Two Limiting Cases

The first limiting case was suggested originally by Monod, Wyman, and Changeux (MWC) in 1965. This model requires that the two subunits be either in the L or in the 

H state. The conformations of the two subunits change in a concerted way. This is equivalent to the consideration of the first and fourth columns in Fig. 3.9. Mathematically, we can obtain this limiting case by taking r = 0 in Eq. (3.3.14), which essentially means that the equilibrium concentrations of x h and x l are negligible. 

This is essentially the same PF as of the model treated in section 3.2 with the replacement of Qa [in Eq. (3.2.9)] by QlQaa In essence, the MWC model is equivalent to two state polymers with energies corresponding to lE + Ei l and 2Eh+ Ehh The fact that we have two subunits does not affect the formalism, except for the redefinitions of the energy levels. 

Usually the MWC is applied when there are no direct interactions between the ligands (i.e., S = 1) in which case (3.3.87) reduces to 

If we choose the L energy level lE + Ell as our zero energy (or, equivalently, define a new PF by = IQIQll), we may rewrite (3.3.88) in the more familiar form as 

This is essentially the same PF as that of the model treated in Section 4.5 with the replacement of everywhere by 2 aa- essence, the MWC model is equivalent 

The second extreme case, suggested by Koshland, Nemethy, and Filmer (KNF) (Koshland et al., 1966), is also known as the sequential model. The mathematical conditions required to obtain this limiting case are quite severe. First, it is assumed that, in the absence of a ligand, one of the conformations is dominant, say the LL form. In addition, it is assumed that a ligand binding to any subunit will change the conformation of that subunit into the H form. These assumptions lead to the consideration of only the four diagonal states of Fig. 4.18, for which the PF is 

The empty state is the LL state on the top left comer of Fig. 4.18. The binding of a ligand on any of the subunits will shift its conformation completely from L to H without affecting the conformation of the second subunit. Binding of the two ligands will shift the entire polymer to the state HH. Thus, in each binding process there is a total change of conformation of one subunit hence the term sequential model. 

The mathematical requirements necessary to obtain the KNF model from the general one can be stated as follows. Let K be the equilibrium constant for the H L conversion of each subunit when it is known to be empty. Likewise, let K be the equilibrium constant when the subunit is known to be occupied. Both of these equilibrium constants are functions of X, i.e.. 

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Detemiining compositions is possible if the distribution of elements over the outer layers of the sample and the surface morphology is known. Two limiting cases are considered, namely a homogeneous composition tliroughout the outer layers and an arrangement in which one element covers the other. 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

It is also interesting to examine the relative importance of thermal transpiration and thermal diffusion in the two limiting cases. From equations (A. 1.12) and (A. 1.13)... 

This method provides a reasonable estimate of the piQ, provided that the weak acid is neither too strong nor too weak. These limitations are easily appreciated by considering two limiting cases. For the first case let s assume that the acid is strong enough that it is more than 50% dissociated before the titration begins. As a result the concentration of HA before the equivalence point is always less than the concentration of A , and there is no point along the titration curve where [HA] = [A ]. At the other extreme, if the acid is too weak, the equilibrium constant for the titration reaction... 

The assumption that k values are constant over the entire duration of the reaction breaks down for termination reactions in bulk polymerizations. Here, as in Sec. 5.2, we can consider the termination process—whether by combination or disproportionation to depend on the rates at which polymer molecules can diffuse into (characterized by kj) or out of (characterized by k ) the same solvent cage and the rate at which chemical reaction between them (characterized by kj.) occurs in that cage. In Chap. 5 we saw that two limiting cases of Eq. (5.8) could be readily identified ... 

Based on considerations we have encountered earlier in this chapter, we can anticipate two limiting cases of this function P(0) approaches unity both in the limit of small particles and in the limit of small angles of observation. Interference is absent in both of these cases. 

The following considerations are based on the assumption that the part of coating that is free from holidays acts as an insulator. Two limiting cases are considered a single circular defect and an almost porous coating with numerous small and equally distributed defects. 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

The ionization and direct displacement mechanisms can be viewed as the extremes of a mechanistic continuum. At the 8 1 extreme, there is no covalent interaction between the reactant and the nucleophile in the transition state for cleavage of the bond to the leaving group. At the 8 2 extreme, the bond formation to the nucleophile is concerted with the bondbreaking step. In between these two limiting cases lies the borderline area, in which the degree of covalent interaction between the nucleophile and the reactant is intermediate between the two limiting cases. The concept of ion pairs is important in the consideration of... 

Equation 6.5 can be solved in an analytical form for two limiting cases in which besides nucleation only either (1) crystal growth or (2) particle agglomeration occurs. 

For these two limiting cases let us write Br nsted-type relationships for variation in the nucleophile, namely. 

No new pnnciples ate involved in describing the bonding in these complexes and appropriate combinations of ihe 4p ottoilals on the diene system can be used lo construct MOs with the metal-based orbitals for donation and back donation of electron density.As with ethene, two limiting cases can be envisaged wbicb can be represented schematically as in Fig. 19.25. Consistent with... 

Two limiting cases of the Langmuir isotherm are of interest. When 0 is very small, as when the pressure (or concentration) is low, or the constant a is small, then equation 20.18 reduces to... 

Two limiting cases for gasification at the fuel surface were considered. In case 1, the fuel concentration was assumed constant and independent of time, i.e., f(Cf) = Cf and in case 2, it was assumed that the fuel mass flux was constant and independent of time or pressure, i.e.,/(Cy) = — D 8Cf/ dx = rfi. Case 1 was identified with a condensed phase behaving as a boiling liquid or subliming solid, and case 2 with a polymer undergoing irreversible decomposition at constant temperature. 

Compressibility of a gas flowing in a pipe can have significant effect on the relation between flowrate and the pressures at the two ends. Changes in fluid density can arise as a result of changes in either temperature or pressure, or in both, and the flow will be affected by the rate of heat transfer between the pipe and the surroundings. Two limiting cases of particular interest are for isothermal and adiabatic conditions. 

In this result one can easily see two limiting cases, for small and large jumps, correspondingly ... 

In this context, two limiting cases of interest are as follows ... 

This expression has two limiting cases. If the rate constant of the second step is small we can ignore it in the denominator of Eq. (75), and obtain... 

Thus we see again that the system can conveniently be described by the Thiele diffusion modulus Op. It is not difficult to see that two limiting cases exist. There are no diffusion limitations when D <... 

This information is sufficient to analyze the qualitative behaviour of E(zg). Indeed, two limiting cases may be considered. 

The reaction (Eqn. 5.4-65) takes place in the liquid phase. The molecules are transferred away from the interface to the bulk of the liquid, while reaction takes place simultaneously. Two limiting cases can be envisaged (1) reaction is very fast compared to mass transfer, which means that reaction only takes place in the film, and (2) reaction is very slow compared to mass transfer, and reaction only takes place in the liquid bulk. A convenient dimensionless group, the Hatta number, has been defined, which characterizes the situation compared to the limiting cases. For a reaction that is first order in the gaseous reactant and zero order in the liquid reactant (cm = 1, as = 0), Hatta is ... 

Limiting Cases Equation (20-94) has two limiting cases for a binary system. First, when a > ([) this case, selectivity is no longer very important. 

Solution of equations (1.26) and (1.27) for above two limiting cases differing in the value of the surface concentration of adsorption particles brings about different dependencies of the value of the surface band bending as a function of parameters of the absorbate-adsorbent system. Thus, in case of adsorption of acceptors we obtain from (1.26) that... 

In two limiting cases differing in the values of stationary concentration of chemisorbed radicals and initial electric conductivity of adsorbent the expression (2.94) acquires the following shape ... 

Thus, analyses of how aggregates break have resorted to idealizations inspired by traditional fluid mechanical analysis, two limit cases being per-... 

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