Collision theory constant

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

Gilbert R G, Luther K and Troe J 1983 Theory of thermal unimolecular reactions in the fall-off range. II. Weak collision rate constants Ber. Bunsenges. Phys. Chem. 87 169-77... 

Troe J 1977 Theory of thermal unimolecular reactions at low pressures. II. Strong collision rate constants. Applications J. Chem. Phys. 66 4758... 

Collision theory leads to this equation for the rate constant k = A exp (-EIRT) = A T exp (,—EIRT). Show how the energy E is related to the Arrhenius activation energy E (presuming the Arrhenius preexponential factor is temperature independent). 

Table 11.3 compares observed rate constants for several reactions with those predicted by collision theory, arbitrarily taking p = 1. As you might expect, the calculated k s are too high, suggesting that the steric factor is indeed less than 1. 

We have deduced that, according to collision theory, the rate constant is... 

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... 

As in collision theory, the rate of the reaction depends on the rate at which reactants can climb to the top of the barrier and form the activated complex. The resulting expression for the rate constant is very similar to the one given in Eq. 15, and so this more general theory also accounts for the form of the Arrhenius equation and the observed dependence of the reaction rate on temperature. 

Some of the rate constants discussed above are summarized in Table VI. The uncertainties (often very large) in these rate constants have already been indicated. Most of the rate constants have preexponential factors somewhat greater than the corresponding factors for neutral species reactions, which agrees with theory. At 2000°K. for two molecules each of mass 20 atomic units and a collision cross-section of 15 A2, simple bimolecular collision theory gives a pre-exponential factor of 3 X 10-10 cm.3 molecule-1 sec.-1... 

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. 

Although collision theory provides a useful formalism to estimate an upper limit for the rate of reaction, it possesses the great disadvantage that it is not capable of describing the free energy changes of a reaction event, since it only deals with the individual molecules and does not take the ensemble into consideration. As such, the theory is essentially in conflict with thermodynamics. This becomes immediately apparent if we derive equilibrium constants on the basis of collision theory. Consider the equilibrium... 

Applying collision theory to the forward and the reverse reactions, and taking the ratio, we obtain the equilibrium constant ... 

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. 

In this equation it is the reaction rate constant, k, which is independent of concentration, that is affected by the temperature the concentration-dependent terms, J[c), usually remain unchanged at different temperatures. The relationship between the rate constant of a reaction and the absolute temperature can be described essentially by three equations. These are the Arrhenius equation, the collision theory equation, and the absolute reaction rate theory equation. This presentation will concern itself only with the first. 

The term A in Equation (2.6) is a constant known as the Arrhenius constant and E is the energy of activation derived from collision theory (Atkins, 1978). The enthalpy of activation can be calculated from transition state theory (Jencks, 1969) as... 

Comparison of this equation with the Arrhenius form of the reaction rate constant reveals a slight difference in the temperature dependences of the rate constant, and this fact must be explained if one is to have faith in the consistency of the collision theory. Taking the derivative of the natural logarithm of the rate constant in equation 4.3.7 with respect to temperature, one finds that... 

Consideration of a variety of other systems leads to the conclusion that very rarely does the collision theory predict rate( constants that will be comparable in magnitude to experimental values. Although it is not adequate for predictions of reaction rate constants, it nonetheless provides a convenient physical picture of the reaction act and a useful interpretation of the concept of activation energy. The major short-... 

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). 

Equation (4.21) gives the rate constant for a bimolecular reaction in terms of collision theory. 

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... 

The Arrhenius relation means that the rate constant or the diffusivity increases with temperature. Typically, at low temperatures (0-60°C), a 10-degree increase in temperature results in a doubling of reaction rates. In this section, two theories are introduced to account for the Arrhenius relation and reaction rate laws. Collision theory is a classical theory, whereas transition state theory is related to quantum chemistry and is often referred to as one of the most significant advances in chemistry. 

Hence, collision theory gives the correct reaction rate law (Equation 1-94) and a reaction rate constant of the form... 

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