First-order reactions consumed

The most crucial observation concerning the effects of added species is that nitrate ion anticatalyses nitration without changing the kinetic form of the reaction. This shows that nitrate does not exert its effect by consuming a proportion of the nitronium ion, for, as outlined above, this would tend to bring about a kinetically first-order reaction. Nitrate ions must be affecting the concentration of a precursor of the nitronium... 

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. 

In the presence of excess bromine, the first-order rate coefficient was 10.3 x 10" 5, but kinetic studies here were complicated due to rapid reaction of bromine with the 2,4,6-tribromophenol to give the intermediate (LXII), with SO3 replaced by Br) which slowly decomposed with a rate coefficient k2 to give two products. By analysis in terms of two consecutive first-order reactions, the values of k2 and k2 were determined as 9.2 x 10"5 and 3.75 x 10"4 and the latter rate being the faster means that two moles of bromine were consumed for every mole of sulphonate undergoing substitution in fact, more than two moles were consumed, the reason for this being undetermined. 

Our aim is to find an expression for the concentration of a reactant A at a time t, given that the initial molar concentration of A is [A]() and that A is consumed in the first-order reaction A —> products. 

The concentration of the reactant does not appear in Eq. 7 for a first-order reaction, the half-life is independent of the initial concentration of the reactant. That is, it is constant regardless of the initial concentration of reactant, half the reactant will have been consumed in the time given by Eq. 7. It follows that we can take the initial concentration of A to be its concentration at any stage of the reaction if at some stage the concentration of A happens to be A], then after a further time tv2, the concentration of A will have fallen to 2[AJ, after a further tU2 it will have fallen to [A], and so on (Fig. 13.13). In general, the concentration remaining after n half-lives is equal to (t)" A 0. For example, in Example 13.6, because 30 days corresponds to 5 half-lives, after that interval [A ( = (j)5 A]0, or [A]0/32, which evaluates to 3%, the same as the result obtained in the example. 

Example 1.3 Find the outlet concentration of component A from a piston flow reactor assuming that A is consumed by a first-order reaction. 

Equation (1.45) gives the spatial distribution of concentration, u(z), in a piston flow reactor for a component that is consumed by a first-order reaction. The local concentration can be used to determine the local reaction rate, S Aiz)-... 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

Another characteristic of first-order reactions is that the time it takes for half the reactant to disappear is the same, no matter what the concentration. This time is called the half-life ( 1/2). Applying Equation to a time interval equal to the half-life results in an equation for / i 2 When half the original concentration has been consumed,... 

The half-life of a reactant is the time required for half of that reactant to be converted into products. For a first order reaction, the half-life is independent of concentration so that the same time is required to consume half of any starting amount or concentration of the reactant. On the other hand, the half-life of a second-order reaction does depend on the starting amount of the reactant. 

Surprisingly, despite requiring two analyte molecules to produce one S2 molecule, the kinetics of the chemiluminescent reaction are first order with respect to the sulfur compound. This can be explained if every H2S or CH3SH molecule is consumed in the reaction and every S atom recombines to form S2, through the use of an excess of OCIO to maintain pseudo-first-order reaction conditions [81]. The limit of detection for this analysis was found to be 3 ppbv for H2S. 

Statement (d) is incorrect it implies a constant rate during the first half-life. The rate of a first-order reaction actually decreases as time passes and reactant is consumed. 

The alkylation of toluene with acetylene in the presence of sulphuric acid is carried out in a batch reactor. 6000 kg of toluene is charged in each batch, together with the required amount of sulphuric acid and the acetylene is fed continuously to the reactor under pressure. Under circumstances of intense agitation, it may be assumed that the liquid is always saturated with acetylene, and that the toluene is consumed in a simple pseudo first-order reaction with a rate constant of 0.0011 s-1. 

The toluene is consumed in a simple pseudo first-order reaction with a rate constant of 0.0011 s 1... 

The yield (y) of a biomass production process is defined as the moles of biomass formed per mole of substrate consumed. Aerobic conditions are more conducive to higher biomass formation (and therefore also to biofilm formation) than anaerobic conditions. Empirically, under aerobic conditions, a yield of 0.05 - 0.6mol biomass/mol carbon can be obtained, while under anaerobic conditions the attainable yield falls to 0.04 -0.083mol. The reaction kinetics of biodegradation processes can be approximated by the first-order reaction rate constant k as follows ... 

The system is sketched in Fig. 3.1 and is a simple extension of the CSTR considered in Example 2.3. Product B is produced and reactant A is consumed in each of the three perfectly mixed reactors by a first-order reaction occurring in the liquid. For the moment let us assume that the temperatures and holdups (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup). Density is assumed constant throughout the system, which is a binary mixture of A and B. 

Introductory textbooks in kinetics or chemical engineering describe how to determine the reaction order of a reaction from experimental data. Typically an assumption about reaction order is made, and this assumption is subsequently tested. Imagine that experimental data for the consumption of reactant A as function of time is available from experiments in a batch reactor. Initially we assume that A is consumed according to a first-order reaction,... 

An alternative method to determine the reaction order is the half-life method. The half life of a reaction (t /2) is the time it takes for 50% of the reactant(s) to be consumed. At time t /2 the concentration of A must then be [A]o/2. For a first-order reaction, Eq. 13.15 yields... 

The overall reaction rate may be affected by diffusion if the mass transfer from the bulk fluid to the coal surface and/or the pore diffusion steps are relatively slow. I have made tentative estimates of both these diffusion effects. The following equation can be written (2) for the rate of oxygen uptake of a porous solid such as coal, assuming the oxygen to be consumed in an irreversible first-order reaction. 

In a first approximation a pseudo-first order reaction rate is often assumed. This must be checked against what really happens in the reactor. In semi-batch or nonsteady state oxidation, the concentration of the pollutants as well as the oxidants can change over time. A common scenario initially a fast reaction of ozone with the pollutants occurs, the reaction is probably mass transfer limited, the direct reaction in the liquid film dominates, and no dissolved ozone is present in the bulk liquid. As the concentration of the pollutants decreases, the reaction rate decreases, less ozone is consumed, leading to an increase in the dissolved ozone concentration. Metabolites less reactive with ozone are usually produced. This combined with an increase in dissolved ozone, may also shift the removal mechanism from the direct to the indirect if radical chain processes are initiated and promoted (see Chapter A 2). These changes are often not observed in waste water studies, mostly because dissolved ozone is often not measured. 

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed as a first-order reaction. 

The half-life (ft/2) of a reaction is the time required for half of the original reactant to be consumed. A first-order reaction has a half-life that is related only to k and is independent of the concentration of the reacting species. After one half-life, a - y) equals a12, and Eq. (2.17) can be... 

Click the box under Reactant Components and use the dropdown menu to select component E (ethylene). Set the stoichiometric Coefficient at — 1 (reactant is consumed) and the Exponent at 1 (first-order reaction) (see Fig. 2.50). Repeat for the other reactant B (benzene). In a similar way, define the product as EB with a coefficient of 1... 

Case II Low substrate concentration, unsaturated kinetics In case II, the concentration of substrate is again insufficient to saturate the sites (a< 1), but now the film is thick (c> 1) and, therefore, all the substrate is consumed in a first-order reaction layer (Xk) at the outside of the film and the current is given by,... 

External mass transfer reduces the concentration of reactant gas close to the particle surface and thus reduces the overall process rate. Thus, consider gasification to be a first-order reaction. Then at steady state, the rate of gasification equals the rate of mass transfer. For a nonporous solid, the surface reaction (whose rate constant is k ) consumes the diffusing reactant ... 

In the absence of oxygen and the possible exchange of ammonia on clays, the only reaction affecting sulfate, carbon dioxide and ammonia is that of sulfate reduction vdiereby sulfate is consumed as a reactant and carbon dioxide and ammonia are liberated from the decomposing organic matter. Thus the expressions for CR for these substances will be the first order reaction term K (OC) modified by the appropriate stoichiometric coefficients. These coefficients are determined by the reaction chosen to describe the process of sulfate reduction. A reaction that has been used successfully by Berner (2 and others (2) is... 

Mathematical models [216] for calculating these effectiveness factors involve simultaneous differential equations, which on account of the complex kinetics of ammonia synthesis cannot be solved analytically. Exact numerical integration procedures, as adopted by various research groups [157], [217]-[219], are rather troublesome and time consuming even for a fast computer. A simplification [220] can be used which can be integrated analytically when the ammonia kinetics are approximated by a pseudo-first-order reaction [214], [215], [221], according to the Equation (21) ... 

Luss, D., and Golikeri, S. V., Grouping of many species each consumed by two parallel first-order reactions, AIChE J. 21,865 (1975). 

---