Intermolecular interactions hard sphere model

The p factor could also be a consequence of the physically naive hard sphere model used. This model requires that there be no intermolecular interactions either at close distances or at a distance. This has been shown not to be the case, with evidence coming from many aspects of physical chemistry. Also, vibrations and rotations do affect reaction, as is shown by experiments with molecular beams. 

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ > w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r > a and M(r) = c for r < a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential... 

Its exponential behavior makes this term the dominant one when short atom-atom distances between the interacting molecules are produced. Consequently, this term prevents molecules from getting closer than some limiting distance this is the physical principle behind the hard-sphere model and Kitaigorodsky s close-packing principles [2]. Moreover, the shortest atom-atom distances that one can find in different intermolecular interactions for the same pair of atoms always fall in a restricted range, a fact that allows one to define atomic radii. They differ in ionic and neutral crystals due to the different electronic structure of ionic and neutral species, as easily shown when comparing the contours at 90% probability in electron density maps for isolated atoms and their ions. 

We will now use our knowledge of intermolecular interactions to modify the ideal gas model for situations when potential interactions between the species are important. In this section, we will use the Sutherland potential function to describe the intermolecular interactions, that is, use the hard sphere model to account for repulsive forces and van der Waals interactions to describe attractive forces. This development leads to the van der Waals equation of state. This equation is particularly well suited for illustrating how the molecular concepts we learned about in Section 4.2 can be related to macroscopic property data. However, it should be emphasized that more accurate equations of state have been developed and will be covered next. 

The macroscopic properties of the three states of matter can be modeled as ensembles of molecules, and their interactions are described by intermolecular potentials or force fields. These theories lead to the understanding of properties such as the thermodynamic and transport properties, vapor pressure, and critical constants. The ideal gas is characterized by a group of molecules that are hard spheres far apart, and they exert forces on each other only during brief periods of collisions. The real gases experience intermolecular forces, such as the van der Waals forces, so that molecules exert forces on each other even when they are not in collision. The liquids and solids are characterized by molecules that are constantly in contact and exerting forces on each other. 

For simple approximations to intermolecular interactions, the kinetic theory of gases has been well developed for the computation of transport properties at low densities. Theory and theory-based correlations are reviewed in references [15] and [57]. If the molecules are modeled as hard spheres of diameter o and molar mass M, kinetic theory gives the following relations for the viscosity ti, thermal conductivity X, and diffusivity D of dilute gases ... 

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... 

Rather, flie assignment is more serious wifli intermolecular interaction potential used. For simple molecules, empirical model potential such as fliose based on Lennard-Jones potential and even hard-sphere potential can be used. But, for complex molecules, potential function and related parameter value should be determined by some theoretical calculations. For example, contribution of hydrogen-bond interaction is highly large to the total interaction for such molecules as HjO, alcohols etc., one can produce semi-empirical potential based on quantum-chemical molecular orbital calculation. Molecular ensemble design is now complex unified mefliod, which contains both quantum chemical and statistical mechanical calculations. 

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. 

Similarly to the fluid-fluid intermolecular potential, we split the solid-fluid intermolecular potential into repulsive hard-sphere and attractive interactions. Here Fhs Ps P is the excess free energy of the solid-fluid HS mixture, for which we employ Rosenfeld fundamental m ure functional [26] with the recent modifications that mve an accurate Carnahan-Starling equation of state in the bulk limit [27,28] r-r ) is the attractive part of the solid-fluid intermolecular potential. Since the iM>lid-soIid attraction interaction is not included, the solid is effectively modeled as a compound of... 

While proper treatment of intermolecular interactions is very important, it should be remembered that entropy has a crucial role in the structure of condensed phases. Hydropho-bicity, e.g., is an entropy-driven effect and is intrinsically many-body in nature. One of the simplest solvation models envisages the solute as filling a cavity in a hard-sphere (i.e., billiard-bair like, without any attractive intermolecular interactions) fluid. Entropy-driven packing of the solvent around the solute results in a liquid structure and it is often helpful to represent this structure in terms of a potential of mean force, a fictitious intermolecular interaction constructed to mimic, in a hypothetical entropy-free world, the actual liquid structure. 

The reference interaction site model (RISM) theory of Chandler and Andersen [9, 11, 10] is an extension of the theory of monatomic liquids. In RISM theory, each molecule is envisioned as a collection of spherically symmetric interaction sites. In most applications of RISM theory to small molecule liquids. Chandler and coworkers [9, 11, 12, 13, 10] modeled interaction sites as overlapping hard spheres. The primary difference between RISM theory and monatomic liquid theory is that the correlations can be propagated intramolecularly as well as intermolecularly. Chandler and Andersen generalized the OZ equation as follows for a single component fluid in Fourier space... 

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