Isotherm of a real gas

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. 

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... 

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. 

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 ... 

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. 

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. 

Graph of state. Changes in the state of real gases can be illustrated in a graph of the relationships between p, v, and T. Usually p and v are the coordinate axes, T is a constant, and the resulting curves are called isotherms. Figure 4-1 shows isotherms for a real gas in transition between gas and liquid phases. For an ideal gas, p v is a constant, producing a hyperbola with the coordinate axes as asymptotes. 

At sufficiently high temperatures, isotherms for a real gas are hyperbolic in accordance with (1.14). At lower temperatures, deviations occur due to intermolecular forces between the gas molecules, and below a certain temperature, a horizontal discontinuity of the curve (fig. 1.25) is seen. 

Deviations from ideal behavior are related to the atomic and molecular nature of a gas and to the existence of intermolecular forces. There are fundamentally two assumptions undertaken with the ideal gas equation of state. It is assumed that the gas particles have no volume and that there are no interactions between the gas particles. For instance, at T = 0, the ideal gas law implies PV = 0, which for any fixed pressure means = 0. Atoms and molecules are not hypothetical points in space that can collapse into a zero-volume existence their volume does not approach zero as the temperature approaches absolute zero. Thus, a proper description of a real gas requires a correction to the ideal gas law. As one example of such a correction, we could introduce a constant, Vq, that is the gas volume at T = 0. This means P(y - Vq) = nRT. Another difference is that ideal gas particles exhibit no attraction, whereas real atoms and molecules do. So, in many ways, the PVT behavior of a real gas tends to differ from that of an ideal gas, and this is manifested in different forms for P-V isotherms. 

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.)... 

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. 

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. 

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 . 

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... 

The spreading pressure of a sorbed gas, tt, can be calculated from the sorption isotherm, although it cannot be measured directly. Assuming the sorbent to be thermodynamically inert, and that the surface area is the same at all temperatures for different gases, then the thermodynamic functions for the sorbed phase become analogous to those of a real fluid. 

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 ... 

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

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