Estimation of Heat Capacity of Boiling of Organic Compounds

Ind. Eng. Chem. Res. 2006, 45, 451-453

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Estimation of Heat Capacity of Boiling of Organic Compounds Ritesh Sanghvi* and Samuel H. Yalkowsky College of Pharmacy, The UniVersity of Arizona, Tucson, Arizona 85721

The aim of this study is to develop a widely applicable model for predicting the heat-capacity change associated with boiling at atmospheric pressure (∆Cpb) of organic compounds. A semiempirical model is generated utilizing a flexibility index (τ) and a hydrogen-bonding parameter. This model is based upon experimental apparent ∆Cpb values of 619 organic compounds. The average absolute error for the estimation is 7.1 J/K mol. The proposed model quantitatively establishes the impact of flexibility and hydrogen bonding on the apparent ∆Cpb. It is shown to be more accurate when compared to some of the previously described methods. Introduction The heat capacity of a compound is defined as the sensitivity of its enthalpy to temperature. The heat-capacity change associated with boiling at constant pressure (∆Cpb) is the difference between the heat capacities of the liquid and gas at the boiling point (Tb), i.e.,

enthalpy of vaporization to that of boiling and vice versa. The heat capacity of boiling is also used in the calculation of the ambient vapor pressure of organic compounds. The heat capacity of boiling is analogous to and has the same units as the entropy of boiling (∆Sb). It can be considered to be the sum of three contributions:

∆Cpb ) Cpbgas - Cpbliq

∆Cpb ) ∆Cpbtrans + ∆Cpbconf + ∆Cpbrot

(I)

The value of ∆Cpb is the difference between the enthalpytemperature slopes of the gas phase and the liquid phase. Since liquid-phase enthalpy is more sensitive to temperature, ∆Cpb values are invariably negative. The heat-capacity change can be used to calculate the enthalpy of boiling (∆Hb) from the more commonly available enthalpy of vaporization at 298 K (∆Hv) using Kirchhoff’s equation,

∆Hb ) ∆Hv +

∫298T ∆Cp(L,G) dT b

(II)

where ∆Cp(L,G) is the difference between the heat capacity of the gas phase and the liquid phase. The heat capacity of the liquid and the gas are not strictly linear with temperature over the (Tb - 298) range, and as a consequence, ∆Cp(L,G) is sensitive to temperature. However for the purpose of this work, the latter is assumed to be constant and equal to ∆Cpb and can be approximated by an average value ∆Cpb calculated over the (Tb - 298) range. Myrdal and Yalkowsky1 have shown that the effect of temperature on ∆Cp(L,G) for the estimation of vapor pressure is small. In a separate study (unpublished work), we have used the ∆Hb value calculated from ∆Hv and ∆Cpb for the estimation of the boiling point of 1350 organic compounds. The error associated with the estimation is small and independent of the boiling point of the compounds. Thus, for the purpose of this study eq II can be written as

∆Hb ) ∆Hv + [∆Cpb (Tb - 298)]

(III)

This is particularly important for the calculation of ∆Hb of high-boiling compounds, because many of these compounds decompose at temperatures lower than their boiling points. Knowledge of ∆Hb is useful for prediction of the boiling point (Tb) as well as the solubility parameter of organic compounds. The above relation is very useful for the conversion of the * To whom correspondence should be addressed. E-mail: [email protected]; phone: 520-626-4308.

(IV)

∆Cpbtrans results from the increase in translational freedom of molecules upon boiling and is typically constant for rigid nonhydrogen-bonding compounds. ∆Cpbconf results from the increase in conformational freedom of molecules upon boiling. This term may be considered analogous to the Cb(T) term used by Bondi2 as the contribution due to the difference in the work expended in cooperative motions of many molecules in the liquid and gas phases. Since flexible molecules tend to be somewhat more aligned in the liquid phase, the impact of ∆Cpbconf increases with the number of torsional units in the molecule. ∆Cpbrot results from the increase in rotational freedom upon boiling. This term can be considered analogous to the Chr(T) used by Bondi2 and quantifies the contribution due to the difference in the hindrance experienced by the molecules in external rotation in the liquid and gas phases. Unlike melting, boiling is associated with a small increase in rotational freedom and, thus, makes a negligible contribution on the ∆Cpb for noninteracting molecules. Hydrogen bonding, though, can result in a higher change in rotational freedom upon boiling as it restricts molecular rotation in condensed phases. A number of methods, both empirical and semiempirical, are available for the estimation of ∆Cpb. According to Sidgwick’s Rule,3 ∆Cpb is constant at -54.4 J/(K mol), while Mackay et al.4 found that it is -0.6 to -0.8× the entropy of boiling of the compound. The equations proposed by Mishra and Yalkowsky5 and Myrdal and Yalkowsky1 incorporate a coefficient for flexibility. However, none of the existing methods account for the possible role played by the hydrogen bonding. Hydrogen bonding can be considered to contribute significantly to the ∆Cpbrot term. A few methods to estimate the individual heat capacities of liquids (Cpliq) and gases (Cpgas) have also been proposed (see refs 2 and 6-8). The heat capacity of boiling can be calculated as the difference between these two predicted heat capacities. Chickos et al.9 proposed that ∆Cp(L,G) is related to Cpliq. In the current study, we have developed a semiempirical equation for

10.1021/ie050675g CCC: $33.50 © 2006 American Chemical Society Published on Web 11/16/2005

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Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006

the estimation of ∆Cpb based on apparent experimental data for 619 compounds. Experimental Section The following molecular parameters have been used in this study: Intercept. An intercept is used to generate the ∆Cpb value for rigid, non-hydrogen-bonding compounds. Flexibility Index (τ). Flexibility results in conformational freedom of a molecule. It arises from the ability of the atoms to torsionally rotate about single bonds. The effective number of torsional bonds has been defined earlier by Dannenfelser and Yalkowsky10 as

τ ) (SP3 + 0.5SP2 + 0.5Ring - 1)

(V)

where SP3 and SP2 are the total number of nonring, nonterminal sp3 and sp2 atoms. Ring indicates the number of independent single, fused, or conjugated ring systems in the molecule. Since terminal atoms and sp hybrid atoms do not contribute to flexibility, they are not counted. The rotation of hydrogen atoms is also ignored. Aliphatic cyclic compounds are counted as a single ring system. Compounds with a negative value for τ are assigned a value of zero. Hydrogen Bonding. The effect of hydrogen bonding was considered as a function of the number of different hydrogenbonding groups [-OH, -COOH, -NH (includes -NH and -NH2), and -SH] present on the molecule. The hydrogenbonding parameter (HBP) is defined as

HBP ) f[(#OH) + (#COOH) + (#SH) + (#NH)]

(VI)

where # represents the number of respective groups and f is the function to be determined. Data. The apparent experimental heat capacity of boiling was calculated for 610 organic compounds using the following form of Kirchhoff’s equation:

∆Cpb ) (∆Hb - ∆Hv)/(Tb - 298)

(VII)

The enthalpy of boiling, enthalpy of vaporization at 298 K, and boiling-point data were collected from various sources.8,11-13 For the entire data set, ∆Cp(L,G) was assumed to be independent of the temperature. For 9 additional compounds, known experimental heat capacity of boiling values were used.8 The data set includes compounds containing between 1 and 20 carbons with boiling points ranging from 184 to 627 K. Of the total compounds, 88 have hydrogen-bonding groups. Statistical Analysis. The data were analyzed by multiple linear regression using SPSS. Result and Discussion On the basis of statistical analysis, the following empirical equation was obtained for HBP:

HBP ) x[(#OH) + (#COOH) + 0.0625(#NH)] (VIII) According to this equation, the effect of hydrogen-bonding groups increases with the number of such groups. This increase is not linear though, as the influence of the second hydrogenbonding group is less than that of the first group and, similarly, the influence of the third hydrogen-bonding group is less than that of the second group. This explains the square root term associated with the equation. The feasibility for a compound to

Figure 1. Predicted vs apparent experimental ∆Cpb. Table 1. Summary of the Average Absolute Errors for Different Classes of Compounds average absolute errora (J/(K mol)) compounds

n

eq IX

ref 3

ref 4

ref 5

ref 1

rigid-non-H-bonding flexible-non-H-bonding rigid-H-bonding flexible-H-bonding

149 382 19 69

5.6 6.2 13.4 13.5

6.0 14.8 16.1 35.8

12.7 9.3 12.8 18.5

7.5 8.0 17.2 33.4

32.3 28.7 24.5 24.7

total data set

619

7.1

15.1

11.2

11.0

29.0

Average absolute error ) ∑|(apparent experimental value - predicted value)| n. a

undergo the second hydrogen bond reduces because of geometric restrictions imposed by the first hydrogen bond. Noteworthy is the smaller coefficient associated with the -NH- group as compared to the -OH and -COOH groups, which can be explained on the basis of the stronger hydrogenbonding nature of the -COOH and -OH groups as compared to that of the -NH- group. The omission of the -SH group from the above equation is because of its statistically insignificant effect, which can be explained using the same reasoning. The effect of intramolecular hydrogen bonding was not investigated quantitatively because of the small number of compounds in the data set capable of undergoing such bonding. In theory, the formation of an intramolecular hydrogen bond can be expected to reduce the chances for the formation of intermolecular hydrogen bonds and, consequently, will reduce the influence of hydrogen bonding on ∆Cpb. Regression between the apparent experimental heat capacity of boiling and the parameters τ & HBP results in the following relation:

∆Cpb ) -56 - 4τ - 40HBP

(IX)

The average absolute error using eq IX is 7.1 J/(K mol) for data ranging from -158 to -33 J/(K mol). A correlation coefficient (r) of 0.82 was obtained with the standard deviation of 8.6 J/(K mol). A plot of the predicted versus apparent experimental heat capacity is given in Figure 1. The scatter in Figure 1 could not be explained, because any common structural feature leading to this deviation was not observed. Table 1 summarizes the results obtained using eq IX and some previously published methods. For convenience, the compounds

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are divided into four classes: rigid-non-hydrogen-bonding, flexible-non-hydrogen-bonding, rigid-hydrogen-bonding, and flexible-hydrogen-bonding. It is evident from Table 1 that eq IX is more accurate than all the other methods for predicting ∆Cpb, particularly for flexible and/or hydrogen-bonding compounds. Sidgwick’s Rule3 proposes a constant value for ∆Cpb and does not consider the effect of flexibility or hydrogen bonding. The equations by Mishra & Yalkowsky5 and Myrdal & Yalkowsky1 incorporate the coefficient for flexibility but not for hydrogen bonding. Therefore, the errors associated with these equations are particularly high for hydrogen-bonding compounds. Mackay’s relationship4 takes into consideration factors that affect the entropy of boiling. However, the quantitative effect of these factors appears to be different for the entropy of boiling and the heat capacity of boiling. The intercept of -56 J/(K mol) represents the predicted value of ∆Cpb for rigid non-hydrogen-bonding compounds. This value is in accordance with -54.4 J/(K mol) proposed by Sidgwick’s Rule. Conformational and rotational restrictions of molecules in the liquid state produced by flexibility and hydrogen bonding make ∆Cpb more negative. The negative sign associated with the coefficients of hydrogen bonding and molecular flexibility demonstrates the temperature sensitivity of the phase-change enthalpy, i.e., the heat capacity is higher for hydrogen-bonding liquids and increases with the flexibility of the compound. Hydrogen bonding restricts the external motion of the molecules in the condensed phase. Thus, the transition of a hydrogen-bonding liquid to the gas phase results in a big increase in external freedom of the molecules. This increase is higher in magnitude at temperatures lower than the boiling point since the strength of the interaction decreases with increasing temperature. Thus, hydrogen bonding results in an increased heat capacity for the liquid and, consequently, a more negative ∆Cpb. Similarly, flexible molecules are associated with a higher degree of alignment in the liquid phase, and the transition to the gas phase is associated with an increase in the conformational freedom of the molecule. Conclusion A semiempirical model for the estimation of the heat-capacity change associated with boiling has been developed using apparent experimental data for 619 organic compounds. The equation contains an intercept and coefficients for molecular flexibility and hydrogen bonding. The associated average absolute error is 7.1 J/(K mol), which is significantly (R ) 0.01) less than those of existing methods. Although this may be considered slightly high for some practical purposes, simplicity and applicability for quick estimation are the appealing attributes of the model. Example An example is given below of the calculation of the heat capacity of boiling for 1,5-pentanediol using eqs VII, VIII, and IX:

(OH)CH2CH2CH2CH2CH2(OH) SP3 ) 5, SP2 ) 0, Ring ) 0; #OH ) 2, #COOH ) 0, #NH ) 0 Predicted ∆Cpb ) -56 - 4(5+ (0.5 × 0) + (0.5 × 0) 1) - 40 x[2 + 0 + 0] ) -128.6 J/(K mol) Apparent experimental ∆Cpb ) -122 J/(K mol) Supporting Information Available: The experimental boiling points, enthalpies of boiling and vaporization, and calculated values of τ and HBP, along with the apparent experimental and predicted heat capacities of boiling, are available for 610 compounds. For the remaining 9 compounds, calculated values of τ and HBP along with the experimental and predicted heat capacities of boiling are available. The absolute error associated with the prediction is available for all compounds. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Myrdal, P. B.; Yalkowsky, S. H. Estimating Pure Component Vapor Pressures of Complex Organic Molecules. Ind. Eng. Chem. Res. 1997, 36, 2494-2499. (2) Bondi, A. Estimation of the Heat Capacity of Liquids. Ind. Eng. Chem. Fundam. 1966, 5, 442-449. (3) Sidgwick, N. K. The CoValent Link in Chemistry; Cornell University Press: Ithaca, NY, 1933; p104. (4) Mackay, D.; Bobra, A.; Chan, D. W.; Shlu, W. Y. Vapor Pressure Correlations for Low-Volatility Environmental Chemicals. EnViron. Sci. Technol. 1982, 16, 645-649. (5) Mishra, D. S.; Yalkowsky, S. H. Estimation of Vapor Pressure of Some Organic Compounds. Ind. Eng. Chem. Res. 1991, 30, 1609-1612. (6) Coniglio, L.; Daridon, J. L. A Group Contribution Method for Estimating Ideal Gas Heat Capacities of Hydrocarbons. Fluid Phase Equilib. 1997, 139, 15-35. (7) Coniglio, L.; Rauzy, E.; Berro, C. Representation and Prediction of Thermophysical Properties of Heavy Hydrocarbons. Fluid Phase Equilib. 1993, 87, 53-88. (8) Shaw, R. Heat Capacities of Liquids Estimation of Heat Capacity at Constant Pressure and 25 °C, Using Additivity Rules. J. Chem. Eng. Data 1969, 14, 461-465. (9) Chickos, J. S.; Hosseini, S.; Hesse, D. G.; Liebman, J. F. Heat Capacity Corrections to a Standard State: A Comparison of New and Some Literature Methods for Organic Liquids and Solids. Struct. Chem. 1993, 4, 271-278. (10) Dannenfelser, R. M.; Yalkowsky, S. H. Predicting the Total Entropy of Melting: Application to Pharmaceutical and Environmentally Relevant compounds. J. Pharm. Sci. 1999, 88, 722-724. (11) Lide, D. R. Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1991-1992; Vol. 72, pp 6-100-6-107. (12) Chickos, J. S.; Acree, W. E. Enthalpies of Vaporization of Organic Compounds, 1880-2002. J. Phys. Chem. Ref. Data 2003, 32, 519-878. (13) Myrdal, P. B.; Krzyzaniak, J. F.; Yalkowsky, S. H. Modified Trouton’s Rule for Predicting the Entropy of Boiling. Ind. Eng. Chem. Res. 1996, 35, 1788-1792.

ReceiVed for reView June 9, 2005 ReVised manuscript receiVed October 19, 2005 Accepted October 21, 2005 IE050675G