Mixtures, gases, ideal fugacity

The heart of the question of non-ideality deals with the determination of the distribution of the respective system components between the liquid and gaseous phases. The concepts of fugacity and activity are fundamental to the interpretation of the non-ideal systems. For a pure ideal gas the fugacity is equal to the pressure, and for a component, i, in a mixture of ideal gases it is equal to its partial pressure yjP, where P is the system pressure. As the system pressure approaches zero, the fugacity approaches ideal. For many systems the deviations from unity are minor at system pressures less than 25 psig. 

The fugacity coefficient is a function of pressure, temperature, and gas composition. It has the useful property that for a mixture of ideal gases (Pi = 1 for all i. The fugacity coefficient is related to the volumetric properties of the gas mixture by either of the exact relations (B3, P5, R6) ... 

If the gas mixture is considered to be an ideal gas mixture then all fugacity coefficients are 1 and since K is a constant, the effect of increasing pressure is an increase of the equilibrium mole fraction of ammonia and a decrease of the mole fractions of nitrogen and hydrogen. However, since the ammonia synthesis is a high pressure process the gas mixture is not an ideal gas and the fugacity coefficients have to be taken into account. 

The differential heat of adsorption for each component in the mixture is estimated using the Clapeyron equation, extended to multicomponent mixtures and assuming ideal behavior of the gas phase (fugacity of i-th components Pi.)> that is,... 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

The equilibrium vapor behaves as an ideal-gas mixture, so the fugacity coefficient 9, of component i in the equilibrium vapor equals 1. 

Unfortunately, very few mixtures are ideal gas mixtures, so general methods must be developed for estimating the thermodynamic properties of real mixtures. In the dis-, cussion of phase equilibrium in a. pure fluid of Sec. 7.4, the fugacity function was especially useful the same is true for mixtures. Therefore, in an analogous fashion to the derivation in Sec. 7.4. we start from... 

If the gas-phase reactive mixture behaves ideally at low to moderate pressures, all fugacity coefficients are very close to unity and the fugacity of each component in the mixture can be approximated by its partial pressure. Hence,... 

The alternative motivation of definition (4.437) for (real) gas mixtures comes from a statement that a mixture is ideal if Amagat s law (4.440) is valid at any T, P. Indeed, Amagat s laws means Va = v and then by (4.454), (4.458) below, for fugacity coefficients also = v therefore by (4.463), this is an ideal mixture. 

If we assume the gas mixture is ideal, the fugacities are the same as the partial pressures, and the Duhem-Margules equation then becomes... 

Solution Deviations. The corrections so far considered have been limited to those associated with the fact that the vapor does not obey the perfect-gas law. A large number of mixtures, in fact most of them, do not obey the ideal solution laws even at very low pressure, and the deviations cannot be predicted by the use of gas-phase fugacity corrections. The deviations are the residt of the forces between the molecules in the liquid phase, and these forces can Be ery Targely diie... 

For ideal solutions, it is then possible to compute the entire vapor-liquid equilibria from the vapor pressures of the pure substances. For pressures too high to apply the ideal-gas law, fugacities are used instead of pressures [29,42). It is also possible to compute the equilibria for the special class of solutions known as regular [21]. For all other mixtures, however, it is necessary to obtain the data experimentally. 

This example shows the interrelations between fugacity, total pressure, vapor pressure, mol fraction, and activity coefficient. If we dealt only with ideal gas mixtures and ideal liquid solutions, we would scarcely have bothered to define fugacity, activity, or activity coefficient, because for ideal gases the fugacity is equal to the partial pressure fy,- P) and for ideal solutions of liquids and solids the fugacity is equal to the mol fraction times the vapor pressure (Xf-pi) making y=1.00 for both. However, Table 7.D (and the experimental data on which it is based) show that this liquid is not an ideal solution, because the activity coefficients are not unity. (The activity coefficient of ethanol = 1.007 1.00, but that of water is 2.31 ) This is an important industrial system, which we will speak about more in the next chapter. 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Unfortunately, the ideal-gas assumption can sometimes lead to serious error. While errors in the Lewis rule are often less, that rule has inherent in it the problem of evaluating the fugacity of a fictitious substance since at least one of the condensable components cannot, in general, exist as pure vapor at the temperature and pressure of the mixture. 

Thus the fugacity of species / in an ideal gas mixture is equal to its partial pressure. 

The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a flmd mixture ... 

Thermodynamic consistency tests for binary vapor-liquid equilibria at low pressures have been described by many authors a good discussion is given in the monograph by Van Ness (VI). Extension of these methods to isothermal high-pressure equilibria presents two difficulties first, it is necessary to have experimental data for the density of the liquid mixture along the saturation line, and second, since the ideal gas law is not valid, it is necessary to calculate vapor-phase fugacity coefficients either from volumetric data for... 

The fugacity coefficients in Equation (7.29) can be calculated from pressure-volume-temperature data for the mixture or from generahzed correlations. It is frequently possible to assume ideal gas behavior so that = 1 for each component. Then Equation (7.29) becomes... 

When dealing with an ideal mixture of gases (that do not behave ideally themselves), the fugacity of gas X is given by... 

In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, fA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal ... 

Assuming ideal gas behavior, the equilibrium partial pressure, ph of a compound above a liquid solution or liquid mixture is a direct measure of the fugacity, fu, of that compound in the liquid phase (see Fig. 3.9 and Eq. 3-33). 

The fugacity coefficients are a function of pressure, temperature and the equilibrium mole fractions, so at given pressure and temperature eq. (2.4-20) can be solved for s and the equilibrium mole fractions can be calculated. Table 2.4-1 gives the calculated equilibrium composition of the reaction mixture at different pressures for an ideal gas mixture and in case the gas is described with the Redlich-Kwong equation of state. 

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 ... 

Thus, for an ideal solution the fugacity coefficient of a species in solution is equal to the fugacity coefficient of the pure species at the mixture T and P and in the same physical state (liquid or gas). 

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