Real gases isotherm

This idea is a consequent transfer of the three-dimensional van der Waals equation into the interfacial model developed by Cassel and Huckel (cf. Appendix 2B.1). The advantages of Frumkin s position is a more realistic consideration of the real properties of a two-dimensional surface state of the adsorption layer of soluble surfactants. This equation is comparable to a real gas isotherm. This means that the surface molecular area of the adsorbed molecules are taken into consideration. Frumkin (1925) additionally introduced, on the basis of the van der Waals equation, the intermolecular interacting force of adsorbed molecules represented by a . 

FIG. 4 ir—A isotherms measured for DSPC at water-1,2-DCE (O) and water-air ( ) interfaces from Ref. 41 and simulated with a real gas model [40] ideal gas with A = 0 and Ug = 0 (thin solid line), hard disks gas with A = 40 and ug = 0 (thick solid line), vdW gas with = 40 and Ug/kT = 3 (thin dashed line), and vdW gas with = 40 A and UgjkT = 7 (thick dashed line). The inset shows part of the thick dashed line. (Reproduced from Ref 40 with permission from Elsevier Science.)... 

Isothermal. The procedure used to calculate the work and energy quantities in an isothermal reversible expansion of a real gas is similar to that used for the ideal gas. 

From Equation (10.29), we can see that the change in the Gibbs function for the isothermal expansion of a real gas is... 

Using the a Function. A typical molar volume-pressure isotherm for a real gas is illustrated in Figure 10.6, together with the corresponding isotherm for an ideal gas. From Equation (10.34) we can write... 

The ratio of the fugacity/2 at the pressure P2 to the fugacity/i at the pressure Pj can be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as Pi approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. 

Figure 10.6. Comparison of molar volume-pressure isotherms for a possible real gas and an ideal gas.

It has already been mentioned that the state of an ideal gas at the temperature of the system and the pressure of 1 atmosphere is most frequently chosen as the standard state of gases. The idea of such an ideal gas can be explained by the imagination of a real gas which is first expanded to zero pressure and then by means of isothermal compression compressed to 1 atm. into the region of the ideal gas. As with an ideal gas pressure equals fugacity, we can substitute in equation (V-8a) p° = f° — 1, whereby the following equation is obtained ... 

Calculate the injection pressure for a 50-50 mixture of hydrogen sulfide and carbon dioxide. The reservoir is at a pressure of 2000 kPa, is at a depth of 750 m, and is isothermal at 20°C. Assume the acid gas will remain gaseous throughout the injection. Further assume (a) the gas is an ideal gas and (b) the gas is a real gas with properties described by the generalized compressibility chart. Take the properties of hydrogen sulfide and carbon dioxide from table 2.1. 

Rudzinski W. and Panczyk T., Phenomenological Kinetics of Real Gas-Adsorption-Systems Isothermal Adsorption, Journal of Non-EqmHbrium Thermodynamics, 27 (2002)pp.l49-204. 

Cy can be measured calorimetrically or, for the ideal gas, calculated from spectroscopic data. For a real gas or liquid, it can be obtained by combining the ideal-gas value with equation-of-state calculations to be discussed in Section 4.2.7, where the isothermal variation of U with V will also be discussed. 

For an ideal gas, z = 1. All real-fluid isotherms approach z = 1 asp, —> 0. The high T, isotherms are close to being horizontal and stay close to the line z = 1 at all values of p,. This is the behavior of common gases such as air, hydrogen, and helium at ambient and higher temperatures. As T,... 

Isochoric approach to equilibrium compared to a real isobaric isotherm for the LaNi5-H2 system in which the gas pressure was increased then decreased by a few kPa under active pressure control to within a few Pa. The point at which the isochore is tangent to the isobaric curve defines the isochoric isotherm, which therefore lies below the isobaric isotherm. The experimental data are from Gray (1992). 

The two-dimensional pressure isotherm characterizing the condensation of surfactant molecules in the adsorption layer (i.e., the transition from gaseous films to condensed ones), has a shape similar to that of a three-dimensional real gas given by the van der Waals equation ... 

Comparison of the van der Waals isotherms with those of a real gas shows similarity in certain respects. The curve at 7 in Fig. 3.7 resembles the curve at the critical temperature in Fig. 3.5. The curve at T2 in Fig. 3.7 predicts three values of the volume, V, V", and F ", at the pressure p. The corresponding plateau in Fig. 3.5 predicts infinitely many volumes of the system at the pressure p. It is worthwhile to realize that even if a very complicated function had been written down, it would not exhibit a plateau such as that in Fig. 3.5. The oscillation of the van der Waals equation in this region is as much as can be expected of a simple continuous function. 

Figure 18.14(b) shows the behavior of the surface pressure at very high areas and very low surface pressures F. The curves look very much like the isotherms of a real gas. In fact, the uppermost curve follows a law that is much like the ideal gas law. 

Example 2.7. We consider the isothermal expansion of a gas. Purposely we do not deal with an ideal gas, but with a real gas. At fixed mol number, the energy of the ideal gas is governed by two relevant variables, i.e., (S, V). Before the expansion, at start, we mark the variables by the subscript s. At start, the energy is Us = U(Ss, Vs). After the isothermal expansion, the relevant variables have been changed by A 5" and Ay. Therefore, at the end of expansion, which we indicate by the subscript e we have = 5s + A5 and Ve = Vs + Ay. In order to have an isothermal expansion we need the condition... 

In extending the isotherms of the path of integration shown in Fig. 1 so that AB approaches the zero-pressure isobar, it is clear that the representation of the P-V-T surface for the real gas must be valid to these low pressures. (The value of PF from the real-gas equation of state must approach RT as P approaches zero.) As a consequence, it now also becomes convenient to choose the reference state on the ideal gas surface. Since a reference point at zero pressure would result in infinite entropies at any finite pressure on the real or ideal gas surface, the standard reference state is usually chosen at 1 atm and To on the ideal gas surface this is equivalent to choosing the standard reference values of enthalpy and internal energy at zero pressure and To, since and C/° are functions of temperature alone for the ideal gas. 

But, generally, such a cycle with adiabatic and isothermal irreversible processes may be realized with real gas (or even liquid). Those with real gas approximate the reversible Carnot cycle with ideal gas by a double limiting process as follows (i.e., we form the ideal cyclic process from set A (and also B and C), see motivation of postulate U2 in Sect. 1.2) running this cycle slower and slower... 

The three isothermal virial coefficients can also be determined experimentally from P-V-T data and assuming that a real gas approaches ideality as (1/VJ —> 0 and as P —> 0 then A = RT. Actually, writing the virial equation of state as follows ... 

A thermodynamic consistent extension of the BET-isotherm to multicomponent systems (N > 1) and to real gas adsorptives will be presented in Sect. 4 of this Chapter. 

Phenomenological Kinetics of Real Gas Adsorption Systems Isothermal Kinetics and Kinetics of Thermodesorption,... 

The behavior of any real gas approaches ideal-gas behavior when the gas is expanded isothermally. As the molar volume Fm becomes large and p becomes small, the average distance between molecules becomes large, and intermolecular forces become negligible. 

Isothermal change of the real gas at pressure / vap to the hypothetical ideal gas at pressure p°. Table 7.5 has the relevant formulas relating molar quantities of a real gas to the corresponding standard molar quantities. 

---