Pseudo-first-order rate equation

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

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. 

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... 

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... 

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. 

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 ... 

The overall study showed that the rate of reaction has a first order dependence on both hydrogen concentration and total [Ir] (up to certain limiting hydrogen and catalyst concentrations) and an inverse dependence on the nitrile concentration. The observed kinetic dependence of the pseudo first order rate constant (k ) for the hydrogenation of C=C in NBR may be summarized by the expression show in Equation (1). 

If a plot of the left side of equation C versus time is linear, the reaction is first order, and the slope of this plot is equal to — k. A plot of the data indicates that the reaction is indeed first order with respect to benzoyl chloride. From the slope of the line the pseudo first-order rate constant is found to be 4.3 x 10 3sec 1. 

Kinetic theory indicates that equation (32) should apply to this mechanism. Since the extent of protonation as well as the rate constant will vary with the acidity, the sum of protonated and unprotonated substrate concentrations, (Cs + Csh+), must be used. The observed reaction rate will be pseudo-first-order, rate constant k, since the acid medium is in vast excess compared to the substrate. The medium-independent rate constant is k(), and the activity coefficient of the transition state, /, has to be included to allow equation of concentrations and activities.145 We can use the antilogarithmic definition of h0 in equation (33) and the definition of Ksh+ in equation (34) ... 

In the simplest case a 1 1 complex is formed between the host in solution and the guest immobilized on the surface. The response of the SPR sensor, R, is proportional to the concentration of the complex formed, and thus pseudo-first-order rate equations can be used to analyze the data.73 If no host is initially bound the function for R... 

What is the value of the pseudo first-order rate constant k l We calculate the pseudo first-order rate constant k by assuming that the reaction obeys first-order kinetics. Accordingly, we write from Equation (8.24) ... 

The pseudo first-order rate constants, obs, are obtained by least-squares fits of the measured peak height increase of the relevant coalesced oxygen-17 signal as a function of time to the modified McKay equation (7, 67, 68). The kinetics can be studied by manipulation of... 

In a typical experiment, the sample is a solution (e.g., in benzene) of both the ferf-butoxyl radical precursor (di-tert-butylperoxide) and the substrate (phenol). The phenol concentration is defined by the time constraint referred to before. The net reaction must be complete much faster than the intrinsic response of the microphone. Because reaction 13.23 is, in practical terms, instantaneous, that requirement will fall only on reaction 13.24. The time scale of this reaction can be quantified by its lifetime rr, which is related to its pseudo-first-order rate constant k [PhOH] and can be set by choosing an adequate concentration of phenol, according to equation 13.25 ... 

Various rate constants which enter into the expression for k i, Equation 14.14, have now been discussed. kunj as defined in Equation 14.13 has the appearance of a first order rate constant for the disappearance of A molecules but it is actually only a pseudo first order rate constant since it explicitly depends on the concentration of M, the species involved in the activation and deactivation of A molecules. In the limit of high concentration, [M] oo, kuni reduces to an apparent first order process, lim (kuni j oo) = ka(E)(8ki(E)/k2)[A] = kt(Apparent)[A], while at low concentration the reduction is to an apparent second order process, lim(kunij[M]->.o) = 8k (E)[A][M] = k2 (Apparent) [A] [M],... 

Fig. 2 Effect of added NaBr on the pseudo-first-order-rate constants, A obsd. for solvolysis of 1-Br in water at 25 °C and 7 = 6.00 (NaC104). The inset shows the linear replot of the data according to equation (3 A) of the text. [Reprinted with permission of the American Chemical Society from Ref. 32].

The data are then plotted as a function of cumulative MTS exposure and fit with an exponential equation to obtain the pseudo first-order rate (ki) and plateau. The plateau achieved with lower MTS concentrations should approach that obtained with the application (2 min) of a high MTS concentration. 

In which kg Is the pseudo-first-order rate constant for OP hydrolysis In the absence of PB. With PB In great excess. In buffered solution, kg and [HO2 ] are constant. Then pseudo-first-order kinetics result and we get equation (8),... 

In the first type of study, pseudo first-order kinetics were observed in both the sediment and aqueous phases from t=0 through two half-lives in overall chlorpyrifos disappearance (total time -8 days). For these studies, computer calculations using the model illustrated in equations 7 were again used to calculate values for kj, k and kg, assuming a value of k equal to the pseudo first-order rate constant in distilled water buffered to the same pH. Values were also calculated for Obfi assuming kg 0 (equation 10) for comparison to the experimental kg values. The results of these calculations are shown in Table VII. 

Ho + log[H+]) + pW and log([SH+]/[S]) + Ho = cf>(Ho + log[H"]) + p/i sH Here, represents the response of the S + H+ SH+ equilibrium to changes in the acid concentration. A corresponding equation can be written for kinetic data log k + Ho = (f>(Ho + log[H+]) -i-log 2° where is the pseudo-first-order rate constant for the reaction of a weak base in an acidic solution and ki° is the second-order rate constant at infinite dilution in water. In this case, 4> represents the response of the reaction rate to changing acid concentrations. See also Acidity Function... 

The reactions obeyed pseudo-first-order kinetics consistent with a rapid reversible protonation of the substrate, S, at the ester carbonyl followed by a rate-determining decomposition to acetic acid and nitrenium ion according to Scheme 19. In accordance with equation 13, the pseudo-first-order rate constant, k, was shown to be proportional to acid concentration and inversely proportional to the activity of the water/acetonitrile solvent . 

The optical rotation of the mixture approaches zero (a racemic mixture) over time, with apparent first-order kinetics. This observation was supported by the semi-log plot [ln(a°D/ aD) vs time], which is linear (Figure 1). It has been shown that this racemization process does in fact follow a true pseudo-first-order rate equation, the details of which have been described by Eliel.t30 Therefore, these processes can be described by the first-order rate constant associated with them, which reflects precisely the intrinsic rate of racemization. Comparison of the half-lives for racemization under conditions of varying amino acid side chain, base, and solvent is the basis for this new general method. 

The pseudo-first-order rate constants for decay of the transients were dependent on the concentrations of added alcohols or H2O according to the equation A obs = + nuc[ROH], with in the range... 

If we assume that the activity coefficients of X- and H20 are independent of the X- concentration at any given ionic strength, then the usual steady state treatment leads, without further approximation, to Equation 3, a relationship between the pseudo first-order rate constant and the other kinetic parameters. 

In Equation 7 the symbols kf and k are used to represent the pseudo first-order rate constant for the I3 " system, the pseudo first-order rate constant for the 0.5Af solution not containing I8 Inspection of Equation 8, obtained by rearranging Equation 7, shows that kz/kb[k2/kz + (I )] may be evaluated from the ratio of slope to intercept in a plot of /(kf — k) vs. l/(I3—). 

Two alternate S rl reactions must be considered for the reaction in basic solution. The first alternative assumes that Co(CN)5OH 3 is completely unreactive and that at any given pH, Nz reacts only with that fraction of the complex present as Co(CN)5OH2-2. In this formulation the pH and azide ion dependence of the pseudo first-order rate constant is given by Equation 10. 

The potentially greater toxicity of peroxynitrite can be readily visualized by comparing the mean diffusion distances that various nitrogen and oxygen-centered species may traverse in one lifetime. The definition of lifetime (t) is the time required for 67% of the initial concentration to decompose, and is readily calculated as the reciprocal of the pseudo-first-order rate constant for the disappearance of the species in question. Distances were calculated from the following equation, which is readily derived from the Fick s laws of diffusion (Nobel, 1983 Pryor, 1992). 

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