Quasi-equilibrium treatment

In this section, we use another chain reaction to show the relation between the steady-state treatment and the quasi-equilibrium treatment. The former is more general than the latter, and leads to more complete but also more complicated results. Ozone, O3, is present in the stratosphere as the ozone layer, and in the troposphere as a pollutant. Ozone production and destruction in the atmosphere is primarily controlled by photochemical reactions, which are discussed in a later section. Ozone may also be thermally decomposed into oxygen, O, although... 

O2], which is the same as one special case of Equation 2-126 when [O2] is high, but does not cover the situation when [O2] is low. Hence, the quasi-equilibrium treatment may be viewed as a special case of the steady-state treatment. 

Quasi-equilibrium treatments, on the other hand, assume that all elementary steps prior to the rds are almost in equilibrium, i.e. they can occur sufficiently fast not to alter significantly their equilibrium conditions under net charge flow at the interface. For this assumption to be valid, the elementary step rate coefficients must be at least ten times larger than that of the rds. 

The general rule for writing the rate equation according to the quasi-equilibrium treatment of enzyme kinetics can be exemplified for the random bisubstrate reaction with substrates A and B forming products P and Q (Figure 7.1), where KaKab = KbKba and KpKpq = KqKqp. 

The steady-state treatment of enzyme kinetics assumes that concentrations of the enzyme-containing intermediates remain constant during the period over which an initial velocity of the reaction is measured. Thus, the rates of changes in the concentrations of the enzyme-containing species equal zero. Under the same experimental conditions (i.e., [S]0 [E]0 and the velocity is measured during the very early stage of the reaction), the rate equation for one substrate reaction (uni uni reaction), if expressed in kinetic parameters (V and Ks), has the form identical to the Michaelis-Menten equation. However, it is important to note the differences in the Michaelis constant that is, Ks = k2/k1 for the quasi-equilibrium treatment whereas Ks = (k2 + k3)/k i for the steady-state treatment. 

The latter theory was first suggested by Becker and Doering (B2), who applied a quasi-equilibrium treatment and developed the following equation for the nucleation rate of condensing vapors ... 

The flow rate is extremely slow on the time scale of molecular motions. Therefore, a quasi-equilibrium treatment is valid. The retention factor - the difference between the retention time for a peak of interest and a standard um-etained reference, divided by the retention time for the reference - can then be taken as... 

The steps after the rds (as they are written in Scheme 1, i.e., as reductions) will be rate limited by the rds only in the reverse, oxidative reaction direction. The effect of these steps following the rds will be observed for applied positive overpotentials. In the case of oxidation the steps following the rds will be in quasi-equilibrium and those preceding the rds will now run (backward) to reactants without influencing the rate in the reverse direction. Ignoring the opposite reaction direction is a convenient simplication that, as we will see (Section IV.6) does not seriously affect the validity of the quasi-equilibrium treatment. 

In most of the simulated curves in Fig. 4 two linear regions in both the cathodic and anodic branches can be clearly seen that have slopes of 1391 and 11181 mV dec (at 298 K), whereas the quasi-equilibrium treatment would have predicted only a single linear Tafel region of slope 1391 mV dec for this mechanism. When the 1 2 ratio increases, varying from 10 (solid line) to... 

The stoichiometric number concept does not, however, demonstrate in a simple way how v becomes incorporated into and affects the theoretical rate expressions that describe polarization behavior for multistep reactions that involve a stoichiometric number. This has been one of the problems in understanding its significance. In the theoretical evaluation of transfer coefficients based upon the quasi-equilibrium treatment of Bockris and Reddy (B R) in their monograph (Ref. 3, p. 1005), v forms... 

The quasi-equilibrium treatment yields an identical rate equation if represented in the Cleland form. However, the quasi-equilibrium condition assumes ks and Ict being rate-limiting, which yields different expressions for the maximum velocities, Vi = ksEt, V2 = IctEt and Michaelis-Menten constants, Ka = ka/ki, Kp = kg/ks respectively. 

Obviously, because of condition b) this treatment must be characterized as a quasi-equilibrium treatment. It can only hold for a sufficiently slow process. However, by the application of this approach one obtains the so-called square-root-law for the time dependence of the layer thickness. But this law has an infinite slope at zero time. In other words, it does not represent a slow process initially. An estimate of the error made in this way cannot be given without the introduction of the physical conceptions of nucleation and supercooling. This fact has been overlooked at the time. 

The classical approach to heat transfer problems complicated by the occurrence of phase changes is, from the point of view of thermodynamics, a quasi-equilibrium treatment. It has been used by Stefan in 1889 for the description of... 

THE RAPID-EQUILIBRIUM TREATMENT. The first rate equation for an enzyme-catalyzed reaction was derived by Henri and by Michaelis and Menten, based on the rapid-equilibrium concept. With this treatment it is assumed that there is a slow catalytic conversion step and the combination and dissociation of enzyme and substrate are relatively fast, such that they reach a state of quasi-equilibrium or rapid equilibrium. 

CONCLUDING REMARKS. In this entry, the derivation of initial-velocity equations under steady-state, rapid-equilibrium, and the hybrid rapid-equilibrium and steady-state conditions has been covered. Derivation of initial velocity equation for the quasi-equilibrium case is quite straightforward once the equilibrium relationships among various enzyme-containing species are defined. The combined rapid-equilibrium and steady-state treatment can be reduced to the steady-state method by treating the equilibrium segments as though they were enzyme intermediates. 

Theoretical treatments for the analysis of complex electrode reactions in terms of elementary steps can be made, as in general chemical kinetics, by the steady state [3, 5, 7] or the quasi-equilibrium methods [4, 5, 47]. 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

In conventional heating the sample is brought linearly to a given temperature since the sample is far from equilibrium, it is kept at that temperature for a ce time (commonly between 2 and 10 hours). This results in a mass-temperature c which is often far removed from the characteristic curve for the adsorbent (cf. p 1 and 2 in Figure 3.21). In some cases, this can invalidate any discussion of the nificance of the results. However, since quasi-equilibrium can be reached at any during a CRTA experiment, it is possible to stop the heat treatment (and evi quench the sample) at any point of the curve common to all samples (cf. point ... 

In his treatment of the photoresponse in the semiconductor, Wilson assumed, as had Memming, that quasi-equilibrium conditions obtained across the depletion layer, i.e. the product np is a constant. 

The strictly mathematical basis of the quasi-equilibrium rate theory is absent from other, qualitative theories of mass spectrometric fragmentation which are based mainly on empirical rules and are discussed more fully in Section VI on the classification of mass spectra. One empirical classification, the charge-localization treatment, has achieved an aura of theoretical respectability through the use of curved arrows and analogies with the formalism of resonance theory in solution chemistry (Budzikiewicz et al., 1967a). The bonds in a molecular ion vibrate with excess energy but since an electron has been removed on ionization, the vibration frequencies are not the same as in the original intact molecule. 

Using the simplified quasi-equilibrium rate equation (Section IIF) and a thermochemical argument, Bentley et al. (1969) showed that ZylZff) correlations for fragmentation processes may be solely due to the correlation of ionization potentials with a (see also Ward et al., 1969). The treatment was applied to simple cleavage processes where the substituent was lost in the neutral fragment and the appearance... 

In order to avoid such complicated expressions, the quasi-equilibrium method is used, although the above steady-state approach can become reduced to the same result if various limiting assumptions about relative values of rate constants, referred to earlier, are introduced. This approach assumes a rate-limiting step so that all other steps are supposed to have much larger rate constants (in both directions) and hence are all in virtual or quasi -equilibrium. Bockris " applied a similar treatment in a less general way to the kinetics of various pathways of the oxygen evolution reaction. With the application of a negative overpotential that drives the forward, supposed reductive direction of the reaction, each of the steps prior to the rds (which are hence limited by it) will be at quasi-equilibrium. 

In the case of complex reactions several steps are possible and rigorous treatment is required, as the quasi-equilibrium approach cannot be applied. Such cases will be treated in Chapter 4. 

Eq. (6.36) can also be derived from the general equation for the three-step sequence. In the treatment above, the quasi-equilibrium steps were defined as the ratio of reactants to the ratio of products following the bio logic to have an analogy with the Michaelis Menten constant. In the Lineweaver-Burk coordinates, eq. (6.36) becomes... 

The steady-state kinetic treatment of random reactions is complex and gives rise to rate equations of higher order in substrate and product terms. For kinetic treatment of random reactions that display the Michaelis-Menten (i.e. hyperbolic velocity-substrate relationship) or linear (linearly transformed kinetic plots) kinetic behavior, the quasi-equilibrium assumption is commonly made to analyze enzyme kinetic data. 

If the potential energy surface has a col, then a separation of the reaction coordinate is possible also in the vicinity of the saddle-point. Supposing that the system remains there sufficiently long time for a stationary state to be established, we may assume the existence of a quasi-equilibrium energy distribution in that transition state, too. Then, we can apply the above statistical treatment of reactants also to the transition state of the reacting system. However, we will not introduce at this stage the above restrictive assumptions, since our aim is to derive a collision theory rate expression of a possibly general validity. 

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