Fugacity coefficient pure gases

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

Furthermore, 5 vapor-liquid phase equilibria are involved, e.g. for molecular NH3, C02, H2S and S02 and for water. Applying the concept of Henry s constant Hi for the solution of a gas i in pure water and fugacity-coefficient for describing the influence of inter-molecular forces in the vapor phase, the resulting equations are ... 

If we adopt as the standard state for gaseous components the state of pure perfect gas at P = 1 bar and T = 298.15 K = f% = 1) and neglect for simphcity the fugacity coefficients, equation 5.304 combined with equation 5.297 gives... 

The vapour pressures as function of temperature are known. If we consider the vapour phase as an ideal gas =, the fugacity coefficient of pure i at the equilibrium conditions, which can be calculated by an equation of state. ye is given by... 

The fugacity coefficient ratio J can be estimated by assuming that the Lewis and Randall rule11 applies, at least approximately, for the mixture, so that each component has the same fugacity coefficient that it would have if it were a pure gas at the same total pressure. The Principle of Corresponding States can then be used to compare the fugacity coefficients of the three components. At p = 60 atm (61 bar) and in the temperature range from 900 to 1600 K, the reduced temperatures and pressures for the components of the equilibrium... 

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). 

Although we have omitted an identifying subscript in the preceding equations, their application so far has been to the development of generated correlations for pure gases only. In the remainder of this section we show how the virial equation may be generalized to allow calculation of fugacity coefficients < , of species in gas mixtures. 

In this definition, the activity coefficient takes account of nonideal liquid-phase behavior for an ideal liquid solution, the coefficient for each species equals 1. Similarly, the fugacity coefficient represents deviation of the vapor phase from ideal gas behavior and is equal to 1 for each species when the gas obeys the ideal gas law. Finally, the fugacity takes the place of vapor pressure when the pure vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor-phase association or dissociation. Methods for calculating all three of these follow. 

Solubilitiesattemperaturesand pressures above the critical values of the solvent liave important applications for supercritical separation processes. Examples are extraction of caffeine from coffee beans and separation of asplraltenes from heavy petroleum fractions. For a typical solid/vapor equilibrium (SVE) problem, tire solid/vapor saturation pressure P is very small, and the saturated vapor is for practical purposes an ideal gas. Hence 0 for pure solute vapor at this pressure is close to unity. Moreover, exceptfor very low values of the system pressure P, the solid solubility yj is small, and can be approximated by j, the vapor-phase fugacity coefficient of the solute at infinite dilution. Finally, since is very small, the pressure difference P — in the Poyntingfactor is nearly equal to P at any pressure where tins factor... 

The fugacities are evaluated at tire pressures indicated iir parentheses, where P is the equilibrium nrixed-gas pressure and P° is the equilibrium pure-gas pressure that produces the same spreading pressure. If the gas-phase fugacities are elinrinated hr favor of fugacity coefficients [Eqs. (11.32)snd (11.48)], then ... 

The value of the parameter 2 13 in a gas mixture can he calculated from PVT data using any traditional EOS. Eor the mixtures that obey the Lewis-Randall rule [16] (the fugacity of a species in a gaseous mixture is the product of its mole fraction and the fugacity of the pure gaseous component at the same temperature and pressure), the fugacity coefficients of the components of the mixture are independent of composition. In such cases, the KB equation [13] for the binary mixtures 1-3 ... 

The ratio of fugacity to partial pressure,// / ), called the fugacity coefficient, ([) , is in common use as a measure of departure of a real fluid/ from its ideal-gas value. For a pure fluid, the fugacity coefficient is simply < >=f/p. For an ideal-gas mixture, ([) = 1 for all i for a pure ideal gas, ([) = 1. For a real fluid, by rearranging Equation (4.305),... 

Gas-liquid system (GLC) The solute concentrations in both phases will be again expressed in mole fractions, the standard concentration and standard physical state of the solute in the stationary (liquid) phase will be defined in the same way as with the liquid-liquid system, and a hypothetical pure solute in a state of ideal gas at a unit pressure and at the temperature of the system will be chosen as a standard state for the solute in the mobile (gaseous) phase. Thus, and may be written as /is Yis is is 3 d /iiy = J iM/ - CiM where is the fugacity coefficient (mean value) of the solute in the mixture with carrier gas, and p is the mean pressure in the column. The corresponding standard fugacities (jc s = 1 and Yis = ffw = T = 1, and = 1) are f° = and = 1, so that, according to equation 43,... 

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only Tc, Pc, and o> were required for the calculation. The proposed equation successfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. 

Equations (1.3-14) and (1.3-15) thus give the prediction from transition-state theory for the rate of a reaction in terms appropriate for an SCF. The rate is seen to depend on (i) the pressure, the temperature and some universal constants (ii) the equilibrium constant for the activated-complex formation in an ideal gas and (iii) a ratio of fugacity coefficients, which express the effect of the supercritical medium. Equation (1.3-15) can therefore be used to calcu-late the rate coefficient, if Kp is known from the gas-phase reaction or calculated from statistical mechanics, and the ratio (0a 0b/0cO estimated from an equation of state. Such calculations are rare an early example is the modeling of the dimerization of pure chlorotrifluoroethene = 105.8 °C) to 1,2-dichlor-ohexafluorocyclobutane (Scheme 1.3-2) and comparison with experimental results at 120 °C, 135 °C and 150 °C and at pressures up to 100 bar [15]. 

In eq (2.3-3) and q) indicate the HC fugacity coefficients in the reference state (pure vapor) and in the vapor phase, respectively, v is the molar volume of the solid phase, T the temperature and R the universal gas constant. As can be reasonably set equal to 1 and the exponential term (known as the Poynting correction) is not markedly greater than 1 for pressures below 500 bar, the enhancement effect is almost completely due to q> , that is to the nonideality of the supercritical solution. 

Ideality seems like a reasonable assumption for this gas mixture at the temperature and pressures of operation since each pure-component fugacity coefficient is not much different from unity. Total pressure will decrease below 350 atm during the course of the reaction because 8 = -2. 

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