First-order reactions process

This equation is that of a first-order reaction process, and thus the fraction of material electrolysed at any instant is independent of the initial concentration. It follows that if the limit of accuracy of the determination is set at C, = 0.001 C0, the time t required to achieve this result will be independent of the initial concentration. The constant k in the above equation can be shown to be equal to Am/ V, where A is the area of the pertinent electrode, V the volume of the solution and m the mass transfer coefficient of the electrolyte.20 It follows that to make t small A and m must be large, and V small, and this leads to the... 

We should realize that certain chemical/biochemical problems can have no multiplicities of their steady states over their entire range of parameters. Consider, for example, a simple first-order reaction process A => B with the rate equation... 

AIPO4 and AlPO -metal oxide catalysts exhibited high activity and selectivity in aniline alkylation to produce N-alkylated products in a consecutive first-order reaction process. N-methylaniline was easily formed at low temperatures/contact times and converted to N,N-dimethylaniline as temperature/contact time increased. N-alkylated products remained 100 mol% at 523-623 K. At 673 K, N,N-dimethyltoluidine (p->o-) also appeared although in very small amounts. The results obtained for pyridine adsorption at 373 (weak acidic) and 573 K (medium acidic) generally agreed with catalytic activity. 

There are different types of behaviour for reactions for example, in zeroth-order reactions, the change in the amount of a reactant or product is essentially linear over time in first-order reactions, it is exponential. The first-order reaction process is extremely common in coordination chemistry, and this is the type that we shall restrict ourselves to here. In its simplest form, the concentration of a reacting species varies over time as (5.33) ... 

Vapor-phase Beckmann rearrangement of cyclohexanone oxime over AIPO4 (AP) and AlP04-Ti02 (APTi, 25-75 wt%) catalysts was investigated. Apparent rate constants and activation parameters were calculated in terms of the kinetic model of Bassett and Habgood for first order reaction processes. In all cases the selectivity to e-caprolactam increased with reaction temperature and,... 

In the absence of a boundary layer and internal and external diffusional influences, the cyclohexanone oxime conversion follows the requirements of Bassett-Habgood kinetic treatment [20] for first order reaction processes in which the rate determining step is the suiiface reaction ... 

ILLUSTRATIVE EXAMPLE 8.3 The author annually hosts a party (his family refer to it as the animal party) that primarily consists of former basketball players he coached during the 1950s and 1960s and old friends from Astoria, Queens (the area where most of the attendees grew up). It is usually held on the first Sunday in January during which there is an NFL playoff double header. Several of the players have been known to drink excessively. On one occasion, an attendee s blood alcohol concentration was 0.21%. The player knew he should not drive home until his blood alcohol concentration dropped to 0.02%. If his blood alcohol drops to 0.17% in 45 minutes and he stops drinking at 8 00 PM, at what time can he leave the party and drive home The reader may assume that the elimination of alcohol in one s body is a first order reaction process that is elementary. 

In the normal process ( ), step (J) occurs very rapidly and step (/) is the rate-determining step, whereas in the inhibition process (B), step (3) occurs very slowly, generally over a matter of days, so that it is rate determining. Thus it has been demonstrated with AChE that insecticides, eg, tetraethyl pyrophosphate and mevinphos, engage in first-order reactions with the enzyme the inhibited enzyme is a relatively stable phosphorylated compound containing one mole of phosphoms per mole of enzyme and as a result of the reaction, an equimolar quantity of alcohoHc or acidic product HX is hberated. 

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. 

Note that if sticking is controled by site-exclusion only, i.e., if S 6,T) = 5 o(P)(l — 0), this rate is that of a first-order reaction at low coverage. This simple picture breaks down when either the sticking coefficient depends dilferently on the coverage, as it does for instance for precursor-mediated adsorption, or when lateral interactions become important. It then does not make much physical sense to talk about the order of the desorption process. 

Thus, if Ca and Cb can both be measured as functions of time, a plot of v/ca vs. Cb allows the rate constants to be estimated. (If it is known that B is also consumed in the first-order reaction, mass balance allows cb to be easily expressed in terms of Ca-) The rate v(Ca) is the tangent to the curve Ca = f(t) at concentration Ca-This can be determined graphically, analytically, or with computer processing of the concentration-time data. Mata-Perez and Perez-Benito show an example of this treatment for parallel uncatalyzed and autocatalyzed reactions. 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

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... 

The number calculated in (b) for the concentration of H+ in blood, 4.0 X 10-8 Af, is very small. You may wonder what difference it makes whether [H+] is 4.0 X 10-8M,4.0 X 10-7Af, or some other such tiny quantity. In practice, it makes a great deal of difference because a large number of biological processes involve H+ as a reactant, so the rates of these processes depend on its concentration. If [H+] increases from 4.0 X 10-8M to 4.0 X 10-7M, the rate of a first-order reaction involving H+ increases by a factor of 10. Indeed, if [H+] in blood increases by a much smaller amount, from 4.0 X 10-8 Af to 5.0 X 10-8 M (pH 7.40----- 7.30),... 

The kinetic effect of increased pressure is also in agreement with the proposed mechanism. A pressure of 2000 atm increased the first-order rates of nitration of toluene in acetic acid at 20 °C and in nitromethane at 0 °C by a factor of about 2, and increased the rates of the zeroth-order nitrations of p-dichlorobenzene in nitromethane at 0 °C and of chlorobenzene and benzene in acetic acid at 0 °C by a factor of about 559. The products of the equilibrium (21a) have a smaller volume than the reactants and hence an increase in pressure speeds up the rate by increasing the formation of H2NO. Likewise, the heterolysis of the nitric acidium ion in equilibrium (22) and the reaction of the nitronium ion with the aromatic are processes both of which have a volume decrease, consequently the first-order reactions are also speeded up and to a greater extent than the zeroth-order reactions. 

For an unsteady-state process, equation 10.170 may be solved analyticaly only in the case of a first-order reaction n = 1). In this case ... 

In a thin flat platelet, the mass transfer process is symmetrical about the centre-plane, and it is necessary to consider only one half of the particle. Furthermore, again from considerations of symmetry, the concentration gradient, and mass transfer rate, at the centre-plane will be zero. The governing equation for the steady-state process involving a first-order reaction is obtained by substituting De for D in equation 10.172 ... 

It may be noted that, in the absence of a chemical reaction, equation 10.203 reduces to equation 10.146. For a steady-state process dCA/dt = 0, and for a first-order reaction n = l. Thus ... 

RT) and ks - 3.11.10 exp(-13639/RT) m. kmof. s. The value of ki, obtained in this research are almost the same as that obtained by Venugopal. Venugopal neglected the side reactions. The value of E and Hatta Number -/m were greater than 3, so that the reaction system can be considered as pseudo-first order reaction with respect to oxygen and the process was controlled by mass transfer aspect. 

Equation does not contain the concentration of A, so the half-life of a first-order reaction is a constant that is independent of how much A is present. The decomposition reactions of radioactive isotopes provide excellent examples of first-order processes, as Example illustrates. We describe the use of radioactive isotopes and their half-lives to determine the age of an object in more detail in Chapter 22. 

This may be understood more fully by reference to Fig. 11.2. Curve A shows the type of response which would be obtained if the lethal process followed precisely the pattern of a first-order reaction. Some experimental curves do, in fact, follow this pattern quite closely, hence the genesis of the original theory. 

The more usual pattern found experimentally is that shown by B, which is called a sigmoid curve. Here the graph is indicative of a slow initial rate of kill, followed by a faster, approximately linear rate of kill where there is some adherence to first-order reaction kinetics this is followed again by a slower rate of kill. This behaviour is compatible with the idea of a population of bacteria which contains a portion of susceptible members which die quite rapidly, an aliquot of average resistance, and a residue of more resistant members which die at a slower rate. When high concentrations of disinfectant are used, i.e. when the rate of death is rapid, a curve ofthe type shown by C is obtained here the bacteria are dying more quickly than predicted by first-order kinetics and the rate constant diminishes in value continuously during the disinfection process. 

If the pressure for the process is lowered, the reaction (R3) will shift from a first-order reaction (high-pressure limit) to a second-order reaction (low-pressure limit). If (R3) is now considered a second-order reaction and assuming that the other pressure dependent reactions do not shift regime, determine expressions for d[C2H6]/dt, d[CH3]/dt, d[C2Hs]/dt and d[H]/dt. 

When only taking into account the concentration polarization in the pores (disregarding ohmic potential gradients), we must use an equation of the type (18.15). Solving this equation for a first-order reaction = nFhjtj leads to equations exactly like (18.18) for the distribution of the process inside the electrode, and like (18.20) for the total current. The rate of attenuation depends on the characteristic length of the diffusion process ... 

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

Preliminary work showed that first order reaction models are adequate for the description of these phenomena even though the actual reaction mechanisms are extremely complex and hence difficult to determine. This simplification is a desired feature of the models since such simple models are to be used in numerical simulators of in situ combustion processes. The bitumen is divided into five major pseudo-components coke (COK), asphaltene (ASP), heavy oil (HO), light oil (LO) and gas (GAS). These pseudo-components were lumped together as needed to produce two, three and four component models. Two, three and four-component models were considered to describe these complicated reactions (Hanson and Ka-logerakis, 1984). 

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