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Thermodynamics of Polymer Mixtures
Published in Timothy P. Lodge, Paul C. Hiemenz, Polymer Chemistry, 2020
Timothy P. Lodge, Paul C. Hiemenz
where B is called the second virial coefficient: it has units of cm3 mol/g2. The quantity B3, the third virial coefficient, reflects ternary or three-body interactions, and is important when c is large enough and when B is small enough; we will not consider it further here. Equation 7.4.7a is the central result of this section. The quantity on the left-hand side is measurable; of the quantities on the right, c is determined by solution preparation, and M and B are determined by examining the c dependence of Π. In Figure 7.6a Π/RT is plotted against c for three different values of B: B > 0, B = 0, and B < 0. In Figure 7.6b we choose a second format, obtained by dividing Equation 7.4.7a through by c: ΠcRT=1M+Bc+⋯
Vapor Liquid Equilibrium Properties
Published in Satish G. Kandlikar, Masahiro Shoji, Vijay K. Dhir, Handbook of Phase Change: Boiling and Condensation, 2019
Here, B, C, and D are functions of temperature and are called the second, third, and fourth virial coefficients, which are related to the extent of interactions on the molecular level. In the limiting case of very low densities, i.e., very large molar volumes where the gas molecules are assumed to behave independently, the virial equation of state reduces to the ideal gas equation of state. Virial coefficients are calculable in principle if a model formulating the interaction forces between gas molecules is known. However, such calculations are successful at present only for the first two or three coefficients for gases with relatively simple molecules. As an alternative means, the first few virial coefficients are empirically determined by fitting P-v-T data but with limited success. Such a truncated virial equation of state based on a single set of virial coefficients is unable to describe the P-v-T relationship in the liquid and vapor phases simultaneously. Therefore, the P-v-T behavior of pure substances at higher densities and in liquid phase is usually expressed by empirical equations of state. Such empirical equations range from relatively simple expressions (including a few arbitrary constants) to complex equations suitable only for computerized calculations and containing as many as twenty or more constants.
Engineering Thermodynamics
Published in Raj P. Chhabra, CRC Handbook of Thermal Engineering Second Edition, 2017
Michael J. Moran, George Tsatsaronis
These expressions are known as virial expansions, and the coefficients B^,C^,D^,… and B, C, D, … are called virial coefficients. In principle, the virial coefficients can be calculated using expressions from statistical mechanics derived from consideration of the force fields around the molecules. Thus far only the first few coefficients have been calculated and only for gases consisting of relatively simple molecules. The coefficients also can be found, in principle, by fitting p–v–T data in particular realms of interest. Only the first few coefficients can be found accurately this way, however, and the result is a truncated equation valid only at certain states.
On the virial expansion of model adsorptive systems
Published in Molecular Physics, 2022
William P. Krekelberg, Vincent K. Shen
The virial expansion provides a systematic means to incorporate increasingly complex (two-body, three-body, etc) interactions, and describe thermodynamic properties beyond the infinite-dilution limit. As such, the virial expansion plays an important role in the metrology of dilute bulk gases [6], The virial expansion provides a direct link between the intermolecular interactions in a fluid and its thermodynamic properties [7,8], and therefore virial coefficients can be used as a metric for characterising fluids. In the case of a bulk fluid, the second virial coefficient is an experimentally measurable quantity that is directly related to the pairwise intermolecular forces between its constituent molecules. Thus, the second virial coefficient can be used to test the accuracy of intermolecular potentials. The reduced second virial coefficient can also be used as a parameter in corresponding states models for fluids with short-ranged attractions [9]. All of these characteristics make the virial expansion an attractive framework to describe adsorptive thermodynamics.
Artificial neural network for the second virial coefficient of organic and inorganic compounds: An ANN for B of organic and inorganic compounds
Published in Chemical Engineering Communications, 2018
Giovanni Di Nicola, Gianluca Coccia, Mariano Pierantozzi, Sebastiano Tomassetti, Roberta Cocci Grifoni
The ideal gas equation of state does not represent the pressure–volume–temperature (PVT) behavior of real gases, therefore the knowledge of a more complex PVT relation is needed. Many equations of state have been proposed to represent the PVT relationship of real gases, but the only equation of state known with a thoroughly sound theoretical foundation is the virial equation (Mason and Spurling, 1969). The virial equation of state (in Leiden form) shows the deviations from the perfect-gas equation as an infinite power series in the molar volume, V:where B is the second virial coefficient, C is the third virial coefficient, D is the forth virial coefficient, etc. Virial coefficients depend on temperature and on the particular gas under consideration, and are independent of density and pressure.