Property Estimation Using Analogous Series - Industrial & Engineering

May 12, 2011 - Brian K. Peterson*. Computational Modeling Center, Air Products and Chemicals, Inc., Allentown, Pennsylvania, United States. Ind. Eng. ...
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Property Estimation Using Analogous Series Brian K. Peterson* Computational Modeling Center, Air Products and Chemicals, Inc., Allentown, Pennsylvania, United States ABSTRACT: A previously introduced method for predicting the properties of physical substances is described in detail and is given a mathematical justification. The method, here called the method of Analogous Series, requires only that the molecular structure of the target species be known and that the physical property of interest be known for several other species of known structure. It is shown to be very general and is successfully applied to the accurate prediction of pure-component properties (critical temperatures, liquid heat capacities), mixture properties (solution Henry constants, adsorption in zeolites, gas solubility in liquids, GC retention times), and to complex properties such as flash points. The relationship of the method to other predictive methods is discussed. The reference substance method is shown to be a special case of the Analogous Series method.

’ INTRODUCTION Values of the physical properties of substances are useful for process design and for selecting or designing materials for particular applications.1 These properties can be measured or estimated. Estimations are necessary when measurements have not been made because the substance has not yet been synthesized, or when the experiments are considered too difficult, unsafe, time-consuming, or expensive. We would like to explore a simple method for producing property estimates that bears some resemblance to methods based on homologous series of compounds, to group contribution methods, to the reference substance method of Othmer,2,3 and to other methods based on the principle that “similar materials have similar properties.” A typical application uses as input, data on two separate sets of molecules that are related in a particular way. If the data requirements for these two sets of molecules can be met, the method requires no property information for the target substance, obviates the need for parametrizing an explicit structure/property relationship, produces accurate property estimates, and also provides information on confidence in the estimates. We recently demonstrated4 that this method provides interesting and useful correlations between the critical temperatures of several families of organic compounds and between the normal boiling temperatures of two families containing highly fluorinated groups. The method has been little used in the past, but is not completely novel (vide infra): recent examples by Mathias et al.5,6 (the property leapfrog method) and Drefahl7 have appeared, and a similar correlation was mentioned by Golovanov and Zhenodarova.8 Smith9 presented a graphical method that is similar in some respects to the present work. Here we give a mathematical basis for the method, establish generality by showing more examples, and discuss aspects of its relationship to some other techniques for property prediction.

a complete specification of the system. For a system of molecules, the elements of t might consist of all the parameters in the Hamiltonian of the system. More generally, they might indicate the presence or absence of structural features, and/or they might represent parameters such as partial charges, sizes, or interaction energies, and they might also contain imposed variables such as mole fraction, temperature, or pressure. A relevant example might be the set of all of the parameters in an intermolecular potential function plus the external variables determining a statistical mechanical ensemble. Another example could be all of the parameters in an equation of state which has parameters with physical meaning. Different systems will have different values for the elements of t and those elements should be selected so that they can be continuously varied to encompass all of the systems that are required to be represented. Some t will describe physical systems while others might correspond to plausible physical systems that do not actually exist. An example of one of these latter systems might be a collection of CH3Cl molecules except that the dipole moment and size have been modified by some small amount. In this way one can posit a continuum of systems between, say, CH3Cl and CH3Br. Now consider a mapping from t to a property p: p = p(t). The property might be anything of interest, such as the critical temperature, the standard state enthalpy of formation, or the refractive index. Let us assume that the function is continuous with derivatives existing to whatever order we require. It may be useful to think of the function as representing the combination of an accurate and flexible intermolecular potential function and a molecular simulation10 that produces the desired property. The function might also be a universal equation of state as mentioned above. Note well that we will not require knowledge of the actual form of the property function: only the fact that it exists and a few of its general properties. Similarly, while we make use of the

’ THEORY Consider a vector, t, that characterizes a particular system. The elements of t can be any consistent set of quantities that provides

Received: January 29, 2011 Accepted: May 2, 2011 Revised: April 18, 2011 Published: May 12, 2011

r 2011 American Chemical Society

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assumption that there exist vectors t that sufficiently describe and differentiate between any systems of interest, we will never actually need to name or identify the parameters in t. Instead, the dependence of p on t will (eventually) be replaced by the dependence of p for one material on p for another material that is analogous in some way. We assume that the parameters in t are linked to the nature of the system in such a way that we can identify the needed analogues without necessarily specifying t. Define two separate sets of systems (denoted A for analogue and T for target) that have the property p and also divide the elements of t into two separate sets represented by the vectors r and s. The members of set A have common values for s: sAm = sAn = sA. The members of set T share a different s: sTm = sTn = sT 6¼ sA. The vector r is different for each member of A and for each member of T, but each member of T has an analogue in A with which it shares all of the elements of r: rAm = rTm. Thus, A and T are different families of systems, A being identified by sA and T by sT. The situation is depicted schematically in Figure 1. As an example, A might be a set of alkyl amines and T a set of alkanols each of which is otherwise isomorphic to one of the amines. The parameters in r describe the alkyl portions of the molecules, and the parameters in s distinguish between the amine and alcohol groups, with sA representing amine functionality and sT representing alcohol functionality. If ethyl amine exists in A, then ethanol exists in T. We wish to find a relationship between the property values of the systems in T and of those in A when sT and sA are not too different. The relationship may then be used to make a prediction for a specific member of the target family, the target itself. Let us exploit the supposed analycity of the p(t) function. Form a multivariable Taylor series expansion11 of p in terms of the elements of r and s around the point r0, sA: pðr, sÞ ¼ fpðr0 , sA Þg (   Nr Dp þ ðri  ri0 Þ Dr i r 0 sA i¼1 ! 1 Nr D 2 p ðri  ri0 Þ2 þ ::: þ 2i ¼ 1 Dri2 Nr

Nr

∑ ∑ i ¼ 1 j ¼ 1ðj6¼ iÞ

D2 p Dri Drj

! ðri  ri0 Þðrj  rj0 Þ þ ::: r 0 sA

9 =

∑

r 0 sA

þ :::þ

Ns

Ns

∑ ∑

i ¼ 1 j ¼ 1ðj6¼ iÞ

D2 p Dsi Dsj

8 ! < Nr Ns D2 p þ : i ¼ 1 j ¼ 1 Dri Dsj

∑∑

! ðsi  sAi Þðsj  sAj Þ þ ::: r 0 sA

ðri  ri0 Þðsj  sAj Þ þ ::: r 0 sA

∑

∑

∑ ∑

r 0 sA

9 = ;

pðr, sT Þ ¼ pðr, sA Þ   Ns Dp þ ðsTi  sAi Þ Ds i r 0 sA i¼1 ! 1 Ns D 2 p þ ðsTi  sAi Þ2 þ ::: 2i ¼ 1 Ds2i r0 sA ! Ns Ns D2 p þ ðsTi  sAi ÞðsTj  sAj Þ þ ::: i ¼ 1 j ¼ 1ðj6¼ iÞ Dsi Dsj r 0 sA ! Nr Ns D2 p þ ðri  ri0 ÞðsTj  sAj Þ þ ::: ð3Þ i ¼ 1 j ¼ 1 Dri Dsj

∑

;

8 ! < Ns Dp 1 Ns D2 p þ ðs  sAi Þ þ ðsi  sAi Þ2 :i ¼ 1 Dsi r0 sA i 2 i ¼ 1 Ds2i

∑

pðr, sA Þ ¼ pðr0 , sA Þ (   Nr Dp þ ðri  ri0 Þ i ¼ 1 Dri r0 sA ! 1 Nr D2 p ðri  ri0 Þ2 þ ::: þ 2i ¼ 1 Dri2 r 0 sA ! Nr Nr D2 p þ ðri  ri0 Þðrj  rj0 Þ þ ::: i ¼ 1 j ¼ 1ðj6¼ iÞ Dri Drj

This equation can describe the behavior of a homologous series as well as a more general series: those parameters that do not vary among the members of the series are contained in s, while those that do vary are contained in r. Also create an instance of eq 1 for the target family (with parameter vector (r,sT)) and subtract eq 2 from this:

∑

þ

members of the analogue family, eq 1 simplifies to

ð2Þ

∑

r 0 sA

Figure 1. The analogue and target families and their relationship to the defining parameter vectors, r and s.

9 = ;

∑

∑ ∑

∑∑

r 0 sA

ð1Þ

Of the four bracketed terms on the right-hand-side of eq 1, the first is a constant, the second varies only with the elements of r, the third varies only with the elements of s, and the last potentially varies with all of the elements of r and s. For the

Significant simplification has been achieved: all of the terms that vary solely with r have canceled. The terms involving variations sT  sA are present, but represent constants since sA and sT are considered fixed. The final set of terms, representing interactions or cross terms between the elements of r and s, varies only with r since sA and sT are fixed. The relationship between p(r,sT) and p(r,sA) is linear except for a correction due to the 7697

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Figure 2. The relationship between the properties of the analogue family and the target family. The symbols represent structures as in Figure 1. The filled stars denote (analogue, target) pairs of property values, (p(rj,sA), p(rj,sT)). The specific target and analogue systems are represented by the shaded symbols, and the open star represents the analogue property and the prediction for the property of the target, (p(rk,sA), p(rk,sT)). The function g is here represented by the best-fit straight line through all of the points other than (p(rk,sA), p(rj,sT)).

interaction terms: pðr, sT Þ ¼ pðr, sA Þ þ δðsT  sA Þ þ f ðr, sT  sA Þ

ð4Þ

where δ(sT  sA) is a constant. If f (r,sT  sA) is negligible, a plot of points for instances of p(r,sT) versus p(r,sA) will be linear with slope 1. Note that f (r, sT  sA) will be smaller and simpler as the ri  ri0 are smaller and/or as the sTi  sAi are smaller. If the r parameters relate to portions of molecules denoted by “R” (e.g., various alkyl groups) and the s parameters relate to two functional groups X and Y (e.g., amines and alcohols), this corresponds to the fact that the corrections are smaller when either the R groups are more similar to one another and/or the X and Y groups are more similar to each other. Note that any parameters that do not vary at all over the systems under consideration may be contained in either r or s and drop out of the problem all together. Supposing we can, at least in some local region (i.e., for some classes of systems), invert p(r,sA) to find r, we can write eq 4 as pðr, sT Þ ¼ pðr, sA Þ þ δðsT  sA Þ þ f 0 ðpðr, sA Þ, sT  sA Þ ð5Þ in which p(r,sT) is a function solely of p(r,sA) and the fixed factor sT  sA. If f 0 (p(r,sA),sT  sA) is adequately characterized by a linear function of p(r,sA), then a linear relationship between p(r, sT) and p(r,sA) still holds, but both the slope and the intercept may be modified. If higher order terms are needed to characterize f 0 (p(r,sA),sT  sA), then the overall linear form no longer holds, but it is still possible that p(r,sT) versus p(r,sA) will be relatively simple to approximate and to fit using available data. The magnitude of f 0 (and the number of numerically significant terms in the expansion) will still be smaller as the ri  ri0 are smaller and/or as the sTi  sAi are smaller. When might f be invertible or approximately invertible? First, if only a single parameter in r significantly affects p; this would define A and T as effectively homologous series with simple p(r) relationships. Also if r could be transformed (e.g., through linear or nonlinear principal component analysis) into a different set of parameters, only one of which significantly affects p. The consequence of the noninvertibility (nonuniqueness of p f r) will be scatter in the p(r,sT) versus p(r,sA) relationship. We suggest that the experimental data can often be used to determine the adequacy of this approximation.

Figure 3. Critical temperatures of alkanols related to those of analogous ethers.

On the basis of the above, we propose to estimate a property of a target system from the property of its analogue and from the relationship between the properties of the target family and the analogue family as established by available data. The quality of the fit between the target family and the analogue family is expected to be a guide to the quality of the prediction from the analogue to the target. Explicitly, if an estimate of property, p, for a physical system, k (the target), is desired, the following steps are taken: 1. Find an analogue of k that shares some features (described by rk) with k but that differs from k in other features (described by sA instead of sT) and for which the property p is also known (p(rk,sA)). 2. Find other members of the target family, indexed by j, that share the features sT with the target and differ by having other features rj, and for which the property values (p(rj, sT)) are known. 3. Find other members of the analogue family (also indexed by j) that share the features sA with the analogue of k and differ from one another by having other features rj, j 6¼ k, where each member of the analogue family shares features with one member of the target family such that rAj = rTj. The property values of the analogue family (p(rj,sA)) must also be known. 4. Find a relationship, p(r,sT) = g(p(r,sA)), between the property values for the target and analogue families. This relationship will often be adequately represented by a straight line or a low order polynomial. 5. If this relationship can be adequately characterized with low scatter, use it to estimate the property of the target from that of the analogue, p(rk,sT) = g(p(rk,sA)). The scheme is depicted in Figure 2.

’ RESULTS AND DISCUSSION Let us explore the use of the method and its relationship to other methods through a series of examples. Example 1. Critical Points of Ethers and Alcohols. In the previous publication4 it was shown that the critical temperatures of polar organic molecules of the form R1XR2 (where R1 and R2 are linear or branched alkyl groups and X indicates carbonyl, amine, ether, or alkanol groups) are related linearly when plotted as Tc(R1X1R2) versus Tc(R1X2R2). As an example, Tc for the alkanols versus the ethers is shown in Figure 3. The linear 7698

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regression line shown in the plot has slope 0.6910, intercept 233.7 (K), and coefficient of determination, R2 = 0.9938. The mean and maximum unsigned deviation of the nine alkanol critical temperatures from the line are 2.2 and 3.4 K, respectively. The accuracy of this relationship compares favorably to other methods of predicting Tc, even those that use measured boiling points for the target molecule.12 The level of fit and the predictive capability of this relationship are typical of those found for any of the pairs of families of polar molecules.4 In the language of the Theory section, we have related property values for a target family to those of an analogue family, and we found that the resulting correlation is strongly linear. Comparing the result to eq 5 and noting that the slope of the correlation is significantly different than unity, we can determine that the f 0 (p(r,sA),sT  sA) term is not negligible and that it is adequately described as a linear function of p(r,sA). It is important to note that we are not claiming that all properties of these materials are analytic functions of any other property at the critical point—clearly the free energy is not an analytic function of the temperature;13 only that, for some classes of materials, the critical temperatures (and apparently the critical pressures4) themselves are approximately analytic functions of variables characterizing the structure of the molecules in the system. Example 2. Linearly Additive Group Contribution Methods. Let us continue with a simple and intuitive analysis. Suppose that a linearly additive group contribution method is applicable to a set of molecules and a property of interest. That is, the property, p, is an additive function: p¼

N

∑ ai ni i¼1

ð6Þ

where the integers ni represent the number of group i in the molecule and the weights ai describe the contribution of group i to p. Divide the group numbers, ni, into two vectors: r = {ni; 1 e i e N, i 6¼ x 6¼ y} and s = {nx,ny}. Consider an analogue family; all members of which contain functional group X and are therefore described by sA = {1,0}. For a target family, all members instead contain functional group Y and are described by sT = {0,1}. We choose the members of each family so that rAj = rTj; each member has an analogue in the other set that contains identical groups except for the presence or absence of the defining functional groups X and Y. We can rewrite eq 6 as N

pjx ¼ ax þ

∑ ai nij i ¼ 1, i6¼ x, y

pjy ¼ ay þ

∑ ai nij i ¼ 1, i6¼ x, y

N

ð7Þ

ð8Þ

for the analogue and target sets, respectively. By subtracting these equations directly or by noting that the linear nature of eq 7 and eq 8 (treating the ni as continuous) causes all but the first two terms in eq 3 to vanish, we find pjy ¼ pjx þ ðay  ax Þ

ð9Þ

the anticipated result. If a linear group contribution method (GCM) is valid for these sets of molecules, a plot of pjy versus pjx will yield points that fall on a straight line with unit slope and an intercept equal to the difference in the contributions of the

groups characterizing the two families. Even without knowledge of any of the ai, one can make a property prediction for a given molecule if data for several other (analogue, target) pairs are known so the intercept can be determined, and data for the analogue of the specific target is also known. This result can also be used for determining the group parameter ay when ax is known and data for several members of the X-containing and Y-containing families are known. It is also useful for determining at the outset whether two families of molecules might or might not adequately be represented by a linear GCM. Alternatively, the pjy versus pjx plot suggested by eq 9 can be used to suggest and test transformed properties p0 (e.g., p0 = ln(p)) that are amenable to a linear GCM while p itself might not be; for a linear GCM to be valid, the slope on this plot should be unity. That the slope in Figure 3 is different than unity suggests that deviations from a first-order, linear, group-contribution method for Tc, with single parameters for the alkanol and ether functionalities, will be observed for this set of materials (eq 9). Other examples will not have the simplicity exhibited by a system well-described by a linear GCM, but many will share the fact that useful predictions can be made by establishing the relationship between the properties of one family of systems and those of another. Example 3. The Reference Substance Technique. As in the first example, our primary purpose is to develop predictions for cases where nothing is known about the target except its molecular structure. A closely related procedure turns out to be the reference substance technique of Othmer3 which is used to extrapolate from a small amount of property data for the target to an extended range of a property curve as a function of, e.g., temperature. Let sA describe the molecules of some reference substance, for example, water, and sT describe the molecules of a target substance such as benzene. We wish to predict the value of some property of the target as a function of a variable (e.g., the temperature). Temperature is then included as the only element of r. Given data for the property of the reference substance at different temperatures, p(Tj,sA), and two or more data for the property of the target at the same temperatures, p(Tj,sT), the latter is related to or plotted against the former. The result is often linear or at least a simple curve, and predictions for the target (p(Tk,sT)) can be made from the known properties of the reference (p(Tk,sA)). Othmer3 gives many examples of the correlative power of this technique for a wide variety of properties and substances. When the properties and the variable are related by known thermodynamic expressions, the resulting relationship of p(r,sT) versu p(r,sA) may be in terms of other identifiable properties:2,3 for example, when the logarithm of the vapor pressure of the target is plotted versus that of the reference, the slope is the ratio of the latent heats. This additional knowledge is not necessary for use of the method. Note that r consisting of a single scalar variable (e.g., temperature), is exactly the case where p(r,sA) is most likely to be invertible over some useful range. Hence f 0 (p(r,sA),sT  sA) (eq 5) is likely to be a function only of p(r,sA), leading to low scatter. Also as suggested by the development in the Theory section, it was found3 that the more similar are the target and the reference substances (the smaller is sT  sA), the simpler (more linear) are the correlations between their properties and the lower is the scatter. Let us emphasize that the reference substance technique is used to extrapolate or interpolate between data that already exists for the target substance, rather than to produce predictions 7699

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Figure 4. Closed-Cup flash points of dienes and alkenes as reported by Catoire and Naudet.14 The four identified outliers are represented with diamonds while the other nine pairs of materials are represented with circles. The upper solid line is a linear regression fit using all of the points and the lower solid line is a linear regression fit to the nonoutliers. The dashed line is a plot of the equation FP(diene) = FP(alkene).

where no prior data exists as is the primary goal of the present work. The reference substance technique is seen to be a special case of the general method of analogous series. Example 4. Flash Points. Flash points are useful in establishing safe operating conditions for process/material combinations. It is often of interest to predict flash points when they have not been measured and it is helpful to know approximate flash points to save time while doing measurements. Catoire and Naudet14 tabulate many closed-cup flash point measurements for organic compounds from various sources and present an equation for predicting these flash points that uses as input the normal boiling temperature, the enthalpy of vaporization, and the number of carbon atoms for the material. Using unfavorable comparisons between the data and their predictions, several compounds were suggested as likely having erroneous data. Five of these were dienes. In Figure 4 we plot the flash points of the dienes quoted in that work against the flash points of analogous alkenes for all 13 cases where both an alkene and the diene were available. Three points are clear outliers on this plot with deviations from a best-fit line through all of the points of 2741 C. The estimated reproducibility of the data (95% confidence) is better than (10 C.14 Removing the three points and refitting the line, one more point is found to be ∼20 C from the new line; two times the maximum deviation of any of the other points. The four compounds were 1,3-cyclohexadiene, 1,4-pentadiene, 1,3-cyclopentadiene, and trans-1,3-pentadiene, which are four of the five suggested by Catoire and Naudet to be erroneous. The fifth material, 4-methyl-1,3-pentadiene (with measured flash point of 34 C), seems to fit the trend in Figure 4 very well. In the analysis here, using no other measured information for these dienes, and building no explicit structureproperty model, we come to largely the same conclusions as do Catoire and Naudet. Of course, with the present method, we can only say that the olefin/diene pairs are anomalous, not specifically that the diene data are incorrect. The best-fit line through the nine nonoutlier points has slope = 0.924, intercept = 2.26, and R2 = 0.978. The mean and maximum absolute deviations of the nonoutlier dienes from this line are 4.1 and 8.1 C, respectively, comparing well with the reported

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Figure 5. Solubility (Ostwald Coefficients) for N2 and O2 in nine solvents.

experimental uncertainty. 4-Methyl-1,3-pentadiene lies only 3 C from the line while its measured flashpoint was 15 C higher than the prediction of Catoire and Naudet. Since the slope of the bestfit line is near unity and the intercept is small, we also show in Figure 4 a dashed line for FP(diene) = FP(alkene) and this line is seen to represent the (nonoutlier) data reasonably well. In light of eq 9, this can be interpreted to mean that a linear group contribution method might work well for this set of alkenes and dienes (predicting their difference) and that the flash-point group parameters for the second double bond and for a single bond would be nearly equal. As did Catoire and Naudet, let us emphasize that the four identified outliers probably err on the hazardous side: their reported flash points are over 20 C higher than those indicated by the analysis presented here. Drefahl7 has recently presented an equivalent analysis of the flash points of tetraalkylgermanium complexes in terms of the analogous tetaalkylsilicon complexes. The resulting expression, called in that work a “quantitative property-property relationship” (QPPR), has the expected linear form. Example 5. Gas Solubility in Liquids. Much of the evaluated data for gas-solubility in liquids is contained in the IUPAC Solubility Series.15 If the analogous series method applies to this property, one can make predictions for a larger number of systems by using the solubility of one gas to predict that of another—if solubilities in several other solvents are available for the two gases or if solubilities are available for several other gases in two solvents. As an example, Kretschmer et al.16 measured the solubility, at several temperatures, of N2 and O2 in nine solvents: 95% ethanol/water, n-butanol, isopropyl alcohol, ethanol, 50% acetone/ethanol, methanol, acetone, 50% iso-octane/ethanol, isooctane (in order of increasing N2 solubility). The 25 C solubility of O2 versus that of N2 for these solvents is plotted in Figure 5. The solubility is presented as the Ostwald coefficient (the ratio of the concentration of the gas in the liquid to its concentration in the gas phase). As can be seen in Figure 5, the (N2,O2) points lie near a straight line (with slope 1.556, intercept 0.0097, and R2 = 0.965). The average and maximum unsigned deviations of the O2 Ostwald coefficients from the line are 3.0 and 7.3%, respectively. This effect has recently been noticed elsewhere. Referencing data from the IUPAC series, Golovanov and Zhenodarova8 state (without demonstration) that, “Indeed, the correlation 7700

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Figure 6. Relative GC retention times for analogous series of ether derivatives of phenols. The points are the ordered pairs: (PFB, DNT; squares), (DNT, DNP; triangles), and (PFB, DNT; diamonds). The lines are simple linear regression fits to the points.

coefficient for the solubilities of oxygen and argon in the same set of solvents exceeds 0.99. This means that the solubility of a gas can be estimated from the solubility of another gas.” Example 6. GC Retention Times. Chemical derivatives are often used in gas chromatography (GC); especially to increase the volatility of polar compounds. Seiber et al.17 prepared three different derivatives for each of the following six phenols: phenol, p-chlorophenol, 3,4,5-trimethylphenol, carbofuranphenol, p-nitrophenol, and 1-naphthol. The derivatives were the pentafluorobenzyl (PFB), 2,6-dinitro-4-trifluoromethylphenyl (DNT), and 2,4-dinitrophenyl (DNP) ethers. In Figure 6 relative (to aldrin) retention times for all three pairs of families of these compounds are plotted as analogous series. The coefficients of determination of the three linear regression lines shown are R2PFB, DNT = 0.98, R2DNT, DNP = 0.97, and R2PFB, DNT = 0.99. The three sets of points are all seen to lie near straight lines with low scatter. The six points in each set are differentiated by parent phenols which differ in both the number and nature of substituents. The phenols do not constitute a homologous series. Example 7. Solution Thermodynamics: Henry’s Law Constants and Infinite Dilution Activity Coefficients. In Figure 7 are shown logarithms of Henry’s law constants18 for a series of very similar alkanes and alkenes in 2-butanol versus the same compounds in 1-butanol. The linear regression line has coefficient of determination, R2 = 0.999. The slope of this line is very near unity (0.996) and the intercept is small (4.07) as might be expected for a set of very similar hydrocarbons in two very similar solvents. In this case, the fit is better than for the related infinite dilution activity coefficients (not shown) with R2 = 0.981. While this paper was being prepared, Mathias5,6 pointed out to us published plots of ln(γ¥) for several hydrocarbon compounds in different solvents. The resulting points lie very near straight lines. The method we are calling Analogous Series is called by Mathias “property leapfrog” or “relative properties” and was equated with the method of Othmer. Example 8. Using Further Approximations: The Heat Capacity of Norbornadiene. The standard state molar heat capacity of norbornadiene (NBDE or BCHD: bicyclo[2.2.1]hepta-2,5diene) has been reported by Watanabe and Kato19 as 147.6 J/ mol-K and by Steele20 as 161.2 J/mol-K. The difference (∼ 9%) is somewhat larger than expected by the reported precision of at least

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Figure 7. Henry’s law constants for hydrocarbons in 2-butanol and 1-butanol. The hydrocarbons (from low to high H) are propane, butane, isobutane, propene, 1-butene, and isobutene.

the former measurements. It was of interest to determine which, if any, value was to be preferred. We could not find a complete set of substances and data useful for the direct application of the analogous series method. One can however, make useful progress by carefully adding assumptions as guided by the method. There do exist measurements21 at a few temperatures for norbornene (NB) which differs from NBDE by the absence of just one double bond, rendering it a possible analogue. Guided by eq 4 and assuming both δ(sT  sA) and f(r,sT  sA) are small, one might expect the heat capacity of these two substances to be very similar and the Cp for NB (150.0 J/mol-K after a short extrapolation to 298.15 K) might be a good first approximation. This is 1.6% from the Watanabe and Kato (WK) value. We can do slightly better. If the hexene and 1,4-hexadiene rings define analogue and target families, respectively, one would look for such six-membered rings containing other functional groups to be the different members of each family. We found only one such pair: cyclohexene (CHE) and 1,4 cyclohexadiene (CHDE) themselves are reported in the WK work, with Cp values of 146.3 and 144.0 J/mol-K, respectively. With only one pair, the AS plot cannot be made. In this case however, the similarity of the structures and of the property values of all four substances suggests that perhaps we can assume directly that f(r, sT  sA) ≈ 0, while estimating δ(sT  sA) from the single pair (CHE, CHDE). For both the structures and the properties: NBDE = NB þ (CHDE  CHE). Performing this operation we find the estimate for Cp(NBDE) = 147.7 J/mol-K. This differs from the measured value of WK by only ∼0.1% and from that of Steele by ∼9%. It is also the case that the measurements for NB by Lebedev et al.21 and by Steele differ by more than 13%. The independent measurements for NB and the WK measurements for NBDE, CHE, and CHDE are entirely consistent, while it seems likely the accuracy of the heat capacity in the 1978 work of Steele is no better than ∼(()10% or larger. For doing analyses of this type, a useful reminder from the analysis in the Theory section is to use materials as similar as possible (small r  r0 and sT  sA) to the target and analogue of interest. This is important in the current case because the 1,4 double-bonded system within rings has different contributions to the heat capacity than do double bonds in other systems (e.g., Cp for CHE and 1,3-CHDE differ by 5.0 J/mol-K instead of 2.3 J/ mol-K for CHE and 1,4-CHDE). Of course, high quality data are needed for the difference in very close property values to have any numerical significance. 7701

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Example 9. Henry’s Law Constants for the Adsorption of Alkanes in Zeolites. In Figure 8 are shown Henry’s law

constants, H, for the adsorption of linear, singly, and multiply methyl-branched C5C8 alkanes in different zeolites.22 In Figure 8A, H for the alkanes in mordenite are plotted against H for those in beta. It can be seen that all of the data lie reasonably near a single, perhaps slightly nonlinear, curve in accordance with expectation based on the previous examples. In Figure 8B, however, a similar plot for H in ZSM-22 versus H in ZSM-5 shows that data for the linear and the branched alkanes fall near two clearly different straight lines. That the alkanes in zeolites follow the analogous series trends is not terribly surprising as the trends for H with carbon number are quite regular in themselves.22 The difference in behavior between the large- and small-pore zeolites is, however, interesting. A plausible explanation, that accords with intuition, can be found in the presence of the final term shown explicitly in eq 3; the interaction term. This term is nonzero when the property of interest (H) is sensitive to a combination of one variable from r and one from s: that is when it is sensitive to a combination of one variable that varies between the members of a given family and one variable that distinguishes the two families shown in the plot. In this case, these can be taken to be one variable indicating the presence or absence of methyl branches and a variable establishing the identity of the zeolite, or nearly equivalently, the diameter of the relevant pores. In the case of the large-pore, 12-ring zeolites Mordenite and Beta, the methyl branches do not appreciably interact with the zeolite in a way significantly different than do the methylene groups contained by all the alkanes, and the interaction term is consequently absent or small. In the case of the medium-pore, 10-ring zeolites ZSM-22 and ZSM-5, the methyl groups do interact with the pore walls in fundamentally different ways than do the methylene groups of the molecular backbones and an interaction term is necessarily present. A largely equivalent way of looking at this is that a single function can describe H as a function of the variables describing all of the alkanes in Mordenite and Beta, but that a single function cannot be found that describes all of the alkanes in all four of the zeolites. That the data naturally fall on one or two curves or show more or less scatter is a simple intuitive guide to which data to select in order to make predictions from such a set of data.

’ SUMMARY AND CONCLUSIONS We have presented a method with some generality for predicting properties of target substances directly from the known properties of carefully selected, related substances. A novel feature of the method is that one makes use of data for substances that are intentionally chosen to be functionally different than the target. The method can be applied graphically and/or through numerical regression. We have justified the method by assuming a homotopic continuation between representations of the molecular systems and that an analytic function exists mapping the representations to a property value. Experimental data are selected which is structured in such a way as to maximize the cancellation of terms in the function relating the property values of two families of materials. Similar kinds of cancellation are a general feature of methods that characterize differences23 in the properties of related materials. If the property function could be exactly inverted to uniquely determine the species of the analogue family corresponding to a particular property value, the analogous series data would exactly

Figure 8. Henry’s law constants for alkanes in zeolites. (A) Linear and branched alkanes in zeolites mordenite and beta. The solid line is a bestfit quadratic. (B) Linear (upper curve) and branched (lower curve) alkanes in zeolites ZSM-22 and ZSM-5. The solid curves are linear bestfit lines through the subsets of data.

fall on a single curve, apart from the effects of measurement error. This ideal can be approached in cases where principal components analysis would show that one dominant variable determines the property. A related procedure is the search for a single dominant descriptor, of whatever type, that relates the property of interest to the structure or other properties of similar species. In the case of Shacham,24 the relationship sought is a linear one. If this procedure is successful for a set of systems which also meet the other requirements for the analogous series, the AS method should correlate the data particularly well. Note that in the AS method, these underlying relationships need not be linear and that the relationship and the single correlating variables do not need to be identified for the successful use of the method. Previously published applications for activity coefficients by Mathias et al.5,6 and for vapor pressures by Drefahl7 have appeared, and a linear property/property correlation for gas solubility in different solvents was mentioned by Golovanov and Zhenodarova.8 From the point of view of the Theory section, the reference substance method of Othmer3 turns out to be a special case of the general procedure presented here. There are several requirements that must be met in order to apply the method. One must be able to select the properties and systems in such a way that discontinuities are avoided within either series or in the relationship between the series. As a simple example, if some members of one of the families undergo a phase 7702

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Industrial & Engineering Chemistry Research transition in the range of the data, the method will fail. The full method (some shortcuts are possible) requires data to be known for at least five systems. If assessment of the quality of the prediction is desired, more data pairs are needed. In general, the method will work better the more similar are the members of each family and the more similar are the analogue and target families. The method overlaps somewhat and is complementary to group contribution methods. GCMs impose a specific functional form on the relationship between structure and property while this is not needed for the analogous series method (ASM). GCMs, once parametrized, can make predictions for a wider range of compounds, while the ASM, if the requisite data exists, is expected to provide more focused and accurate predictions. The ASM is expected to be able to distinguish between isomers of various sorts if the analogues are known. The ASM can be applied to some data sets that are not sufficiently extensive for extraction of group contributions, such as when the available substances contain a large number of functional groups compared to the number of available data. The method, as applied in eq 9, can be used to extract contributions for new functional groups and can be used to suggest and test transformations of properties that are amenable to a linear GCMs. The ASM has been shown here to apply to pure-component and multiple-component systems, to systems containing simple molecules and to complex molecules which exhibit polarity and/ or hydrogen-bonding. Equilibrium and transport properties can be predicted as can complex properties like flash points which are determined by more fundamental equilibrium, transport, and chemical-kinetic properties as well as by aspects of the experimental apparatus. As shown by the general nature of the development in the Theory section, the method is actually not limited to molecular systems at all and can be applied to systems of many types. Several areas for further exploration are evident. A related application might be the prediction of parameters in an equation of state or an intermolecular potential function based on parameters for other materials which have been determined in the usual ways from experimental data and molecular simulations. The results of quantum chemical calculations may also be predicted from results on species similar to a target molecule or system. The relationship between the ASM and GCMs, as exemplified in Example 2, might be used to help understand the limited accuracy of some GCMs and to point the way toward their improvement. It is possible that automated extensions combined with large databases may prove to be useful for property prediction and outlier identification. If a property value required by the method is not available, perhaps it can be estimated using AS or another method with the result then used in the estimation for the original target substance, while taking into account the likely increase in uncertainty. While the accuracy of the method benefits from possible smoothing in the correlation between the target series and the analogue series, a drawback is that any error in the property of the specific analogue of the target propagates directly to the property of the target. It may be interesting to optimize predictive accuracy and precision by further developing systematic choices of single or multiple analogue families, based on the number of data available, measures of molecular similarity, and the estimated measurement errors. The method is expected to work best when an underlying structure/property relationship is continuous and invertible over a useful range of similar systems. Whether or not this situation is present for any particular materials need not be

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determined by analysis, but can be determined directly from the data and the level of its adherence to a simple curve on a property/property plot.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Present Addresses

ExxonMobil Research and Engineering Company, Annandale, NJ.

’ ACKNOWLEDGMENT The author thanks Air Products and Chemicals, Inc. for permission to publish some of the material presented here and Paul Mathias for helpful discussions and for first introducing us to refs 5, 6, and 9. ’ REFERENCES (1) O’Connell, J. P.; Gani, R.; Mathias, P. M.; Maurer, G.; Olson, J. D.; Crafts, P. A. Thermodynamic Property Modeling for Chemical Process and Product Engineering: Some Perspectives. Ind. Eng. Chem. Res. 2009, 48, 4619–4637. (2) Othmer, D. F. Correlating Vapor Pressure and Latent Heat Data. A New Plot. Ind. Eng. Chem. 1940, 32, 841–856. (3) Othmer, D. F.; Chen, H.-T. Correlating and Predicting Thermodynamic Data. Ind. Eng. Chem. 1968, 60, 39–61. (4) Peterson, B. K. Relationships between the Properties of Families of Materials. Ind. Eng. Chem. Res. 2010, 49, 3492–3495. (5) Mathias, P. M.; Elliott, J. R., Jr.; Klamt, A. Butadiene Purification Using Polar Solvents. Analysis of Solution Nonideality Using Data and Estimation Methods. Ind. Eng. Chem. Res. 2008, 47, 4996– 5004. (6) Mathias, P. M. The Role of Experimental Data in Chemical Process Technology. Pure Appl. Chem. 2009, 81, 1727–1743. (7) Drefahl, A. Quantitative Property-Property Relationships (QPPRs) and Molecular-Similarity Methods for Estimating Flash Points of Si-Organic and Ge-Organic Compounds. Proceedings of the IEMS Third Biennial Meeting, “Summit on Environmental Modelling and Software.” International Environmental Modelling and Software Society, University of Vermont, Burlington, USA, July 2006. (8) Golovanov, I. B.; Zhenodarova, S. M. Quantitative StructureProperty Relationship: XXIII. Solubility of Oxygen in Organic Solvents. Russ. J. Gen. Chem. 2005, 75, 1792–1797. (9) Smith, A. L. Family Plots for Evaluating Physical Properties of Organosilicon Compounds. AIChE J. 1994, 40, 373–377. (10) Stefanovic, J.; Pantelides, C. C. Molecular Dynamics as a Mathematical Mapping. I. Differentiable Force Functions. Mol. Simul. 2001, 26, 237–271. (11) Jenson, V. G.; Jeffreys, G. V. Mathematical Methods in Chemical Engineering; Academic Press: London, 1977. (12) Yan, X.; Dong, Q.; Hong, X. Reliability Analysis of GroupContribution Methods in Predicting Critical Temperatures of Organic Compounds. J. Chem. Eng. Data 2003, 48, 374–380. (13) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, 1971. (14) Catoire, L.; Naudet, V. A Unique Equation to Estimate Flash Pints of Selected Pure Liquids. Application to the Correction of Probably Erroneous Flash Point Values. J. Phys. Chem. Ref. Data 2004, 33, 1083–1111. (15) Gamsj€ager, H.; Lorimer, J. W.; Salomon, M.; Shaw, D. G.; Tompkins, P. T. The IUPAC-NIST Solubility Data Series: A Guide to Preparation and Use of Compilations and Evaluations (IUPAC Technical Report). J. Phys. Chem. Ref. Data 2010, 39, 023101-1–023101-13. 7703

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