Collision number, calculation

The vibrational relaxation of simple molecular ions M+ in the M+-M collision (where M = 02, N2, and CO) is studied using the method of distorted waves with the interaction potential constructed from the inverse power and the polarization energy. For M-M collisions the calculated values of the collision number required to de-excite a quantum of vibrational energy are consistently smaller than the observed data by a factor of 5 over a wide temperature range. For M+-M collisions, the vibrational relaxation times of M+ (r+) are estimated from 300° to 3000°K. In both N2 and CO, t + s are smaller than ts by 1-2 orders of magnitude whereas in O r + is smaller than t less than 1 order of magnitude except at low temperatures. 

Since there are no experimental values of Z for M+-M collisions available at present, we first calculate Z for the M-M collisions for N2, CO, and 02 whose experimental values are well established 7,21). After establishing the usefulness of this approach for the M-M collisions, we then calculate the collision numbers for the M +-M systems, Z+. 

Figure 4. Calculated values of the reduced collision number Z as a function of temperature for N2+-N2, 02+-02, and CO+-CO...

As we are particularly interested in surface reactions and catalysis, we will calculate the rate of collisions between a gas and a surface. For a surface of area A (see Fig. 3.8) the molecules that will be able to hit this surface must have a velocity component orthogonal to the surface v. All molecules with velocity Vx, given by the Max-well-Boltzmann distribution f(v ) in Cartesian coordinates, at a distance v At orthogonal to the surface will collide with the surface. The product VxAtA = V defines a volume and the number of molecules therein with velocity Vx is J vx) V Vx)p where p is the density of molecules. By integrating over all Vx from 0 to infinity we obtain the total number of collisions in time interval At on the area A. Since we are interested in the collision number per time and per area, we calculate... 

Rate constant temperature dependence Processing threshold Calculation of rate constants at different temperatures, including collision numbers and concentrations of species in steady state Calculation of the rate of photodissociation and cosmic ray-induced molecular processing from photon and particle fluxes... 

Calculate the collision number ZAb (m-3 s-1) for the collision between two hydrogen molecules and the collisions per molecule for ... 

The first maj or extension of the stochastic particle method was made by O Rourke 5501 who developed a new method for calculating droplet collisions and coalescences. Consistent with the stochastic particle method, collisions are calculated by a statistical, rather than a deterministic, approach. The probability distributions governing the number and nature of the collisions between two droplets are sampled stochastically. This method was initially applied to diesel sprays13171... 

An arithmetical error in Lewis s calculation of the collision number has been corrected here, which appeared also in the first edition of this book in several places. 

The theoretical treatment of vibration-vibration transfer was outlined in Section 3. Sufficient data for a priori theoretical calculations are only available for the simpler molecules. It is interesting first to discuss the general pattern revealed by the collision numbers in Tables 5 and 6 in terms of equations (18) and (19). 

CALCULATED COLLISION NUMBERS FOR INTERMOLECULAR VIBRATIONAL ENERGY TRANSFER IN OXYGEN MIXTURES AT 300 °K... 

Table 4.2 Comparison of experimental A factors with collision numbers, Z, calculated from collision theory...

Taking the collision diameter to be 400 pm, calculate the collision number, Z, for collisions between the molecules CH2=CH-CH=CH2 and CH2=CH-CHO at 500 K, and from this find the pre-exponential factor, A. 

There are several rough experimental values for the decomposition of chemically activated CH4. Some older data on the reaction D + CH3 — CH3D, studied at 25°C. by Taylor and co-workers,38 correspond to (tf) 3 keal. (the zero-point energy difference for C—H and C—D is 2 keal.). These experiments correspond to the low-pressure limit, for which the calculated value in Table XI is kao 1.7 X 1010 sec.-1 at this energy. Marcus14 analyzed these data to obtain ka = 8 X 108 sec.-1 which, if we correct for the presence of a primary and secondary isotope effect due to the D atom, would be ka 1.5 X 109 sec.-1. He estimated a possible error of a factor of 5-10 in these values. We believe that the collision number used by Marcus in the calculation of ka is too low by a factor of 3-5 which would raise the experimental value to >5 X 10 sec.-1. The agreement is adequate, but the desirability of redoing these experiments with improved techniques is evident. 

Marcus14 earlier analyzed some older data of Darwent and Steacie44 and obtained a value of ka = 1.1 X 107 sec.-1. Again, it appears that the collision number used in this calculation is too low by a factor of 2-4, and the deduced value should be raised accordingly. 

Comparison of Collision Numbers for V-V Transfer with Values Calculated Using SSH Theory, Method B [219]. 

Theoretical values for V-V transfer calculated by these authors using SSH theory (method B) were higher than experiment for the HI case and lower for the DI case, with a greater temperature dependence than that observed. Presumably, long-range forces play a more important role than the theory permits. Table 3.3 gives a comparison of collision numbers at three different temperatures, calculated from the relation Z J — tab ab> where the pressure is at 1 atm of B molecules, and where AB is the collision rate per A molecule with B molecules. 

Thus for constant vessel diameter and surface it would be expected that the product Phj P02 constant, while at constant composition the result should be that Pi xd is constant. Both these relationships are observed to good approximation considering the sensitivity of the limits to vessel surface. A further interesting feature of the lower limit phenomenon, which has not yet been discussed, is the effect of inert gases. In the presence of inert gases the collision numbers in the above calculation are altered, and the necessary modification of the calculation leads to the limit condition... 

Fig. 12 Variation of modal temperatures of OH, initially in (v n) = (8 3) with number of collision cycles (collision number) for a 1 10 mixture of OH (8 3) in a 4 1 mixture of N2 (0 10) and 02 (0 12) at 250 K. Total number of molecules is (nominally) 8,000. Tv (squares) represents vibrational temperature, Tt (circles) rotational temperature and Tt (triangles) translational temperature throughout. For OH and N2 the symbols are black, red and green for Tv, Tt and Tt respectively and are solid for OH and open for N2. The symbols for 02 follow this same pattern but are blue for all three modal temperatures. As described in the text, the primary data are the quantum state populations. Modal temperatures are calculated assuming a Boltzmann distribution and thus Tv and Tr will not be meaningful at the outset and in the early stages of ensemble evolution...

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann energy (or momentum or internal coordinate) distribution of reactants is postulated to persist during a reaction. In the collision theory, mainly due to Hinshelwood,7 the number of energetic, reaction producing collisions is calculated under the assumption that the molecular velocity distribution always remains Maxwellian. In the absolute... 

As far as we are aware, no a priori calculations, have even been carried out to evaluate Pc from the molecular properties of the collision partners, its order of magnitude has only been found a posteriori for reaction systems. It is, therefore, impossible to make a direct numerical comparison between the frequency factors A found for the process of stepwise activation Eq. VII.36 and for the "all or nothing" kinetic theory activation Eq. VII.41. The following indirect comparison is, however, instructive. Measurements on the rate of activation of J2 at about 300°K in various inert gases such as He, Ne, A, Kr, and N2 have shown22 that the frequency factor A is of the order of about 5x 1016 cm3/mole /sec. Since the collision number Z is only about 1014 cm3/mole/sec at 300°K, even a value of Pc = 1, which corresponds to unit efficiency in direct collisional activation, could not raise the calculated A-value for the standard collision.theory (Eq. VII.41) to the observed one. This is, of course, one of the old and vexing problems in chemical kinetics,17... 

If we take a = a2 = 5A. for all of our molecules, and hence r = I0A., we obtain A = 16 kcal./mole from Equation 12. If w << A/4 + AF° /2 + (AF0,)2/4a, we may obtain AF directly from Equation 11. We may then obtain an absolute value of kbi from Equation 10 if a value is assumed for the collision number, Z. Or alternatively the theory may be tested by examining the internal consistency of the calculated values, compared with our data, of the ratio (kbi)n/(kbi)1 for a series of rate constants relative to one reference constant, assuming, as a first approximation, that the collision number Z is the same for all our pairs, and therefore cancels out. 

Let us consider again a plot of the standard free energy " of species O and R as a function of reaction coordinate (see Figure 3.3.2), but we now give more careful consideration to the nature of the reaction coordinate and the computation of the standard free energy. Our goal is to obtain an expression for the standard free energy of activation, AG as a function of structural parameters of the reactant, so that equation 3.1.17 (or a closely related form) can be used to calculate the rate constant. In earlier theoretical work, the pre-exponential factor for the rate constant was written in terms of a collision number (37, 38, 58, 59), but the formalism now used leads to expressions like ... 

---