The Rigid Sphere Model

2 Brief Description of the Modees 10.3.2.1 The Rigid Sphere Model 

A rigorous treatment of the rigid sphere model was presented by Mason and McDaniel here, we give an abbreviated version. According to this model, the collision between an ion and a gas molecule is treated as that of rigid spheres in which the ion is eqnaUy likely to be scattered in any direction (in the center-of-mass coordinate system). Mason and McDaniel described the mean ion energy (14 mv ) in Equation 10.12  

FRAME 10.2 ON MASON S FORMULAS The following is the Mason-Schamp equation  

The equation is in centimeter-gram-second (cgs) units (Mason)  

Consider in Nj at 300 K and 1 atm and suppose that K = 2.0 cm V- S i. Calculate the cross-section ft with the assumption that at low field = the 

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Experimental data on nitrogen obtained from spin-lattice relaxation time (Ti) in [71] also show that tj is monotonically reduced with condensation. Furthermore, when a gas turns into a liquid or when a liquid changes to the solid state, no breaks occur (Fig. 1.17). The change in density within the temperature interval under analysis is also shown in Fig. 1.17 for comparison. It cannot be ruled out that condensation of the medium results in increase in rotational relaxation rate primarily due to decrease in free volume. In the rigid sphere model used in [72] for nitrogen, this phenomenon is taken into account by introducing the factor g(ri) into the angular momentum relaxation rate... 

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

In view of the failure of the rigid sphere model to yield the correct isochoric temperature coefficient of the viscosity, the investigation of other less approximate models of the liquid state becomes desirable. In particular, a study making use of the Lennard-Jones and Devonshire cell theory of liquids28 would be of interest because it makes use of a realistic intermolecular potential function while retaining the essential simplicity of a single particle theory. The main task is to calculate the probability density of the molecule within its cell as perturbed by the steady-state transport process. 

Table 19.2 Equations for the rigid sphere model of LLPTC (after Yang. 1998)... 

Several models have been proposed to account for the overall effect of these three forces on the motion of the ion, and some of the classical models are discussed here in brief, and their usefulness in predicting the mobility of polyatomic ions in different drift gases is examined. Two simple models are considered first the rigid sphere model and the polarization limit model. Next, a more refined yet relatively simple-to-use model is described in which a 12,4 hard-core potential represents the ion-neutral interaction. The more complex three-temperature model is not discussed because ions in linear IMS are traditionally regarded as thermalized. This is the one-temperature assumption, in which ion temperature is assumed to be equal to the temperature of the drift gas. 

The preceding treatment was based on considerations of the conservation of momentum and energy, as well as a few approximations, such as the one-temperature assumption. The rigid sphere model qualitatively describes some aspects of observed mobility measurements For a given effective temperature, the mobility coefficient is inversely proportional to the neutral gas density, and the drift velocity of the ions Vj depends on EIN. 

According to the rigid sphere model, the collision cross section given in Equation 10.10 may be written per Equation 10.16 as... 

In the rigid sphere model, the sum of the radii of the ion and the neutral molecule d will increase slightly as the chain length and ion mass in the homologous series increase. In the polarization limit model, the ion size is totally neglected, whereas in the hard-core potential model, (the minimum in the interaction potential) depends on the ion mass, as shown in Equation 10.22 ... 

FIGURE10.3 The measured inverse mobility of protonated acetyl compounds in air at 200°C as a function of ion mass. Curve a was calculated according to the rigid sphere model with Tq= 2.60 A curve b according to the polarization limit model curve c according to the hard-core model with a = 0.2, z = 0 A/amu, and Tq = 2.40 A curve d with a = 0.2, z = 0.0013 A/amu, and Tq= 2.20 A. (From Berant and Karpas, Mass-mobility correlation of ions in view of new mobility data, /. Am. Chem. Soc. 1989, 111, 3819-3824. With permission.)... 

As an example. Figure 2.34 presents g(r) for Lennard-Johnes intermolecular potential and for the rigid sphere model (Stanley, 1971). Hence, the theory of liquids replaces calculation of the statistical sum and the configurative integral by consideration of the probability of configuration groups of 2, 3, and more particles. Therefore, computation of the correlation functions leads to the description of all the thermodynamic functions (Croxton, 1974). 

Relationship 14 is an approximation, valid for regular solutions and van der Wtials interaction forces, in particular, for the rigid sphere model (s —> 00 when r < d) with the attraction potential... 

For the rigid sphere model, the coefficient of thermal conductivity is... 

According to the rigid sphere model in the kinetic theory of gases, the mean free path X is given by (Kennard, 1938)... 

We are finally ready to use the angular distribution as a probe for the potential. As usual, we begin with the rigid-sphere model. Using Eq. (4.8) for the deflection function needed in Eq. (4.9), Eq. (4.10) yields, for the hard-sphere scattering,... 

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