Cross section elastic hard spheres

If the projectile and target atoms interact like colliding billiard balls (elastic hard-spheres), the interatomic potential that represents this condition is called a hard-sphere potential. For a hard-sphere potential, the power-law cross-section parameter m in (4.19) is equal to 0. Derive the total cross-section, a (It), for a hard-sphere potential. 

For elastic hard spheres with weak central attractive forces, the collision cross section is enhanced by the attractive force, since two particles can... 

The simplified-kinetic-theory treatment of reaction rates must be regarded as relatively crude for several reasons. Numerical calculations are usually made in terms of either elastic hard spheres or hard spheres with superposed central attractions or repulsions, although such models of molecular interaction are better known for their mathematical tractability than for their realism. No account is taken of the internal motions of the reactants. The fact that every combination of initial and final states must be characterized by a different reaction cross section is not considered. In fact, the simplified-kinetic-theory treatment is based entirely on classical mechanics. Finally, although reaction cross sections are complicated averages of many inelastic cross sections associated with all possible processes by which reactants in a wide variety of initial states are converted to products in a wide variety of final states, the simplified kinetic theory approximates such cross sections by elastic cross sections appropriate to various transport properties, by cross sections deduced from crystal spacings or thermodynamic properties, or by order-of-magnitude estimates based on scientific experience and intuition. It is apparent, therefore, that the usual collision theory of reaction rates must be considered at best an order-of-magnitude approximation at worst it is an oversimplification that may be in error in principle as well as in detail. 

In Equation 7 /ab, Xr, and mixture component (R ), the mole hraction of Ri, and the hard sphere cross section for A vs. R elastic collisions. 

Mean hard sphere elastic intercollision lifetimes, , have been calculated using cross sections derived from gas-liquid critical data, leading to a [/] ratio of 3600 for the C2H6/C2F6 experiment... 

Although the scattering cross sections that underlie these and values are subject to uncertainty, the above conclusions are believed to be valid for many atom transfer reactions. The replacement of hard sphere by mean elastic intercollision lifetimes corresponding to a realistic potential description of the intermolecular forces, for example, would reinforce the present arguments (4). 

Several conclusions follow from the present results (i) The per-bond nonthermal F-to-HF reactivities for Ci-Ce alkanes are roughly equivalent. Steric and/or bond strength eflFects in these substances may give rise to 10-15% reactivity diflFerences, (ii) The deuterium kinetic isotope eflFects for the per-bond nonthermal F-to-HF (DF) reactivities are quite small for cyclopentane and C2-C5 alkanes, (iii) The nonthermal corrections to the MNR H F yields for low-reactivity hydrogen donors are negligibly small, and (iv) For reactive hydrocarbons the uniform per-bond reactivity model may be combined with the simple collision fraction mixture law and hard sphere elastic cross sections obtained from gas-liquid critical data to estimate the nonthermal H F yield corrections in MNR experiments. The simple mixture law should provide a good description of the trace nonthermal yields in experiments in which the total thermal competitor concentration is held constant. 

If two colliding particles can be considered hard elastic spheres of radii r and r2, their collisional cross section is equal to 7r(ri +r2). Obviously, interaction radius and cross section can exceed corresponding geometric sizes because of the long-distance nature of forces acting between electric charges and dipoles. On the other hand, if only a few out... 

In both equations [3] and [4], the collisions of ions with neutrals are considered to be a completely elastic process. Thus, the collision cross section obtained is termed the hard-sphere collision cross section. When compared to molecular simulations, these collision cross section measurements can provide detailed structural information about the analyte (31-34). 

Hard sphere elastic cross sections [o.-] have been obtained from averaged molecular force constant as determined from experimental equation of state and transport property data (t6.77). The 2h.2 value for represents the self-collision elastic cross section for Ne. The mixed,values for vs. Hg, Ar and... 

In both Eqs. (12.4) and (12.5), the colhsions of ions with neutrals are considered to be a completely elastic process. Thus, the collision cross section obtained is termed the hard sphere collision cross section because it emulates the scattering process of billiard balls that is, only momentum is transferred between the two collision partners. By comparing empirically determined cross sections with theoretical results, it has been shown that the hard-sphere approximation is best suited for analytes larger than 1000 Da, which is typically the size range in which many bioanalytical measurements are made. However, as the size of the analyte approaches the size scale of the drift gases used for separation. 

Instead of assuming hard sphere behaviour, we can refine the collision theory by using cross sections calculated on the basis of microscopic interaction potentials of species A and B during their collision, Before we develop the problem for reactive collisions, we will start by treating the simpler situation of an elastic collision between two bodies subjected to a central force, which only leads to scattering. 

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