Collision theory energy dependence

Effect of Solvent on Arrhenius Plots. If water is a substrate, then the presence of an organic solvent, which may disrupt the structure and/or orientation of water, may alter the Arrhenius plot. For example, a linear plot is seen with fumarate hydratase in the presence of 10% methanol. However, the plot is biphasic in the presence of 10% ethanol . See Boltzmann Distribution Collision Theory Temperature Dependency, Transition-State Theory Energy of Activation On... 

All approaches discussed thus far rest on stationary-state collision theory. Time-dependent scattering theory has also been applied, making use of different potential-energy surfaces, for a description of the H + Hg reaction by BffAZUR and RUBIN /94/, Mc.GULLOUGH and WYATT /95/, and ZURT, KAMAL and ZOLIGKE /96/. The wave function in the initial state, chosen at a moment t = t far before the collision, is represented by the product (102.11) where is a trans-... 

Collision theory is mute about the value of fji. Typically,1, so that the number of molecules colliding is much greater than the number reacting. See Problem 1.2. Not all collisions have enough energy to produce a reaction. Steric effects may also be important. As will be discussed in Chapter 5, fji is strongly dependent on temperature. This dependence usually overwhelms the dependence predicted for the collision rate. 

In experimental practice, we usually ignore the temperature dependence of the prefactor and extract the activation energy by making an Arrhenius plot, as discussed in Chapter 2. The consequence of collision theory, however, is that a curved plot, rather than a straight line, will result if the activation energy is of the same order of k T. 

The rate of reaction in collision theories is related to the number of successful collisions. A successful reactive encounter depends on maw things, including (1) the speed at which the molecules approach each other (relative translational energy), (2) how close they are to a head-on collision (measured by a miss distance or impact parameter, b, Figure 6.10), (3) the internal energy states of each reactant (vibrational (v), rotational (/)), (4) the timing (phase) of the vibrations and rotations as the reactants approach, and (5) orientation (or steric aspects) of the molecules (the H atom to be abstracted in reaction 634 must be pointing toward the radical center). 

The Bohr criterion k I depends on the projectile speed rather than its kinetic energy. This, together with the fact that Zi — l, implies that for electrons or positrons the validity of semiclassical collision theory becomes... 

By comparison of (M) and (F), it can be seen that the preexponential factor A in the Arrhenius equation can be identified with PaA]i(8kT/Tr/ji,y/2 and the activation energy, a, with the threshold energy Eu. It is important to note that collision theory predicts that the preexponential factor should indeed be dependent on temperature (Tl/2). The reason so many reactions appear to follow the Arrhenius equation with A being temperature independent is that the temperature dependence contained in the exponential term normally swamps the smaller Tl/Z dependence. However, for reactions where E.t approaches zero, the temperature dependence of the preexponential factor can be significant. 

Elementary reactions are initiated by molecular collisions in the gas phase. Many aspects of these collisions determine the magnitude of the rate constant, including the energy distributions of the collision partners, bond strengths, and internal barriers to reaction. Section 10.1 discusses the distribution of energies in collisions, and derives the molecular collision frequency. Both factors lead to a simple collision-theory expression for the reaction rate constant k, which is derived in Section 10.2. Transition-state theory is derived in Section 10.3. The Lindemann theory of the pressure-dependence observed in unimolecular reactions was introduced in Chapter 9. Section 10.4 extends the treatment of unimolecular reactions to more modem theories that accurately characterize their pressure and temperature dependencies. Analogous pressure effects are seen in a class of bimolecular reactions called chemical activation reactions, which are discussed in Section 10.5. 

Perrin s argument that the very nature of a unimolecular reaction demands independence of collisions, and therefore dependence on radiation, is adequately met both by the theory of Lindemann and by that of Christiansen and Kramers. Both these theories have the essential element in common that the distribution of energy among the molecules is not appreciably disturbed by the chemical transformation of the activated molecules thus the rate of reaction is proportional simply to the number of activated molecules and therefore to the total number of molecules, sinc in statistical equilibrium the activated molecules are a constant fraction of the whole. Thus the radiation theory is not necessary to explain the existence of reactions which are unimolecular over a wide range of pressures. 

As the temperature increases, the distribution of collision energies broadens and shifts to higher energies (Figure 12.15), resulting in a rapid increase in the fraction of collisions that lead to products. At 308 K, for example, the calculated value of / for the reaction with Ea = 75 kj/mol is 2 x 10-13. Thus, a temperature increase of just 3%, from 298 K to 308 K, increases the value of / by a factor of 3. Collision theory therefore accounts nicely for the exponential dependence of reaction rates on reciprocal temperature. As T increases (1 /T decreases), / = e E RT increases exponentially. Collision theory also explains why reaction rates are so much lower than collision rates. (Collision rates also increase with increasing temperature, but only by a small amount—less than 2% on going from 298 K to 308 K.)... 

The observed energy dependence is indistinguishable from that predicted by RRKM theory, suggesting that increased translational energy is redistributed statistically in the collision complex. 

The development of a theory of unimolecular reactions proceeded rapidly in the mid-1920s, initiated by Hinshelwood with an A whose collision-free lifetime for reaction was approximated by an energy-independent one. The analysis was much elaborated by Rice and Ramsperger [60] and Kassel [61], known later as the RRK theory, where now the lifetime was, as it is in modern times, energy-dependent [62]. These theoretical works of the 1920s stimulated many measurements of the unimolecular rates of dissociation of organic compounds as a function of the gas pressure. Within a few years, however, this entire field collapsed or, more precisely, evolved into a new field It was shown experimentally that the unimolecular reactions , assumed originally to consist of only one chemical step, in-... 

The reaction rate mainly depends on the concentration of reactants and products. According to the collision theory, frequent collisions and rapid conversions occur at high concentrations. Yet not all collisions cause conversions, a certain position of the molecules to each other as well as a certain threshold energy are required. Besides the concentration, pH, light, temperature, organics, presence of catalysts, and surface-active trace substances can have a significant influence on reaction rates. 

An overall view shows the CID process as a sequence of two steps. The first step is very fast (10 14 to 10 16 s) and corresponds to the collision between the ion and the target when a fraction of the ion translational energy is converted into internal energy, bringing the ion into an excited state. The second step is the unimolecular decomposition of the activated ion. The collision yield then depends on the activated precursor ion decomposition probability according to the theory of quasi-equilibrium or RRKM. This theory is explained elsewhere. Let us recall that it is based on four suppositions ... 

The chemical reaction is characterized on the one hand by the kinetic mechanism, that is to say the dependence on the concentrations of the participants in the reaction, on the other hand by the reaction (velocity) constant. This latter in the simplest form is k — Ae EIRT in which E is the energy of activation and A the frequency factor. The latter is in the classical collision theory equal to where Z the collision number ( io11) and P the probability factor or steric factor. The latter can be much larger than unity if the activation energy is divided over several internal degrees of freedom (mono-molecular reactions) but it can also be as low as io 8, e.g., in cases where steric hindrance plays a role. 

The C atom is a member of the graphite grain and serves to chemisorb the X atom until molecule formation has taken place. The theory used in the calculations of formation rates on grains is collision theory (e.g. Polanyi, 1962). As always assumed it is dependent on an activation energy A, which may be regarded as the energy the incoming atom requires to overcome the repulsion of the chemically bound molecule. The reaction rate is then proportional to... 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

Both A and E may also be temperatme dependent. We can estimate the activation energy from either potential energy smfaces or various empirical relationships and the frequency factor from either collision theory, transition state theory or from computational chemistry software (see Appendix J). 

Classical complex formation such as outlined above has been observed in a number of classical Monte Carlo trajectory studies,45 and Brumer and Karplus46 have recently reported an extensive study of alkali halide-alkali halide reactions which involve long-lived collision complexes. These purely classical studies cannot, of course, describe the resonance structure in the energy dependence of scattering properties, but rather give an average energy dependence the resonance structure, a quantum effect, is described only by a theory which contains the quantum principle of superposition. 

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