Specular collision

Mathematically HkH accoimts for the porosity (e) and the tortuosity (x) for gas permeation dominated by Knudsen diffusion (see Eq. (9.16)). P[-] is used to correct for behaviour deviating from the ideal Knudsen behaviour, e.g., due to reflection conditions deviating from elastic specular collisions with the pore wall. 

In the case when the molecules make specular collisions with the wall, the distribution function approaches a Maxwell-Boltzmann distribution function but with a temperature that is not determined by the walls but by the total energy of the gas. 

It is very satisfactory from a macroscopic point of view that the Boltzmann equation, through the H-theorem, predicts the approach to equilibrium of an initial nonequilibrium state of the gas. However, one can raise serious objections to the //-theorem, and to the Boltzmann equation, from a microscopic point of view. The fundamental difficulty is that the Boltzmann equation is inconsistent with the laws of mechanics. The laws of mechanics require that any equation of motion describing the gas be invariant under time reversal if the particles make specular collisions with the walls. Otherwise any dynamical processes that do not involve collisions with the walls must be time reversal invariant. That is, the form of the equations of motion must be invariant if v-> —V and t —t. It is clear from an inspection of the Boltzmann equation for points far from the walls. 

We now want to show that our model gas of point-mass particles obeys the ideal gas law. We assume that the box confining our model system is rectangular with walls that are perpendicular to the coordinate axes and that the walls are smooth, slick, flat, and impenetrable. A collision of a molecule with such a wall is called a specular collision, which means (1) It is elastic. That is, the kinetic energy of the molecule is the same before and after the collision. (2) The only force exerted on the particle is perpendicular to the wall. 

Particles are moved along their current velocity vectors without undergoing interactions for a time At which is chosen smaller than the mean collision time. If a particle hits the domain boundary, its velocity vector is modified according to the corresponding boundary condition (for example specular or diffuse reflection if a particle hits a wall) ... 

Thus, for Knudsen cosine scattering, / = 1, and for specular reflection, / = 0. Equation (59) may be solved for the drift velocity of the scattered molecule to give Uf = (1 - f)ut. The viscous force transmitted to the wall during gas collisions is the product of the number of collisions per second and the momentum change per collision,... 

When the particle concentration is high, the shear motion of particles leads to interparticle collisions. The transfer of momentum between particles can be described in terms of a pseudoshear stress and the viscosity of particle-particle interactions. Let us first examine the transfer of momentum in an elastic collision between two particles, as shown in Fig. 5.8(a). Particle 1 is fixed in space while particle 2 collides with particle 1 with an initial momentum in the x-direction. Assume that the contact surface is frictionless so that the rebound of the particle is in a form of specular reflection in the r-x-plane. The rate of change of the x-component of the momentum between the two particles is given by... 

So far, in this model, we assume that all the particles are elastic and the collision is of specular reflection on a frictionless smooth surface. For inelastic particles, we may introduce the restitution coefficient e, which is defined as the ratio of the rebound speed to the incoming speed in a normal collision. Therefore, for a collision of an inelastic particle with a frictionless surface as shown in Fig. 5.9, we have... 

Figure 17 Angular distributions for direct scattering of preferentially oriented NO from Pt(l 11), presented in a polar plot. E = 0.18 eV, Ts = 573 K, i = 50°. The hnes through the angular distributions are drawn to guide the eye. The arrows indicate the angle of incidence and the specular angle. In case of N-end collisions less molecules are directly scattered. Molecules with are directly scattered after an N-end collision come off closer to the surface normal than molecules with an O-end collision. From Kuipers et al. [94].

Furthermore in the derivation of (9.4) it is assumed that after collisions with the wall the molecules are specularly reflected. This is usually not the case. The walls have a certain roughness and this causes diffuse reflections. This effect is accounted for by a factor 1/0k for smooth walls 0 equals unity, otherwise it is larger. 

Any interaction with another system must be such as to leave p, quantized that is, to change it by the amount Ap, = Anji/d or nh/d, in which = An, is an integer. One such type of interaction is collision with a photon of frequency v, represented in Figure 6-4 as impinging at the angle and being specularly reflected. Since the momentum of a photon is hv/c, and its... 

Experimental evidence for the existence of intrinsic precursor states is rather more difficult to come by. The common observation that the initial sticking probability, s0, often decreases with increaing substrate temperature is consistent with the existence of such a state, as discussed here. Indirect evidence is also provided by molecular beam studies, for example, Hayward and Walters [401] (for H2 on W 001 ) and Engel [402] (for 02 on Pd 111 ) have observed scattered particle intensity distributions which, even at a fractional coverage in the chemisorbed layer close to zero, exhibit a strong directional lobe in the specular direction superimposed on a cosine law distribution. The specular lobe clearly contains molecules scattered at the first collision, while the cosine law component is most readily attributed to the particles which are trapped in the precursor state and then scattered back into the gas phase. Of... 

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