Collision theory rate constants from

Compare the collision theory rate with the database value and calculate the steric factor, that is, the ratio between the measured rate constant and the rate constant estimated from collision theory. 

For parameterization of the adsorption/desorption steps, we choose the adsorption direction. Accordingly, we define the adsorption rate constant from collision theory as... 

A general expression taking into account the rotational energy was derived from RRKM theory.29 If the intermediate C is sufficiently short-lived (or the total pressure is sufficiently low) that it is not stabilized by collisions, the rate constant k for the formation of the product(s) can be written as... 

Expression (67.Ill) can be considered as a "statistical formulation of the rate constant in that it represents a formal generalization of activated complex theory which is the usual form of the statistical theory of reaction rates. Actually, this expression is an exact collision theory rate equation, since it was derived from the basic equations (32.Ill) and (41. HI) without any approximations. Indeed, the notion of the activated complex has been introduced here only in a quite formal way, using equations (60.Ill) and (61.Ill) as a definition, which has permitted a change of variables only in order to make a pure mathematical transformation. Therefore, in all cases in which the activated complex could be defined as a virtual transition state in terms of a potential energy surface, the formula (67.HI) may be used as a rate equation equivalent to the collision theory expression (51.III). 

Now that we have a model, we must check its consistency with various experiments. Sometimes such inconsistencies result in the complete rejection of a model. More often, they indicate that we need to refine the model. In the present case, the results of careful experiments show that the collision model of reactions is not complete, because the experimental rate constant is normally smaller than predicted by collision theory. We can improve the model by realizing that the relative direction in which the molecules are moving when they collide also might matter. That is, they need to be oriented a certain way relative to each other. For example, the results of experiments of the kind described in Box 13.2 have shown that, in the gas-phase reaction of chlorine atoms with HI molecules, HI + Cl — HC1 I, the Cl atom reacts with the HI molecule only if it approaches from a favorable direction (Fig. 13.28). A dependence on direction is called the steric requirement of the reaction. It is normally taken into account by introducing an empirical factor, P, called the steric factor, and changing Eq. 17 to... 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

The pre-exponential factor for the H -i- H2 reaction has been determined to be approximately 2.3 x lO " mol cm s . Taking the molecular radii for H2 and H to be 0.27 and 0.20 nm, respectively, calculate the value of the probability factor P necessary for agreement between the observed rate constant and that calculated from collision theory at 300 K. 

A pre-exponential factor and activation energy for each rate constant must be established. All forward rate constants involving alkyne adsorption (ki, k2, and ks) are assumed to have equal pre-exponential factors specified by the collision limit (assuming a sticking coefficient of one). All adsorption steps are assumed to be non-activated. Both desorption constants (k.i and k ) are assumed to have preexponential factors equal to 10 3 sec, as expected from transition-state theory [28]. Both desorption activation energies (26.1 kcal/mol for methyl acetylene and 25.3 kcal/mol for trimethylbenzene) were derived from TPD results [1]. 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

The reactions of the bare sodium ion with all neutrals were determined to proceed via a three-body association mechanism and the rate constants measured cover a large range from a slow association reaction with NH3 to a near-collision rate with CH3OC2H4OCH3 (DMOE). The lifetimes of the intermediate complexes obtained using parameterized trajectory results and the experimental rates compare fairly well with predictions based on RRKM theory. The calculations also accounted for the large isotope effect observed for the more rapid clustering of ND3 than NH3 to Na+. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

A comparison of equations (4.54) and (4.55) shows that the rate constant for a complex reaction differs from that obtained in simple atomic reaction by a factor of (qjqr)5. Since qv is nearly unity, while qr varies from 10 to 100 for a complex molecule, the ratio qv/qr, therefore, varies from 10 I to 10 2 and (qv/qT)5 varies from 10 5 to 10 10. This factor may link to steric factor p. On comparing equation (4.55) with collision theory and Arrhenius equation, we get... 

The value of first order rate constant (k1), i.e. k ko/ki (say k,J at high concentration is determined from experiments and has also been calculated from simple collision theory (i.e. rate constant = Ze E ,c IRT). In all cases it has been observed that rate constant dropped at much higher concentration than is actually observed. Since there can be no doubt about kwhich is an experimental quantity, the error must be in the estimation of rate constant. 

Compare the rate constant obtained from collision theory and transition state... 

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

In Table 1 (pp. 251-254), IM rate constants for reaction systems that have been measured at both atmospheric pressure and in the HP or LP range are listed. Also provided are the expected IM collision rate constants calculated from either Langevin or ADO theory. (Note that the rate constants of several IM reactions that have been studied at atmospheric pressure" are not included in Table I because these systems have not been studied in the LP or HP ranges.) In general, it is noted that pressure-related differences in these data sets are not usually large. Where significant differences are noted, the suspected causes have been previously discussed in Section IIB. These include the reactions of Hcj and Ne with NO , for which pressure-enhanced reaction rates have been attributed to the onset of a termolecular collision mechanism at atmospheric pressure and the reactions of Atj with NO and Cl with CHjBr , for which pressure-enhanced rate constants have been attributed to the approach of the high-pressure limit of kinetic behavior for these reaction systems. 

Arrhenius recognized that for molecules to react they must attain a certain critical energy, E. On the basis of collision theory, the rate of reaction is equal to the number of collisions per unit time (the frequency factor) multiplied by the fraction of collisions that results in a reaction. This relationship was first developed from the kinetic theory of gases . For a bimolecular reaction, the bimolecular rate constant, k, can be expressed as... 

Because a is a parameter that cannot be calculated from first principles. Equation 1-95 cannot be used to calculate reaction rate constant k from first principles. Furthermore, the collision theory applies best to bimolecular reactions. For monomolecular reactions, the collision theory does not apply. Tr3dng to calculate reaction rates from first principles for all kinds of reactions, chemists developed the transition state theory. 

Collision theory is based on the concept that molecules behave like hard spheres during a collision of two species, a reaction may occur. To estimate a rate constant for a bimolecular reaction between reactants A and B based on this theory, one needs first to calculate the number of collisions occurring in a unit volume per second (ZA1 ) when the two species, A and B, having radii rA and ru, are present in concentrations jVa and Aru, respectively. From gas kinetic theory, this can be shown to be given by Eq. (I) ... 

The preexponential factor involves the entropy change in going from reactants to the transition state the more highly ordered and tightly bound is the transition state, the more negative A S° will be and the lower the preexponential factor will be. Transition state theory thus automatically takes into account the effect of steric factors on rate constants, in contrast to collision theory. 

Let the flow of molecules into the Knudsen cell be F (molecules s l). In the absence of the reactive surface, these molecules are removed when they strike the escape aperture into the mass spectrometer. Let kCM. be the effective first-order rate constant (s ) for escape of the gas from the cell through this orifice, which can be measured experimentally. Alternatively, kcsc can be calculated from kinetic molecular theory since the number of collisions per second, Js, of a gas on a... 

The rate constant for adsorption, A , is also temperature dependent, but the dependence is small compared to that for k r The value and temperature dependence of k, are determined by the rate of gas-solid collisions, which from kinetic molecular theory is given by... 

Schweikert s theory differs radically from the conventional thermohydrodynamic Chapman-Jouguet theory in that it provides for a continuous transition from burning to deton. In Section I entitled "Introduction , the author criticizes the validity of the C-J theory for condensed expls. In Section II the burning rate constants of a colloidal propint are related to fundamental parameters such.as specific surface vol of the powd, the most probable molecular vel, and the collision efficiency c. Schweikert arrives in Section III at the conclusion that burning deton differ primarily in the magnitude of c i.e. c l in a deton and is a much. smaller value in a burning process A surprisingly simple relation is derived in Section IV for the upper boundary of the deton vel Dm of a condensed expl ... 

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, k m, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann-Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A by collision with some other molecules in the system, M. In this picture, collisions between M and A are assumed to transfer energy in the other direction, that is, returning A to A ... 

The least resolved measurement is determination of the isothermal rate constant k(T), where T is the isothermal temperature. Although conceptually simple, such measurements are often exceedingly difficult to perform for activated process without experimental artifact (contamination) because they require high pressures to achieve isothermal conditions. For dissociative adsorption, k(T) = kcol (T) [S (Tg = TS = T)), where kcol(T) is simply the collision rate with the surface and is readily obtainable from kinetic theory and Tg and T, are the gas and surface temperatures, respectively [107]. (S ) refers to thermal averaging. A simple Arrhenius treatment gives the effective activation energy Ea for the kinetic rate as... 

This result is identical to the hard-sphere rate constant Eq. 10.82 derived from the simple collision theory introduced in Section 10.2. 

The high-pressure limit of the rate constant Um,oo is readily measured. From the assumptions in the model, molecular collision theory should be adequate to predict the excitation-reaction rate constant ke, using Eq. 10.76 ... 

Collins and Jameson11 found that for small air bubbles (20 to 100 jzm), varying the particle zeta potential from +30 mV to +60 mV resulted in an order of magnitude change in the observed rate constants for each drop size. Table 9 shows the values of the calculated and observed first-order rate constants for the data of Collins and Jameson obtained when their particles (polystyrene) had the minimum stability (zeta potential + 30 mV). The observed rate constants are much smaller than those calculated from collision theory. Their data indicate that between 1 in 40 to I in 100 collisions results in the particles sticking to bubbles. This is consistent with the particle-collision removal mechanism. 

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