Collision cylinder

Figure A3.1.2. A collision cylinder for particles with velocity v striking a small region of area A on the surface of a contamer within a small time interval 5f Here is a unit nomial to the surface at the small region, and pomts into the gas.

Notice that each collision is counted twice, once for the particle with velocity v and once for the particle with velocity v We also note that we have assumed that the distribution fiinctions/do not vary over distances which are the lengths of the collision cylinders, as the interval 6t approaches some small value, but still large compared with the duration of a binary collision. 

Figure A3.1.3. The collision cylinder for collisions between particles with velocities v and v. The origin is placed at the centre of the particle with velocity v and the z-axis is in the direction of v - v. The spheres indicate tire range, a, of the intennolecular forces.

The number of (v, v)-collision cylinders in the region 8r8v is equal to the number of particles with velocity v in this region,/(r,v,0 r5v. 

Each (VpV)-collision cylinder has the volume given above, and the total volume of these cylinders is equal to the product of the volume of each such cylinder with tire number of these cylinders, that is / (r,v,0 vj - v jt<3 8r8v8t. 

Stosszahlansatz. The total number of (Vj, v)-collisions taking place in bt equals the total volume of the (Vj, v)-collision cylinders times the number of particles with velocity per unit volume. 

The gas has to be dilute because the collision cylinders are assumed not to overlap, and also because collisions between more than two particles are neglected. Also it is assumed that/hardly changes over 8r so that the distribution fimctions for both colliding particles can be taken at the same position r. 

Figure A3.1.7. Direct and restituting collisions in the relative coordinate frame. The collision cylinders as well as the appropriate scattering and azimuthal angles are illustrated.

Thus die increase of particles in our region due to restituting collisions with an impact parameter between b and b + Ab and azimuthal angle between e and e + de (see figure A3.1.7 can be obtained by adjusting the expression for the decrease of particles due to a small collision cylinder ... 

Now we are in the correct position to compute F, using exactly die same kinds of arguments as in the computation of r, namely, the constmction of collision cylinders, computing the total volume of the relevant cylinders and again making the Stosszahlansatz. Thus, we find that... 

In the relative coordinate system in which the first sphere is at rest, a collision can occur only if the center of the second sphere lies within the collision cylinder" as shown in figure 9.6. Now, the volume of the cylinder is equal to b d4> db u 5t. Prom figure 9.6 it should be clear that... 

Thereafter, one such collision cylinder is associated with each of the /i(r, Cl, t)dcidr molecules of type 1 within the specified velocity range dc about Cl in the volume element dr about r. 

Fig. 2.10. Collision cylinder. Scattering in the ci molecular frame, where the C2 molecules have velocity g2i.

To deduce the formula for the dense gas collision frequency a modified relation for the volume of the collision cylinder is required. As mentioned in chap 2, it is customary to consider the motion of particles 2 relative to the center of particles 1 (see Fig 2.2). For a binary molecular collision to occur the center of particle 2 must lie on the sphere of influence with radius di2 about the center of particle 1, see Fig 2.7. The radius of the sphere of influence is defined by (2.152). Besides, since the solid angle dk centered about the apse line k is conveniently used in these calculations in which the billiard ball molecular model is adopted, it is also necessary to specify the direction of the line connecting the centers of the two particles at the instant of contact [86]. The two angles 6 and 4> are required for this purpose. Moreover, when the direction of the apse line lies in the range of 0, 4> and 6 - - dO, 4> + d4>, at the instant of collision, the center of particle 2 must lie on the small rectangle da cut out on the sphere of influence of particle 1 by the angles dO and d< >. The area of this rectangle is ... 

Now, again, we use a probabilistic argument to say that the number of particles with velocity Vj in this total volume is given by the product of the total volume and the number of particles per unit volume with velocity Vj, that is, 5v(v,Oj)y(r,Vj,0. To complete the calculation, we suppose that the gas is so dilute that each of the collision cylinders has either zero or one particle with velocity Vj in it, and that each such particle actually collides with the particle with velocity v. Thus the total number of collisions suffered by particles with velocity v in time St is... 

For particles that are approaching, require that they lie in the collision cylinder, viz.,... 

Fig. 3. The collision cylinder for a (vi, V2) collision taking place in time 8t. The sphere of radius a about particle 1 is the action sphere.

We now turn to a computation of T. We first consider (vi, V2) collisions, which in the relative coordinate frame take place with impact parameter in the range b to b+ db, and azimuthal angle in the range e to e + de. We see that such a (vi, V2) collision will be initiated in the time interval St if there is a molecule with velocity V2 situated within the little collision cylinder of volume v2—Vi f> db de St attached to the action sphere as shown in Fig. 4. We imagine... 

That is, that each particle with velocity V2 in a (vi, V2, ft, e) collision cylinder actually leads to a (vi, V2) collision. According to Boltzmann, this number is found by combining... 

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