Consumption equation, first order

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. 

It was found that the substrate consumption rate followed first-order kinetics with respect to substrate concentration.21,22 The expression of substrate consumption with time is written in a first-order differential equation ... 

The reaction rate for simple fermentation systems is normally given by the Monod equation. This model indicates that the specific conversion rate is constant when applied to an immobilised cell system (Table 8.7). If a first-order rate equation for sugar consumption is used, (8.7.4.2) yields ... 

Scheme B. Oxidation occurs as a chain reaction in scheme A. However, hydroperoxide formed is decomposed not by the reaction with free radicals but by a first-order molecular reaction with the rate constant km [3,56]. This scheme is valid for the oxidation of hydrocarbons where tertiary C—H bonds are attacked. For km 3> k i[RH] the maximum [ROOH] is attained at the hydroperoxide concentration when the rate of the formation of ROOH becomes equal to the rate of ROOH decay fl[RH](kj [ROOH][RH])l/2 km[ROOH] therefore, [ROOH]max = a2kn km 2 [RH]3. The kinetics of ROOH formation and RH consumption are described by the following equations [3],...

Figure 4.15 Rates of oxygen consumption by shaken suspensions of anaerobic soils. Points are measured data, lines are fits to two first-order rate equations. The apparent rate constant for the initial reaction is common to all sods that for the main reaction varies 30-fold between the soils and is well correlated with [Fe +] (Reddy et al., 1980). Reproduced by permission of Soil Sci. Soc. Am.

Hydrolysis of alkoxysilanes in water is reported to follow a first-order rate law [8, 13]. In the water-acetone mixture required for solubility of TMMS, there is a 15-fold molar ratio of water to TMMS. However, a plot of In ([TMMS],) vs. time does not result in a linear plot. This is because the consumption of water is significant. Therefore, a bimolecular, second-order rate law was employed as shown in equation (1) ... 

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 reaction is assumed to be first order in steam partial pressure and nth order (to be determined from the kinetic data) with CaBr2 concentration. Similarly, the rate equation for consumption of H20 is given by ... 

Here the terms are in order of statistical significance. It is reasonable to suspect that if 02 consumption were a first-order function of its concentration, growth would be likewise. The regression equation above would be dominated by the first-order 02 term. In fact, in Eq. (18), growth is a relatively weak function of 02. This suggests that metabolic activity was not a first-order function of 02 concentration in the concentration range we investigated. 

In this way, the diffusion/reaction equations are reduced to trial and error algebraic relationships which are solved at each integration step. The progress of conversion can therefore be predicted for a particular semi-batch experiment, and also the interfacial conditions of A,B and T are known along with the associated influence of the film/bulk reaction upon the overall stirred cell reactor behaviour. It is important to formulate the diffusion reaction equations incorporating depletion of B in the film, because although the reaction is close to pseudo first order initially, as B is consumed as conversion proceeds, consumption of B in the film becomes significant. 

This type of solution method is possible for reactions where deactivation is slow, and a pseudo steady-state assumption can be made when solving the mass balance equations. Thus, these equations are applicable to reactions where the activity loss is first-order in both the poison and the active sites, and where deactivation is slow compared to the main reaction. A similar type of approach was taken by Johnson et al. (5), for oxygen consumption and carbon content during catalyst regeneration and by Bohart and Adams (6), for chlorine consumption and absorbence capacity of charcoal. 

A final modification to these equations is necessary if the substance under consideration is radioactive, or subject to some first-order concentration-dependent consumption or removal process. Thus in general (11) can be rewritten as... 

An example would be the dehydration of ethanol to ethylene and its dehydrogenation to acetaldehyde. If both reactions are first order, selectivity is unaffected by internal mass transport the ratio of the rates of reactions, 1 and 2 is k jkj at any position within the pellet. Equation (11-89) cannot be applied separately to the two reactions because of the common reactant A. The development of the effectiveness-factor function would require writing a differential equation analogous to Eq. (11-45) for the total consumption of A by both reactions. Hence k in Eq. (11-89) would be k- + k2 and Fp would be (Tp) -1- (rp)2- Such a development would shed no light on selectivity. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

The first-order rate constants J , and associated with the reactions producing A, and A3, respectively, are determined by measuring the rate of consumption of the reactant Ag and the production of A3. The rate constant, k2, for the reaction producing A2 is determined by solving a nonlinear equation that is derived as follows. 

In Equation 58, the time-dependent terms between the braces contain the decay constant A. Therefore, the rate of change in Ra-226 concentration at any depth (dC/dt) depends on the decay rate constant. Thus, in the case of a first-order reaction (radioactive decay), the rate of change in concentration depends on the reaction rate constant, whereas it has been shown in the preceding section that for a zero-order reaction (oxygen consumption), the rate of change in concentration (dC/dt) is independent of its rate constant. 

Upon substitution of an appropriate kinetic expression for the rate of generation or consumption of solute within the tissue space, Equation 3-50 can be solved to determine concentration as a function of time and position. Full analytical solutions are generally difficult to obtain, unless both the kinetic expression and the geometry of the system are simple. For example, consider the linear diffusion of solute from an interface where the concentration is maintained constant (as in Figure 3.4d). If the diffusing solute is also eliminated from the tissue, such that the volumetric rate of elimination is first order with a characteristic rate constant k, Equation 3-51 can be reduced to ... 

The model pollutant, phenol, is photo-converted via a first order or pseudo-first order reaction consistent with equation(l-14) considering in this manner all possible sources of phenol consumption. 

Now, all of the tools required to calculate the molar density of reactant A on the external surface of the catalyst are available to the reactor design engineer. It is important to realize that Ca, surface is the characteristic molar density, or normalization factor, for all molar densities within the catalyst. Hence, Ca, surface only appears in the expression for the intrapellet Damkohler number (i.e., excluding first-order kinetics) when isolated pellets are analyzed. Furthermore, intrapellet Damkohler numbers are chosen systematically to calculate effectiveness factors via numerical analysis of coupled sets of dimensionless differential equations. Needless to say, it was never necessary to obtain numerical values for Ca, sur ce in Part IV of this textbook. Under realistic conditions in a packed catalytic reactor, it is necessary to (1) predict Ca, surface and Tsurface, (2) calculate the intrapellet Damkohler number, (3) estimate the effectiveness factor via correlation, (4) predict the average rate of reactant consumption throughout the catalyst, and (5) solve coupled ODEs to predict changes in temperature and reactant molar density within the bulk gas phase. The complete methodology is as follows ... 

Pyrolysis of hydrocarbons is a first order type reaction (see chapter 3.3.1, Eqs. 3-7,3-8, and 3-9) whereas oxidation does not obey first order. But it has been found experimentally that it may be treated mathematically as a first order reaction with respect to the consumption of fuel, provided there is an excess of air (oxygen). The relation of the reaction rate to the temperature is described by the Arrhenius equation (Eq. 3-7). 

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