Pseudo-first order conditions

Under pseudo first-order conditions (excess 2.5) k is given by d[2.Al... 

The concentration of aluminum in serum can be determined by adding 2-hydroxy-1-naphthaldehyde p-methoxybenzoyl-hydrazone and measuring the initial rate of the resulting complexation reaction under pseudo-first-order conditions.The rate of reaction is monitored by the fluorescence of the metal-ligand complex. Initial rates, with units of emission intensity per second, were measured for a set of standard solutions, yielding the following results... 

Flooding and Pseudo-First-Order Conditions For an example, consider a reaction that is independent of product concentrations and has three reagents. If a large excess of [BJ and [CJ are used, and the disappearance of a lesser amount of A is measured, such flooding of the system with all components butM permits the rate law to be integrated with the assumption that all concentrations are constant except A. Consequentiy, simple expressions are derived for the time variation of A. Under flooding conditions and using equation 8, if x happens to be 1, the time-dependent concentration... 

Using pseudo-first order conditions with [SjO I ] = 1.8x10 and [Fe(CN)g ] = 6.5x10 M, the following absorbances were recorded at 25°C ... 

Table 2-4 gives data for the alkaline hydrolysis of phenyl cinnamate under pseudo-first-order conditions, with calculations made in order to apply the Guggenheim method. The plot according to Eq. (2-55) is shown in Fig. 2-9. From the slope the pseudo-first-order rate constant is 3.37 x 10 s . ... 

If pseudo-first-order conditions can be established, for example, by setting Cb Ca, then Scheme II collapses to Scheme III,... 

If pseudo-first-order conditions do not apply, the Scheme II rate equation is... 

Scheme VII constitutes an equivalent system if the concentration of reagent R is much larger than the reactant concentrations, so that we have pseudo-first-order conditions.

Any combination of first-order reactions can be simulated by extension of this procedure. Reversible reactions add only the feature that reacted species can be regenerated from their products. Second-order reactions introduce a new factor, for now two molecules must each be independently selected in order that reaction occur in the real situation the two molecules are in independent motion, and their collision must take place to cause reaction. We load the appropriate numbers of molecules into each of two grids. Now randomly select from the first grid, and then, separately, randomly select from the second grid. If in both selections a molecule exists at the respective selected sites, then reaction occurs and both are crossed out if only one of the two selections results in selection of a molecule, no reaction occurs. (Of course, if pseudo-first-order conditions apply, a second-order reaction can be handled just as is a first-order reaction.)... 

Considering the attention that we have given in this chapter to concentrationtime curves of complex reactions, it may seem remarkable that many kinetic studies never generate a comprehensive set of complicated concentration-time data. The reason for this is that complex reactions often can be studied under simplified conditions constituting important special cases for example, whenever feasible one chooses pseudo-first-order conditions, and then one studies the dependence of the pseudo-first-order rate constant on variables other than time. This approach is amplified below. 

First-order and second-order rate constants have different dimensions and cannot be directly compared, so the following interpretation is made. The ratio intra/ inter has the units mole per liter and is the molar concentration of reagent Y in Eq. (7-72) that would be required for the intermolecular reaction to proceed (under pseudo-first-order conditions) as fast as the intramolecular reaction. This ratio is called the effective molarity (EM) thus EM = An example is the nu-... 

In some cases the alkoxide ions have been used in large excess under pseudo-first-order conditions. ... 

An interesting kinetic study was carried out under pseudo-first-order conditions for the base hydrolysis of the three isomeric N-methyl-cyanopyridinium salts, a reaction that leads partly to CN replacement and partly to the formation of a carboxamido derivative. ... 

Reactions with uncharged species such as amines, alcohols, and water offer frequent opportunities for investigations under pseudo-first-order conditions since many of these reagents are suitable solvents. However, the reactions with amines have often been investigated in alcohols and in non-hydroxylic solvents 27-29a have been found to follow second-order kinetics. 

Kinetic measurements were performed on a Hitachi 150-20 UV/VIS spectrophotometer. Dehydrobrominations were studied in DMF solution using cyclohexyl amine (CHA) as the base. Applied CHA concentrations were 2, 2.5, 3, 3.5, 4 and 5 10 3 mole.dm-3, initial concentration of 1 was 5 10 5 mole.dm-3 in every case (pseudo-first-order conditions). Ionic strength was adjusted to lO l mole.dm 3 with potassium nitrate. Kinetic curves / D(t) / were recorded at fix wavelength, X = 290 ran and the temperature was maintained at 30, 35.5, 40°C. Stock solutions were made daily for la and freshly for every measurement of Ih. The reaction was started by injection of solution of 1 to the thermostated solution of CHA. 

Kinetic studies of the hydride cluster [W3S4H3(dmpe)3] with acids in a non-coordinating solvent, i.e., dichloromethane, under the pseudo-first-order condition of acid excess, show a completely different mechanism with three kineti-cally distinguishable steps associated to the successive formal substitution of the coordinated hydrides by the anion of the acid, i.e., Ch in HCl [37]. The first two kinetic steps show a second-order dependence with the acid concentration. 

Figure A1.5 Concentration of [El] (A) and of free inhibitor, [/]f, (B) as a function of time for a binding reaction run under pseudo—first-order conditions.

In potentiometry, the variation of the potential of a Pt electrode relative to a calomel reference electrode represents the time-dependent bromine concentration. Available [Br2] is about 2 x 10-5-10-4 m pseudo-first-order conditions ([Ol] [Br2]) have to be used. Rate constants up to 104 5 m- 1 s 1 can thus be obtained (Atkinson and Bell, 1963 Dubois et ai, 1968). 

Activation rate constants (k) in ATRP/ATRA are typically determined from model studies in which copper complex is reacted with alkyl halide in the presence of radical trapping agents such as TEMPO [127,128,129], Rates are determined by monitoring the rate of disappearance of alkyl halide in the presence of large excess of the activator (Cu X/L) and TEMPO. Under such pseudo-first order conditions, the activation rate constant can be calculated ln([RX]0/[RX]() vs.t plots (slope =-k) Cu C/... 

To circumvent complex mathematical evaluations, stopped flow experiments are measured preferentially under pseudo first-order conditions. The rate of complex formation between protein A and protein or ligand B is described by... 

As described for stopped flow experiments above, all commercially available SPR systems work under (pseudo) first-order conditions as well. This is realized either by a large excess of free ligand (in the large volume of the cuvette) compared with a nanoliter volume of the sensor layer [156] or by continuous replacement of free ligand in a flow injection system (e.g.,BIAcore [157]). 

The quantity and quality of experimental information determined by the new techniques call for the use of comprehensive data treatment and evaluation methods. In earlier literature, quite often kinetic studies were simplified by using pseudo-first-order conditions, the steady-state approach or initial rate methods. In some cases, these simplifications were fully justified but sometimes the approximations led to distorted results. Autoxidation reactions are particularly vulnerable to this problem because of strong kinetic coupling between the individual steps and feed-back reactions. It was demonstrated in many cases, that these reactions are very sensitive to the conditions applied and their kinetic profiles and stoichiometries may be significantly altered by changing the pH, the absolute concentrations and concentration ratios of the reactants, and also by the presence of trace amounts of impurities which may act either as catalysts and/or inhibitors. 

By lifting the simplifying restrictions, the kinetic observations can be examined in more detail over much wider concentration ranges of the reactants than those relevant to pseudo-first-order conditions. It should be added that sometimes a composite kinetic trace is more revealing with respect to the mechanism than the conventional concentration and pH dependencies of the pseudo-first-order rate constants. Simultaneous evaluation of the kinetic curves obtained with different experimental methods, and recorded under different conditions, is based on fitting the proposed kinetic models directly to the primary data. This method yields more accurate estimates for the rate constants than conventional procedures. Such an approach has been used sporadically in previous studies, but it is expected to be applied more widely and gain significance in the near future. 

The rate of a reaction and its dependency on the concentrations of the reactants can be measured in several ways. A simple method involves the measurement of the rate at zero to low conversion at different concentrations of one of the substrates, keeping the concentration of other substrates constant. The latter can be done by using an excess of the other substrates (e.g. tenfold excess), which means that we can assume that the concentrations of the latter ones are constant under so-called pseudo-first-order conditions. Secondly we can monitor the reaction rate over a longer period of time taking into account the change in concentration for this one substrate. Alternatively, one can monitor the concentrations of all species and analyse the results numerically. 

Another example in which a pseudo-first-order condition can arise in evaluating experimental data is the case in which one of the reactants (generally the oxidizer in a combustion system) is in large excess. Consider the arbitrary process... 

Second-order reactions between two dissimilar molecules A and B are invariably studied under pseudo first-order conditions (Sec. 1.4.4) because this is by far the simpler procedure. If however this condition cannot be used because, for example, both reactants absorb heavily... 

When a = b = 1 in (1.42) the overall reaction is second-order. Even a quite small excess of one reagent (here B) can be used and pseudo first-order conditions will still pertain. As the reaction proceeds, the ratio of concentration of the excess to that of the deficient reagent progressively increases so that towards the end of the reaction, pseudo first-order conditions certainly hold. Even if [B] is maintained in only a two-fold excess over [A], the error in the computed second-order rate constant is 2% for 60% conversion. ... 

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