First-order system concentration variables

These four rate equations, along with the mass balance, can be solved for the desired yields of the products (in terms of the amount of benzene reacted) by eliminating time as a variable. The expressions cannot be solved directly to give compositions as a function of time because the magnitudes of the individual rate constants are not known (only their ratios are known). Although the rate equations are second order, the chlorine concentration appears in all the expressions and cancels out. Hence the reaction system is equivalent to the consecutive first-order system aheady considered, except that three reactions are involved. 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

Experimental. In order to study the nucleophilic properties of 13 it was necessary to add excess I " to the solutions to prevent precipitation of I2. The rate of formation of CoCCN I-3 was followed spectrophotometrically after the I3 " in aliquots of the solution taken at suitable time intervals was reduced to I by arsenite ion. A typical set of experiments was carried out at 40°C. and unit ionic strength, with all solutions containing 0.5/1/ 1 and variable I3 " at a maximum concentration of 0.28M, the approximate upper limit imposed by solubility restrictions. The results are presented in Figure 3 as a plot of k the symbol used for the pseudo first-order rate constant for this system, vs. l/(lf). It is apparent that 13 is a remarkably efficient nucleophile, with a reaction rate considerably greater than that found for I at comparable concentrations. The points in Figure 3 also show detectable deviation from linearity, despite the limited range of 13 " concentration which was available. 

The reaction rates in this system are presumably first-order in catalyst concentration, as implied by the scaling of product formation rates proportionately to rhodium concentration (90, 92, 93). Responses to several other reaction variables may be found in both the open and patent literature. Fahey has reported studies of catalyst activity at several pressures in tet-raglyme solvent with 2-hydroxypyridine promoter at 230°C (43). He finds that the rate to total products is proportional to the pressure taken to the 3.3 power. A large pressure dependence is also evident in the results shown in Table VII. Analysis of these results indicates that the rate of ethylene glycol formation is greater than third-order in pressure (exponents of 3.2-3.5), and that for methanol formation somewhat less (exponents of 2.3-2.8). The pressure dependence of the total product formation rate is close to third-order. A possible complicating factor in the above comparisons is the increased loss of soluble rhodium species in the lower-pressure experiments, as seen in Table VII. Experiments similar to those of Fahey have also been... 

A variety of pulsed techniques are particularly useful for kinetic experiments (Mclver and Dunbar, 1971 McMahon and Beauchamp, 1972 Mclver, 1978). In these experiments, ions are initially produced by pulsing the electron beam for a few milliseconds. A suitable combination of magnetic and electric fields is then used to store the ions for a variable period of time, after which the detection system is switched on to resonance to measure the abundance of a given ionic species. These techniques allow the monitoring of ion concentration as a function of reaction time. Since the neutrals are in large excess with respect to the ions, a pseudo first-order rate constant can be obtained in a straightforward fashion from these data. The calculation of the rate constant must nevertheless make proper allowance for the fact that ion losses in the icr cell are not negligible. 

A general systematic technique applicable to second-order differential equations, of which (11.31) is a particular example, is that of phase plane analysis. We have seen this approach before (chapter 3) in the context of systems with two first-order equations. These two cases are, however, equivalent. We can replace eqn (11.31) by two first-order equations by introducing a new variable g, which is simply the derivative of the concentration with respect to z. Thus... 

The system is shown in Fig. 21.7. It is described by two concentrations (state variables), CA and CB, by two zero-order input functions, JA and JB (input per volume and time), by two first-order output functions, kACA and kBCB (output per volume and time), and by the first-order transformations from A to B and vice versa. The inputs and outputs can be the sum of two or more processes, for instance, the sum of the input through different inlets and from the atmosphere (as in Eq. 21-7a), or the sum of the output at the outlet and by exchange to the atmosphere (as in Eq. 21-7b.). 

The ability to separate the removal rates due to air bubbles from drop aggregation/coalcscencc for each oil drop size permitted a detailed study of the system variables. These variables and their ranges of variation are shown in Table I. Note that the first-order removal rate constants were independent of residence time and oil droplet population in the feed and effluent. The variables which may influence the rate constants are air flowrate, temperature, NaCI concentration, bubble diameter, cationic polymer concentration, and oil drop diameter. 

Substrate concentration is yet another variable that must be clearly defined. The hyperbolic relationship between substrate concentration ([S ) and reaction velocity, for simple enzyme-based systems, is well known (Figure C1.1.1). At very low substrate concentrations ([S] ATm), there is a linear first-order dependence of reaction velocity on substrate concentration. At very high substrate concentrations ([S] A m), the reaction velocity is essentially independent of substrate concentration. Reaction velocities at intermediate substrate concentrations ([S] A"m) are mixed-order with respect to the concentration of substrate. If an assay is based on initial velocity measurements, then the defined substrate concentration may fall within any of these ranges and still provide a quantitative estimate of total enzyme activity (see Equation Cl. 1.5). The essential point is that a single substrate concentration must be used for all calibration and test-sample assays. In most cases, assays are designed such that [S] A m, where small deviations in substrate concentration will have a minimal effect on reaction rate, and where accurate initial velocity measurements are typically easier to obtain. 

When a reaction does not follow first-order kinetics and if experimental conditions lie Avithin the range of diffusion effects, changes in gas pressure such as result from a reaction in a static system will cause a change in the diffusion effect rj during the course of the experiment because the concentration c is contained in the modulus (p. This will result in an apparently variable order of reaction. This order will, at all times, be smaller than the real order. 

A dramatic reduction in dimensionality is often possible by converting a design equation from dimensioned to dimensionless form. Equation 1.62 contains the dependent variable a and the independent variable z. The process begins by selecting characteristic values for these variables. By characteristic value we mean some known parameter that has the same dimensions as the variable and that characterizes the system. Eor a PER, the variables are concentration and length. A characteristic value for concentration is flin and a characteristic value for length is L. These are used to define the dimensionless variables a = ala-m and zIL. The governing equation for a first-order reaction in an ideal PER becomes... 

Fig. 24a. Phase diagram of the asymmetric polymer mixture (A. = 2.0, NA = NB = N = 32, 4, = 0.5) in the plane of variables reduced temperature and relative concentration 4a/(4a + 4b) of component A. The dashed lines are the histogram extrapolations for three simulated system sizes, the full line denotes the binodal, and the circle denotes the critical point. From Deutsch and Binder [93]. b Phase diagram of asymmetric polymer mixtures for NA = NB = N = 32, 4 = 0.5 in the (T, Ap) plane. Three choices of the asymmetry parameter A are shown as indicated. The first order transitions are shown as a full line, the critical points as circles. Temperature is normalized such that in the Flory-Huggins-approximation the critical temperature would occur for the same abscissa value. From Deutsch [266]...

ERA The reaction was performed in a batch reactor and in the gas phase. 10% N2 was introduced in the reactor at 2 atm and 450°C. After 50 min, the pressure was 3.3 atm. The reaction is irreversible and of first order. Calculate the specific reaction rate constant. If the same reaction would be performed in a piston system with variable volume, how volume changes keeping pressure constant at 2 atm and considering the same conversion as before Calculate the initial concentration ... 

As we did with rate equations, we start with the simplest possible systems—that is, those described by a single concentration variable. Consider such a system with a first-order irreversible reaction ... 

An exothermic first-order reactive system in a sequence of jacketed CSTRs is considered. Several alternative process designs are constructed and studied with respect to their static and dynamic controllability properties to multiple and simultaneous process disturbances. The same system has been studied by numerous researchers (Ref 14, 15, 44) and served as an illustrative example of process design and control interactions. The reaction is carried out in either a single reactor or two reactors in series (Fig. 1). The dynamic model (see Ref 14, 15) contains four state variables per reactor namely the reactor s volume, concentration and temperature and the jacket temperature. Model parameters for the system are shown in Table... 

Transitions in two-component systems have one more degree of freedom, — i.e., one more variable of state must be specified. The additional variable is the concentration. The phase rule of Fig. 3.7 shows the relationship between number of phases, degrees of freedom and number of components. The equation can be verified using reasoning analogous to that for the one-component system. In this section the thermodynamics of the first-order transition in systems involving one pure and one mixed phase for small molecules will be treated. Other systems, especially those involving macromolecules, are treated in Chapters 4 and 5. Under certain conditions of temperature, pressure, and concentration, the transitions can be sharp and thermometry can yield useful information. 

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