Kinetics first-order rate laws

Case 2. The particles rotate in small packets ( coherently or in phase ). Obviously, the first-order rate law no longer holds. In chapter B2.1 we shall see that this simple consideration has found a deeper meaning in some of the most recent kinetic investigations [21]. 

The isomerization of A to B yielded kinetic data that conformed to a first-order rate law. but the apparent first-order rate constant depended on the initial concentration of A. The authors propose competing unimolecular and bimolecular processes, and they show that the system reduces to a first-order expression when the equilibrium constant K is unity that is,... 

It may be noted that the pseudo-first-order rate law for an Sn2 reaction in the presence of a large excess of Y [Eq. (10.2)] is the same as that for an ordinary SnI reaction [Eq. (10.3)]. It is thus not possible to tell these cases apart by simple kinetic... 

Kinetic 37, 50-52, 97, 168) and stereochemical 54, 191, 192) investigations on the carbonylation of manganese alkyls and the decarbonylation of manganese acyls were already discussed in Sections III-V. The original finding (50) that the rate of CO insertion follows second-order kinetics has now been qualified 192). At higher pressures of CO (>15 atm) the first-order rate law [Eq. (23)] is obeyed. 

As a result, ammonium carbonate is conveniently studied by mass loss techniques such as TGA. In one study, the decomposition of particles having different size distributions (302 80, 98 36, and 30 10 pm, respectively) was studied by carrying out a large number of kinetic runs. It was found that decomposition of the largest particles almost always followed either a first-order or a three-dimensional diffusion rate law. The samples consisting of particles having intermediate size decomposed by a first-order rate law, and samples containing the smallest particles decomposed by a three-dimensional diffusion rate law. 

In the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

Assuming a first-order rate law with respect to hydrogen, with a kinetic constant kc, the maximum rate of chemical reaction (mol s-1 mL3) is obtained when the hydrogen concentration reaches equilibrium (Ch,l=C ,i) and the corresponding maximum reaction flux ( m(mol s 1) results in Eq. (19). 

Therefore, in order to obtain information about the nature of the brominating species present in the reaction mixture, and on its stability, spectroscopic measurements were carried out in the absence of olefin on methanolic Br2 solutions containing increasing amount of NaN3. (14) When bromine (4.3 x 10 3 M) and methanolic solution of NaN3 (between 4.7 x 10 2 to 2.37 xlO 1 M) were rapidly mixed in a stopped-flow apparatus, at 25 °C, no kinetic of disappearance of Br2 could be observed, but only the presence of a new absorption band (> ax 316 nm) and its subsequent decrease could be measured. The disappearance of the absorption band followed a first order rate law. The observed kinetic constants are reported in Table I. 

It was found that CO exchange in (diphosphine)Rh(CO)2H complexes proceeds via the dissociative pathway [60], The decay of the carbonyl resonances of the (diphosphine)Rh(13CO)2H complexes indeed followed simple first-order kinetics. The experiments with ligand 20 at different 12CO partial pressure show that the rate of CO displacement is independent of the CO pressure. Furthermore, the rate is also independent of the (diphosphine)Rh(13CO)2H complex concentration, as demonstrated by the experiments with ligand 18. It can therefore be concluded that CO dissociation for these complexes obeys a first-order rate-law and proceeds by a purely dissociative mechanism. 

If the process of APIO is properly described by Equation (19), which infers the presence of a soluble Fe(III) intermediate species, it will be difficult to analyze this species directly, given the low levels that are expected. We must therefore develop mathematical approaches to estimating the isotopic composition of this component, as was done for DIR. The equations used in the previous chapter (Chapter lOA Beard and Johnson 2004) to describe abiotic Fe(II) oxidation are useful for illustrating possible isotopic fractionations that may occur during APIO. We will assume that the overall oxidation process occurs through a series of first-order rate equations, where relatively slow oxidation of FefTI) to a soluble Fe(III) component occurs, which we will denote as Fe(III)jq for simplicity. The oxidation step is followed by precipitation of Fe(III)jq to ferrihydrite at a much faster rate, which maintains a relatively low level of Fe(III)jq relative to Fe(II)jq. The assumption of first-order kinetics is not strictly valid for the experiments reported in Croal et al. (2004), where decreasing FefTI) contents with time do not closely follow either zeroth-, first-, or second-order rate laws. However, use of a first-order rate law allows us to directly compare calculations here with those that are appropriate for abiologic Fe(II) oxidation, where experimental data are well fit to a first-order rate law (Chapter lOA Beard and Johnson 2004). 

Figure 15. Illustration of possible variations in isotopic fractionation between Fe(III),q and ferric oxide/ hydroxide precipitate (Aje(,n),q.Fenicppt) and precipitation rate. Skulan et al. (2002) noted that the kinetic AF (ni)aq-Feiricppt fractionation produced during precipitation of hematite from Fe(III), was linearly related to precipitation rate, which is shown in the dashed curve (precipitation rate plotted on log scale). The most rapid precipitation rate measured by Skulan et al. (2002) is shown in the black circle. The equilibrium Fe(III),-hematite fractionation is near zero at 98°C, and this is plotted (black square) to the left of the break in scale for precipitation rate. Also shown for comparison is the calculated Fe(III),q-ferrihydrite fractionation from the experiments of Bullen et al. (2001) (grey diamond), as discussed in the previous chapter (Chapter lOA Beard and Johnson 2004). The average oxidation-precipitation rates for the APIO experiments of Croal et al. (2004) are also noted, where the overall process is limited by the rate constant ki. As discussed in the text, if the proportion of Fe(III),q is small relative to total aqueous Fe, the rate constant for the precipitation of ferrihydrite from Fe(III), (Ai) will be higher, assuming first-order rate laws, although its value is unknown.

A number of studies have been performed investigating the kinetics of the decrease in mechanical properties. Stamm (1956), in a study of the heat treatment of wood under a variety of conditions, plotted the logarithm of strength loss against linear treatment time and found linear relationships (as with weight loss), showing that the decrease in mechanical properties also obeyed first order rate laws. 

Detailed kinetic studies revealed that glycine methyl ester and phenylalanine methyl ester in glycine buffer at pH 7.3 undergo a facile hydrolysis catalyzed by cupric ion (11). Under these conditions the reactions closely follow a first-order rate law in the substrate. Using these kinetic data it is possible to compare the rates of hydrolysis of DL-phenylalanine ethyl ester as catalyzed by hydronium, hydroxide, and cupric ion (see Table III). 

In any case what is usually obtained is a graph showing how a concentration varies with time. This must be interpreted46 to obtain a rate law and a value of k. If a reaction obeys simple first- or second-order kinetics, the interpretation is generally not difficult. For example, if the concentration at the start is A0, the first-order rate law... 

It may be noted that the pseudo-first-order rate law for an Sn2 reaction in the presence of a large excess of Y [Eq. (2)) is the same as that for an ordinary SnI reaction [Eq. (3)]. It is thus not possible to tell these cases apart by simple kinetic measurements. However, we can often distinguish between them by the common-ion effect mentioned above. Addition of a common ion will not markedly affect the rate of an Sn2 reaction beyond the effect caused by other ions. Unfortunately, as we have seen, not all SnI reactions show the common-ion effect, and this test fails for f-butyl and similar cases. 

Formal kinetic investigations (performed only with acidic ion exchange catalysts) revealed, in most cases, the first-order rate law with respect to the alkene oxide [285,310,312] or that reaction order was assumed [309,311]. Strong influence of mass transport (mainly internal diffusion in the polymer mass) was indicated in several cases [285,309, 310,312,314]. The first-order kinetics with respect to alkene oxide is in agreement with the mechanism proposed for the same reaction in homogeneous acidic medium [309,315—317], viz. 

Kinetics. Kinetic measurements using [Rh((5)-binap)(CH30H)2]C104 and geranylamine substrate indicate that (1) The initial phase of the reaction obeys the first-order rate law, but as the initial substrate concentration, [substratelo, is increased, the rate starts to deviate from the first-order plots at a relatively early stage of the reaction, implying a product inhibition. (2) The dependence of the initial rate Rq on the initial... 

More recently, Fujiwara has investigated the racemization of anhydrous salts of the type [Co(phen)3 n(en)n]X3 where n = 0, 1 or 2 and X- = Cl-, Br- or I-.29 All compounds exhibit anion effects. For [Co(phen)3]3+ the rates varied Cl- > Br-, whereas for [Co(phen)2(en)]3+ and [Co(phen)(en)2]3+ the sequence I- > Br- > Cl- was observed. The anion effects were interpreted in terms of the chemical interactions due to the donicity and hydrogen bonding ability of the anions. However, the kinetic analysis is open to criticism because a first-order rate law was assumed. 

In Figure 16-6 b, the interface at = 0 controls the reaction kinetics. If L denotes the interface conductivity coefficient, the rate of A uptake is given by L-A//a( = 0). For long times, the sensor registers a first order rate law E(t) e /T, r = (c°A-A )/(L-R T). This result is obtained for the linear geometry of Figure 16-6. In this context, we mention the a->P transformation of Ag2S as discussed in Sec-... 

From a structural point of view, mechanism in a single crystal can be much closer to a set of identical atomic trajectories than to the kind of fuzzy statistical average with which one must be content in solution. It is not surprising that with this kind of structural uniformity the site problems that plague kinetic studies in rigid glasses disappear. Adherence to first-order rate laws can be as close in single crystals as it is in fluids, and equally valid activation parameters can be obtained for thermal unimolecular reactions of reaction intermediates [12]. 

It is convenient at this juncture to introduce a concept that, in electro analytical chemistry, sometimes is referred to as the reaction order approach. Consider first the half-life-time, t1/2> which in conventional homogeneous kinetics refers to the time for the conversion of half of the substrate into product(s). From basic kinetics, it is well known that t /2 is independent of the substrate concentration for a reaction that follows a first-order rate law and that 1/t j2 is proportional to the initial concentration of the substrate for a reaction that follows a second-order rate law. Similarly, in electro analytical chemistry it is convenient to introduce a parameter that reflects a certain constant conversion of the primary electrode intermediate. In DPSCA, it is customary to use ti/2 (or to.s), which is the value of (f required to keep the value of Ri equal to 0.5. The reaction orders (see Equation 6.30) are then given by Equations 6.35 and 6.36, where Ra/b = a + b, and Rx = x (in reversal techniques such as DPSCA, in which O and R are in equilibrium at the electrode surface, it is not possible to separate the... 

First-Order Kinetics, K[A] Unimolecular processes, such as ligand dissociation from a metal center or a simple homolytic or heterolytic cleavage of a single bond, provide a straightforward example of a first-order reaction. The kinetics of this simple scheme, Equation 8.5, is described by a first-order rate law, Equation 8.6, where A stands for reactants, P for products, [A]0 for initial concentration of A, and t for time. The integrated form is shown in Equation 8.7 and a linearized version in Equation 8.8. 

Several examples of first-order rate laws in mechanistically interesting reactions are given below. However, in none of the examples in this chapter does the mechanistic assignment rely on a single type of information such as the rate law or rate constant. Rather, various tools and approaches are typically combined to unravel the mechanism. In the examples chosen, the kinetic information was crucial, but never exclusive. 

The kinetics and mechanisms of gas-phase elimination of ethyl 1-piperidinecarboxyl-ate, ethyl pipecolinate, and ethyl 1-methylpipecolinate has been determined in a static reaction system.9 The reactions proved to be homogeneous, unimolecular, and obey a first-order rate law. The first step of decomposition of these esters is the formation of the corresponding carboxylic acids and ethylene. The acid intermediate undergoes a very fast decarboxylation process. The mechanism of these elimination reactions has been suggested on the basis of the kinetic and thermodynamic parameters. 

The kinetics of the gas-phase elimination of 3-hydroxy-3-methylbutan-2-one have been investigated in a static system, seasoned with allyl bromide, and in the presence of the free chain radical inhibitor toluene.14 The reaction was found to be homogeneous, unimolecular and to follow a first-order rate law. The products of elimination are acetone and acetaldehyde. Theoretical estimations suggest a molecular mechanism involving a concerted non-synchronous four-membered cyclic transition state process. 

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