Virial temperature dependence

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations ... 

CALCULATE THE TEMPERATURE DEPENDENT SECOND VIRIAL COEFFICIENTS. 

Higher virial coefficients are defined analogously. AH virial coefficients depend on temperature and composition only. The pressure series and density series coefficients are related to one another ... 

This equation of state applies to all substances under all conditions of p, and T. All of the virial coefficients B, C,. .. are zero for a perfect gas. For other materials, the virial coefficients are finite and they give information about molecular interactions. The virial coefficients are temperature-dependent. Theoretical expressions for the virial coefficients can be found from the methods of statistical thermodynamic s. 

This stipulation of the interaction parameter to be equal to 0.5 at the theta temperature is found to hold with values of Xh and Xs equal to 0.5 - x < 2.7 x lO-s, and this value tends to decrease with increasing temperature. The values of = 308.6 K were found from the temperature dependence of the interaction parameter for gelatin B. Naturally, determination of the correct theta temperature of a chosen polymer/solvent system has a great physic-chemical importance for polymer solutions thermodynamically. It is quite well known that the second viiial coefficient can also be evaluated from osmometry and light scattering measurements which consequently exhibits temperature dependence, finally yielding the theta temperature for the system under study. However, the evaluation of second virial... 

The higher the pressure, the larger the number of terms we have to consider in equations 2.17 and 2.18. Let us assume that the pressure is such that only the second term needs to be considered. Then because the virial coefficients depend only on the temperature, we have ... 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

A2 from equation (5.16) or the cross second virial coefficient from equation (5.17). In turn, this knowledge of the second virial coefficients and their temperature dependence allows calculation of the values of the chemical potentials of all components of the biopolymer solution or colloidal system, as well as enthalpic and entropic contributions to those chemical potentials. On the basis of this information, a full description and prediction of the thermodynamic behaviour can be realised (see chapter 3 and the first paragraph of this chapter for the details). 

Figure 6.5 Temperature dependence of the characteristics of sodium k-carrageenan particles dissolved in an aqueous salt solution (0.1 M NaCl). The cooling rate is 1.5 °C min-1, (a) ( ) Weight-average molar weight, Mw, and (A) second virial coefficient, A2. (b) ( ) Specific optical rotation at 436 nm, and ( ) penetration parameter, y, defined as tlie ratio of the radius of the equivalent hard sphere to the radius of gyration of the dissolved particles (see equation (5.33) in chapter 5). See the text for explanations of different regions I, II, III and IV.

Equations of this type are known as virial equations, and the constants they contain are called the virial coefficients. It is the second virial coefficient B that describes the earliest deviations from ideality. It should be noted that B would have different but related values in Equations (26) and (27), even though the same symbol is used in both cases. One must be especially attentive to the form of the equation involved, particularly with respect to units, when using literature values of quantities such as B. The virial coefficients are temperature dependent and vary from gas to gas. Clearly, Equations (26) and (27) reduce to the ideal gas law as p - 0 or as n/V - 0. Finally, it might be recalled that the second virial coefficient in Equation (27) is related to the van der Waals constants a and b as follows ... 

This is a virial expansion form of the osmotic pressure analogous to the van der Waals fluid. Dusek and Patterson examined this equation and predicted the presence of two phases, i.e. collapsed and swollen phases. % is temperature dependent and is given by,... 

For the temperature dependence of the virial coefficients, we assumed the following formula by analogy with real gas systems ... 

The modification of the Van der Waals equation by Redlich and Kwong [29], who introduced a different temperature dependence and a slightly different volume dependency in the attractive term, is very important since it opened the way to a better description of the temperature dependent properties like virial coefficients. 

Few determinations of gas-solid virial coefficients have been made. Halsey and coworkers (32,33) used the temperature dependence of the first gas-solid virial coefficient to calculate the potential energy curve for a single molecule in the presence of a solid. Hanlan and Freeman (34) showed this coefficient may be... 

This measure, however, pertains to the normal boiling point rather than to ambient conditions. The deficit of the entropy of the liquid solvent relative to the solvent vapour and to a similar non-structured solvent at any temperature, such as 25 °C, has also been derived (Marcus 1996). An alkane with the same skeleton as the solvent, i.e., with atoms such as halogen, O, N, etc. being exchanged for CH3, CH2, and CH, etc., respectively, can be taken as the non-structured solvent. Since the vapour may also be associated, the temperature dependence of the second virial coefficient, B, of the vapour of both the solvent and the corresponding alkane, must also be taken into account. The entropy of vaporization at the temperature T, wherep P°, is given by ... 

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

The temperature-dependent second and third virial coefficient describe the increasing two- and three-particle collisions between the gas molecules and their accompanying increase in gas density. The virial coefficients are calculated using a suitable intermolecular por-tential model (usually a 12-6 Lennard-Jones Potential) from rudimentary statistical thermodynamics. 

Zcoii/y is the sum of two Gaussians, the first centered at y = 0 to ensure the correct behavior in the limit of zero density, and the second centered at the rectilinear diameter yT to provide the correct value of residual Helmholtz energy at high density. The temperature dependence of yu is given universal [2], The parameters b and c, which determine the second and third virial coefficients, are given universally for non-polar substances [3]. The pre-exponential factor w has a dominating influence on the vapor pressure. Its temperature dependence requires two substance-specific correction parameters wi and w2. 

Considering the applied high temperature gradients of up to 10000 K/cm,the proper calculation of v(x) very much depends on the proper description of the temperature dependence of the viscosity r (x,T). This dependence can be described by a virial expansion of the form ... 

A further issue for review is the treatment of attractive interactions. The treatment here was limited to consideration of the second virial coefficient as in Eq. (4.46), and this implies the composition and temperature dependences exhibited in Eq. (4.49). Those composition and temperature dependences are certainly the leading factors, but a more general evaluahon of first-order perturbahon theory could result in subtle corrections to those dependences. Additionally, some implicit temperature and pressure dependence is implied by the variations of the pure liquid properties in those factors. Finally, the limitation of the hrst-order perturbation theory must also be borne in mind there are experimental cases where hrst-order perturbation theory appears to be unsatisfactory (Lefebvre et al, 1999). 

In the expressions (184) and (184b) the second, temperature-dependent term defines the Born effect due to superposition of the two non-linear processes of second-order distortion and reorientation of permanent dipole moments in the electric field. Buckingham et al. determined nonlinear polarizabflities If and c for numerous molecules by Kerr effect measurements in gases as a function of temperature and pressure. It is here convenient to use the virial expansion of the molar Kerr constant, when the first and second virial coefficients Ak and Bk result immediately from equations (177), (178), and (184). Meeten et al. determined nonlinear molecular polarizabilities by measuring K in liquids as a function of temperature. 

Note Virial coefficients depend on temperature and are found by fitti ng experimental data to the virial equation. This equation is more general than the van der Waals equation but is more difficult to use to make predictions. 

We calculated the temperature dependence of the second virial coefficient for some models and compared them to the experimental data". (See Figure 1) Each model underestimated the correct value. The size of the discrepancies varied by the size of their dipole moment, i.e., TIP4P (2.18D), TIP4P-m (2.29D), TIP4P-EW (2.32D), average... 

Figure 1 Temperature dependence of the second virial coefficient of water as predicted by six models in comparison with experimental data...

Indeed, data on the temperature dependence of second virial coefficient... 

Measurement of the temperature dependence of second virial coefficient A2 for polymers with known molar mass M and Kuhn length b allows estimation of the number of thermal blobs per chain A /gj using Eq. (3.109). 

Use the following light scattering data for the temperature dependence of the second virial coefficient of a linear polytmethyl methacrylate) with... 

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