Equilibrium state approximation

Based on the principle of equilibrium state approximation, only one elementary step is the determining step for the three kinetic models listed in Table 2.5. The overall rate of a reaction is determined by the rate of the rate determining step (RDS) with slowest rate, while other steps are approximately in chemical or adsorption equilibrium. There are three opinions or possibilities for the rate determining step of ammonia synthesis reaction on fused iron catalyst (i) RDS is the dissociation of adsorbed dinitrogen N2(ad) 2N (ad) (ii) the surface reaction of adsorbed species N (ad) + H (ad) NH (ad) (iii) the desorption of adsorbed ammonia NH3(ad) NHs(g). 

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.)... 

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. 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

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. 

The adsorption of gases on solid surfaces proceeds to such an extent that approximately 10 7 gr. is present per cm.2 in the equilibrium state. This is of the same order of magnitude as the strength of the limiting capillary layer of a liquid ( 184), hence it is not improbable, as suggested by Faraday (9) (1884), that the adsorbed gas is sometimes present in the liquid state. The adsorbed amount increases with the pressure and diminishes with rise of temperature. The first effect does not follow a law of simple proportionality, as in the case of the absorption of gases by liquids, rather the adsorbed amount does not increase so rapidly, and the equation ... 

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

The rate law of an elementary reaction is written from the equation for the reaction. A rate law is often derived from a proposed mechanism by imposing the steady-state approximation or assuming that there is a pre-equilibrium. To be plausible, a mechanism must be consistent with the experimental rate law. 

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism, (a) For the following mechanism, determine the rate law by the steady-state approximation, (c) Under what conditions do the two methods give the same answer (d) What will the rate law become at high concentrations of Br ... 

For a given molecule and a given intemuclear separation a would have a definite value, such as to make the energy level for P+ lie as low as possible. If a happens to be nearly 1 for the equilibrium state of the molecule, it would be convenient to say that the bond is an electron-pair bond if a is nearly zero, it could be called an ionic bond. This definition is somewhat unsatisfactory in that it does not depend on easily observable quantities. For example, a compound which is ionic by the above definition might dissociate adiabatically into neutral atoms, the value of a changing from nearly zero to unity as the nuclei separate, and it would do this in case the electron affinity of X were less than the ionization potential of M. HF is an example of such a compound. There is evidence, given bdow, that the normal molecule approximates an ionic compound yet it would dissociate adiabatically into neutral F and H.13... 

It is important to avoid saturation of the signal during pulse width calibration. The Bloch equations predict that a delay of 5 1] will be required for complete restoration to the equilibrium state. It is therefore advisable to determine the 1] values an approximate determination may be made quickly by using the inversion-recovery sequence (see next paragraph). The protons of the sample on which the pulse widths are being determined should have relaxation times of less than a second, to avoid unnecessary delays in pulse width calibration. If the sample has protons with longer relaxation times, then it may be advisable to add a small quantity of a relaxation reagent, such as Cr(acac) or Gkl(FOD)3, to induce the nuclei to relax more quickly. 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

In solving the kinetics of a catalytic reaction, what is the difference between the complete solution, the steady-state approximation, and the quasi-equilibrium approximation What is the MARI (most abundant reaction intermediate species) approximation ... 

We assume that, on formation of B- XY, a fraction 5j (i = intermolecular) of an electronic charge is transferred from the electron donor atom of Z of the Lewis base B to the npz orbital of X and that similarly a fraction 5p (p = polarisation) of an electronic charge is transferred from npz of X to n pz of Y, where z is the XY internuclear axis and n and n are the valence-shell principal quantum numbers of X and Y. Within the approximations of the Townes-Dailey model [187], the nuclear quadrupole coupling constants at X and Y in the hypothetical equilibrium state of B- -XY can be shown [178] to be given by ... 

Instead of using the steady-state approximation in the manipulation of the individual rate expressions, the same result may be reached by assuming that a pseudo equilibrium condition is established with respect to reaction B and that reaction C continues to be the rate limiting... 

One useful trick in solving complex kinetic models is called the steady-state approximation. The differential equations for the chemical reaction networks have to be solved in time to understand the variation of the concentrations of the species with time, which is particularly important if the molecular cloud that you are investigating is beginning to collapse. Multiple, coupled differentials can be solved numerically in a fairly straightforward way limited really only by computer power. However, it is useful to consider a time after the reactions have started at which the concentrations of all of the species have settled down and are no longer changing rapidly. This happy equilibrium state of affairs may never happen during the collapse of the cloud but it is a simple approximation to implement and a place to start the analysis. 

The second issue is how to explain the observation of both left- and right-handed helices in the phosphonate material. While Thomas et al. found both helical senses in the early stages of formation of DCggPC tubules, they found both helical senses even in the equilibrium state of the phosphonate. In the previous section, we attributed their results on tubule formation kinetics to a biased chiral symmetry-breaking in which the molecular packing has two possible states which are approximately mirror images of each other. The... 

The H20 exchange mechanism was studied by Hunt et al. (32) who reported that exchange between aqueous solvent and Fein(TPPS)(H20)2 occurs with a first-order rate constant (kex = 1.4xl07s-1 in water at 298 K) far exceeding the k0 s values determined at any [NO]. If the steady state approximation was applied with regard to the intermediate Fem(Por)(H20), the kohs for the exponential relaxation of the non-equilibrium mixture generated by the flash photolysis experiment would be,... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

King et a/.54,138,155 applied the steady state approximation to both systems. In the case of Fe(CO)5, as shown in Scheme 7b, both C02 and H2 production rates should be the same, and so 2[Fe(CO)5][OH-] = fc4[H2Fe(CO)4]. Reactant concentrations are far from equilibrium, and the reactions are assumed to be driven to the right. 

Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158 160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by vrbb = k+ [EAB — k EPQ. Using the conservation of total enzyme... 

A short induction period is typically followed by an oscillatory phase, visible by the alternating colour of the aqueous solution due to the different oxidation states of the metal catalyst. Addition of a coloured redox indicator, such as the Fe,llZ,hl) phen)n couple, results in more dramatic colour changes. Typically, several hundred oscillations with a periodicity of approximately one minute, gradually die out within a couple of hours and the system slowly drifts towards its equilibrium state. 

In general, the effective overall rate constant associated with loss of reactants can be expressed in terms of the individual rate constants a-e in eq 3 by use of the steady state approximation. Simpler expressions can be obtained if the species related by diffusion (a,b) and activation (c,d) processes are assumed to be in thermal equilibrium. In such a case one finds straightforwardly that the effective first order rate constant, k (r), for electron transfer at separation r can be written as... 

Writing these two equations equal to zero does not imply that equilibrium conditions exist, as was the case for Eq. (2.28). It is also important to realize that the steady-state approximation does not imply that the rate of change of the radical concentration is necessarily zero, but rather that the rate terms for the expressions of radical formation and disappearance are much greater than the radical concentration rate term. That is, the sum of the positive terms and the sum of the negative terms on the right-hand side of the equality in Eqs. (2.33) and (2.34) are, in absolute magnitude, very much greater than the term on the left of these equalities [3],... 

Consequently, it is seen, from the measurement of the overall reaction rate and the steady-state approximation, that values of the rate constants of the intermediate radical reactions can be determined without any measurement of radical concentrations. Values k exp and xp evolve from the experimental measurements and the form of Eq. (2.31). Since (ki/k5) is the inverse of the equilibrium constant for Br2 dissociation and this value is known from thermodynamics, k2 can be found from xp. The value of k4 is found from k2 and the equilibrium constant that represents reactions (2.2) and (2.4), as written in the H2 Br2 reaction scheme. From the experimental value of k CX(l and the calculated value of k4, the value k3 can be determined. 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

If one invokes the steady-state approximation described in Chapter 2 for the N atom concentration and makes the partial equilibrium assumption also described in Chapter 2 for the reaction system... 

Any finite expansion that occurs in a finite time is irreversible. A reversible expansion can be approximated as closely as desired, and the values of the thermodynamic changes can be calculated for the limiting case of a reversible process. In the limiting case, the process must be carried out infinitely slowly so that the pressure P is always a well-defined quantity. A reversible process is a succession of states, each of which is an equilibrium state, in which the temperature and pressure have well-defined values such a process is also called a quasi-static process. 

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