Lennard-Jones equation approximation

Figure 4. Susceptibility of the energy moment at To — 2. The symbols are static Monte Carlo results [1] and the curve is obtained from a local thermodynamic approximation [1] using the bulk susceptibilities from a Lennard-Jones equation of state [90], (From Ref. 1.)...

Cook equation of state, using covolume approximation) 63-4 (Other coyolume equations of state) 65 (Jones, Jones-Miller and Lennard-Jones equations of state) 66 (Cot-trell-Paterson equation of state) 12aj) W. Fickett W.W. Wood, Physics of Fluids 1, 528(1958 (A Detonation-Product Equation of State Observed from Hydrodynamic Data)... 

This is in contrast to the soft-sphere approximation where some distance-dependent function such as the second term in the Lennard-Jones equation (3.166) is used. 

The contribution of the dispersion forces to the total potential can be well approximated by the Lennard-Jones equation for pairwise interaction. 

An approximation of the Lennard-Jones equation was made by assuming a potential barrier with a thickness of D, to express the repulsive force. Equation 6.28 then becomes... 

An approximate value for dc in the equation for tire Lennard-Jones potential, quoted above, may be obtained from the van der Waals constant, b, since... 

Obviously, the mean-field treatment of the attraetive van der Waals inter-aetion results in negleet of the influenee of the interpartiele eorrelations on as well as the influenee of attraetive forees on assoeiation effeets (ef. the definition of Eq. (78)). To obtain a more adequate approximation, Johnson and Gubbins (see, e.g., [114]), have developed an aeeurate equation of state for assoeiating Lennard-Jones fluids, or more preeisely for the following nonassoeiative potential... 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

This approximation amounts to truncating the functional expansion of the excess free energy at second order in the density profile. This approach is accurate for Lennard-Jones fluids under some conditions, but has fallen out of favor because it is not capable of describing wetting transitions and coexisting liquid-vapor phases [105-107]. Incidentally, this approximation is identical to the hypemetted chain closure to the wall-OZ equation [103]. 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

FIGURE 3.9 Comparison of the actual interatomic potential for two iodine atoms (solid line) with the approximate Lennard-Jones potential, Equation 3.32. The Lennard-Jones potential... 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

Within the model represented by equations (1) and (2), the intermolecular potential energy function is fully determined by the set of -1 charges and n n +1) Lennard-Jones parameters, where n is the number of different types of atoms in the system. For example, the water intermolecular potential in this approach requires 5 different parameters. In practice, this is modified in two ways First, one may wish to add additional point charges to provide more flexibility in modeling the molecular charge distribution. In this case, the locations of the point charges are not necessarily identified with the equilibrium positions of the atoms. Second, a major simplification can be achieved if one uses the following approximation[13] ... 

---