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Ind. Eng. Chem. Res. 1996, 35, 1788-1792
CORRELATIONS Modified Trouton’s Rule for Predicting the Entropy of Boiling Paul B. Myrdal,† Joseph F. Krzyzaniak, and Samuel H. Yalkowsky* College of Pharmacy, University of Arizona, Tucson, Arizona 85721
Simple modifications of Trouton’s rule for estimating the entropy of boiling of organic compounds are presented. An empirical equation is developed which expresses the entropy of boiling as a function of molecular flexibility and hydrogen bonding. This equation, which enables the simple yet reasonable estimate of entropy of boiling directly from molecular structure, was generated from a training set of 850 structurally diverse compounds. The average absolute percent error for a test set of 88 compounds is 2.5% which is more accurate than Trouton’s rule. Introduction There are several methods for estimating the entropy of vaporization of an organic compound at its normal boiling point, i.e., the entropy of boiling. The first and most widely recognized method was proposed by Trouton (1884). Trouton observed that the ratio of the enthalpy of vaporization at the boiling point and the normal boiling point is nearly constant at 88 J/deg mol. Hildebrand (1915) demonstrated that it is more appropriate to compare the entropies of vaporization of normal liquids at temperatures in which their vapors have equal molar volumes. On this basis, several methods have been proposed which express the entropy of boiling as a function of normal boiling temperature (Reid et al., 1977; Reid et al., 1987). Other methods relate the entropy of boiling to critical temperatures and pressures (Hoshino et at., 1983). There are comparatively few methods which enable the estimation of entropy of boiling on the basis of molecular structure alone. The group contribution method of Hoshino et al. (1983) was one of the first significant advancements in relating the entropy of boiling to molecular structure. Recently, Ma and Zhao (1993) developed a group contribution method which utilizes a more elaborate description of functional groups. Although the method of Ma and Zhao yields good results, it is somewhat cumbersome and has limited applicability. In this paper, modifications to Trouton’s rule are presented for estimating the entropy of boiling of organic compounds. The proposed method, which takes into account molecular flexibility and hydrogen bonding, provides a simple, non-group contribution approach, for estimating entropy of boiling directly from molecular structure. Background Entropy is a measure of the degree of disorder in a system. The entropy of a system is related to the * To whom correspondence should be addressed. Department of Pharmaceutical Sciences, College of Pharmacy, University of Arizona, Tucson, AZ 85721. Telephone: (520) 6261289. FAX: (520) 626-4063. E-mail: YALKOWSK@TONIC. PHARM.ARIZONA.EDU. † Present address: 3M Pharmaceuticals, 3M Center, Building 270-4S-17, St. Paul, MN 55144-1000.
number of ways, W, in which its molecules can be arranged. The entropy of boiling, ∆Sb, is therefore proportional to the ratio of the number of ways in which the molecules can be arranged in the gas and the liquid phase, i.e.,
∆Sb ) R ln
Wgas Wliq
(1)
The entropy of boiling can be further expanded into the sum of the entropies corresponding to the changes in translational, rotational, and conformational motion of the molecules, e.g.,
∆Sb ) ∆Strans + ∆Srot + ∆Sconf
(2)
It is assumed that the internal entropy (i.e., the entropy corresponding to intramolecular stretching and bending vibrations) of gases and liquids are nearly identical. The translational entropy gain is the largest contributor to ∆Sb and is the major reason for the success of Trouton’s rule. In general, the increase in ∆Strans can be understood in terms of free volume change, i.e.,
∆Strans = R ln
f Vgas f Vliq
(3)
f f and Vliq are the free volumes in the gas and where Vgas liquid, respectively. The free volume is the volume into which the center of a molecule can move. The volume of virtually all gas molecules at standard temperature and pressure (STP) is approximately the same as that of an ideal gas (22.4 l/mol). Since the van der Waals molar volume of the molecules is negligible compared to this number, it is assumed that the free volume in all gases, at STP, is constant. It is also reasonable to assume that the free volume of most liquids at STP is constant. This is because the free volume is related to the average distance of separation between the van der Waals shells of the molecules, and at constant temperature this value is essentially constant. If the free volume of most liquids is constant and the free volume of gases is constant, then the ratio of the free volumes will be constant. This constancy of free volumes at a constant temperature and pressure is responsible for the constant entropy of boiling.
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1789 Table 1. Examples of the Effective Number of Torsional Bonds, τ, and Hydrogen Bond Number, HBN name
τ
HBN (×10-3)
propane butane butanol ethyl acrylate pentylbenzene 2-methyl-1-hexanol aniline 1-pentyne
0 1 2 2 3.5 4 0 1
0 0 13.5 0 0 8.61 3.54 0
τ)
Since the normal boiling points of organic compounds range over several hundred degrees, it follows from the above reasoning that the corresponding entropies of boiling will vary. Hildebrand (Hildebrand, 1915, 1978; Hildebrand and Scott, 1950) has extensively discussed the importance of comparing Trouton’s ratios at the same saturated vapor volumes. From his work several methods have been proposed which express the entropy of boiling as a function of boiling temperature. While these methods attempt to correct for the differences in vapor volumes, they cannot entirely account for the fact that the free volume of the liquid is also dependent upon temperature. Even at constant vapor volumes it can be observed (Rice, 1937; Halford, 1940; Hermsen and Prausnitz, 1961; Hildebrand and Scott, 1950) that the entropy of boiling is not entirely constant. Conformational and rotational restrictions in the liquid that are not present in the vapor will also contribute to the overall entropy. These restrictions are a consequence of molecular structure, and therefore, a simple constant as described by Trouton’s rule cannot be expected to be universally applicable. Experimental Section The following molecular descriptors and experimental setup was used in this study. Hydrogen Bonding. Preliminary studies were performed to assess the effect of hydrogen bonding on the entropy of boiling. Three significant factors were observed. First, only alcohols, carboxylic acids, and primary amines significantly changed the entropy of boiling with the contribution of the primary amines approximately one-third that of an alcohol or acid. Secondly, the effect of hydrogen bonding tends to decrease with increasing molecular size. Finally, there is a diminishing change in the entropy of boiling for compounds with more than one hydrogen-bonding group. From these preliminary findings an empirical parameter called the hydrogen bond number, HBN, was developed. It is defined by the following equation
HBN )
xOH + COOH + 0.33xNH2 MW
different conformations. Butane on the other hand has one torsional bond, pentane has two, hexane has three, etc. The effective number of torsional bonds has been defined by Dannenfelser and Yalkowsky (1996) by
(4)
where OH, COOH, and NH2 represent the number of alcohols, carboxylic acids, or primary amines, respectively, and MW is the molecular weight of the compound. The calculated HBN for some representative compounds are given in Table 1. Flexibility. The conformational freedom, or flexibility, of a molecule arises from the ability of its atoms to torsionally rotate about single bonds. Note that the rotation of hydrogen atoms is ignored. For example, ethane and propane have no torsional bonds since the rotation about any of the single bonds will not produce
∑(SP3 + 0.5SP2 + 0.5RING) - 1
(5)
where SP3 and SP2 are the number of nonring, nonterminal sp3 (including NH, N, O, and S) and sp2 atoms, respectively. RING indicates the number of independent ring systems in the compound. Since sp hybrid atoms and tert-butyl groups do not contribute to flexibility, they are not counted. Note that τ is set equal to zero if eq 5 gives a negative value. The calculated τ for some sample compounds are given in Table 1. Dannenfelser and Yalkowsky (1996) have successfully used τ for predicting the entropy of melting of a molecule. Since the entropy of melting considers the conformational entropy change from the solid to the liquid, it follows that this variable will also be applicable to the entropy of boiling. Intercept. An intercept is used in the regression to describe rigid molecules which do not hydrogen bond. Training Set Data. The 850 compounds used in this study were taken from Majer and Svoboda (1985), Zwolinski and Wilhoit (1971), and Reid et al. (1977). Of the compounds compiled only four (cyclohexanol, methanol, dibutyl o-phthalate, and propionic acid) were deleted from the final regression. These outliers were found to deviate by at least 4 standard deviations when using the proposed model and significantly altered the final equation. Test Set Data. The 88 test compounds are from Ma and Zhao (1993). None of these compounds were used in the training set. Statistical Analysis. The data were analyzed by multiple linear regression using Statistical Analysis System (SAS). Results The relationship between the entropy of boiling and the parameters τ and HBN was found by regression analysis to be
∆Sb ) 86 + 0.4τ + 1421HBN n ) 850
(6)
RMSE ) 3.0
Table 2 gives a summary of how eq 6 performs on different types of compounds. For comparison, the results using Trouton’s rule are also included in the table. For the 850 compounds used in the training set the average absolute percent error for eq 6 is 2.6%. Surprisingly, the overall error for Trouton’s rule is only slightly worse at 3.7%. In fact, for the less flexible hydrocarbons (τ < 10), halocarbons, and non-hydrogenbonding compounds the errors for both eq 6 and Trouton’s rule are comparable. However, significant differences between the models are observed for both the very flexible hydrocarbon compounds (τ < 10) and hydrogenbonded compounds. In Table 2, the hydrocarbons have been divided into two classes, those with τ less than 10 and those having a τ greater than or equal to 10. For the very flexible compounds it can be seen that Trouton’s rule systematically underestimates the entropy of boiling. Although the average percent error is not large
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1790 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Table 2. Summary of the Percent Errorsa for Different Types of Compounds Used in the Training Set equation 6 average errors compound type hydrocarbons halocarbons non-hydrogen-bonding (O, N, S) alcohols and 1° amines acids multiple hydrogen-bonding total data set a
τ < 10 τ g 10
Trouton’s rule average errors
n
|%|
%
|%|
%
405 31 78 276 48 4 8 850
2.4 0.9 1.8 2.9 3.5 5.2 3.5 2.6
-1.7 -0.3 0.4 1.9 0.5 -0.4 -0.3 -0.1
3.2 3.2 2.3 2.8 11.4 13.2 19.0 3.7
-3.0 3.2 -1.4 0.7 11.4 13.2 19.0 -0.3
Percent error calculated by [(observed - predicted)/observed] × 100.
(3.2%), it illustrates that the entropy does vary systematically with flexibility. Equation 6, which takes into account flexibility, performs very well on very flexible hydrocarbons having average percent errors of less than 1%. More dramatic systematic deviations from Trouton’s rule are observed for hydrogen-bonding compounds. Trouton’s rule systematically underestimates the entropy of boiling for the alcohols and primary amines (11.4%), acids (13.2%), and multiple hydrogen-bonding compounds (19.0%). The average absolute percent error for the proposed model is 3.5% for the alcohols, primary amines, and multiple hydrogen-bonding compounds and 5.2% for the acids with a molecular weight greater than 75. It is important to note that there are special cases in which hydrogen bonding can cause the entropy of boiling to be lower than expected. If hydrogen bonding results in dimers which are stable upon vaporization, then the number of moles is effectively reduced and the entropy of boiling calculated on the basis of monomeric species will be less than expected. This phenomenon is observed for small carboxylic acids which can dimerize extensively in both the liquid and the vapor state (Alberty, 1987). As a result, the proposed method overestimates the entropy of boiling for formic acid and acetic acid by 55.1 and 49.1 J/deg mol, respectively. Therefore, these compounds were excluded from the training set. Since the overestimation of the entropy of boiling is inversely related to the size of the molecule and directly related to its ability to intermolecularly hydrogen bond, those rare compounds that can exist as dimers in the vapor state should not be estimated using this method. Discussion The regression generated intercept of eq 6 (86 J/deg mol) is almost equal to Trouton’s rule (88 J/deg mol), supporting the applicability of the latter to relatively rigid, non-hydrogen-bonding compounds. And since 759 of the 850 compounds used in the training set fit this description, it explains the relative success of Trouton’s rule. As discussed previously, the intercept (Trouton’s rule) can be primarily attributed to the gain in translational freedom which occurs as a result of the large increase in free volume upon boiling. Since the boiling points of the compounds used in this study range over several hundred degrees, the effect of the molar volumes of the vapor must be considered. However, if boiling point is included as an additional variable in eq 6, the resulting equation is not significantly more accurate in estimating the entropy of boiling. Furthermore, residual errors from eq 6 did not correlate with boiling point. The fact that boiling point does not significantly improve the
proposed model is not surprising since the parameters used, HBN and τ, are collinear with boiling point (r ) 0.75). The elimination of boiling point as a variable allows the prediction of the entropy of boiling directly from molecular structure. The positive coefficient of HBN in eq 6 indicates that hydrogen bonding significantly increases the entropy of boiling for alcohols and acids and, to a lesser extent, for amines. It can be reasoned that hydrogen bonding inhibits rotational freedom in the liquid phase whereas hydrogen bonds have no effect over the large intermolecular distances of the gas phase. (The low molecular weight acids and water are notable exceptions.) Hydrogen bonds in a liquid increases the magnitude of loss of order that occurs with boiling. Hydrogen bonding also increases the normal boiling point which will, in turn, increase the translational freedom gain. Hence, the entropy of boiling for compounds that hydrogen bond is greater than that of non-hydrogen-bonding liquids. The effect of a hydrogen-bonding group was found to decrease as the size of the compound increases. As molecular size increases there are fewer hydrogenbonding groups per unit volume, and the probability of one hydrogen-bonding group interacting with another is reduced. As a result any effect of hydrogen bonding on rotational or translational freedom will decrease with increasing molecular size. To account for this, molecular weight is included in the denominator of the HBN term (eq 4). Although the addition of a second hydrogen-bonding group increases the entropy of boiling, the relationship between the number of hydrogen-bonding groups and the increase in the entropy of boiling is not linear. Empirically it was found that the square root of the number of hydrogen-bonding groups effectively describes the multiple hydrogen-bonding compounds used in this study. Another significant component of the entropy of boiling is the molecular flexibility. The coefficient of 0.40 for τ reconfirms the observations of Mishra and Yalkowsky (1990) that there is a dependency of entropy of boiling upon flexibility. This is believed to be due to the fact that the conformational and rotational freedom of long chain molecules is somewhat restricted in the liquid phase as compared to those in the gas phase. In addition, the boiling points increase with chain length which in turn increases translational freedom. It is interesting to note that the entropy of boiling for the normal paraffins does not show a simple linear dependence upon carbon number, rather it is slightly concave. This trend was also observed by Screttas and Micha-Screttas (1991). The decrease in entropy gain with increasing chain length may be due to the fact that boiling points converge with chain length thus dimin-
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1791 Table 3. Summary of the Percent Errorsa for Different Types of Compounds Used in the Test Set equation 6 ave errors compound type hydrocarbons halocarbons non-hydrogen-bonding (O, N, S) alcohols and 1° amines acids multiple hydrogen-bonding total test set a
τ < 10 τ g 10
Trouton’s rule ave errors
Ma and Zhao ave errors
n
|%|
%
|%|
%
|%|
%
57 1 6 11 13 0 0 88
2.4 0.8 1.5 3.2 3.0
-1.4 -0.8 0.5 2.6 1.6
3.2 1.6 2.1 2.7 13.3
-2.5 1.6 -1.5 0.9 13.3
2.2 4.0 0.8 1.5 3.9
0.5 4.0 -0.5 -1.5 -1.6
2.5
-0.3
4.6
0.4
2.3
-0.1
Percent error calculated by [(observed - predicted)/observed] × 100.
ishing differences in free volume. Additionally, the conformational freedom is not simply a linear function of chain length since some conformations are forbidden because they would require two atoms to occupy the same space (Smith, 1965, 1966). A final observation can be made regarding differences between aliphatic and aromatic compounds. In general, for the current data set, the proposed model slightly overestimates the entropy of boiling for aliphatic and alicyclic compounds and slightly underestimates the aromatic compounds. While this trend seems to be relatively systematic, additional parameters to account for these differences did not substantially increase the accuracy of the method. Hence, in order to preserve the simplicity of the model no additional parameters were included. Validation of the Method The proposed model was applied to the 88 compound test set developed by Ma and Zhao (1993) to evaluate their estimation scheme. A summary of the results is given in Table 3. Also included in Table 3 are the predictions of Trouton’s rule and the method of Ma and Zhao. It is important to note that Ma and Zhao did not consider acids or multiple hydrogen-bonding compounds, and as a result none are included in the test set. The results of the test set for eq 6 and Trouton’s rule closely mirror those observed for the training set (Table 2). Both methods do equally well for the relatively rigid hydrocarbons (τ < 10), halocarbons, and nonhydrogen-bonding compounds having average errors less than 3.2%. Significant differences are observed for the very flexible hydrocarbons and the hydrogen-bonding compounds. Trouton’s rule systematically underestimates the entropy of boiling for the very flexible hydrocarbon (1.6%) and hydrogen-bonding compounds (13.3%), whereas the average absolute errors for eq 6 are only 0.8% and 3.0%, respectively. Equation 6 is also comparable in accuracy to the group contribution method of Ma and Zhao which consists of nearly 100 molecular fragments. For the total test set the average absolute percent error for eq 6 is 2.5%, while the average absolute error for the method of Ma and Zhao is 2.3%. In summary, a constant such as that defined by Trouton is shown to be applicable to relatively rigid, non-hydrogen-bonding compounds. Deviations from the constant can be effectively taken into account by considering molecular flexibility and hydrogen bonding. The proposed method gives a simple means of estimating the entropy of boiling for structurally diverse organic compounds directly from molecular structure.
Acknowledgment This work was supported through a grant from the Environmental Protection Agency (R-817475-01). The contents of this paper do not necessarily reflect the views and policies of the EPA. Supporting Information Available: Tabulated data listing the entropies of boiling used in this study along with the calculated values of equation 6. Also included are the data for the 88 compounds used in the test set along with the predictions from eq 6, Trouton’s rule, and the method of Ma and Zhao (24 pages). Ordering information is given on any current masthead page. Nomenclature ∆Sb ) entropy of vaporization at the normal boiling point, J/deg‚mol ∆Strans ) translational entropy, J/deg‚mol ∆Srot ) rotational entropy, J/deg‚mol ∆Scon ) conformational entropy, J/deg‚mol f Vgas ) free volume in the gas, L/mol f Vliq ) free volume in the liquid, L/mol Wgas ) number of ways a molecule can be arranged in the gas Wliq ) number of ways a molecule can be arranged in the liquid HBN ) hydrogen bond number MW ) molecular weight, dalton Greek Symbols τ ) effective number of torsional bonds
Literature Cited Alberty, R. A. Physical Chemistry, 7th ed.; Wiley: New York, 1987. Dannenfelser, R-M.; Yalkowsky, S. H. Estimation of Entropy of Melting from Molecular Structure: A Non-Group Contribution Method. Ind. Eng. Chem. Res. 1996, 35, 1483-1486. Halford, R. S. Entropy of Vaporization and Restricted Molecular Rotation in Liquids. J. Chem. Phys. 1940, 8, 496-499. Hermsen, R. W.; Prausnitz, J. M. Entropies of Vaporization of Hydrocarbons and the Hildebrand Rule. J. Chem. Phys. 1961, 34, 1081-1083. Hildebrand, J. H. The Entropy of Vaporization as a Means of Distinguishing Normal Liquids. J. Am. Chem. Soc. 1915, 37, 970-978. Hildebrand, J. H. Theories and Facts About Liquids. Faraday Discuss. Chem. Soc. 1978, 66, 151-152. Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes, 3rd ed.; Reinhold Publishing Corp.: New York, 1950. Hoshino, D.; Nagahama, K.; Hirata, M. Prediction of the Entropy of Vaporization at the Normal Boiling Point by the Group Contribution Method. Ind. Eng. Chem. Fundam. 1983, 22, 430433. Ma, P.; Zhao, X. Modified Group Contribution Method for Predicting the Entropy of Vaporization at the Normal Boiling Point. Ind. Eng. Chem. Res. 1993, 32, 3180-3183.
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1792 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Majer, V.; Svoboda, V. Enthalpies of Vaporization of Organic Compounds, A Critical Review and Data Compilation; Blackwell Scientific: Oxford, 1985. Mishra, D.; Yalkowsky, S. H. Estimation of Entropy of Vaporization: Effect of Chain Length. Chemosphere 1990, 21, 111117. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill, Inc.: New York, 1977. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill, Inc.: New York, 1987. Rice, O. K. Internal Volume and the Entropy of Vaporization of Liquids. J. Chem. Phys. 1937, 5, 353-358. Screttas, C. G.; Micha-Screttas, M. Some Properties and Trends of Enthalpies of Vaporization and of Trouton’s Ratios of OrganicCompounds. Correlation of Enthalpies of Vaporization and Enthalpies of Formation with Normal Boiling Points. J. Org. Chem. 1991, 56, 1615-1622. Smith, R. P. Polymethylene Chains and Rings on a Diamond Lattice with Atom Overlap Excluded. J. Chem. Phys. 1965, 42, 1162-1166.
Smith, R. P. Configurational Entropy of Polyethylene and Other Linear Polymers. J. Polym. Sci., Part A-2 1966, 4, 869880. Trouton, F. On Molecular Latent Heat. Philos. Mag. 1884, 18, 54-57. Zwolinski, B. J.; Wilhoit, R. C. Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds; Thermodynamic Research Center/API, Texas A&M: Fort Worth, TX, 1971.
Received for review February 28, 1995 Revised manuscript received January 30, 1996 Accepted January 30, 1996X IE950147C
X Abstract published in Advance ACS Abstracts, April 1, 1996.
