Stockmayer potential

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

For polar molecules, the most widely used intermolecular potential energy is the Stockmayer Potential ... 

Stockmayer potential is considered as a superposition of a Lennard-Jones (6-12) potential and the interaction of two point dipoles. Many of the properties of gases and liquids have been calculated in terms of these two potential functions. It should be borne in mind, however, that Lennard-Jones and Stockmayer potentials are idealizations of the true energy of interaction and that they are reasonably accurate for a number of simple molecules. The interaction of long molecules, molecules in excited states, free radicals, and ions cannot be described by these two potential functions (Ref 8a, pp 23 35)... 

Another useful intermolecular potential is the Stockmayer potential [178,379], which can be used to describe the interaction between polar molecules. The functional form of the Stockmayer potential is... 

The first term on the right-hand side of Eq. 12.9 or 12.14 describes the short-range, repulsive interaction between molecules as they get very close to one another. The second term accounts for the longer-range, attractive potential (i.e the dispersion interaction between the molecules). The final term is the longest-range interaction, between the dipole moments JTj and JTj of the two molecules. In the case where one or both of the dipole moments are zero, the Stockmayer potential reduces to the Lennard-Jones potential discussed in Sec 12.2.1. 

The contributions from the short-range repulsive potential and the long-range attractive potential are shown explicitly in Fig. 12.6. Also shown are the full Stockmayer potential for three different orientations of the dipole moments. The curve listed as no dipole is for orientation angles ft = Gj = rjr — n/2. In this case the x term in Eq. 12.9 is zero. The potential has a minimum at r, - = 21/6a,y, with an attractive well depth of e,y. The curve listed as attractive dipole has orientation angles ft = 6j = 0. Thus x = 2, and this orientation has the maximum (attractive) contribution from the dipole-dipole term. The well depth in this case is almost a factor of 6 deeper due to the dipole interaction. The repulsive... 

Monchick and Mason [289] have given tables of the collision integrals and transport properties for the Stockmayer potential. These table were calculated by integrating the potential over all orientations of the dipoles. The tables are actually presented as a function... 

Convenient empirical fits of 1and Q-2,2 as a function of the reduced temperature T for the Lennard-Jones interaction potential were given in Eqs. 12.6 and 12.7. These expressions can be generalized for the Stockmayer potential (S . 0) through an additional... 

A useful compendium of formulas for estimating the interaction parameters e and cr in the Lennard-Jones or Stockmayer potentials has been presented by Svehla [389]. His work... 

The Stockmayer potential, Section 12.2.2, accounts for the dipole-dipole interaction if both molecules i and j are polar. The parameter needed was given earlier as Eq. 12.18,... 

The Stockmayer potential, Eq. 12.9, describes the molecular interactions between two polar molecules. For HC1, the Stockmayer parameters are a = 3.305 A, t/kg = 360 K, and JZ = 1.03 Debye, and for HI, the Stockmayer parameters are a = 4.123 A, t/kB = 324 K, and JZ = 0.38 Debye. Find the well depth for the Stockmayer interaction between the molecules (1) when they are aligned in their most attractive orientation and (2) when they are aligned in the most repulsive orientation. 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent they range from 7 x 10"31 to 36 x 10"28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are 19 x 10 28 and 21 x 10 28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO. 

Note that for the weak noncentral Stockmayer potential Aj(t) changes very little and is positive during the time it is observed, while for the stronger noncentral potentials there are regions in which Aj(t) is negative. 

Fig. 47. Intermediate scattering functions for the C.M. motion of a CO molecule from the modified Stockmayer potential and from Eqs. (376) and (396).

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

Since we are basically interested in polar systems here we make the same estimate with a simple binary potential function which accounts for polar interactions. A Stockmayer potential, which describes the interaction between two dipoles, in addition to a Lennard-Jones potential is most suitable for this purpose... 

For technical reasons explained in Section 5.2.2, the GCEMC simulations have been performed using a slightly modified Stockmayer potential defined... 

For a potential in which a clearly defined measure of particle diameter d is available [i.e., a hard-core particle in which (12)=oo for rLennard-Jones a is the measure of core size, etc.], it is also often convenient (although not necessary) to satisfy (2.41) by requiring... 

Prausnitz and Myers have used the Kihara potential for the calculation of 5j2 s for mixtures of simple molecules, and Blanks and Prausnitz employed the Stockmayer potential in studies of mixtures involving polar molecules. A variety of specific non-central interactions were included in the potentials that Lichtenthaler and Schafer employed in their analysis of their measurements of virial coefficients of Ar with polar and non-polar molecules. 

More recently, Khoury and Robinson calculated virial coefficients for ethane + hydrogen sulphide mixtures from an effective Lennard-Jones 6—12 potential that includes a contribution from the dipole-induced-dipole interaction. The parameters were evaluated by fitting their measurements on CgHg and HgS to a 6—12 potential and to a Stockmayer potential respectively. Bradley and King have used several potential functions to analyze interactions of phen-anthrene with a number of small polyatomic molecules. 

Rowlinson (1951) used a modified Stockmayer potential to calculate the second virial coefficient for water [see Ben-Naim (1974), page 247, Table 6.1.]... 

The interactions of diatomic molecules have been treated in two fashions. In one case, orientation-dependent dipolar and quadrupolar interactions are superimposed on a spherically symmetric potential. Computer simulations of N2 and CO have been carried out using the Stockmayer potential, which is a sum of a center-to-center Lennard-Jones potential and a number of multipole interaction terms. Alternately, the N2 molecule can be envisioned as two bound force centers, each of which interacts isotropically with force centers on other molecules. The total potential of two nitrogen molecules is thus the sum of four terms. 

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