Hinshelwoods Treatment

The first difficnlty of the Lindemann-Christiansen mechanism is associated with the fact that first-order rates are maintained down to lower concentrations than appear to be 

Maxwell and Boltzmann, in their treatment of the distribntion of molecnlar speeds and energies, demonstrated that the fraction of molecules having energies between 

This relationship applies to a molecule moving in the x direction. 

The fraction of molecules, /, that have energies above a certain specified quantity e is 

The result of the integration cannot be expressed in a closed form. However, if one considers molecular motions along two directions, x and y, the probability of finding the molecule with a translational energy between and fix + de,. in the jc direction and e and ej, + dey along the y direction is 

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The Langmuir-Hinshelwood treatment of the kinetics of surface catalyzed reactions affords a useful representation of some of the characteristics of catalytic hydrogenation. It is a limiting form of more exact equations which recognize that, even though the elementary steps are reversible, few if any will be at equilibrium (ref. 15). Not surprisingly, alternative assumptions regarding the relative rates of the forward and reverse elementary reactions can lead to approximate equations of the same form. 

The Langmuir-Hinshelwood treatment has also been modified to account for the more complicated conditions presented by competitive adsorption between two or more species for a single adsorption site such as might occur within multicomponent systems or with intermediates formed during multi-step oxidation processes [3, 151-153]. This modification yields the following expression for 0sub ... 

The correct treatment of the mechanism (equation (A3.4.25), equation (A3.4.26) and equation (A3.4.27), which goes back to Lindemann [18] and Hinshelwood [19], also describes the pressure dependence of the effective rate constant in the low-pressure limit ([M] < [CHoNC], see section A3.4.8.2). 

The treatment of Hinshelwood and further modifications may be considered in terms of the following scheme ... 

In Hinshelwood s treatment, the molecule A is allowed to acquire an amount of energy El at an enhanced rate. The rate at which A converts to A is independent of that energy. Let us take the expression for first order rate constant given by Lindemann s theory, i.e. 

In case of Hinshelwood s modification both k2 and (k lk ) have been treated as independent of E, i.e. amount of energy in the energized molecule. In Hishelwood s treatment the critical energy El is involved, not E. ... 

However, the following difficulties still remain, which could not be explained on the basis of Hinshelwood s treatment. 

The first of the shortcomings of the Lindemann theory—underestimating the excitation rate constant ke—was addressed by Hinshelwood [176]. His treatment showed that ke can be much larger than predicted by simple collision theory when the energy transfer into the internal (i.e., vibrational) degrees of freedom is taken into account. As we will see, some of the assumptions introduced in Hinshelwood s model are still overly simplistic. However, these assumptions allowed further analytical treatment of the problem in an era long before detailed numerical solution was possible. 

The overall model, too complex, can be converted in the case of limited H2 pressure, where Langmuir-Hinshelwood kinetics simplify to 1st order [47] (more complicated mathematical treatments can nevertheless be made, as shown by Aris [48]). 

In deriving the kinetic equations of heterogeneous catalytic reactions, the surface concentrations are assumed to be steady state (or stationary), as has been done by Langmuir in the previously mentioned study of the reactions of CO and H2 with 02 on platinum (22). The treatment of surface reactions as including adsorption equilibria widely used by Hinshelwood and other authors is a particular case of this more general approach of Langmuir. 

More sophisticated treatments of Lindemann s scheme by Lindemann— Hinshelwood, Rice—Ramsperger—Kassel (RRK) and finally Rice— Ramsperger—Kassel—Marcus (RRKM) have essentially been aimed at re-interpreting rate coefficients of the Lindemann scheme. RRK(M) theories are extensively used for interpreting very-low-pressure pyrolysis experiments [62, 63]. 

The preceding treatment is, undoubtedly, an oversimplification. For example, many diatomic molecules dissociate upon adsorption (e.g., H2, SiH, GeH). Each atom from the dissociated molecule then occupies its own distinct surface site and this naturally changes the rate law expression. When these types of details are accounted for, the Langmuir-Hinshelwood mechanism has been very successful at explaining the growth rates of a number of thin-film chemical vapor deposition (CVD) processes. However, more important, our treatment served to illustrate how crystal growth from the vapor phase can be related to macroscopic observables namely, the partial pressures of the reacting species. 

Besides volume processes wall collisions of hydrogen particles can contribute to the vibrational population. A direct process is the interaction of already vi-brationally excited molecules with the surface (v) +wall —> ff2(w) mostly depopulating the vibrational levels. Further fundamental mechanisms are the Langmuir-Hinshelwood and the Eley-Rideal mechanism. They are based on recombining hydrogen atoms or ions Hads/gas + Hads —> H2(v). In the first case an adsorbed particle at the surface recombines with another adsorbed particle (Langmuir-Hinshelwood mechanism). In the second case one particle from the gas phase recombines with an adsorbed particle (Eley-Rideal mechanism). For these processes the data base is scarce and often not determined from plasma material interaction experiments. A dependence on particle densities, surface material and surface treatment as well as surface temperature can be expected. 

These observations all point clearly to the occurrence of a branched chain reaction with chain breaking at the vessel surface. The expression for the rate of a chain reaction is shown by the formal treatments of Semenov and Hinshelwood to have the general form... 

Developing the argument further by means of the treatment of Semenov [18] and Hinshelwood and Williamson [1], let us suppose that in the hydrogen—oxygen system near the lower limit we have two kinds of active particles, Xq and Xh, and let Xq react with hydrogen and Xh with oxygen thus... 

In contrast to the work of Hinshelwood et al., Kassel found the reaction order to be y and concluded that the mechanism is complex. It has been pointed out ° , however, that the treatment employed as well as the interpretation given by Kassel are both disputable. 

This study presents the kinetic parameters and reactivity profiles for steam gasification of birch and beech char. The inhibition effect of hydrogen is also studied using Langmuir-Hinshelwood kinetics. In addition, the influence of the treatment of the experimental results is analysed by comparing the kinetic parameters differently obtained from the same experiments. 

The analysis of Langmuir [/. Am. Chem. Soc. 40 1361 (1918)] and Hinshelwood (Kinetics of Chemical Change, Oxford, 1940) form the basis for the simplified treatment of kinetics on heterogeneous catalysts. For a solid catalyzed reaction between gas phase reactants A and B, the postulated mechanism consists of the following steps in series ... 

In the treatment above it was assumed that two adsorbed species react forming a product, the so-called case of Langmuir-Hinshelwood reaction (Figure 3.10). 

In the case that KAPA the rate is first order in both reactants similar to the Langmuir-Hinshelwood mechanism. However, the rate cannot have negative order dependence as was visible from LH treatment. 

The rate V of the reaction is now expressed by the current treatment, in terms of the mass action law, as proportional to 0(H2)0(CO2) for the Langmuir-Hinshelwood mechanism and similarly to (7 0(CO2) or C ° 0(H2) for the Rideal-Eley mechanism, depending on the premised adsorption state of the initial system. The rate law is thus obtained according to Eqs. (11.24) as... 

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