Pseudo equilibrium constant

The material balance was calculated for EtPy, ethyl lactates (EtLa) and CD by solving the set of differential equation derived form the reaction scheme Adam s method was used for the solution of the set of differential equations. The rate constants for the hydrogenation reactions are of pseudo first order. Their value depends on the intrinsic rate constant of the catalytic reaction, the hydrogen pressure, and the adsorption equilibrium constants of all components involved in the hydrogenation. It was assumed that the hydrogen pressure is constant during... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

If one takes the ratio of the pseudo rate constant for adsorption to that for desorption as an equilibrium constant for adsorption (K), equation 6.2.4 can be written as... 

The first indication that A-acyloxy-A-alkoxyamidcs reacted by an acid-catalysed process came from preliminary H NMR investigations in a homogeneous D20/ CD3CN mixture, which indicated that A-acetoxy-A-butoxybenzamide 25c reacted slowly in aqueous acetonitrile by an autocatalytic process according to Scheme 4 (.k is the unimolecular or pseudo unimolecular rate constant, K the dissociation constant of acetic acid and K the pre-equilibrium constant for protonation of 25c).38... 

Upon addition of a solution of sulfuric acid in D20 the reaction of A-acetoxy-A-alkoxyamides obeys pseudo-unimolecular kinetics consistent with a rapid reversible protonation of the substrate followed by a slow decomposition to acetic acid and products according to Scheme 5. Here k is the unimolecular or pseudo unimolecular rate constant and K the pre-equilibrium constant for protonation of 25c. Since under these conditions water (D20) was in a relatively small excess compared with dilute aqueous solutions, the rate expression could be represented by the following equation ... 

From Eyring, the rate constant of reaction k depends on a pseudo equilibrium constant AT, relating to the formation of a transition-state complex, TS. Clearly, AT will always be virtually infinitesimal. 

We start with the case where the initial electron transfer reaction is fast enough not to interfere kinetically in the electrochemical response.1 Under these conditions, the follow-up reaction is the only possible rate-limiting factor other than diffusion. The electrochemical response is a function of two parameters, the first-order (or pseudo-first-order) equilibrium constant, K, and a dimensionless kinetic parameter, 2, that measures the competition between chemical reaction and diffusion. In cyclic voltammetry,... 

Let us consider the possible events following excitation of an acid AH that is stronger in the excited state than in the ground state (pK < pK). In the simplest case, where there is no geminate proton recombination, the processes are presented in Scheme 4.6, where t0 and Tq are the excited-state lifetimes of the acidic (AH ) and basic (A- ) forms, respectively, and ki and k i are the rate constants for deprotonation and reprotonation, respectively, kj is a pseudo-first order rate constant, whereas k i is a second-order rate constant. The excited-state equilibrium constant is K = k /k 7 ... 

Dankwerts and McNeil ( 3) have employed the method of Van Krevelen et al. to predict the partial pressure of carbon dioxide over carbonated alkanolamine solutions. The central feature of this model is the use of pseudo-equilibrium constants and their dependence on ionic strength. The ratio of the pseudo-equilibrium constant at a certain ionic strength to that at zero ionic strength has been termed the "ionic characterization factor". However, ionic strength alone is insufficient to determine the ionic characterization factors. As well the ionic characterization factors are sometimes not a simple linear function of ionic strength. 

Kent and Eisenberg (5) also correlated solubility data in the system S+CC +alkanoleimines+ O using pseudo-equilibrium constants based on molarity. Instead of using ionic characterization factors, they accepted published values of all but two pseudoequilibrium constants and found these by fitting data for MEA and DEA solutions. They were able to obtain excellent fits by this approach and also discovered that the fitted pseudo-equilibrium constants showed an Arrhenius dependence on temperature. 

The most straightforward of the various models describing micellar kinetics is the Menger-Portnoy model for (pseudo) unimolecular reactions.The Menger-Portnoy model assumes rapid equilibration of the reactant of interest over bulk water and the micellar pseudophase with equilibrium constant K. The reaction then proceeds in both pseudophases with rate constants and in bulk water and the micellar pseudophase, respectively (Scheme 4). 

The Henry s law and equilibrium constants are from Sillen and Martell, 1964.) In such cases, the Henry s law constant reflects only the physical solubility (i.e., reaction (1, -1). A pseudo-Henry s law constant, H, is often defined to take into account the increased uptake compared to that expected based on simple dissolution of the gas without further reaction. With C02 as an example, this pseudo-Henry s law constant is defined by Eq. (A) ... 

Ion-exchange equilibrium can be considered to be analogous to chemical equilibrium. From that point of view, the mass-action law can be used to express the state of equilibrium despite the fact that this law is defined exclusively for homogeneous systems. Derived this way, the so-called pseudo-equilibrium constant Ke is not really a constant, since it depends on the total concentration ... 

However, if the valences of the exchanging cations are equal, the selectivity coefficient or pseudo-equilibrium constant is not affected by concentration. As already mentioned, one isotherm corresponds to a specific temperature in the case of adsorption or ion exchange of equal valence ions, whereas additionally, the same normality is required for the existence of only one isotherm in the case of ion exchange of different valence ions, due to the concentration-valence effect (Helfferich, 1962). The determination of the true equilibrium constant should be based on the thermodynamic activities (activity coefficients) of the species rather than concentrations. It is clear that the difficulties in the determination of activity coefficients also complicate the determination of the true equilibrium constant (Culfaz and Yagiz, 2004). 

A number of studies of H-atom transfer from hydrogen halides to free radicals, R + HX - RH + X, have been done by FPTRMS in which R was detected by photoionization, and its decay was monitored as a function of [HX] under pseudo-first-order conditions. When the rate coefficient is combined with determinations of the rate coefficient of the reverse reaction to obtain the equilibrium constant, the enthalpy of formation of the radical can be deduced. If the kinetics are accurately measured in isolation, this is a direct kinetic method which can be used to confirm (or otherwise) thermodynamic data obtained by classical, indirect kinetic methods which depend on correct mechanistic interpretation. In a number of instances free radical enthalpies of formation by these two different approaches have not been in good agreement. It is not the purpose of this short survey to discuss the differences, but rather to briefly indicate the extent to which the FPTRMS method has contributed to the kinetics of these reactions and to free radical thermochemistry. 

If the adsorption of A is the rate determining step in the sequence of adsorption, surface reaction and desorption processes, then equation 3.71 will be the appropriate equation to use for expressing the overall chemical rate. To be of use, however, it is first necessary to express CA, Cv and Cs in terms of the partial pressures of reactants and products. To do this an approximation is made it is assumed that all processes except the adsorption of A are at equilibrium. Thus the processes involving B and P are in a state of pseudo-equilibrium. The surface concentration of B can therefore be expressed in terms of an equilibrium constant KB for the adsorption-desorption equilibrium of B ... 

The Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, f3. The reaction isdd... 

In buffered solutions, the term k2KsKi [S]/[SH+] is constant, so the expected overall rate law is again second order (i.e. pseudo first order in [Y ]) but the correspondence of fcQbs with mechanistic rate constants is different. Of course, if the equilibrium constant Ki is appreciable, the phenolate concentration must be taken into account in the mass balance for the total phenol, i.e. [ArOH]T [ArOH]free + [ArOH- -S] + [ArO-], whereupon the mechanistic rate equation becomes more complicated. 

Equation (3.243) shows that for pseudo-first-order catalytic mechanism, the normalized current varies linearly with the radius for any value of the applied potential and the term k + k2) can be calculated from the slope provided the equilibrium constant and the diffusion coefficients are known. 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

The stoichiometric rather than the equilibrium concentrations of A and B are often known, unless one can measure cA and cB directly or the equilibrium constant for the reaction is known independently (Bernasconi, 1976). Thus, Eq. (4.18) may have little utility. However, there are several ways to increase its usefulness (Bernasconi, 1976). One can maintain the concentration of one of the reactants quasi-constant by having it in large excess over the other reactant (pseudo-first-order conditions). 

A transition state theory formalism (Equation 9.10) was used to calculate the lsO KIE from the reaction coordinate frequency (rrc), defined as the mode that converts the 0—0 vibration into a translation, and the pseudo-equilibrium constant for forming the transition state (KTs). In the expression for the latter (Equation 9.11), the one vibration that becomes the reaction coordinate has been removed. 

In unier tu cuklcuuy prevent proteolysis, the inhibitor should fulfill at least two requirements (1) inhibit the target enzyme in an irreversible or pseudo-irreversible manner and (2) suppress the enzyme activity before significant proteolysis has occurred. From a kinetic viewpoint, the classical example of vn irreversible protein inhibitor of NE is ai-PI. Proteolysis can also be prevented by reversible inhibitors if the in vivo inhibitor concentration is much greater than Kj, the equilibrium constant far inhibition (Ujvta >>Kj), resulting in a pseudo-irreversible behavior. 

A two-site model was used as well by Shih et al. (102) for the HDN of quinoline. They assumed not only pseudo-first-order kinetics but also that all nitrogen-containing species, including ammonia, have the same adsorption equilibrium constant this is clearly not the case (77). 

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