Rate expression Forms

Then, the next step is the selection of the appropriate rate expression form and, subsequently, the rate parameters should be estimated by minimizing the differences between the values predicted from the selected rate expression and the data experimentally found. 

The units for k (at 25°C) depend on the rate expression form. Rate constant is dependent on the pH. 

The kinetics of reactions in which a new phase is formed may be complicated by the interference of that phase with the ease of access of the reactants to each other. This is the situation in corrosion and tarnishing reactions. Thus in the corrosion of a metal by oxygen the increasingly thick coating of oxide that builds up may offer more and more impedance to the reaction. Typical rate expressions are the logarithmic law,... 

The algebraic form of the expression (9.24) for the enhancement factor is specific to the particular reaction rate expression we have considered, and corresponding results can easily be obtained for other reactions in binary mixtures, for example the irreversible cracking A—2B. ... 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

To deal with the case of termination by combination, it is convenient to write some reactions by which an n-mer might be formed. Table 6.5 lists several specific chemical reactions and the corresponding rate expressions as well as the general form for the combination of an (n - m)-mer and an m-mer. On the assumption that all kj values are the same, we can write the total rate of change of [M -] ... 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

The term dqljdt represents the overall rate of mass transfer for component / (at time t and distance averaged over a particle. This is governed by a mass transfer rate expression which may be thought of as a general functional relationship of the form... 

The two dashed lines in the upper left hand corner of the Evans diagram represent the electrochemical potential vs electrochemical reaction rate (expressed as current density) for the oxidation and the reduction form of the hydrogen reaction. At point A the two are equal, ie, at equiUbrium, and the potential is therefore the equiUbrium potential, for the specific conditions involved. Note that the reaction kinetics are linear on these axes. The change in potential for each decade of log current density is referred to as the Tafel slope (12). Electrochemical reactions often exhibit this behavior and a common Tafel slope for the analysis of corrosion problems is 100 millivolts per decade of log current (1). A more detailed treatment of Tafel slopes can be found elsewhere (4,13,14). 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Timoshenko et al (1967) recommended running a set of experiments in a CSTR on feed composition (now called feed-forward study), and then statistically correlating the discharge concentrations and rates with feed conditions by second order polynomials. In the second stage, mathematical experiments are executed on the previous empirical correlation to find the form and constants for the rate expressions. An example is presented for the dehydrogenation of butane. 

C THE REACTION IS NOW CALCULATED USING THE EXPLICIT FORM OF C THE RATE EXPRESSION GIVEN IN THE PAPER C... 

The flow coefficient is the capacity of the flow rate expressed in dimensionless form... 

What is the reason for this dependence of the form of the rate expression on the reactivity of the aromatic compound ... 

Generally, all praetieal reaetions oeeur by a sequenee of elementary steps that eolleetively eonstitute the meehanism. The rate equation for the overall reaetion is developed from the meehanism and is then used in reaetor design. Although there are eases where experimental data provide no information about intermediate ehemieal speeies, experimental data have provided researehers with useful guidelines in postulating reaetion meehanisms. Information about intermediate speeies is essential in identifying the eorreet meehanism of reaetion. Where many steps are used, different meehanisms ean produee similar forms of overall rate expression. The overall rate equation is the result... 

Reaetions that oeeur in systems that are far removed from equilibrium give the rate expressions in the form ... 

Table 3-11 gives the initial rate data [-d(B2Hg)/dt] reported for the gas phase reaetion of diborane and aeetone at 114°C BjHg -i-dMejCO —> 2(Me2CHO)2BH. If a rate expression is of the form Rate = PmcjCO determine n, m, and k. 

Prediction and analysis of crystallizer performance is achieved by constructing models based on conservation equations and rate expressions respectively, as follows. In general form, the conservation equation is given by... 

Tliis rate expression is conslstenl widi die reaction sclieme shown in Eq. 10.6, fot-niLilaled on die basis of die Ktauss-Smidi paper. Tlius, die inilially formed cuptate diniet/enone complex widi lidiium/catbonyl and coppet/olefin cootdinalions [71, 72] Itansfbrms inlo die product via an intermediate ot inlermediales. A lidiium/ carbonyl complex also forms, bul diis is a dead-end intermediate. Tliougli detailed... 

As we will see shortly, the rate expression can take various forms, depending on the nature of the reaction. It can be quite simple, as in the N2Os decomposition, or exceedingly complex. 

Notice from the rate expressions just written that the rate of an elementary step is equal to a rate constant k multiplied by the concentration of each reactant molecule. This rule is readily explained. Consider, for example, a step in which two molecules, A and B, collide effectively with each other to form C and D. As pointed out earlier, the rate of collision and hence the rate of reaction will be directly proportional to the concentration of each reactant. 

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

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

In earlier sections, we described various single-term rate expressions in which v = fc[A]n, where n has the values 0, j, 1, , and 2. One can decide which, if any, of these forms is correct by fitting the data with any of several approaches. 

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]. 

Since the slow step involves only the substrate, the rate should be dependent only on the concentration of that. Although the solvent is necessary to assist in the process of ionization, it does not enter the rate expression, because it is present in large excess. However, the simple rate law given in Eq. (10.3) is not sufficient to account for all the data. Many cases are known where pure first-order kinetics are followed, but in many other cases more complicated kinetics are found. We can explain this by taking into account the reversibility of the first step. The X formed in this step competes with Y for the cation and the rate law must be modified as follows (see Chapter 6) ... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

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... 

This definition for reaction order is directly meaningful only for irreversible or forward reactions that have rate expressions in the form of Equation (1.20). Components A, B,... are consumed by the reaction and have negative stoichiometric coefficients so that m = —va, n = —vb,. .. are positive. For elementary reactions, m and n must be integers of 2 or less and must sum to 2 or less. 

Equation (1.20) is frequently used to correlate data from complex reactions. Complex reactions can give rise to rate expressions that have the form of Equation (1.20), but with fractional or even negative exponents. Complex reactions with observed orders of 1/2 or 3/2 can be explained theoretically based on mechanisms discussed in Chapter 2. Negative orders arise when a compound retards a reaction—say, by competing for active sites in a heterogeneously catalyzed reaction—or when the reaction is reversible. Observed reaction orders above 3 are occasionally reported. An example is the reaction of styrene with nitric acid, where an overall order of 4 has been observed. The likely explanation is that the acid serves both as a catalyst and as a reactant. The reaction is far from elementary. 

An irreversible, elementary reaction must have Equation (1.20) as its rate expression. A complex reaction may have an empirical rate equation with the form of Equation (1.20) and with integral values for n and w, without being elementary. The classic example of this statement is a second-order reaction where one of the reactants is present in great excess. Consider the slow hydrolysis of an organic compound in water. A rate expression of the form... 

Studies on similar catalysts have suggested a rate expression of the form... 

For enzymatic and other heterogeneously catalyzed reactions, there may be competition for active sites. This leads to rate expressions with forms such as... 

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

As given above, the statements that adjust the exponents m and n have been commented out and the initial values for these exponents are zero. This means that the program will fit the data to. = k. This is the form for a zero-order reaction, but the real purpose of running this case is to calculate the standard deviation of the experimental rate data. The object of the fitting procedure is to add functionality to the rate expression to reduce the standard deviation in a manner that is consistent with physical insight. Results for the zero-order fit are shown as Case 1 in the following data ... 

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