Pre-collision velocities

The nine-fold integration for the gain term is over both pre-collision velocities and over the second (all but the first) post-collision velocity. Both pre-collision states are folded with their corresponding distribution function. 

Because the collisions are elastic, the pre-collision velocities before inverse collisions are the same as the values after a direct collision (which is not the case for inelastic collisions, as will be discussed below). As a result, conservation of momentum and energy follows directly from the definitions of the pre-collision velocities since... 

N number of particles, a range of inter-particle forces). If the collision process is binary and non-reactive (post-collision species i, j remain the same as pre-collision species i and j), these indices do not appear in the collision integral, and we can adopt the standard notations of a binary collision turning the two velocities v, v into v, v, with the corresponding abbreviations for the distribution functions /, /i and /, /, respectively. Let W(v, vi —> v, v ) denote the probability for such a transition, then... 

The potential deflects the colliding molecules from their original, pre-collision paths. The deflection caused by the collision is defined as the angle between the final and initial relative velocity vectors. Section 4.1 showed that the initial collision energy and the impact parameter specify a imique collision trajectory. It follows that the deflection is a unique function of the initial collision energy and impact parameter and can be computed. Section 4.2.3, once we have determined the colhsion trajectory R t). 

In fact (98) is model independent to order u, since only the linear term in u c, contributes. To close the hydrodynamic equations for the mass and momentum densities [(74) and (75)] we need expressions for the pre-collision and post-collision momentum fluxes, and n . From (76) we can obtain an expression for in terms of the velocity gradient. 

Consider a sphere-sphere collision. Let the incoming and outgoing distributions be given by /i, /2 and /i, /2, respectively. Also, assume that the pre- and postcollision velocity-space regions occupied by hard-spheres 1 and 2 are given by cPv, cPv and cPv2, cPv2, respectively. 

In this equation, r is a unit vector separating the particles 1 and 2, so that (r v)u(r v) is the velocity of approach of the particles together and is zero if the particles are separating. The first delta function describes the post -collision motion, which leads to a gain of the density f2(vl5 v2,rlT r2, f)> while the second delta function term describes the pre -collison motion which leads to a loss of the density f2 (v, v2, r , r2, f), and (a = 1, b = 2)... 

Thus, the pre-exponential factor in the rate constant of a reaction following collision of a molecule with a surface has the dimensions of a linear velocity. The probability factor, which, as seen in a previous section, might be unity, for simple condensation of an atom on a surface, will in general be smaller than unity. It will be small if the reactant loses freedom upon reaching the transition state. A calculation of P would be tantamount to a calculation of A5 . ... 

Equation 5.33 shows that pre K QK by 5 ol ijJ i tR Sl mean square velocity of the individual particles. The physical meaning of this dependence is that the larger the average velocity, the more frequent the collisions and the larger the force of collision. These two independent terms give us the quantity u in the kinetic theory expression for the pressure. 

There are two key side effects of the velocity dependent force. First, kinematic cooling results in real cooling, not just a rotation of position-momentum phase space, yielding an increased phase space for the cold molecules. Second, since there is dissipation, if the collisions occur in a region containing a trap for the molecules, the trap can be continuously loaded without the worry of how to load pre-cooled molecules into a conservative potential well. 

Figure 6 Newton diagram showing pre- and postcollision velocity vectors in the laboratory and centre-of-mass reference frames for the collision of and M2 having velocities of /i and...

The numerical value of the collision theory value rate constant, previously calculated for the NOj + CO reaction (3.11 dm moL sec 0 corresponds to the typical values calculated for the pre-exponential factors of gas-phase bimolecular reactions. Dividing this value by the molar volume of an ideal gas at standard temperature and pressure (22.421 dm moL ), we obtain a relaxation rate, which is approximately = 10 ° atm sec at TTi K. This implies that the mean free time between collisions of molecules of an ideal gas is 0.1 nsec, and that the corresponding collision frequency is 10 ° see . Some caution must be made in using these values, since molecules interact with a range of velocities and instantaneous inter-molecular distances, such that there is no single collision frequency, but... 

Equations (135)-(138) can be solved to relate the pre- and post-collision stresses to the velocity gradient ... 

---