Hypothetical pure component pressure

The set of equations presented in the previous section (5.3.3) in general can not be solved analytically hence it must be solved numerically. Even if the spreading pressure equation (5.3-21) can be integrated analytically, the inverse of the hypothetical pure component pressure versus spreading pressure is not generally available in analytical form with the exception of the Langmuir, Freundlich and Sips equations (see Table 5.3-3). 

At the mixture reduced spreading pressure, the hypothetical pure component pressure is given by (Table 5.3-3)... 

Substituting this hypothetical pure component pressure into eq. (5.3-3la) to solve for the reduced spreading pressure, we get... 

Knowing this reduced spreading pressure, eq. (5.3-45) can be solved for the hypothetical pure component pressure in terms of the gas phase partial pressures... 

Substituting this hypothetical pure component pressure into the Raoult s law equation (5.3-20a), we get the mole fraction of the adsorbed phase... 

The method essentially makes use of the analytical expression for the reduced spreading pressure in terms of the hypothetical pure component pressures. Since the reduced spreading pressure, z, is the same for all components, we let ... 

Both the Fast IAS and IAS theories require the numerical computation for the solution. So what are the difference and the advantage of the FastlAS The difference is that the Fast IAS involves the solution of N variables of pure component pressure, bjP while the solution for the spreading pressure is sought in the IAS theory (see Section 5.3.3). Once the spreading pressure is known in the IAS theory, the hypothetical pure component pressures can be obtained as the inverse of... 

A. 8 The Initial Guess for the Hypothetical Pure Component Pressure... 

Once the hypothetical pure component pressures are obtained, the mole fractions in the adsorbed phase are obtained from the Raoult law (Eq. 5.4-7) and then the total adsorbed concentration can be calculated from... 

For multicomponent systems obeying the ideal adsorption solution theory, the spreading pressure of the adsorbed mixture is n. The partial pressure of the species i in the gas phase is related to the hypothetical pure component pressure which gives the same spreading pressure n as that of the mixture according to the Raoult s law analogy ... 

The initial guess for the hypothetical pure component pressure 230... 

Expected errors for this method are 4-5 percent. At higher pressures, a pressure correction using Eq. (2-130) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5), (2-8), and (2-17) and with the molecular weight being mole-averaged. 

According to Radke and Prauznitz the total load of the mixture, qtot, is only a function of the load qf for single-component adsorption of component i (Eq. 2.51a) at the hypothetical pure component concentration c (Eq. 2.51b). In Eq. 2.51 c is a hypothetical concentration of component i of a single-component system that causes a spreading pressure n in the adsorbed phase, which is equal to the spreading pressure of the mixture (Eq. 2.52). 

Develop an expression for the Henry s law constant as a function of the A parameter in the Marguies expression, the vapor pressure, and composition. Compare the hypothetical pure component fugacity based on the Henry s law standard state with that for the usual pure component standard state. 

Figure 5.3-2a Plot of the reduced spreading Figure 5.3-2b Plots of the hypothetical pressure versus the gas mole fraction pure component pressure vs gas mole fraction...

PURE calculates pure liquid standard-state fugacities at zero pressure, pure-component saturated liquid molar volume (cm /mole), and pure-component liquid standard-state fugacities at system pressure. Pure-component hypothetical liquid reference fugacities are calculated for noncondensable components. Liquid molar volumes for noncondensable components are taken as zero. 

FO(I) Vector (length 20) of pure-component liquid standard-state fugacities at zero pressure or hypothetical liquid standard-... 

Amagat S Law. The total volume of a gaseous mixture equals the sum of the pure-component volumes. By definition, the pure-component volume of a component gas in a mixture is the hypothetical volume that the component would occupy at the same temperature and total pressure of the mixture. By Amagat s law,... 

Fig. 1.9 Vapor pressure for a hypothetical regular solution for which cAh(RT = 1 plotted against the mole fraction of component B. The vapor pressure of pure component B is 26.7 kPa, and that of component A, 20.0 kPa. The broken lines show Raoult law behavior.

The standard state for a pure liquid or solid is taken to be the substance in that state of aggregation at a pressure of 1 bar. This same standard state is also used for liquid mixtures of those components that exist as a liquid at the conditions of the mixture. Such substances are sometimes referred to as liquids that may act as a solvent. For substances that exist only as a solid or a gas in the pure component state at the temperature of the mixture, sometimes referred to as substances that can act only as a solute, the situation is more complicated, and standard states based on Henry s law may be used. In this case the pressure is again fixed at 1 bar, and thermal properties such as the standard-state enthalpy and heat capacity are based on the properties of the substance in the solvent at infinite dilution, but the standard-state Gibbs energy and entropy are based on a hypothetical state.of unit concentration (either unit molality or unit mole fraction, depending on the form of Henry s law used), with the standard-state fugacity at these conditions extrapolated from infinite-dilution behavior in the solvent, as shown in Fig. 9.1-3a and b. Therefore just as for a gas where the ideal gas state at 1 bar is a hypothetical state, the standard state of a substance that can only behave as a solute is a hypothetical state. However, one important characteristic of the solute standard state is that the properties depend strongly upon the solvent. used. Therefore, the standard-state properties are a function of the temperature, the solute, and the solvent. This can lead to difficulties when a mixed solvent is used. 

The fie lines on the plot in Figure 12.16 are the same tie lines shown on the plot in Figure 12.15. Note that each pure component has the same value of for its vapor and liquid phases, but the two phases in VLB have different values for (that is, the tie lines are not horizontal). For these mixtures, each liquid has a much more negative value of g than does the vapor at the same T and P that is, compared to the vapor phases, much more work must be done to convert these liquids into ideal gases (see 6.3.2). Note that since P 22 bar at 330 K, the pure liquid alkane (component 1) is a hypothetical state at the lower pressures of the mixtures at this T the phi-phi approach can readily handle this situation, but use of gamma-phi with a model would be awkward. 

Hypothetical states appear because the ideal solution (and the ideal-gas) equations call for the properties of the pure liquid (and pure ideal-gas component) at the pressure and temperature of the solution. Specifying pressure, temperature and phase for a pure component represents an over-specification of state and thus may lead to conflict between the actual and specified state. The same is true with the ideal-mixture equations, which require the properties of pure components as gases at the temperature and pressure of the mixture. These hypothetical states are mathematical, not physical, states and their properties are calculated by the equations that apply to each phase. 

The situation can be analyzed more clearly on the enthalpy/pressure chart of pure component (Figure 11-2). The more volatile component (acetonitrile in the previous example) is a hypothetical liquid at the bubble temperature. It is shown by point H, located on the liquid portion of the isotherm at Tbubwe but extrapolated into the vapor-liquid region. Since isotherms in the compressed liquid region are almost vertical, the enthalpy of state is to a very good... 

N values of the hypothetical pressure of the pure component, P , which gives the same spreading pressure as that of the mixture. 

Knowing the mole fractions in the adsorbed phase (xj) and the hypothetical pressure of the pure component () that gives the same spreading pressure as that of the mixture (ti), the total amount adsorbed can be calculated from the equation ... 

Eq. (5.3-26a) is known as the Lewis relationship. It has been tested with a number of systems involving silica gel and activated carbon. This equation relates the amounts adsorbed in multicomponent system (C j) to those for pure component systems (C j) evaluated at the hypothetical pressure P . 

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