Virial pressure

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

For atomic gases the intermolecular potential most used in calculations of B, has been the Lennard-Jones 6-12, with the parameters e and r determined from gas viscosities and pressure virial coefScients. For non-dipolar gases possessing higher moments, most authors have used the 6-12 potential together with the appropriate terms from equations (27) and (28), while for dipolar gases some form of equations (26)—(28) is used with a shape-dependent term added to uq. 

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... 

There is another commonly used series equation of state, sometimes called the pressure virial equation of state ... 

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... 

Hill, T. L. 1959. Theory of solutions. II. Osmotic pressure virial expansion and hght scattering in two component solutions. Journal of Chemical Physics. 30, 93. 

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). 

The evidence just cited illustrates potential usefulness of dielectric virial data when coupled with pressure virial and viscosity data. Similar information for CH3I and CH2C12 among others should provide much useful information. The case of CH3I seems particularly interesting because of the anomalously low static permittivities of it and other iodo compounds in the liquid state. Unfortunately measurements of the necessary precision (a few parts per million in and 1 part in 10 for relative density with the former easier than the latter) are not a popular activity. 

Baranowska et reported a high-level investigation of the potential energy surface (PES) of the Ne-CO interaction. The calculations were performed at the CCSD(T) level of theory with the d-aug-cc-pVTZ basis set supplemented with 3s3p2dlflg midbond functions. The minimum of the PES corresponds to a nearly T-shaped structure with the Ne atom located at a distance of 3.383 A from CO and an angle of 79.4°. The interaction-induced electric dipole moment and polarizability was determined at the CCSD/d-aug-cc-pVTZ-33211 level of theory. In addition, the authors determined the dielectric and refractivity virial coeflicients and the pressure virial coeflicients of the CO-Ne complex. 

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... 

An equation of state that is a power series in P is called the pressure virial equation... 

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. 

Derive the expression for the entropy change for an isothermal pressure change of a gas described by this truncated pressure virial equation of state. 

Calculate AS for the expansion of 1.000 mol of argon from 10.00 atm to 1.000 atm at 298.15 K, assuming the truncated pressure virial equation of state. Compare your result with that obtained... 

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