First-order reversible reactions, rate

Exercise 7,7.2, The first order reversible reaction rate... 

A first order reversible reaction, A B, is carried out in a plug flow reactor, starting with pure A. The specific rate and equilibrium constants are functions of temperature, k = A exp (-E/T)... 

Substitution of the rate equation for a first-order reversible reaction (102) in the design equation (130) gives... 

In order to assess the feasibility of any nuclear waste disposal concept, mathematical models of radionuclide sorption processes are required. In a later section kinetic descriptions of the three common sorption isotherms (3) are compared with experimental data from the mixing-cell tests. For a radionuclide of concentration C in the groundwater and concentration S on the surface of the granite, the net rate of sorption, by a first-order reversible reaction, is given by... 

Figure 6-28. Rate of reaction profile for a first order reversible reaction A <-> R.

Consider a first order reversible reaction A R, where the rate expression is (-rA) = kxCA - k2CR. Therefore, /= = t>R = 1. For... 

The rate expression for the first order reversible reaction is... 

Formally, the overall rate of a first-order reversible reaction A 5 B can be... 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

Figure 18. Effectiveness factor rj of a first-order reversible reaction versus the Weisz modulus ip (related to the forward rate constant k+). Influence of intraparticle diffusion on the effective reaction rate (isothermal reaction in a sphere, equal diffusivitics i,e = Die, equilibrium constant as a parameter).

The effectiveness factor versus the Weisz modulus according to Kao and Satterfield [61] is shown in Fig. 21 for C = 0.5 and different values of B. From this diagram, a similar behavior is seen as in the case of a simple, first order, reversible reaction (see Fig. 18) with decreasing value of B, the effectiveness factor is reduced. A decline of the effectiveness factor is also observed for a rise of the parameter C, which corresponds to a shift towards the chemical equilibrium, and hence to a reduction of the net reaction rate [91]. 

The example of this reaction demonstrates another important facet of kinetics. Figure 5.4 shows side by side the experimental data plotted as a first order-first order reversible reaction and as an irreversible reaction of order 1.5. Over a limited conversion range (here about two thirds of the way to equilibrium) the second plot is linear within the scatter of the data points. Although evaluation of the full conversion range leaves no doubt that the reaction is indeed reversible and first order-first order, its rate up to a rather high conversion is approximated surprisingly well by the equation for an irreversible reaction of higher order, in this instance of order 1.5 ... 

In first order-first order reversible reactions, the rate of approach to equilibrium is proportional to the fractional distance from equilibrium, measured in terms of any quantity that is a linear function of the concentrations. The same rule holds true for any participant in reactions with first-order parallel steps. 

A palladium-hydrogen-mordenite catalyst with a 10.8/1 silica/alumina mole ratio was evaluated for the hydroisomerization of cyclohexane. The rate of reaction followed a first-order, reversible reaction between cyclohexane and methylcyclopentane. The energy of activation for this reaction between 400° and 500°F was 35.5 it 2.4 kcal/mole. Cyclohexane isomerization rates decreased with increasing hydrogen and cyclohexane-plus-methylcyclopentane partial pressure. These effects are compatible with a dual-site adsorption model. The change of the model constants with temperature was qualitatively in agreement with the expected physical behavior for the constants. 

Initial variable studies had shown that gas-to-particle mass transfer and intra-particle diffusion were not rate limiting. The reaction mechanism was assumed to follow a first-order reversible reaction. After confirming this assumption, the effect of temperature and pressure on this reaction was investigated by determining the effect on the rate constant. 

In acidic conditions the tetracyclines undergo epimerisation at carbon atom 4 to form an equilibrium mixture of tetracycline and the epimer, 4-epi-tetracycline (Scheme 4.7). The 4-epi-tetracycline is toxic and its content in medicines is restricted to not more than 3%. The epimerisation follows the kinetics of a first-order reversible reaction (see equation (4.24)). The degradation rate is pH-dependent (maximum epimerisation occurring... 

If the integral in Eq. (9.6.2) can be worked out explicitly, then it may be possible to obtain an equation for the optimal temperature. For example, consider the first order reversible reaction A B, with rate law r = ka — taking place in an isothermal reactor whose feed is pure A. If the required fractional conversion is Y, then the feed concentration can be written the current concentrations are a = Gq — and 6 = f, and... 

Based on this information, it is possible to proceed with development of a first-order reversible reaction. As suggested by Harter (1989), it is convenient to convert sorbate concentration in solution to fraction remaining at time, t. Using this approach, the data were adequately modeled by a single reaction (Fig. 6-2b) with apparent rate coefficients k = 3.26 A / = 1.848 and the apparent thermodynamic equilibrium constant, K 1.76. Considering that the reaction appears to be completed within 2 min, cation exchange is the probable sorption mechanism. 

The individual rate constants of the reaction can be evaluated from the slope of a plot, providing the equilibrium constant is available. Many distribution processes between immiscible liquid phases of noncharged species, as well as distribution of solute ions (e.g., metal ions) performed at very low solute concentrations, can be treated as first-order reversible reactions when the value of the equilibrium (partition) constant is not very high. 

Amide CTI is a first-order reversible reaction (unimolecular process) characterized by a kinetic constant kobs = k( >c + kr >(. In secondary amides, kobs is entirely determined by kc >(, suggesting that a stabilized cis isomer corresponds to a decelerated cis —r trans isomerization rather than an accelerated trans —> cis reaction, whereas both rate constants usually contribute in a similar way for tertiary amides. The kinetic constant kc >( was determined for a set of Gly- and Ala-con-... 

The zone shape in a first-order reversible reaction in non-linear chromatography was examined by Cremer and Kramer [67]. The rate of conversion of ortho and para isomers of hydrogen, calculated on the basis of their findings, agrees with static data. 

Similarly, for the first order reversible reaction, we have the net rate as the difference between forward and backward reaction. 

The set of rate equations for first-order reversible reactions between the N components of a mixture can be written... 

Substituting (1.148) into equation (1.144), we will find the common rate of the first order reversible reaction, expressed through the correlation of chemical affinity ... 

This latter equation is formally the same as that encountered previously for a first-order reversible reaction and can be easily integrated [cf. Eq. (1-25)]. Simplification of rate equations via the steady-state approximation is often adopted, but evidence for the validity of this approximation should be critically examined in each case where it is used. 

The reaction was followed by observing the imine formation spectrophotometrically under conditions where the total glycine concentration was much greater than that of pyridinealdehyde. In this case, the reaction could be described by the rate law for a first-order reversible reaction. (This is an example of a pseudo first-order reaction.) The assumption was then made that the reaction in the forward direction was also first order with respect to glycine, and a second-order rate constant was obtained by dividing the first-order constant by the total glycine concentration. This second-order rate constant varied with the glycine concentration in the manner shown in Table P7-7 [35]. 

The formulation of Wei and Prater offers many advantages. Its limitations are not as confining as suggested by the single application discussed in this chapter, namely, the analysis of first-order reversible reactions. The method can be extended readily to networks of first-order reactions that are not all reversible. Furthermore, as discussed in Chapter 5, rate equations for the coupled sequences of a network are very frequently of a type that can still be handled by the analysis. They are commonly, though not always, of the form ... 

The area-based determination of rate constants from first-order reversible reactions is possible owing to the fact that molecules of Rj produced by R2 do not return under the primary peak of Ri, and vice versa. This means that the area of the primary peaks diminishes like in a simple irreversible reaction. The internal standard method together with the peak area correction gives appropriate results for these reactions too in case of small conversions. ... 

For a first-order reversible reaction A B, r r — r = k Cx — k c-Q, where and k are the rate coefficients of the forward and reverse reaction. In accordance with the law of mass conservation, assuming the reactor initially only contains A, Ca+Cb = Cao-... 

A first-order reversible reaction A = B is carried out in an ideal CSTR. For a space time of 10 min, 40% conversion of A is achieved in the reactor. Conversion dropped to 30% level when the feed flow rate is doubled. Calculate the rate constant k and the equilibrium constant K. 

Calculate the space time required to achieve 95% of the equilibrium conversion of a first-order reversible reaction A B carried out in an ideal PER. The feed concentration of A is 1 kmol/m. The rate constant of the forward reaction is ki = 0.1 S" and the equilibrium constant is X = 5. 

The rate of the metal-ion exchan for the trivalent ions (Am, Cm, Bk, Cf and Eu) withEuEDTA" (ChoppinandWilUams 1973, Williams andChoppin 1974,D Oheslager et al. 1970) and (Ce, Eu and Am) with LaEDTA (D Olieslager and Choppin 1971) was studied using radiotracer techniques. The exchange was found to be a first-order reversible reaction and both the forward (kf) and the reverse (fc,) rate constants contained an acid-dependent and an acid-independent term,... 

For kinetics not described by power law equations, the calculation of >jpore is complicated and we refer to the literature (Hong, Hecker, and Fletcher, 2000 Leven-spiel, 1999 BischofF, 1965 Valdman and Hughes, 1976 Roberts and Satterfield, 1965,1966). Here we only consider the reaction of a species adsorbed according to Langmuir and a first-order reversible reaction (Example 4.5.6). For a Lattgmuir type reaction [Eq. (4.5.2)] the rate related to the mass of catalyst is ... 

For a first-order reversible reaction, the reaction rate is... 

Example 9.4 Effectiveness for a first-order reversible reaction Consider a first-order reversible reaction R < IP. The rate of reaction is... 

After that, the user has to decide how the results should be visualized. It is possible to print the answer in the form of individual values of the desired function, an array, etc. However, the most visual output form is the graphical one. The plotting of the results is provided by the command odeplot from the graphical library plots. Figures 3.11 and 3.12 show a solution of the differential equation set, which describes the kinetics of the first-order reversible reaction B with arbitrary rate constant values. 

The sum of the two rate constants appears in place of the single one of equation (3.2.1). Integration provides again a logarithmic or an exponential equation. Using the same reasoning for the identification of the integration constant from the initial conditions we can express the equation for a first order reversible reaction in the form... 

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