Other Orders of Reaction

The development of an analytical expression for tj in Example 8-4 is for a first-order reaction and a particular particle shape (flat plate). Other orders of reaction can be postulated and investigated. For a zero-order reaction, analytical results can be obtained in a relatively straightforward way for both tj and flat plate and 8-15 for a sphere). Corresponding results can be obtained, although not so easily, for an nth-order reaction in general an exact result can be obtained for and an approximate one for tj. Here, we summarize the results without detailed justification. 

That is, /or a first-order reaction, the two stages must be of equal size to minimize V. The proof can be extended to an N-stage CSTR. For other orders of reaction, this result is approximately correct. The conclusion is that tanks in series should all be the same size, which accords with ease of fabrication. 

Although, for other orders of reaction, equal-sized vessels do not correspond to the minimum volume, the difference in total volume is sufficiently small that there is usually no economic benefit to constructing different-sized vessels once fabrication costs are considered. 

For an isothermal first-order reaction taking place in a constant-volume BR but at varying density in a PFR, it can be shown that the times are also equal this is not the case for other orders of reaction (see problem 17-8). 

Here we have listed three alternate reaction paths whereby a substance A may disappear. It is clear that this scheme can be indefinitely extended to include other orders of reaction of A with other substances than B and finally the reactions of B itself with other substances as illustrated by... 

Other orders of reaction have been deduced from the results reported [48] by Flory, including second order [49], second order followed by third order [49], and a 2.5-order scheme [50]. These conflicting interpretations may be a consequence of ignoring or inadequately taking into consideration the complexities of kinetics under typical conditions of... 

The situation is more complicated for other orders of reaction. Luss [128] shows that for order n > 1, there is less likelihood of multiplicity, and the converse is true for n < 1. As might be expected, the situation for reaction orders approaching zero or native order behavior could combine the complications of possible concentration and thermal instalnlity for example, see Smith, Zahradnik, and Carberry [129]. 

Other orders of reaction are also possible and can give more complicated relationships. But these three cases cover the great majority of the reactions we will encounter. Let s determine the rate law for such a reaction involving only one reactant. 

Similar amendments are used for other orders of reactions. 

The previous example shows that the term half-life can be applied to any order of reaction, not just first-order reactions. However, only for first-order reactions is the half-life independent of the initial amount, and a characteristic of the reaction. For any other order of reaction, a half-life can be defined, but will always include the initial amoimt in the expression. For example, for second-order reactions, the half-life ti/2 can be defined as... 

Units in SI system Si Stanton number h/Cf,pu Dimensions depend ort order of reaction. Suffixes 0 Value in bulk of phase 1 Phase 1 2 Phase 2 A Component A B Component B AB Of A in B b Bottom of column equilibrium with bulk of other phase G Gas phase / Interface value. L Liquid phase u Overall value (for height and number of transfer units) value in bulk of phase i Top of column Dimensions in in M. N, 1. T. 

MgATP hydrolysis and electron transfer between the two proteins seems not to be direct and the order of reactions may depend on the precise conditions of the experiment at low temperature, electron transfer seems to be reversible (see Ref. 12) for a discussion). One innovation is incorporation of data in which the release of inorganic phosphate was monitored. With other MgATP hydrolyzing enzymes, this step is often the work step in which the energy released by MgATP hydrolysis is utilized. With nitrogenase this step takes place before the dissociation of the two proteins 106). 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

To determine the order of reaction. It is always good research strategy to change only one variable at a time. That way, the measured response (if any) can be attributed unambiguously to the change in that variable. And the variable of choice in this example will be one or other of the two initial concentrations. 

All the other reactants will be ignored here to make the analysis more straightforward, even if steps (1) and (2) are, in fact, bimolecular. We again write the reaction number within brackets to avoid confusion we do not want to mistake the subscripted number for the order of reaction. We call the rate constant of the first reaction k([) and the rate constant of the second will be k(2). 

This well-known kinetic expression for a drained equilibrium implies that at high values of m the reaction is of zero order, at low values of first order, with respect to m. Few other examples of this type have been reported. However, orders of reaction less than unity with respect to m may also be due to the sequestration of a metal halide initiator by complexation with the monomer [4], Which, if any, of these two causes is responsible in any particular case for a low or varying kinetic order with respect to m may be determined by suitable experiments, and there seems no reason why both may not occur in the same system. 

There are also other types of reactions, besides first-order and second-order reactions. For example, consider the decomposition of ammonia on... 

Having obtained the exponent a in (1.15) by monitoring the concentration of A in deficiency we may now separately vary the concentration of the other reactants, say B and C, still keeping them however in excess of the concentration of A. The variation of the pseudo rate-constant k with [B] and [C] will give the order of reaction b and c with respect to these species, leading to the expression... 

For orders of reaction other than one, a knowledge of the molar absorptivities is necessary. See Fig. 1.3. 

Conversion in a reactor with nonideal flow can be determined either directly from tracer information or by use of flow models. Let us consider each of these two approaches, both for reactions with rate linear in concentration (the most important example of this case being the first-order reaction) and then for other types of reactions where information in addition to age distributions is needed. 

For zeroth-order reaction steps, we still have linear equations, but for other orders of kinetics we have nonlinear simultaneous equations, which generally have no closed-form analytical solutions. We must solve these sets of equations numerically to find C/i(t), Cfi(r), and Cc(t). 

Obviously, since the gas-carbon reactions have different activation energies and orders of reaction, the relative rates of these reactions will be a function of the temperature and pressure selected for the correlation. Unfortunately, the authors can find no reactivity data for all four of the gas-carbon reactions using the same carbon. Furthermore, data for even two of the gas-carbon reactions on the same carbon are limited. The available data will be taken and extrapolated, where necessary, to give at least a qualitative idea of relative rates of the gas-carbon reactions at 800° and 0.1 atm. gas pressure. Extrapolation of these relative reactivities to other temperatures and pressures by the reader will require the assumption of activation energies and orders of reaction. 

For n t- 1 it is important to keep in mind that the half-life is a function of the initial concentration c(A)a. Knowing the order of reaction and the rate constant allows the half-life to be calculated. Or it can be determined experimentally and used to calculate the other parameters, e. g. by the trial and error method. 

They claim that their data indicate that the order of reaction with respect to NO decreases as [NO] increases. This is inconsistent with other findings. 

Formaldehyde is formed as an intermediate product. This suggested two-step reaction would indicate two second-order reactions occurring in sequence. The over-all apparent order of reaction of Equation 3, if expressed in a form suggested by Equation 2, could be of noninteger order. If one of the reactions suggested by Equations 4 or 5 was extremely fast compared to the other, the reaction would essentially be controlled by the slower one and would appear to be of the second order. Similarly, the standard combustion equation of hydrocarbons with air is usually written as... 

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