Collision theory deviations

In the case of bunolecular gas-phase reactions, encounters are simply collisions between two molecules in the framework of the general collision theory of gas-phase reactions (section A3,4,5,2 ). For a random thennal distribution of positions and momenta in an ideal gas reaction, the probabilistic reasoning has an exact foundation. Flowever, as noted in the case of unimolecular reactions, in principle one must allow for deviations from this ideal behaviour and, thus, from the simple rate law, although in practice such deviations are rarely taken into account theoretically or established empirically. 

Sometimes forms deviating from simple linear relations were used in this early work, particularly the relation of log A to E (31,36), which was derived theoretically (29) but later rejected (37, 38). Otherwise, the quantity log (kM ) was plotted instead of log k, M" being the sum of reciprocal masses of reacting particles according to collision theory (33), and this correction was also later abandoned (34). The two modifications mentioned, in practice, have little influence on the shape of the graph, and the simple plot of E versus log A (or AH versus AS) is now preferred. 

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.]... 

A useful comparison between the predictions of simple collision theory and experiment can be made, since if the activation energy is determined, the experimental frequency factor can be directly compared with that predicted by Eq. (2-33). The hard-sphere diameter can be estimated from transport properties, although the choice of this parameter is somewhat arbitrary. In Table 2-1 a comparison between theory and experiment is presented for several well-studied bimolecular reactions (cf. Benson [10] for a more complete compilation). The tabulated steric factor is that value which makes the experimental and theoretical values coincide. In view of the assumptions involved, many of the steric factors are surprisingly close to unity. However, marked deviations in the form of unreasonably small steric factors do occur, especially if polyatomic molecules are involved. This often indicates that quantum-mechanical effects may be important or that a different classical theory may be required. 

However, later work showed that rather large deviations from experiment are obtained for reactions in which the reacting molecules are more complicated. This collision theory is evidently too simple and unlikely to be generally reliable. One weakness is the assumption that molecules are hard spheres, which implies that any collision with sufficient energy will lead to reaction if the molecules are more complicated, this is not the case. A more fundamental objection to the treatment is that when applied to forward and reverse reactions it cannot lead to an expression for the equilibrium constant that involves the correct thermodynamic parameters. More recent work has involved a similar approach but has treated molecular collisions in a more realistic and detailed way. 

In order to account for deviations from collision theory, the rate equation is written has been written as... 

In many cases, the observed rate does not agree with the value calculated on the basis of Eq.(9.17). In order to account for deviation from the collision theory, Eq.(9.15) is modified to... 

Meanwhile, the pre-exponential factor A in the Arrhenius Eq. (2.39) is the temperature independent factor related to reaction frequency. Comparing the Eq. (2.33) for the collision theory and Eq. (2.38) with the transition state theory, the pre-exponential factors in these theories contain temperature dependences of T and T respectively. Experimentally, for most of reactions for which the activation energy is not close to zero, the temperature dependence of the reaction rate constants are known to be determined almost solely by exponential factor, and the Arrhenius expression holds as a good approximation. Only for the reaction with near-zero activation energy, the temperature dependence of the pre-exponential factor appears explicitly, and the deviation from the Arrhenius expression can be validated. In this case, an approximated equation modifying the Arrhenius expression can be used. 

Lucretius (Titus Lucretius Caras, 99 BCE - 55 CE) of Rome wrote a poem, De Rerum Natura (On the Nature of Things) (24) in which he described the atomic theory of Epicums of Samos (342-271 BCE). For Epicurus, atoms were indivisible, invisible, and indestmctible, and they differ in size, shape and weight. He believed that a void exists because there can be no motion of the atoms without it. The motions of atoms included the downward motion of free atoms because of their weight, swerve, the deviation of atomic motion from straight downward paths, and blow, which results from collisions and motion in compoimd bodies. Lucretius called atoms poppy seeds, bodies, principals, and shapes (25). 

When the deviation from the elastic state of the material surface is small, the Hertzian theory can estimate the force of impact, contact area, and contact duration for collisions between spherical particles and a plane surface using Eqs. (2.132), (2.133), and (2.136), respectively. To account for inelastic collisions, we may introduce r as the ratio of the reflection speed to the incoming speed, V. Therefore, we may write... 

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). 

Within transition-state theory, Eq. (8.2) is an exact expression for the rate constant. We observe that the pre-exponential factor deviates from the simple interpretation, as being related to the collision frequency Zab via Z, due to the presence of internal degrees of freedom. Typically, the calculated value of Z is of the order of 1011 dm3 mol-1 s 1 10 16 m3 molecule-1 s 1 (see Example 4.1). The magnitude of the partition functions in Eq. (8.2) is typically small compared to this number. Thus, if we neglect the internal degrees of freedom of the reactants and the activated complex, except for rotational degrees of freedom of the activated complex (AB), and assume that the associated partition function can be approximated by QAB, we will get a pre-exponential factor given by Z. 

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiem-pirical equation, based on hard-sphere theory, to estimate self-diffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. 

Inelastic collisions have also been dealt with recently in a detailed paper by Bethe. The results, however, refer only to fast electrons and to the diffraction caused by a single atom. The continuous background formed by the electrons which are scattered inelastically falls off much more steeply, according to this theory, than the coherently scattered de Broglie waves. Thus when the angle of deviation is large we are more or less in a position to neglect it in experiments. 

The extension of the kinetic theory approach to include large values of a (and hence large deviations from equilibrium) requires higher order perturbations for the solution of the Boltzmann equation. It is probably unprofitable to proceed in this difficult and laborious direction until one understands the detailed analytical dependence of the transition probability a on the mechanism of molecular energy exchange and redistribution on collision. Currently available information on intermolecular forces is insufficient to establish this dependence. 

On the basis of the model employed here, it has been shown that the rate of a chemical reaction will deviate from the equilibrium rate, as calculated from collision or absolute rate theory, when E Gt/kT < 10. The calculated deviation of about 20 per cent for Ea.ctjkT = 5 (see Fig. 4) is in good agreement with the results obtained previously by other authors. For E t/kT 10 the error in the rate as calculated by equilibrium theory is < 10 per cent. 

---