Hard sphere model with attractive forces

With regard to the imperfection of gases, we are limited to the forces between two molecules only (two body forces) which give the expressions of the second coefficient of the virial. Researchers have endeavored to calculate the third, fourth and fifth coefficient of the virial. Here the three body forces are involved for the third coefficient, four body forces for the fourth and compact packing of spheres for the fifth coefficient. As for the second coefficient, the authors initially stuck to the hard-sphere model without attraction force (see section 7.3.3.1 and Figure 7.6), and as in the case of the second coefficient, they obtained coefficients practically independent of temperature, which allowed Hirshfelder and Roseveare to propose a state equation in the form ... 

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. 

FIGURE 3.5 Procedure to form a molecule in the SAFT model, (a) The proposed molecule, (b) Initially the fluid is a hard-sphere fluid, (c) Attractive forces are added, (d) Chain sites are added and chain molecules appear, (e) Association sites are added and molecules form association complexes through association sites. (From Fu, Y.-H. and Sandler, S.I., Ind. Eng. Chem. Res., 34, 1897, 1995. With permission.)... 

Figure 6.3. Hard sphere model with force of attraction...

Figure 7.9. Hard-sphere model with force of attraction To integrate expression [7.76], we develop the exponential in series ...

Air at room temperature and pressure consists of 99.9% void and 0.1% molecules of nitrogen and oxygen. In such a dilute gas, each individual molecule is free to travel at great speed without interference, except during brief moments when it undertakes a collision with another molecule or with the container walls. The intermolecular attractive and repulsive forces are assumed in the hard sphere model to be zero when two molecules are not in contact, but they rise to infinite repulsion upon contact. This model is applicable when the gas density is low, encountered at low pressure and high temperature. This model predicts that, even at very low temperature and high pressure, the ideal gas does not condense into a liquid and eventually a solid. 

If an elastic-hard-sphere model is not used, S(i y) in Eq. (2-26) must be properly expressed as a function of for the specified model. For example, for elastic hard spheres with superposed central attractive forces, the maximum value of To for contact collisions is di2. and if the attractive forces are weak,... 

Because there are no attractive interactions in the potential, the hard sphere model does not describe the forces between molecules very well. More realistic potentials include an attractive contribution, which usually varies as — 1/r (as discussed shortly) as well as a repulsive term. The latter is chosen to vary as 1 /r , with n > 6, to ensure that repulsions dominate at short distances, w = 12 often being assumed. This combination of attractive and repulsive terms defines the Lennard-Jones (12,6) potential ... 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

ID. S3rmmetrical Molecule with Central Forces. If we discard the idea of a hard sphere and replace it by a molecule that is capable of exerting both attractive and repulsive forces but acts centrally, we have the closest approach yet to real molecules, and also the model that is most difficult to treat. Such a molecule is characterized completely by the function chosen to represent its force field. A function commonly used is the Lennard-Jones function... 

This model is characterized by three parameters the well depth C/o, the range of the attractive forces o-a, and the hard sphere radius Cr. Probably because of this it is able to represent many equilibrium and transport properties of real molecules with semiquantitative accuracy. 

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 ... 

Elastic hard spheres with superposed central attractive forces This is the so-called van der Waals model for which the equation of state is... 

Fig. 2-1 Models of intermolecular potentials, (a) Forceless mass points (b) elastic hard spheres (c) elastic hard spheres with superposed central attractive forces (d) molecules with central finite repulsive and attractive forces (e) square-well model (f) point centers of inverse-power repulsion or attraction.

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... 

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