Hard sphere molecules model

Equation (4) is a variable covolume departure from the hard-sphere-molecule Eq. (1), for if /3=0.625, k=1, and a—0 the K-W equation would be identical with the Boltzmann equation to the third virial term, and the ki s would be just the bi of the hard-sphere-molecule model. If 0 were 0.625, with a about 0.25 to 0.5, one might consider the K-W equation to be a soft-sphere equation of state. In applying Eq. (4) to the calculation of detonation velocities it was quickly found, however, that 0 could not be as large as 0.625,... 

In addition, assuming a model for the molecular interaction we can obtain explicit expressions for the transport coefficients of the momentum and energy equation. For example, for the hard sphere molecules model, we have... 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

From the second of these derivatives, evaluate 0C as predicted by this model. Use this value of 6C and the first of these derivatives to evaluate the relationship between Tc and the two-dimensional a and b constants. How does this result compare with the three-dimensional case The van der Waals constant b is four times the volume of a hard -sphere molecule. What is the relationship between the two dimensional b value and the area of a haid-disk molecule ... 

The perturbed hard sphere fluid model used in this study is an extension of the the work of Chandler and coworkers (23 J2). In this model the chemical potential of the solute molecule and the resulting Raman frequency shift are separated into two terms. 

Most free-volume models for diffusion in polymers follow the phenomenological basis set in (55) where the self-diffusion of an ideal liquid of hard spheres ( molecules ) has been analysed. These molecules are confined - for most of the time - in a cage formed by their immediate neighbours. A local fluctuation in density may open a hole within a cage, large enough to permit a considerable displacement of the sphere contained by it. This displacement gives rise to diffusion only if another sphere jumps into the hole before the first sphere returns to its initial position. Diffusion occurs not as a result of an activation process in the ordinary sense but rather as a result of the redistribution of the free-volume within the liquid of hard spheres. 

Table 1 summarizes a few values of the dipole moment fx and of the electrosorption valency l of halide and alkali metal ions adsorbed on mercury. The experimental values of l are relative to low coverages near the potential of zero charge and are taken from Schultze and Koppitz,50 while the corresponding // values were calculated from Eq. (75). The theoretical values in the last column are from a hard sphere electrolyte model. Further data can be found in the article by Schmickler49 Note that, in the electrochemical environment, the dipole moments are much smaller than in vacuo, where they can reach values of the order of 7 D for the alkali metal ions. No doubt, this difference is caused by the screening of the adsorbate dipole by the solvent molecules. 

IC. Hard Sphere Model with Central Attractive Forces. A great improvement over the hard sphere model physically, but one th( t is much more difficult to deal with mathematically, is the hard sphere molecule which is capable of exerting attractive forces, centrally directed (i.e., the force between two molecules depends only on the distance between them and is directed along their line of centers). The molecule is still spherical, and the closest distance of approach of two molecules is given by their mean diameter. This model can now account for the properties of condensed states. 

By using the hard sphere collision model we can compute a collision frequency for three molecules A, B, and C by first computing the stationary concentration of the three possible binary complexes AB, BC, and CA. If we call tab, tbc, and tca the mean lifetime of these binary complexes/ their stationary concentrations are approximately given by... 

The simplest molecular model capable of describing a liquid is the hard sphere molecule with an attractive potential well (Sec. VII. 1). There are... 

With such a definition of a collision and the hard sphere well model we must then conclude that each molecule is in a constant state of collision with all of its Z nearest neighbors. If we consider a dilute solution of two species A and B in a third solvent S, then the crowded condition of the system will make it likely that if A and B ever do collide and become nearest neighbors, they may remain so for some time. The mean lifetime of such a collision pair A-B can be estimated crudely by considering the time /ab that it will take for A and B initially tab apart to diffuse apart to a distance 1.7/ ab out of range of each other s attractive forces. This time ab is of the order of magnitude of ... 

Such a model is meaningful only for hard sphere molecules. For real force fields, the molecules are always in a state of collision, i.e., interaction which causes momentum exchange. 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

Thus, we first consider Eq. (8.10) for hard-core chain models, specifically tangent hard-sphere chain models (Dickman and Hall (1986) Yethiraj and Hall, 1991). Models and theories of the packing problems associated with hard-core molecules have been treated in Sections 4.3, 6.1, 7.5, and 7.6. We recall... 

The volume restriction effect as discussed in this paper was proposed several years ago by Asakura and Oosawa (12,13). Their theory accounted for the instability observed in mixtures of colloidal particles and free polymer molecules. Such mixed systems have been investigated experimentally for decades (14-16). However, the work of Asakura and Oosawa did not receive much attention until recently (17,18). A few years ago, Vrij (19) treated the volume restriction effect independently, and also observed phase separation in a microemulsion with added polymer. Recently, DeHek and Vrij (20) have reported phase separation in non-aqueous systems containing hydrophilic silica particles and polymer molecules. The results have been treated quite well in terms of a "hard-sphere-cavity" model. Sperry (21) has also used a hard-sphere approximation in a quantitative model for the volume restriction flocculation of latex by water-soluble polymers. 

The Cohen and Turnbull free-volume model [30] assumes a liquid composed of hard-sphere molecules and voids in which diffusion occurs whenever a void larger than some minimum volume V forms in the body of the liquid and a molecule jumps into it. The equation for the diffusion coefficient is... 

At first glance it seems paradoxical to treat unimolecular reactions, in which a single molecule is apparently involved in reaction, in terms of a collision theory based on pairwise interactions. Indeed, we have developed a rather specific picture of a chemical reaction from the hard-sphere collision model, in which bonds are formed rather than broken and in which the energetics of reaction are represented in terms of relative kinetic energy. 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

Other models for ternary amphiphilic systems are based on mixtures of hard spheres and ellipsoids with Lennard-Jones interactions [58] or on mixtures of hard spheres and diatomic hard-sphere molecules [59]. Such models have been studied by molecular dynamics simulations. 

The values of the dimensionless parameters 2 and CO for the most classic collision models are given in Table 1. The Maxwell molecules (MM) model assumes a linear relationship between viscosity and temperature, although for the hard sphere (HS) model, the viscosity is proportional to the square root of the temperature. These models could be roughly considered as limits for the real behavior of gases, and the variable hard sphere (VHS) model proposed by Bird [2] is much more accurate. Another sophistication has been proposed by Koura and Matsumoto who developed the variable soft... 

The simplest behaviour is of course expected for reactions between neutral molecules in non-polar solvents, which will correspond most closely to hard-sphere theoretical models. Complications may be expected when the reactants are polar or ionic, and when the solvent molecules interact strongly with them or with each other. Solvent-solvent interactions have been introduced into Smoluchowski-type diffusion theory [32,c], by assuming that the solvent molecules interact according to a Lennard-Jones potential. In general, however, models that treat the solvent as a continuum must be abandoned when short-range interactions are considered. Kinetic theory and computer simulation are more flexible means of taking account of these and other additional factors (Section 2.6 below). 

The important theoretical expressions for fcc above have been derived from a model of hard-sphere molecules in a continuous medium. Intermolecular forces between reactant molecules have been neglected. When the reactants are ionic or polar, there will be long-range Coulombic interactions between them. For reactions between ions, we stated in Chapter 2 (Section 2.5.3) an expression for the value of the rate constant at low concentrations, and noted some reactions between oppositely-charged ions that have rate constants in approximate agreement with it. We also noted that for several such reactions the effect of added inert ions follows approximately the Debye-Hiickel limiting law. 

Cohen and Turnbull developed a statistical mechanics model of diffusion in a liquid of hard spheres. This model has been widely adapted to describe the diffusion of small molecules in polymers (25), It provides the following expression for the diffusion coefficient (3 ... 

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