Reacting first-order reaction

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The stabilization of chloromethoxycarbene (234) was intensively studied. It is formed from diazirine (233) in a first order reaction with fi/2 = 34h at 20 C. It reacts either as a nucleophile, adding to electron poor alkenes like acrylonitrile with cyclopropanation, or as an electrophile, giving diphenylcyclopropenone with the electron rich diphenylacetylene. In the absence of reaction partners (234) decomposes to carbon monoxide and methyl chloride (78TL1931, 1935). 

Scheme 10. Mechanislic possibililies for PF condensalion. Mechanism a involves an SN2-like attack of a phenolic ring on a methylol. This attack would be face-on. Such a mechanism is necessarily second-order. Mechanism b involves formation of a quinone methide intermediate and should be Hrst-order. The quinone methide should react with any nucleophile and should show ethers through both the phenolic and hydroxymethyl oxygens. Reaction c would not be likely in an alkaline solution and is probably illustrative of the mechanism for novolac condensation. The slow step should be formation of the benzyl carbocation. Therefore, this should be a first-order reaction also. Though carbocation formation responds to proton concentration, the effects of acidity will not usually be seen in the reaction kinetics in a given experiment because proton concentration will not vary.

To see the connection between this stochastic process and a chemically reacting system, consider the first step of Scheme IX. Each (real) molecule of A has an equal and constant probability of reacting in time t. In the simulation, each position in the grid has an equal and constant probability (p) of being selected. For this first-order reaction, the chemical system is described by... 

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.)... 

A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Pick s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, 1), the first order reaction rate constant k. the film thickness L and the concentration Cas of solute in a saturated solution. The reaction is initially carried our at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K7... 

For an isothermal, first-order reaction, the probability that a particular molecule reacts depends only on the time it has spent in the system ... 

The simplest form of a physicochemical reaction takes place when one species simply changes to another. This can be written in a general way as A B. The rate of such a reaction is defined as the amount of reactant (the reacting species, A, in this case) or equivalently the product (B) that changes per unit time. The key feature here is the form of the rate law, i.e., the expression for the dependence of the reaction rate on the concentrations of the reactants. For a first-order reaction... 

The apparent reaction rate constant for the first order reaction, k, was calculated from the conversion of CO2. Since the gas-volume reduction rate increased with k, a poor fluidization was induced by high reaction rate. We investigated the effect of the rate of the gas-volume change on the fluidization quality. The rate of the gas-volume change can be defined as rc=EA(dxA/dt), where Sa is the increase in the number of moles when the reactants completely react per the initial number of moles. This parameter is given by 7-1. When the parameter, Ea, is negative, the gas volume decreases as the reaction proceeds. 

Recall also from Chapter 15 that for first-order reactions, the time required for exactly half of the substance to react is independent of how much material is present. This constant time interval is the half-life, Equation... 

For many electrochemical reactions the reaction rate is proportional to the concentration of the reacting species (first-order reaction) ... 

The ions or cluster ions are thermalized by collisions with an inert carrier gas (usually helium), although often argon or even nitrogen is employed. Neutral reactant gas is added through a reactant gas inlet at an appropriate location downstream in the flow tube, and allowed to react with the injected ions. Rate coefficients, k, are determined by establishing pseudo-first-order reaction conditions in which the reactant ion concentration is small compared to the reactant neutral concentration. Bimolecular rate coefficients, k, are obtained from the slope of the natural logarithm of the measured signal intensity, /, of the reactant ion versus the flow rate (2b of reactant gas 45,48-50... 

First-Order Reactions in Constant Volume Systems. In a first-order reaction the reaction rate is proportional to the first power of the concentration of one of the reacting substances. 

As was discussed earlier in this chapter, the concept of a reaction order does not apply to a crystal that is not composed of molecules. However, there are numerous cases in which the rate of reaction is proportional to the amount of material present. We can show how this rate law is obtained in a simple way. If the amount of material at any time, t, is represented as W and if we let W0 be the amount of material initially present, the amount of material that has reacted at any time will be equal to (W0 — W). In a first-order reaction the rate is proportional to the amount of material. Therefore, the rate of reaction can be expressed as... 

Basolo and Wojcicki26 have reported that at 0 °C in toluene solution Ni(CO)4 exchanges with radioactive CO by a first-order reaction with k = 7.5 x 10-4 sec-1. Heck24 reports that under the same conditions nickel carbonyl reacts with triphenyl phosphine with k = 4.3xl0-4 sec-1. The fact that both processes are first-order and have approximately the same rate coefficient was originally in-... 

Two possible approaches are indicated in Schemes 4 and 5. In the first, a reactive radical R> is spin-trapped in competition with its pseudo-first order reaction with a substrate SH, which occurs at a known rate to give RH and S. The growth of both spin-adducts (ST—R ) and (ST—S ) is monitored, and simple analysis leads to the trapping rate constant kT. In the second approach, R-does not react with a substrate, but undergoes unimolecular rearrangement or fragmentation at a known rate to give a new species R. This latter procedure... 

Let us start with an example the Matlab function Data AB. m models the absorption spectra of a reacting solution as a function of time. They are stored as rows of the matrix Y. The reaction is a simple first order reaction A - B as introduced in Chapter 3.4.2, Rate Laws with Explicit Solutions. 

We develop the idea using a kinetic example. Any reaction scheme that consists exclusively of first order reactions, results in concentration profiles that are linear combinations of exponentials. There is no limit to the number of reacting components nc. 

In most chemical reactions the rates are dominated by collisions of two species that may have the capability to react. Thus, most simple reactions are second-order. Other reactions are dominated by a loose bond-breaking step and thus are first-order. Most of these latter type reactions fall in the class of decomposition processes. Isomerization reactions are also found to be first-order. According to Lindemann s theory [1, 4] of first-order processes, first-order reactions occur as a result of a two-step process. This point will be discussed in a subsequent section. 

The half-life of a certain first-order reaction is 120 s. How long do you estimate that it will take for 90% of the original sample to react ... 

As indicated earlier, the units of the specific reaction rate k depend on the order of the reaction. This is because the overall reaction rate 31 always has the same units (moles per unit time per unit volume). For a first-order reaction of A reacting to form B, the overall reaction rate 31, written for component A, would have units of moles of A/min ft. ... 

Resins (19) ( 30 mg each) reacted with 5% TFA in DCM. Droplet of suspension was taken at various time intervals for single bead FTIR (Fig. 12.15) and kinetics analysis (Fig. 12.16). The data was also fitted to a first order reaction rate equation and rate constants were determined to be 4.8x10 (5% TFA). Cleavage of carbamides (18), (20), (21), ureas (22-25), amides (26-29), and sulfonamides (30-33) were studied in the same way. 

In a first-order reaction, the rate-determining step involves a transformation where one reactant reacts to give one product, that is, A — B. In first-order reactions, there is an exponential decrease in the reactant concentration, so that at any given time, the transformation rate is dependent on the corresponding concentration of the reactant at the same time. This can be expressed in the following way ... 

Reaction rates are influenced not only by the activation energy and the temperature, but also by the concentrations of the reactants. When there is only one educt, A (1), v is proportional to the concentration [A] of this substance, and a first-order reaction is involved. When two educts, A and B, react with one another (2), it is a second order reaction (shown on the right). In this case, the rate v is proportional to the product of the educt concentrations (12 mM at the top, 24 mM in the middle, and 36 mM at the bottom). The proportionality factors k and k are the rate constants of the reaction. They are not dependent on the reaction concentrations, but depend on the external conditions for the reaction, such as temperature. 

In the absence of an enzyme, the reaction rate v is proportional to the concentration of substance A (top). The constant k is the rate constant of the uncatalyzed reaction. Like all catalysts, the enzyme E (total concentration [E]t) creates a new reaction pathway, initially, A is bound to E (partial reaction 1, left), if this reaction is in chemical equilibrium, then with the help of the law of mass action—and taking into account the fact that [E]t = [E] + [EA]—one can express the concentration [EA] of the enzyme-substrate complex as a function of [A] (left). The Michaelis constant lknow that kcat > k—in other words, enzyme-bound substrate reacts to B much faster than A alone (partial reaction 2, right), kcat. the enzyme s turnover number, corresponds to the number of substrate molecules converted by one enzyme molecule per second. Like the conversion A B, the formation of B from EA is a first-order reaction—i. e., V = k [EA] applies. When this equation is combined with the expression already derived for EA, the result is the Michaelis-Menten equation. 

Consider a CSTR of constant volume V, operated in the steady state, with a volumetric throughflow Q in which a first-order reaction, with rate coefficient k, is occurring. The inlet concentration of the reacting species is Ca and the outlet concentration is Ca. Writing the conventional mass balance for a CSTR... 

First consider a single cylindrical pore of length L, with reactant A diffusing into the pore, and reacting on the surface by a first-order reaction... 

When Cg (i.e., concentration of B which reacts with A) is much larger than C, Cg can be considered approximately constant, and k Cg) can be regarded as the pseudo first-order reaction rate constant (T ). The dimensionless group y, as defined by Equation 6.23, is often designated as the Hatta number (Ha). According to Equation 6.22, if y > 5, it becomes practically equal to E, which is sometimes also called the Hatta number. For this range. 

In the general case of a first order reaction which does not occur from the primary excited state and whose quantum yield is not unity, experimental determination of the rates of radiationless processes and of the quantum yield of formation of reacting electronic state becomes necessary. 

Similarly, the thioaldehyde complexes 66a,b were formed via addition of thiols to the vinylidene complex 64 and elimination of ethene. The vinylidene complex 64 reacted with benzyl mercaptan to yield the isolable intermediate 65. On heating 65 lost ethene in a first-order reaction to give 66b.188 The complexes 66a-d could also be prepared from the benzyne-(hydrido) complex 67 and thiols RCH2SH (Scheme 18).188... 

Depending on the relative gains and losses in internal rotation, the intramolecular reaction is favored entropically by up to 190 J/deg/mol (45 cal/deg/mol) or 55 to 59 kJ/mol (13 to 14 kcal/mol) at 25°C. Substituting 190 J/deg/mol (45 cal/deg/mol) into the exp (ASVR) term of equation 2.7 gives a factor of 6 X 109. Taking into account the difference in molecularity between the second-order and first-order reactions, this may be considered as the maximum effective concentration of a neighboring group, i.e., 6 X 109 M. In other words, for B in equation 2.22 to react with the same first-order rate constant as A B in equation 2.23, the concentration of A would have to be 6 X 109 M. 

In a first-order reaction the concentration of the reacting species is specified at one particular time (usually at the start of the reaction). 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

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