Approximation steady-state

This is known as the steady-state approximation, since the concentration of B is assumed to be constant throughout the reaction. This is valid except at the initial and final stages of many kinetic schemes. Applying the steady-state approximation, the above kinetic scheme is rednced to a first-order rate law, where the observed psendo-first-order constant is given by k k2l(k. i+k.  

It is reasonable that the variation of [B] with time will be small if [B] is also a small quantity. Thus, the condition for application of the steady-state approximation can be given as 

At the start of the reaction, when the intermediate B is being formed, the change in its concentration with time is not negligible, and the steady-state approximation does not hold. However, as it is assumed that the concentration of B is small, that is [B] . [A], we can also formulate the conditions for the application of the steady-state approximation in terms of the rate constants for the formation and decay of B 

One of the most important examples of the application of this approximation is enzyme catalysis [11]. Enzyme catalysis can be described in its simplest form by the scheme 

As the total concentration of enzyme present [E], is normally known and can be expressed as 

This is an assumption used to derive the Michaelis-Menten equation in which the velocity of ES formation is assumed to be equal to the velocity of ES breakdown. 

As with most assumptions and approximations, those professors who do not ignore this entirely will undoubtedly think that you should at least know that it s an assumption. What the steady-state assumption actually 

The 3 is just a constant. What we don t know is [ES], mainly because we don t know how much of our enzyme is present as [Ej. But we do know how much total enzyme we have around—it s how much we added. 

But we still don t know [ES]. We need another equation that has [E] and [ES] in it. Here s where the steady state approximation comes in handy. At steady state, the change in the concentration of [ES] is zero, and the velocity of [ES] formation equals the velocity of [ES] breakdown. 

The rest is simple algebra. Solve the preceding equation for [E] and stick the result in the equation for [E]totai- The E and ES should disappear at 

A mechanism often invokes an unstable intermediate of some defined structure, and a general mechanism might take the form of 

The total concentration of reagents can be defined as [T] and will remain constant. Since B is a reactive intermediate, its concentration will always 

The rate of disappearance of A is given as follows (note that the mechanism has speciEed that all the steps have Erst-order or pseudo-Erst-order rate constants)  

Note that the right-hand side of Eq. (2.13) is the same as the coefficient for [A] on the right-hand side of Eq. (2.7). Therefore, it really was not necessary to go through the equilibrium conditions in order to find the expression for It is always true that once one has an integratable equation with only one concentration variable in first-order form, then the coefficient of the concentration variable in that equation will be the expression for k.  

A limiting form of Eq. (2.13) that is often encountered assumes that k k. Then, the k is given by 

When the intermediate Q in the consecutive reaction scheme of Section 1.3.2 is very reactive, meaning that k2 ki, the equations (1.3.2-6) reduce to 

For (1.4.1-l)-(1.4.1-3) to be a useful approximation of the complete scheme, the induction time should be very short, meaning that the concentration of the intermediate must be very small. From (1.3.2-6) it is seen how the maximum in the curve Cq versus t moves towards t = 0 as k2 k. Quite frequently the existence of an intermediate is chemically logical, but it is difficult or impossible to measure its concentration. The pseudo steady state approximation is then a very useful tool. Examples will be encountered in Section 1.5 on bio-kinetics and Section 1.6 on complex reactions. 

The concentrations are the solutions of two differential equations and one algebraic equation 

---

It is assumed that irreversible aggregation occurs on contact. The rate of coagulation is expressed as the aggregation flux J of particles towards a central particle. Using a steady-state approximation, the diffusive flux is derived to be... 

The rate equation for the S z form is, from the steady-state approximation, as follows ... 

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. 

Comparing this to the theoretical expression based on the steady state approximation suggests that the mechanism is not straightforward ... 

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. 

Following the steady-state approximation, both propagation steps must proceed at the same rate otherwise, the concentration of A- or C- would build up. By substituting for the concentration of the intermediate C-, we obtain ... 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

The concentration of the reaction intermediate AB may be determined by using the steady state approximation for intermediates,... 

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.)... 

Proceeding in the usual way to apply the steady-state approximation to the complex ES ... 

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 ... 

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. 

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. 

Apply the steady-state approximation to Scheme XXII for ester hydrolysis to find how the experimental second-order rate constant qh is related to the elementary rate constants. 

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... 

Suppose in Scheme XVI that the steady-state approximation is applicable to the intermediate A. ... 

Either step could be rate determining. Study of many reactions has shown that most occur with a significant isotopic effect, but for some reactions the isotope effect is absent.If we apply the steady state approximation to the intermediate, this reaction scheme leads to... 

Because the cationic intermediate is unstable, it will be permissible to apply the steady-state approximation, leading to Eq. (8-65) for the reaction rate. 

---