The rate expression

MICROSCOPIC EXPRESSIONS FOR THE RATE CONSTANT 4.3.1. The Rate Expression 

The rate constant is given by the number of collisions between the molecules multiplied by the probability of reaction. The number of collisions follows from the average velocity of the molecules and their effective cross section for collision. The energy distribution over the different degrees of freedom of the molecules is equal to the Boltzmann distribution as long as the number of collisions between the molecules is large compared to their probability of reaction. 

An expression will be derived for the rate constant kf Clearly, once k is known, k follows from the relation between equilibrium constant and rate constants (see (2.3)). The rate constant can be computed from the cross section Of. or a collision of molecule A with B, and the probability that the collision leads to a reaction. 

In general, A and B are molecules which consist of two or more atoms. Each molecule has vibrational and rotational degrees of freedom, denoted with quantum numbers i and j. In addition, the centers of mass of A and B have velocities u and ub, and a relative velocity between A and B can be defined as 

When A and B react, they necessarily collide. The energy involved in the collision will be redistributed over the translational (kinetic), vibrational and rotational energy of the product molecules, in a way that total energy and momentum are conserved. 

The functional dependence of the rate on system parameters is called the rate expression. This is frequently referred to as the rate law, but the term law ought to be reserved for fundamental principles, and not used to describe an empirical equation. Important parameters are the concentrations of the reactants and (sometimes) the products, and the temperature. Other quantities may influence the reaction rate. In gas phase reactions total pressure may play a key role (see Chapter 4), while for reactions in solution bulk properties such as ionic strength and dielectric constant may influence the rate. Rate expressions for stoichiometric reactions must be empirically determined, taking any mathematical form that fits the data. The rate will be expressed as a function of temperature, concentrations of species present, and any other parameters that are found to influence the rate 

A form that is frequently found for reactions where the rate depends only on concentrations and temperature is 

For other functional forms of the rate expression the overall order may not be defined. In addition, the rate expression may vary as the reaction proceeds. The rate expression found from experiments done at an early stage of the reaction, where the reactants are not appreciably converted to products, may be different from the rate expression found at later stages of the reaction. 

Although the rate expressions for stoichiometric reactions are empirical, the mathematical form of the rate expression for the reaction proceeding in the reverse direction is constrained by thermodynamics and the rate expression in the forward direction for reactions of the form of equation (8). Denbigh [1] has shown that the reaction orders with respect to each component in the reaction mixture are related by the equation 

There is a large literature describing experimental methods for finding empirical rate expressions and analyzing rate data. Comprehensive treatments can be found in volume 8 of the Weissburger series [2] and in volume 1 of the Comprehensive Chemical Kinetics series [3]. In addition, almost every textbook on kinetics has material on this subject. See, for example, the texts by Espenson [4], Laidler [5], and Steinfeld et al [6]. This book is devoted to detailed chemical kinetic modeling of chemical reactions, and is concerned primarily with elementary reactions. 

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A rather different method from the preceding is that based on the rate of dissolving of a soluble material. At any given temperature, one expects the initial dissolving rate to be proportional to the surface area, and an experimental verification of this expectation has been made in the case of rock salt (see Refs. 26,27). Here, both forward and reverse rates are important, and the rate expressions are... 

Rate fonrrulations that treat tire iinrer-sphere nrode(s) quairtum nrechairically aird tire outer sphere modes classically are used ratlrer widely. The rate expression for a single hanrronic quairtum mode is... 

Low temperatures strongly favor the formation of nitrogen dioxide. Below 150°C equiUbrium is almost totally in favor of NO2 formation. This is a slow reaction, but the rate constant for NO2 formation rapidly increases with reductions in temperature. Process temperatures are typically low enough to neglect the reverse reaction and determine changes in NO partial pressure by the rate expression (40—42) (eq. 13). The rate of reaction, and therefore the... 

Collecting all of the rate expressions for these five reactions to yield one equation for the formation of HBr it was found that the reaction rate could be described by... 

At low temperatures, when the bath is quantum (icoj P 1), the rate expression, expanded in series over the coupling strength, breaks up into the contributions from the various processes involving the bath phonons... 

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 UCKRON AND VEKRON kinetics are not models for methanol synthesis. These test problems represent assumed four and six elementary step mechanisms, which are thermodynamically consistent and for which the rate expression could be expressed by rigorous analytical solution and without the assumption of rate limiting steps. The exact solution was more important for the test problems in engineering, than it was to match the presently preferred theory on mechanism. 

If [D] is much greater than k i, the rate expression simplifies to... 

Because proton-transfer reactions between oxygen atoms are usually very fast, step 3 can be assumed to be a rapid equilibrium. With the above mechanism assume4 let us examine the rate expression which would result, depending upon which of the steps is rate-determining. 

If step 1 is rate-controlling, the rate expression would be rate = it,[PhCOCH3][OEt]... 

Under these conditions, the concentration of the second reactant, benzaldehyde, would not enter into the rate expression. 

If step 1 is an equilibrium and step 2 is rate-controlling, we obtain the rate expression rate = i [PhCOCH2 ][PhCHO]... 

The addition of hydrogen chloride to olefins in nitromethane follows the rate expression... 

The rates of the reactions of several aromatic ketones with alkyllithium reagents have been examined. The reaction of 2,4-dimethyl-4 -(methylthio)benzophenone with methyl-lithium in ether exhibits the rate expression ... 

The hydrolysis of the lactone A shows a signifieant catalysis by acetate ion in acetate buffer, with the rate expression being... 

Molecular bromine is believed to be the reactive brominating agent in uncatalyzed brominations. The brominations of benzene and toluene are first-order in both bromine and the aromatic substrate in trifluoroacetic acid solution, but the rate expressions become more complicated when these reactions take place in the presence of water. " The bromination of benzene in aqueous acetic acid exhibits a first-order dependence on bromine concentration when bromide ion is present. The observed rate is dependent on bromide ion concentration, decreasing with increasing bromide ion concentration. The detailed kinetics are consistent with a rate-determining formation of the n-complex when bromide ion concentration is low, but with a shift to reversible formation of the n-complex... 

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

Structure-reactivity relationships can be probed by measurements of rates and equiUbria, as was diseussed in Chapter 4. Direct comparison of reaction rates is used relatively less often in the study of radical reactions than for heterolytic reactions. Instead, competition methods have frequently been used. The basis of competition methods lies in the rate expression for a reaction, and the results can be just as valid a comparison of relative reactivity as directly measured rates, provided the two competing processes are of the same kinetic order. Suppose that it is desired to compare the reactivity of two related compounds, B—X and B—Y, in a hypothetical sequence ... 

The slope of a concentration-time curve to define the rate expression can be determined. However, experimental studies have shown the reaction cannot be described by simple kinetics, but by the relationship ... 

This confirms the half order of the rate expression. 

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 ... 

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

Case 2 involves a system with a single reaetant. In this ease, the rate at any instant is proportional to the square of the eoneentration of A. The reaetion meehanism is 2A—produets. The rate expression for a eonstant volume bateh system (i.e., eonstant density) is... 

Reaction Rate Expression Linearizing the rate expression gives... 

Since the reaction is carried out in a batch system of constant volume, the rate expression for a second order rate law is... 

For the synthesis of ammonia, Nj -i- 3H2 —> 2NH3, over an iron catalyst, develop the rate expression for the following mechanism... 

The rate expression for die above mechanism is based on die following assumptions ... 

Sinee step 2 is the rate-determining step, tlien the rate expression is ... 

Substituting Equation 3-351 into the rate expression of Equation 3-340 gives... 

Adesina has shown that it is superfluous to carry out the inversion required by Equation 5-255 at every iteration of the tri-diagonal matrix J. The vector y"is readily computed from simple operations between the tri-diagonal elements of the Jacobian matrix and the vector. The methodology can be employed for any reaction kinetics. The only requirement is that the rate expression be twice differentiable with respect to the conversion. The following reviews a second order reaction and determines the intermediate conversions for a series of CFSTRs. 

Assume that the reaction between A and B is second order and is represented by A -i- B —> products where A is the limiting reactant. The rate expression is... 

The design equations for a CFSTR with perfeet mixing, eonstant fluid density, and steady state operation are as follows. If u is the volumetrie flowrate and K = kj/kj, relative reaetion rate eonstant, where kj, kj, and kj are the speeifie reaetion rate eonstants for reaetions 5-357, 5-358, and 5-359. The rate expressions of A, B, R, S, and T are... 

CH3COOH or A -h B —> 2C, where component A is the acetic anhydride. The rate expression is first order in terms of component A. 

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