And steady state approximation

Of course it is also possible for a reaction system not to belong to any of these classes of approximate description.) Only in class III can equilibrium be said to be a special case of the steady-state treatment. Note that, for class III systems, the steady-state concentration of intermediate is very large,whereas for class I it is very small. Zuman and Patel have discussed the equilibrium and steady-state approximations in terms similar to the present treatment. 

Volmer relationship and steady-state approximations, the kinetic scheme of Fig. 13 leads to Eq. (38) where kq is the... 

Example 1,4-3 Rate Determining Step and Steady-State Approximation... 

To be analytically useful equation 13.16 needs to be written in terms of the concentrations of enzyme and substrate. This is accomplished by applying the steady-state approximation, in which we assume that the concentration of ES is essentially constant. After an initial period in which the enzyme-substrate complex first forms, the rate of formation of ES... 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

For a sequenee of reaetion steps two more eoneepts will be used in kinetics, besides the previous rules for single reaetions. One is the steady-state approximation and the seeond is the rate limiting step eoneept. These two are in strict sense incompatible, yet assumption of both causes little error. Both were explained on Figure 6.1.1 Boudart (1968) credits Kenzi Tamaru with the graphical representation of reaction sequences. Here this will be used quantitatively on a logarithmic scale. 

A useful approach that is often used in analysis and simplification of kinetic expressions is the steady-state approximation. It can be illustrated with a hypothetical reaction scheme ... 

Assume that the steady-state approximation can be applied to the intermediate TI. Derive the kinetic expression for hydrolysis of the imine. How many variables must be determined to construct the pH-rate profile What simplifying assumptions are justified at very high and very low pH values What are the kinetic expressions that result from these assumptions ... 

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction ... 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

Modeling of Chemioal Kinetios and Reaotor Design Using the steady state approximation,... 

Applying the rate expressions to Equations 1-222, 1-223, 1-224, 1-225 and 1-226, and using the steady state approximation for CH3, C2H5, and H, for a eonstant volume bateh reaetor yields ... 

This also accounts for the production of the small amount of butane. If the reaction mechanism were steps 1, 2, 3, 4, 5a, and 5b, then applying the steady state approximations would give the overall order of reaction as 1/2. 

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. 

Consider further Scheme XIV and rate equations (3-139) to (3-141). Evidently Cb will be small relative to (Ca + Cc) if ( -i + 2) i- Then B plays the role of a reactive intermediate in the overall reaction A— C. This is the usual condition that is taken as a warrant for the application of the steady-state approximation. If Cb is small, it is reasonable that Cb will be small throughout most the reaction, so it is set equal to zero. As Wong (53) has pointed out, however, the condition Cb = 0 is a sufficient but unnecessary condition for Eq. (3-142) to hold. Erom Eq. (3-140) we obtain... 

The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

Several features of this treatment are of interest. Compare the denominators of Eqs. (3-147) and (3-149) Miller has pointed out that the form of Eq. (3-147) is usually seen in chemical applications of the steady-state approximation, whereas the form of Eq. (3-149) appears in biochemical applications. The difference arises from the manner in which one uses the mass balance expressions, and this depends upon the type of system being studied and the information available. 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

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-... 

In the preceding subsection we described the preequilibrium assumption. Let us now see how that assumption is related to the steady-state approximation. Scheme XIV will serve for the discussion. The equilibrium and steady-state expressions for the intermediate concentration are... 

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 ... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

For Scheme XIV, and for each of the following sets of rate constants, calculate the exact relative concentration cb/ca as a function of time. Also, for each set, calculate the approximate values of cb/ca using both the equilibrium assumption and the steady-state approximation. 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

Application of a steady state approximation (that / t = / j, eq. 2) and a long chain approximation (negligible monomer consumption in the initiation or reinitiation steps) provides a number of useful relationships. ... 

Using the steady-state approximation, and taking into account the fact that the values of kQn are relatively insensitive to the ratio of radii rx/rB (Debye, 1942), so that one can set rx/rB l, one gets the expression in Scheme 3-33 as a good approximation for the encounter rate coefficient (R = gas constant). 

Applying the steady-state approximation to both [II] and [IV] individually leads to... 

Applying the steady-state approximation to both [VI] and [VIII] one obtains (29) for the steady-state concentration of VI and (30) for the steady-state concentration of VIII, viz. 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

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,... 

THE PRIOR-EQUILIBRIUM AND IMPROVED STEADY-STATE APPROXIMATIONS... 

Steady-state. An erroneous rate law is shown below for the reaction scheme believed to represent the reaction between Fe3+ and I-, in that an extraneous denominator term appears. In the scheme shown, I2 and Fel2+ obey the steady-state approximation. Show what the incorrect part of the expression is. Suggest a simple derivation of the correct equation that avoids extensive algebraic manipulations. 

Making the steady-state approximation for [PFe], derive the rate law. Next, repeat the derivation including the reverse step with k-2. If [CO] and [02] are s> [PFe(O2)]0, what is the expression for ke, as defined in Chapter 3 ... 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

Steady-state mechanism. Derive the expression for -d[hR]/dt in this scheme, making the steady-state approximation for [A] and [B]. The answer must contain no concentration other than [AB],... 

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