Successful collisions

A typical value of the collision number is 10 °s in gases at one atmosphere pressure and room temperature, and the number of successful collisions which can bring about the chemical reaction is equal to this number multiplied by the Anhenius or probability factor, exp(— /f 7 ), where E is the activation energy, the critical collision energy needed for reaction to occur. 

To carry out this method, values are chosen for Tq, the desired temperature, and v, the mean frequency with which each particle experiences a stochastic collision. If successive collisions are uncorrected, then the distribution of time intervals between two successive stochastic collisions, P(v, t), is of the Poisson fonn. 

Higher time moments of Km(0 are negative as well. This is physically accounted for by the anticorrelated nature of successive collisions in a gas [44], A collision from the front is usually followed by a collision from the back (Fig. 1.5). Owing to the opposite direction of collisions the product of the corresponding moments (M(O)M(t)) is negative, and it provides the main contribution to Km for times of order tj. This is a manifestation of the correlated character of the interaction between a molecule and perturbers. 

For forbidden transitions in atoms and molecules this phenomenon may be experimentally observed in spectra induced by collisions. As is known, the selection rules on some transitions may be cancelled during collision. The perturbers are able to induce a dipole moment of transition having the opposite direction in successive collisions due to intercollisional correlation. Owing to this, the induced spectra do involve the gap (Fig. 1.7), the width of the latter being proportional to the gas density [46, 47], Theorists consider intercollisional correlation to be responsible for the above phenomenon [48, 49, 50]. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

We consider a collection of molecular dipoles in thermal equilibrium. It is assumed that all the molecules are identical and they can take on any orientation. Because of thermal energy each molecule undergoes successive collisions with the surrounding molecules. In the absence of an applied electric field, the collisions tend to maintain a perfectly isotropic statistical orientation of the molecules. This means that for each dipole pointing in one direction there is statistically a corresponding dipole pointing in the opposite direction, as described by Fig. 1.2. 

It was also assumed that a successful collision in flocculation can only occur if a polymer-free area on one floe hits a polymer-covered area on another floe or vice versa. The complete dimensionless flocculation rate equation is given by Equation (5) below... 

Reactions may be either direct , where an energetic particle has such a small wavelength that it only sees one nucleon of the target, or compound nucleus reactions, where the projectile s energy is shared among many nucleons in successive collisions within a compound nucleus which can then decay into one of a number of exit channels (Fig. 2.6). The first case is the more common one in reactions between light nuclei, whereas the second dominates for heavier ones. 

On increasing concentration, there are more reactant particles per unit volume (1) do not simply put more reactant particles an increase in the frequency of the collisions (1) therefore the greater the probability of a collision with energy more than the activation energy (1) also known as a successful collision . 

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). 

Heating a reaction mixture increases the number of successful collisions between the reactant species, increasing the amount of product formed per unit time. 

Fig. 21. Example of successive collisions giving rise to long-range effects.

In going from Eq. (A.37) to Eq. (A.38), we have explicitly taken into account that the same solvent molecule does not appear in two successive collisions when the thermodynamic limit (N — oo,... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

Figure 3.4 is a schematic illustration of the fact that dipoles induced in successive collisions tend to be more or less antiparallel. This anticorrelation of dipoles induced in subsequent collisions leads to the absorption defect and causes the breakdown of the pair behavior illustrated in Fig. 3.3, if the product fx 2 is of the order of unity or less. If, on the other hand, fxn 1, superposition occurs with widely varying, random phase differences which render an interference effect inefficient. 

Fig. 3.4. The dipoles induced in successive collisions of a light particle (small circle) with massive ones the induced dipoles pn and pn tend to be more or less antiparallel.

Long-time behavior of correlation functions. The dipoles induced in successive collisions are correlated as Fig. 3.4 on p. 70 suggests. As a consequence, the dipole autocorrelation function has a negative tail of a duration comparable to the mean time between collisions, Fig. 5.3. Furthermore, the area under the negative tail is of similar order of magnitude as the area under the positive (or intracollisional) part of C(r). If the neg-... 

The leading term on the right-hand side of Eq. 5.88 is, of course, the intracollisional profile, g(co), that is the spectrum that would be observed if no correlations existed between the dipoles induced in successive collisions. The remaining term describes the intercollisional process, which is the central theme of this subsection. 

This minimum amount of energy is known as the activation energy, Ez (Figure 7.2). Collisions which result in the formation of products are known as successful collisions. 

An increase in the surface area of a solid reactant results in an increase in the number of collisions, and this results in an increase in the number of successful collisions. Therefore, the increase in surface area of the limestone increases the rate of reaction. 

---