Fugacity limiting cases

The grand canonical formalism provides expansions of the osmotic pressure and of the concentrations in powers of the fugacities but, in principle, it is possible to eliminate these fugacities so as to obtain an expansion of the osmotic pressure in powers of the concentrations moreover, in practice, the calculations are nearly always performed in the limiting case V - go. ... 

From this point of view, the fugacity can be regarded as a "corrected pressure." In the limiting case of ideal gas behavior, pressure and fugacity are identical. In the... 

In the liquid phase, just as in the vapor phase, we need to choose a suitable reference state with a corresponding reference chemical potential and reference fugacity to complete the definition provided by Equation (7.3). We then adjust for the difference between the reference phase and the real system. However, while there is an obvious reference case for gases—the ideal gas—there is no single suitable choice for the liquid phase. There are two common choices for the reference state (1) the Lewis/Randall rule and (2) Henry s law. The choice of reference state often depends on the system. Both these reference states are limiting cases that result from a natural idealization for condensed phases the ideal solution. 

In the intermediate concentration ranges in Figure 7.5, we see that the fugacity of species a in the liquid phase is between the two limiting cases given by the Lewis/ Randall rule and Henry s law. We expect this behavior because, at intermediate concentrations, molecule a sees some other a molecules (characteristic of the Lewis/Randall rule) and some b molecules (characteristic of Henry s law). Thus, the fugacity of species... 

When transfers of molecules from media where the microorganisms do not occur (e.g., solids, NAPLs, gases) to phases where they are present (e.g., in water) are not rate-limiting, then it is possible that uptake by the microorganisms can be the slowest step in the sequence. Such a situation implies that the chemical fiigacity of the chemical of interest in that system is not equal to the chemical fugacity inside the cells where the relevant enzymatic apparatus occurs. Now we have a case in which the rate of biodegradation is expressed ... 

In this case, the cluster integral of Zimm and Lundberg (10) for water-water interactions is high enough to approach the vapor pressure of pure water as a limit and fugacity equal to 1. 

A pure suhetance i may be considered a special case of either a mixture or of a component in solution, in the limit as mole fraction x, approaches unity, Thus, formulas for the fugacity / of pure i are recovered as special cases or Eqs, (1.2-21)-(1.2-22), In psrticular,... 

Henry s law was introduced as a way of calculating the fugacity of a component in solution when the component is above its critical temperature at the temperature of the solution. Nonetheless, Henry s law maybe used even when the component is below its critical point. There is a certain symmetry between the Lewis-Randall rule, which applies in the limit x, 1, and Henry s law, which applies in the limit x, o. The relationship is demonstrated in Figure 13-12. which shows the fugacity of carbon dioxide in n-pentane, plotted as a function of the mol fraction of carbon dioxide at constant temperature. In this case carbon dioxide is below its critical temperature (304.2 K) and forms a liquid solution at all compositions between o and 1. The Lewis-Randall rule gives the fugacity of component by the linear relationship... 

The activity coefficients may be regarded as a measure of the deviation of a real system from the idealized behaviour of an arbitrarily chosen reference state. For a solute of limited solubility, infinite dilution is chosen as the reference state, for completely miscible liquids the single components and for gases and vapours the fugacity / = 1 at standard temperature. In all these cases y< = 1 for the reference state [19]. 

A simple equation of state should be chosen. The deterioration of accuracy of fugacity coefficients obtained will be negligible. Moreover, in the effort at elucidating the chemical equilibrium in a broad temperature and pressure range, extrapolation outside the limits of validity may be subject to greater risk in the case of multi-constant equations as opposed to simple relationships. 

These equations are similar to those for miscible liquids except that the mol fraction in the liquid is omitted. It is to be expected that the fugacity relationship would give satisfactory results up to a pressure of approximately one-half of the critical pressure. At higher pressures the Lewis and Randall fugacity rule for the vapor mixture would tend to be less satisfactory. Actually it is doubtful whether absolute immiscibility ever occurs. However, there are cases in which the miscibility is so limited that each phase would act as essentially a pure material, e.g, mercury and water. 

Equation (89) defines the change of the relative free energy of the surface, Af.(P), in the pressure domain P-P - Equation (89) is thermodynamically correct if, in the pressure domain P-Pm, the ideal-gas law is applicable and the supposition tf is vahd. The applicability of Eq. (89) may be extended if instead of pressures, the fugacities are apphed (i.e., the limits of integration are / and, corresponding to pressures P and P, respectively). This extension of Eq. (89) is supported by the fact that the supposition if in most cases is still valid when instead of the ideal-gas state equation the relationship (56) should be apphed. 

The requirement to specify the polymer density may represent a serious limitation for the practical application of the NELF model. Indeed, the dilation of the polymer matrix at high penetrant pressure could be significant and difficult to estimate without specific experimental data. On the other hand, the use of the non-equilibrium polymer (tensity is actually a powerful tool to represent complex non-equilibrium phenomena. It has been shown, for example, that it allows a description of sorpdon-desorption hysteresis (7) as well as the influence of pretreatments on the solubility isotherms of gases in glassy polymers. For such cases, the different pseudo-equilibrium solubility values at the same prevailing temperature and penetrant fugacity are satisfactorily accounted for by considering die different pseudo-equilibrium polymer densities. 

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