Equations pseudo-first-order

Herein k js is the observed pseudo-first-order rate constant. In the presence of micelles, analogous treatment of the experimental data will only provide an apparent second-order rate constant, which is a weighed average of the second-order rate constants in the micellar pseudophase and in the aqueous phase (Equation 5.2). 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

The value for the pseudo-first-order rate constant is determined by solving equation 13.6 for k and making appropriate substitutions thus... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

We know from equation 13.6 that for a pseudo-first-order reaction, the concentration of picrate at time t is... 

Equation 13.14 shows how [A]o is determined for a two-point fixed-time integral method in which the concentration of A for the pseudo-first-order reaction... 

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... 

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. 

The numerical solution of these equations is shown in Fig. 23-28. This is a plot of the enhancement fac tor E against the Hatta number, with several other parameters. The factor E represents an enhancement of the rate of transfer of A caused by the reaction compared with physical absorption with zero concentration of A in the liquid. The uppermost line on the upper right represents the pseudo-first-order reaction, for which E = P coth p. 

The unit of the veloeity eonstant k is see Many reaetions follow first order kineties or pseudo-first order kineties over eertain ranges of experimental eonditions. Examples are the eraeking of butane, many pyrolysis reaetions, the deeomposition of nitrogen pentoxide (NjOj), and the radioaetive disintegration of unstable nuelei. Instead of the veloeity eonstant, a quantity referred to as the half-life iyj is often used. The half-life is the time required for the eoneentration of the reaetant to drop to one-half of its initial value. Substitution of the appropriate numerieal values into Equation 3-33 gives... 

If there is exeessively large aldehyde, its eoneentration remains eonstant and the reaetion ean be referred to as pseudo-first order kineties. Equation 3-192 ean be rewritten as... 

The isolation technique showed that the reaction is first-order with respect to cin-namoylimidazole, but treatment of the pseudo-first-order rate constants revealed that the reaction is not first-order in amine, because the ratio k Jc is not constant, as shown in Table 2-2. The last column in Table 2-2 indicates that a reasonable constant is obtained by dividing by the square of the amine concentration hence the reaction is second-order in amine. For the system described in Table 2-2, we therefore find that the reaction is overall third-order, with the rate equation... 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

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

This consists of two consecutive irreversible first-order (or pseudo-first-order) reactions. The differential rate equations are... 

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

As the second step in Scheme 5-1 is much faster than the first, the observed pseudo-first-order rate constant ( obs) is related to kx in Scheme 5-1 as described by Scheme 5-7. The second term in this equation arises from the fact that k x cannot... 

This peculiar form applies when a dimeric molecule dissociates to a reactive monomer that then undergoes a first-order or pseudo-first-order reaction. This scheme is considered in Section 4.3. Unless one can work at either of the limits, the form is such that a numerical solution or the method of initial rates will be needed, since the integrated equation has no solution for [A]r. 

Thus, we may write the pseudo first-order rate constant for disappearance of CD4 as n(CH5 iD -+) = 4.40 X 10 4 sec.-1 Appropriate rate equations are... 

Example 11.12 Solve Equations (11.31) and (11.32) for the simple case of constant parameters and a pseudo-first-order reaction occurring in the liquid phase of a component supplied from the gas phase. The gas-phase film resistance is negligible. The inlet concentration of the reactive component is... 

Since MeOH was used in excess amount compared to EC, Cm can be assumed constant during the reaction. Therefore, the reaction rate equation can be written as a pseudo first order with respect to the concentration of EC. 

In mild alkaline conditions, highly methylated pectin was demethylated following a (pseudo)-first order kinetics with respect to the concentration of methoxylated galacturonate moieties. Investigation in this pH range, where the initial concentration of methylesters was higher than the initial concentration of OH ions, was complicated by the necessity to use a buffer. This led to deviations from the theoretical behavior as the concentration of OH ions still varied in proportions which could not be neglected in the equations of the kinetics. However these deviations could be accounted for be the pH variation, and the pH variation itself predicted from the amount of liberated methanol. The constant we found was similar to previously reported data (Scamparini Bobbio, 1982). 

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... 

Pseudo-first order kinetics was assumed to interpret the experimental data. Fig. 5.4-36 shows that the fit of the experimental data using first order-kinetics is acceptable. However, a systematic deviation is observed in the curve obtained at 110 C. This indicates inadequacy of the first-order kinetics, which is inappropriate from the view of theory. On the other hand, the kinetic equation seems to describe the... 

The absorbance data enabled the determination of extraction rate constants. For a pseudo-first-order reaction, the following equation describes the extraction process ... 

Using the pseudo-first-order equation A obsd = 0 + co2 [COiKwhere kcoi is the second-order rate constant for the reaction of carbene with CO2 and ko is the rate of carbene decay in the absence of CO2), solution-phase values of kcoi for phenylchlorocarbenes 9 and 12, and diphenylcarbenes 14 and 15 in dichloromethane were estimated (Table 4.1). (The concentration of CO2 in saturated dichloromethane solution at 25°C and 1 atm is 196mmol/L. ) The trend of these estimated second-order order rate constants agrees with that observed in low-temperature matrices by Sander and co-workers. ... 

The successive equilibria are characterized by K12 and K23, respectively, and when Kl2 (often denoted K0) cannot be directly determined, it may be estimated from the Fuoss equation (3), where R is the distance of closest approach of M2+ and 1/ (considered as spherical species) in M OH2 Um x) +, e is the solvent dielectric constant, and zM and zL are the charges of Mm+ and Lx, respectively (20). Frequently, it is only possible to characterize kinetically the second equilibrium of Eq. (2), and the overall equilibrium is then expressed as in Eq. (4) (which is a general expression irrespective of mechanism). Here, the pseudo first-order rate constant for the approach to equilibrium, koba, is given by Eq. (5), in which the first and second terms equate to k( and kh, respectively, when [Lx ] is in great excess over [Mm+]. When K0[LX ] <11, koba - k,K0[Lx ] + k.it and when K0[LX ] > 1, fc0bs + k l. Analogous expressions apply when [Mm+] is in excess. 

Since all the kinetic characteristics of the disappearance of a drug from plasma are the same as those for the pseudo-first-order disappearance of a substance from a solution by hydrolysis, the same working equations [Eqs. (11) and (13)] and the same approach to solving problems can be used. 

If one were to mix fixed concentrations E and / at time zero and then measure the concentration of El complex as a function of time after mixing, the data would appear to be described by the pseudo-first-order rate equation ... 

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