Molecular billiard ball model

Adopting the molecular billiard ball model considering molecules that are smooth and symmetric rigid elastic spheres not surrounded by fields of force, the molecules affect each other motion only at contact. 

If we take the billiard-ball, or hard-sphere, model literally, we can calculate the excluded volume constant, b, from the diameter of the molecular billiard balls, ct. The centers of two billiard balls, each of radius a, can come no closer than r = a. Therefore, we can consider that around each molecule there is a... 

It is therefore remarkable that 100 years or so before the laws of thermodynamics were formulated, Daniel Bernoulli developed a billiard ball model of a gas that gave a molecular interpretation to pressure and was later extended to give an understanding of temperature. This is truly a wonderful thing, because all it starts with is the assumption that the atoms or molecules of a gas can be treated as if they behave like perfectly elastic hard spheres—minute and perfect billiard balls. Then Newton s laws of motion are applied and all the gas laws follow, together with a molecular interpretation of temperature and absolute zero. You have no doubt... 

Physical chemists are well aware of the usefulness of models. An understanding of the fundamental properties of matter can hardly be gained from watching reality, requiring instead the posing of if-then questions that can be answered only by models. The nature of pressure or temperature of a gas as a collective property of its individual atomic or molecular constituents became obvious only through the billiard ball models of Clausius, Maxwell, and Boltzmann, despite our later insights that true atoms or molecules have quantized motion. 

Figure 2-1. We consider a surface 5 drawn in a fluid that is modeled as a billiard-ball gas. Initially, when viewed at a macroscopic level, there is a discontinuity across the surface. The fluid above is white and the fluid below is black. The macroscopic (volume average) velocity is parallel to S so that u n — 0. Thus there is no transfer of black fluid to the white zone, or vice versa, because of the macroscopic motion u. At the molecular (billiard-ball) level, however, all of the molecules undergo a random motion (it is the average of this motion that we denote as u). This random motion produces no net transport of billiard balls across S when viewed at the macroscopic scale because u n = 0. However, it does produce a net flux of color. On average there is a net flux of black balls across S into the white region and vice versa. In a macroscopic theory designed to describe the transport of white and black fluid, this net flux would appear as a surface contribution and will be described in the theory as a diffusive flux. The presence of this flux would gradually smear the initial step change in color until eventually the average color on both sides of. S would be the same mixture of white and black.

Onsager Dupuis (i960) and Jaccard (1959, 1964, 1965). The essence of the theory can be seen from a simple billiard-ball model of the molecular processes involved, while a detailed consideration of some of the mechanisms requires, as we shall... 

IB. Hard Sphere Model. Here the molecule is assumed to be the equivalent of a billiard ball. That is, the molecule is presented as a rigid sphere of diameter or, mass m (the molecular weight), and the capability... 

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 ... 

In the discussion of dilute gases in sect 2.4.2 the corresponding surface area element is determined by the product da = bdbd(f>, as illustrated in Fig 2.10. For the billiard ball molecular model the link between the two surface element formulas when centered about the apse line is defined analogous to (2.159). 

In accordance with the ideas presented in sect 2.4.3, the corresponding dilute gas collision operator can be expressed analogous to (2.185). However, the operator is reformulated and defined in terms of k because the billiard ball molecular model is adopted. The details of the transformation is explained in sect. 2.11. The result is ... 

A fundamental principle of science is that simpler models are more useful than complex ones—as long as they explain the data. You can certainly appreciate the usefulness of the kinetic-molecular theory. With simple postulates, it explains the behavior of the ideal gases in terms of particles acting like infinitesimal billiard balls, moving at speeds governed by the absolute temperature, and experiencing only perfectly elastic collisions. 

How, indeed, should one interpret the symbols in a molecular formula Dalton thought that each of the symbols in his formulas must signify an actual "atom," in the sense of an absolutely unsplittable entity, much like an invisibly small but very real billiard ball— which is why he chose to represent his atoms by distinctive iconic circles, or spherical wooden models. Few chemists thereafter took such an unre-flectively realist position. At the other extreme, some regarded chemical formulas purely conventionally, as a mere aid to memory in representing the empirical facts of chemical analysis and having no real referent in the microworld at all. 

In many cases, such as in most of the nanochaimels found in biological systems, the channel diameter is so small that the continuum model would be clearly inappropriate. There are even nanochannels that are too small to permit the passage of even a single molecule of water. In such cases, one is forced to recognize the underlying molecular structure of matter and perform what is called a molecular dynamics (MD) simulation. It is important to recognize, just like the continuum approximation, the MD approach is also an approximation to reality but at a different level. In the MD approach, one ignores the fact that the water molecule, for example, contains protons, neutrons, and electrons which interact with the protons, neutrons, and electrons of every other water molecule via quantum mechanical laws. Such a description would be enormously complicated Instead, each molecule is treated as a discrete indivisible object and the interaction between them is described by empirically supplied pair interaction potentials. For example, the simplest MD model is the hard sphere model where each molecule is modeled by a sphere, and the molecules do not interact except when they touch in which case they rebound elastically like billiard balls. 

The SCT model considers reaction between chemical species A and B, each considered to be structureless, spherical masses that interact according to the hard sphere potential V(r) — 0, r>dAB V(r) = CO, r — dAB, and all collisions result in reaction. The last may be restated as a reaction probability the probability of chemical reaction, F(r), is 1 when r — <7ab and 0 otherwise. The collision diameter, <7ab — (<5 a + <5 b)/ 2, where <7a and <7b are the molecular diameters of A and B, respectively, defines the interaction distance for these billiard ball-like collisions. The collision rate, Zab, is... 

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