Equations competitive equilibrium between

Nelson et al(28) have developed an equation assuming competitive equilibrium between Pu(IV) and soluble complexing ligands and between Pu(IV) and a solid adsorber ... 

In the simplest case of a competitive uptake of two metals (or a metal and proton) for an identical uptake site under equilibrium conditions, the reduction of the uptake flux of the solute can be quantitatively predicted using the respective equilibrium formation constants (equations (38) (41)). As can be seen in Table 3, for a given study, constants among the trace metals, protons and alkaline earth metals are often sufficiently similar for competition to be important. Nevertheless, competition is likely to be negligible under most environmentally relevant conditions where competition occurs between low concentrations of metals, such that the free carrier concentration remains approximately equal to the total receptor concentration. 

Reversible inhibition is characterized by an equilibrium between enzyme and inhibitor. Many reversible inhibitors are substrate analogues, and bear a close relationship to the normal substrate. When the inhibitor and the substrate compete for the same site on the enzyme, the inhibition is called competitive inhibition. In addition to the reaction described in equation 1, the competing reaction described in equation 3 proceeds when a competitive inhibitor I is added to the reaction solution. 

If the system is well represented by Langmuir-Hinshelwood kinetics, then the relative values of the adsorption equilibrium constants of various unsaturated compounds can be obtained from the relative rates determined individually and in competition. The expression for the competitive reaction between two compounds A and B results from the division of the corresponding rate expressions for each compound (equation 3) to give equation (4). The ratio of the individual rates of hydrogenation, k/Jk, permits the abstraction of the ratio K/JK from the competitively determined ratio kkK/Jk K (equation 5). The assumption that the adsorption of the unsaturated compound is rapid and reversible relative to the rates of the surface-catalyzed reactions may not be correct. In that case, KfJK is an apparent relative adsorption equilibrium constant (see Section 3.1.3.2.2). 

The theory of the hodograph transform and the relationship derived between the equations of the two lines given by this transform in the case of a binary mixture and those of the competitive equilibrium isotherms were briefly presented in Section 8.1.2. The theory is easily extended to multicomponent mixtures, although in this case we must represent the hodograph transform in an n-dimensional coordinate system, Ci, C2, , C , or in its planar projections. If the solution presents a constant state (Figure 8.1), it is a simple wave solution, and there is a relationship between the concentrations of the different components in the eluent at the column exit (Figure 8.2). This result is valid for any convex-upward isotherm. In the particular case in which the competitive Langmuir isotherm apphes, these relationships are linear. 

Finally, we need the isotherm equations that relate the concentration of each of the solutes in the stationary phase and the concentrations of all the solutes in the mobile phase. In general, these adsorption isotherms are competitive, meaning that the amount of component i adsorbed at equilibrium between phases from a solution with a constant concentration Q decreases with increasing concentrations of any one of the other components (Chapter 4). 

Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the n-dimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4). 

Where competitive inhibition is observed between two solutes (i.e. binding to a single, identical carrier), it is also possible to estimate carrier concentrations using a steady-state treatment [193-195], In that case, data from the competing solutes are used to generate a sufficient number of equilibrium expressions (e.g. equations (38) and (39)) and corresponding mass balance equations (e.g. equations (40) and (41)) to resolve for the total carrier concentration. 

FIGURE 2.1. EC reaction scheme in cyclic voltammetry. Kinetic zone diagram showing the competition between diffusion and follow-up reaction as a function of the equilibrium constant, K, and the dimensionless kinetic parameter, X. The boundaries between the zones are based on an uncertainty of 3 mV at 25°C on the peak potential. The dimensionless equations of the cyclic voltammetric responses in each zone are given in Table 6.4. 

In the ID limit, Eqs. (7) and (8) and related equations have been used to analyze the relaxation of non-equilibrium step profile - and in a variety of other application We will not review this work here, but instead turn directly to two cases where characteristic 2D step patterns and step bunching are found as a result of the competition between the step repulsions and a driving force favoring step bunching. Perhaps the simplest application arises as a result of surface reconstmction. 

This expression is known as the Stern-Volmer equation and Ksv as Stern-Volmer constant. Ksv is the ratio of bimolecular quenching constant to unimolecular decay constant and has the dimension of litre/mole. It implies a competition between the two decay pathways and has the ch".acter of an equilibrium constant. The Stern-Volmer expression is linear in quencher concentration and Ksv is obtained as the slope of the plot of 4>f°If vs [Q], if the assumed mechanism of quenching is operative. Here, t is the actual lifetime of the fluorescer molecule in absence of bimolecular quenching and is expressed as... 

If CL, CM and pH are kept the same for a series of different cations, the position of equilibrium in equation (6a) would depend on the value of the formation constant, (3 , for the metal complex in question and the extent of precipitation on its solubility product, K. The more stable the complex and the lower its solubility in water, the greater the extent of precipitation. Essentially there is a competition between cations M"+, and protons, H+, for the free ligand anion Ox. ... 

Simple comparisons of stability constants in this way have been criticized because no allowance is made for the differences in ligand basicities. Increased basicity as a rule parallels increased metal binding. Also, in the majority of cases competition between hydrogen ion and metal ion occurs (equation 3). The equilibrium constant (K) for reaction (3) is then given by equation (4). 

This equation can be interpreted in terms of Langmuir adsorption isotherms. It is assumed (see 1.5.4) that both reactants must be adsorbed in order to react and that KA and KB are the respective Langmuir adsorption equilibrium constants. The denominator allows for competition for sites between... 

Reactions were studied under the pseudo first-order condition of [substrate] much greater than [initial dihydroflavin]. Under these conditions, the reactions are characterized by a burst in the production of Flox followed by a much slower rate of Flox formation until completion of reaction. The initial burst is provided by the competition between parallel pseudo first-order Reactions a and b of Scheme 3. These convert dihydroflavin and carbonyl compound to an equilibrium mixture of carbinolamine and imine (Reaction a), and to Flox and alcohol (Reaction b), respectively. The slower production of Flox, following the initial burst, occurs by the conversion of carbinolamine back to reduced flavin and substrate and, more importantly, by the disproportionation of product Flox with carbinolamine (Reaction c followed by d). Reactions c and d constitute an autocatalysis by oxidized flavin of the conversion of carbinolamine back to starting dihydroflavin and substrate. In the course of these studies, the contribution of acid-base catalysis to the reactions of Scheme 3 were determined. The significant feature to be pointed out here is that carbinolamine does not undergo an elimination reaction to yield Flox and lactic acid (Equation 25). The carbinolamine (N(5)-covalent adduct) is formed in a... 

Changes with time in the isotopic exchange reaction rate between H2 and HDO(v) over a hydrophobic Pt-catalyst induced by the addition of HN03 were studied experimentally The HN03 poisoning was found to be reversible and was well explained in terms of the competitive adsorption of HNO, with H2 or HDO onto the catalytic active sites. The adsorption equilibrium for HN03 could be expressed by the Frumkin-Temkin equation and the time evolution of the activity was well expressed by the Zeldovich rate equation. 

The equilibrium constant, K, thermodynamically could be described as the exponent of the Gibbs free energy of the analyte s competitive interactions with the stationary phase. In hquid chromatography the analyte competes with the eluent for the place on the stationary phase, and resulting energy responsible for the analyte retention is actually the difference between the analyte interaction with the stationary phase and the eluent interactions for the stationary phase as shown in equation (1-5)... 

A steady-state treatment yields the same final equations with Kmg and Km, replacing Ks and JCp. In place of the usual [S] in the numerator of the above equation, we have the difference between [S] and the equilibrium value of [S]. The Ks term in the denominator is modified in a manner consistent for the product acting as a competitive inhibitor with respect to the substrate. In other words, the initial net velocity depends on the displacement of the system from equilibrium (i.e, the thermodynamic driving force) and the amount of enzyme tied up with product. A more detailed account of competitive inhibition is given in a later section. 

We obtain the same final velocity equation for steady-stale conditions, except replaces Kg. This is not surprising since the steady-state assump. tion does not change the form of the velocity equation for the uninhibited reaction while the reaction between E and 1 to yield El must be at equilibrium, (There is nowhere for El to go but back to E-H.) The velocity equation differs from the usual Michaelis-Menten equation in that the K term is multiplied by the factor [1 4- ([I]/J i)]- The above derivation confirms our original prediction that is unaffected by a competitive inhibitor, but that the apparem K value is increased. The increase in the value does. loi mean that the El complex has a lower affinity for the substrate. El has no affinity at all for the substrate, while the affinity of E (the only form that can bind substrate) is unchanged. The apparent increase in results from a distribution of available enzyme between the full affinity and "no affinity forms. The factor [14-([I]/ffi)] may be considered as an [I]-dependent statistical factor describing the distribution of enzyme between the E and El forms. Figure 4-21 shows the effect of a competitive inhibitor on the v versus [S] plot. 

The mechanism given above places no restrictions on the source of the reversible poison. Consequently, the poisoning can be due not to an adsorption competition between the reactant and a diluent but to an adsorption competition between the reactant and one or more of the reaction products. When this occurs the products will determine the kinetics in the flow type and static systems where appreciable conversion is allowed. Under these conditions the kinetics may be expressed by equations similar to equation (6), and the order will be determined by the magnitude of constants similar to H which depend upon the various velocity constants and adsorption equilibrium constants of the heterogeneous reaction. 

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