Internally Consistent Prediction of Vapor Pressure and Related

Jan 29, 2000 - Rose, Arthur; Supina, Walter R. Journal of .... Pitzer, K. S.; Lippmann, D. Z.; Curl, R. F., Jr.; Huggins, C. M.; Petersen, D. E. The V...
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Ind. Eng. Chem. Res. 2000, 39, 813-820

813

Internally Consistent Prediction of Vapor Pressure and Related Properties Hans Korsten† Mobil Technology Company, 600 Billingsport Road, Paulsboro, New Jersey 08066

A new vapor pressure model containing one parameter is presented. The vapor pressure curves of all hydrocarbons that may contain heteroatoms merge together at a common point. Thus, the vapor pressure curve can be predicted using one known data point. If no data are available, the characteristic parameter can be obtained using structural information about the component. Since vapor pressure curves end in their critical points, the equation allows internally consistent prediction of critical pressure from known critical temperature, or vice versa, and for predicting the acentric factor. For calculating the heat of vaporization at any temperature using the Clapeyron equation, knowledge of the nonideal behavior of a component along the vapor pressure curve is needed. The equations presented for this purpose show excellent agreement for all components for which experimental data have been found. Introduction Many vapor pressure equations have been published. Some of them, the multiparameter methods, show a high degree of accuracy if applied to pure components. But all these equations require the knowledge of too many data for parameter estimation as to be applied to heavy hydrocarbon components and their mixtures. Methods with fewer parameters show sufficient accuracy only within relatively small ranges. Group contribution methods are generally useful for predicting properties of components that are similar to those used to develop the method. Application to components far away from these standard data can yield large errors. For predicting heats of vaporization, different strategies have been proposed. One strategy uses empirical correlations for calculating the heat of vaporization at a specified temperature, usually at the normal boiling point. Another procedure utilizes the Clapeyron equation in combination with a vapor pressure correlation. The effect of temperature on heat of vaporization is commonly depicted by the equation proposed by Watson.1 Finally, corresponding states methods can be employed. For application of corresponding states methods, the knowledge of critical data and the acentric factor is required. The latter property represents a characteristic location on the vapor pressure curve. According to the investigation of Majer et al.,2 the expected errors of the different strategies for calculating heats of vaporization are very similar. However, for certain types of components huge errors can be expected. A new method for predicting vapor pressure curves is presented, which contains only one parameter for each chemical species. Demonstrated by using data of several types of hydrocarbons, fuel ethers, methyl ethers of fatty acids, and S, N, CL, F-containing hydrocarbons, all these vapor pressure curves end in one single common point. The vapor pressure curves of polar components such as alcohols and water have a common point, too, but it is different from that of the other components. The vapor pressure parameter can be † Phone: (856)224-2766. Fax: [email protected].

(856)224-3843. E-mail:

determined from any known point on the vapor pressure curve. Alternatively, it is possible to predict the parameter because it is a function of structural characteristics of the components. Knowing the vapor pressure parameter together with critical properties enables us to calculate the volume change during vaporization as a measure of the nonideal behavior. With this information, the heat of vaporization along the temperature scale can be predicted by using the Clapeyron equation. The Vapor Pressure Vapor Pressure Equations. Most of the proposed vapor pressure equations find their origin in the Clapeyron equation

∆V H d(ln p) )1 R(∆VZ) d T

()

(1)

where p is the pressure, T is the temperature, and R represents the gas constant. ∆VH and ∆VZ are changes in enthalpy and compressibility factor associated with vaporization. Integration of the Clapeyron equation assuming a constant ratio ∆VH/∆VZ leads to the simplest vapor pressure equation that shows reasonable accuracy within a limited low-pressure range:

ln p ) A +

B T

(2)

In eq 2, the parameter B equals the right-hand side of eq 1, and A is an integration constant. A and B have to be determined from experimental vapor pressure data. To improve the quality of vapor pressure representations, several modifications of eq 2 have been proposed. Cox3 expanded the equation by adding two additional terms containing the parameters C and D that represent an empirical correction for deviations from the ideal gas law:

ln p ) A +

B + CT + DT2 T

10.1021/ie990579d CCC: $19.00 © 2000 American Chemical Society Published on Web 01/29/2000

(3)

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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000

On the basis of the few data that were available in 1923, Cox found that the vapor pressure curves can be represented by straight lines that seem to converge to a common point if the temperature axis is plotted as a proper function of T. For most of the components investigated only the normal boiling point was available, and the heaviest molecule was the C16 n-paraffin. By introducing a third parameter into the denominator of eq 2, the range of reliable representation of vapor pressure curves at low pressure can be improved. This equation is known as the Antoine equation:

ln p ) A +

B C+T

(4)

Zwolinski4

presented the three coefKudchadker and ficients A, B, and C of the Antoine equation for calculating vapor pressures of the n-paraffins C21-C100. The parameters of about 700 pure components of different structures are given by Yaws and Yang.5 In general, within a homologous series the parameters A and B increase with increasing carbon number, whereas the parameter C decreases. However, a reliable procedure for predicting these parameters is not available, so that for each component a set of vapor pressure data is required for determining best parameters. Maxwell and Bonnell6 assumed similar vapor pressure curve structures for all components, and n-hexane with its accurately known vapor pressure curve was used as a reference. The vapor pressure curve of another chemical species can be predicted by using this reference curve and additional two coefficients that are characteristics of each component. For hydrocarbons other than paraffins, Maxwell and Bonnell proposed a correction using the Watson characterization factor, which is calculated using the normal boiling point and the specific gravity at 60 °F. Today we know that the Watson characterization factor provides only a poor characterization. For instance, paraffins are characterized by values in the range of approximately 12-13.5. Following the procedure suggested by Maxwell and Bonnell, paraffins with Watson factors other than 12 would require corrections of several tens of kelvins. High accuracy can be expected if the triple point and the critical point are included in the vapor pressure correlation. This has been demonstrated by IglesiasSilva et al.7 for some light hydrocarbons and other simple fluids. The application of a model containing triple points is limited by the fact that triple points are not known for all the components that might be of interest, especially in the hydrocarbon industry. To increase the accuracy of representing vapor pressure data, more and more parameters have been added, and several combinations of temperature exponents have been proposed in polynomial type equations. An equation of this type was published by Wagner:8

Aτ + Bτ1.5 + Cτ3 + Dτ6 ln pr ) Tr

(5)

τ ) 1 - Tr

(6)

where

Tr and pr are the reduced temperature and pressure, respectively. In contrast to components with known vapor pressure behavior, there are usually not sufficient data of more complex, high-boiling hydrocarbons avail-

able for parameter estimation. Willman and Teja9 proposed a set of empirical, polynomial type equations for predicting the four coefficients of a modified Wagner equation as well as the required critical temperature and pressure as function of an effective carbon number. The empirical nature of the proposed equations contains the risk of poor prediction if applied to components that are not reasonably similar to the data used for developing the correlations. Group Contribution Methods. Abrams et al.10 developed the following vapor pressure correlation based on the kinetic theory of fluids

ln p ) A +

B + C ln T + DT + ET2 T

(7)

in which the five parameters, A-E, can be expressed as functions of the number of equivalent oscillations per molecule, the enthalpy of vaporization of the hypothetical liquid at T ) 0 K, and the hard-core van der Waals volume. For obtaining the latter value, the authors recommend the group contribution method by Bondi.11 The former two parameters can be determined from known vapor pressure data, or alternatively they can be predicted by the group contribution method published by Macknick and Prausnitz.12 Additional group contribution values of this model are given for sulfur- and nitrogen-containing components (Edwards and Prausnitz13), for five- and six-membered hydrocarbon rings (Ruzˇicˇka14), and for halogenated aromatic hydrocarbons (Burkhard15). According to Abrams et al.,10 eq 7 (with A-E) is not suitable for applications close to the critical point. As shown below, the application of the vapor pressure equation at the critical point is very important in order to obtain internal consistency between the thermodynamic properties. A New Vapor Pressure Equation. From the Clapeyron equation, eq 1, it has generally been concluded that ln p would be a function of (1/T). This is not the right functionality because heat and compressibility of vaporization are functions of temperature, too. If ln p is plotted against 1/T1.30, the vapor pressure curves of all chemical components appear as straight lines. This is shown in Figure 1a,b for the homologous series of n-paraffins between C4 and C41 and n-alkylbenzenes between C6 and C49; the experimental data have been taken from the TRC Thermodynamic Tables.16 Similar figures can be drawn for the homologous series of n-alkylcyclopentanes, n-alkylcyclohexanes, monoolefins, alkanethiols, and chloroalkanes, for which many vapor pressure data are reported in the literature. Equation 8 shows the new vapor pressure equation:

ln p ) A +

B T1.30

(8)

The two parameters, A and B, can be obtained from two known vapor pressure data. As is obvious from Figure 1, the vapor pressure curves of all the components merge together at a common point. This information is useful for eliminating one parameter, and the final equation is

ln p ) ln pR + B

(

)

1 1 T1.30 T1.30 R

(9)

The single parameter B is a characteristic of each component, and the common point, depicted by the

Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 815

Figure 2. Vapor pressure curves of miscellaneous components.

Figure 1. Vapor pressure curves of hydrocarbons: (a, top) n-paraffins; (b, bottom) n-alkylbenzenes.

index R, is (TR, pR) ) (1994.49 K, 1867.68 bar) for all the hydrocarbons and most other chemical species investigated. This common point has been determined by applying a Marquardt optimization routine to the vapor pressure data. The temperature TR represents the upper boiling temperature limit, which cannot be exceeded by any component at any pressure. At pressures far below 0.05 bar, the deviations between experimental vapor pressure data and the new method seem to increase. Besides a possible weakness of the model in this region, it should be considered that the measurement of vapor pressures at low pressures usually contains errors of far more than 50%. Therefore, low-pressure data presented in the literature are often the result of smooth extrapolation from equations that were thought to be appropriate for extrapolation. For instance, Mosselman et al.17 obtain a vapor pressure correlation from exact integration of the Clapeyron equation. The required information are the virial coefficients, heat capacities of vapor and liquid as function of temperature, and the temperature-dependent liquid volume. The ethanol vapor pressure curve from the critical point down to 10-7 bar calculated by Mosselman et al.’s method is substantially identical with that calculated by eq 8 utilizing the normal boiling point and the critical point only. At low pressures, an increasing deviation by 1 order of magnitude from experimental data was reported, whereas experimental data from another source seemed to be in good agreement with the prediction. As is obvious from Figure 1, the TRC data are smoothed data, i.e., they were interpolated between the original experimental data in order to report the data of all components at the same temperature or pressure in tabular format. Since different multiparameter vapor pressure equations were used in the low- and the high-pressure region, the errors can

be neglected. In Figure 2, vapor pressure curves of three polynuclear aromatic hydrocarbons are shown. The source of the experimental data was the compilation by Egloff,18 who collected vapor pressure data from many different research articles. These nonsmoothed data indicate that the predictions by the new vapor pressure correlation lay within experimental errors. In the oil-refining business we are mainly dealing with hydrocarbons of any complexity, which may contain sulfur and nitrogen. Vapor pressure curves of those components can be predicted by eq 9 using the same common point. Figure 2 shows the vapor pressure curves of quinoline (experimental data from Steele et al.19) and dibenzothiophene (experimental data from Sivaraman and Kobayashi,20 as examples of aromatic nitrogen- and sulfur-containing hydrocarbons, respectively. The vapor pressure curves of most oxygen components, excluding those containing OH groups, can also be predicted by eq 9. This is demonstrated for the fuel ethers MTBE, ETBE, and TAME (experimental data from Kra¨henbu¨hl and Gmehling21 and Semar et al.22), and for two methyl esters of fatty acids (experimental data from Rose and Supina23). The vapor pressure curves of other oxygen components such as alcohols and water can be described by eq 9 as well, but they merge together at another common point. It has been observed that the vapor pressure curve of propane shows an inflection point at a reduced temperature of about 0.8 (Reid et al.24), which would demand that the second derivative of ln p as function of 1/T should change signs. None of the components that are present in liquid petroleum fractions show any significant curvature associated with an inflection point (cf. Figures 1 and 2). Clearly, consistency of the single parameter within entire homologous series is more important than a small improvement in accuracy that could be obtained by trying to describe a slight S-shape in the high-pressure region. The Volume Change during Vaporization In the following section an equation is presented for calculating the heat of vaporization as a function of temperature. This correlation has been derived from the Clapeyron equation requiring information about the difference between compressibility factors ∆VZ or molar volumes ∆V of saturated vapor and liquid at saturation conditions. For this purpose, the molar volume of the vapor can be calculated using an equation of state, and the Rackett equation (Reid et al.24) provides us with the

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equation (Reid et al.24)

(

∆VHb ) 1.093RTc Tbr

)

ln pc - 1.013 0.930 - Tbr

(12)

where R, Tc, and pc are in units of J/(mol‚K), K, and bar, respectively. The Riedel equation represents a modification of the Giacalone equation (Reid et al.24)

(

∆VHb ) RTc(∆VZb) Tbr

)

ln(pc/pb) 1 - Tbr

(13)

which is the result of inserting the derivative

d(ln pbr) B ) 1 Tc d Tbr

( )

(14)

of the Clapeyron equation solved for the critical point and for the boiling point with

ln pbr B ) Tbr Tc 1 - Tbr

Figure 3. Vaporization volume change of hydrocarbons: (a, top) n-paraffins; (b, bottom) n-alkylbenzenes.

liquid volume, for instance. Alternatively, the following simple equation can be used to predict the change of compressibility factors associated with vaporization

(

T0.30 1 - Tr r ∆VZ(T) ) 0.87 0.46 Tbr 1 - Tbr

)

0.38

(10)

where Tbr is the reduced normal boiling temperature. Equation 10 fulfills the constraint ∆VZ ) 0 at the critical point, and it reflects the observation that ∆VZb at the normal boiling point follows the expression

∆VZb ) ∆VZ(Tb) )

0.87 T0.16 br

(11)

The volume change upon vaporization can now be calculated by inserting eq 10 into the gas law. Figure 3a,b shows a comparison between experimentally obtained volume changes during vaporization of nparaffins and n-alkylbenzenes with those predicted. At the critical point, the volume changes are zero. The Heat of Vaporization The thermodynamic functionality of the heat of vaporization as function of temperature is given by the Clapeyron equation, eq 1. Most equations proposed are developed to predict the heat of vaporization at the normal boiling point. A compilation of experimental heats of vaporization at the normal boiling point is presented by Yaws et al.26 When no experimental value of a component was available, its value in units of J/mol was predicted by the Riedel

(15)

into the Clapeyron equation. The difference in the compressibilities of saturated vapor and liquid at the boiling point is usually set to ∆vZb ) 1. In methods utilizing a vapor pressure correlation, information about critical data and the boiling point is usually required because these data can be used to eliminate two parameters of the vapor pressure equation. If the data are not available, an empirical correlation might be used to obtain the heat of vaporization. Ibrahim and Kuloor27 presented the following equation for predicting the heat of vaporization at the normal boiling point as function of the molecular weight M:

∆ VH b )

C Mn

(16)

The parameters C and n are characteristics for each homologous series. In Figure 4, the heat of vaporization at the normal boiling point is shown for the homologous series of the n-paraffins, n-alkylbenzenes, and 1-alkylthiols. In all cases the deviation between experimental data and the values predicted by eqs 12, 13, and 16 increases with increasing molecular weight. As shown in Figure 4c, the heat of vaporization of all the 1-alkylthiols, as examples for sulfur-containing hydrocarbons, is poorly predicted by the Giacalone and the Riedel equation. The most widely known expression for predicting the effect of temperature on heat of vaporization is the equation proposed by Watson:1

∆VH(T) ) ∆VH(Tb)

(

)

Tc - T Tc - T b

0.38

(17)

New Generalized Equation. From the new vapor pressure equation, eq 8, follows

d(ln p) )B 1 d 1.30 T

( )

(18)

Inserting this derivative into the Clapeyron equation,

Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 817

Figure 5. Temperature effect on heat of vaporization.

Figure 4. Standard heat of vaporization: (a, top) n-paraffins (b, middle) n-alkylbenzenes; (c, bottom) 1-alkylthiols.

∆VH(T1)

eq 1, we obtain the relationship

∆VH(T) ) -1.30RB

∆VH(T2)

∆VZ(T)

(19)

T0.30

where the constant slope of the vapor pressure curve, B, and the temperature-dependent difference between gas and liquid compressibilities, ∆VZ(T), are readily available in eqs 9 and 10, respectively. Any two known pairs (T, p) of a vapor pressure curve can be used in order to calculate the parameter B from eq 9. If we combine eqs 1, 9, and 19 using the normal boiling point and the critical point, we obtain the following expression for the heat of vaporization at the normal boiling point:

(

∆VHb ) 1.131RTc T0.84 br

of vaporization at the normal boiling point calculated by eq 20 shows excellent agreement with experimental data, as shown by the solid lines in Figure 4. For solving eq 20 even for heavy components, the normal boiling temperature was predicted by the equations proposed by Korsten,28 and the critical pressure was obtained by the method by Korsten29 expanded in order to account for the effect of sulfur. Finally, Tc was predicted by inserting pc into eq 9, after B was obtained by applying eq 9 to the normal boiling point. Concerning the experimental data of the n-alkylbenzenes shown in Figure 4b, it should be mentioned that the first four data taken from the TRC Tables16 are reported to be revised data, whereas the data of the heavier molecules are not. Considering the good agreement between all the other experimental data and eq 20, as shown in Figure 4, it can be expected that higher experimental heats of vaporization of n-alkylbenzenes with carbon number greater than 9 will be found in the future. Combination of eqs 10 and 19 allows the calculation of heat of vaporization as a function of temperature. The experimental data shown in Figure 5 are taken from the compilations by Smith and Srivastava25 and the TRC Thermodynamic Tables.16 The data of 1-octene reported by Smith and Srivastava25 show a strange temperature behavior that does not appear for the other members of its homologous series. It is suspected that these data are faulty. The ratio of the heats of vaporization at two temperatures, each expressed by eq 19, gives:

)

ln(pc/1.01325) 1 - T1.30 br

(20)

In eq 20 the same units are used as in eq 12. The heat

)

()

∆VZ(T1) T2 ∆VZ(T2) T1

0.30

(21)

If we insert eq 10 for both temperatures T1 and T2, the Watson equation, eq 17, appears. The Acentric Factor The acentric factor is used as third parameter in cubic equations of state (EOS) such as the Soave-RedlichKwong30 and the Peng-Robinson31 EOS, or their modifications. To obtain satisfying results from any application of EOS to wide-boiling hydrocarbon mixtures, a sound method for extrapolating the acentric factor is needed. In this aspect, the acentric factor obtained from the vapor pressure curve by its defining equation is the best value to use in the EOS. The acentric factor is defined as (Pitzer et al.32)

ω ) -log pr(Tr)0.7) - 1

(22)

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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000

ω)

0.3(Tbr + C/Tc) log pc (1 - Tbr)(0.7 + C/Tc)

-1

(24)

For estimating the parameter C of the Antoine equation, Chen et al. proposed the following equation:

C/Tc ) 0.2803 - 0.5211Tbr

(25)

Lee and Kesler36 obtain the acentric factor from the reduced vapor pressure equation

ln pr ) f(0)(Tr) + ωf(1)(Tr)

(26)

with the functions Figure 6. Acentric factor of homologous series.

f (0) ) 5.92714 -

which can be derived from writing eq 2 in the reduced form. The acentric factor of simple fluids, such as hydrogen or methane, is zero. As shown in Figure 6, the acentric factor increases with the size of a molecule, and it represents a measure of the deviation from its intermolecular potential from that of a simple fluid. The heavier the components, the smaller the differences in their values. The decomposition of hydrocarbons at temperatures above 620 K is a further problem in applying the concept of acentric factors to heavy components because the required critical data of those components cannot be measured. Thus, in petroleum technology we have to rely on extrapolation techniques. In Figure 6, experimental data of the nparaffin C10H22 are shown. This is the heaviest nparaffin for which the acentric factor can be calculated on the basis of experimental data. All the lines shown in Figure 6 are predicted by the new method, as outlined below. Prediction of the Acentric Factor. As other properties such as critical data, the acentric factor of hydrocarbon mixtures is usually predicted as a function of other properties. Some of these correlations are fully empirical, whereas others take into account the type of component. For instance, the equation proposed by Riazi and Al-Sahhaf33 predicts acentric factors of four homologous series as a function of molecular weight. For each homologous series four parameters are needed. Edmister and Lee34 used the Clapeyron equation to derive an expression for predicting the acentric factor. Since the Clapeyron equation contains two parameters, two points on the vapor pressure curve must be specified. By using the critical point and the normal boiling point, Edmister and Lee derived the following equation

ω)-

(

)

3 log pbr -1 7 T-1 - 1 br

(23)

where Tbr is the reduced normal boiling temperature and pbr is the reduced, saturated vapor pressure at the normal boiling temperature Tb. The errors of values predicted by eq 23 tend to increase with increasing molecular weight of the components because with increasing molecular size the vapor pressure at which the acentric factor is defined decreases rapidly and is far below 1 atm. The Clapeyron equation is not capable of handling such a wide extrapolation. Chen et al.35 inserted the more accurate Antoine equation, eq 4, into eq 22 and derived the following equation for calculating the acentric factor:

6.09648 - 1.28862 ln Tr + Tr 0.16934T6r (27)

f (1) ) 15.2518 -

15.6875 - 13.4721 ln Tr + Tr 0.43577T6r (28)

This set of equations represents an improved version of the tables published by Pitzer et al.32 The acentric factor can be calculated by solving eqs 26-28 for the normal boiling point or another known point on the vapor pressure curve. In turn, if the critical properties and the acentric factor of a component are known, the vapor pressure curve can be calculated. A similar procedure was published by Riedel,37 who used a parameter R instead of the acentric factor ω as a third parameter. If we combine eqs 8 and 22 and insert the critical point and the normal boiling point, we obtain the following equation for predicting the acentric factor:

ω)

ln pbr 10 1.30 1-1 1 7 (ln 10) 1.30 - 1 Tbr

(

[ ( ) ]

)

(29)

By using the information about the common R-point of the vapor pressure curve, one of the two critical properties in eq 29 can be eliminated. This procedure may fail if the common point is different from the standard value. Then, the generally valid eq 29 should be preferred. The boiling point in eq 29 does not necessarily need to be the normal boiling point; any other data point on the vapor pressure curve can be used as well. Prediction of the Parameter B As shown in Figure 7, the characteristic slope B of the vapor pressure equation, eq 9, is a linear function of the molecular weight M0.65 within each homologous series. Since all these straight lines are parallel, the parameter B can be predicted as function of the structural parameter ΘB

B ) B0(ΘB) + B1M0.65

(30)

with the parameters B1 ) -1162.8 and B0 as

B0 ) 4669.09 - 321.42ΘB

(31)

The parameter ΘB contains information about the

Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 819

Figure 7. Characteristic vapor pressure parameter.

Figure 8. Prediction of the vapor pressure coefficient.

structural characteristics of the component. As shown in Figure 8, ΘB is a function of naphthenic and aromatic ring content, aromatic and olefinic double bonds, and heteroatom content. Thus, the development of a group contribution method for calculating this parameter would be a valuable topic for future research. With eq 30 the vapor pressure curve of components, for which no experimental data are available, can be predicted. For instance, the compilation by Boublik et al.38 contains data of C1-C16 chloroalkanes except for those with 5, 13, and 15 carbon atoms. Suppose we are interested in these three compounds, but we cannot find any vapor pressure data in the literature. The vapor pressure curves of these components can be predicted by interpolation using the data of other members of the same homologous series (cf. Figure 7). The vapor pressure curves of only the first two members of the alkylfluorides are available in Boublik et al.38 The missing data of the remaining species can be predicted by extrapolation in a similar way. Literature Cited (1) Watson, K. M. Thermodynamics of the Liquid State. Ind. Eng. Chem. 1943, 35, 398. (2) Majer, V.; Svoboda, V.; Pick, J. Heats of Vaporization of Fluids; Elsevier: Amsterdam, 1989. (3) Cox, E. R. Pressure-Temperature Chart for Hydrocarbon Vapors. Ind. Eng. Chem. 1923, 15, 592. (4) Kudchadker, A. P.; Zwolinski, B. J. Vapor Pressures and Boiling Points of Normal Alkanes, C21 to C100. J. Chem. Eng. Data 1966, 11, 253. (5) Yaws, C. L.; Yang, H.-C. To Estimate Vapor Pressure Easily. Hydrocarbon Proc. 1989, 68 (Oct.), 65. (6) Maxwell, J. B.; Bonnell, L. S. Derivation and Precision of a New Vapor Pressure Correlation for Petroleum Hydrocarbons. Ind. Eng. Chem. 1957, 49, 1187.

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(37) Riedel, L. A New Universal Vapor Pressure Equation. Chem. Ing. Tech. 1954, 26, 83. (38) Boublik, T.; Fried, S. V.; Hala, E. The Vapor Pressure of Pure Substances, 2nd ed.; Elsevier: Amsterdam, 1984.

Received for review August 3, 1999 Revised manuscript received November 5, 1999 Accepted December 10, 1999 IE990579D