Reaction rate collision model

In this section we will introduce a model that can be used to account for the observed characteristics of reaction rates. This model, the collision model, is built around the central idea that molecules must collide to react. We have already seen that this assumption can explain the concentration dependence of reaction rates. Now we need to consider whether this model can also account for the observed temperature dependence of reaction rates. 

As it has appeared in recent years that many hmdamental aspects of elementary chemical reactions in solution can be understood on the basis of the dependence of reaction rate coefficients on solvent density [2, 3, 4 and 5], increasing attention is paid to reaction kinetics in the gas-to-liquid transition range and supercritical fluids under varying pressure. In this way, the essential differences between the regime of binary collisions in the low-pressure gas phase and tliat of a dense enviromnent with typical many-body interactions become apparent. An extremely useful approach in this respect is the investigation of rate coefficients, reaction yields and concentration-time profiles of some typical model reactions over as wide a pressure range as possible, which pemiits the continuous and well controlled variation of the physical properties of the solvent. Among these the most important are density, polarity and viscosity in a contimiiim description or collision frequency. 

The collision model of reaction rates just developed can be made quantitative. We can say that the rate constant for a reaction, k, is a product of three factors ... 

Reaction quotient (Q) An expression with the same form as Kbut involving arbitrary rather than equilibrium partial pressures, 333-334 Reaction rate The ratio of the change in concentration of a species divided by the time interval over which the change occurs, 285 catalysis for, 305-307 collision model, 298-300 concentration and, 287-292,314q constant, 288 enzymes, 306-307 egression, 288... 

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... 

Relationships having the same form as eq 14 can also be written for the enthalpic and entropic contributions to the intrinsic free energy barriers (10). Provided that the reactions are adiabatic and the conventional collision model applies, eq 14 can be written in the familiar form relating the rate constants of electrochemical exchange and homogeneous self-exchange reactions (13) ... 

An understanding of reaction rates can be explained by adopting a collision model for chemical reactions. The collision theory assumes chemical reactions are a result of molecules colliding, and the rate of the reaction is dictated by several characteristics of these collisions. An important factor that affects the reaction rate is the frequency of collisions. The reaction rate is directly dependent on the number of collisions that take place, but several other important factors also dictate the speed of a chemical reaction. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

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 ... 

Such a reaction may be considered to approximate a simple collision model. The rate of the reaction is faster than cyanide exchange for either reactant so we consider the process to consist of electron transfer from one stable complex to another with no breaking of Fe—CN or Mo—CN bonds. 

To understand why reaction rates depend on temperature, we need a picture of how reactions take place. According to the collision theory model, a bimolecu-lar reaction occurs when two properly oriented reactant molecules come together in a sufficiently energetic collision. To be specific, let s consider one of the simplest possible reactions, the reaction of an atom A with a diatomic molecule BC to give a diatomic molecule AB and an atom C ... 

The physical interpretation of this result is, relatively, simple. The reaction rate predicted by the model is equal to the collision frequency, Eq. (4.16), times the factor exp(—E /ksT). This factor is clearly related to the Boltzmann distribution.2 To that end, let us evaluate the probability of finding a relative velocity, irrespective of its direction, corresponding to a free translational energy EtI = (1 /2)/. v that exceeds i tr = E (see Problem 1.3) ... 

On a microscopic scale, atoms and molecules travel faster and, therefore, have more collisions as the temperature of a system is increased. Since molecular collisions are the driving force for chemical reactions, more collisions give a higher rate of reaction. The kinetic theory of gases suggests an exponential increase in the number of collisions with a rise in temperature. This model fits an extremely large number of chemical reactions and is called an Arrhenius temperature dependency, or Arrhenius law. The general form of this exponential relationship is... 

Based on the molecular collision model, that describes successfully experimental data for a large number of bimolecular reactions, the rate of the reaction can be calculated as the number of collisions of molecules having energy higher than the required value E [7] ... 

Much more can be said about the magnitude of pre-exponcntial factors and activation energies of elementary processes based on statistical thermodynamics applied to collision and reaction-rate theory [2, 61], but in view of the remark above one should be cautious in their application and limit it to well-defined model reactions and catalyst surfaces. 

The kinetic molecular theory of gases predicts that an increase in temperature increases molecular velocities and so increases the frequency of in-termolecular collisions. This agrees with the observation that reaction rates are greater at higher temperatures. Thus there is qualitative agreement between the collision model and experimental observations. However, it is found that the rate of reaction is much smaller than the calculated collision frequency in a given collection of gas particles. This must mean that only a small fraction of the collisions produces a reaction. Why ... 

Equation (15.10) is a linear equation of the type y = mx + b, where y = ln( ), m = —EJR = slope, x = 1/T, and b = ln(A) = intercept. Thus, for a reaction where the rate constant obeys the Arrhenius equation, a plot of n(k) versus 1/T gives a straight line. The slope and intercept can be used to determine the values of a and A characteristic of that reaction. The fact that most rate constants obey the Arrhenius equation to a good approximation indicates that the collision model for chemical reactions is physically reasonable. 

Collision model a model based on the idea that molecules must collide to react used to account for the observed characteristics of reaction rates. (15.8)... 

Figure 11 shows a typical example of the temperature-dependent behavior for the reactions of OH radical with aromatic compounds. The measured bimolecular rate constants of OH radical with nitrobenzene showed distinctly non-Arrhenius behavior below 350°C, but increased in the slightly subcritical and supercritical region. Feng a succeeded in modeling these data with a three-step reaction mechanism originally proposed by Ashton et while Ghandi etal. claimed to have developed a so-called multiple collisions model to predict the rates for the reactions of OH radical in sub- and super-critical water. 

The mutual diffusion constant obtained from Stokes-Einstein equation shows a large difference with the values obtained from the measurements of fluorescence quenching reaction and also empirical equations especially in viscous solvents. By using the values of R and D, we estimated the reaction rate constant, kR, and reaction probability in collision complex, rp, when we assumed the yan der W s radii of CNA and CHD estimated from space-fitting molecular model to be 5.0 A and 2.0 A, respectively, that is R=7.0 A in eq.(6).The obtained rp, ko and kR values are listed in Table 2. 

---