Of pure real gases

For species present as gases ia the actual reactive system, the standard state is the pure ideal gas at pressure F°. For Hquids and soHds, it is usually the state of pure real Hquid or soHd at F°. The standard-state pressure F° is fixed at 100 kPa. Note that the standard states may represent different physical states for different species any or all of the species may be gases, Hquids, or soHds. 

In this chapter we will apply the concepts developed in Chapter 11 to gaseous systems, first to mixtures of ideal gases, then to pure real gases, and finally to mixtures of real gases. 

The van der Waals equation applies strictly to pure real gases, not to mixtures. For a mixture like the one resulting from the reaction of part (a), it may still be possible to define effective a and b parameters to relate total pressure, volume, temperature, and total number of moles. Suppose the gas mixture has a = 4.00 atm moU and b = 0.0330 L moU. Recalculate the pressure of the gas... 

Purely phenomenological as well as physically based equations of state are used to represent real gases. The deviation from perfect gas behaviour is often small, and the perfect gas law is a natural choice for the first term in a serial expression of the properties of real gases. The most common representation is the virial equation of state ... 

Supermolecular spectra could perhaps be studied with state-selection using adequate molecular beam techniques. That would not be easy, however, because of the smallness of the dipole moments induced by in-termolecular interactions. For the purpose of this book, we will mostly deal with bulk spectra, or interaction-induced absorption of pure and mixed gases. A great variety of excellent measurements of such spectra exists for a broad range of temperatures, while state-selected supermolecular absorption beam data are virtually non-existent at this time. Furthermore, important applications in astrophysics, etc., are concerned precisely with the optical bulk properties of real gases and mixtures. 

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. 

Two further crude approximations have been used for the virial equation of state. The first is that the virial coefficients combine linearly. This combination of constants results in an equation of state that is additive in the properties of the pure components. In such a mixture Dalton s and Amagat s laws still hold, and the mixture may be called an ideal mixture of real gases. The assumption is probably the crudest that can be used and is... 

Real gases are usually non-ideal. Thermodynamics describes both ideal and non-ideal gases with the same type of formulas, except that for non-ideal gas mixtures the fugacity f is substituted in place of the pressure pi and that the activity at is substituted in place of the molar fraction xi or concentration c, of constituent substance i. We have already seen that in the ideal gas of a pure substance the chemical potential is expressed by Eq. 7.5. By analogy, we write Eq. 7.9 for the non-ideal gas of a pure substance i ... 

This ratio of / to p for a non-ideal gas of a pure substance may be calculated from the equation of state for real gases such as the virial equation and the van der Wools equation. 

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. 

In summary, one can see that separation selectivity for gas and vapor molecules depends on the category of pores (mesopores, supermicropores, and ultramicropores) and on the related transport mechanisms. Either size effect or preferential adsorption effect (irrespective of molecular dimension) is involved in selective separation of multicomponent mixtures. The membrane separation selectivity for two gases is usually expressed either as the ratio between the two pure gas permeation fluxes (ideal selectivity) or between each gas permeation flux measured from the mixture of the two gases (real selectivity). More detailed information on gas and vapor transport in porous ceramic membranes can be found in Ref. [24]. 

The definition of tlie fugacity of a species in solution is parallel to the definition of tire pure-speciesfugacity. For species i m a mixture of real gases or in a solution of liquids, the equation analogous to Eq. (11.28), tire ideal-gas expression, is ... 

Real gases do not obey the ideal gas law because attractive and repulsive forces between molecules introduce potential energy that competes with the purely kinetic energy of translation from which the ideal gas law arises. The van der Waals equation of state is the simplest two-parameter equation of state for real gases. A simple physical model underlies the form of the equation and its two parameters. 

Experimental evidence indicates that the heat capacity of a substance is not constant with temperature, although at times we may assume that it is constant in order to get approximate results. For the ideal monoatomic gas, of course, the heat capacity at constant pressure is constant even though the temperature varies (see Table 4.1). For typical real gases, see Fig. 4.7 the heat capacities shown are for pure components ... 

As discussed above, the ideal separation factor a in the case of pure Knudsen diffusion is given by Eq. (9.37) and is equal to the permselectivity provided that surface diffusion is not present (high temperature). As can be seen from (9.37) the highest ideal separation factors are obtained for mixtures of light and heavy gases. Back-diffusion effects are taken into account by Eq. (9.38) to give the real separation factor. 

For real pure gases, we cannot apply the (V = nRT/P) expression and we cannot integrate Equation (153). However, we need to preserve the form of expressions that have been derived for the ideal thermodynamic system. In order to adapt Equation (156) for real gases, the replacement of the true measurable pressure, P, with another effective pressure term called jugacity, f, was carried out in classical thermodynamics ... 

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