Gas-solid virial coefficients

The determination of gas-solid virial coefficients can be a useful technique to explain the interaction between an adsorbed gas and a solid surface. The terms are defined so that the number of adsorbate molecules interacting can be readily ascertained. For example, the second order gas-solid interaction involves one adsorbate molecule and the solid surface the third order gas-solid interaction involves two adsorbate molecules and the surface, and so on. The number of adsorbed molecules under consideration is expanded in a power series with respect to the density of the adsorbed phase. 

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... 

Rudzinski and coworkers (35,36) first used the second and third gas-solid virial coefficients obtained from GC data to estimate surface areas. The surface area of silica gel determined using virial expansion data was greater than that obtained using the BET method (Section 11.1). The discrepancy was explained by noting the BET method does not take the lateral interactions into account. These interactions have an effect of decreasing the effective area of the adsorbent, thus making the calculated BET area less than it should be (Table 11.7). 

It is clear that the calculation of gas—solid virial coefficients is very difficult, so that only the first few of them could be evaluated. This means that the model will be useful only at low values of the adsorbed phase density. But on the other hand, the most important effects of heterogeneity can be seen for the low-pressure part of the adsorption isotherm. 

Figure 10.2 Normalized gas-solid virial coefficients for a Lennard-Jones potential, as a function of the reduced temperature T/T, for different values of the correlation length Tq and for a given value of the standard deviation of the adsorptive potential kg T,. Adapted from Ref. 25.

Low-coverage volumetric isotherm data [41, 42] were used to extract the isosteric heat of adsorption extrapolated to zero coverage, the gas-solid virial coefficient, and the two-dimensional second virial coefficient. It is found that fitting the two-dimensional virial coefficient obtained from the measurements in the vicinity of the surface in terms of Lennard-Jones inter-molecular potentials reduces the well depth obtained from the bulk by about 20%. In addition, the effective quadmpole moment of CO needs to be significantly reduced [42] by as much as about 50%. These fitted parameters are believed to account for various substrate mediation effects in some effective way (see Refs. 41 and 42 for more details concerning these param-eterizations). It is also concluded [41] that the asymmetric empirical parameterization of Ref. 238 should be replaced by models in which the similarity between the isoelectronic CO and N2 molecules is exploited for the non-electrostatic contributions as in Ref. 17. 

Gas-solid virial coefficients, characterizing the interaction of the adsorbate molecule with the surface of the solid, can be used to estimate surface area and for quantitative studies of the energetic heterogeneity of a surface. 

Methods in which the specific surface area is calculated from the second (B2s) and third gas-solid virial coefficients are valid if the values of Bas [=V g(r>] and are known. In such cases, the specific surface area can be calculated from the following equation ... 

Knowledge of the adsorbate-adsorbent interaction is fundamental in any statistical mechanics theory of adsorption. As indicated earlier, the comparison between experimental Henry s constants or gas-solid virial coefficients and theory [8,33] permits one to test the validity of a given model for the gas-solid potential. As a first approximation, the potential f/sf( ,) is considered to be a function only of the perpendicular distance z for monolayer mobile adsorption on homogeneous surfaces [29,33,43,219]. The analytical forms used are similar to the Lennard-Jones potential, but replacing r by z and considering different (10-4 or 9-3, for example) powers than the 12-6 case expressed in Eq. (12). In each case, the gas-surface molecular parameters, Sjf and cTsf, can be determined by comparison with experimental results. This procedure must be considered as semiempirical and thus not fidly theoretical. 

Comparison between the experimental adsorption isotherms (adsorption pressure as a function of the amotmt of matter adsorbed) and the theoretical expression for the adsorption pressure as a function of the density of the monolayer, obtained from Steele s two-dimensional approximation, Eq (23). In this case, Henry s constant must be previously calculated from experimental data or through the gas-solid virial coefficient [210,211], Eq. (4). 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The Clapeyron equation can be simplified to some extent for the case in which a condensed phase (liquid or solid) is in equilibrium with a gas phase. At temperatures removed from the critical temperature, the molar volume of the gas phase is very much larger than the molar volume of the condensed phase. In such cases the molar volume of the condensed phase may be neglected. An equation of state is then used to express the molar volume of the gas as a function of the temperature and pressure. When the virial equation of state (accurate to the second virial coefficient) is used,... 

Very accurate potentials for rare gases, based on a large amount of experimental data in gas phase (second virial coefficient, molecular beam scattering cross sections, spectroscopic data) and in solid phase, have been proposed by Maitland and Smith [80] with the functional form proposed by Barker and Pompe [81]. This function, with its larger number of parameters, is much more flexible than the LJ or Buckingham potential and the attractive part is correctly described as in equation (36) while the repulsive one also includes an exponential function, multiplied by a fifth order polynomial. 

Fig. 21. Entropy versus log-temperature diagram for the hard-sphere model. The solid curves give the computer simulation values for the supercooled fluid, glass, and crystal. The dashed curves have the following bases (a) a calculation from the virial equation using the known first seven coefficients and higher coefficients obtained from the conjectured closure (the plot corresponds quite closely with that calculated from the Camahan-Starling equation ) and (i>) an extrapolation of higher temperature behavior such as that used by Gordon et al., which implies a maximum in the series of virial coefficients. The entropy is defined in excess of that for the ideal gas at the same temperature and pressure. Some characteristic temperatures are identified 7, fusion point 7 , upper glass transition temperature T/, Kauzmann isoentropic point according to closure virial equation.

It is important to note that most treatments using the matrix method for the lattice dynamical treatment of molecular solids adjust the parameters of the potential model to fit observed frequencies. In this case, the formal error in the treatment is minimized in its importance. It may be of importance only inasmuch as the same potential model is assumed to be useful to interpret other physical properties such as gas phase second virial coefficients. It may also be important when one attempts to determine parameters for intermolecular potentials in terms of atom-atom interactions which are general to a class of molecules rather than specific to one substance. It is also evident that the error will be very serious in every case involving low-frequency librations (as for a-Ng). The reason for the small effect of the first derivative term in many cases is as follows. Both repulsive and attractive terms of the potential usually contribute significantly to the first derivative but their contributions have opposite sign and cancel. By comparison, the contribution to the second derivative of the potential is usually much larger for the repulsive potential term than for the attractive term (Shimanouchi, 1970),... 

Suzuki and Schnepp (1971) have shown that the specific heat of solid a-N2 can be calculated in good agreement with experiment from the results of Schnepp and Ron (1969). These authors also have shown that the potential model of Kuan and others gives good results for second virial coefficients of N2 gas. They used the statistical mechanics results for the diatomic model given by Sweet and Steele (1967). 

Fig. 2.1. A modem version of the intermolecular pair potential for argon. Solid line a representation of the full potential symbols A inversion of gas viscosity inversion of second virial coefficient.

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