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Znd. Eng. Chem. Res. 1991,30,1609-1612
Estimation of Vapor Pressure of Some Organic Compounds Dinesh S. Mishra Department of Pharmaceutical Sciences, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
Samuel H. Yalkowsky* Department of Pharmaceutical Sciences, University of Arizona, Tucson, Arizona 85721
An accurate and thermodynamically sound equation for the estimation of vapor pressure of organic solids and liquids is developed. This equation utilizes improved estimation schemes for the entropies of vaporization and melting over Trouton's rule and Walden's rule. It also utilizes improved estimations of the heat capacity changes associated with vaporization and melting. The equation, which has no adjustable parameters, is shown to successfully estimate the vapor pressures of a large number of organic compounds.
Introduction There are a large number of empirical relationships between vapor pressure and boiling point (for liquids) and melting point (for solids). These relationships are usually presented in the form of polynomials in various functions of temperature, T, containing between two and four terms. The coefficients of the various temperature functions are generally designated by empirically determined alphabetic characters. Since there is no way of predicting these values, they are invariably determined by regression analysis of temperature versus vapor pressure. Those equations that have only two coefficients are easier to use but they are also less accurate than the equations that utilize more terms. The Rankine-Kirchoff and Frost-Kalkwarf equations, which are in the form of an integration of the van't Hoff equation, are the most theoretically sound. The Frost-Kalkwarf equation differs from the Rankine-Kirchoff equation in that the former accounts for the nonideality of the gas. In practice this correction is small, and the two equations are nearly identical. The Antoine equation is the most commonly used. Antoine coefficients for many compounds are available in a number of handbooks. Mackay (1982) developed an equation in the form of the Rankine equation that accounts for the melting point of solids. Attempts to eatimate the coefficientsof the above equations by group contribution approaches have been largely unsuccessful except when applied to a very limited series of compounds. In this study we have developed a vapor pressure equation for which the coefficients are related to chemical structure and do not have to be determined by regression. This will enable the estimation of the vapor pressure for compounds for which it is not possible or practical to measure experimentally. It will facilitate the assessment of the exposure hazard, volatilization rate, and other parameters that are related to vapor. Theory Vapor Pressure Equation (Yalkowsky et al. 1990). Vapor pressure, P,can be calculated from the total enthalpy of vaporization or sublimation, AHm, and temperature, T, by the Clausius-Clapeyron equation: lnP= S
s
d
T
Table I. Thermodynamic Characterization Vaporization P" from to enthalpy change 1 solid at T solid at TM 2 solid at TM liquid at TM 3 liquid at TM liquid at T 4 liquid at T liquid at TB 5 liquid at TB gas at TB 6 gas at T gas at TB
can be described by a series of six well-defined, reversible processes as illustrated in Table I, where TM and TB are the melting point and boiling point of the substance in degrees kelvin. Since it is possible to characterize each of these reversible processes thermodynamically in terms of pressure as a function of temperature, it is possible to characterize the enthalpy of the whole process. The expressions for the enthalpy change for each step are given in Table I. It can be seen that the summation of the six resulting expressions for the total enthalpy, AHTOT, is AHTW
=AHM
+ A H B + (C,S - CpL)(TM- T ) + (CpL- Cpc)(T~ - T ) (2)
If the heat capacities of the phases are assumed to be independent of temperature, the heat capacity differences that accompany melting, ACm, and boiling, AC,, can be defined as (3)
AC,, = ' ,C
- CpL
(4)
Thus the total enthalpy of vaporization or sublimation becomes AHTOT
=
AHM + AHB+ ACm(T - TM) + AC,(T
- TB) (6)
Substituting eq 6 into the Clausius-Clapeyron equation gives S,'"d In P =
There ie currently no means available for the determination of A H v However, since enthalpy is a state function, the irreversible vaporization of a solid at temperature T
*Towhom correapondence should be addressed. 0888-5885/91/2630-1609$02.50/0 Ca 1991 American Chemical Society
1610 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 Table 11. Observed and Predicted V a m r Pressure Data for Some Owanic Communds" name ob8 propylcyclo-1.79 pentane tert-butylbenzene -2.54 2-ethyl-4.47 naphthalene naphthalene -.396 2-methyl-4.14 naphthalene biphenyl -4.88 acenaphthalene -5.23 fluorene -6.05 p-dichlorobenzene -3.05 1,2,3-trichloro-3.55 benzene 1,3,5-trichloro-3.11 benzene 1,2,3,4-tetrachloro- -4.28 benzene 1,2,3,5-tetrachloro- -4.01 benzene 1,2,4,5-tetrachloro- -5.14 benzene p-dibromo-3.67 benzene 1,4-bromochloro- -3.46 benzene octane -1.73 3-methylheptane -1.59 2,3,4-trimethyl-1.44 pentane -2.24 nonane 2,2,5-trimethyl-1.66 hexane decane -2.76 4-methyloctane -2.05 undecane -3.28 dodecane -3.80 hexadecane -5.04 l-octene -1.64 methylcyclo-1.21 hexane ethylcyclopentane -1.27 ethylcyclohexane -1.77 cis-l,2-dimethyl- -1.72 cyclohexane Error 1 and error 2 are posed equation.
log p eq 12 eq 16 -1.13 -1.21
log P error1 error2 name obs eq 12 eq 16 (es 12) (eq 16) trarr-l,4-di-1.52 -1.37 -1.49 0.14 0.03 methylcyclo-2.10 -2.34 0.44 0.20 hexane -3.39 -3.96 1.07 0.50 1,1,3-trimethyl-1.28 -1.16 -1.25 0.11 0.03 cyclopentane -3.36 -3.77 0.59 0.18 -2.21 -2.03 -2.27 -0.05 cumene 0.18 -3.25 -3.74 0.89 0.39 -2.48 -2.04 -2.28 l-ethyl-2-methyl0.20 0.44 benzene -3.81 -4.37 l-ethyl-4-methylbenzene 1.06 0.51 -2.41 -1.99 -2.22 0.41 0.18 -4.42 -5.07 0.80 0.15 n-butylbenzene -2.86 -2.30 -2.59 0.56 0.27 -4.89 -5.62 sec-butylbenzene 1.15 0.43 -2.62 -2.15 -2.41 0.46 0.20 -2.45 -2.71 isobutylbenzene 0.59 0.33 -2.56 -2.15 -2.41 0.41 0.15 -3.11 -3.52 0.44 0.03 0.34 1,2,4,5-tetramethylbenzene -3.18 -2.50 -2.84 0.68 l-isopropyl-4-methylbenzene-2.69 -2.21 -2.49 0.47 0.20 -3.03 -3.41 0.07 -0.29 n-pentylbenzene -3.36 -2.63 -2.99 0.73 0.36 I-methylnaphthalene -4.03 -3.20 -3.71 0.31 0.83 0.71 1-ethylnaphthalene -3.57 -4.12 -4.60 -3.40 -3.95 1.19 0.16 0.62 -1.43 -1.24 -1.33 toluene 0.19 0.09 -3.53 -4.04 0.48 ethylbenzene -1.88 -1.61 -1.77 0.26 0.11 -0.03 p-xylene -1.93 -1.64 -1.80 0.29 0.13 -4.33 -4.84 0.81 0.30 m-xylene -1.96 -1.66 -1.82 0.29 0.13 0.32 o-xylene -2.06 -1.73 -1.91 0.15 -3.45 -3.86 1,2,4-trimethylbenzene 0.47 -0.19 -2.57 -2.10 -2.34 0.22 0.22 -2.70 -2.20 -2.47 1,2,3-trimethylbenzene 0.49 0.22 -2.94 -3.28 0.18 -2.49 -2.02 -2.25 0.52 1,3,5-trimethylbenzene 0.46 0.23 -2.34 -1.95 -2.17 propylbenzene 0.39 0.17 -1.46 -1.68 1,1,2-trichloroethane 0.10 0.04 -1.39 -1.29 -1.39 0.26 0.00 -1.31 -1.46 0.27 0.12 0.19 -1.73 -1.53 -1.68 1,1,1,2-tetrachloroethane 0.05 -1.29 -1.39 -2.06 -1.76 -1.94 1,1,2,2-tetrachloroethane 0.15 0.05 0.29 0.11 0.22 1,1,2,2,2-pentachloroethane -2.22 -1.99 -2.22 0.00 0.41 tetrachloroethene -1.61 -1.40 -1.51 -1.83 -2.19 0.21 0.05 0.09 -1.44 -1.60 0.21 0.06 trichloropropane 0.46 0.25 -2.38 -1.92 -2.13 1,2-dibromomethane -2.57 -2.07 -2.31 0.49 0.25 -2.17 -2.70 0.58 0.33 0.14 0.05 bromoform -2.14 -1.81 -2.00 -1.71 -1.99 0.24 0.05 chlorobenzene -1.80 -1.56 -1.70 0.33 0.10 -2.46 -3.17 0.44 0.11 o-dichlorobenzene -2.71 -2.26 -2.55 0.82 0.16 -2.79 -3.74 m-dichlorobenzene -2.51 -2.15 -2.41 1.01 0.35 0.10 0.06 -3.82 -5.89 0.47 -0.85 1,2,4-trichlorobenzene -3.22 -2.75 -3.14 1.22 0.07 -1.40 -1.55 a-chlorotoluene -2.76 -2.25 -2.53 0.51 0.08 0.23 0.23 -1.10 -1,M a,a,a-trifluorotoluene -1.30 -1.12 -1.20 0.10 0.18 0.10 0.03 bromobenzene -2.26 -1.91 -2.11 0.35 0.14 -1.14 -1.22 m-dibromobenzene -3.24 -2.81 -3.22 0.13 0.05 0.43 0.02 -1.53 -1.67 0.67 (2-bromoethyl)benzene -3.49 -2.81 -3.22 0.24 0.10 0.26 -1.52 -1.66 iodobenzene -2.88 -2.38 -2.69 0.19 0.05 0.50 0.19 trichlorohydrin 0.25 -2.38 -1.92 -2.13 0.46 the differences between the observed and the predicted vapor pressures from Mackay's equation and the proerror 1 (eq 12) 0.65
error 2 (es 16) 0.57
where P and PB are the vapor pressures at temperatures T and TB,respectively. Integration of the above equation, replacing AHMITM with ASM and AHBlTg with A& and assuming that PB is unity, gives
L
. I
Equation 7 describes the vapor pressure of a substance in terms of ita melting point and boiling point and four transition parameters: ASM, ASB, ACp , and AC Equation 7 has the same form as the Ranfine-Kirchgf equation (Bondi, 1968). However, it is derived rigorously by using only the assumptions that the heat capacity of each phase is independent of temperature and that the vapor behaves ideally. Mackay (1982) assumed that the heat capacities are given by ACpB= 0 (8) ACpM= 0
.
He also assumed that the entropy of vaporization is given by Trouton's rule: ASB/R = 10.75 ( 10) and that the entropy of melting is given by Walden's rule: ASM:/R= 6.75 (11) On the basis of the above assumptions Mackay showed that In P = [-10.75(T~- T ) - 6 . 7 5 ( T ~- T ) ] / T (12) Although eq 12, which is a form of the Clausius equation, is theoretically sound, it is not very accurate. Experimental Section The parameters required for estimation of vapor pressure using eq 7 are as follows: entropies of melting and vaporization, heat capacity change on melting and vaporization, and melting and boiling temperatures. In this study we used melting and boiling temperatures from the literature. AU other properties were estimated as described below.
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1611
Entropy of Melting. Walden’s rule has been shown by Tsakanikae and Yalkowsky (1988) to be inappropriate for compounds that have either a high rotational symmetry or a high conformationd flexibility. They developed the following empirical equation for the estimation of the entropy of fusion: A S M / R = 8.5 - 5.0 log u + 2.3 log 4 (13) where u is the rotational symmetry number and 4 is the conformational flexibility number of the molecule. (For alkyl homologues log 4 is proportional to (N - 51, where N is the number of carbons in the longest chain.) The rotational symmetry number, u, of a molecule is equal to the number of ways that the molecule can be positioned that are identical with a reference position. The values for the compounds considered in this study are given in Table 111. The molecular flexibility number is defined as the number of reasonable conformations in which the molecule can exist. For rigid m o l d e a 4 = 1. For the linear alkanes 4 = 3N-8,where N is the number of carbons in the molecule. The molecular flexibility values for the compounds considered in this study are given in Table 111. Entropy of Boiling. Mishra and Yalkowsky (1990b) have recently found that Trouton’s rule, like Walden’s rule, must be modified for highly flexible and highly symmetrical molecules. They found that ASB/R = 10.0 + 0.08 log 4 (14) where (0 is as described above. Heat Capacity Change on Boiling. Shaw (1969) determined the heat capacities of a large number of organic compounds. Shaw’s data can be described by AC,/R = -6 - 0.9 log 4 (15) which is analogous to the equations developed above for entropy of boiling but without the symmetry term. Heat Capacity Change on Melting. MacKay’s assumption (eq 9) that the heat capacity change upon melting of organic compounds is negligible appears to be justified as a reasonable first approximation. Our studies on the solubility also suggest that this assumption is reasonable (Mishra and Yalkowsky, 1990a). Data Analysis. All data were analyzed with the aid of a SAS (statistical analysis system) software package.
Results and Discussion MacKay developed a data set of 72 organic compounds. The equations developed here were tested by using this data set. The observed vapor pressures of these compounds are given in Table I1 along with the predictions of eqs 12 and 16 and the error of the predictions. All data are expressed in base 10 logarithmic units. MacKay found that this data was better described by an empirical equation than by eq 12. Substituting the entropy and heat capacity expressions (eqs 13-16) into eq 7 gives In P -([TM - TJ/T)(8.5 - 5.0 log u + 2.3 log 4) ([TB- TJ/T)(lO + 0.08 log 4) + ([TB- TJ/T - 1x1 [TB/TI)(-S- 0.9 log 6) (16) This equation describes the vapor pressure in terms of T,T M , TB, and only two molecular descriptors, u and 4. Equation 16 gives a significantly better estimate of vapor pressure for Mackay’s data set. The slope and intercept of the regression of observed log vapor pressure vs predicted from eq 12 and 16 are given in Table IV. The value
Table 111. Physicochemical Properties of the Compounds Used in the Study‘ name mp,OC bp,OC Q, u 155.7 215.2 256.6 353.2 307.6 344.0 369.2 389.0 326.1 326.0 336.0 320.5 1,2,3,4-tetrachlorobenzene 327.5 1,2,3,5-tetrachlorobenzene 413.0 1,2,4,5-tetrachlorobenzene 360.3 p-dibromobenzene 341.0 1,4-bromochlorobenzene octane 216.4 152.5 3-methylheptane 163.8 2,3,4-trimethylpentane nonane 222.0 167.2 2,2,54rimethylhexane 243.3 decane 159.8 4-methyloctane 247.4 undecane 263.4 dodecane 291.7 hexadecane 171.3 1-octane 146.4 methy lcyclohexane 134.6 ethylcyclopentane ethylcyclohexane 161.7 222.9 cis-1,2-dimethylcyclohexane trans-1,4-dimethylcyclohexane 236.0 258.8 1,1,3-trimethylcyclopentane cumene 176.4 192.2 1-ethyl-2-methylbenzene 210.6 1-ethyl-4-methylbenzene 185.0 n-butylbenzene 197.5 sec-butylbenzene 221.5 isobutylbenzene 193.8 1,2,4,5-tetramethylbenzene 205.1 1-isopropyl-4-methylbenzene 198.0 n-pentylbenzene 251.0 1-methylnaphthalene 259.2 1-ethylnaphthalene toluene 178.0 178.0 ethylbenzene 286.2 p-xylene 225.1 m-xylene o-xylene 247.8 229.2 1,2,44rimethylbenzene 247.6 1,2,3-trimethylbenzene 228.3 1,3,5-trimethylbenzene propylbenzene 171.4 1,I ,2-trichloroethane 236.5 1,1,1,2-tetrachloroethane 202.8 237.0 1,1,2,2-tetrachloroethane 1,1,2,2,2-pentachloroethane 244.0 tetrachloroethene 254.9 trichloropropane 258.3 l,2-dibromomethane 238.3 bromoform 264.7 chlorobenzene 227.4 o-dichlorobenzene 256.0 m-dichlorobenzene 248.3 1,2,44richlorobenzene 289.9 a-chlorotoluene 234.0 a,a,a-trifluorotoluene 243.9 242.2 bromobenzene m-dibromobenzene 266.0 (2-bromoethyl)benzene 205.5 241.8 iodobenzene trichlorohydrin 258.3
propylcyclopentane tert-butylbenzene 2-ethylnaphthalene naphthalene 2-methylnaphthalene biphenyl acenaphthalene fluorene p-dichloro benzene 1,2,3-trichlorobenzene 1,3,5-trichlorobenzene
376.0 442.0 530.9 491.0 514.1 528.9 550.5 568.0 447.0 491.0 481.0 527.0 519.0 516.0 492.0 469.0 398.6 388.0 386.5 423.8 397.1 447.1 415.4 466.9 489.3 560.0 394.3 373.9 376.4 402.9 402.7 392.4 377.9 437.7 438.2 436.0 456.0 446.0 445.8 469.8 450.1 478.4 517.6 531.7 383.0 409.0 411.0 412.0 417.4 442.4 449.1 437.7 432.2 386.8 403.5 419.2 435.0 394.0 429.9 440.3 422.5 405.0 453.5 446.0 486.5 462.3 375.1 429.0 491.0 491.0 461.3 429.9
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 27 9 1
81 3 243 18 729 2187 177147 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 4 1 4 2 2 4 2 6 2 2 4 4 2 2 1 1 2 1 2 1 2 2 2 1 1 1 1 2 2 1 1 1 2 1 1 1 4 1 1 1 1 2 1 4 2 2 2 2 6 1 1 1 2 1 4 1 2 3 2 2 2 1 1 2 2 2 1 2 1
‘All the data in this set taken from: Mackey, et al. Enuiron. Sei. Technol. 1982, 16,646449.
Ind. Eng. Chem. Res. 1991,30, 1612-1617
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Table IV. Summary of Regmrrion of Observed Vapor Pmrrure VI Predicted from Ear 12 and 16 eq R' slope av error MacKay (eq 12) 0.99 1.21 (*0.008) 0.47 (10.27) this work (eq 16) 0.99 1.04 (*0.008) 0.18 (10.15)
It is believed that the improved estimates of ASB,ASM, ACm,and ACm will lead to a better understanding of the vaporization process and to better estimates of vapor
of the slope is closer to the theoretical value of unit for eq 16 compared to 1.2 for MacKay's equation. The prediction is improved for each of the 72 compounds. The mean of the absolute error, lobserved minus predictedl, is reduced to 0.18 log units compared to 0.47 log units for eq 12. It should be noted that, unlike empirical equations described above, eq 16 uses no adjustable parameters. Furthermore, all numerical coefficients are obtained from non vapor pressure data. Although eqs 13-15 are at present only approximate, they clearly show that symmetry and flexibility are important factors in the evaluation of the parameters that govern vapor pressure, These equations will be finalized as more reliable data are acquired.
Literature Cited
pressure for complex molecules.
Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Gases; Wiley: New York, 1968. Mackay, D.;Bobra, A.; Chan, D. W.; Shiu, W . Y. Environ. Sci. Technol. 1982,16,645-649. Mishra, D.S.;Yalkowsky,S. H.Znd. Eng. Chem. Res. 199Oa, in press. Mishra, D.S.; Yalkowsky, S . H. Chemosphere 1990b, 21,111-117. Shaw, R. J . Chem. Eng. Data 1969,14,461-465. Tsakaniias, P.; Yalkowsky, S.H.Toxicol. Enuiron. Chem. 1988,17, 19.
Yalkowsky, S.H.;Mishra, D.S.; Morris, K. Chemosphere 1990,21, 107-110.
Received for review May 14, 1990 Accepted November 26,1990
Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 3. Thermal Properties. A New Significant Characterizing Parameter m Marek &galski* Laboratoire de Thermodynamique Chimique et Appliquie, ZNPL-ENSZC, 54001 Nancy Cedex, France
Bruno Carrier and Andr6 Pheloux Laboratoire de Chimie-Physique, Faculti des Sciences de Luminy, 13228 Marseille Ceder 9,France
Enthalpies of vaporization and isobaric liquid heat capacities of hydrocarbons were calculated by using a previously developed cubic equation of state. The enthalpy of vaporization predictions were in good agreement with the available experimental data and compared favorably with those obtained by using predictive methods published in the literature. The results obtained with liquid heat capacities were satisfactory in most cases. Characterizing parameters proposed by Carrier et al. were used to develop general correlations for enthalpies of vaporization and vapor pressures of hydrocarbons. On the basis of the results obtained, m can be said to be a significant parameter that can be useful for correlating thermodynamic properties.
Introduction Recently Carriet et al. (1988) proposed a modified version of the Peng-Robinson equation (CRP model) with which it is possible to accurately represent the vapor pressures of hydrocarbons between the triple and the normal boiling point. The encouraging results obtained with vapor pressures led us to study the thermal property predictions yielded by this model. In the present study, predictions of enthalpies of vaporization and isobaric liquid heat capacities obtained with the CRP model were studied and the results were compared with experimental data. In the second part of this paper, a component characterization proposed by Carrier et al. (1988) is considered. According to the CRP model, the Peng-Robinson equation of state (EOS),which is valid in the low-pressure range, can be established for every hydrocarbon for which the normal boiling temperature and the parameter m are known. The physical significance of the parameter m is
discussed with regard to the possibility of using it as an alternative to the acentric factor. Expressions giving the enthalpy and the isobaric heat capacity derived from the CRP model are given in Table I. Full details concerning the formulation and the use of eq T.1 were given in the paper by Carrier et al. (1988).
Enthalpies of Vaporization and Isobaric Liquid Heat Capacities All the experimental enthalpies of vaporization used here were from the recent compilation by Majer and Svoboda (1985),covering data published in the literature up to that date. The selected data base contained the enthalpies of vaporization of 80 hydrocarbons between 260 and Tb+ 20 K. The liquid isobaric heat capacities of 35 hydrocarbons that are below the normal boiling temperature were from TRC tables (1975-1986). Calculations were performed by using eqs T.l-T.12 established with the covolume calcu-
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