Second Virial Coefficients from Potential Functions

SECOND VIRIAL COEFFICIENTS FROM POTENTIAL FUNCTIONS... 

Second Virial Coefficients from Potential Functions... 

It follows that atoms or molecules interacting with the same pair potential s( )(rya), but with different s and cj, have the same thermodynamic properties, derived from A INkT, at the same scaled temperature T and scaled density p. They obey the same scaled equation of state, with identical coexistence curves in scaled variables below the critical point, and have the same scaled vapour pressures and second virial coefficients as a function of the scaled temperature. The critical compressibility factor P JRT is the same for all substances obeying this law and provides a test of the hypothesis. Table A2.3.3 lists the critical parameters and the compressibility factors of rare gases and other simple substances. 

Values for the virial coefficients are derived from experimental measurements which can be conveniently classified as follows low pressnre p-V-T measnrements high pressnre p-V-T measnrements speed of sonnd measurements vaponr pressnre and enthalpy of vaporization measnrements refractive index/dielectric constant measurements and Jonle-Thomson experiments. These will be discussed in Chapter 1.2, and methods of data evalnation described in Chapter 1.5. Much attention has been paid to the correlation of virial coefficient data and the more satisfactory methods are considered in Chapter 1.3, together with a brief discussion of the theoretical calculation of the second virial coefficient from pair potential energy functions which have been derived a priori or by consideration of other dilute gas properties. So far, this calculation is only applicable to molecules with a spherically symmetric intermolecular potential energy function, for which... 

Intermolecular potential functions have been fitted to various experimental data, such as second virial coefficients, viscosities, and sublimation energy. The use of data from dense systems involves the additional assumption of the additivity of pair interactions. The viscosity seems to be more sensitive to the shape of the potential than the second virial coefficient hence data from that source are particularly valuable. These questions are discussed in full by Hirschfelder, Curtiss, and Bird17 whose recommended potentials based primarily on viscosity data are given in the tables of this section. 

The possibility of occurrence of instability of colloidal dispersions in the presence of free polymer was first predicted by Asakura and Oosawa (5), who have shown that the exclusion of the free polymer molecules from the interparticle space generates an attractive force between particles, DeHek and Vrij (1) have developed a model in which the particles and the polymer molecules are treated as hard spheres and rederived in a simple and illuminating way the interaction potential proposed by Asakura and Oosawa. Using this potential, they calculated the second virial coefficient for the particles as a function of the free polymer concentration and have shown that... 

The potential for the water dimer that was just described is not yet very accurate. Since, however, both the functional form and calibration were derived from ab initio calculations, there is room for well controlled improvements that would follow future more accurate ab initio data. This model provided the geometry of the global minimum in very good agreement with experiment, and a fair account of the second virial coefficient. It should be mentioned that the well-depth of this potential is smaller than the experimental value of 5.4 0.7 kcal/mol. However, the best ab initio calculations on the water dimer also consistently predict smaller values. 

Although Eq. (16.12) is based on an intemiolecular potential function that is in detail unrealistic, it nevertheless often provides an excellent fit of second-virial-coefficieiitdata. An example is provided by argon, for which reliable data for B are available over a wide temperature range, from about 85 to 1000 K. The correlation of these data by Eq. (16.12) as shown in Fig. 16.3 results from the parametervaluese/k = 95.2 K,/ = 1.69,andJ = 3.07 x 10 cm. This empirical success depends at least in part on the availability of three adjustable parameters, and is no more tlian a limited validation of the square-well potential. Use of tins potential does illustrate by a very simple calculation how the second virial coefficient (and hence tire vohnne of a gas) may be related to molecular parameters. 

One can use any specific model potential function to estimate the right-hand side of (2.7.59). Fortunately, since the combination [exp( — 8hb) — IJAX/Stt has already been estimated from the second virial coefficient, we need not know the values of each factor. Using the value of 298.25 cm mol given in (2.7.21), we estimate for T = 298 K, the solvation Helmholtz energy per arm as... 

The Nath, Escobedo, and de Pablo (NERD) force field [100,131-133] was developed to provide accurate predictions of thermodynamic properties. It is currently available for linear [100] and branched alkanes [131,133] as well as for alkenes [132]. It has a similar functional form as the TraPPE-UA force field, but bond stretching is included. This interaction and angle bending are represented by harmonic potentials [(20) and (22)]. The torsional potential is of the form of (25), neglecting cross terms. The U 12-6 potential (6) is used to describe the intermolecular and intramolecular interactions between sites that are separated by more than three bonds. The LJ parameters were obtained from fits to experimental values of liquid density and second virial coefficient. Saturated liquid densities from the NERD force field are in good... 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

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