Vapor pressure, corresponding-state reduced

Fig. 13.1 Reduced vapor pressure and molar density vs. reciprocal reduced temperature for HoO, CH4, H2, and 4He. In each case, were simple corresponding states theory adequate, all data would lie on a single master curve. Using extended CS the curves are fit to acceptable precision, (a) (top) = reduced vapor pressures, (b) (bottom) = reduced liquid molar densities... 

When the value of the vaporization enthalpy is known at one temperature (conunonly at normal boiling point), it is desirable to be able to evaluate it at another temperature. In such procedures for extrapolating volumes or enthalpies, a corresponding state procedure is often used. This is frequently based on the reduced temperature 7), reduced vapor pressure Pr, and reduced molar volume Vr, defined by... 

Corresponding states have been used in other equations. For example, the Peng-Robinson equation is a modified RedHch-Kwong equation formulated to better correlate vapor—Hquid equiHbrium (VLE) vapor pressure data. This equation, however, is not useful in reduced form because it is specifically designed to calculate accurate pressure data. Reduced equations generally presuppose knowledge of the pressure. 

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

Figure 13.1a shows reduced vapor pressures and Fig. 13.1b reduced liquid molar densities for the parent isotopomers of the reference compounds. Such data can be fit to acceptable precision with an extended four parameter CS model, for example using a modified Van der Waals equation. In each case the parameters are defined in terms of the three critical properties plus one system specific parameter (e.g. Pitzer acentric factor). Were simple corresponding states theory adequate, the data for all... 

Figure 9,4 (A) Reduced isotherms of N2, CO2, and H2O vapor according to reduced pressure. (B) Comparison between experimental isotherms of H2O and f predicted by principle of corresponding states.

The explicit formula pxr — I = (1 — Pr)0 for reduced saturation density as a function of reduced pressure is proposed for the entire liquid-vapor saturation boundary. The expression A 1 depends on Pr p 0.35 depends weakly on Pr, corresponding at Pr = 1 to the critical exponent pc. The parameters A and ft can be related to the Pitzer factor o>. Special cases include the power law pr — 1 = C(1 — Tr)0c. . . and the low-pressure vapor equation prx0 = p0Pr The function A — Ac = g(Pr) is found from data to be a universal function for nonpolar substances. If Ac is correlated with o>, the formula takes on the corresponding-states form pr = /o,(Pr, to). This form predicted the density of saturated liquid and vapor with 0.4% and 0.9% accuracy, respectively, for 38 substances. 

The theorem of corresponding states goes back to van der Waals, who formulated the principle of corresponding states based on his equation of state. In addition, van der Waals deduced straightforwardly that in reduced coordinates, the vapor pressure curve and the coexistence curve must be the same for all fluids [3]. An extensive treatment of the corresponding state principle has been published by Xiang [4]. 

If the plot of the reduced vapor pressure against the reciprocal reduced temperature fits a master curve, it does not mean that the rule of Trouton is valid. To meet the rule of Trouton, also the rule of Hildebrand must be valid. In this case, all the boiling points will be found at r /r = Tc/Tb = 3/2 = 1.5. On the other hand, if the corresponding states are sound, then all substances should have a common critical pressure, because In Pb/Pc = C in this case (since Pb = 1 atm). 

These forms of a generalized equation of state only require the critical temperature and the critical pressure as substance-specific parameters. Therefore, these correlations are an example for the so-called tsvo-parameter corresponding-states principle, which means that the compressibility factor and thus the related thermodynamic properties for all substances should be equal at the same values of their reduced properties. As an example, the reduced vapor pressure as a function of the reduced temperature should have the same value for all substances, provided that the regarded equation of state can reproduce the PvT behavior of the substance on the basis of the critical data. In reality, the two-parameter corresponding-states principle is only well-suited to reflect the properties of simple, almost spherical, nonpolar molecules (noble gases as Ar, Kr, Xe). For all other molecules, the correlations based on the two-parameter corresponding-states principle reveal considerable deviations. To overcome these limitations, a third parameter was introduced, which is characteristic for a particular substance. The most popular third parameter is the so-called acentric factor, which was introduced by Pitzer ... 

A modification of the corresponding-state principle by introducing a parameter related to the vapor pressure curve is reasonable, because experimental vapor pressure data as a function of temperature are easy to retrieve. Furthermore, the vapor-liquid equilibrium is a very sensitive indicator for deviations from the simple corresponding-state principle. The value Tr = 0.7 was chosen because this temperature is not far away from the normal boiling point for most substances. Additionally, the reduced vapor pressure at Tr = 0.7 of the simple fluids has the value = 0.1 (log = —1). As a consequence, the acentric factor of simple fluids is 0 and the three-parameter correlation simplifies to the two-parameter correlation. 

For molecules more complex than the noble gases, a third parameter is introduced (Pitzer et al. 1955 Pitzer 1955) to form a three-parameter theory of corresponding states. This is the acentric factor a, which is a function of the acentricity or noncentral nature of the intermolecular forces. It is completely empirical. For simple fluids, it was observed that at a reduced temperature of 0.7, the saturation pressure (i.e., the vapor pressure of the liquid-gas equilibrium) divided by the critical pressure is very close to 1/10, or... 

Corresponding-states correlations of Z based on this theorem are referred to as two-parameter correlations. They require the use of two reducing parameters, and P. These correlations are nearly exact when used to describe noble gases such as argon, krypton, and xenon. When complex fluids were encountered, a three-parameter corresponding states parameter was found to be needed. An acentric factor, o), can be defined as introduced by Pitzer [11]. The acentric factor for a pure substance is defined with respect to its vapor pressure ... 

Evaporation of water is prevented, maintaining a solvent like density, which is essential to make it a homogeneous reaction medium, reactant, and catalyst precursor. Typically, pressures not to far above the corresponding vapor pressure is adjusted in subcritical applications. In the supercritical state, the density is reduced by a factor of 5-10 compared to water at ambient conditions. 

When two phases of a single substance are at equilibrium, the pressure is a function only of the temperature. A phase diagram for a pure substance contains three curves representing this dependence for the solid-liquid, solid-gas, and liquid-gas equilibria. These three curves meet at a point called the triple point. The liquid-vapor coexistence curve terminates at the critical point. Above the critical temperature, no gas-liquid phase transition occurs and there is only one fluid phase. The law of corresponding states was introduced, according to which all substances obey the same equation of state in terms of reduced variables... 

Dew-point Temperature (DPT). DPT is the temperature at which the condensation of water vapor in a space begins for a given state of humidity and pressure as the temperature is reduced. It is the temperature corresponding to saturation (100% rh) for a given absolute humidity at constant pressure. 

The small spheres are fluid molecules, and the large spheres are immobile silica particles. The top visualizations are for a disordered material and the bottom visualizations are for an ordered material of the same porosity. The visualizations on the left are for the saturated vapor state, and those on the right are for the corresponding saturated liquid state, (b) Simulated adsorption and desorption isotherms for Lennard-Jones methane in a silica xerogel at reduced temperature kT/Sfi = 0.7. The reduced adsorbate density p = pa is plotted vs the relative pressure X/Xo for methane silica/methane methane well depth ratios ejf/Sff = 1- (open circles) and 1.8 (filled circles) [44]. (Reproduced with permission from S. Ramalingam,... 

For a given temperature, pressure, and composition, the corresponding density and Z factor given by the equation of state is needed. The vapor root is found by solving the truncated virial equation of state (see Table 14-8) for Z, and for small values of BP/RT, the expression so obtained reduces to Z = 1 + BP/RT, that is,... 

Despite widespread use of the ideal K-value concept in industrial calculations, particularly during years prior to digital computers, a sound thermodynamic basis does not exist for calculation of the fugacity coefficients for pure species as required by (4-85). Mehra, Brown, and Thodos discuss the fact that, for vapor-liquid equilibrium at given system temperature and pressure, at least one component of the mixture cannot exist as a pure vapor and at least one other component cannot exist as a pure liquid. For example, in Fig. 4.3, at a reduced pressure of 0.5 and a reduced temperature of 0.9, methane can exist only as a vapor and toluene can exist only as a liquid. It is possible to compute vl or f v for each species but not both, unless vl = vy, which corresponds to saturation conditions. An even more serious problem is posed by species whose critical temperatures are below the system temperature. Attempts to overcome these difficulties via development of pure species fugacity correlations for hypothetical states by extrapolation procedures are discussed by Prausnitz. ... 

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