Pressure fugacity relationship

As in the case for adsorption (see Section 2.2), in equilibrium, the quantity N of a given solute which is dissolved in a given solvent depends on its gas phase partial pressure (fugacity) P and on the temperature T, and a basic phenomenological description of the equilibrium is specification of the functional relationship between N, P, and T. At sufficiently low pressures, it is expected that the pressure dependence is linear (Henry s law) ... 

Deviation from ideality of real liquid solutions manifests itself by departure of the various characteristics such as partial pressure, fugacity, activity, and activity coefficient, from the simple linear relationships previously shown. Of these, perhaps the most convenient to study is the activity coefficient 7. Thus, for a binary solution of A and 5,... 

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 (1) is of little practical use unless the fuga-cities can be related to the experimentally accessible quantities X, y, T, and P, where x stands for the composition (expressed in mole fraction) of the liquid phase, y for the composition (also expressed in mole fraction) of the vapor phase, T for the absolute temperature, and P for the total pressure, assumed to be the same for both phases. The desired relationship between fugacities and experimentally accessible quantities is facilitated by two auxiliary functions which are given the symbols (f... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

We used the system (.vic-Q,H 1CH3 +. vic-CeH ) as an example of a system that closely approximates ideal behavior. Figure 6.5 showed the linear relationship between vapor pressure and mole fraction for this system. In this Figure, vapor pressure could be substituted for vapor fugacity, since at the low pressure involved, the approximation of ideal gas behavior is a good one, and... 

Since the standard state fugacity, f°, can be approximated by saturation vapor pressure, p , then equation (2) reduces to the well known relationship ... 

The temperature is high enough for the gases to be considered ideal, so the equilibrium constant is written in terms of partial pressure rather than fugacity, and the constant will not be affected by pressure. Mol fraction can be substituted for partial pressure. As the total mols in and out is constant, the equilibrium relationship can be written directly in mols of the components. 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

As was discussed earlier in Section 1.2.8 a complication arises in that two of these properties (solubility and vapor pressure) are dependent on whether the solute is in the liquid or solid state. Solid solutes have lower solubilities and vapor pressures than they would have if they had been liquids. The ratio of the (actual) solid to the (hypothetical supercooled) liquid solubility or vapor pressure is termed the fugacity ratio F and can be estimated from the melting point and the entropy of fusion. This correction eliminates the effect of melting point, which depends on the stability of the solid crystalline phase, which in turn is a function of molecular symmetry and other factors. For solid solutes, the correct property to plot is the calculated or extrapolated supercooled liquid solubility. This is calculated in this handbook using where possible a measured entropy of fusion, or in the absence of such data the Walden s Rule relationship suggested by Yalkowsky (1979) which implies an entropy of fusion of 56 J/mol-K or 13.5 cal/mol-K (e.u.)... 

Most applications in materials science are carried out under pressures which do not greatly exceed 1 bar and the difference between/and/ is small, as can be seen from the fugacity of N2(g) at 273.15 K [15] given in Figure 2.11. Hence, the fugacity is often set equal to the partial pressure of the gas, i.e./ p. More accurate descriptions of the relationship between fugacity and pressure are needed in other cases and here equations of state of real, non-ideal gases are used. 

That is, as the pressure approaches zero, the fugacity approaches the pressure. Figure 10.5 indicates the relationship between P and/for ideal and real gases. The standard state for a real gas is chosen as the state at which the fugacity is equal to 0.1 MPa, 1 bar, along a line extrapolated from values off at low pressure, as indicated in Figure 10.5. The standard state for a real gas is then a hypothetical 0.1 MPa standard state. 

Because of this relationship between (TT — and p-j x.. the former quantity frequently is referred to as the Joule-Thomson enthalpy. The pressure coefficient of this Joule-Thomson enthalpy change can be calculated from the known values of the Joule-Thomson coefficient and the heat capacity of the gas. Similarly, as (H — is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule-Thomson coefficient. As the fugacity and the Joule-Thomson coefficient are both measures of the deviation of a gas from ideahty, it is not surprising that they are related. 

Because is adimensional, it is evident that fugacity has the dimension of pressure. It is important to emphasize that it is conceptually erroneous to consider the fugacity of a gaseous species as the equivalent of the thermodynamic activity of a solid or liquid component (as is often observed in literature). The relationship between activity a, and fugacity f of a gaseous component is given by... 

T is temperature, P is pressure, and / is the fugacity of the component. In Equation 3 subscript k refers to each component of the system. In the present discussion the fugacity 42) is employed in preference to the chemical potential 21). Earlier in the history of the petroleum industry, Raoult s 55) and Dalton s laws were applied to equilibrium at pressures considerably above that of the atmosphere. These relationships, which assume perfect gas laws and additive volumes in the gas phase and zero volume for the liquid phase, prove to be of practical utility only at low pressures. Henry s law was found to be a useful approximation only for gases which were of low solubility and at reduced pressures less than unity. 

The first term on the right-hand side is the idea gas limit, and the remaining -logarithmic terms express the successive virial corrections for the real gas behavior. It is evidently most convenient for this problem to choose the standard state pressure as P° = 0, where all gases are ideal. With this choice, we can write the relationship between fugacity and pressure as... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

No wonder the vapour pressure assumes a high importance in thermodynamic study of metallurgical systems. In the above derivation, the applicability of ideal gas law has been assumed. It would be very nearly true for vapours of metals and their oxides, as their pressures are extremely low. For a more general applicability, the above relationships have been restated in terms of fugacity or escape tendency, which has been qualitatively introduced in the beginning of this chapter. Quantitatively, fugacity,/ is defined as... 

The partial pressures of CH4 and H2O, like the overall fluid pressure (Pf), depend on the fugacity of hydrogen arriving from deep sources under a certain pressure. The quantitative relationships between these gases are determined by the reactions ... 

It can be seen from Table XX that this relationship is approximately true for oxygen at 0° C up to pressures of about 100 atm., and so equation (29.16) may be utilized to give reasonable values of the fugacity at moderate pressures. For gases which depart from ideal behavior to a greater extent than does oxygen, however, equation (29.17), and hence (29.16), does not hold up to such bigh pressures, except at hi er temperatures. 

The solubility increases with increase in pressure at a hxed temperature, owing to enhanced solvation due to greater attractive forces between the solute and carbon dioxide. A fundamental relationship for phase equilibrium (Prausnitz et al. 1999) can be used to relate fugacities of the solute in the solid and fluid phases as follows ... 

If a volumetric equation of state (that is, the relationship among the molar volume, temperature, and pressure) is applicable to pure liquid i at T and P, the pure component fugacity can be computed from... 

When the distribution coefficients are composition-dependent, the above method must be modified to account for the effect of composition. A search for the unknown bubble point or dew point temperature or pressure is started on the basis of some composition-independent relationship between the X-values and the temperature and pressure, such as Equations 2.20 and 2.21. Component fugacities are then calculated for the vapor phase and the liquid phase, and the /f-values are updated using Equation 2.15. The calculations are repeated until Equation 2.16 or 2.17, as well as Equation 2.12, are satisfied. The iterative scheme for the bubble point pressure calculation may proceed along the following steps ... 

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