SIMPOL . 1 : A simple group contribution method for predicting vapor pressures and enthalpies of vaporization of multifunctional organic compounds

1 Department of Environmental and Biomolecular Systems, OGI School of Science & Engineering, Oregon Health & Science University, Beaverton, Oregon 97006-8921, USA 2 Air-Sea Interaction and Remote Sensing Department, Applied Physics Laboratory, University of Washington, Seattle, Washington 98105, USA Received: 3 July 2007 – Accepted: 18 July 2007 – Published: 13 August 2007 Correspondence to: J. F. Pankow (pankow@ebs.ogi.edu)


Introduction
For organic compound i, knowledge of the liquid vapor pressure p o L, i at the system temperature (T ) is required whenever phase equilibrium of i between a liquid phase and the gas phase is of interest. This type of partitioning arises frequently in many disciplines, and so the need for reliable p o L, i values is considerable. And, since the T dependence of p o L, i is determined by the compound-dependent enthalpy of vaporization H vap,i , the same need extends to H vap,i values. In our case, the topic of interest is gas/particle partitioning in atmospheric and smoke aerosol systems (e.g., Pankow, 1994aPankow, , 1994bPankow, , 2001Pankow et al., 2001Pankow et al., , 2004Barsanti and Pankow, 2004. Given the infinite structural variety possible with organic compounds, laboratory measurements will never keep pace with the need for new p o L, i information. Consequently, there is continuing interest in the development of reliable methods for predicting p o L, i and H vap,i values. In the case of the behavior and formation of organic particulate matter (OPM) in the atmosphere, there is growing interest in a Published by Copernicus Publications on behalf of the European Geosciences Union. Table 1a. Non-oxygenated, hydroxyl, ketones, aldehydes, and carboxylic acid saturated compounds in the basis set for the initial fit.
Quantum-mechanical calculations are making steady progress in the theater of predicting p o L, i values for any structure of interest (Diedenhofen et al., 2007;Verevkin et al., 2007;Banerjee et al., 2006;Tong et al., 2004). However, prediction efforts for more complicated structures can now only be based on either a complex consideration of the interaction forces between molecules (i.e., dispersion, induction, dipole and H-bonding) as in the SPARC model discussed by Hilal et al. (1994), or by empirical group-contribution means.
In the group contribution approach to prediction of molecular properties, it is hypothesized that the value of a property of interest for compound i can be predicted based on empirically determinable contributions from the structural fragments that comprise i. As a function of temperature T , the result is often an equation of the type log 10 Z i (T ) = b 0 (T ) + k ν k,i b k (T ) where: Z i (T )is the property of interest, e.g., p o L, i (T ); the parameter b 0 (T ) is a T -dependent constant; ν k,i is the number of groups of type k in i; the index k may take on the values 1,2,3, etc.; and b k (T ) is the group contribution term for group k. Values for b 0 (T ) and the set of b k (T ) are usually determined by fitting (i.e., optimizing) Eq. (1) using laboratory-based measures of Z i (T ) for a large number of compounds that contain the groups of interest. For example, for both 2,3-and 2,4-dihydroxypentane it can be considered that ν OH, i = 2, ν CH 3 , i =2, ν CH 2 ,i =1, and ν CH,i =2. In this approach, four b k (T ) values are required, and Eq. (1) will give the same prediction for Z i (T ) for both isomers. However, the vicinal nature of the two OH groups in 2,3dihydroxypentane allows greater intramolecular interaction of the OH groups (and less intermolecular interaction) than in the 2,4 isomer, causing differences in molecular properties. In the case of vapor pressure, p o L, i (T ) will be higher for the 2,3 isomer than for the 2,4 isomer. Accounting for such property differences among isomers can be accomplished by consideration of additional, "higher-order" groups. Thus, for 2,3-dihydroxypentane a "second-order"
In the most general application of a group contribution model, the fitting takes place over a broad range of compound types, e.g., simple alkanes, functionalized alkanes, aromatics, functionalized aromatics, etc. In that case, b 0 (T ) serves as a general fitting constant. Alternatively, the fitting can take place within a particular class of compounds, as in the study by Lee et al. (2000) of substituted benzene compounds wherein for predicting p o L, i (298.15) the value of b 0 (298.15) was not obtained from the fitting process. Rather, it was defined that b 0 (298.15)= log 10 p o L, benzene (298.15). A secondorder group contribution model was then fit to log 10 p o L, i (298.15)= log 10 p o L, benzene (298.15)+ k ν k,i b k (298.15) The summation accounts for how the presence of the various first-and second-order groups cause p o L, i (298.15) to differ from p o L, benzene (298.15).
In a generalization (though first order) of the Lee et al. (2000) approach, Capouet and Müller (2006) allowed that a range of parent structures would be of interest, and so existing p o L,i (T ) data for a range of compounds were fit to where p o L,hc−i (T ) is the known vapor pressure for the nonfunctionalized hydrocarbon (hc) compound that possesses the skeletal structure underlying compound i, and the τ k (T ) are conceptually equivalent to the b k (T ). Application of Eq.
(3) to a particular i requires knowledge (or an independent prediction) of p o L,hc−i (T ); the summation accounts for how the substituents in i cause p o L,i (T ) to differ from p o L,hc−i (T ). In the fitting carried out by Capouet and Müller (2006), multiple different hc-i structures were considered; the corresponding p o L,hc−i (T ) and p o L,i (T ) were taken as the inputs, and the output was a set of τ k (T ) encompassing 10 groups: OH (as bonded to a primary, secondary, and tertiary carbon); C=O (aldehyde or ketone); COOH; hydroperoxy; nitrate (primary, secondary, and tertiary); and peroxyacetylnitrate (PAN). Table 1c. Amides, amines, ethers, and nitrate-group containing compounds in the basis set for the initial fit.
(3) carries accuracy advantages for predicting p o L,i (T ) values because each prediction utilizes important specific knowledge of the vapor pressure of the compound with the underlying hc-i structure. It is not surprising, then, that Capouet and Müller (2006) report generally better prediction accuracies for the Eq. (3) method than with the more general UNIFAC-p o L method of Asher et al. (2002), though the fitting constants in Asher et al. (2002) have been superseded by those given in Asher and Pankow (2006). In any case, as a practical matter, requiring knowledge of p o L,hc−i (T ) can be a significant disadvantage relative to a more general method that can be executed Table 1e. Peroxide, hydroperoxide, and carbonylperoxynitrate-group containing compounds in the basis set for the initial fit.

Hydroperoxides
Peroxyacids 1-methyl-1-phenyl-ethyl-hydroperoxide 1-oxo-ethyl-hydroperoxide methyl-hydroperoxide 1-oxo-propyl-hydroperoxide ethyl-hydroperoxide 1-oxo-butyl-hydroperoxide (1,1-dimethylethyl)-hydroperoxide using fitting constants alone, e.g. the method of Asher and Pankow (2006) or that of Makar (2001). Moreover, for the compounds that actually form OPM in the atmosphere, good knowledge of the underlying structures is lacking, the available information being limited to a general idea of structural characteristics such as the number of carbons, the likely number and types of functional groups, and whether any aromatic or non-aromatic rings are likely to be present. The goal of this work was to develop a simple p o L,i (T ) group contribution method for which that level of information would be sufficient.

General
The groups of interest considered include a range of firstorder group functionalities important for organic compounds involved in OPM formation, and several second order groups. Nevertheless, the total number of groups N G was kept as small as possible while still affording good accuracy of the overall fit: SIMPOL.1 is not intended as a method that employs many second-and third-order groups. The SIMPOL.1 method is based on wherein the role of b 0 (T ) is the same as in Eq. (1), and the index k may take on the values 1,2,3, etc. The units carried by p o L,i (T ) are atm. The form of Eq. (4) is equivalent to log 10 p o L,i (T )=ν 0,i b 0 (T )+ k ν k,i b k (T ) k=1, 2, 3... (5) so that b 0 (T ) can be viewed as pertaining to group "zero", with ν 0,i ≡ 1 for all i. Thus, Eqs. (4) and (5) are equivalent to log 10 p o L,i (T ) = k ν k,i b k (T ) k= 0, 1, 2, 3...
wherein k may take on the values 0,1,2,3, etc., and for k=0, ν 0,i ≡1 for all i.  Perhaps the most important chemical group in SIMPOL.1 is molecular carbon, for which k=1. Thus, ν 1, i denotes the number of carbon atoms in i, and b 1 (T ) denotes the percarbon group contribution to log 10 p o L,i (T ). At ambient temperatures, b 1 (T )≈−0.5 (see Table 6 below) and so within any given compound class, p o L,i (T ) drops by about 1/3 of  an order of magnitude for every unit increase in the carbon number.
By way of comparison with prior work from our group, Asher and Pankow (2006) follow Jensen et al. (1981) and begin with log 10 p o L,i (T )= k ν k,i log 10 ( k,i )+ g k (T ) 2.303RT (7) where: each log 10 ( k,i ) is a UNIFAC "residual term" that accounts for the intramolecule and intermolecular group-group interactions involving group k; R is the gas constant; and g k (T )is the difference between the molar free energy of group k in the pure liquid state and in the perfect gas at 1 atm. After using tabulated values of UNIFAC group interactions parameters compiled in Hansen et al. (1991) to compute k ν k,i log 10 ( k,i ) for the compounds in their basis set, Asher and Pankow (2006) fit p o L,i (T ) data values to Eq. (7) to obtain expressions for g k (T ); a total of 24 groups were considered. Adoption of Eq. (6) in place of Eq. (7) amounts to assuming that each b k (T ) can be fit as a lumped equivalent of [log 10 ( k,i ) + g k (T )/2.303RT ].
In SIMPOL.1, the T dependence in each of the b k (T ) is fit to its own set of B 1,k to B 4,k according to which is the form of the T dependence utilized for the 17 coefficients in the UNIFAC model of Jensen et al. (1981). The goal of this work is to use p o L,i (T ) data for a wide range of compounds to obtain best-fit functional representations of the b k (T ). The temperature dependence of log 10 p o L,i (T ) may be used to estimate H vap,i (T ) according to Thus, by Eq. (6) Eq. (10) may be viewed as a group contribution expression for H vap,i (T ) based on the SIMPOL.1 framework where the group contribution to H vap,i (T ), defined as h vap,i (T ), is given by each term in the summation in Eq. (10). Eq. (10) may also be used to derive the predicted change in H vap,i (T ) as a function of T in the SIMPOL.1 framework. Substitution of the functional form for b k given in Eq. (8) into Eq. (10) and taking the derivative with respect to T results in For any real compound i in the liquid state, H vap,i (T )>0, but d H vap,i (T )/dT <0 because H vap,i decreases monotonically to zero as T approaches the compound's critical temperature T c,i (Reid et al., 1986). (As Table 3a. Non-oxygenated, hydroxyl, phenolic, aldehyde, ketone, and carboxylic acid compounds in the test set for the initial fit.

Fitting the SIMPOL.1 coefficients
All B 1,k − B 4,k sets were determined by an optimization process using a set of compounds with measured p o L (T ) values. See Asher et al. (2002) and Asher and Pankow (2006) for descriptions of this type of process. The optimization used nonlinear regression to minimize a least-squares goodnessof-fit criterion defined as Table 3b. Amide, amine, ester, ether, nitrate, nitro-containing, and peroxide compounds in the test set for the initial fit.

Amides
Ethers where: N c is the number of compounds (=272 for the initial basis set); N G is the total number of groups considered; and each (p o L,i (T j,i )) E is the vapor pressure of i at temperature T as evaluated using a p o L,i =f i (T ) expression (e.g., an Antoine-type equation) fitted to experimentally derived For compounds for which f i (T ) had been fit over that entire range, N T,i =7; for others, N T,i <7. With the initial basis set compounds, the total number of points considered in the optimization was N=1844.

Optimization
There is no general theoretical method for determining whether a local minimum χ 2 value found by optimizing the set of B values for Eq. (1) is the desired global minimum. However, beginning the optimization with a large number of suitably different sets of initial B values provides an equal number of optimized χ 2 values, and selecting the lowest of these local minima provides a measure of confidence that the corresponding optimized B set either is the set for the global minimum, or is nearly as good as the set for the global minimum.
The χ 2 fitting function in Eq. (1) was minimized using the generalized reduced-gradient method (Lasdon et al., 1978) contained in the nonlinear optimization routines LOADNLP and OPTIMIZE from SOLVER.DLL (Frontline Systems, Boulder, Colorado). The optimization was performed in two steps. First, 100 sets of initial B values (with each   set containing 31×4 initial values) were populated randomly (though subject to the condition that the absolute value of all four terms on the right-hand side of Eq. (8) were of order unity). The mean and standard deviation of the 100 χ 2 values were 472 and 23, respectively. The smallest of these χ 2 was 372.
In the second step of the optimization, the set of B values giving χ 2 =372 was subjected to further refinement by running 100 additional optimizations, varying each B by a random amount, with all variations restricted within ±30%. The mean and standard deviation of the resulting 100 χ 2 values were 332 and 3, respectively. The smallest of the χ 2 was 325. Further attempts to refine the coefficients did not produce any significant decrease in χ 2 . When comparing the B set for χ 2 = 372 to the set for χ 2 =325, the median absolute difference is 30%.

Fit accuracy of SIMPOL.1 with initial basis set compounds
The overall agreement between the experimental and predicted values can be assessed in terms of an absolute value form of standard error of the fit: where (p o L,i (T j,i )) P is the predicted vapor pressure for i at temperature T j,i by Eq. (6). For the basis set, N c =272 and N=1844 (see above), and using the set of B giving χ 2 =325 yields σ FIT =0.29 (log units): on average, (p o L,i (T j,i )) E for compounds in the basis set is predicted to within a factor of ∼2. This is evidenced in Fig. 1, which is a plot of log 10 (p o L,i (T j,i )) P vs. log 10 (p o L,i (T j,i )) E for the initial basis set compounds at 333.15 K, the lowest T to which all of the experimentally based p o L,i =f i (T ) expressions extended. It should be noted that although the minimum p o L,i shown in Fig. 1 is 10 −9 atm, there were 24 values of p o L,i included in the optimization that were lower than 10 −9 atm with a minimum p o L,i of 7.90×10 −14 atm. However, these lower values were for compounds at lower temperatures, where the data is not shown on the figure.
Given the multi-functionality possessed by many of the compounds, the 13 major compound class designations used in the figures are somewhat arbitrary. The "saturated" class for example, includes all compounds lacking double bonds and aromatic rings that are not assigned to another class, and so includes simple alcohols, carbonyls, and acids. Similarly, the nitro class contains compounds having only nitro groups, but also compounds with nitro groups and hydroxyl, carbonyl, or acid functionality. Table 2 provides σ FIT for the initial basis set by compound class, i.e., with N c and N in Eq. (13) limited to represent the compounds within a particular class. Figure 2 provides a plot of the corresponding individual σ FIT,i vs. log 10 (p o L,i (T j,i )) E for 333.15 K.
An estimate of the method bias towards over-or underfitting the p o L,i is obtained by a variation of Eq. (13) that does not use absolute values:      For the initial basis set of compounds, the set of B producing χ 2 =325 gives σ SGN =1.4×10 −3 (log units). This indicates that as averaged over all 272 initial basis set compounds and seven temperatures, there is no significant bias in the fitting; the σ SGN values in Table 2 indicate that this result extends down to each of the 13 major compound classes considered. Figure 3 provides a plot of the corresponding individual σ SGN,i vs. log 10 (p o L,i (T j,i )) E for 333.15 K.

Method validation of SIMPOL.1 with a test set of compounds
The ability of the set of B coefficients producing χ 2 =325 to predict values of (p o L,i (T j,i )) E for compounds outside the initial basis set was examined using a test set of 184 compounds (Table 3) atmos-chem-phys.net/ 8/2773/2008/acp-8-2773-2008-supplement.pdf. Averaged over all test set compounds, σ FIT =0.45, and σ SGN =−0.071: the average prediction error is a factor of ∼3, and there is no significant overall bias. Table 4 gives σ FIT and σ SGN values for the test compounds when N c and N are limited to represent the compounds within a particular compound class. Overall, given the wide range of compounds in the test set, SIMPOL.1 does well in predicting (p o L,i (T j,i )) E . However, the individual compounds for which the performance is appears to be poor bear some discussion. In the case of the nitro class, σ FIT and σ SGN are 1.0 and 0.40, respectively. These ostensibly poor results are driven by: 1) the small number of nitro compounds in the test set; and 2) large apparent errors for only two of the nitro compounds, 3-nitrophenol and 4nitrophenol (σ FIT,i =2.64 and 2.24, respectively). The cause of the poor performance for 3-nitrophenol and 4-nitrophenol is not clear. By comparison, for 2-nitrophenol (which is in the initial basis set), σ FIT,i is better (0.42). Thus, there might be a large effect of meta and para substitution on p o L for nitrophenols. Alternatively, it is possible that the (p o L,i (T j,i )) E values for 3-nitrophenol and 4-nitrophenol are in error. Indeed, it is undoubtedly true that some of the experimentally based p o L,i =f i (T ) expressions suffer from significant error: numerous prior parameter prediction studies have identified experimental data that likely are in error, e.g., see the comments by Rathbun (1987) on the likelihood of errors in the p o L data of Stull (1947) for 2-pentanone and other similar ketones.
Besides compounds containing the nitro group, method performance appears to be relatively poor for some compounds in the saturated class, the aromatic class, and some compounds in the ether class. For the saturated class, σ FIT =0.66, due mainly to 2-hydroxy-2-methyl-3-hexanone, 2-ethyl-hexanoic acid, norpinic acid, and the three longchain hydroxyl acids. When these six compounds are removed, σ FIT for the remaining 32 compounds is lowered to 0.51, and the method may be viewed as performing relatively well. Given their relatively simple structures, (i.e., the absence of likely effects from higher order groups), errors in some of the (p o L,i (T j,i )) E values for compounds in the saturated class seem possible. This is especially the case for the hydroxyl acids, where at T =273.15 K SIMPOL.1 underestimates the measured (p o L,i (T j,i )) E by over 3 orders of magnitude. However, SIMPOL.1 overestimates the measured (p o L,i (T j,i )) E by a factor of at least 10 for T =273.15 K, suggesting that the dependence of (p o L,i (T j,i )) E on T is very different from that predicted overall by SIMPOL.1. For the ethers, the overall error is relatively low, σ FIT =0.42, with a relatively large bias, σ SGN =−0.25, but there are not consistent patterns in the results that explain the relatively large bias. However, for trans-2,2,4,6-tetramethyl-1,3-dioxane, for its set of T j,i values, σ FIT,i averages 1.2; removing this compound from the average for the ether class dramatically reduces the magnitudes of σ FIT and σ SGN for the ethers to 0.34 and -0.15, respectively.

Final coefficients for SIMPOL.1 and associated error estimates for p o L values
In the determination of the final set of B coefficients, the basis set compounds in Table 1 were combined with the test set compounds in Table 3. For this combined set (456 compounds), the set of B coefficients determined using the initial basis set gives χ 2 =855. For each of 100 subsequent optimization runs, the initial value of each B coefficient was taken as the final value determined using the initial basis set modified randomly by at most ±30%. The lowest χ 2 value thus obtained was 728 (mean=736, standard deviation=6). Further optimization attempts did not succeed in lowering χ 2 . Table 5 gives the final B coefficients giving χ 2 =736. Table 6 gives the values of the b k (T ) at T =293.15: at that T , adding one carbon, carboxylic acid, alkyl hydroxyl, ketone, or aldehyde groups alters log 10 p o L,i by -0.438, -3.58, -2.23, -0.935, and -1.35, respectively. For comparison, Table 6 also provides the corresponding values of τ k (293.15) from Capouet and Müller (2006); these are generally similar to the b k (293.15) determined here. For the carboxylic acid, primary hydroxyl, and carbonyl (i.e., ketone or aldehyde) groups, Capouet and Müller (2006) give τ k (293.15)=-3.10, -2.76, and -0.91.
Consider the transformation of cyclohexene to adipic acid, an example that has historical significance in the evolution of the understanding of the formation of secondary OPM in the atmosphere (Haagen-Smit, 1952). For cyclohexene, υ 0 = 1, υ 1 =6, υ 4 =1, and υ 5 =1, and by Eq. (6) and the values in Table 6, SIMPOL.1 predicts log 10 p o L (293.15)=−0.94. For adipic acid, υ 0 =1, υ 1 =6, and υ 10 =2, and SIMPOL.1 predicts log 10 p o L (293.15)=−7.99. Overall, for cyclohexene→adipic acid, the SIMPOL.1 method provides a simple parameterization for quantifying how addition of two COOH groups (b 10 =−3.58 at 293.15 K) causes a seven order magnitude change in volatility. The log 10 p o L (293.15) values derived using SIMPOL.1 may be compared with experimental values as follows. For cyclohexene, data in Lister (1941), Meyer and Hotz (1973) and Steele et al. (1996) yield the Antoine fit log 10 p o L (T )=4.814-(1713/(T +0.04870)), which gives log 10 p o L (T )=−1.08 at 293.15 K. For adipic acid, when the p o S (T ) (sublimation) data of Davies and Thomas (1960), Tao and McMurray (1989), Chattopadhyay andZieman (2005), andCappa et al. (2007) are combined with the entropy of fusion data of Roux et al. (2005) and averaged with sub-cooled liquid vapor pressures from Bilde et al. (2003), the resulting value for log 10 p o L is -8.49 at 293.15 K. Figure 7a provides a plot of log 10 (p o L,i (T j,i )) P vs. log 10 (p o L,i (T j,i )) E for all compounds at T =333.15 K and Fig. 7b is a plot of log 10 (p o L,i (T j,i )) P vs. log 10 (p o L,i (T j,i )) E for all compounds at all seven temperatures showing the full lower volatility range of the dataset; Figs. 8 and 9 provide corresponding plots of σ SGN,i and σ FIT,i vs. log 10 (p o L,i (T j,i )) E except in the interest of brevity the data in Figs. 8 and 9 are shown for T =333.15 K only. Table 7 provides σ SGN and σ FIT values by compound class and sub-class. All σ SGN values for the major classes are low (no significant biases). However, among the compounds containing the nitro group, as noted above, p o L is predicted poorly for 3-nitrophenol and 4-nitrophenol. When these two compounds are excluded, σ FIT for the nitro class is reduced from 0.50 to 0.42, but even so prediction for this class seems problematical. As discussed above, this may be due to complexities in the effects of structure on p o L with nitro-containing compounds, or accuracy problems with the experimental data. Figure 10 shows σ FIT at various T by major compound class. For some classes, e.g., amides and peroxides, the mean error is least for T values in the center of the fitted range, and larger at both T <300 K and T >360 K. This type of parabolic behavior in the error is typical of least-squares fitting carried out over a specific data range for the independent variable. The relatively larger errors at lower T for all classes are likely exacerbated due to the increase in experimental difficulty at low p o L . Evidence of this difficulty at low p o L is shown in Fig. 11 using data for nitroethanol. Fig. 12 plots σ FIT vs. log 10 (p o L,i (T j,i )) E , again showing the general tendency in the error to increase with decreasing log 10 (p o L,i (T j,i )) E . The increase in σ FIT with decreasing p o L is most likely a combination of the relatively small number of data points at low vapor pressure, the increase in experimental error with decreasing volatility, and the parabolic error profile for a least-squares type of approach.

3.4
H vap,i prediction using SIMPOL.1 with final coefficients Values of H vap,i may be predicted using Eq. (10) and the final B coefficients in Table 5. Figure 13 shows predicted values of H vap,i at T =333.15 K vs. experimentally based values derived by consideration of the experimental p o L,i =f i (T ) functions and Eq. (9). Table 8 summarizes the quality of the predictions at T =333.15 K based on the following un-normalized (σ ) and normalized (i.e., relative, ρ) error estimates, with each in absolute value and signed form: For all compounds, σ H =8.9 kJ mol −1 , σ H SGN =2.7 kJ mol −1 , ρ H =0.16 (i.e., 16%), and ρ H SGN =0.080 (i.e., 8%). Overall, the fit is reasonably good, especially considering that the fitted quantity was not H vap,i but rather the underlying p o L,i (T ) functionalities.
3.5 Temperature dependence of H vap,i using SIMPOL.1 with final coefficients As noted above, theoretical considerations indicate that d H vap,i /dT <0 for any real compound below its critical temperature T c,i . Examination of values returned by Eq. (10) with Eq. (11) indicate that while imperfect, the results are encouraging in this regard, with 408 of the 456 compounds considered returning d H vap,i /dT < 0 for T =335.15 K. The results by compound class and sub-class are given in Table 9. At any given T <T c,i , though we know that d H vap,i (T )/dT <0 (see above), this does not require for any particular group k that d h vap,k (T )/dT <0, only that the υ k,i − weighted sum is negative. However, since all υ k,i ≥0, by Eq. (11), at least some fraction of the structurally important groups must give d h vap,k (T )/dT <0. Table 6 gives the sign of the d h vap,k (T )/dT values at 293.15 K for the SIMPOL.1 groups based on Eq. (11) and the B values in Table 5. Importantly, for the carbon group (k=1), d h vap,k (T )/dT <0. This result is important in causing d H vap,i (T )/dT <0 to be predicted for many of the compounds in Tables 1 and 3.
For the method of Capouet and Müller (2006), taking the derivative of Eq. (3) with respect to (1/T ) and consideration of Eq. (9) yields The analogous expressions for the SIMPOL.1 representation are Eqs. (10) and (11), respectively. The functionality selected for the b k (T ) as fitted by Capouet and Müller (2006) is τ k (T )=α k +β k T , giving d τ k (T )/d(1/T )= − β k T 2 and (d/dT )d τ k (T )/d(1/T )=−2β k T . In the fitting results reported by Capouet and Müller (2006), all β k >0. Thus in that fitting, the role of forcing d H vap,i (T )/dT < 0 must then be borne entirely by d H vap,hc−i (T )/dT . This is not possible for any real compound i. The latter derivative is only capable of bringing H vap,hc−i (T ) to zero, and for the groups considered by Capouet and Müller (2006), H vap,i (T )> H vap,hc−i (T ). Caution should therefore accompany use of the temperature dependencies given for the τ k in Capouet and Müller (2006).
Overall, regardless of the p o L (T ) prediction method used when modeling the atmospheric behavior of a compound over a particular T interval, when it is correctly predicted over the entire interval that d H vap,i /dT < 0, then the T dependence given by Eq. (10) may be used. However, when d H vap,i /dT >0 over some portion of the T interval of interest, H vap,i should be evaluated at the central T and then assumed to remain constant over the entire interval.

Conclusions
A simple group contribution method has been developed that allows prediction of p o L,i and H vap,i values based on straightforward molecular structure considerations. Extensive error analyses for both parameters provide a detailed understanding of the reliability of the estimates by compound class and sub-class.
One of the implications of this work is related to the information in Figs. 10, 11, and 12. There is an obvious increase in error of the fit at low vapor pressures and temperatures. The reasons for this are related to the difficulty of making accurate measurements of p o L,i for low temperatures and pressures. Improvement in vapor pressure estimation techniques, especially for compounds with p o L,i <10 −10 atm will require additional empirical data.