Derivation of rate expressions

Examples of derivation of rate expressions Rj — R,( T, ck, 6m x) are presented in the following sections, depending on the amount of knowledge on the catalytic chemistry, either empirical or mechanistically consistent forms can be adopted. 

The formulation of rate expressions for surface reactions has been described in detail by Laidler [42]. In this section, we confine our attention to its application to the derivation of rate expressions relevant to hydrogenation reactions. Again, it is necessary to specify the mechanism in order to derive relevant rate expressions. 

In the approach to the derivation of rate expressions two principles are important ... 

In Sections 3.2 and 3.4, the approach based on one rate-determining step with the other steps in quasi-equilibrium was applied to simplify the derivation of rate expressions. A further simplification of the rate expressions is obtained when the product concentrations or partial pressures are negligibly small, i.e., at low conversions of a pure reactant feed stream. Similarly, at high conversions, when nearly only product is present, the rate expressions can also be simplified. Applying e single-site model, Eqns. (3.17)-(3.19), for the reaction A B at low conversion, the following expressions (often called "initial rate expressions") are obtained, provided that the feed contains pure A. 

Apply pseudo-steady-state and pseudoequilibrium approximations for the derivation of rate expressions from the postulated reaction mechanisms. 

See, for example, Poliak E 1986 Theory of activated rate processes a new derivation of Kramers expression J. Chem. Phys. 85 865... 

Flotation process kinetics determine the residence time, the average time a given particle stays in the flotation pulp from the instant it enters the ceU until it exits. One way to study flotation kinetics is to record flotation recoveries as a function of time under a given set of conditions such as pulp pH, coUector concentration, particle size, etc. The data allow the derivation of an expression that describes the rate of the process. 

A detailed study of the solvolysis of L has suggested the following mechanism, with the reactivity of the intermediate M being comparable to that of L. Evidence for the existence of steps ki and k 2 was obtained fiom isotopic scrambling in the sulfonate M when it was separately solvolyzed and by detailed kinetic analysis. Derive a rate expression which correctly describes the non-first-order kinetics for the solvolysis of L. 

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

The account of the formal derivation of kinetic expressions for the reactions of solids given in Sect. 3 first discusses those types of behaviour which usually generate three-dimensional nuclei. Such product particles may often be directly observed. Quantitative measurements of rates of nucleation and growth may even be possible, thus providing valuable supplementary evidence for the analysis of kinetic data. Thereafter, attention is directed to expressions based on the existence of diffuse nuclei or involving diffusion control such nuclei are not susceptible to quantitative... 

Reactions 2 and 3 regulate the balance of O and O3, but do not materially affect the O3 concentration. Any ozone destroyed in the photolysis step (3) is quickly reformed in reaction 2. The amount of ozone present results from a balance between reaction 1, which generates the O atoms that rapidly form ozone, and reaction 4, which eliminates an oxygen atom and an ozone molecule. Under conditions of constant sunlight, which implies constant /i and /s, the concentrations of O and O3 remain constant with time and are said to correspond to the steady state. Under steady-state conditions the concentrations of O and O3 are defined by the equations d[0]/df = 0 and d[03]/df = 0. Deriving the rate expressions for reactions 1-i and applying the steady-state condition results in the equations given below that can be solved for [O] and [O3]. 

Derive the rate expression for the formation of CH4 by using the steady state assumption. 

The assumptions inherent in the derivation of the Hertz-Knudsen equation are (1) the vapor phase does not have a net motion (2) the bulk liquid temperature and corresponding vapor pressure determine the absolute rate of vaporization (3) the bulk vapor phase temperature and pressure determine the absolute rate of condensation (4) the gas-liquid interface is stationary and (5) the vapor phase acts as an ideal gas. The first assumption is rigorously valid only at equilibrium. For nonequilibrium conditions there will be a net motion of the vapor phase due to mass transfer across the vapor-liquid interface. The derivation of the expression for the absolute rate of condensation has been modified by Schrage (S2) to account for net motion in the vapor phase. The modified expression is... 

Since the problem of deriving a rate expression from a postulated set of elementary reactions is simpler than that of determining the mechanism of a reaction, and since experimental rate expressions provide one of the most useful tests of reaction mechanisms, we will now consider this problem. 

ILLUSTRATION 4.3 USE OF THE BODENSTEIN STEADY-STATE APPROXIMATION TO DERIVE A RATE EXPRESSION FROM A CHAIN REACTION MECHANISM... 

Derive a rate expression that is consistent with this mechanism. Treacy and Daniels [J. Am. Chem. Soc., 11 (2033), 1955] have determined that the orders of the reaction with respect to oxygen and nitric oxide are one and two, respectively, at high pressures and less than one and greater than two at low pressures. Is the proposed mechanism consistent with this data ... 

The major problem challenging a quantitative theory of a copolymerization is the derivation of the expressions for the rate of this process and for the statistical characteristics of the chemical structure of its products. Among the latter in the case of multiblock copolymers is the size-composition distribu-... 

Example 7.1 Derivation of the expression [Equation (7.2)]for the maximum specific growth rate, (JLh> for the heterotrophic biomass... 

The plateau currents are thus a function of two dimensionless parameters, Jis/ik and 4/4(1 — k/i )- On this basis, a kinetic zone diagram may be established (Figure 4.19) as well as the expressions of the plateau currents pertaining to each kinetic zone (Table 4.1).17 Derivation of these expressions is described in Section 6.4.4. There are in most cases two successive waves, and the expressions of both limiting currents are given in Table 4.1. The general case corresponds to a situation where none of the rate-limiting factors... 

In accordance with the derivation of an expression for the regulation of the lac operon by Yagil and Yagil (83), the relationships discussed above between the relative rates of enzyme synthesis, a, and effector concentration, E, were evaluated. From... 

L t PJ sents the monomer concentration at time t, as obtained from Fourier treinsform 13C-NMR data. The rate consteint of propagation kp can now be determined by measuring Cm] as a function of polymerization time, t. (For the derivation of this expression, see the Appendix.)... 

Once we have derived a rate expression and carefully checked the derivation, the sequence of steps in the calculation of the rate is quite different from the the sequence of steps in the derivation of the rate expression. 

The application of Absolute Rate Theory to the interpretation of catalytic hydrogenation reactions has received relatively little attention and, even when applied, has only achieved moderate success. This is, in part, due to the necessity to formulate precise mechanisms in order to derive appropriate rate expressions [43] and, in part, due to the necessity to make various assumptions with regard to such factors as the number of surface sites per unit area of the catalyst, usually assumed to be 10 5 cm-2, the activity of the surface and the immobility or otherwise of the transition state. In spite of these difficulties, it has been shown that satisfactory agreement between observed and calculated rates can be obtained in the case of the nickel-catalysed hydrogenation of ethylene (Table 3), and between the observed and calculated apparent activation energies for the... 

The partial -derivative of this expression therefore gives the rate of local entropy production... 

The general forms of rate expressions in heterogeneous systems can have concentration or partial pressure dependences in both numerator and denominator along with various exponents. In heterogeneous reactors, it is not unusual to derive kinetic expressions that are more complicated than just a power-law expression. This, of course, has implications on how the reactor is controlled and the potential for runaway in exothermic systems. In some cases, where kinetics are very fast relative to mass transfer rates, the reactor behavior is governed by mass transfer and the variables that affect it. 

Problem 2 What do you understand by a zero order reaction Derive the rate expression for it. What are the characteristics of zero order reactions Given an example of such a reaction. 

Problem 7 What is a second order reaction Derive the rate expression for a second order reaction. Discuss the characteristics of a second order reaction. Write some examples of second order reactions and study the kinetics of any one reaction. (Meerut 2002,2000)... 

Problem 9 What are third order reactions Derive the rate expressions and discuss their characteristics. Give examples of third order reactions. 

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