First-order dependence

First-order reactions are extremely common and form the bulk of reported kinetic studies. The rate of loss of the reactant A decreases as the concentration of A decreases. The differential form (1.18) leads to a number of equivalent integrated expressions, in addition to (1.19) and (1.19 a)  

Important quantities characteristic of a first-order reaction are the half-life of the reaction, which is the value of t when [A], = [A](,/2, and t, the relaxation time, or mean lifetime, defined as k.  

The latter is invariably used in the relaxation or photochemical approach to rate measurement (Sec. 1.8), rmd is the time taken for A to fall to 1/e (1/2.718) of its initial value. Half-lives or relaxation times are eonstants over the complete reaction for first-order or pseudo first-order reactions. The loss of reactant A with time may be described by a single exponential but yet may hide two or more concurrent first-order and/or pseudo first-order reactions. 

The change in the absorbance at 450 nm when eis-Ni([14]aneN4)(H20) + (for structure see Ligands and Complexes ) is plunged into 1.0 M HCIO4 is first-order (rate constant = k). It is compatible with concurrent isomerization and hydrolysis (1.29) 

Estimation of the amounts of 3 and 4 produced from the final absorbance change allows 

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When the concentration of 2-phenylethanesulphonate anion was >0-5 mol 1, or when 2-mesitylethanesulphonate anion (v), " mesitylene-a-sulphonate anion, or iso-durene-a -sulphonate anion were nitrated, the initial part of the reaction deviated from a first-order dependence on the concentration of the aromatic towards a zeroth-order dependence. 

Unlike the effect of sulphuric acid upon nitration in nitric acid ( 2.2.3 where zeroth-order reactions are unknown), the form of the catalysis of zeroth-order nitration in nitromethane by added sulphuric acid does not deviate from a first-order dependence with low concentrations of catalyst. ... 

Note that this expression is equivalent to the barometric formula which gives the variation of atmospheric pressure ( c) with elevation (oc r). A first-order dependence on the distance variable holds in the barometric equation, since the acceleration is constant in this case. 

The kinetics of initiation reactions of alkyllithium compounds often exhibit fractional kinetic order dependence on the total concentration of initiator as shown in Table 2. For example, the kinetics of the initiation reaction of //-butyUithium with styrene monomer in benzene exhibit a first-order dependence on styrene concentration and a one-sixth order dependence on //-butyUithium concentration as shown in equation 13, where is the rate constant for... 

The first detailed investigation of the reaction kinetics was reported in 1984 (68). The reaction of bis(pentachlorophenyl) oxalate [1173-75-7] (PCPO) and hydrogen peroxide cataly2ed by sodium saUcylate in chlorobenzene produced chemiluminescence from diphenylamine (DPA) as a simple time—intensity profile from which a chemiluminescence decay rate constant could be determined. These studies demonstrated a first-order dependence for both PCPO and hydrogen peroxide and a zero-order dependence on the fluorescer in accord with an earher study (9). Furthermore, the chemiluminescence quantum efficiencies Qc) are dependent on the ease of oxidation of the fluorescer, an unstable, short-hved intermediate (r = 0.5 /is) serves as the chemical activator, and such a short-hved species "is not consistent with attempts to identify a relatively stable dioxetane as the intermediate" (68). 

Molecular bromine is believed to be the reactive brominating agent in uncatalyzed brominations. The brominations of benzene and toluene are first-order in both bromine and the aromatic substrate in trifluoroacetic acid solution, but the rate expressions become more complicated when these reactions take place in the presence of water. " The bromination of benzene in aqueous acetic acid exhibits a first-order dependence on bromine concentration when bromide ion is present. The observed rate is dependent on bromide ion concentration, decreasing with increasing bromide ion concentration. The detailed kinetics are consistent with a rate-determining formation of the n-complex when bromide ion concentration is low, but with a shift to reversible formation of the n-complex... 

This mechanism explains the first order dependence of the reaction. 

Oxidation kinetics over platinum proceeds at a negative first order at high concentrations of CO, and reverts to a first-order dependency at very low concentrations. As the CO concentration falls towards the center of a porous catalyst, the rate of reaction increases in a reciprocal fashion, so that the effectiveness factor may be greater than one. This effectiveness factor has been discussed by Roberts and Satterfield (106), and in a paper to be published by Wei and Becker. A reversal of the conventional wisdom is sometimes warranted. When the reaction kinetics has a negative order, and when the catalyst poisons are deposited in a thin layer near the surface, the optimum distribution of active catalytic material is away from the surface to form an egg yolk catalyst. 

Schmid (1936 a) was the first to observe a third-order reaction in the diazotization of aromatic amines in the presence of sulfuric acid, and he proposed the kinetic equation of Scheme 3-3. In subsequent work (1936b, 1937 Schmid and Muhr, 1937), he investigated the course of the reaction in dilute hydrochloric or hydrobromic acid, which could be described by incorporating an extra term for the halide ion with only a first-order dependence on (HNO2), as in Scheme 3-4. 

The first kinetic study appears to have been that of Martinsen148, who found that the sulphonation of 4-nitrotoluene in 99.4-100.54 wt. % sulphuric acid was first-order in aromatic and apparently zeroth-order in sulphur trioxide, the rate being very susceptible to the water concentration. By contrast, Ioffe149 considered the reaction to be first-order in both aromatic and sulphur trioxide, but the experimental data of both workers was inconclusive. The first-order dependence upon aromatic concentration was confirmed by Pinnow150, who determined the equilibrium concentrations of quinol and quinolsulphonic acid after reacting mixtures of these with 40-70 wt. % sulphuric acid at temperatures between 50 and 100 °C the first-order rate coefficients for sulphonation and desulphonation are given in Tables 34 and 35. The logarithms of the rate coefficients for sulphonation... 

The kinetic data based on the demonstration of specific acid catalysis in buffers, solvent isotope effects and acidity functions all support mechanisms where the proton-transfers are fast. It is possible to write equations which accommodate these facts together with the first-order dependence on hydrazo-compound and the concurrent first and second-order dependence on acidity. These are... 

White et al.1A have obtained similar kinetic results for the acid-catalysed rearrangement of N-nitro-N-methylaniline, i.e. a first-order dependence on the nitroamine with a linear H0 plot of slope 1.19 for phosphoric acid, and a deuterium solvent isotope effect of about three, although the results have only been presented in preliminary form. Further, an excellent Hammett a+ correlation was claimed for thirteen para substituted nitroamines which gave a p value of —3.9. Since it is expected that the rate coefficients would correlate with a (rather than different basicities of the amines, the a+ correlation implies that the amino nitrogen is electron-deficient in the transition state,... 

The rates of many catalyzed reactions depend upon substrate concentrations, as shown in Fig. 4-7. The rate at high substrate concentrations is zeroth-order with respect to [S], falling until it shows a first-order dependence in the limit of low [S], This pattern is that of a rectangular hyperbola, defined by an empirical relation known as the Michaelis-Menten equation. 

The oxidation of mercurous ions by thallium (3+) ion shows an inverse first-order dependence on [Hg2+], which means that one Hg2+ ion must be subtracted out in figuring the composition.3 Thus, we have... 

Owing to its carboxyl group, aspirin exists as a neutral species, AH, at low pH and an ionized one, A-, at high pH. Its pKu is 3.38. In region 1, where AH predominates, the first-order dependence on the H+ concentration suggests the pathway shown, with 105 = k ... 

The actual result is this The chains are broken when Cu24 is added. The reaction slows considerably but does not come to a halt. Kinetic studies show that the order with respect to [RM] drops from f to 1 with Cu2+ added and that the order with respect to [02] rises from 0 to 1. One term shows an inverse first-order dependence on [Cu2+], The rate under these conditions became... 

When initiators are used, the initiation rate has a first-order dependence on monomer concentration. 

Littler has studied the oxidation of cyclohexanone with lead(IV), thallium(III), and mercury(II) salts (84), and found that, with all three reagents the rates of oxidation are independent of the concentration of oxidant. Oxidation by thallium(III) and mercury(II) in 35% aqueous perchloric acid showed first-order dependence on [H" ], and Littler suggested that the results were best interpreted in terms of the reaction sequence shown in Scheme 27. The major product of thallium(III) oxidation of... 

The rate showed a first-order dependence on the concentrations of both hydroxide and the cobalt complex, and Hogenkamp 89) suggested that the initial step was the removal of a proton from the /3-carbon, followed by the elimination of Co(I). Schrauzer, Weber, and Beckham 159) were subsequently able to show the formation of 7r-olefin-Co(I) complexes in... 

The most limiting factor for enzymatic PAC production is the inactivation of PDC by the toxic substrate benzaldehyde. The rate of PDC deactivation follows a first order dependency on benzaldehyde concentration and reaction time [8]. Various strategies have been developed to minimize PDC exposure to benzaldehyde including fed-batch operation, immobilization of PDC for continuous operation and more recently an enzymatic aqueous/octanol two-phase process [5,9,10] in which benzaldehyde is continuously fed from the octanol to the enzyme in the aqueous phase. The present study aims at optimal feeding of benzaldehyde in an aqueous batch system. 

A model developed by Leksawasdi et al. [11,12] for the enzymatic production of PAC (P) from benzaldehyde (B) and pyruvate (A) in an aqueous phase system is based on equations given in Figure 2. The model also includes the production of by-products acetaldehyde (Q) and acetoin (R). The rate of deactivation of PDC (E) was shown to exhibit a first order dependency on benzaldehyde concentration and exposure time as well as an initial time lag [8]. Following detailed kinetic studies, the model including the equation for enzyme deactivation was shown to provide acceptable fitting of the kinetic data for the ranges 50-150 mM benzaldehyde, 60-180 mM pyruvate and 1.1-3.4 U mf PDC carboligase activity [10]. 

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