First Example Chemical Reaction

Although the methods that are discussed in this chapter deal explicitly with the disposition of dmgs in animals and humans, their scope is much wider. In general, these methods can be applied to study the transport of substances within parts of a system provided that these transports can be described by zero or first order kinetics. This applies, for example, when the rate of change of the amount in one part of the system depends linearly on the amounts present in all the various parts of the system. Applications are found commonly in first order chemical reactions. 

A cascade of three continuous stirred-tank reactors arranged in series, is used to carry out an exothermic, first-order chemical reaction. The reactors are jacketed for cooling water, and the flow of water through the cooling jackets is countercurrent to that of the reaction. A variety of control schemes can be employed and are of great importance, since the reactor scheme shows a multiplicity of possible stable operating points. This example is taken from the paper of Mukesh and Rao (1977). 

It is important to understand that the time constant xp of a process, say, a stirred tank is not the same as the space time x. Review this point with the stirred-tank heater example in Chapter 2. Further, derive the time constant of a continuous flow stirred-tank reactor (CSTR) with a first-order chemical reaction... 

A good example of a first-order (pseudo-first-order) chemical reaction is the hydration of CO2 to form carbonic acid. Reaction l-7f, C02(aq) + H20(aq) H2C03(aq). Because this is a reversible reaction, the concentration evolution is considered in Chapter 2. 

However, the reverse process, in going from speed to distance, involves integration of the rate equation (6.2). In chemistry, the concept of rate is central to an understanding of chemical kinetics, in which we have to deal with analogous rate equations which typically involve the rate of change of concentration, rather than the rate of change of distance. For example, in a first-order chemical reaction, where the rate of loss of the reactant is proportional to the concentration of the reactant, the rate equation takes the form ... 

Let us take as a simple example the following set of first-order chemical reactions of simple stoichiometry ... 

Whereas radioactive decay is never a reversible reaction, many first-order chemical reactions are reversible. In this case the characteristic life time is determined by the sum of the forward and reverse reaction rate constants (Table 9.5). The reason for this maybe understood by a simple thought experiment. Consider two reactions that have the same rate constant driving them to the right, but one is irreversible and one is reversible (e.g. k in first-order equation (a) of Table 9.5 and ki in first-order reversible equation (b) of the same table). The characteristic time to steady state tvill be shorter for the reversible reaction because the difference between the initial and final concentrations of the reactant has to be less if the reaction goes both ways. In the irreversible case all reactant will be consumed in the irreversible case the system tvill come to an equilibrium in which the reactant will be of some greater value. The difference in the characteristic life time between the two examples is determined by the magnitude of the reverse reaction rate constant, k. If k were zero the characteristic life times for the reversible and irreversible reactions would be the same. If k = k+ then the characteristic time for the reversible reaction is half that of the irreversible rate. 

Example 7.6 Product B is produced in an isothermal tubular reactor where the following gas-phase, first-order chemical reactions take place ... 

Example 7.12 A heavy hydrocarbon feedstock is being cracked in a tubular reactor placed in a furnace that maintains the wall of the reactor at 980 K. The cracking is represented by the following first-order chemical reactions ... 

Radium-226 in Water. Ra-226 will be considered as an example of a chemical species which is being removed from the oceanic water column by a first-order chemical reaction, radioactive decay. Other possible mechanisms of removal, such as uptake by detrital silicates and organisms, will not be discussed. The supply of Ra-226, however, from organic matter decomposing in the water column will be considered. 

EXAMPLE 8.6 In a first-order chemical reaction with no back reaction, the concentration of the reactant is governed by... 

Example 1.25 Mass Transfer with First-Order Chemical Reaction... 

Sometimes a dissolved species may react chemically, for example, a metal ion produced by anodic dissolution may undergo hydrolysis. In that case, the above equation must be modified by including a term that takes into account the chemical reaction. Por example, for a species being consumed by an irreversible first-order chemical reaction having a rate constant g ts... 

We illustrate the above five variations of weighted residuals with the following example of diffusion and first order chemical reaction in a slab catalyst (Fig. 8.1). We choose the first order reaction here to illustrate the five methods of weighted residual. In principle, these techniques can apply equally well to nonlinear problems, however, with the exception of the collocation method, the integration of the form (8.7) may need to be done numerically. 

Example 12.4 considers a problem of first order chemical reaction in a fixed bed. The boundary condition at the bed entrance does not account for the axial diflhision of mass. To account for this, the boundary conditions at the two ends should be (Danckwerts boundary conditions)... 

One last example should suffice to illustrate the way the point method can be used. If we take the diffusion equation in cylindrical coordinates and add a homogeneous first-order chemical reaction. 

Analogy with a chain reaction. The equivalence between a chain of several poles, making a multipole, and the dipole formed by the two ends of the chain % is a well-known property of chains composed of linear relationships. A repre-S L sentative example in physical chemistry is the chain formed by a sequence of first-order chemical reactions. The poles between the ends are the intermediate species and the poles at the ends are the substrate A and product species Z. In such a chain, the overall reaction rates and equilibrium constants correspond to those of the equivalent dipole A - Z. 

Examples of linear first-order differential equations occur frequently in chemical engineering practice through unsteady state mass balances or first-order chemical reaction problems. 

These features occur both for systems with static and for systems with dynamical disorder. A typical example is a first-order chemical reaction, A —> Products, described by the kinetic equation... 

If the chemical structure of a polymer is altered during a viscoelastic experiment—in particular, if a cross-linked network is subjected to a reaction which increases or decreases the number of network strands while it is being investigated in the rubbery zone of viscoelastic behavior—the apparent mechanical properties will be profoundly influenced. For example, scission of the network strands will cause stress relaxation at constant strain" (Fig. 14-12) or creep under constant stress. Formally, if a single first-order chemical reaction is responsible, the relaxation may be described by a single relaxation time which is the reciprocal of the chemical rate constant, instead of the broad spectra which are characteristic of the usual mechanical processes. 

The main results from the thermodynamics of linear systems are the Onsager relations. To give a simple example of the Onsager relations we consider, following Onsager,a closed system at constant temperature and pressure in which three substances flfi, and M, with concentrations c, and Cj are transformed into each other by first order chemical reactions. We have then... 

Example 12.13. The differential equation for a first-order chemical reaction without back reaction is... 

We are interested in how a first-order chemical reaction alters the mass transfer in industrial equipment. For example, imagine that we are scrubbing ammonia out of air with water, using equipment like that shown in Fig. 10.2-1. To increase our equipment s capacity, we are considering adding small amounts of hydrogen chloride to the water. We want to predict the effect of this acid. However, the a-priori prediction of mass transfer in a scrubber is a tremendously difficult problem, requiring expensive numerical calculation. 

A second requirement is that the rate law for the chemical reaction must be known for the period in which measurements are made. In addition, the rate law should allow the kinetic parameters of interest, such as rate constants and concentrations, to be easily estimated. For example, the rate law for a reaction that is first order in the concentration of the analyte. A, is expressed as... 

Several important points about the rate law are shown in equation A5.4. First, the rate of a reaction may depend on the concentrations of both reactants and products, as well as the concentrations of species that do not appear in the reaction s overall stoichiometry. Species E in equation A5.4, for example, may represent a catalyst. Second, the reaction order for a given species is not necessarily the same as its stoichiometry in the chemical reaction. Reaction orders may be positive, negative, or zero and may take integer or noninteger values. Finally, the overall reaction order is the sum of the individual reaction orders. Thus, the overall reaction order for equation A5.4 isa-l-[3-l-y-l-5-l-8. 

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