Approximate rate expressions

Approximate Rate Expressions for Muitipie-Reaction Systems 181... 

APPROXIMATE RATE EXPRESSIONS FOR MULTIPLE-REACTION SYSTEMS... 

This reaction system is an example of a chain reaction, which we will consider in more detail in Chapter 10. However, here we will just use the pseudo-steadly-state approximation to find the approximate rate expression above. The only reactant is acetaldehyde, and there are six products listed CH4, CO, CH3, CH3O, CHO, and C2H6. The first two, CH4 and CO, are the major products C2H6 is a stable but minor product. The other species, CH3, CH3CO, and CHO, are free radicals that are very reactive and never build up to high concentrations. 

We are interested in obtairiitig approximate rate expressions that do have power-law dependences on partial pressures. We are interested in writing these rates as... 

Hydrogen reacts in the gas phase with chlorine or iodine as well as with bromine, but there are differences between the three reactions. The reaction of hydrogen with chlorine is inhibited by oxygen. An approximate rate expression for the thermally initiated reaction is... 

Also shown are the corresponding curves calculated for the same system assuming a diffusion model in place of the linear rate expression. For intracrystalline diffusion k = 15Dq/v, whereas for macropore diffusion k = 15e /R ) Cq/q ), in accordance with the Glueckauf approximation (21). 

At the other extreme, when the ratio ki /mkc is much smaller than unity, the interfacial concentration of reactant A may be approximated by the equihbrium relation Xi = y/m, and the specific absorption rate expression is... 

Linear Driving Force Approximation Simplified expressions can also be used for an approximate description of adsorption in terms of rate coefficients for both extrapai ticle and intraparticle mass transfer controlling. As an approximation, the rate of adsorption on a particle can be written as ... 

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 ... 

Applying the rate expressions to Equations 1-222, 1-223, 1-224, 1-225 and 1-226, and using the steady state approximation for CH3, C2H5, and H, for a eonstant volume bateh reaetor yields ... 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

Ng et al. [1261] report that dehydration of copper sulphate pentahydrate (- CuS04 3 H20) 320—336 K, obeys the Avrami—Erofe ev equation [eqn. (6), n = 2] with E = 104 kj mole-1. Dehydration of the trihydrate (- CuS04 H20), 343.5—359 K, obeyed the same rate expression with E = 134 kJ mole 1. Activation energies are approximately equal to reaction enthalpies. 

We shall now consider a simplifying approximation for the system A 2P. The reaction proceeds at a rate expressed in terms of S by Eq. (3-33). If the shift resulting from the concentration jump is small, the term AK l82 is negligible in comparison to (1 + 4 l [P](,)S. In that case, the solution is... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

Rate expression. The oxidation of SOj- to SO4- by molecular oxygen is catalyzed by traces of Cu2+ and inhibited by small amounts of alcohols. Derive the expression for -d[SO ]/dt by making the usual approximations, assuming the following mechanism ... 

Derive the rate expression. Make the steady-state approximation for the radical intermediates, and assume that the chains are long. 

There are various approximations (7) to the above expression for the absorption rate Rj that offer further insight into the photon absorption process and form a basis for comparison to the non Bom-Oppenheimer rate expression. The most classical (and hence, least quantum) approximation is to ignore the fact that the kinetic energy operator T does not commute with the potentials Vj f and thus to write... 

Bearing in mind the discussion of the nature of the electronic non BO matrix elements (Q) given in Sec. I. C, the above rate expression can be further approximated by constraining Q and Q to the region Q =Q=Qo where the anion and neutral surfaces approach most closely ... 

Evaluation of F(x) for Second Order Deactivation. As mentioned earlier for the case of second order decay F(x) cannot be derived analytically, however numerical calculation of F(x) or Its evaluation from simulated rate data Indicates that the function defined In Equation 11 provides an excellent approximation. This was also confirmed by the good fit of model form 12 to simulated polymerization data with second order deactivation. Thus for second order deactivation kinetics the rate expression Is Identical to Equation 12 but with 0 replacing 02. 

Note here that at high pressures of M, fe[M]kj. and Eq. (9.3) reduces to the first-order rate expression v (A i 3/ 2)[A] = A [A], whereas at low pressures 2[1V[] -C kj. and the expression becomes v A i[A][M], the normal second-order form. (Approximations such as these are commonly used in many areas of science and mathematics.)... 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory ... 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

The rate expression for each intermediate in Figure 3.2C can be derived based on the Bodenstein approximation of quasi-stationaiy states of trace-level... 

Instead of using the steady-state approximation in the manipulation of the individual rate expressions, the same result may be reached by assuming that a pseudo equilibrium condition is established with respect to reaction B and that reaction C continues to be the rate limiting... 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

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

Since the branching parameter a is greater than unity (usually it is 2), it is conceivable that under certain circumstances the denominator of the overall rate expression could become zero. In principle this would lead to an infinite reaction rate (i.e., an explosion). In reality it becomes very large rather than infinite, since the steady-state approximation will break down when the radical concentration becomes quite large. Nonetheless, we will consider the condition that Mol - 1) is equal to (fst T fgt) to be a valid criterion for an explosion limit. 

Reaction rate expressions for enzymatic reactions are usually derived by making the Bo-denstein steady-state approximation for the intermediate enzyme-substrate complexes. This is an appropriate assumption when the substrate concentration greatly exceeds that of the enzyme (the usual laboratory situation) or when there is both a continuous supply of reactant and a continuous removal of products (the usual cellular situation). 

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