... 

Table 4-1 lists six combinations of rate constants for which an RCS can be defined and two others lacking one. A method has been presented for exploring the concept of the RCS by means of reactant fluxes.11 Consider the case k < (k- + k2), such that the steady-state approximation is valid. One defines an excess rate , for each step i as the difference between the forward rate of that step and the net forward rate v/. Thus, for Step 1,... 

This approximation will be valid if the time scale of the first step is much shorter than that of the second, or k + k i k2. We note that in this approximation [I] is not necessarily a minute fraction of [A], nor is d[ ]/dt = 0, as in the steady-state approximation. Substitution of Eq. (4-91) into the equation... 

One set of experiments was done with both Q and B present at initial concentrations much higher than that of A. With k, kx, and k-j known from other work, the value of k was then estimated, because under these conditions the steady-state approximation for [I] held. To check theory against experiment, one can also determine the products. In the case at hand, meaningful data could be obtained only when concentrations were used for which no valid approximation applies for the concentration of the intermediate. With kinsim, the final amount of each product was calculated for several concentrations. Figure 5-3 shows a plot of [P]o<4R] for different ratios of [B]o/[Q]o the product ratio changes 38-fold for a 51-fold variation in the initial concentration ratio. Had the same ratios of [B]o/tQ]o been taken, but with different absolute values, the indicated product ratios would not have stayed the same. Thus, this plot is for purposes of display only and should not be taken to imply a functional relationship between the quantities in the two axes. 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

Applying the steady state approximation to the partial pressures of the O and N atoms is valid if the average number of propagation cycles prior to termination is large. Assuming this to be the case we find... 

Since the branching parameter a is greater than unity (usually it is 2), it is conceivable that under certain circumstances the denominator of the overall rate expression could become zero. In principle this would lead to an infinite reaction rate (i.e., an explosion). In reality it becomes very large rather than infinite, since the steady-state approximation will break down when the radical concentration becomes quite large. Nonetheless, we will consider the condition that Mol - 1) is equal to (fst T fgt) to be a valid criterion for an explosion limit. 

The assumption made is called the quasi-steady-state approximation (QSSA). It is valid here mainly because of the great difference in densities between the reacting species (gaseous A and solid B). For liquid-isolid systems, this simplification cannot be made. 

In dealing with a complex reaction scheme as the one indicated above, one frequently introduces a so-called steady state approximation for reactive intermediates in order to find simplified rate laws (see, for example, Section 11.2). This approximation is usually sufficiently valid to give rise to useful results most physical chemistry texts discuss and use this application. In the steady state approximation for A, one writes... 

Wong" has pointed out that the steady-state approximation only requires that d[BA]/dt be small compared with A i[A][Eq]. In the early phase of the reaction, if [A] [Eo], the rate of change of [EA] due to diminishing [A] will be relatively slow. It is clear that the validity of steady state is intimately tied to the condition of high substrate to enzyme ratio. 

Textbooks state that the pseudo-steady-state approximation will be valid if the concentration of a species is small. However, one then proceeds by setting its time derivative equal to zero (]/t/f = 0) in the batch reactor equation, not by setting the concentration (CH3CO ) equal to zero. This logic is not obvious from the batch reactor equations because setting the derivative of a concentration equal to zero is not the same as setting its concentration equal to zero. 

The steady-state approximation is often used for the atomic and free radical intermediates occurring in combustion processes. The validity of this approximation has been examined in connection with the theoretical calculation of laminar flame velocities (3, 20, 21) in premixed gaseous systems. The steady-state approximation is occasionally useful for obtaining first-order estimates for flame-propagation velocities but should probably not be used in estimating concentration profiles for reaction intermediates. Some additional observations on the steady-state approximation are contained in Appendix I. 

For the general case, when neither the steady state approximation nor approximations regarding relaxation rates are valid, one has to use computer solutions of a set of rate equations, analogous to those above, including the triplet population. 

The model is able to predict with reasonable accuracy the experimental data if the following hypotheses are made (1) the pre-exponential factors of the rate constants for the formation of the carbonium ions from any of the isomers are the same (2) the pre-exponential factors for the disappearance of the carbonium ions only differ by a statistical factor, which takes into account the fact that in forming the 1-pentene the 2-carbonium ion can lose any of the three hydrogens of Ci, while in forming the 2-pentenes there is the possibility of losing only one hydrogen from the C3 (3) the steady-state approximation is valid for the concentration of adsorbed carbonium ions. The same assumptions were made for the butenes. 

What is the steady-state approximation and under what conditions is it valid ... 

The derivation of the mechanistic rate expression is considerably simplified if the steady state treatment can be used (Sections 3.19, 3.19.1 and 3.20). When intermediate concentrations are not sufficiently low and constant, the steady state approximation is no longer valid. Numerical integration by computer of the differential equations involved in the analysis, or computer simulation, may have to be used. 

To illustrate this approximation, let us consider a pressure flow in which the driving-force pressure drop varies with time. We set dp/dt and d /dt in the equations of continuity and motion, respectively, equal to zero and proceed to solve the problem as if it were a steady-state one, that is, we assume AP to be constant and not a function of time. The solution is of the form v = v(x, A P t), geometry, etc.). Because AP was taken to be a constant, v is also a constant with time. The pseudo-steady-state approximation pretends that the foregoing solution holds for any level of AP and that the functional dependence of v on time is v(x,-, t) = v(x,-, AP(t), geometry, etc.). The pseudo-steady state approximation is not valid if the values of A(pv)/At (At being the characteristic time of fluctuation of AP) obtained using this approximation contribute to an appreciable fraction of the mean value of the applied AP. 

These evaluations are made within the context of the two level model and the steady state approximation. The steady-state approximation is probably valid for this experiment. C is not (electronically, vibronically, or rotationally) a two level system. Other groups, particularly Daily (23) and Berg and Shackleford (18) have developed expressions which allow for the inclusion of more levels and provide for incomplete relaxation. Lucht and Laurendeau (28) have carefully considered the effect of rotational equilibration. There is not time here to discuss these models in detail. The theoretical models which include specifically more than two electronic levels require experimental measurements independently of the radiation coupling the various levels. We have not found a system experimentally tractible for testing the three electronic level model. 

Time Dependence. The pumped level population does not settle to within 90% of its steady state value until 30 nsec after the laser is turned on. Thus, while a steady state approximation is valid for a flashlamp pumped laser with a pulse length of the order of 1 p,sec, the signals using a Nd YAG pumped dye ( 10 nsec) may exhibit an observable time dependence. 

The series model can be extended to longer series and to the inclusion of reversibility to illustrate a variety of fundamental kinetic phenomena in an especially simple and straightforward manner. Depending on the relative rates employed, one can demonstrate the classic kinetic phenomena of a rate-limiting step and preequilibrium,72 and one can examine the conditions needed for the validity of the steady-state approximation commonly used in chemical kinetics.70... 

The same considerations made before are valid for this case and it is very important to have an available validated reaction mechanism. It can be obtained from three main sources (Blelski et al., 1985 Buxton et al., 1988 Stefan and Bolton, 1998) and it is shown in Table 5. With the available information about the constant k2, k, k, fcg, and k-j, it could be possible to solve a system of four differential equations and extract from the experimental data, the missing constants 4> and k (that in real terms is k /Co2)-This method would provide good information about the kinetic constants, but it is not the best result for studying temperature effects if the same information is not available for the pre-exponential factors and the activation energies. Then, it is better to look for an analytical expression even if it is necessary to make some approximations. This is particularly true in this case, where the direct application of the micro steady-state approximation (MSSA) is more difficult due to the existence of a recombination step that includes the two free radicals formed in the reaction. From the available information, it is possible to know that to calculate the pseudo-steady-state... 

The condition expressed by the Bodenstein approximation rx = 0 is often misleadingly called a steady state. It is not. It is not a time-independent state, only a state in which a specific variation with time (or reactor space time) is small compared with the others. In fact, some older textbooks applied what they called the steady-state approximation to batch reactions in order to derive the time dependence of the concentrations, unwittingly leading the incorrect presumption of a steady state ad absurdum. And a continuous stirred-tank or tubular reactor may, and usually does, come to a true steady state, even if the Bodenstein approximation is and remains inapplicable. [The approximation compares process rates r, it is irrelevant for its validity whether or not the reactor comes to a steady state, that is, whether the rates of change, dC /dr, become zero.]... 

When the steady-state approximation of Section B.2.5.2 is applied to nonisothermal systems involving molecular transport, it is sometimes referred to as the extended steady-state approximation. The simplifications that have just been indicated to follow from the application of this approximation underscore the importance of having methods for ascertaining the validity of the approximation. Criteria for the applicability of the extended steady-state approximation have been developed by Giddings and Hirsch-felder [64] and improved by Millan and Da Riva [89]. The discussion... 

The partial-equilibrium approximation differs from the steady-state approximation in that it refers to a particular reaction instead of to a particular species. The mechanism must include the forward and backward steps of any reaction that maintains partial equilibrium, and the approximation for a reaction k is then expressed by setting = 0 in equation (11). It is not always proper to conclude from this that when equations (6), (10), and (11) are employed in equation (14), the terms may be set equal to zero for each k that maintains partial equilibrium partial equilibria occur when the forward and backward rates are both large, and a small fractional difference of these two large quantities may contribute significantly to dcjdt. The criterion for validity of the approximation is that be small compared with the forward or backward rate. 

In the previous section the steady-state approximation was defined and illustrated. It was shown that this approximation is valid after a certain relaxation time that is a characteristic of the particular system under investigation. By perturbing the system and observing the recovery time, information concerning the kinetic parameters of the reaction sequence can be obtained. For example, with A > B ---> C, it was shown that the relaxation time when i 2 was Thus, relaxation methods can be very useful in determining the kinetic parameters of a particular sequence. 

One way to examine the validity of the steady-state approximation is to compare concentration-time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , = 2 s . The period during which the concentration... 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-tejAs. pp. 76-81 si Q s some numerical calculations. Farrow and Edelson and Porter and Skinner used numerical integration to test the validity of the steady-state approximation in complex reactions. 

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