Collision theory problems with

Problem 15. Discuss the mathematical treatment of transition state theory. Compare this theory with collision theory. 

In order to predict the value of the frequency factor, one may assume that all collisions between reactant molecules with sufficient activation energy result in the instantaneous formation of the reaction products. With this simple hypothesis (collision theory), if the activation energy is known, then the problem of computing the reaction rate reduces to the problem of computing the rate of collision between the appropriate reactant molecules in the ideal gas mixture. This last problem is easily solved by the elementary kinetic theory of gases. 

The first problem discovered with the Lindemann model is the use of the simple collision theory, Eq. (1) for the activation rate constant. The collision frequency Zam can be calculated simply from collision theory, leaving the activation energy Eq to be determined. The low pressure limit is often difficult to attain experimentally, because collisions with impurities or collisions between... 

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Model kinetic equation approaches of these types should probably be more thoroughly investigated for such complex systems before more elaborate kinetic theories are constructed. Ultimately, however, difficult problems such as the nature of the friction coefficient or collision frequency associated with an internal coordinate must be solved. What, for instance, is the form of its space and time nonlocality The solution of this problem will involve a more complex calculation than that outlined in Section IX.B for the two-particle friction tensor. 

The problem of termolecular reactions can be treated by collision theory also. A number of such reactions are known reactions of NO with H2, O2, CI2 are famous examples. If we choose the reaction with oxygen. 

Collision theory does not deal directly with unimolecular reactions but touches on the subject through the Lindemann mechanism. Once the molecule has been provided with sufficient energy by collision, the problem is to calculate the rate constant for the unimolecular decomposition,... 

With the distribution laws established, we can now attack the problem that is central to a collision theory of reaction the number of collisions experienced per molecule per second in the Maxwellian gas. Clearly the magnitude of this collision number is a function of temperature (through the constant a), and if we define a total collision number, collisions of all molecules per second per volume, it will also depend on molecular density (i.e., concentration). Thus, the two independent variables of concentration and temperature used in power-law rate equations will appear in the total collision number. 

A collision theory of even gas phase reactions is not totally satisfactory, and the problems with the steric factor that we described earfier make this approach more empirical and qualitative than we would like. Transition state theory, developed largely by Henry Eyring, takes a somewhat different approach. We have already considered the potential energy surfaces that provide a graphical energy model for chemical reactions. Transition state theory (or activated complex theory) refers to the details of how reactions become products. For a reaction fike... 

The major problem is how to calculate the kinetics of these reactions. There are several proposals and one of them is gas collision theory, because they are of the same order of magnitude. Collisions occur repeatedly with high frequency, so they constitute multiple collisions. It is estimated that the distances between the molecules in the liquid phase are approximately equal, whereas in the gas phase are quite different. Moreover, besides the attractive forces, there are also repulsive forces, but the important thing is that for the reaction to occur it is necessary to overcome the energy barrier E. Differently from the reactions in the gas phase, the collisions in the liquid phase are 10-1000 times higher, but depend on other properties such as the viscosity. 

For the calculation of the rate constants of such processes, the transition-state method has been proposed [129, 131, 365, 518]. Its relative simplicity (permitting calculation of the rate constants for many reactions) lies in that it does not attempt taking into account all the dynamical features of the elementary processes. It introduces instead the activated complex concept. However, it does not give unambiguous indication as to how the activated complex properties are connected with those of the reactants, thus leaving aside the dynamical problem. For this reason, the transition-state approach is sometimes opposed to the collision theory, though very often they are correlated. 

Apart from this the physical problem clearly requires fundamenteilly different boundary conditions we do not start from a state V, but instead from tti, and we inquire about the probability of a change into U2 Both u and U2 are states which are not exactly stationary, even for arbitrary large atomic distances. But they do represent stationary states of the isolated atoms. In what follows, we therefore feel prompted to treat the case of sharp resonances with these boundary conditions completely afresh. Here they always represent the assumptions (3) of the usual collision theory, and due to this the results obtained in the two theories are equivalent. 

The Drude model is a crude model, but it contains the accepted mechanism for electrical resistance in solids, which is the effect of collisions with the cores of the crystal. There are a number of more sophisticated theories than the Drude theory. However, the results of these theories are similar in their general form to Eq. (28.4-9). The major differences are in the interpretation of the quantities r, and m One problem with the Drude theory is that the conductivities of most common metals are found experimentally to be approximately inversely proportional to the temperature, instead of being inversely proportional to the square root of the temperature, as in Eq. (28.4-11). One can rationalize this by arguing that the mean free path should decrease as the temperature rises, because of the increased vibrational amplitude of the cores, making them into targets with larger effective sizes at higher temperature. 

Nevertheless, materials such as coal and biomass are important in a practical sense, and scientists and engineers must deal with reactions involving these materials. In these cases, mass concentrations are used instead of molar concentrations. The use of mass concentrations in rate equations is less fundamental, perhaps, than the use of molar concentrations. This is because theories such as collision theory and transition-state thewy teach that reaction rates depend on molar concentrations. Nevertheless, the use of mass concentrations in problems involving complex materials has proven to be a practical approach is solving such problems. 

Chapter 2 is an overview of rate equations. At this point in the text, the subject of reaction kinetics is approached primarily from an empirical standpoint, with emphasis on power-law rate equations, the Arrhenius relationship, and reversible reactions (thermodynamic consistency). However, there is some discussion of collision theory and transition-state theory, to put the empiricism into a more fundamental context. The intent of this chapter is to provide enough information about rate equations to allow the student to understand the derivations of the design equations for ideal reactors, and to solve some problems in reactor design and analysis. A more fundamental treatment of reaction kinetics is deferred until Chapter 5. The discussion of thermodynamic consistency... 

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