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Ind. Eng. Chem. Res. 1997, 36, 2494-2499
CORRELATIONS Estimating Pure Component Vapor Pressures of Complex Organic Molecules Paul B. Myrdal* and Samuel H. Yalkowsky† 3M Pharmaceuticals, 3M Center, St. Paul, Minnesota 55144, and College of Pharmacy, University of Arizona, Tucson, Arizona 85721
Modifications of the entropy of boiling term and heat capacity change upon boiling term are made to the Yalkowsky-Mishra vapor pressure equation. These modifications eliminate a systematic error and enable the pure component vapor pressure estimation of complex organic compounds, including those that are hydrogen bonding. The new vapor pressure equation, which requires only the knowledge of transition temperatures and molecular structure, is shown to be more accurate than the Yalkowky-Mishra equation for a wide variety of compounds over a wide range of temperatures (vapor pressures from 1 atm and below). Introduction Vapor pressure estimation methods are often derived from the integration of the Clausius-Clapeyron equation. The differences between these methods are a result of the assumptions made in their derivation. Mackay et al. (1982) developed an equation which is able to account for the crystal contribution to vapor pressure; however, they assumed that the heat capacity changes between the solid, liquid, and gas phases are negligible. Yalkowsky and Mishra (Yalkowsky et al., 1990; Mishra and Yalkowsky, 1991) have developed an equation that takes into account the difference between the liquid and gas heat capacities. When compared to the data of Mackay et al. (1982), it was found that new equation of Yalkowsky and Mishra (1991) gives significantly better results. In spite of these results, Yalkowsky and Mishra recognized that the parameters within their equation were developed from relatively simple organic compounds and suggested that improved estimates for of the heat capacity changes and entropies of transition would lead to an equation with wider applicability. In this work, modifications are made to the YalkowskyMishra method which enable even more accurate vapor pressure estimations for complex organic compounds.
Background
(
∆Sm(Tm - T) ∆Cp(ls) Tm - T + 2.303RT 2.303R T Tb ∆Sb(Tb - T) ∆Cp(gl) Tb - T Tm + - ln (1) ln T 2.303RT 2.303R T T
log P ) -
)
(
)
where, T ) temperature of interest in Kelvin, Tm ) melting point in Kelvin, Tb ) boiling point in Kelvin, R ) universal gas constant, ∆Sm ) entropy of melting at the melting point, ∆Cp(ls) ) heat capacity change upon melting, ∆Sb ) entropy of boiling at the boiling point, and ∆Cp(gl) ) heat capacity change upon boiling. For relatively simple organic compounds Mishra and Yalkowsky (1991) assumed that the latter four parameters can be approximated by
∆Sm ) 56.5 - 19.2 log(σ) + 9.2 (N - 3) (J/deg mol) (2) ∆Cp(ls) ) 0 (J/deg mol)
(3)
∆Sb ) 83.7 + 0.7 (N - 5) (J/deg mol)
(4)
∆Cp(gl) ) -50.2 - 7.5 (N - 5) (J/deg mol)
(5)
where σ is the molecular symmetry number (defined in the Input Parameter section) and N is the number of carbons in the longest chain. Inserting eqs 2-5 into eq 1 gives the YalkowskyMishra vapor pressure equation
Yalkowsky and Mishra (Yalkowsky et al. 1990) calculate vapor pressure through the Clausius-Clapeyron equation by integrating the total enthalpy of vaporization or sublimation over temperature. Their equation for vapor pressure, P (in atm), of an organic nonelectrolyte is
log P ) [56.5 - 19.2 log(σ) + 9.2 (N - 3)](Tm - T) 2.303RT [83.7 + 0.7 (N - 5)](Tb - T) + 2.303RT Tb [-50.2 - 7.5 (N - 5)] Tb - T - ln (6) 2.303R T T
* Author to whom correspondence should be addressed: 3M Pharmaceuticals, 3M Center, Building 260-4N-12, St. Paul, MN 55144. E-mail:
[email protected]. † College of Pharmacy.
which expresses vapor pressure (atm) in terms of only the boiling point, melting point, molecular symmetry, and chain atoms. For compounds which are liquid or gas at the temperature of interest, the first term of eq
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(
© 1997 American Chemical Society
)
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2495
systematic error for both solid and liquid solutes suggests that the possible error may be found in the second two terms (eqs 4 and 5) which are necessary for both liquids and solids. Based on this, and to reduce the total number of variables, eqs 2 and 3 will be assumed to be appropriate for estimating the vapor pressure of organic compounds which are solid at the temperature of interest. Consequently, only modifications to the entropy of boiling ∆Sb and the heat capacity change upon boiling ∆Cp(gl) are considered in this study. Input Parameters
Figure 1. Residual error using the Yalkowsky-Mishra vapor pressure equation (eq 6) vs the experimental vapor pressures given in the work of Banerjee et al.
6 is eliminated. Note that eq 6 contains no parameters which are determined from, or fitted to, vapor pressure data. Application of the Yalkowsky-Mishra Vapor Pressure Equation. The Yalkowsky-Mishra vapor pressure equation (eq 6) is relatively new and has not been extensively tested, especially on complex organic compounds or those that have low vapor pressures. To further validate eq 6, it was applied to the data of Bannerjee et al. (1990) for 111 environmentally important organic compounds ranging in vapor pressures from 0.7 to 7.2E-12 atm. When eq 6 is applied to the Bannerjee data the results are quite good, giving an overall absolute average error of 0.35 log units (factor error of 2.24). The method of Bannerjee et al., which is based on a correlation of solvatochromic and UNIFAC parameters, has an overall absolute average error of 0.52 log units (factor error of 3.31) for the same compounds. The fact that eq 6 out performs Bannerjee et al. is quite surprising since they used all 111 compounds in the derivation of their equation. Despite these impressive results, the predictive ability of the Yalkowsky-Mishra equation can be improved. Figure 1 is a scatter plot of the residuals (observed vapor pressure minus predicted vapor pressure) vs the experimental vapor pressure. As can be seen by the scatter plot, eq 6 does very well for those compounds that have relatively high volatility; however, as the volatility decreases so does the accuracy of eq 6. In fact, for those compounds that have experimental vapor pressures less than -4.0 atm, the average error is 0.9 log units (factor error of 7.94). The plot clearly illustrates that there is a systematic error which increases as the vapor pressure decreases. This systematic error is one which causes eq 6 to consistently overestimate the vapor pressure of low volatility compounds. Modification of the Yalkowsky-Mishra Vapor Pressure Equation. The purpose of this work is to modify the Yalkowsky-Mishra vapor pressure equation so that systematic errors are not incurred. To determine the cause of the systematic error, it is necessary to reevaluate the assumptions and approximations (eqs 2-5) that went into the derivation of eq 6. The first two equations have already been considered elsewhere (Mishra 1988; Mishra et al., 1992) and have been found to be reasonable for estimating the ideal solubility of a solid. In addition, the fact that eq 6 introduces a
Experimental Data. In order to help evaluate the necessary modifications to eq 6, the vapor pressures of 300 diverse organic compounds at 298 K were compiled. The vapor pressures of the compounds used in this study are given in Supporting Information along with the references and melting and boiling points. An average vapor pressure was used when multiple values were found for a compound. Of the compounds compiled, three were not used in the final analysis. The values for pyrene and o-cresol are used for testing purposes and octacosane was an outlier. The vapor pressures used for testing purposes, except four, were taken from Boublik et al. (1973). The 298 K vapor pressure for pyrene was taken from Bannerjee et al. (1990), the 408 K value for octacosane was taken from Piacente et al. (1994) and the 298 K and 327 K vapor pressures for o-cresol and 1-octanol were taken from Mackay et al. (1982). None of these vapor pressures were used in the generation of the final model. Entropy of Boiling. A new method for estimating the entropy of boiling has been recently developed by Myrdal et al. (1995). They found that the entropy of boiling for a diverse set of compounds could be described quite accurately by
∆Sb ) 86 + 0.4τ + 1421HBN (J/deg mol)
(7)
where τ is the effective number of torsional bonds and HBN is the hydrogen bond number as described below. The effective number of torsional bonds is a measure of overall molecular flexibility and can be determined using the formula
τ)
∑(SP3 + 0.5SP2 + 0.5RING) - 1
(8)
where SP3 and SP2 are the number of nonring, nonterminal sp3 (eg., CH2 and CH and also includes NH, N, O, and S) and sp2 atoms (eg., -Cd), respectively. RING indicates the number of independent ring systems found in the compound. The minus one is needed to give the proper number of torsional bonds. For example, butane has one torsional bond (2 SP3 atoms -1 ) torsional bond). Triple bonds and tert-butyl groups do not contribute to flexibility and are not counted. Thus, τ is equal to 0, 2, and 1.5 for propane, pentane, and 1-pentene, respectively. Note that τ is set equal to zero when equation 5 is -1 or -0.5. The effect of hydrogen bonding is described by the hydrogen bond number HBN which is determined by
HBN )
xOH + COOH + 0.33xNH2 MW
(9)
where OH, COOH, and NH2 represent the number of
2496 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
alcohols, carboxylic acids, or primary amines, respectively, and MW is the molecular weight of the compound. Entropy of Melting. The entropy of melting is estimated by
∆Sm ) 56.5 - 19.2 log(σ) + 9.2τ
(10)
where σ is the rotational symmetry number and τ is the effective number of torsional bonds as described above. The rotational symmetry number is the number of indistinguishable orientations in which a compound may be positioned in space. For example, σ for naphthalene is 4 while benzene has a symmetry number of 12. A rigorous description of symmetry has recently been given by Dannenfelser et. al (1993). Statistical Analysis. All statistical analyses were performed using the Statistical Analysis System (SAS, 1982) on the University of Arizona VAX. The PROC REG subroutine was used for multiple linear regressions. Results and Discussion Before any modifications to eq 6 were considered, it was applied directly to the new data set given in the Supporting Information. The overall root mean square error for the new set is 0.40 log units (factor error of 2.51); however, once again it was observed that the equation systematically overestimates the vapor pressures of low volatility compounds. As a result, modifications to the entropy of boiling ∆Sb and the heat capacity change upon boiling ∆Cp(gl) were considered. Entropy of Boiling. Yalkowsky and Mishra developed eq 6 on compounds which were not hydrogen bonding. However, the data presented from Bannerjee et al., as well as the data contained in the Supporting Information, include compounds which are capable of hydrogen bonding. Consequently, it is necessary to modify the ∆Sb equation so that it takes into account the effects of hydrogen bonding. Recently, Myrdal et al. (1995) developed an equation which takes into account the effect of hydrogen bonding on ∆Sb (see eq 7 in the experimental section). When eq 7 is incorporated into eq 6, the resulting equation is more accurate for the hydrogen bonding compounds. In fact, the overall error for the data compiled in the Supporting Information is reduced from 0.40 (factor error of 2.52) to 0.32 log units (factor error of 2.09). Nonetheless, the modified ∆Sb equation does not significantly affect most of the compounds; a systematic error is still observed and further modification of eq 6 must be considered. Heat Capacity Change upon Boiling. Since the systematic error is not entirely a result of the entropy of boiling term, the remaining error must be accounted for by the heat capacity change upon boiling ∆Cp(gl). Unlike ∆Sb, the heat capacity change upon boiling is not as well defined. This is mainly due to the fact that there is little experimental data on the heat capacity change at a given temperature much less over wide temperature ranges. This is partially the reason for considering the ∆Cp(gl) term as a constant. Yalkowsky and Mishra have described ∆Cp(gl) (eq 5) based on a the data of Shaw (1969). Although the data of Shaw clearly support a constant of -50.2 J/(deg mol), none of the compounds compiled in that report were crystalline at room temperature. As a result, this assumption may not be applicable to crystalline com-
pounds. The heat capacity change upon boiling is very important for solids because their boiling points are generally high. As boiling point increases any error in the heat capacity assumption will be magnified because the error is multiplied by a greater temperature difference. In other words, for compounds with low volatility the estimated vapor pressure is greatly affected by a small error in the heat capacity term. Chickos et al. (1993a) have recently examined a large amount of experimental data for ∆Cp(gl) at 298K. They found a mean ∆Cp(gl) of -64.2 J/(deg mol) for 289 different organic liquids, with a standard deviation of 32.2 J/(deg mol). They also calculated the ∆Cp(gl) for 114 compounds which are solid at 298K. For these compounds, the heat capacity of the liquid was estimated by a group contribution scheme (Chickos et al., 1993b) and experimental values for the heat capacity of the gas were used. Overall the organic solids were found to have larger values of ∆Cp(gl), having a mean of -134 J/(deg mol) with a standard deviation of 71 J/(deg mol). These data of Chickos et al. (1993a) clearly show that the ∆Cp(gl) value for most organic compounds is larger than the value of -50.2 cal/(deg mol) which is used by Yalkowsky and Mishra (eq 5). Since it has been shown that a modification to the ∆Cp(gl) value in eq 6 is appropriate, an effective ∆Cp(gl) value has been determined from the data in the Supporting Information. The new ∆Cp(gl) was calculated from eq 1 using the experimental vapor pressures and melting and boiling points and using eqs 2, 3, and 7 for ∆Sm, ∆Cp(ls), and ∆Sb, respectively. Incorporating these values and solving for ∆Cp(gl) gives
∆Cp(gl) ) -90.0 - 2.1τ J/(deg mol)
(11)
where τ is the identical to that used in the entropy of boiling estimation. Although the value of ∆Cp(gl) (from eq 11) is really an effective ∆Cp(gl) (i.e., it could incorporate errors which are not directly related to the heat capacity term), its magnitude is reasonable. The constant of -90.0 J/(deg mol) is well within the range of ∆Cp(gl) values observed by Chickos et al. (1993a). In fact, if the mean ∆Cp(gl) found for liquids by Chickos et al. (1993a) is averaged with the mean value determined for solids, the resulting value is -99.1 J/(deg mol). In addition, the dependency of ∆Cp(gl) upon molecular flexibility confirms the findings of Mishra and Yalkowsky (1991). It is noteworthy to mention that ∆Cp(gl) was also considered to be a function of temperature. In the derivation of eq 1 it is assumed that the heat capacity of a liquid and the heat capacity of a gas are parallel functions with respect to temperature (i.e., ∆Cp(gl) is constant). However, it is also reasonable to express the heat capacity change as a + bT, where T is temperature. This heat capacity expression was incorporated into the Clausius-Clapeyron equation and integrated with respect to temperature. The resulting function was applied to the vapor pressure data but did not significantly influence the results. This indicates that ∆Cp(gl) is not a strong function of temperature which is consistent with the conclusion of Chickos et al. (1993a). Nevertheless, the dependency of the heat capacity change with temperature should be reconsidered as more data becomes available. Final Vapor Pressure Equation. Incorporating eqs 2, 3, 7, and 11 into eq 1 gives the new modified vapor
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2497 Table 1. Comparison of Equations 12 and 6 for Some Compounds at Different Temperatures eq 12
eq 6
compound
Tb (K)
temp (K)
log P (atm)
predicted
|error|
predicted
|error|
1-undecene 2-methyl-5-ethylpyridine 3,5-diethylphenol cis-decalin R-naphthol isopropylcyclohexane methyl salicylate n-ethylaniline resorcinol valeric acid 1-nonanol 1,4-dioxane 1-hexanethiol 1-tetradecanol 1-chlorotetradecane biphenyl β-pinene 2,4,4-trimethylhexane octacosane average absolute error (factor error)
466 452 521 469 556 428 494 478 549 446 486 374 426 569 570 528 439 404 705
379 325 388 373 415 344 352 323 425 345 365 303 354 425 415 342 300 323 408
-1.20 -2.10 -1.88 -1.26 -1.88 -1.20 -2.25 -2.50 -1.88 -1.86 -2.13 -1.20 -1.02 -2.16 -2.16 -2.99 -2.36 -1.20 -5.40
-1.20 -2.05 -1.96 -1.29 -1.93 -1.22 -2.33 -2.60 -1.76 -1.73 -1.96 -1.17 -1.03 -2.01 -2.11 -2.97 -2.48 -1.28 -5.01
0.00 0.05 0.08 0.03 0.05 0.02 0.08 0.10 0.12 0.13 0.17 0.03 0.01 0.15 0.05 0.02 0.12 0.08 0.39 0.09(1.23)
-1.16 -1.86 -1.61 -1.19 -1.60 -1.13 -1.92 -2.31 -1.36 -1.37 -1.70 -1.08 -0.96 -1.90 -2.12 -2.65 -2.23 -1.18 -6.03
0.04 0.25 0.27 0.07 0.28 0.07 0.33 0.19 0.52 0.49 0.43 0.12 0.07 0.27 0.05 0.34 0.13 0.02 0.63 0.24(1.74)
Figure 2. Plot of the predicted vapor pressure vs experimental vapor pressure for the 297 compounds used in this study.
Figure 3. Residual error using eq 12 vs experimental vapor pressures.
pressure equation, i.e.,
[56.5 - 19.2 log(σ) + 9.2τ](Tm - T) 19.1T [86.0 + 0.4τ + 1421HBN](Tb - T) + 19.1T Tb [-90.0 - 2.1τ] Tb - T - ln (12) 19.1 T T
log P ) -
(
)
The modified equation (eq 12) fits the data in the Supporting Information quite well, having an overall root mean square error of 0.21 log units (factor error of 1.62). In addition, the modifications appear to be appropriate for both solids and liquids. The root mean square error for the liquids (n ) 260) and the solids (n ) 37) are 0.16 (factor error of 1.45) and 0.38 log units (factor error of 2.40), respectively. The predicted vapor pressures using eq 12 are given in the Supporting Information. An example of the use of eq 12 is given in appendix A. A plot of the predicted vapor pressure vs the experimental vapor pressure is given in Figure 2. The plot illustrates the predictive ability of eq 12 over 11 orders of magnitude. As a final check the residuals from eq 12 were plotted against the experimental vapor pressures. The results are given in Figure 3. As can be seen in the scatter
Figure 4. Comparison of eq 6 (- -), and eq 12 (s) for predicting the vapor pressure of pyrene as a function of temperature. The open circles (O) represent experimental data.
plot, the residuals of the new equation is devoid of any systematic trends. Application of Vapor Pressure Equation. To test the new equation (eq 12), 19 structurally diverse compounds were selected and their vapor pressures were estimated at different temperatures. The results are given in Table 1 along with vapor pressures estimated from eq 6. As can be seen from the table, eq 12 is more accurate than eq 6 for 17 of the 19 compounds
2498 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
occurs at the melting point (indicated by Tm on the x-axis) for pyrene and o-cresol. In summary, the modifications to the entropy of boiling and the heat capacity change upon boiling have yielded an improved equation for the estimation of vapor pressure. This equation is shown to provide a reliable means of estimating the vapor pressure of complex organic compounds directly from transition temperatures and molecular structure and can be used over a wide range of temperatures. Acknowledgment This work was supported through a grant from the Environmental Protection Agency (Grant R-817475-01). Figure 5. Comparison of eq 6 (- -), and eq 12 (-) for predicting the vapor pressure of 1-octanol as a function of temperature. The open circles (O) represent experimental data.
Figure 6. Comparison of eq 6 (- -), and eq 12 (-) for predicting the vapor pressure of o-cresol as a function of temperature. The open circles (O) represent experimental data.
tested. The overall average absolute error for eqs 12 and 6 are 0.09 and 0.24 times r, respectively (factor errors of 1.23 and 1.74, respectively). All of the predicted vapor pressures from eq 12, except two, are estimated within 0.15 log units (factor errors of 1.41). None of the vapor pressures tested in Table 1 were used in the development of eq 12. It is also important to note the inclusion of octacosane in Table 1. As previously mentioned, the 298 K vapor pressure for octacosane was omitted as an outlier. Although another room temperature value for octacosane could not be obtained, a vapor pressure at an elevated temperature was found in a recent publication by Piacente et al. (1994) and was used in the test set. As a further validation, the new model was used to estimate vapor pressure as a function of temperature. Three different compounds were selected and their vapor pressures were estimated from (or near) room temperature (indicated by RT above the x-axis) to their corresponding boiling points (indicated by Tb above the x-axis). The data for pyrene, 1-octanol, and o-cresol are given in Figures 4, 5, and 6, respectively. Once again, the predicted vapor pressures of the proposed equation (indicated by the solid line) are closer to the experimental values (open circles) than those of the original Yalkowsky-Mishra equation (dashed lines). These results are quite impressive considering that fact that the compounds tested span at least 3 orders of magnitude and none of the compounds were used in the generation of eq 12. Note the change of slope which
Nomenclature T ) temperature of interest in Kelvin Tm ) melting point in Kelvin Tb ) boiling point in Kelvin R ) universal gas constant ∆Sm ) entropy of melting at the normal melting point ∆Cp(ls) ) heat capacity change upon melting (liquid minus solid) ∆Sb ) entropy of boiling at the normal boiling point ∆Cp(gl) ) heat capacity change upon boiling (gas minus liquid) N ) number of chain atoms HBN ) hydrogen bond number SP3 ) nonterminal, nonring sp3 atoms SP2 ) nonterminal, nonring sp2 atoms RING ) represents the presence of a ring in a compound OH ) represents alcohol functional group COOH ) represents carboxylic functional group NH2 ) represents primary amine MW ) molecular weight Greek Symbols σ ) external rotational symmetry number τ ) an empirical measure of the effective number of torsional bonds
Appendix A Calculation of Vapor Pressure for 3,5-Diethylphenol at 388 K Using eq 12
Tm )
