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Ind. Eng. Chem. Res. 2006, 45, 8744-8747
CORRELATIONS Estimating Pure-Component Vapor Pressures of Complex Organic Molecules: Part II. Kia Sepassi,* Paul B. Myrdal, and Samuel H. Yalkowsky College of Pharmacy, UniVersity of Arizona, Tucson, Arizona 85721
The purpose of this work was to develop an equation to estimate the saturated vapor pressures of organic compounds at ambient temperature. A new equation based on the integrated form of the Clausius-Clapeyron equation was used to estimate the room temperature saturated vapor pressures of 815 organic compounds. It was found to reliably estimate vapor pressures over 15 orders of magnitude with an average absolute error of 0.18 logarithmic units, corresponding to a factor of 1.50. Introduction
Background
The tendency for an environmental contaminant or pesticide to partition into the atmosphere is determined largely by its vapor pressure.1 Thus, knowledge of the vapor pressure will allow for a better understanding of the environmental fate of organic compounds. Recently, Voutsas et al. demonstrated the successful estimation of saturated vapor pressures from knowledge of the normal boiling point.2 This method is based on the Clausius-Clapeyron equation and the use of an empirically fitted parameter obtained from regressing vapor pressure data from the melting point up to the normal boiling point for each compound.2 Coutsikos et al. estimated the saturated vapor pressure of organic compounds from the hypothetical liquid vapor pressure and the entropy of melting.3 The latter was estimated from a group contribution approach. Both of these methods are based on simplifications of the Clausius-Clapeyron equation. However, they are based on specific chemical classes and are not applicable to a wide range of compounds. Myrdal and Yalkowsky4 used the integrated form of the Clausius-Clapeyron equation to estimate the saturated vapor pressure of a wide range of organic compounds with reasonable accuracy. This approach requires the melting point, boiling point, and four transition properties: entropy of melting, entropy of boiling, heat capacity change on melting, and heat capacity change on boiling. Using four empirical relationships for the estimation of these transition properties for each compound, they estimated the saturated vapor pressures of 297 organic compounds with a root-mean-square error of 0.21 log units, corresponding to a factor of 1.61.4 In this work, the equations for the transition properties are reevaluated and a new equation for the estimation of the saturated vapor pressures is generated and validated on a larger data set containing over 800 compounds.
The integrated form of the Clausius-Clapeyron equation allows the estimation of the saturated vapor pressure (VP) in atmospheres at any temperature (T) in Kelvin units and is given by
log VP ) -
[
[
]
where the saturated vapor pressure is estimated from the melting point (Tm), boiling point (Tb), entropy of melting (∆Sm), entropy of boiling (∆Sb), heat capacity change on melting (∆Cp,m), and heat capacity change on boiling (∆Cp,b). For liquid compounds at ambient temperature, eq I is simplified to
log VP ) -
[
]
∆Sb(Tb - T) ∆Cp,b Tb - T Tb (II) + - ln 2.3RT 2.3R T T
where the saturated vapor pressure is estimated from the boiling point, entropy of boiling, and heat capacity change on boiling. Mackay et al. showed that eq I can be further simplified by assuming the heat capacity changes on melting and boiling to be negligible.1 These assumptions lead to
log VP ) -
∆Sm(Tm - T) ∆Sb(Tb - T) 2.3RT 2.3RT
(III)
Equation III was further simplified by assuming the entropies of melting and boiling were constant and given by Walden’s (56.5 J/mol‚K) and Trouton’s rule (88 J/mol‚K), respectively.1 These assumptions along with the universal gas constant (8.314 J/mol‚K) lead to
log VP ) * Corresponding author. College of Pharmacy, The University of Arizona, 1703 East Mabel Street, Tucson, AZ 85721. E-mail:
[email protected]. Phone: (520) 626-4309. Fax: (520) 626-2466.
]
Tm ∆Sm(Tm - T) ∆Cp,m Tm - T + - ln 2.3RT 2.3R T T Tb ∆Sb(Tb - T) ∆Cp,b Tb - T + - ln (I) 2.3RT 2.3R T T
2.95(Tm - T) 4.60(Tb - T) T T
(IV)
where the saturated vapor pressure was estimated from the melting point, boiling point, and reference temperature. Although
10.1021/ie060979i CCC: $33.50 © 2006 American Chemical Society Published on Web 11/10/2006
Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8745
eq IV is a simplified form of the Clausius-Clapeyron equation, it is not very accurate in the estimation of saturated vapor pressures. The assumptions made by Mackay et al.were reevaluated by Mishra and Yalkowsky5 and Myrdal and Yalkowsky.4 These authors improved the estimation of saturated vapor pressures by using empirical structure based equations for the entropies of melting and boiling as well as the heat capacity change on boiling in eq I. Entropy of Melting. The entropy of melting can be approximated by Walden’s rule, which is strictly based on observation. Walden’s rule states that the entropy of melting of rigid organic compounds can be assumed to be a constant value of 56.5 J/mol‚K. For a more accurate estimation, Myrdal and Yalkowsky estimated the total entropy of melting from
∆Sm ) 56.5 - 19.2 log σ + 9.2τ (J/mol‚K)
(V)
where σ and τ denote the molecular symmetry and the flexibility number, respectively.4 The molecular symmetry number is defined as the number of ways a molecule can be superimposed on itself, resulting in an identical structure with respect to a reference position. For example, toluene has a symmetry number of 2. The molecular flexibility number is given by
τ ) SP3 + 0.5SP2 + 0.5RING - 1
(VI)
where SP3 and SP2 denote the number of nonring nonterminal sp3 and sp2 atoms, respectively, and RING denotes the number of independent, single fused ring systems in a molecule.4 For example, hexane would have a molecular flexibility number of 3. Entropy of Boiling. As a first-hand approximation, the entropy of boiling can be approximated by Trouton’s rule. This rule states that the entropy of boiling for non-hydrogen bonded organic compounds is a constant value of 88 J/mol‚K. Myrdal et al. proposed the following equation for the estimation of the entropy of boiling of complex and hydrogen bonded compounds
∆Sb ) 86 + 0.4τ + 1421HBN (J/mol‚K)
(VII)
where the entropy of boiling is estimated from the molecular flexibility (τ) and hydrogen bond density number (HBN).6 The molecular hydrogen bond density number is determined from
xOH + COOH + 0.33xNH2 HBN ) MW
capacity change on boiling
∆Cp,b ) -90 - 2.1τ (J/mol‚K)
(IX)
This equation was used by the authors in the estimation of the saturated vapor pressures.4 Sanghvi and Yalkowsky recently generated an empirical equation for the estimation of the heat capacity change on boiling from the enthalpy of vaporization at 298 K, the enthalpy of vaporization at the boiling point, and Kirchhoff’s equation.7 Their equation states
∆Cp,b ) -56 - 4τ - 40HBP (J/mol‚K)
(X)
where the heat capacity change on boiling was determined from the molecular flexibility number (τ) and the hydrogen bond parameter (HBP). The hydrogen bonding parameter was determined from
HBP ) x(OH) + (CO2H) + 0.0625(NH)
(XI)
where OH, CO2H, and NH denote the number of hydroxyl, carboxyl, and amine groups on a compound.7 Equations IX and X assume that the heat capacity change on boiling is constant and does not change with respect to temperature. Heat Capacity Change on Melting. The heat capacity change on melting is analogous to the heat capacity change on boiling; it is assumed to be constant and independent of temperature. As in the case of the heat capacity change on boiling, there is a minimal amount of experimental data available for organic compounds. For simplicity, Mackay et al.,1 Mishra and Yalkowsky,5 and Myrdal and Yalkowsky4 assumed that the heat capacity change on melting was negligible and could be approximated by
∆Cp,m ) 0 (J/mol‚K)
(XII)
Several other workers, Neau et al.8 and Neau and Flynn,9 suggested that it was more appropriate to assume that the heat capacity change on melting was better approximated by the entropy of melting; thus,
∆Cp,m ) ∆Sm
(XIII)
Methods
(VIII)
where OH, COOH, and NH2 represent the number of hydroxyl, carboxylic acid, and amine groups on a compound, respectively.6 MW is the molecular weight of the compound. The square root accounts for competition among multiple hydrogen bonding groups on a molecule. Heat Capacity Change on Boiling. Mackay et al. proposed that the ratio of the heat capacity change on boiling to the entropy of vaporization for many organic compounds should typically range from -0.60 to -1.0.1 For the estimation of vapor pressure, Mackay et al. found -0.76 to be the best value for small non-hydrogen bonding compounds having boiling points above 100 °C.1 Myrdal and Yalkowsky calculated an effective ∆Cp,b from eq I using experimental vapor pressures, melting points, and boiling points.4 For solid compounds, ∆Cp,m was assumed to be negligible and the entropies of melting and boiling were estimated from eqs V and VII, respectively. These values were used to generate the following empirical equation for the heat
Experimental Data. The experimental melting points, boiling points, and vapor pressures were obtained from MPBPWIN10 version 1.41 provided by the United States Environmental Protection Agency. A total of 815 organic compounds (680 liquid and 135 solid) with reported vapor pressures ranging from 10-15 to 2.16 atm at 298 K were retained. The boiling points ranged from 299 to 809 K, and the melting points ranged from below ambient temperature to 603 K. The experimental entropies of boiling for 291 of the liquid compounds were obtained from the work of Sanghvi and Yalkowsky.7 The boiling points of these compounds ranged from 312 to 596 K. Statistical Analysis. Multiple linear regression analyses were performed using the statistical analysis program SPSS version 10.0 for Windows. Entropy of Melting. Recently, the equation for the estimation of the total entropy of melting (i.e., eq V) was modified by Jain et al.11 The modified equation states
∆Sm ) 50 - 19.1 log σ + 7.4τ (J/mol‚K)
(XIV)
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Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006
Table 1. Log of the Average Absolute Errors and Average Errors for the Estimation of the Ambient-Temperature Vapor Pressure of 680 Liquid Organic Compounds
Table 2. Log of the Average Absolute Errors and Average Errors for the Estimation of the Ambient-Temperature Vapor Pressures of 135 Solid Organic Compounds
heat capacity equation
AAEa
AEb
estimation
∆Cp,m
∆Cp,b
AAEa
AEb
eq X eq XV eq XVI
0.158 0.131 0.176
-0.109 -0.029 -0.148
est 1 est 2 est 3 est 4 est 5 est 6
eq XII eq XII eq XII eq XIII eq XIII eq XIII
eq X eq XV eq XVI eq X eq XV eq XVI
0.56 0.43 0.53 0.60 0.44 0.57
-0.24 0.02 -0.29 -0.32 -0.06 -0.37
a AAE ) Σ |(experimental value - predicted value)|/n. b AE ) Σ (experimental value - predicted value)/n.
This equation was validated on the experimental entropies of 1799 complex organic compounds, resulting in an average absolute error of (15 J/mol‚K.11 This updated equation will be used in the estimation of the saturated vapor pressures. Entropy of Boiling. The proposed equation of Myrdal et al.6 (i.e., eq VII) for the estimation of the entropy of boiling will be used without further modification. Heat Capacity Change on Boiling. The effective heat capacity change on boiling (∆Cp,b) will be back-calculated from the experimental boiling points, entropies of boiling, and ambient-temperature vapor pressure data using eq II. Only the data of liquid compounds will be considered, thus eliminating the effect of assumptions made on the heat capacity change on melting. The resultant values will be used to generate a new empirical equation for the estimation of the heat capacity change on boiling from multiple linear regression analyses. The value of -0.76 recommended by Mackay et al.1 will be used to generate a second equation for the estimation of the heat capacity change on boiling strictly from the experimental entropies of boiling [i.e., -0.76∆Sb]. The equation will be generated using multiple linear regression analyses. Thus, saturated vapor pressures will be estimated using eq X and two newly developed empirical equations for the heat capacity change on boiling. Heat Capacity Change on Melting. The room-temperature saturated vapor pressures of the solid compounds will be estimated by combining either eq XII or eq XIII with each of the three different equations for the heat capacity change on boiling. If eq XIII is used, then the heat capacity change on melting is estimated from eq XIV. Results Discussion The following empirical heat capacity change on boiling equation was developed from the experimental data of the liquid organic compounds
∆Cp,b ) -91 - 1.2τ (J/mol‚K)
(XV)
This equation differs from eq IX in that it was obtained entirely from experimental data. In other words, this equation was generated from back-calculating the heat capacity change on boiling from eq II using the experimental boiling points, entropies of boiling, and room-temperature vapor pressures. From the proposed ratio of Mackay et al.,1 a third empirical equation for the estimation of the heat capacity change on boiling was generated
∆Cp,b ) -68 - 0.31τ (J/mol‚K)
(XVI)
This equation was obtained entirely from the experimental entropies of boiling (-0.76∆Sb) of the liquid compounds. Note that both eqs XV and XVI were generated from two different approaches utilizing the same set of liquid compounds.
a AAE ) Σ |(experimental value - predicted value)|/n. b AE ) Σ (experimental value - predicted value)/n.
Table 3. Log of the Average Absolute Errors and Average Errors for the Estimation of the Ambient-Temperature Vapor Pressure of 815 Solid and Liquid Organic Compounds estimation
∆Cp,m
∆Cp,b
AAEa
AEb
est 1 est 2 est 3 est 4 est 5 est 6
eq XII eq XII eq XII eq XIII eq XIII eq XIII
eq X eq XV eq XVI eq X eq XV eq XVI
0.23 0.18 0.24 0.23 0.18 0.24
-0.13 -0.03 -0.17 -0.14 -0.04 -0.19
a AAE ) Σ |(experimental value - predicted value)|/n. b AE ) Σ |(experimental value - predicted value)|.
Effect of the Heat Capacity Change on Boiling. Table 1 provides a summary of the estimation of the saturated vapor pressures of 680 liquid organic compounds at ambient temperature. The boiling points of these compounds ranged from 299 to 690 K. The vapor pressures were estimated from eq II by using the empirical equation for the entropy of boiling and the three different equations for the heat capacity change on boiling. The error in the estimation of the liquid vapor pressures is the lowest with the use of eq XV. Although all three equations estimate the heat capacity change on boiling from molecular flexibility, the constant in eq XV (-91 J/mol‚K) is significantly higher than those in eqs X and XVI. This is due to the different methods used for the generation of these equations. Effect of the Heat Capacity Change on Melting. Table 2 depicts the average absolute errors and average errors for the estimation of the ambient-temperature vapor pressures of 135 solid compounds having melting points ranging from 299 to 603 K. The vapor pressures were estimated from eq I using the empirical equations for the entropies of boiling and melting along with the three different equations for the heat capacity change on boiling. In estimations 2 and 5, the heat capacity change on melting was assumed to be negligible (eq XII) or equal to the entropy of melting (eq XIII). The results of these estimations demonstrate that both assumptions result in nearly identical errors. This is because of the small contribution of the term [(Tm - T)/T (ln (Tm/T))] in eq I to the overall estimation of vapor pressure. The two terms in the bracket are similar in magnitude but have opposite signs. The assumption of the heat capacity change on melting becomes significant for solid compounds having high melting points. In the estimations illustrated in Table 2, the use of eq XV provides the most accuracy in the estimation of the roomtemperature vapor pressures of solid compounds. Tables 1 and 2 show that eqs X and XVI do not estimate liquid or solid vapor pressures very well. Final Vapor Pressure Equation. Table 3 depicts the errors in the estimation of the ambient-temperature vapor pressures of all compounds in the data set. The saturated vapor pressures
Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8747
where Tm < 298 K, Tb ) 532 K, T ) 298 K, σ ) 1.0, τ ) 0.5, MW ) 156.23 g/mol, and HBN ) 0. The calculated transition properties are as follows:
∆Sm ) 50 - 19.1 log σ + 7.4τ ) 50 - 7.4(0.5) ) 53.7 J/mol‚K ∆Sb ) 86 + 0.4τ + 1421HBN ) 86 + 0.4(0.5) + 1421(0) ) 86.2 J/mol‚K Figure 1. Log of the experimental and predicted vapor pressures in atmospheres.
were estimated by eqs XII and XIII and three empirical equations for the heat capacity change on boiling. From Table 3, it can be seen that the use of eq XV provides the best overall estimation of ambient-temperature vapor pressures. Thus, the final vapor pressure equation can be illustrated by incorporating eqs VII, XII, XIV, and XV into eq I, leading to
(50 - 19.1 log σ + 7.4τ)(Tm - T) 2.3RT (88 + 0.4τ + 1421HBN)(Tb - T) + 2.3RT Tb (-91 - 1.2τ) Tb - T (atm) (XVII) - ln 2.3R T T
∆Cp,b ) - 91 - 1.2τ ) - 91 - 1.2(0.5) ) - 91.6 J/mol‚K Substituting into eq XVII yields the following:
log VPest ) -4.51 log VPobs ) -4.60 Supporting Information Available: The experimental and estimated vapor pressures along with the average absolute errors are available for all the organic compounds used in this study. This material is available free of charge via the Internet at http://pubs.acs.org.
log VPest ) -
Literature Cited
Figure 1 depicts the logarithm of the experimental and predicted vapor pressures for all compounds in the data set. The line in the figure is the line of identity. The data presented in Figure 1 demonstrate estimation of saturated vapor pressures at 298 K ranging over 15 orders of magnitude with reasonable accuracy using eq XVII. The average absolute error is 0.18 log units, corresponding to a factor of 1.5. These errors are well within the normal range of error of vapor pressure measurements. According to Coutsikos et al.,3 experimental vapor pressures for a compound can vary by as much as 30-100%. Also, Spencer and Cliath12 noted that experimental vapor pressures obtained from various references and experimentalists can vary by a factor of 2-3.
(1) Mackay, D.; Bobra, A.; Chan, D. W.; Shiu, W. Y. Vapor Pressure Correlations for Low-Volatility Environmental Chemicals. EnViron. Sci. Technol. 1982, 16, 645. (2) Voutsas, E.; Lampadariou, M.; Magoulas, K.; Tassios, D. Prediction of vapor pressures of pure compounds from knowledge of the normal boiling point. Fluid Phase Equilib. 2002, 198, 81. (3) Coutsikos, P.; Voutsas, E.; Magoulas, K.; Tassios, D. Prediction of vapor pressures of solid organic compounds with a group contribution methd. Fluid Phase Equilib. 2003, 207, 263. (4) Myrdal, P. B.; Yalkowsky, S. H. Estimating Pure Component Vapor Pressures of Complex Organic Molecules. Ind. Eng. Chem. Res. 1997, 36, 2494. (5) Mishra, D. S.; Yalkowsky, S. H. Estimation of Vapor Pressure of Some Organic Compounds. Ind. Eng. Chem. Res. 1991, 30, 1612. (6) Myrdal, P. B.; Krzyaniak, J. F.; Yalkowsky, S. H. Modified Trouton’s Rule for Predicting the Entropy of Boiling. Ind. Eng. Chem. Res. 1996, 35, 1788. (7) Sanghvi, R.; Yalkowsky, S. H. Estimation of the Normal Boiling Point of Organic Compounds. Ind. Eng. Chem. Res. 2006, 45, 2856. (8) Neau, S. H.; Flynn, G. H.; Yalkowsky, S. H. The influence of heat capacity assumptions on the estimation of solubility parameters from solubility data. Int. J. Pharm. 1989, 49, 223. (9) Neau, S. H.; Flynn, G. L. Solid and liquid heat capacities of n-alkyl para- aminobenzoates near the melting point. Pharm. Res. 1990, 7, 1157. (10) MPBPWIN, v1.41; U.S. Environmental Protection Agency: Washington, DC, 2000 (www.epa.gov). (11) Jain, A.; Yang, G.; Yalkowsky, S. H. Estimation of Total Entropy of Melting of Organic Compounds. Ind. Eng. Chem. Res. 2004, 43, 4376. (12) Spencer, W. F.; Cliath, M. M. Measurement of Pesticide Vapor Pressure. Residue ReV. 1983, 85, 57.
[
]
Conclusion New equations for the entropy of melting and heat capacity change on boiling were used to obtain a new equation (eq XVII) for the estimation of saturated vapor pressures. This new equation is shown to estimate the room-temperature saturated vapor pressures of a large structurally diverse data set with reasonable accuracy. Example. This example involves calculating the vapor pressure of 1-ethylnaphthalene at 298 K using eq XVII,
ReceiVed for reView July 26, 2006 ReVised manuscript receiVed September 29, 2006 Accepted October 3, 2006 IE060979I
