Hard sphere collision diameter

Notes Calculated from modified direct count procedures. Estimated from experimental values of P1/2, the pressure where the pseudounimolecular rate constant is 1/2 its high pressure limiting value, assuming hard-sphere or modified hard-sphere collision diameters. ... 

In this experiment the mutual diffusion coefficients for the Ar—CO2 and He—CO2 systems are to be measured using a modified Loschmidt apparatus. These transport coefficients are then compared with theoretical values calculated with hard-sphere collision diameters. 

Experimental values of can be compared with Eq. (4-23) by obtaining at a given temperature directly from experiment and from (rfln/ )/ [d(l/T)]. A preliminary value of n is then selected which is less than the maximum possible value of 6(s — 2), where s is the number of atoms in the (nonlinear) decomposing molecule. The hard-sphere collision diameter is estimated from viscosity measurements or similar sources. Now if, in Eq. 

Figure A3.1.9. Hard sphere collision geometry in the plane of the collision. Here a is the diameter of the spheres.

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

A. Mechanics of a Collision. The mechanical properties of a collision between two hard spheres of diameters ai and 0-2 and masses mi and m2 are obtained from the laws of conservation of energy and momentum. A scheme of such a collision is shown in Fig. VII.4. 

G. Duration of a Collision. The hard sphere model is very useful because it permits us to describe molecular collisions in terms of a single, simple, molecular parameter, the collision diameter. It is, however, insufficient to permit a detailed description of a chemical reaction, which is an event that transpires during a collision between two molecules, because the duration in time of a hard sphere collision is precisely zero. 

To extend the usefulness of the model to permit a description of chemical reactions, we must introduce another parameter, the effective duration of a collision. The rectangular well or central force models do this automatically by permitting molecular interaction over a range of distances. However, they are both more complex than the hard sphere model. We can rescue the hard sphere model by specifying a parameter era, the effective diameter for chemical interaction, while keeping hard sphere core diameter. When the centers of two identical molecules are a distance effective reaction volume is 7r([Pg.155]

In the first approximation the collision phenomena are described in terms of hard sphere molecular diameters, which are independent of temperature. Actually, the diameters decrease with higher temperature, approaching individual limits [1]. Let us consider a single molecular entity with the mass m and the diameter afmj, which diffuses through a gas consisting mainly of more abundant dissimilar molecules of the mass m2, the diameter dm.2 and the concentration no. If the collision diameter is mi.2 = (dm, 1 + <7m.2)/2, the tracer molecule must collide each second with the host molecules contained in a volume of about nm 2um. Because the host molecules also move, the mean relative speed u 1,2 is... 

Figure 6.1. A direct collision between two hard spheres with diameters d. ...

Chapter 6 is devoted to the topic of hard-sphere collision models (and related simpler kinetic models) in the context of QBMM. In particular, the exact source terms for integer moments due to collisions are derived in the case of inelastic binary collisions between two particles with different diameters/masses, and the use of QBMM to overcome the closure problem is illustrated. 

In the simplest version of the kinetic theory of gases, molecules are treated as hard spheres of diameter d which make binary collisions only. In this approximation the mean distance traveled by a molecule between successive collisions, the mean free path I, is related to the collision diameter by ... 

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).

To a first approximation (i.e. if the molecules behave like hard spheres with no other intermolecular forces), the virial coefficients are simply related to the rigid-sphere collision diameter, [Pg.226]

We now proceed to reduce (292) for the case of hard-core interaction. The dynamical description of a collision between two hard spheres of diameter cr is depicted in Fig. 5. The initial relative separation and momentum are r and p. It is evident that there will be no collision unless r p < 0 and where b is the... 

These ideas on energy transfer can be formalized by considering the rate of formation and the rate of destruction of excited molecules with energies between and c + de. Excited molecules are formed via collisions. If n is the concentration of unexcited A molecules, the total collision rate as computed in (2.21) is yn l2, where y, assuming the molecules to be hard spheres of diameter rf,... 

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 ... 

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