## Estimation of Second Cross Virial Coefficients

Our treatment of cross virial coefficients is the same as for pure coefficients in Sec. 4-5. All of the methods there can be used here if the parameters are suitably adjusted. As before, the formulation is in CSP for all pairs of components in the mixture, i and j. where V* is a characteristic volume for the pair, the amij are strength parameters for various pair intermolecular forces described in Sec. 4-5, and the (m) are sets of universal functions of reduced temperature, T T*, with T* a...

## Diffusion In Electrolyte Solutions

When a salt dissociates in solution, ions rather than molecules diffuse. In the absence of an electric potential, however, the diffusion of a single salt may be treated as molecular diffusion. The theory of diffusion of salts at low concentrations is well developed. At concentrations encountered in most industrial processes, one normally resorts to empirical correlations, with a concomitant loss in generality and accuracy. A comprehensive discussion of this subject is available (Newman and...

## References

Fundam., 15 78 (1976). Akgerman, A., and J. L. Gainer J. Chem. Eng. Data, 17 372 (1972). Albright, J. G., A. Vernon, J. Edge, and R. Mills J. Chem. Soc., Faraday Trans. 1, 79 1327 (1983). Alvarez, R., I. Medlina, J. L. Bueno, and J. Coca J. Chem. Eng. Data, 28 155 (1983). Amdur, I., and L. M. Shuler J. Chem. Phys, 38 188 (1963). Amdur, I., J. Ross, and E. A. Mason J. Chem. Phys., 20 1620 (1952). Amourdam, M. J., and G. S. Laddha J. Chem. Eng. Data, 12 389 (1967)....

## Xzw

Goletz and Tassios (1977) have used this form (for the kinematic viscosity) and report values of A, B, and C for many pure liquids. Equation (9-10.1) requires at least two viscosity-temperature datum points to determine the two constants. If only one datum point is available, one of the few ways to extrapolate this value is to employ the approximate Lewis-Squires chart (1934), which is based on the empirical fact that the sensitivity of viscosity to temperature variations appears to depend...

## H

Experimental and smoothed data are given for the vapor pressures of pure substances in the normal and low-pressure region Antoine constants are reported. A thorough compilation of polymerization reactions and of solution and physical properties of polymers and their associated oligomers and mon- TABLE 8-1 b Some Useful Books on Fluid-Phase Equilibria (Continued) Broul, M., J. Nyult, and O. S hnel Solubility in Inorganic Two-Component Systems, Elsevier, Amsterdam, 1981. Caruthers, J. M., K. C....

## Info

Note that this model predicts that the component with the lower surface tension, n-propanol, is 17 times more concentrated in the surface than the bulk. Using the UNIFAC method in Chap. 8 to calculate activity coefficients at both the bulk and surface concentrations would have predicted a value of fm of 43.40 dyn cm for an error of 3.8 . If one just takes both bulk and surface activity coefficients equal to 1 in Example 12-5, the error is 13 . The Suarez method gives an error of -0.4 for the...

## D ln aA d ln xA

Where the activity aA xAyA and yA, the activity coefficient is described in Chap. 8. By virtue of the Gibbs-Duhem equation, a is the same regardless of whether activities and mole fractions of either A or B are used in Eq. (11-2.8). For gases, a is almost always close to unity (except at high pressures), and this correction is seldom used. For liquid mixtures, however, it is widely adopted, as will be illustrated in Sec. 11-10. 11-3 DIFFUSION COEFFICIENTS FOR BINARY GAS SYSTEMS AT LOW PRESSURES...

## Fluid Phase Equilibria In Multicomponent Systems

Solution There are three unknowns the bubble-point temperature and two vapor-phase mole fractions. To find them we use three equations of equilibrium yfatP x ytfoL i 1, 2, 3 (8-9.26) where y is the vapor-phase mole fraction and x is the liquid-phase mole fraction. Fu-gacity coefficient fa is given by the truncated virial equation of state where subscript M stands for mixture Bm y 2B11 + y 2B22 + y 2B33 + 2yi y2 B12 + 2yi y3 Ba + 2y2 y3 B33 (8-9.28) All second virial coefficients By are found...

## Ar Ar AOtr xrh

Where A low-pressure gas thermal conductivity, W (m K) and T is defined in Eq. (10-3.9). (AT)tr 8.757 exp(0.0464Tr) - exp(-0.2412Tr) (10-3.11) Relations for f(Tr) are shown in Table 10-1. The constant C is specific for each material, and it is estimated by a group contribution technique as shown below. Estimation of Roy-Thodos Constant C. In the discussion to follow, one identifies carbon types as shown TABLE 10-1 Recommended f(Tr) Equations for the Roy-Thodos Method TABLE 10-1 Recommended...

## Analytical Equations Of State For Mixtures

As discussed in Sec. 4-6, analytical EoS models allow the solution for the density of a fluid from a specified T, P, and for mixtures, y , to be noniterative. Thus, the formulation must be cubic or quartic in V. As discussed in Sec. 5-2, the common way to include the effects of composition in the parameters is with mixing rules and combining rules as in the CSP treatments of Sec. 5-3. Commonly, the one-fluid approach is used where the mixture is assumed to behave as a pure component with...

## Notation

A parameter in Eq. (10-9.1), W (m K) Aij Wassiljewa coefficient, Eq. (10-6.1) Ci electrolyte concentration, mol L, Eq. (10-12.20) C heat capacity, J (mol K) Cv, at constant volume Cp, at constant pressure Ctat, due to internal degrees of freedom Ctr, due to translational motion C group contribution constant in Eq. (10-3.12) scaling parameter in Eqs. (10-5.13) and (10-7.13) tr translational factor in Eq. (10-3.1) tat internal energy factor in Eq. (10-3.1) FK parameter defined in Eq. (10-5.15)...

## Concentration Dependence Of Binary Liquid Diffusion Coefficients

The concentration dependence of binary diffusion coefficients is not simple. In some cases it varies linearly between the two limiting diffusion coefficients, while in others strong positive or negative deviations from linearity are observed. In Sec. 11-2, it was suggested that the diffusion coefficient DAB in a binary mixture may be proportional to a thermodynamic correction a ( ln aA ln xA) TP aA and xA are the activity and mole fraction of species A respectively. From the Gibbs-Duhem...

## Compressed Liquid Densities

Critical evaluations of the literature have been recently published by Cibulka and coworkers (see Cibulka and Takagi 1999 and references to earlier work). Sun et al. (1990) describe many different relations that have been used for the effect of pressure on liquid volumes and densities. They are generally of the forms P - Po f(T, P Po0 (4-12.1b) where the reference density p0> is the density at T and P0> often picked as atmospheric. Since the desired quantity is p at T and P. Eq. (4-12.1b)...

## Van der Waals and Density Dependent Mixing Rules

As described above, the basic approach of van der Waals mixing rules is in Eqs. (5-2.2) and (5-2.4), while density-dependent mixing rules add functions to the van der Waals expressions. The simplest justification of the van der Waals concept is to match the composition dependence of the second virial coefficient from an EoS model to the rigorous relationship (5-4.2a). For the general cubic equation of Eq. (4-6.1), and therefore all of the models of Tables 4-6 and 4-7, this is (Abbott, 1979)

## Diffusion Coefficients At Infinite Dilution

For a binary mixture of solute A in solvent B, the diffusion coefficient D AB of A diffusing in an infinitely dilute solution of A in B implies that each A molecule is in an environment of essentially pure B. In engineering work, however, D AB is assumed to be a representative diffusion coefficient even for concentrations of A of 5 to 10 mole . In this section, several estimation methods for D AB are introduced the effect of concentration for mutual diffusion coefficients is covered in Sec....

## EoS Models for the Near Critical Region

Conditions near the vapor-liquid critical point of a substance show significantly different behavior from simple extrapolations of the EoS models described so far in this chapter. The shape of the critical isotherm, the variations of CV, the isothermal compressibility, and other properties and the vapor-liquid coexistence curve are all different than that given by most EoS models. This is because the molecular correlations are much longer ranged and fluctuate differently in this region. The...

## Diffusion Coefficients

A MC-SD 3.84 3.83 1 2 3.84 A To determine nD, T kT e MC-SD 323 399 0.810. With Eq. 11-3.6 , nD 1.60. Then with Eq. 11-3.7 , n 16 0.19 0.49 2 1 66 d L6 0.810 L66 With Eq. 11-3.2 and M MC 50.49, M SD 64.06, and MAB 2 1 50.49 1 64.06 -1 56.47 1 56.47 1 2 3.84 2 1.66 The experimental value is 0.078 cm2 s Brokaw, 1969 and the error is 8 . A comprehensive review of the theory and experimental data for gas diffusion coefficients is available Marrero and Mason, 1972 . There have been many studies...

## 1

COO esters or HCOO formates NH2 TABLE 9-3 Values of the Group Contributions Ci for the Estimation of a in Eq. 9-4-22 Reichenberg, 1971 Continued A comparison between calculated and experimental low-pressure gas viscosity values is given in Table 9-2, and the method is illustrated in Example 9-3. Example 9-3 Use Reichenberg's method to estimate the viscosity of ethyl acetate vapor at 125 C and low pressure. The experimental value is reported to be 101 P Landolt-Bornstein Tabellen, 1955 ....

## Liquid Mixture Viscosity

Essentially all correlations for liquid mixture viscosity refer to solutions of liquids below or only slightly above their normal boiling points i.e., they are restricted to reduced temperatures of the pure components below about 0.7. The bulk of the discussion below is limited to that temperature range. At the end of the section, however, we suggest approximate methods to treat high-pressure, high-temperature liquid mixture viscosity. At temperatures below Tr 0.7, liquid viscosities are very...

## Aqueous Solutions Of Electrolytes

Physical chemists have given much attention to aqueous mixtures containing solutes that ionize either completely e.g., strong salts like sodium chloride or partially e.g., sulfur dioxide and acetic acid . The thermodynamics of such mixtures is discussed in numerous references, but the most useful general discussions are by Pitzer, 1995 and by Robinson and Stokes, 1959 . Helpful surveys are given by Pitzer, 1977, 1980 , Rafal, 1994 , and Zemaitis, 1986 . Unfortunately, however, these discussions...

## BWR and MBWR Models

The BWR expressions are based on the pioneering work of Benedict, Webb and Rubin 1940, 1942 who combined polynomials in temperature with power series and exponentials of density into an eight-parameter form. Additional terms and parameters were later introduced by others to formulate modified Benedict-Webb-Rubin MBWR EoS. The general form of BWR MBWR correlations is the T-dependent functions f T can contain more than 30 parameters in addition to m, n, a, and y. Until very recently, this...

## Heat Capacities Of Real Gases

The heat capacities of ideal gases have been discussed in Chap. 3. In this section we discuss the behavior of real gases, including methods for estimating their behavior using equations of state. The focus is on the constant pressure heat capacity, Cp dH dT p, since it is normally obtained from calorimetric measurements and tabulated. The constant volume heat capacity, Cv dU dT V, is also of interest and the two are related by the thermodynamic equation Equation 6-5.1 shows that the difference...

## Solubility of Gases

At modest pressures, most gases are only sparingly soluble in typical liquids. For example, at 25 C and a partial pressure of 1.01 bar, the mole fraction solubility of nitrogen in cyclohexane is x 7.6 X 10 4 and that in water is x 0.18 X 10 4. Although there are some exceptions notably, hydrogen , the solubility of a gas in typical solvents usually falls with rising temperature. However, at higher temperatures, approaching the critical temperature of the solvent, the solubility of a gas usually...

## Chapter Eight

It has three advantages 1 it has only two rather than three adjustable parameters, 2 UNIQUAC's parameters often have a smaller dependence on temperature, and 3 because the primary concentration variable is a surface fraction rather than mole fraction , UNIQUAC is applicable to solutions containing small or large molecules, including polymers. Simplifications One-parameter Equations It frequently happens that experimental data for a given binary mixture are so fragmentary that it is not possible...

## Dipole Moments

Dipole moments of molecules are often required in property correlations for polar materials such as for virial coefficients Chap. 4 and viscosities Chap. 9 . The best sources of this constant are the compilations by McClellan 1963 and Nelson, et al. 1967 . There also is a large number of values in the compilation of Lide 1996 . For the rare occasions when an estimated value is needed, vector group contribution methods such as summarized by Minkin, et al. 1970 can be used. In addition, all of...

## Excess Free Energy GE Mixing Rules

As discussed above, to treat more complex solutions, mixing rules based on expressions more appropriate for the composition dependence of liquids have been developed, especially for phase equilibrium calculations. The first widely recognized analysis of this approach was by Huron and Vidal 1979 . Since that time, a very large literature has arisen with many different expressions reviews have been given, for example, by Orbey 1994 , Orbey and Sandler 1995 , Kalospiros, et al. 1995 , and...

## Phase Envelope Construction Dew and Bubble Point Calculations

Phase-equilibrium calculations at low pressures are normally easy, but high-pressure calculations can be complicated by both trivial-root and convergence difficulties. Trivial-root problems can be avoided by starting computations at a low pressure and marching toward the critical point in small increments of temperature or pressure. When the initial guess of each calculation is the result of a previous calculation, trivial roots are avoided. Convergence difficulties are avoided if one does dew...

## Viscosity

Where k is Boltzmann's constant and e is the minimum of the pair-potential energy. The working equation for y must have as many parameters as were used to define the original r relation. While many potential models have been proposed Hirschfelder, et al., 1954 , the Lennard-Jones 12-6 was the first and has most often been applied for ideal gas viscosity. In Eq. 9-4.2 , a is like a molecular diameter and is the value of r that causes r to be zero. With this potential, the collision integral has...

## Thermodynamic Properties Of Pure Components And Mixtures 619

In a molecule contribute a definite value to the total molar heat capacity that is independent of other groups present. Such methods include those of Chueh and Swanson 1973ab and Missenard 1965 which are described in the 4th Edition of this book and the Chueh and Swanson method is also described in Perry's Handbook Perry and Green, 1997 . More recently, methods have been presented that account for differing contributions depending on what a particular atom is bonded to in the same way as...

## B 2 nb d 2 nd

Where nt is the number of groups of type i, k is the total number of different kinds of groups, and the parameters ai, bi, and di are listed in Table 6-5 for 114 different groups and Table 6-6 for 36 different ring strain corrections rsc . Twenty-one more groups can be accommodated by the method with the group equivalency table, Table 6-7. See Chap. 3 or Ruzicka and Domalski, 1993 for discussion and examples of group assignments in a molecule. The nomenclature in Ruzicka and Domalski has been...

## Wagner Models

Setzmann and Wagner 1989, 1991 describe a computer-intensive optimization strategy for establishing highly accurate EoS models by a formulation for the residual Helmholtz energy, where A T, V is the ideal gas Helmholtz energy at T and V. See Eq. 3-1.14 . The model expressions contain large numbers of parameters whose values are ob- Z 1 T V 2 V2 T Vn 4 T a y V2 Vm exp -y V2 tained by regression on data for many properties over wide ranges of conditions. Recently, the trend is that empirical EoS...

## Thermodynamic Properties Of Pure Components And Mixtures 615

For hydrocarbons and hydrocarbon gas mixtures including light gases such as N2, CO2, and H2S calculate Hig - H from the Soave, Peng-Robinson, or Lee-Kesler equation. Errors should be less than 4 J g-1 Dillard, et al., 1968 Peng and Robinson, 1976 Tarakad and Daubert, 1974 West and Erbar, 1973 . The truncated virial equations, Eq. 4-5.1 are simple but can be used only for the vapor phase. Temperatures and pressures for which this condition applies are given in Table 4-4 and generally in the...

## B

Log10 A t C - 273.15 7-3J where T is in kelvins. When C 0, Eq. 7-3.1 reverts to the Clapeyron equation 7-2.3 . Simple rules have been proposed Fishtine, 1963 Thompson, 1959 to relate C to the normal boiling point for certain classes of materials, but these rules are not reliable and the only reliable way to obtain values of the constants A, B, and C is to regress experimental data. Values of A, B, and C are tabulated for a number of materials in Appendix A with Pvp in bars and T in K....

## Estimation Methods

Since the properties of most of the species treated in this chapter are for ideal gases, intermolecular forces play no role in their estimation and, as a result, the law of corresponding states is inapplicable. Rather, AH T , S T , AG f T , and Cp T are estimated from schemes related to the molecular structure of the compound. Benson 1968 and Benson and Buss 1969 have pointed out a hierarchy of such methods. The most simple methods would use contributions based on the atoms present in the...

## Riedel Correspondingstates Method

Riedel 1954 proposed a vapor pressure equation of the form ln Pvp A B C ln T DT6 7-5.1 The T6 term allows description of the inflection point of the vapor pressure curve in the high-pressure region. To determine the constants in Eq. 7-5.1 , Riedel defined a parameter a From a study of experimental vapor pressure data, Plank and Riedel 1948 showed that Using Eq. 7-5.3 as a constraint on 7-5.1 , Riedel found that ln P A - T C ln Tr D Tf 7-5.4 where A -35Q B -36 Q C 42 Q ac D -Q Q K 3.758 - ac...

## Method of Elbro et al 1991

Elbro, et al. 1991 have presented a group contribution method for the prediction of liquid densities as a function of temperature from the triple point to the normal boiling point. In addition to being applicable to simple organic compounds, the method can also be used for amorphous polymers from the glass transition temperature to the degradation temperature. The method should not be used for cy-cloalkanes. To use the method, the volume is calculated by where ni is the number of group i in the...

## J k

Where vkj is the number of interaction groups k in molecule j. Activity coefficient yf is given by ln yi 2 vk ln Tk - 2 vk ln T 8-10 . 57 where Tk activity coefficient of group k in the mixture of groups T activity coefficient of group k in the standard state This standard state depends on molecule i Activity coefficient Tk is given by Wilson's equation ln Tk -ln 2 XlAla l - 2 y A 8-10 . 58 where the summations extend over all groups present in the mixture FLUID PHASE EQUILIBRIA IN...

## J

Activity Coefficients in Electrolyte Solutions. I amp II. Boca Raton. Fla. CRC Press. 1979. Seidell. A. Solubilities of Inorganic and Organic Compounds, 3d ed Van Nostrand. New York. 1940 1941. 1952 . Linke. W. L. Solubilities of Inorganic and Organic Compounds, 4th ed Van Nostrand. Princeton. N.J 1958 and 1965. vols. 1 and 2. Silcock. H., ed. Solubilities of Inorganic and Organic Compounds, translated from Russian. Pergamon, Oxford. 1979. Stephen. H. and T. Stephen eds...

## Method Of Constantinou And Gani Cg

Choosing the same first and second order atomic and molecular groups as for the properties in Chap. 2, Constantinou and Gani 1994 obtained group contributions for AG 298.15 K and AH 298.15 K . Following this approach and the initial developments of Coniglio and Daridon 1997 for hydrocarbons, Nielsen 1998 developed correlations for polynomial coefficients to obtain C T for all classes of organic substances. The group values are shown in Appendix Tables C-2 and C-3 with sample assignments in...

## The Virial Coefficients Of Pure Gases And Mixtures Critical Compilation

Fluid Phase Equil., 141 1 1997 . Aalto, M., K. I. Keskinen, J. Aittamaa, and S. Liukkonen Fluid Phase Equil., 114 1 1996 . Abbott, M. M. Adv. in Chem. Ser., 182 47 1979 . Abbott, M. M., and J. M. Prausnitz Fluid Phase Equil., 37 29 1987 . Akinfiev, N. N. Geochem. Internat., 35 188 1997 . Aly, G. and I. Ashour Fluid Phase Equil., 101 137 1994 . Anderko, A Fluid Phase Equil., 61 145 1990 . Anderko, A Fluid Phase Equil., 65 89 1991 . Anderko, A, I. G. Economou, and M. D. Donohue IEC...

## Additive Methods

Schroeder Partington, 1949 suggested a simple additive method for estimating molar volumes at the normal boiling point. His rule is to count the number of atoms of carbon, hydrogen, oxygen, and nitrogen, add one for each double bond, two for each triple bond and multiply the sum by seven. Schroeder's original rule has been expanded to include halogens, sulfur, and triple bonds. This gives the volume in cubic centimeters per gram mole. This rule is surprisingly accurate, giving results within 3...

## T

Properties in the method of Constantinou and Gani, Eq. 2-2.4 functions in Pitzer's correlation for Pvp, Eqs. 2-3.3 and 7-4.2 to 7-4.4 molecular weight of substance Avogadro's Number, 6.022142 X 1023 molecules mor1 number of atoms in substance number of groups of type k in a molecule Nk for First-Order groups in all methods and Mj for Second-Order groups in Constantinou Gani, Eq. 2-2.4 , and Wilson Jasperson methods, Eqs. 2-2.8 and 2-2.9b number of rings in substance, Eqs. 2-2.8 and 2-2.9b...

## Tyn and Calus Method 1975

In this method, Vb is related to the critical volume by where both Vb and Vc are expressed in cubic centimeters per gram mole. Comparisons with the substances of Table 4-11 shows that this method is somewhat more accurate and has greater reliability since only 1 substance has an error of more than TABLE 4-10 Group Atom Contributions for Schroeder Eq. 4-10.1 and Le Bas Methods TABLE 4-10 Group Atom Contributions for Schroeder Eq. 4-10.1 and Le Bas Methods

## Z 8p

Z B i b B 2A 2B Z Z - A Z P 3Z2 - 2Z A - B - B 2 where a and bt are given in Tables 4-7 and 4-8. fv 0.48 1.574 w - 0.176 s2 We next describe dew-point and flash calculations. 8-12.7 8-12.8 8-12.9 8-12.10 Example 8-20 Use the Soave 1972 equation to calculate the dew-point temperature at 40 bar for a 26.54 ethane-73.46 heptane mixture. solution By using the pure-component properties in Example 8-19, dew points may be calculated at 5-bar intervals from 5 to 35 bar and then at 1-bar intervals from...

## Method Of Joback

Choosing the same atomic and molecular groups as for the properties in Chap. 2, Joback 1984 1987 used the values given in Stull, et al. 1956 1969 to obtain group contributions for AGf 298.15 K , AHf 298.15 K and polynomial coefficients for Cp T . His group values are shown in Appendix Table C-1, and they are to be used in Eqs. 3-3.1 to 3-3.3 . Distinctions are made among nonring, nonaromatic ring and aromatic ring groups as well as different atoms and bonding. AGf 298.15 K 53.88 NkAgfk kJ mol1...

## Vapor Pressures And Enthalpies Of Vaporization Of Pure Fluids 733

Lichtenthaler, and E. G. de Azevedo Molecular Thermodynamics of Fluid-Phase Equilibria, 3d ed. Prentice Hall, Englewood Cliffs, New Jersey, 1999, pp. 638641. Riedel, L. Chem. Ing. Tech., 26 679 1954 . RUZiCka, K., and V. Majer J. Phys. Chem. Ref. Data, 23 1 1994 . Schreiber, D. R., and K. S. Pitzer Fluid Phase Equil., 46 113 1989 . Smith, G., J. Winnick, D. S. Abrams, and J. M. Prausnitz Can. J. Chem. Eng., 54 337 1976 . Smith, J. M., H. C. Van Ness, and M. M. Abbott...

## Hv At The Normal Boiling Point

A pure-component constant that is occasionally used in property correlations is the enthalpy of vaporization at the normal boiling point AHvb. Any one of the correlations discussed in Sec. 7-8 or 7-9 can be used for this state where T _ Tb, P _ 1.01325 bar. Some additional techniques are discussed below. Several special estimation methods are also suggested. In Table 7-4, we show equations for , _ AHv RTc AZv as determined from a few of the more accurate vapor-pressure equations. Each can be...

## Thermodynamic Properties Of Pure Components And Mixtures 617

FIGURE 6-2 Reduced density variations at constant temperature of cv and cp for propane from the EoS of Younglove, et al. 1987 as calculated by Konttorp 1998 . FIGURE 6-2 Reduced density variations at constant temperature of cv and cp for propane from the EoS of Younglove, et al. 1987 as calculated by Konttorp 1998 .

## Determination Of Activity Coefficients

As discussed in Secs. 8-5 and 8-6, activity coefficients in binary liquid mixtures can often be estimated from a few experimental vapor-liquid equilibrium data for the mixtures by using some empirical or semiempirical excess function, as shown in Table 8-3. The excess functions provide a thermodynamically consistent method for interpolating and extrapolating limited binary experimental mixture data and for extending binary data to multicomponent mixtures. Frequently, however, few or no mixture...

## Fugacity Of A Pure Liquid

To calculate the fugacity of a pure liquid at a specified temperature and pressure, we may use an equation of state capable of representing the liquid phase and first calculate L See Chap. 6 and then use Eq. 8-2.2b . Alternatively, we may use the two primary thermodynamic properties the saturation vapor pressure, which depends only on temperature, and the liquid density, which depends primarily on temperature and to a lesser extent on pressure. Unless the pressure is very large, it is the vapor...

## Chemical Theory EoS

In many practical systems, the interactions between the molecules are quite strong due to charge-transfer and hydrogen bonding see Sec. 4-7 for a brief introduction and Prausnitz, et al., 1999 for a fuller description of the origin and characteristics of these interactions . This occurs in mixtures if alcohols, carboxylic acids, water, HF, etc. are present. It can lead to quite complex behavior of vapors of these PRESSURE-VOLUME-TEMPERATURE RELATIONSHIPS OF MIXTURES 5.21 substances though not...

## Rt

A TT ru 12 ln i xi 12 12 ln y2 12 8-5.8 Upon differentiating Eq. 8-5.5 as indicated by Eqs. 8-5.3 and 8-5.4 , we find FLUID PHASE EQUILIBRIA IN MULTICOMPONENT SYSTEMS 8.15 With these relations we can now calculate activity coefficients y1 and y2 at any desired x even though experimental data were obtained only at one point, namely, This simplified example illustrates how the concept of excess function, coupled with the Gibbs-Duhem equation, can be used to interpolate or extrapolate experimental...

## Discussion And Recommendations For Enthalpy Of Vaporization

If experimental values, or constants that correlate experimental values for a particular fluid are available in Majer, et al., 1989 for example, use those. Otherwise, values of AHv may be estimated by one of the three techniques described above. The first is based on Eq. 7-2.1 and requires finding dPvpldTfrom a vapor-pressure-temperature correlation Sec. 7-8 . A separate estimate of AZv must be made before AHv can be obtained. This procedure can give results as accurate as those measured...

## Vapor Pressures And Enthalpies Of Vaporization Of Pure Fluids

Wagner 1973, 1977 used an elaborate statistical method to develop an equation for representing the vapor pressure behavior of nitrogen and argon over the entire temperature range for which experimental data were available. In this method, the actual terms as well as their coefficients were variables i.e., a superfluity of terms was available and the most significant ones were chosen according to statistical criteria. The resulting equation is ln Pvpr ar br15 cr3 dr6 Tr 7-3.2 Pvpr is the reduced...

## Thermodynamic Properties Of Pure Components And Mixtures 633

Hearing J. Phys. Chem. Ref. Data, 22 805 1993 . Domalski, E. S., and E. D. Hearing J. Phys. Chem. Ref. Data, 25 1 1996 . Djordjevic, B. D., M. LJ. Kijevcanin, S. P. Serbanovic Fluid Phase Equil., 155 205 1999 . Fuller, G. G. Ind. Eng. Chem. Fundam., 15 254 1976 . Gambill, W. R. Chem Eng., 64 5 263, 64 6 243, 64 7 263 64 8 257 1957 . Gmehling, J., D. D. Liu, and J. M. Prausnitz Chem. Eng. Sci., 34 951 1979 . Gregorowicz, J., J. P. O'Connell, and C. J. Peters Fluid...

## S

Presents experimental data for vapor pressure, critical properties, enthalpies M of formation and phase equilibria. A critical compilation of data for virial coefficients of pure gases and binary mixtures published through 1979. A comprehensive compilation of azeotropic compositions, temperatures and pressures. Hao, W., H. Elbro, and P. Alessi Polymer Solution Data Collection, 3 vols., DECHEMA. Frankfurt. Germany. 1992. Hicks. C. P K. N. Marsh. A. G. Williamson. I. A. McLure. and C. L. Young...

## True Critical Points Of Mixtures

In Chap. 5, emphasis was placed upon the estimation of pseudocritical constants for mixtures. Such constants are necessary if one is to use most corresponding-states correlations to estimate mixture PVT y or derived properties. However, these pseudocritical constants often differ considerably from the true critical points for mixtures. Estimation techniques for the latter can be evaluated by comparison with experimental data. A summary of experimental values is given by Hicks and Young 1975...

## Enthalpy Of Fusion

The enthalpy change on fusion or melting is commonly referred to as the latent heat of fusion. References for literature values are listed in Tamir, et al. 1983 . Domalski and Hearing 1996 is an extensive tabulation of experimental values. The enthalpy of fusion is related to the entropy change on fusion and the melting point temperature by Reliable methods to estimate the enthalpy or entropy of fusion at the melting point have not been developed. However, as first suggested by Bondi 1963 ,...

## Corresponding States Principle

Equations 4-2.2a and 4-2.2b , Figs. 4-1 to 4-3, and Equations of State are all formulations of a general principle of dimensionless functions and dimensionless variables called the corresponding states principle CSP or sometimes the Law of Corresponding States.'' It asserts that suitably dimensionless properties of all substances will follow universal variations of suitably dimensionless variables of state and other dimensionless quantities. Its general and specific forms can be derived from...

## Ambrosewalton Correspondingstates Method

Equation 7-2.4 is a two-parameter corresponding-states equation for vapor pressure. To improve accuracy, several investigators have proposed three-parameter forms. The Pitzer expansion is one of the more successful ln Pvpr 0 2 2 7-4.1 Although a number of analytical expressions have been suggested for 0 , 1 , and 2 Brandani, 1993 Schreiber and Pitzer, 1989 Twu, et al., 1994 we recommend the following developed by Ambrose and Walton 1989 . 5.97616t 1.29874t15 0.60394t25 1.06841t5 5.97616t...

## Vapor Pressures And Enthalpies Of Vaporization Of Pure Fluids 713

Parameters are applicable, it is very accurate. Of all the methods shown in Table 7-3, the Wagner equation has the largest number of correlated parameters and is the most accurate. Both the Ambrose-Walton and Riedel estimation methods perform well at higher temperatures. At low temperatures the Ambrose-Walton method does better for the hydrocarbon, -tetradecane, while the Riedel method does better for the alcohol, 1-octanol. The estimation methods presented in this chapter require the critical...

## R

Two methods for estimating liquid heat capacities have been described. The Ruz-icka-Domalski method, Eq. 6-6.2 , is a group contribution method that is applicable between the melting and boiling points. At higher temperatures, it generally underpredicts the heat capacity. Figure 6-3 illustrates this for propylene. Equation 6-6.4 is a CSP correlation that works well for all compounds except those that associate. Equation 6-6.4 requires Tc, c, and C . Table 6-8 illustrates the capabilities and...

## Vapor Pressures And Enthalpies Of Vaporization Of Pure Fluids 731

Thermodynamics, 18 45 1986 . Ambrose, D., and R. H. Davies J. Chem. Thermodynamics, 12 871 1980 . Ambrose, D., and N. B. Ghiassee J. Chem. Thermodynamics, 19 505 1987 . Ambrose, D., and N. B. Ghiassee J. Chem. Thermodynamics, 19 903 1987a . Ambrose, D., and N. B. Ghiassee J. Chem. Thermodynamics, 19 911 1987b . Ambrose, D., and N. B. Ghiassee J. Chem. Thermodynamics, 20 765 1988 . Ambrose, D., and N. B. Ghiassee J. Chem. Thermodynamics, 20 1231 1988a . Ambrose, D., and N....

## Pure Component Constants

Joback's values for contributions to the critical properties are in Table C-1. Example 2-10 shows the method. Example 2-10 Estimate Tfp and Tb for 2,4-dimethylphenol by using Joback's group method. solution 2,4-dimethylphenol contains two CH3, three CH ds , three C ds , and one OH phenol . From Appendix Table C-1 solution 2,4-dimethylphenol contains two CH3, three CH ds , three C ds , and one OH phenol . From Appendix Table C-1

## Compressibility Charts

PVT RELATIONSHIPS OF PURE GASES AND LIQUIDS 4.3 PVT RELATIONSHIPS OF PURE GASES AND LIQUIDS 4.3 FIGURE 4.1 Generalized compressibility chart at low Pr. Vr V RTc Pc . Nelson and Obert, 1954 ' FIGURE 4.1 Generalized compressibility chart at low Pr. Vr V RTc Pc . Nelson and Obert, 1954 ' FIGURE 4.2 Generalized compressibility chart at moderate Pr. Vr V RTc Pc . Nelson and Obert, 1954 . ' FIGURE 4.2 Generalized compressibility chart at moderate Pr. Vr V RTc Pc ....

## Acentric Factor

Along with the critical properties of Sec. 2-2, a commonly used pure component constant for property estimation is the acentric factor which was originally defined by Pitzer, et al. 1955 as The particular definition of Eq. 2-3.1 arose because the monatomic gases Ar, Kr, Xe have to 0 and except for quantum gases H2, He, Ne and a few others e.g., Rn , all other species have positive values up to 1.5. To obtain values of t from its definition, one must know the constants Tc, Pc, and the property...

## Vaporliquid Critical Properties

Vapor-liquid critical temperature, Tc, pressure, Pc, and volume, Vc, are the pure-component constants of greatest interest. They are used in many corresponding states correlations for volumetric Chap. 4 , thermodynamic Chaps. 5-8 , and transport Chaps. 9 to 11 properties of gases and liquids. Experimental determination of their values can be challenging Ambrose and Young, 1995 , especially for larger components that can chemically degrade at their very high critical tem- peratures Teja and...

## The Estimation Of Physical Properties

In this book, the various estimation methods are correlations of experimental data. The best are based on theory, with empirical corrections for the theory's defects. Others, including those stemming from the law of corresponding states, are based on generalizations that are partly empirical but nevertheless have application to a remarkably wide range of properties. Totally empirical correlations are useful only when applied to situations very similar to those used to establish the...