Hard sphere collision frequency

The pre-exponential factor of a bimolecular reaction is related to the reaction cross-section (see Problem 2.3). A relation that is fairly easy to interpret can be obtained within the framework of transition-state theory. Combining Eqs (6.9) and (6.54), we can write the expression for the rate constant in a form that gives the relation to the (hard-sphere) collision frequency ... 

Here the partition functions refer to internal degrees of freedom (subscript int for internal), QAB = 8i 2(ii,d2)kBT/ h2, that is, a rotational partition function where A and B are considered as point masses separated by the distance d, and Z = itd2 v) is related to the (hard-sphere) collision frequency Zab defined in Eq. (4.16), that is, Zab = Z[A][B. ... 

From equation (2.30) it can be seen that p E, t) is dependent on cj, the form of P(EIE ) and k E). o is most often taken to be the Lennard-Jones collision frequency i.e., the hard sphere collision frequency which is rectified for the effects of intermolecular forces by the inclusion of a collision integral factor. 

Figure 5.3 Free jet center line beam velocity (u/uj, temperature T/TJ, gas density (n/nj, and hard sphere collision frequency (v/vj as a function of the distance from the nozzle, in units of nozzle diameters for the case of 7 = 5/3. Note that none of these parameters depends upon the stagnation pressure. Taken with permission from D.R. Miller (1988).

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. 

By using the hard sphere collision model we can compute a collision frequency for three molecules A, B, and C by first computing the stationary concentration of the three possible binary complexes AB, BC, and CA. If we call tab, tbc, and tca the mean lifetime of these binary complexes/ their stationary concentrations are approximately given by... 

This expression has a particular physical meaning. If we set equal to zero and replace nd with the hard-sphere collision cross section of the kinetic theory of gases then we obtain the collision frequency Z (Hirschfelder et ai., 1959). 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

If we assume that molecules can be considered as billiard balls (hard spheres) without internal degrees of freedom, then the probability of reaction between, say, A and B depends on how often a molecule of A meets a molecule of B, and also if during this collision sufficient energy is available to cross the energy barrier that separates the reactants, A and B, from the product, AB. Hence, we need to calculate the collision frequency for molecules A and B. 

At high-particle number densities or low coefficients of normal restitution e, the collisions will lead to a dramatical decrease in kinetic energy. This is the so-called inelastic collapse (McNamara and Young, 1992), in which regime the collision frequencies diverge as relative velocities vanish. Clearly in that case, the hard-sphere method becomes useless. 

Consider a volume containing c A molecules of A (mass mA) and c B molecules of B (mass mB) per unit volume. A simple estimate of the frequency of A-B collisions can be obtained by assuming that the molecules are hard spheres with a finite size, and that, like billiard balls, a collision occurs if the center of the B molecule is within the collision diameter dAB of the center of A. This distance is the arithmetic mean of the two molecular diameters dA and dB ... 

Frequency of collisions. The mean frequency of collisions is similarly expressed in the hard spheres approximation as... 

This result has a simple physical interpretation. The collision frequency between two hard spheres is given by1... 

The relative efficiency per collision for deactivation of excited molecules in thermal reactions increases with the number of atoms in the collider, but reaches a constant limit when this number exceeds about 12. This has been demonstrated for many thermal reactions by studying the low pressure fall-off. It may be noted from eqn. (10) that plots of k /k against pressure for different inert gases should comprise a set of curves dispersed along the pressure axis according to the various efficiencies of deactivation per unit pressure. The relative efficiency per collision can be derived by calculating the collision frequency, Z, with a hard sphere or Lennard—Jones model. 

From the concept of a root-mean-square speed we can estimate the collision frequency Z between successive elastic collisions between molecules in a gas and the mean free path X. We assume an effective diameter d of two molecules (assumed to be hard spheres, so that each molecule will collide with another within an area nd1) the collision frequency z is given by... 

F. Bhnolecular Collision Frequencies. We can immediately generalize our results to include the case of collisions between molecules that have different hard sphere diameters. If the diameters of the two species are [Pg.154]

Collision frequencies were calculated using the adjusted hard-sphere expression given by Curtiss, Hirschfelder and Bird. The collision diameter of HNO3 has not apparently been measured and was assumed to be 6.7 A, as for NO2CI the values for He, Ha and SFe were taken to be 2.6, 2.9 and 5.0 A. It was then assumed that SFe deactivates HONOf at every collision so that o) = Z Fe] but that co = PZ[M] for other gases. From the ratio of experimentally determined efficiencies, P was found to be 0.15 for He. 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

To compare the computed thermal rate of cyclohexane ring inversion with experimental results, we need to match the collision frequency with pressure. We compare with vapor- and liquid-phase measurements in CS2 solvents at a temperature of 263 K. A hard-sphere model gives ft 10 6 ps 1 torr 1 in the gas phase. We find that the experimental data in the gas phase compare well with the calculations if a somewhat lower collision frequency corresponds to the same... 

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