Kinetics, molecular collision theory

According to the molecular collision theory of kinetics, only collisions between molecules with high enough energy can result in reaction(s). In other words, a chemical reaction occurs only when the two conditions are met (1) the molecules to take part in the reaction collide with each other and (2) the colliding molecules have a sufficiently high energy level. 

We are concerned with bimolecular reactions between reactants A and B. It is evident that the two reactants must approach each other rather closely on a molecular scale before significant interaction between them can take place. The simplest situation is that of two spherical reactants having radii Ta and tb, reaction being possible only if these two particles collide, which we take to mean that the distance between their centers is equal to the sum of their radii. This is the basis of the hard-sphere collision theory of kinetics. We therefore wish to find the frequency of such bimolecular collisions. For this purpose we consider the relatively simple case of dilute gases. 

Simple collision theory does not provide a detailed interpretation of the energy barrier or a method for the calculation of activation energy. It also fails to lead to interpretations in terms of molecular structure. The notable feature of collision theoiy is that, with very simple means, it provides one basis for defining typical or normal kinetic behavior, thereby directing attention to unusual behavior. 

Z, the collision frequency, which gives the number of molecular collisions occurring in unit time at unit concentrations of reactants. This quantity can be calculated quite accurately from the kinetic theory of gases, but we will not describe that calculation. 

The science of reaction kinetics between molecular species in a homogeneous gas phase was one of the earliest fields to be developed, and a quantitative calculation of the rates of chemical reactions was considerably advanced by the development of the collision theory of gases. According to this approach the rate at which the classic reaction... 

Too little attention is generally paid to the concentrations of the reactants in preparative organic work. With the exception of rare cases (e.g. in intramolecular rearrangements) we are concerned with reactions of orders higher than the first, and in these several kinds of molecules—usually two—are involved. Since, according to the kinetic molecular theory, the velocity of bimolecular reactions is proportional to the number of collisions between the various dissolved molecules and therefore to the product of the concentrations,... 

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. ... 

Transport of the gas to the surface and the initial interaction. The first step in heterogeneous reactions involving the uptake and reaction of gases into the liquid phase is diffusion of the gas to the interface. At the interface, the gas molecule either bounces off or is taken up at the surface. These steps involve, then, gaseous diffusion, which is determined by the gas-phase diffusion coefficient (Dg) and the gas-surface collision frequency given by kinetic molecular theory. 

We shall treat the individual processes in terms of the rate of transfer of gas across a surface of unit area per second. However, this rate will be expressed relative to the number of gas-surface collisions per second, given according to kinetic molecular theory by... 

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... 

Using gas kinetic molecular theory, show that under typical atmospheric conditions of pressure and temperature corresponding to an altitude of 5 km (see Appendix V) collisional deactivation of a C02 molecule will be much faster than reemission of the absorbed radiation. Take the collision diameter to be 0.456 nm and the radiative lifetime of the 15-/rm band of C02 to be 0.74 s (Goody and Yung, 1989). 

All gas particles have some volume. All gas particles have some degree of interparticle attraction or repulsion. No collision of gas particles is perfectly elastic. But imperfection is no reason to remain unemployed or lonely. Neither is it a reason to abandon the kinetic molecular theory of ideal gases. In this chapter, you re introduced to a wide variety of applications of kinetic theory, which come in the form of the so-called gas laws. ... 

Instead, we must turn to the kinetic molecular theory of gases for an estimate of the frequency with which molecules collide with a solid surface. We shall not be misled, however, if we anticipate that this pressure is low. Example 9.6 is a numerical examination of gas collisions with walls. 

EXAM PLE 9.6 Rate of Atomic Collisions as a Function of Pressure. Assuming 1019 atoms per square meter as a reasonable estimate of the density of atoms at a solid surface, estimate the time that elapses between collisions of gas molecules at 10 6 torr and 25°C with surface atoms. Use the kinetic molecular theory result that relates collision frequency to gas pressure through the relationship Z = 1/4 vNIV, for which the mean velocity of the molecules v = (BRTI-kM) 12 and NIV is the number density of molecules in the gas phase and equals pNJRT. Repeat the calculation at 10 8 and 10 10 torr. 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

Collision Theory oi Reaction. A theory to account for observed kinetics of reaction in terms of the molecular behavior of the reacting systems. For interaction, this theory requires that the molecules must collide and, in addition, have sufficient energy to be activated (See also Absolute Rate Theory in Vol 1 of Encyclopedia, pA4-R)... 

Experimental evidence for the notion of an activation energy barrier comes from a comparison of collision rates and reaction rates. Collision rates in gases can be calculated from kinetic-molecular theory (Section 9.6). For a gas at room temperature (298 K) and 1 atm pressure, each molecule undergoes approximately 109 collisions per second, or 1 collision every 10 9 s. Thus, if every collision resulted in reaction, every gas-phase reaction would be complete in about 10-9 s. By contrast, observed reactions often have half-lives of minutes or hours, so it s clear that only a tiny fraction of the collisions lead to reaction. 

The Van der Waal s equation takes into account the deviations of real gases from the kinetic molecular theory of gases (nonzero molecular volume and nonelastic collisions). 

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... 

The kinetic molecular theory assumes that the number of collisions per second in a gas depends on the number of particles per litre. The rate at which N02 and C1NO are formed in this reaction should, therefore, be directly proportional to the concentrations of both C1N02 and NO. 

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. 

Another very important concept in kinetic theory is the average distance a molecule travels between collisions—the so-called mean free path. On the basis of a very simple conception of molecular collisions, the following equation for the mean free path A can be derived ... 

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