First-order reaction equation

The same boundary conditions will be used as for the first-order reaction. Equation 10.171 may be re-arranged to give ... 

We have examined the effects of concentration, temperature, solvent and added electrolyte on the kinetics of this structural interconversion. In all instances, the kinetics are well described by the rate law for a reversible first-order reaction [Equation 1] ... 

Referring back to the rate equation for a first-order reaction (Equation A1.2), we have a differential equation for which the derivative of the variable ([S]) is proportional to the variable itself. Such a system can be described by an infinite series with respect to time ... 

Nitronate(47a) is not the only oxazete derivative. For example, sterically hindered nitroalkenes (42b-d) can be prepared by nitration and halogenation of readily available allenes (48). Compounds (42b-d) are rather smoothly isomerized into the corresponding four-membered cyclic nitronates (47b-d) by the first-order reaction equation (168). Storage of nitronate (47c) is accompanied by its slow transformation into acid chloride (47e) from which amide (47f) can be easily synthesized. 

For a first-order reaction (equation 3.4-10), and from the definition of /a>... 

In molar notation, and referencing to the equilibrium concentration cf assuming n = 1 (i.e., first-order reaction), equation 8.275 can be translated into... 

First-Order Reaction. Equation 18 has been solved analytically by Wehner and Wilhelm (1956) for first-order reactions. For vessels with any kind of entrance... 

Figure 7.10. Time course for reaction 11. All spectra were normalized by making the intensity of the polystyrene band at 1945 cm 1 equal. The area integration for the hydroxyl band from 3181 to 3637 cm-1 (circles), aldehyde C-H band from 2664 to 2766 cm-1 (squares), and aldehyde carbonyl band from 1641 to 1765 cm-1 (triangles) for spectra at various times are plotted against time. Lines were calculated from the best fit to a first-order reaction equation with the rate constant shown. The amount of catalyst TPAP was 0.2 eq in A, 0.1 eq in B and 0.05 eq in C.

Biodegradation can be described as a second order reaction (Equation la) which, under conditions of constant competent biomass, simplifies to a pseudo first-order reaction (Equation lb). 

Substituting Equation 3-197 into the integrated first order reaction Equation 3-33 gives the corresponding equations expressed in terms of the solution absorbance ... 

Plotting log to 5 against log [A]0 results in a straight line with a slope of (1 - n) as shown in Figure 5.11. If n = 1 (i.e., first-order reaction), Equation (5.14a) should be used instead. The order of the reaction should be rounded off to the integer closest to n, especially in pharmaceutical applications. An exact integrated equation is then used to determine the reaction rate constant. 

This is the general equation for the half-life of a first-order reaction. Equation (15.3) can be used to calculate tV2 if k is known or k if t1/2 is known. Note that for a first-order reaction the half-life does not depend on concentration. 

For isothermal/isobaric CVI processes there is no forced flow, i.e. the convection term cannot be taken into consideration. Assuming the first-order reaction, Equation (5.2) becomes... 

To establish the order of thermal decomposition reactions, the kinetic data were verified using the first order reaction equation ... 

The relation between volumes required in stirred-tank and tubular-flow reactors can be illustrated by reference to a constant-density first-order reaction. Equation (4-6) is applicable for the stirred-tank reactor and gives... 

In the special case of first order reaction, Equation 16 becomes linear and after some manipulation gives for the exit concentration a result identical to that of Equation 10 thus, for first order reaction, calculations for complete and minimum segregation yield a single result in the transient case. Inputting the reactant as a slug of amount m, c0(t) — (m/V)S(t), and Equation 10 gives for the output of either system... 

Examples on How to Ese Table 8-1. Wc now couple the energy balance equations in Table 8-1 with the appropriate reactor mole balance, rate law. stoichiometry algorithm to solve reaction engineering problems with heat effects. For example, recall rate law for a first-order reaction. Equation (E8-1.5) in Example 8-1. 

For the second-order reaction, Equation 7.101 show s that varies with the molar flow, which means and r vary along the length of the reactor as Na decreases. We are asked to estimate the catalyst mass needed to achieve a conversion of A equal to 7S%. So for this particular example, 4> decreases from 6.49 to 3.24. As shown in Figure 7,9, we can approximate the effectiveness factor for the second-order reaction using the analytical result for the first-order reaction, Equation 7.42,... 

Thus, decomposition of OM in phytoplankton can be approximately described by a first-order reaction equation. Proceeding from these observations, the approximate amount of regenerated biogenic elements can be estimated at a certain time, and also the time required for mineralisation of a given amount of organic N and P in the phytoplanktonic OM. The values... 

In both experiments, during the first 6—10 days, when the labile organic compounds were mostly subject to degradation, the process was described by a first-order reaction equation, the rate constants of which were less than 0.1 day". The rate constants according to organic carbon were 0.041 day" at the fourth day and 0.039 day" at the seventh in the aerobic experiment, and 0.024 day" at the fifth and 0.022 day" at the twelfth in the anaerobic experiment. The decrease in their value in later periods is attributed to the increasing influence from the decomposition of more stable organic constituents. 

Depending on the conditions in which a process is conducted and its features, any of the five steps may be the slowest one. Hence, the rate of the catalytic process may be limited by one of them. An interfacial chemical reaction may proceed only with continuous molecular or convective diffusion of the reactants to the surface on which the given reaction is proceeding, and also with continuous reverse diffusion of the products. The rate of a process as a whole will be determined by the rate of its slowest step. If the rate of a reaction on the surface of a catalyst is greater than that of diffusion, the rate of the process as a whole will be determined by the rate of diffusion. The observed macroscopic kinetics of the reaction will obey equations that can be obtained by considering only processes of diffusion and will not reflect the true rate of the chemical reaction at the interface. Such a process is a diffusion-controlled one. It is most frequently described by a first-order reaction equation, since the rate of diffusion is directly proportional to the concentration. 

For the sake of simplicity, the chemical reaction is treated as a first-order reaction (Equation 2.1-10). 

For the boundary conditions of an ideal batch stirred-tank reactor and a first-order reaction Equations (2.2-51) and (2.2-52) result from Equations (2.2-44) and (2.2-45) ... 

Three-phase reactors are operated in either the semibatch or continuous mode, and batch operation is almost never used because the gas phase is invariably continuous. The general principles of design are the same for all types of reactors for a given mode of operation, semibatch or continuous. They differ with respect to their hydrodynamic features, particularly mass and heat transfer. Thus, for simple first-order reactions. Equation 17.8 is valid for any reactor. The rate constant ky,i would be the same for all of the reactors, but specific to each reactor type is the mass transfer term k/. Hence we consider first the design of... 

Zimmermann and Bauer [34,35] investigated the drying of baker s yeast (granulated yeast in a fluidized bed) and developed a mathematical model in which thermal inactivation of a product was based on the first-order reaction equation and the Arrhenius equation. Taking into account... 

For a first-order reaction. Equation 14.12 or 14.13 can be used in several ways. Given any three of the following quantities, we can solve for the fourth k, t, [A]q, and [A],. Thus, you can use these equations to determine (1) the concentration of a reactant remaining at any time after the reaction has started, (2) the time interval required for a given fraction of a sample to react, or (3) the time interval required for a reactant concentration to fall to a certain level. 

Applying Danckwerts boundary conditions [2] Equation 3.66 can be solved for an irreversible first order reaction (Equation 3.67) [6]. 

The rate of a first order reaction (Equation 2.3) is proportional to the concentration of a single species and is described by a first order rate constant with the units... 

At equilibrium (steady state), the rates of the various steps depicted in Figure 2.2 should be equal. For a simple first-order reaction, Equation 2.16 gives the rates of the three steps in series ... 

For an intrinsic first-order reaction, Equation (15.25) becomes 1... 

For first order reactions, Equation (8) can be written as follows ... 

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