Pressure virial coefficients

The classical expressions for ai, pf , and up are similar to the pure-gas expressions with the slight added complication of having two sets of isolated-molecule parameters o oo, fio,. Fo, The determination of the mixed-pair parameters involved in uq is much more difficult. The determination of these parameters from mixed viscosities and pressure virial coefficients is hampered by a shortage of experimental data, and the usual procedure is to employ a set of empirical combining rules which relate the mixed parameters to those of the pure gases. A number of such rules have been proposed for the 6-12 potential, the most widely used for dielectric calculations being... 

Dielectric and pressure virial coefficients of NzO have been measured at 6.5, 30.1, and 75.1 °C. The dipole moment, polarizability, and molecular quadrupole moment were determined to be 0.18 D, 3.03 x 1CT24 cm3, and 3.4 xlO 26 e.s.u. cm2, respectively.91 A lower limit of —0.15 0.1 eV has been calculated for the molecular electron affinity of N20, using molecular beam studies.92 The enthalpy-pressure behaviour for N20 along eleven isotherms in the vapour phase has been determined from measurements of the Joule-Thomson effect.91... 

This section discusses how spectroscopy, molecular beam scattering, pressure virial coefficients, measurements on transport phenomena and even condensed phase data can help determine a potential energy surface. 

Again, this expansion is performed along an isotherm T and each derivative is evaluated in the ideal-gas limit. These derivatives are defined to be the pressure-virial coefficients... 

Heltmann R, Bich E, Vogel E (2008) Ab initio intermolecular potential energy surface and second pressure virial coefficients of methane. J Chem Phys 128 214303... 

The parameters 5, C,. .. are called the second, third,. .. virial coefficients, and the parameters Bp, Cp,. .. are a set of pressure virial coefficients. Their values depend on the substance and are functions of temperature. (The first virial coefficient in both power series is 1, because pV must approach RT as 1/Fm or p approach zero at constant T) Coefficients beyond the third virial coefficient are small and rarely evaluated. 

Because of orientation-dependent terms in both the moments and the Boltzmann factor values of B are much siore sensitive to molecular anisotropies than the pressure virial coefficient or the gas shear viscosity as a function of temperature. For nonpolar molecules quadrupole moment effects are large in the case of CO2 for example demonstrating the importance of quadrupole moments Q s 4.2 X 10 esitcii)> inferred from B while octopole and even hexadecapole effects can be recognized for more symmetrical molecules e.g. CH and SFg. For polar molecules permanent dipole interactions also come into play and anisotropy of repulsive forces (shape) is also important. The result is a very wide range in magnitudes and sign of B even for relatively simple molecules and comparison of calculated values with experiment is a sensitive test of multipole moments and anisotropies of used in the calculation. All these matters are discussed in detail by Sutter (21). 

This procedure based on the Rainwater-Friend theory is valid only for reduced temperatures T > 0.7 (T = 175 K for ethane). This lower limit will not be exceeded by the viscosity representation of ethane in the vapor phase, since the range of validity of its zero-density contribution has a lower limit of T = 2(X) K. Nevertheless, experimental data in the liquid phase are available at much lower temperatures extending to T = 100 K. In order to use a single overall viscosity correlation it must be ensured that the initial-density contribution extrapolates satisfactorily to low temperatures. For this purpose, for temperatures below T = 0.7, the second viscosity virial coefficient has been estimated by use of the modified Enskog theory (see Chapter 5), which relates B,f to the second and third pressure virial coefficients. Although this method enables Brj to be evaluated, it is cumbersome for practical applications. Therefore, the calculated B, values using both methods have been fitted to the functional form... 

The coefficients A2, A3, etc., are called pressure virial coefficients and also must depend on the temperature. It can be shown that A2 and B2 are equal. 

The pressure virial equation of state was shown in Eq. (1.3-4), and it was shown in an example that A2, the second pressure virial coefficient, is equal to B2, the second virial coefficient. Find an expression for (dS/dP)j- for a gas obeying the pressure virial equation of state truncated at the A2 term. 

Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute... 

To use Equation (10b), we require virial coefficients which depend on temperature. As discussed in Appendix A, these coefficients are calculated using the correlation of Hayden and O Connell (1975). The required input parameters are, for each component critical temperature T, critical pressure P, ... 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

The leading correction to the classical ideal gas pressure temi due to quantum statistics is proportional to 1 and to n. The correction at constant density is larger in magnitude at lower temperatures and lighter mass. The coefficient of can be viewed as an effective second virial coefficient The effect of quantum... 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Extensive tables and equations are given in ref. 1 for viscosity, surface tension, thermal conductivity, molar density, vapor pressure, and second virial coefficient as functions of temperature. 

Tables 2,3, and 4 outline many of the physical and thermodynamic properties ofpara- and normal hydrogen in the sohd, hquid, and gaseous states, respectively. Extensive tabulations of all the thermodynamic and transport properties hsted in these tables from the triple point to 3000 K and at 0.01—100 MPa (1—14,500 psi) are available (5,39). Additional properties, including accommodation coefficients, thermal diffusivity, virial coefficients, index of refraction, Joule-Thorns on coefficients, Prandti numbers, vapor pressures, infrared absorption, and heat transfer and thermal transpiration parameters are also available (5,40). Thermodynamic properties for hydrogen at 300—20,000 K and 10 Pa to 10.4 MPa (lO " -103 atm) (41) and transport properties at 1,000—30,000 K and 0.1—3.0 MPa (1—30 atm) (42) have been compiled. Enthalpy—entropy tabulations for hydrogen over the range 3—100,000 K and 0.001—101.3 MPa (0.01—1000 atm) have been made (43). Many physical properties for the other isotopes of hydrogen (deuterium and tritium) have also been compiled (44).

Parameters B, C, D,..., are density series virial coefficients, and B (V, are pressure series virial coefficients. The second virial coefficients are defined... 

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 ... 

Because virial coefficients beyond the third are rarely available, equations 23 and 30 are usually tmncated after two or three terms. For low pressures, the two-term tmncation of equation 30 is preferred ... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Second virial coefficients, B, are a fnncBon of temperature and are available for about 1500 compounds in the DIPPR compilaOond The second virial coefficient can be regressed from experimental PX T data or can be reasonably and accurately predicted. Tsonoponlos proposed a predicOon method for nonpolar compounds that requires the criOcal temperature, critical pressure, and acentric factor Equations (2-68) through (2-70) describe the method. 

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... 

---