Chemical reactions, kinetics pressure-independent

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

If a chemical reaction is operated in a flow reactor under fixed external conditions (temperature, partial pressures, flow rate etc.), usually also a steady-state (i.e., time-independent) rate of reaction will result. Quite frequently, however, a different response may result The rate varies more or less periodically with time. Oscillatory kinetics have been reported for quite different types of reactions, such as with the famous Belousov-Zha-botinsky reaction in homogeneous solutions (/) or with a series of electrochemical reactions (2). In heterogeneous catalysis, phenomena of this type were observed for the first time about 20 years ago by Wicke and coworkers (3, 4) with the oxidation of carbon monoxide at supported platinum catalysts, and have since then been investigated quite extensively with various reactions and catalysts (5-7). Parallel to these experimental studies, a number of mathematical models were also developed these were intended to describe the kinetics of the underlying elementary processes and their solutions revealed indeed quite often oscillatory behavior. In view of the fact that these models usually consist of a set of coupled nonlinear differential equations, this result is, however, by no means surprising, as will become evident later, and in particular it cannot be considered as a proof for the assumed underlying reaction mechanism. 

To summarize, the conditions under which Equations 11.3-12 and 11.3-13 are valid are (a) negligible kinetic and potential energy changes, (b) no accumulation of mass in the system, (c) pressure independence of U and / , (d) no phase changes or chemical reactions, and (e) a spatially uniform system temperature. Any or all of the variables T, T, , Q and (or IV) may vary with time, but the system mass, M, the mass throughput rate, m, and the heat capacities, C and Cp, must be constants. 

It is noteworthy that even a separate treatment of the initial data on branched reactions (1) and (2) (hydrogenation of crotonaldehyde to butyr-aldehyde and to crotyl alcohol) results in practically the same values of the adsorption coefficient of crotonaldehyde (17 and 19 atm-1)- This indicates that the adsorbed form of crotonaldehyde is the same in both reactions. From the kinetic viewpoint it means that the ratio of the initial rates of both branched reactions of crotonaldehyde is constant, as follows from Eq. (31) simplified for the initial rate, and that the selectivity of the formation of butyraldehyde and crotyl alcohol is therefore independent of the initial partial pressure of crotonaldehyde. This may be the consequence of a very similar chemical nature of both reaction branches. 

Model formulation. After the objective of modelling has been defined, a preliminary model is derived. At first, independent variables influencing the process performance (temperature, pressure, catalyst physical properties and activity, concentrations, impurities, type of solvent, etc.) must be identified based on the chemists knowledge about reactions involved and theories concerning organic and physical chemistry, mainly kinetics. Dependent variables (yields, selectivities, product properties) are defined. Although statistical models might be better from a physical point of view, in practice, deterministic models describe the vast majority of chemical processes sufficiently well. In principle model equations are derived based on the conservation law ... 

Figure 3.6. Example of the type of kinetic information available for the catalytic reduction of NO on rhodium single-crystal surfaces under atmospheric conditions. The data in this figure correspond to specific rates for C02, N20, and N2 formation over Rh(l 11) as a function of inverse temperature for two NO + CO mixtures PNO = 0.6 mbar and Pco — 3 mbar (A), and Pno — Pco = 4 mbar (B) [55]. The selectivity of the reaction in this case proved to be approximately constant independent of surface temperature at high NO pressures, but to change significantly below Pno 1 mbar. This highlights the dangers of extrapolating data from experiments under vacuum to more realistic pressure conditions. (Reproduced with permission from the American Chemical Society, Copyright 1995).

In the range of temperatures and pressures where the reaction is substantially reversible, the kinetics is much more complicated. There is no grounds to consider chemical changes described by (272) and (273) as independent, not interconnected, reactions. Conversely, if processes (272) and (273) occur on the same surface sites, then free sites will act as intermediates of both processes. Thus one must use the general approach, treating (272) and (273) as overall equations of a certain single reaction mechanism. But if a reaction is described by two overall equations, its mechanism should include at least two basic routes hence, the concept of reaction rate in the forward and reverse directions can be inapplicable in this case. However, experiments show that water-gas equilibrium (273) is maintained with sufficient accuracy in the course of the reaction. Let us suppose that the number of basic routes of the reaction is 2 then, as it has been explained in Section VIII, since one of the routes is at equilibrium, the other route, viz., the route with (272) as overall equation, can be described in terms of forward, r+, and reverse, r, reaction rates. The observed reaction rate is then the difference of these... 

For rate processes in which the Arrhenius parameters are independent of reaction conditions, it may be possible to interpret the magnitudes of A and ii, to provide insights into the chemical step that controls the reaction rates. However, for a number of reversible dissociations (such as CaCOj, Ca(OH)2, LijSO Hp, etc.) compensation behaviour has been foimd in the pattern of kinetic data measured for the same reaction proceeding under different experimental conditions. These observations have been ascribed to the influence of procedural variables such as sample masses, pressure, particle sizes, etc., that affect the ease of heat transfer in the sample and the release of volatile products. The various measured values of A and cannot then be associated with a particular rate controlling step. Galwey and Brown [52] point out that few studies have been specifically directed towards studying compensation phenomena. However, many instances of compensation behaviour have been recognized as empirical correlations applicable to kinetic data... 

The models in chemical kinetics usually contain a number of unknown parameters, whose values should be determined from experimental data. Regression analysis is a powerful and objective tool in the estimation of parameter values. The task in regression analysis can be stated as follows the value of the dependent variable (y) is predicted by the model a function (/), contains independent variables (x) and parameters (/ ). The independent variable is measured experimentally, at different conditions, i.e. at different values of the independent variables (x). The goal is to find such numerical values of the parameters (/ ) that the model gives the best possible agreement with the experimental data. Typical independent variables are reaction times, concentrations, pressures and temperatures, while molar amounts, concentrations, molar flows... 

In large part, the chemistry we meet in practice takes place in a solution of some kind, but a quantitative description of the chemical kinetics involved is much more complex than for gaseous reactions. The key difference lies in the interparticle distances. In a gas at atmospheric pressure, the particles occupy less then 1 % of the total volume and, effectively, move independently of each other. In a solution the solute and solvent molecules, with the latter being in the majority, take up more than 50% of the available space, the distances between the various species are relatively small, and each particle is in continuous contact with its neighbours. It is these interactions which greatly complicate the formulation of a satisfactory theory of chemical kinetics in solution. Indeed, the rate of an elementary reaction and for that matter a composite reaction, can be significantly influenced by the choice of solvent. 

Kinetics, specifically studying rate laws and measurement of rate constants, can only be done under laboratory conditions, whereas reaction conditions should be simulated in special reactors ( smog chambers ) closely resembling the atmospheric one. Once established, the k-value of an elementary reaction is universally applicable, or in other words, pure chemistry is independent of meteorological and geographical specifics but the conditions for reactions (pressure, temperature, radiation, humidity) and the concentration field depends from location. This is the difference to air chemistry . For more detailed information on chemical kinetics, see Zumdahl (2009), Atkins (2008) and Houston (2006). 

Commonly used reaction mechanisms for atmospheric or combustion systems contain a significant fraction of unimolecular and chemically activated reactions. Each of these reactions is in principle both temperature and pressure dependent, although the pressure dependence might vanish under certain conditions. Consequently, in order to achieve accurate kinetic predictions of complex chemical systems, it is necessary to incorporate this pressure and temperature dependence into kinetic models. This leads to the need to develop tools which allow the kineticist to analyze these types of reactions and which yield apparent time-independent rate constants that can be used in modeling studies. 

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