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Relationships between the Properties of Families of Materials Brian K. Peterson* Computational Modeling Center, Air Products and Chemicals, Inc., Allentown, PennsylVania 18195-1501
A general and useful regularity exists between the properties of families of chemical species: the properties of the members of one family are shown to be simply, and approximately linearly, related to the properties of analogous materials in similar families. As examples, the phenomenon is shown to enable accurate predictive correlations from small amounts of data for the liquid/vapor critical temperatures of several families of organic molecules and for the normal boiling temperatures of SF5- and CF3-containing compounds. Introduction Typical approaches to understanding and predicting the physical properties of materials relate the properties to structural or other features of the atoms and molecules comprising the material. These approaches include theoretical and computational ones based in quantum and/or statistical mechanics1 and numerical/statistical ones parametrized by data, such as group contribution methods2 and quantitative structure/property relationships.3 Perhaps the simplest method is to use a homologous series if one exists: the property is related to some single feature that varies within a family of molecules, and a smooth curve fitting the data is used to predict the property for members for which data are missing. Emphasis is usually placed on finding an explicit relationship between molecular structure and the property of interest. We have found a quite general effect that leads in many cases to an alternative method of analysis and prediction and that does not require any explicit structure/property relationship. When a property of members of one family of materials is compared to the same property of analogues in related families of materials, a simple, usually linear, relationship is found. We introduce the effect and prediction method by examples showing the relationship between the liquid/vapor critical temperatures and the normal boiling points of different families of compunds. Critical Points of Organic Compounds Because of their use in treatments based on the principal of corresponding states, critical properties have been cited4 as the most important properties to be able to predict for the analysis of liquid/vapor systems. Relatively few direct measurements of critical properties have been made, and so it is important to validate the existing data and to develop accurate prediction methods based on them. Let us first establish a structural correspondence between two families of materials. A simple and useful correspondence is to pair materials that differ by a single functional group. Table S1 in the Supporting Information contains evaluated and recommended experimental values5-8 of the critical temperatures of several families of organic molecules (R1XR2) containing different functional groups X: ketones (CdO), ethers (-O-), alkanols (C(H)OH), amines (NH), and methyl-branched alkanes (CH2CH3), with R1 and R2 representing either branched or linear alkyl groups. Every such structure from the data set is included in Table S1 in the Supporting Information if data exist for at * To whom correspondence should be addressed. E-mail: petersbk@ yahoo.com.
least two of the families. In Figure 1 are shown critical temperatures for the ketones on the ordinate and critical temperatures for the analogous ethers, alkanols, amines, and alkanes on the abscissa. The plots in Figure 1A-D therefore consist of the ordered pairs {Tc(R1OR2), Tc(R1(CdO)R2)}, {Tc(R1(C(H)OH)R2), Tc(R1(CdO)R2)}, {Tc(R1NHR2), Tc(R1(CdO)R2)}, and {Tc(R1C(H)CH3R2), Tc(R1(CdO)R2)}, respectively. The lines are the results of linear regression fits to the data points. The points in Figure 1A-C are seen to lie near straight lines with apparent predictive ability. Table 1 contains the results of the linear regression fits shown in Figure 1 and also those for every nontrivial pairing of the five families of compounds. Several features are evident: Tc for any of the polar groupcontaining families is nearly linearly related to Tc of any of the others (R2 > 0.99), and a predictive correlation can be made in each case. The mean unsigned error (MUE) for these regressions is in the range of 1-3 K, which is a little above the reported uncertainties in the measurements (0.1-2 K)5-8 and is smaller than the mean errors reported for group contribution methods for Tc, even those that also use a measured boiling point to make the predictions.9 Correlations between polar compounds and alkanes (such as that shown in Figure 1D) exhibit larger deviations from linearity and larger scatter than those for the other families with 0.94 < R2 < 0.975 and 4.0 < MUE (K) < 11.5. A second-order polynomial provides a better fit (not shown) to the ketone versus alkane data in Figure 1D, but the deviations are still larger than those for the linear fits for any of the pairs of polar families. Several other features of the analysis are of interest. The obvious outlier shown in Figure 1C, which was not included in the regression analysis, has an analogue in each of the data sets that contained the amines, and the same compound was involved in each instance: N-methyl-1-propanamine, with reported Tc ) 550 K. Correspondence with the authors10 of the review confirmed that this compound and this property value are not actually associated with one another and that the datum was included inadvertently. That relatively few data are available for the amines is partially due to their instability at the critical point.6 One value of correlations such as those shown in Figure 1C is that more data are available for the more stable families, such as the ketones. If the amine/ketone relationship holds for these other analogues, the critical points of more than 30 amines of structure R1NHR2 can be predicted from just the ketone data5 without developing a detailed structure/property model. As an example, the predicted Tc of N-methyl-1-propanamine
10.1021/ie901721w 2010 American Chemical Society Published on Web 02/24/2010
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and 502 K, respectively, demonstrating also that correlations based on different families of materials give similar results. As is evident in Table 1, the relationship of the ketones and alkanols is more than just linear: there is near-numerical coincidence. This perhaps reflects the importance of keto-enol tautomerism at the high temperatures characteristic of the critical point, leading to very similar molecular and fluid structures. A demonstration of the effect discussed here on critical pressures, Pc, is made difficult by several factors. Among these are that there are fewer data available than for Tc and that the magnitude of the reported uncertainties is larger (by almost an order of magnitude) relative to the property values themselves and to the range of the property values within a given family. However, as for Tc, the relationship of Pc for the ketones5 versus alkanols7 is linear and also is nearly an equality (N ) 8, R2 ) 0.9981, slope ) 0.9971, and intercept ) 0.0444 MPa). A plot (not shown) of Pc for the ketones5 versus ethers5 (N ) 9) exhibits a useful correlation but shows significantly less scatter around a quadratic expression (R2 ) 0.9879) than a linear one (R2 ) 0.9537). The reported uncertainties in critical densities, Fc, for these families of materials are of a magnitude similar to that of the total variation of the property value within a family and therefore obscure any meaningful trend. Normal Boiling Temperatures
Figure 1. Critical temperatures of ketones related to those of analogous ethers (A), alkanols (B), amines (C), and alkanes (D).
from the regression fit of the amines to the ketones is 503 K. The predictions from the ether and alkanol fits are 507
While we have found simple or linear relationships for a wide variety of other properties and a wide variety of families of materials, we here show just one more example. The pentafluorosulfanyl (-SF5) functional group is of recent interest (a) because of its similarity to and differences from the trifluoromethyl (-CF3) group, (b) because of its potential use in biochemically active substances,11 and (c) because of new synthetic techniques for introducing the group to organic molecules.12 Because there are relatively few known materials containing the pentafluorosulfanyl group, its effects on the physical properties of substances containing it are relatively unknown. The trifluoromethyl group is present in many more known materials, and it would be useful to understand the properties of the SF5-containing compounds based on knowledge of the properties of the CF3-containing compounds. In the literature, we found 12 pairs of stable compounds, RX, where R represents a variety of molecular structures and X represents either functional group SF5 or CF3 and for which the normal (1 atm) boiling temperatures or vapor-pressure curves were reported. Table S2 in the Supporting Information contains the normal boiling points, Tb, of the species, and in Figure 2, the boiling temperatures are plotted as the ordered pairs: [Tb(RCF3), Tb(RSF5)]. As in the case of critical temperatures, the normal boiling temperatures lie remarkably close to a straight line. This relationship holds over the entire temperature range shown with Tmax - Tmin > 200 °C. A regression line fit through all of the points yields slope ) 0.929, intercept ) 52.0 (°C), a coefficient of determination, R2 ) 0.9983, and MUE ) 2.1 (°C). It is evident that a line fit through several of the points could be used to predict Tb for SF5-containing compounds in cases where Tb is known for CF3-containing analogues. When points to be predicted are left out of the regression analyses, the average unsigned prediction error for the 12 SF5 compounds is 2.5 (°C). It is important to note that there is no simple regularity between the members of either of these families by themselves: the R groups contain phenyl, various halides, oxyhalides, and fully or partially halogenated alkanes, alkenes, and alkynes and
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Table 1. Properties of Linear Regression Fits for the Critical Temperatures (Tc) of Analogous Ethers, Alkanols, Amines, Alkanes, and Ketonesa independent variable ether
alkanol
amineb
alkane
ketone
dependent variable
N
slope
intercept (K)
R2
MUE (K)
MaxUE (K)
MUE (%)
MaxUE (%)
alkanol amineb alkane ketone ether amineb alkane ketone ether alkanol alkane ketone ether alkanol amineb ketone ether alkanol amineb alkane
9 4 10 11 9 4 12 22 4 4 4 5 10 12 4 15 11 22 5 15
0.6910 0.9211 1.0460 0.7115 1.4382 1.2759 1.4329 1.0040 1.0854 0.7834 1.1396 0.7844 0.9092 0.6804 0.8514 0.6778 1.3952 0.9945 1.2732 1.3880
233.73 68.77 3.40 223.86 -333.17 -212.50 -306.48 -2.12 -74.49 166.74 -79.70 166.77 20.58 222.94 83.37 223.51 -308.70 3.05 -211.60 -279.17
0.9938 0.9997 0.9510 0.9926 0.9938 0.9995 0.9750 0.9984 0.9997 0.9995 0.9702 0.9987 0.9510 0.9750 0.9702 0.9408 0.9926 0.9984 0.9987 0.9408
2.18 0.94 8.72 2.32 3.11 1.18 6.09 1.52 1.01 0.92 11.43 1.69 7.76 4.10 9.54 6.01 3.15 1.53 2.15 8.81
3.36 1.70 15.32 6.63 4.93 1.79 14.07 3.18 1.85 1.40 14.85 2.67 17.03 10.11 13.28 14.15 9.28 3.08 3.45 19.19
0.38 0.17 1.74 0.40 0.64 0.23 1.21 0.25 0.20 0.16 2.27 0.29 1.60 0.71 1.83 1.03 0.64 0.25 0.41 1.71
0.60 0.31 3.48 1.16 1.06 0.36 3.45 0.58 0.35 0.25 2.94 0.48 3.12 1.61 2.66 2.24 1.87 0.55 0.69 4.48
a N ) number of observations. R2 ) coefficient of determination. MUE ) mean unsigned error. MaxUE ) maximum unsigned error. correlations involving the amines leave out one obvious outlier (see the text).
b
All
Figure 2. Normal boiling temperatures of pentafluorosulfanyl- (-SF5) compounds related to those of analogous trifluoromethyl- (-CF3) containing compounds.
yet the points lie near a single straight line. The two functional groups that define the families are similar but not so similar that there is a close numerical correspondence between Tb(RCF3) and Tb(RSF5): they differ by δT ∼ 50 °C. Summary We have shown several cases where an interesting effect occurs: a property of one family of materials is simply related to the same property of analogous members of another family of materials. For similar families, the relationships shown are nearly linear and have little scatter, while for dissimilar families (e.g., polar vs alkanes), there is less useful correlation. Within a family, the members need not be related homologously. On the basis of this effect, predictive correlations can be developed without positing any specific model for the structure/property relationship, as long as the property value for an analogue is known. The correlations, or simply the property/property plots, are very effective in aiding the detection of outliers or anomalous data.
The effect demonstrates that information useful for directly determining the value of a property of a material is to be found not only within the same family of materials but also in a different family of materials and in the relationship between the two families. In future work, we will explore a mathematical justification for the effect and the relationship of the correlative method based on it to other methods of producing predictive correlations. Acknowledgment The author thanks Air Products and Chemicals, Inc., for permission to publish this work. Supporting Information Available: Tables of evaluated critical temperatures from the literature for a series of related alkanols, ethers, amines, ketones, and alkanes and also normal boiling temperatures for a series of related SF5- and CF3containing compounds and literature citations for all of the data. This material is available free of charge via the Internet at http:// pubs.acs.org.
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ReceiVed for reView November 1, 2009 ReVised manuscript receiVed January 24, 2010 Accepted February 5, 2010 IE901721W
