Liquid, fugacity ideal

Raoult s law is the simplest quantitative expression for vapor-liquid equilibrium. This law is based on the vapor phase being an ideal gas and the liquid phase being an ideal solution. Therefore, the vapor-phase fugacity is given by Eq. (2). The effect of pressure on the liquid phase is very small, and since the vapor is ideal, the liquid fugacity may be written as in Eq. (4). Here, is the vapor pressure of component i at the temperature of the solution. 

For liquids that cannot be represented by equations of state, the liquid fugacities are expressed in terms of activity coefficients, discussed in Section 1.3.3. For ideal solutions. Equation 1.24 reduces to Raoult s law, presented in Section 1.3.2. 

Thus, to compute K values for ideal solutions, it is necessary only to evaluate the fugacities f and f at the temperature T and pressure P of the mixture. Except for the special case where the vapor forms a perfect gas which is treated below, the calculation of the K values is divided into two parts (1) the calculation of the vapor fugacities for pure components, and (2) the calculation of the liquid fugacities for pure components. 

Another reference-state for the solute / may be its pure liquid fugacity, fj. This state is a virtual one, because in practice x, 1. If the actual liquid mixture has as reference an ideal solution obeying the Lewis-Randall rule, we may define the reference-state f- as the limit of component fugacity at jc, 1 ... 

Ideal gas enthalpy, Hfv> is obtained from (4-60). The derivative of pure component liquid fugacity coefBcient with respect to temperature leads to the following relation for combined effects of pressure and latent heat of phase change from vapor to liquid. 

Although pure-component standard states are the ones most commonly used, situations arise in which a pure-liquid fugacity is unknown or difficult to determine. These situations occur, for example, when the mixture temperature T is above the critical temperature of the pure component (the gas solubility problem) and when T is below the pure-component melting temperature (the solid solubility problem). In such cases, we seek alternatives to the pure-component standard state. One way is to exploit any data available for mixtures that contain only small amounts of the component however, we emphasize that this approach does not require the real mixture to be dilute in that component. We are merely seeking an alternative to pure-component data to use as a basis for defining an ideal solution. 

At low pressures the vapor will be ideal, so q) =1 then, on using (5.5.8) for the liquid fugacity coefficient, neglecting the Poynting factor at low pressures, and taking the pure-i vapor pressure as the standard-state pressure, (12.1.5) reduces to... 

As for the liquid phase, in an ideal liquid solution the liquid fugacity of each component in the mixture is directly proportional to the mole fraction of the component as ... 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

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 ... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

The values of the thermodynamic properties of the pure substances given in these tables are, for the substances in their standard states, defined as follows For a pure solid or liquid, the standard state is the substance in the condensed phase under a pressure of 1 atm (101 325 Pa). For a gas, the standard state is the hypothetical ideal gas at unit fugacity, in which state the enthalpy is that of the real gas at the same temperature and at zero pressure. 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

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. 

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... 

E6.2 The fugacity of liquid water at 298.15 K is approximately 3,17 kPa. Take the ideal enthalpy of vaporization of water as 43.720 TmoD1, and calculate the fugacity of liquid water at 300 K. 

Phase Equilibria Models Two approaches are available for modeling the fugacity of a solute in a SCF solution. The compressed gas approach includes a fugacity coefficient which goes to unity for an ideal gas. The expanded liquid approach is given as... 

For most of the situations encountered in solvent extraction the gas phase above the two liquid phases is mainly air and the partial (vapor) pressures of the liquids present are low, so that the system is at atmospheric pressure. Under such conditions, the gas phase is practically ideal, and the vapor pressures represent the activities of the corresponding substances in the gas phase (also called their fugacities). Equilibrium between two or more phases means that there is no net transfer of material between them, although there still is a dynamic exchange (cf. Chapter 3). This state is achieved when the chemical potential x as... 

The first method, which is the more flexible, is to use an activity coefficient model, which is common at moderate or low pressures where the liquid phase is incompressible. At high pressures or when any component is close to or above the critical point (above which the liquid and gas phases become indistinguishable), one can use an equation of state that takes into account the effect of pressure. Two phases, denoted a and P, are in equilibrium when the fugacity / (for an ideal gas the fungacity is equal to the pressure) is the same for each component i in both phases ... 

The aqueous solubility of a gaseous compound is commonly reported for 1 bar (or 1 atm = 1.013 bar) partial pressure of the pure compound. One of the few exceptions is the solubility of 02 which is generally given for equilibrium with the gas at 0.21 bar, since this value is appropriate for the earth s atmosphere at sea level. As discussed in Chapter 3, the partial pressure of a compound in the gas phase (ideal gas) at equilibrium above a liquid solution is identical to the fugacity of the compound in the solution (see Fig. 3.9d). Therefore equating fugacity expressions for a compound in both the gas phase and an equilibrated aqueous solution phase, we have ... 

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). 

Let us now consider two special cases. In the first case, we assume that the compound of interest forms an ideal solution or mixture with the solvent or the liquid mixture, respectively. In assuming this, we are asserting that the chemical enjoys the same set of intermolecular interactions and freedoms that it has when it was dissolved in a liquid of itself (reference state). This means that Yu is equal to 1, and, therefore, for any solution or mixture composition, the fugacity (or the partial pressure of the compound i above the liquid) is simply given by ... 

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