Hougen-Watson rate equations

Example 9.3. Hougen-Watson rate equation for rate control by a reversible bi-molecular surface reaction. In the previous example, a rate equation was derived long-hand for rate control by a surface reaction... 

Table 9.1. Groups in Hougen-Watson rate equations (compiled from a table by Yang and Hougen [4]).

As already mentioned, the denominator of Langmuir-Hinshelwood-Hougen-Watson rate equations is composed of additive terms, each being proportional to one state of coverage the leading " 1" to the number of vacant sites, and each other term K, pt to the number of sites occupied species i. Thus, if a macs exists, all terms but one are negligible and if any lacs exist, their terms are negligible. 

Hougen-Watson rate equations reduce to simpler and more typical forms for initial rates, that is, if the terms involving product concentrations are omitted. This may allow the rate-controlling step to be identified or at least an incorrect one to be ruled out. Table 9.2 shows the equations for the initial rates and their dependences on initial pressure for the four Yang-Hougen stoichiometries and the three possible rate-controlling steps. 

Langmuir-Hinshelwood-Hougen-Watson rate equations... 

By comparing experimental and estimated parameters three elementary reactions have been identified to be the rate-determining steps (RDSs) for methanol synthesis. For the three RDS the rate equations [according to the general Langmuir-Hinshelwood-Hougen-Watson rate equation, Eq. (4.5.8)] have been derived ... 

This is the Michaelis-Menten equation for the rate of a simple enzymatic reaction and Km = K lk is known as the Michaelis-Menten constant. At high reactant concentration Ca much larger than iQ, the rate levels off and becomes zero order with respect to the reactant, =k Cl- At low Q (1.5.1-8) degenerates into a first order rate equation. This equation is entirely similar to the Hougen-Watson rate equations that will be derived in Chapter 2 for reactions catalyzed by solids. 

Some Eurther Thoughts on the Hougen-Watson Rate Equations... 

Hinshelwood or Hougen-Watson rate equations. For the sake of brevity in this text they will be referred to as Hougen-Watson rate equations. 

The validity of the approach is also discussed in a paper by Boudart [1986]. Certainly, the nonuniformity of catalytic surfaces, revealed by data on the heats of chemisorption, is a reality, but does this mean that a reaction necessarily senses this nonuniformity — that the reaction is structure sensitive That depends on the reaction itself, but also on the operating conditions. It may be that the reaction requires only one (or perhaps two) metal atoms or actives sites to proceed, but also that the operating conditions lead to a surface which is almost completely covered by species, so that the nonuniformities are no longer felt. In such a case the use of Hougen-Watson rate equations, based on the Langmuir isotherm, is "... not only useful, but it is also correct. In all cases their use provides physical intuition, improvable rate equations and mechanistic insight unattainable through empirical rate laws [Boudart, 1986]. Since then, further support for this point of view has been published. 

The Hougen-Watson rate equations go further than the mass action kinetic equations in that they account exphcitly for the interaction of the reacting species with the catalyst sites, but as to the mechanism they don t go very far beyond what is expressed by the stoichiometric equation, hi Sections 2.4.2 and 2.4.3 on the other hand, the reaction was decomposed in elementary steps. This is now illustrated by means of an example. 

There are only a few examples of the application of the Hougen-Watson formalism to more than a single reaction. De Deken et al. [1982] developed Hougen-Watson rate equations for steam reforming, described in terms of two parallel reactions. Marin and Froment [1982] developed a set of rate equations... 

Even with an isothermal particle and without including the film, multiplicity is theoretically possible when the rate increases with conversion as may occur for some type of Hougen-Watson rate equations and some particular parameter values. According to Luss, the sufficient condition for uniqueness of the concentration profile in the particle is... 

Such rate expressions are often termed Langmuir-Hinshelwood-Hougen-Watson (LHHW) equations and are widely used in chemical engineering [see Froment and BischofT (79)]. The usual procedure is to postulate plausible mechanisms without considering cycles, as in Example 1. In such cases it may be desirable to develop the complete list of possible direct mechanisms even if further considerations can rule out some as being unlikely. The following example illustrates a typical case. 

This Hougen-Watson type equation considers the inhibiting influence of H2O on the reaction rate, whereby the intrinsic reaction rate constant k, h2,hw and the coefficient Khw are given by... 

The quasi-equilibrium assumption is frequently used in the heterogeneous catalysis, since the surface reaction steps are often rate-Hmiting, while the adsorption steps are rapid. This is not necessarily true for large molecules. Here we consider the application of the quasi-equilibrium hypothesis on two kinds of reaction mechanisms, an Eley-Rideal mechanism and a Langmuir-Hinshelwood mechanism. The rate expressions obtained with this approach are referred to as Langmuir-Hinshelwood-Hougen-Watson (LHHW) equations in the literature, in honor of the pioneering researchers. 

Firstly, is a kinetic expression, a rate law, such as, e.g., the Langmuir-Hinshelwood-Hougen-Watson rate expressions in heterogeneous catalysis, and as such has no universal applicability. It is derived on the basis of mass action kinetics and does reduce to the fundamental thermodynamic Nemst equation for i = 0, thus q = 0. ° Nevertheless, experimental deviations can be expected as with any other, even most successful, rate expression. 

This approach was applied to data obtained by Hausberger, Atwood, and Knight (17). Figure 9 shows the basic temperature profile and feed gas data and the derived composition profiles. Application of the Hougen and Watson approach (16) and the method of least squares to the calculated profiles in Figure 9 gave the following methane rate equation ... 

The analyses developed in this section are readily extended to reactions with different stoichiometries. Regardless of whether an adsorption or a desorption process is rate limiting, the resulting rate expressions may be written in the typical Hougen-Watson fashion represented by equation 6.3.30. A comprehensive summary of such relations has been developed by Yang... 

I. Derive equations relating the initial reaction rate (7 0) to the total pressure (71) for each of the above cases when the sulfur dioxide and oxygen are initially present in equimolar amounts. Do this using the Hougen-Watson mechanistic models. Show your derivations. 

Hydrogenation of octenes occurs with surface reaction controlling (Hougen Watson, Chemical Process Principles, p 943, 1947). The rate equation is... 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

Each detailed mechanism of reaction with its controlling factor has its corresponding rate equation, involving anywhere from three to seven arbitrary constants, the K values. For reasons to be made clear, we do not intend to use equations such as these. Consequently, we do not go into their derivations. These are given by Hougen and Watson (1947), Corrigan (1954, 1955), Walas (1959), and elsewhere. 

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