Gases virial coefficients

Non-ideal Gases Virial Coefficient Isotope Effects... 

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... 

Corner J 1948 The second virial coefficient of a gas of non-spherical molecules Proc. R. Soc. A 192 275... 

Figure A2.1.7. The second virial coefficient 5 as a fiinction of temperature T/T. (Calculated for a gas satisfying the Leimard-Jones potential [8].)...

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

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... 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative ... 

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. 

The next level of approximation is valid to higher pressures. It assumes that the gas mixture obeys the virial equation of state, with the third, fourth and higher, virial coefficients equal to zero. Thus... 

Although the virial equation can be used to make accurate predictions about the properties of a real gas, provided that the virial coefficients are known for the temperature of interest, it is not a source of much insight without a lot of advanced analysis. An equation that is less accurate bur easier to interpret was proposed by the Dutch scientist Johannes van der Waals. The van der Waals equation is... 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

An alternative approach to quantifying the interactions and deviations from the ideal-gas equation is to write Equation (1.13) in terms of virial coefficients ... 

Equation (2.3) becomes the ideal-gas equation if both B and C are tiny. In fact, these successive terms are often regarded as effectively fine-tuning the values of p or Vm. The C coefficient is often so small that we can ignore it and D is so minuscule that it is extremely unlikely that we will ever include a fourth virial coefficient in any calculation. Unfortunately, we must exercise care, because B constants are themselves a function of temperature. 

Several techniques are available for measuring values of interaction second virial coefficients. The primary methods are reduction of mixture virial coefficients determined from PpT data reduction of vapor-liquid equilibrium data the differential pressure technique of Knobler et al.(1959) the Bumett-isochoric method of Hall and Eubank (1973) and reduction of gas chromatography data as originally proposed by Desty et al.(1962). The latter procedure is by far the most rapid, although it is probably the least accurate. 

The analysis providing interaction second virial coefficients from chromatography rests upon three principal assumptions 1) vapor-liquid equilibrium exists in the column 2) the solute (component 1) is soluble in both the carrier gas (component 2) and the stationary liquid phase (component 3) 3) the carrier gas and stationary liquid are insoluble. Under assumption 1, we can write... 

Laub, R.J. Pecsok, R.L. "Determination of Second-Interaction Virial Coefficients by Gas-Liquid Chromatography," J. 

Young, C.L. Cruickshank, A.J.B. Gainey, B.W. Hicks, C.P. Letcher, T.M. Moody, R.W. Gas-Liquid Chromatographic Determination of Cross-Term Second Virial Coefficients using Glycerol, Trans. Far. Soc., 65, 1014(1969). 

In Fig. 5.1 we see that the intermolecular interactions accounting for VCIE (upper curve) and VPIE (lower curve) differ not in kind, only in degree. The well depth for gas-gas interaction is available from analysis of the virial coefficient of the parent isotopomer, that for the condensed phase can be obtained by combining the energy of vaporization and the zero point energies of the condensed and ideal vapor phases. 

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