Ind. Eng. Chem. Res. 2005, 44, 3799-3806
3799
Estimation of the Differential Molar Heat Capacities of Organic Compounds at Their Melting Point Georgia D. Pappa,* Epaminondas C. Voutsas, Kostis Magoulas, and Dimitrios P. Tassios Thermodynamics and Transport Phenomena Laboratory, School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zographou Campus, 157 80 Athens, Greece
Methods for the estimation of the differential molar heat capacity, the difference between the heat capacity of the solid and the liquid form of organic compounds at their melting point ∆cp(Tm), are presented. Three schemes are considered: the first involves use of group contribution methods for the prediction of solid heat capacity (cpS) and liquid heat capacity (cpL); the other two, empirical correlations through the entropy of fusion at the melting point ∆Sf(Tm). Recommendations for the different categories of organic compounds are made that provide substantial improvement over the commonly used assumption of ∆cp ) 0, in the prediction of ideal solid solubility and solid vapor pressure. Introduction Differential molar heat capacity (∆cp) data are required for the estimation of properties of solid organic compounds such as solubility and vapor pressure. The knowledge of such properties is crucial for a number of processes. For example, a key process for pharmaceutical, agrochemical, food, and other industries is crystallization. The design and optimization of the crystallization process requires knowledge of the solubility of the compound of interest.1 Also vapor pressure is a crucial property for evaluating environmental and safety hazards of chemicals.2 For compounds for which experimental ∆cp data are not available, there are two alternative assumptions commonly employed for the estimation of their solubility: the first one is that ∆cp at Tm is equal to zero3-6 and the second that it is equal to the entropy of fusion (∆Sf) at the melting temperature.7,8 Both are based on empirical observations, and their success depends on the chemical structure of the compound, its physical properties, the temperature of the system, etc. For example, Yalkowsky5 concluded that the assumption ∆cp ) 0 provides a better fit to solubility data of naphthalene, phenanthrene, fluorene, and anthracene in benzene and is thus a better choice compared to ∆cp ) ∆Sf. On the other hand, Neau and co-workers,9,10 having compared a number of differential heat capacity data to the corresponding ∆Sf values, concluded that, except for flat rigid molecules, ∆cp is better approximated by the entropy of fusion than by a value of zero. Finally Mishra and Yalkowsky11 stated that, unless there are compelling reasons to do otherwise, the choice between the two assumptions should be based upon convenience and ease of use, rather than on the presumed theoretical superiority of a particular approximation. In this work, we demonstrate first the importance of ∆cp in the prediction of ideal solubility (xid) of solids in liquids and vapor pressure (Ps) of solids. We evaluate next the prediction of liquid heat capacity (cpL) and solid heat capacity (cpS) with group contribution (GC) meth* To whom correspondence should be addressed. Tel.: +30 210 772 3274. Fax: +30 210 772 3155. E-mail: gepappa@central. ntua.gr.
ods and proceed to evaluate the methods for ∆cp prediction: (a) the GC methods for the prediction of cpL and cpS, (b) the equality of ∆cp with ∆Sf, and (c) empirical correlations of ∆cp with ∆Sf we developed in this study. We close with a discussion of the resultssand their effect on xid and Ps predictionssand our conclusions. Importance of ∆cp for Ideal Solubility and Vapor Pressure Calculations The solubility of a crystalline organic compound in a solvent is dependent upon those physical properties that are related to the energy of disengagement of molecules from the crystal: the melting point (Tm), the enthalpy of fusion (∆Hf), and the difference in the heat capacity (cp) between the hypothetical liquid and the solid form of the organic compound (∆cp ) cpL - cpS) at constant pressure.9 The solubility (xs) of a solute in a solvent, expressed in mole fraction, is calculated with the following expression:
xsγ ) exp
(
(
)
Tm ∆Hf(Tm) 1+ RTm T
∫T
T
∫TT (cpL - cpS) dT
m
m
RT2
)
dT (1)
where γ is the activity coefficient of the solute in the saturated solution and ∆Hf(Tm) is its enthalpy of fusion at the melting point, ∆Hf(Tm) ) Tm∆Sf(Tm). It is noted that eq 1 is obtained when the solid does not undergo a solid transition or the transition temperature is lower than that of the system. As suggested by eq 1, the solubility of a solid solute in a solvent depends not only on its thermodynamic properties but also on the activity coefficient of the solute. It is, thus, evident that the reliable estimation of the actual solubility of a solute in a solvent requires the availability of an accurate activity coefficient model. Activity coefficient values can be estimated using, for example, predictive group contribution models such as UNIFAC, although for complex molecules such models do not always provide satisfactory results as has been pointed out in the literature.4,12
10.1021/ie048916s CCC: $30.25 © 2005 American Chemical Society Published on Web 04/08/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
The right-hand part of eq 1 is the ideal solubility (xid) of the solute, i.e., its solubility when its activity coefficient equals to unity:
(
(
)
∆Hf(Tm) Tm 1+ RTm T
xid ) exp
∫T
T
∫TT (cpL - cpS) dT m
RT2
m
)
dT (2)
The heat capacity of the solid, cpS, generally increases with temperature at a rate similar or greater than that of the liquid, cpL. Little is known of the heat capacity of the subcooled liquid form of the organic compounds.9 However, to simplify eq 2, ∆cp is assumed to be independent of temperature and equal to ∆cp at the melting point, ∆cp(Tm). So, the following expression for ideal solubility is obtained:
(
xid ) exp
(
)
Tm ∆Sf(Tm) 1R T Tm Tm ∆cp(Tm) ln +1 R T T
(
))
(3)
Similarly, the expression for the vapor pressure of a solid is the following:
Ps ) Psls exp
(
(
)
(
Tm ∆cp(Tm) Tm ∆Sf(Tm) 1ln R T R T Tm + 1 (4) T
))
where Psls is the vapor pressure of the subcooled liquid, usually found by extrapolating the liquid-vapor pressure curve to the required temperature (T). As indicated by eqs 3 and 4 the percentage errors obtained in the ideal solubility and vapor pressure calculation by an error introduced in ∆cp are equal and can be calculated using the following expression:
{
% error in xid or Ps ) 100 1 -
[
(
∆cp Tm Tm ln +1 exp (% error in ∆cp) 100 T T
)]}
(5)
As shown by eq 5, the error obtained in ideal solubility and solid vapor pressure depends not only on the ∆cp value but also on the system temperature as compared to the Tm of the solid compound. The error is relatively low when the temperature of the system is close to Tm because the sum in the last parenthesis approaches zero but becomes larger as we move to lower temperatures, and consequently, the deviation from Tm becomes significant. This is graphically demonstrated in Figure 1 where for the four compounds listed in Table 1 the error in the prediction of ideal solubility is plotted versus the T/Tm ratio, using the assumption that ∆cp ) 0. In the same figure, the errors at T ) 298 K, a common temperature at which solubility data are usually needed, are also shown for each compound. p-Cresol, which has the largest heat capacity difference at the melting point (17.4 cal/mol‚K) of all four compounds, shows also the largest error in xid for the same T/Tm. However, at 298 K, a temperature near the Tm of p-cresol (307.93 K), its
Figure 1. Error in ideal solubility vs the T/Tm ratio for the assumption ∆cp ) 0. The symbols correspond to T ) 298 K. Table 1. Melting Temperature, Entropy of Fusion, and Heat Capacity Difference at the Melting Point for the Organic Compounds of Figure 1 compound
Tm (K)
∆Sf (cal/mol‚K)
∆cp,exp (cal/mol‚K)
p-cresol p-hydroxybenzoic acid triphenylmethane naphthalene
307.9 487.2 365.3 353.4
9.9 15.4 14.4 12.8
17.4 15.1 12.6 2.3
ideal solubility is underestimated by only 0.5%. On the other hand, when comparing the errors in ideal solubility of triphenylmethane and naphthalene, two compounds with close melting temperatures, significantly larger errors are obtained for triphenylmethane since its ∆cp value is more than 5 times larger than that of naphthalene. Finally, for p-hydrobenzoic acid, the error in ideal solubility at 298 K rises up to 67% as a result of both its high Tm as compared to 298 K and its high ∆cp value. The same applies, of course, also for the vapor pressure. From these examples it is concluded that the assumption of ∆cp ) 0 should be avoided for compounds with high Tm’s such as the common pharmaceuticals or organic pollutants when we are interested in their solubility or vapor pressure at ambient temperatures. Group Contribution Methods for cpL and cpS Prediction Ruzicka-Domalski (R-D) GC Method for Liquid Heat Capacity (cpL). Ruzicka and Domalski13,14 developed a group contribution method to predict the liquid heat capacities of organic compounds from the melting to the boiling point. The group assignment is able in most cases to satisfactorily distinguish between isomers since it employs different contributions depending on the kind of atoms bonded to a particular group. There are, however, exceptions such as the simple case of the xylene isomers, which are all described by the same groups. According to the authors, the method has an average absolute percent deviation (AAD) equal to 1.9% from the melting to normal boiling point for 4000 exp. points that correspond to hydrocarbons (C2-C48)
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3801 Table 2. Absolute Percent Deviations (% AD) in the Heat Capacities Predicted with the Group Contribution Methods compound
Tm (K)
cpL exp (cal/mol‚K)
cpL pred R-D method
% ΑDb
cpS exp (cal/mol‚K)
cpS pred PL method
%ΑD
cpS pred PF method
%ΑD
ref
alkanes propane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane n-tetradecane n-hexadecane isobutane 2-methylbutane 2-methylpentane 3-methylpentane 2,2-dimethypentane 2,4-dimethylpentane 2-methylhexane 2,2,3-trimethyl butane 2-methylheptane 3-methylheptane 2,2,4-trimethylpentane 2,3,4-trimethylpentane 3,3-diethylpentane 2,2,4,4-tetramethylpentane 5-methylnonane cyclohexane methylcyclohexane
85.5 143.4 177.9 182.6 216.4 219.7 243.5 263.6 279.0 291.3 113.5 113.3 119.6 110.3 149.3 153.9 154.9 248.6 164.2 152.6 165.8 164.0 240.1 207.0 185.5 279.7 146.6
20.3 33.6 40.0 47.8 54.8 63.0 70.3 86.7 102.4 118.6 24.0 29.5 35.1 34.6 39.1 40.0 42.9 45.9 50.2 48.5 44.0 44.8 58.3 53.9 63.4 34.7 33.4
20.3 32.4 39.4 45.7 53.8 60.4 68.9 84.5 100.3 116.4 23.4 29.1 35.0 34.7 39.7 40.0 42.3 45.7 48.8 48.1 44.5 44.5 59.1 52.3 62.3 35.7 33.2
0.1 3.6 1.5 4.3 1.8 4.1 1.9 2.6 2.0 1.9 2.5 1.3 0.5 0.1 1.5 0.1 1.3 0.3 3.0 0.8 1.3 0.6 1.3 3.0 1.7 2.9 0.6
12.7 24.2 29.6 35.0 42.9 49.9 57.8 71.8 86.0 101.5 20.3 21.8 22.4 21.7 32.5 32.8 32.2 45.8 34.9 33.6 36.9 33.6 51.8 52.1 55.9 32.4 22.0
12.0 22.5 29.6 33.4 42.1 46.7 55.3 69.5 84.5 100.1 16.9 18.9 22.1 20.7 30.0 30.5 30.2 45.3 35.1 33.1 36.9 36.1 54.0 50.1 47.0 34.4 23.6
5.1 6.9 0.0 4.6 1.9 6.5 4.3 3.2 1.8 1.4 16.8 13.0 1.2 4.4 7.6 6.9 6.1 1.0 0.4 1.5 0.1 7.4 4.2 3.9 16.0 6.2 7.5
11.0 23.6 31.9 36.0 45.5 49.6 59.0 71.5 85.2 97.9 16.9 19.5 22.8 21.5 30.7 32.2 31.4 47.6 35.8 33.9 37.5 38.2 55.6 46.1 45.9 38.7 25.1
13.0 2.4 7.5 2.8 6.2 0.6 2.1 0.3 1.0 3.5 16.8 10.3 1.9 1.0 5.7 1.7 2.5 4.0 2.6 1.0 1.6 13.8 7.3 11.6 18.0 19.2 14.1
17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17
% AADc aromatics benzene toluene 1,2,3-trimethylbenzene 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene o-xylene m-xylene p-xylene ethylbenzene n-propylbenzene biphenyl diphenylmethane triphenylmethane
1.7 278.7 178.2 247.8 229.3 228.4 247.8 225.3 286.4 178.2 173.6 342.2 298.4 365.3
31.5 32.2 47.7 46.1 44.0 41.4 38.8 42.3 37.3 43.0 64.2 63.4 106.5
30.3 32.3 46.9 45.9 45.9 41.3 38.6 42.4 37.1 43.0 65.8 62.5 105.7
% AAD alcohols ethanol propanol 2-propanol butanol isobutanol 2-butanol pentanol hexanol heptanol
159.1 147.0 185.3 183.9 165.2 158.5 195.6 228.6 239.2
21.0 25.8 26.7 32.2 30.2 30.5 38.4 47.4 56.4
20.6 26.7 25.6 32.6 30.3 31.2 39.0 47.4 54.9
% AAD
29.6 23.8 40.5 38.1 38.0 35.4 32.8 39.7 26.7 29.2 48.7 49.1 77.6
2.0 3.5 4.3 1.2 0.4 2.4 1.6 0.1 2.6
289.8 252.5 268.0 239.2 309.1 285.6
29.2 34.0 40.4 45.3 48.4 76.0
28.5 33.1 40.9 45.3 49.6 77.1
2.3 2.5 1.2 0.1 2.6 1.6
15.6 19.3 24.4 23.2 21.5 21.7 29.9 36.6 41.4
17.8 18.7 22.6 24.9 23.2 22.4 28.9 36.2 41.3
23.5 28.9 27.4 36.9 34.9 42.7
23.7 23.6 27.5 36.9 32.3 39.1
0.9 18.5 0.5 0.2 7.5 8.3 6.0
29.1 23.9 42.2 39.2 38.4 36.8 33.6 41.6 27.3 29.8 53.5 52.5 81.7
14.1 2.9 7.4 7.0 7.6 3.3 3.2 1.2 0.3
22.7 27.6 36.9 34.4 45.4 66.1
28.5 28.6 33.5 34.1 43.1 57.9
25.5 3.7 9.1 0.9 5.2 12.4
15.9 18.4 22.7 26.4 24.1 23.6 31.8 40.4 46.6
16.7 19.3 20.8 21.7 20.9 26.8
16.8 3.7 8.0 5.1 9.3 10.9 9.0
9 9 9 9 9 9 9 9 9 9 9 17 9
1.8 4.5 6.9 13.6 11.8 8.7 6.4 10.5 12.6
17 17 17 17 17 17 17 17 17
8.5 24.3 27.9 35.0 37.0 48.9 62.1
9.5 14.3 18.6 22.6 22.9 23.0 30.0
3.8 11.3 2.6 7.6 9.1 6.5 12.9 10.6 7.2 3.1 2.4 5.5 12.9 7.3
5.2
1.7 179.8 181.0 156.1 188.4 178.9 206.2
2.1 10.9 6.5 4.4 7.8 2.3 10.2 5.5 4.7 1.1 11.0 11.6 17.3
6.4
7.4
2.0
% AAD amines methylamine dimethylamine trimethylamine n-propylamine isopropylamine tert-butylamine
30.3 21.5 43.3 36.5 35.2 34.6 29.8 37.6 25.5 28.9 54.7 55.6 93.9
1.3
% AAD acids acetic propanoic butanoic pentanoic 2,2-dimethylpropanoic nonanoic
3.7 0.4 1.6 0.3 4.2 0.1 0.4 0.3 0.4 0.1 2.6 1.4 0.7
5.2
7.1 1.1 5.2 7.6 7.8 6.2
17 17 17 17 17 17
5.8 13.5 17.1 20.0 22.7 21.9 27.1
5.4 8.3 11.6 0.7 4.8 9.7 6.8
9 9 9 17 17 17
3802
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
Table 2. (Continued) compound amines methylamine dimethylamine trimethylamine n-propylamine
Tm (K)
cpL exp (cal/mol‚K)
cpL pred R-D method
% ΑDb
cpS exp (cal/mol‚K)
cpS pred PL method
%ΑD
cpS pred PF method
%ΑD
ref
179.8 181.0 156.1 188.4
23.5 28.9 27.4 36.9
23.7 23.6 27.5 36.9
0.9 18.5 0.5 0.2
14.3 18.6 22.6 22.9
16.7 19.3 20.8 21.7
16.8 3.7 8.0 5.1
13.5 17.1 20.0 22.7
5.4 8.3 11.6 0.7
9 9 9 17
% AAD ketones, aldehydes acetone MEK 2-pentanone 3-pentanone MiPK 2-hexanone 3-hexanone 3,3-dimethyl-2-butanone diisopropyl ketone 2-octanone butyl aldehyde hexyl aldehyde heptyl aldehyde
6.0 178.5 186.5 196.3 234.2 181.2 217.4 217.5 221.2 204.8 252.9 176.8 217.2 230.2
28.0 35.6 41.2 43.7 39.0 48.3 49.1 45.1 49.6 63.1 35.2 45.2 56.0
27.5 35.1 41.3 42.5 38.6 48.7 48.7 45.0 50.3 62.7 32.7 47.7 55.0
%AAD ethers dimethyl ether methyl propyl ether methyl isopropyl ether diethyl ether methyl butyl ether ethyl propyl ether methyl tert-butyl ether diisopropyl ether dipropyl ether
131.7 134.0 127.9 156.9 157.5 145.7 164.6 187.7 150.0
23.5 33.7 32.4 35.1 40.5 39.9 36.8 44.0 45.8
22.0 33.6 32.4 34.3 40.0 39.8 36.8 44.3 45.8
21.9 25.4 29.5 34.0 27.9 35.7 35.7 37.3 38.8 49.2 22.8 33.2 38.4
6.3 0.1 0.1 2.4 1.2 0.1 0.1 0.8 0.1
353.4 488.9 366.6 387.9 372.4 383.3 423.8 271.7
52.1 86.6 62.8 70.4 71.6 87.3 87.7 42.6
53.1 87.8 58.4 68.0 72.8 79.9 83.9 40.0
2.0 1.4 7.0 3.3 1.6 8.4 4.3 6.1
16.8 21.2 21.1 24.5 26.7 25.1 32.2 34.3 27.2
16.1 20.4 19.8 23.2 25.9 29.2 27.6 33.9 27.6
26.6 23.8 89.7 154.1 128.1 123.4 74.0 59.9 44.7 55.7 52.3 54.3
27.1 23.9 86.1 149.3 191.8 120.0 97.5 60.9 45.5 54.4 52.7 54.7
1.7 0.5 3.9 3.1 49.7 2.8 31.7 1.7 1.9 2.3 0.8 0.7
20.0 25.4 30.7 35.8 35.1 37.5 37.9 39.9 43.0 51.8 23.7 37.7 44.1
4.2 3.4 5.9 5.5 3.3 16.6 14.2 1.1 1.4
49.7 85.1 61.8 70.3 67.8 86.6 83.3 35.7
43.7 66.3 48.6 53.1 53.4 56.1 60.8 35.7
12.1 22.1 21.4 24.5 21.2 35.2 27.1 0.0
14.5 21.3 20.8 24.2 27.9 26.3 30.0 36.9 29.9
18.1 22.3 60.1 128.3 68.9 89.1 83.2 45.2 32.4 37.7 35.9 38.1
3.6 13.6 14.4 10.6 22.5 8.6 32.5 1.4 4.1 0.3 1.2 3.0
% AAD
1.9
5.9
overall
2.1
8.0
46.0 71.6 50.7 55.1 57.4 62.4 63.9 34.9
13.8 0.8 1.5 1.2 4.3 5.0 6.8 7.5 10.1
17 17 17 17 17 17 17 17 17
7.6 16.0 17.9 21.6 15.3 27.9 23.3 2.3
9 9 9 9 9 17 17 17
16.5 18.6 17.8 71.9 133.9 89.0 96.9 85.1 53.3 34.0 39.6 37.2 40.1
a Not included in the estimation of average and overall errors. b % AD ) |c c pi,calc - cpi,exp|/cpi,exp × 100. % AAD ) 1/N cpi,exp|/cpi,exp × 100. N: number of compounds.
and 2.9% for 5772 exp. points of organic compounds that contain halogens, nitrogen, oxygen, and sulfur (C2-C42). Goodman et al. GC Methods for Solid Heat Capacity (cpS). Goodman et al.15 presented two group contribution methods for the estimation of solid heat capacities of organic compounds. The first one (PL) employs a power law form for the temperature dependency of the heat capacity and is quite simple. The
17 17 17 17 17 17 17 17 17 17 17 17 17
5.7
20.4 17.5 19.7 70.2 143.6 89.0 97.5 62.8 45.8 33.8 37.6 35.4 37.0
11.4 2.1 9.0 6.0 2.9 14.5 11.9 3.2 5.3 1.5 4.9 1.9 5.7 6.2
6.2
4.3 115.0 187.5 455.4 441.8 438.7 428.5 347.2 395.5 257.9 304.2 285.4 307.9
2.9 2.0 12.4 10.7 18.2 18.8 17.2 3.6 14.5 3.5 8.3 10.5 18.0
6.8
10.8
1.2
% AAD miscellaneous chlorotrifluoroethene furan anisic acid diethylstilbestrol mannitola naproxen ibuprofena benzoic acid benzyl alcohol o-cresol m-cresol p-cresol
22.5 24.9 33.7 38.0 34.1 43.9 43.0 38.6 45.4 51.0 24.9 37.1 46.8
1.9
% AAD polycyclic aromatics (fused) naphthalene anthracene acenaphthene fluorine phenanthrene fluoranthrene pyrene indene
1.5 1.5 0.3 2.6 0.9 0.9 0.7 0.1 1.4 0.6 7.1 5.4 1.8
9.0
6.1 9.4 2.4 6.7 0.1 0.6 35.6 16.4 0.4 5.3 5.1 8.5
9 9 10 10 10 10 1 17 9 17 17 17
6.1 7.4 N ∑i)1 |cpi,calc
-
second (PF), based on the Einstein-Debye partition function, is more complicated than the first and also requires knowledge of the radius of gyration. The group assignment is simple and common for both methods, but unfortunately it does not distinguish between isomers. According to the authors, correlation of 7967 exp. points of 455 compounds gave AADs equal to 6.8% for the PL and 8% for the PF method while prediction with AADs
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3803 Table 3. ∆cp and Percent Deviations Estimated with the Three Schemes Examined in This Work scheme I(a)
scheme I(b)
scheme II
scheme III
compound
Tm (K)
∆cp, exp (cal/mol‚K)
∆cp
% devb
∆cp
% dev
∆cp ()∆Sf)
% dev
∆cp
% dev
alkanes propane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane n-tetradecane n-hexadecane isobutane 2-methylbutane 2-methylpentane 3-methylpentane 2,2-dimethypentane 2,4-dimethylpentane 2-methylhexane 2-methylheptane 3-methylheptane 2,2,4-trimethylpentane 2,3,4-trimethylpentane 3,3-diethylpentane 2,2,4,4-tetramethylpentane 5-methylnonane methylcyclohexane
85.5 143.4 177.9 182.6 216.4 219.7 243.5 263.6 279.0 291.3 113.5 113.3 119.6 110.3 149.3 153.9 154.9 164.2 152.6 165.8 164.0 240.1 207.0 185.5 146.6
7.6 9.4 10.4 12.7 11.9 13.1 12.5 15.0 16.4 17.1 3.6 7.7 12.8 13.0 6.6 7.2 10.7 15.3 14.9 7.1 11.2 6.5 1.7 7.4 11.4
8.3 9.9 9.8 12.3 11.8 13.7 13.6 15.0 15.9 16.4 6.5 10.1 12.9 14.0 9.7 9.5 12.1 13.7 15.0 7.7 8.5 5.0 2.2 15.3 9.5
8.8 4.9 -5.6 -3.5 -1.4 5.1 9.1 0.2 -3.0 -4.4 77.1 31.6 0.9 7.8 46.4 32.2 12.8 -10.8 0.8 8.2 -24.5 -22.3 24.5 105.7 -16.2
9.3 8.8 7.6 9.7 8.3 10.8 9.9 13.0 15.2 18.5 6.5 9.6 12.2 13.2 9.1 7.8 10.9 12.9 14.2 7.1 6.3 3.5 6.2 16.4 8.1
21.9 -6.6 -27.0 -23.9 -30.2 -17.1 -20.6 -13.4 -7.3 8.0 77.4 24.2 -4.6 1.9 37.2 8.3 2.0 -15.6 -4.8 -0.5 -43.7 -46.8 251.8 120.2 -29.0
9.9 14.0 17.6 18.4 22.9 16.8 28.2 33.4 38.6 43.8 9.6 10.9 12.5 11.5 9.3 10.6 14.2 17.3 18.2 13.3 13.5 10.0 11.2 21.4 11.0
29.3 48.2 68.9 44.4 91.7 28.7 125.5 123.0 135.8 155.7 162.1 41.2 -1.9 -11.2 41.1 47.6 32.3 12.8 22.4 86.7 20.4 54.6 540.3 187.5 -3.1
7.5 9.8 11.3 11.6 13.0 11.0 14.4 15.5 16.4 17.2 7.3 8.1 9.1 8.5 7.1 8.0 9.9 11.2 11.5 9.4 9.5 7.6 8.3 12.6 8.2
-1.8 3.6 8.3 -9.2 8.8 -16.0 14.9 3.3 0.3 0.7 99.7 5.6 -29.1 -34.4 7.8 10.8 -8.0 -27.2 -22.7 32.7 -14.9 17.1 375.6 68.7 -27.8
% AAD aromatic hydrocarbons benzene toluene 1,2,3-trimethylbenzene 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene o-xylene m-xylene p-xylene ethylbenzene n-propylbenzene biphenyl diphenylmethane triphenylmethane
18.7 278.7 178.2 247.8 229.3 228.4 247.8 225.3 286.4 178.2 173.6 342.2 298.4 365.3
1.2 10.7 4.3 9.6 8.8 6.8 8.8 4.8 11.8 14.2 9.4 7.9 12.6
0.7 8.5 6.4 7.8 7.9 5.9 5.8 2.7 10.4 13.8 17.1 13.4 28.1
% AAD alcohols ethanol propanol 2-propanol butanol isobutanol 2-butanol pentanol hexanol heptanol
159.1 147.0 185.3 183.9 165.2 158.5 195.6 228.6 239.2
5.4 6.6 2.3 9.0 8.7 8.8 8.5 10.8 15.0
2.8 8.0 3.0 7.7 7.1 8.8 10.1 11.2 13.6
% AAD
-48.7 22.5 27.9 -13.8 -17.5 0.3 18.5 4.5 -8.9
289.8 252.5 268.0 239.2 309.1 285.6
6.5a 6.4 3.6 10.9 3.0 9.8
0.1 4.5 7.4 11.3 6.6 19.2
-98.9 -29.0 108.0 3.3 121.3 95.5
4.7 8.3 2.9 6.2 6.2 7.7 7.2 7.0 8.3
9.2 10.3 4.8 14.0 11.9 12.7
7.0 4.3 6.7 15.2 11.4 12.4
-24.0 -58.4 40.4 8.9 -4.1 -2.3 23.0
8.5 8.9 7.9 12.0 9.9 13.1 12.3 14.3 12.3 12.8 13.0 14.6 14.4
-13.1 27.3 23.4 -30.8 -28.0 -13.0 -15.3 -35.2 -44.6
4.3 5.3 5.9 8.3 0.7 15.1
-34.8 -17.7 67.1 -23.6 -76.1 53.8
7.4 8.7 7.0 12.2 9.1 16.4 12.0 16.1 18.2
10.6 -36.7 57.7 1.7 -12.7 -5.2 20.8
6.5 6.8 6.0 8.8 7.5 9.3 8.9 9.9 8.9 9.2 9.3 10.1 10.0
37.7 33.3 200.6 36.0 5.6 86.1 40.3 49.7 21.4
9.7 10.1 10.3 14.2 1.8 16.6
48.3 58.2 191.0 29.7 -41.2 69.0
5.4 6.4 5.1 8.9 6.7 11.9 8.7 11.7 13.2
-11.3 -24.0 109.4 6.2 -17.2 -91.9 43.3
0.1 -3.1 118.5 -1.2 -23.2 35.3 2.0 8.8 -11.8 22.7
7.0 7.3 7.5 10.3 1.3 12.1
72.9 8.1 7.8 10.0 14.8 9.8 1.0
440.4 -36.3 39.2 -8.8 -14.5 38.5 1.0 107.7 -24.2 -35.2 -1.5 28.1 -21.2 61.3
56.7
45.5 10.2 6.5 7.6 14.2 10.4 12.0
605.0 -16.8 82.2 24.8 12.7 94.2 39.0 199.4 4.4 -9.9 37.8 86.0 13.9
34.0
94.3
25.6
76.0 179.8 181.0 156.1 188.4 178.0 206.2
-0.5 -21.6 7.9 -30.5 -15.3 -33.2 -43.6 -83.7 -16.7 -6.6 31.5 27.8 90.0
84.7
31.5
18.1
% AAD amines methylamine dimethylamine trimethylamine n-propylamine isopropylamine tert-butylamine
1.2 8.4 4.7 6.7 7.5 4.5 5.0 0.8 9.8 13.2 12.4 10.0 24.0
40.0
% AAD acids acetic propanoic butanoic pentanoic 2,2-dimethyl propanoic nonanoic
-43.6 -20.8 47.4 -18.4 -10.2 -11.9 -34.5 -43.2 -11.5 -2.7 81.8 71.0 122.5
33.8
7.8 15.0 111.5 -5.7 -57.2 22.8 36.7
5.9 5.7 7.3 10.8 7.2 0.7
-35.5 -44.8 52.2 -22.8 -39.8 -94.1 48.2
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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
Table 3. (Continued) scheme I(a)
scheme I(b)
scheme II
compound
Tm (K)
∆cp, exp (cal/mol‚K)
∆cp
% devb
∆cp
% dev
∆cp ()∆Sf)
ketones, aldehydes acetone MEK 2-pentanone 3-pentanone MiPK 2-hexanone 3-hexanone 3,3-dimethyl-2-butanone diisopropyl ketone 2-octanone butyl aldehyde hexyl aldehyde heptyl aldehyde
178.5 186.5 196.3 234.2 181.2 217.4 217.5 221.2 204.8 252.9 176.8 217.2 230.2
5.4 10.7 7.5 5.7 4.9 4.4 6.0 6.4 4.2 12.1 10.3 8.2 9.2
5.7 9.7 11.8 8.6 10.7 13.1 13.1 7.8 11.5 13.5 9.9 14.5 16.6
4.2 -9.5 57.4 51.4 121.0 197.7 117.2 20.7 172.2 11.4 -4.4 77.3 80.0
7.6 9.7 10.7 6.8 3.5 11.2 10.8 5.2 7.3 10.9 9.0 9.9 10.9
39.2 -9.8 42.3 19.7 -27.8 154.5 79.4 -19.9 72.8 -9.7 -12.6 21.4 17.9
7.6 10.8 12.9 11.8 12.3 16.4 14.8 12.2 13.0 20.2 15.0 20.8 24.5
% AAD ethers dimethyl ether methyl propyl ether methyl isopropyl ether diethyl ether methyl butyl ether ethyl propyl ether methyl tert-butyl ether diisopropyl ether dipropyl ether
71.1 131.7 134.0 127.9 156.9 157.5 145.7 164.6 187.7 150.0
6.7 12.5 11.3 10.6 13.7 14.8 4.6 9.7 18.6
-11.4 5.4 10.7 4.7 2.9 -28.3 100.2 7.8 -1.8
5.9 13.2 12.5 11.1 14.1 10.6 9.2 10.4 18.2
% AAD polycyclic aromatics (fused) naphthalene anthracene acenaphthene fluorene phenanthrene fluoranthrene pyrene indene
7.5 12.3 11.6 10.1 12.1 13.5 6.8 7.5 15.9
19.2
353.4 488.9 366.6 387.90 372.4 383.33 423.8 271.7
2.3 1.5 1.1 0.04 3.8 0.7 4.3 6.9
9.3 21.5 9.9 14.9 19.3 23.8 23.1 4.3
7.1 16.2 7.7 12.9 15.3 17.5 20.0 5.1
562.9 115.0 187.5 455.4 441.8 438.7 428.5 347.2 395.5 257.9 304.2 285.4 307.9
9.2 4.1 19.5 10.5 39.1 26.0 11.3 14.1 10.9 18.0 16.9 17.4
9.0 1.6 26.1 20.9 122.8 30.9 14.3 15.7 13.1 16.7 16.9 16.6
12.5 -1.6 2.5 -5.2 -11.9 -8.7 48.1 -22.7 -14.5
208.2 997.3 632.2 32150a 305.3 2400a 361.9 -25.5
9.0 13.7 10.9 11.0 16.5 13.8 11.0 14.1 17.2
8.5 6.1 14.3 15.3 102.7 23.1 12.4 7.5 11.6 14.8 15.5 14.6
-6.7 47.8 -26.8 -46.7 162.4a -11.1 10.3 -46.5 6.4 -18.0 -8.3 -15.9
∆cp
% dev
40.3 0.8 72.8 109.2 153.9 272.5 146.1 89.9 208.8 67.1 45.3 154.3 165.2
3.6 5.1 6.1 5.6 5.8 7.8 7.0 5.8 6.2 9.6 7.1 9.9 11.6
-33.5 -52.2 -18.1 -0.8 20.3 76.5 16.6 -10.0 46.4 -20.8 -31.1 20.5 25.7
33.7 9.4 -3.4 3.1 20.0 -7.1 140.2 45.5 -7.6
12.8 14.4 14.0 12.1 10.6 11.7 9.8 9.0
11.5 4.9 14.6 15.6 27.6 17.6 17.6 10.9 8.3 12.4 9.0 9.9
26.0 17.7 -25.0 48.7 -29.6a -32.3 55.9 -22.3 -23.7 -31.1 -46.9 -43.2
26.2
22.2
33.9
66.4
54.2
98.1
Not included in the estimation of average and overall errors. N 1/N∑i)1 |∆cpi,calc - ∆cpi,exp|/∆cpi,exp × 100. N: number of compounds.
equal to 13 and 20%, respectively, was obtained for 1800 exp. points of 45 compounds. Finally, the PL method was found to be more accurate at temperatures below 250 K and the PF method at temperatures above 250 K. Results for cpL and cpS predictions at Tm with the R-D and the Goodman et al. methods, respectively, are shown in Table 2 for a variety of organic compounds: simple linear and branched hydrocarbons, alcohols, ketones, acids, amines, esters, polycyclic aromatic hydrocarbons, and some multifunctional compounds. Ac-
19.0 -2.7 -14.1 -8.2 6.8 -17.3 113.8 29.5 -17.8 25.5
482.3
overall
b
8.0 12.2 9.7 9.8 14.7 12.3 9.8 12.5 15.3
455.4 869.6 1232.4 30150a 179.6 1571a 126.4 30.3
% AAD a
28.7
30.0
421.7
-1.9 -62.0 34.0 99.9 213.8a 19.0 27.3 11.7 20.5 -7.6 -0.2 -4.3
% dev
117.4
14.2
304.4 1350.7 840.0 37150a 411.4 3300a 433.9 -37.2
% AAD miscellaneous chlorotrifluoroethene furan anisic acid diethylstilbestrol mannitol naproxen ibuprofen benzoic acid benzyl alcohol o-cresol m-cresol p-cresol
40.5
scheme III
14.2 6.0 18.0 19.1 33.9 21.6 21.6 13.4 10.2 15.3 11.0 12.1
55.0 44.8 -7.7 82.9 -13.4 -16.7 91.7 -4.5 -6.1 -15.3 -34.7 -30.2 35.4 36.4
% deviation ) (∆cpi,calc - ∆cpi,exp)/∆cpi,exp × 100. % AAD ) c
cording to the DIPPR quality code, the estimated accuracy of the experimental heat capacity data is better than 5%. When the radius of gyration, which is necessary for the PF method, was not experimentally available it was calculated using the expression proposed by Hayden and O’Connell:18
P ) 50 + 7.6R + 13.75R2 where R is the radius of gyration and P the parachor.
Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3805
Parachor values were predicted with the group contribution method of Quayle.19 The comparison of the predicted values with the experimental ones indicates the following: (1) The accuracy of the R-D method in the prediction of cpL at Tm is very good with an overall error equal to 2% and AADs lower than 4% for more than the 85% of the compounds examined here. There are only a few exceptions for rather complicated compounds such as mannitol and ibuprofen. Interestingly, high error is also obtained for dimethylamine. Given that for the rest of the amines the method performs very well, this high error is attributed to the group contribution value of the CH3 group connected to a N atom. Actually, while in most cases the R-D method employs different contributions for a particular atom depending on the kind of atoms it is bonded to, for the case of a carbon atom in a CH3 group, the contribution is considered common for all compounds. This probably explains the high errors obtained for dimethyl ether (6.3%), methanol (17%), and of course dimethylamine (18.5%) compared to those for diethyl ether (2.4%), ethanol (2.0%), and diethylamine (1.05% at 248.3 K) where the contribution of the carbon atom in the CH2 group differs depending on the functional group it is bonded to. (2) With the exception of the polycyclics for which the error often exceeds 20%, the PL and PF methods perform reasonably well with average absolute errors equal to 8.0 and 7.4%, respectively. In contrast to the observation of Goodman et al., no advantage for one of the methods with respect to the temperature involved was observed. ∆cp Prediction The following schemes were examined for the prediction of ∆cp at Tm:
Table 4. Empirical Correlations of ∆cp with ∆Sf: Scheme III alkanes + aromaticsa alcohols + acids + amines ethers ketones, aldehydes polycyclic aromatics fused miscellaneous a
correlation
% AAD
) 6.54 ln(∆Sf) - 7.48 ) 0.73 ∆Sf ) 0.89 ∆Sf ) 0.47 ∆Sf no correlation exists ) 1.23 ∆Sf
43.4 34.0 25.5 28.7 35.4
For ∆Sf >3.2. Below that use ∆cp ) 0.
Figure 2. Error obtained in ideal solubility when ∆cp ) 0, ∆cp ) ∆cp,exp ( 30% and ∆cp ) ∆cp,exp ( 50% vs the T/Tm ratio for (a) ∆cp,exp ) 5 cal/mol‚K, (b) ∆cp,exp ) 10 cal/mol‚K, (c) ∆cp,exp ) 15 cal/mol‚K, and (d) ∆cp,exp ) 20 cal/mol‚K.
Scheme I: ∆cp(Tm) ) cpL(Tm) - cpS(Tm) using predicted cpL and cpS The R-D method was employed for the prediction of cpL(Tm) while for cpS(Tm) both the PL and PF methods of Goodman et al. were examined. The two schemes are referred to as scheme Ia and Ib, respectively. Prediction results for the two schemes are presented in Table 3.
correlation of ∆cp(Tm) as a function of ∆Sf(Tm) was obtained as shown in Table 4. For polycyclic aromatic hydrocarbons, experimental ∆cp values are very low, independently of ∆Sf. Consequently, the assumption ∆cp ) 0 was adopted here, which is in agreement with the findings of Yalkowsky.5 The results obtained using this approach are also presented in Table 3. Discussion and Conclusions
Scheme II: ∆cp(Tm) ) ∆Sf (Tm) The assumption ∆cp(Tm) ) ∆Sf(Tm) (here referred to as scheme II) has already been used in the literature.9,10 It is based on the observation by Hildebrand and Scott16 that, in the case of ideal solubility, ln as (as is the activity of the solute which is equal to the ideal solubility) and ln T are linearly dependent. Results with this method are presented in Table 3.
Scheme III: empirical correlation of ∆cp(Tm) with ∆Sf (Tm) In an effort to improve the results of scheme II, we classified the compounds of the database considered here in six categories. For each category, an empirical
On the basis of the results of Table 3, the recommendations for the prediction of ∆cp(Tm) presented in Table 4 can be made. To evaluate the effect that these ∆cp values have on the prediction of the solute ideal solubility (xid) and solid vapor pressure (Ps), we consider the results shown in Figure 2, which demonstrates the effect of a given percent error in ∆cp on xid or Ps for four typical ∆cp values as a function of the distance of the temperature from Tm. These four ∆cp values and the temperature range 0.65 < T/Tm < 1 considered cover practically all compounds that may be encountered in practice for a typical temperature of interest equal to 298 K. Notice that for ∆cp errors as high as (50%, the corresponding errors in the predicted xid and Ps values are lower (about half at a 50% error for ∆cp) than those obtained with the commonly used assumption that ∆cp ) 0. Actually,
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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005
Literature Cited
Figure 3. Performance of three approaches in ∆cp(Tm) prediction: ∆cp ) 0, ∆cp ) ∆Sf, and the recommended ones (Table 5) in the prediction of ideal solubility and solid vapor pressure at T ) 0.65Tm. Table 5. Recommendations for the Prediction of ∆cp category
scheme
alkanes, alcohols acids, ketones, aldehydes aromatics, ethers, amines miscellaneous polycyclic aromatics fused
Ia III Ib Ib ∆cp ) 0
for negative errors, improved results over ∆cp ) 0 are obtained. On the basis of these observations, the advantage of using the recommended Table 5 schemes for ∆cp estimation in the prediction of xid and Ps, over the commonly employed assumptions of ∆cp ) 0 or ∆cp ) ∆Sf, is clearly demonstrated in Figure 3: errors lower than 10% are obtained in xid or Ps for the majority of compounds (∼60%) while for 90% of them the errors are lower than 20% as compared to 20 and 40%, and 7 and 15%, for ∆cp ) ∆Sf and ∆cp ) 0, respectively. Only for 3% of the compounds are the errors higher than 40% as compared to more than 30% using the assumption ∆cp ) ∆Sf and 50% of them using ∆cp ) 0. It should be noted that these errors refer to the worst case, i.e., T/Tm ) 0.65, and lower ones are obtained in all cases as T approaches Tm. In conclusion, the recommended schemes for ∆cp prediction provide substantial improvement in the prediction of solid ideal solubility and solid vapor pressure over the commonly used assumption of ∆cp ) 0 or ∆cp ) ∆Sf. In addition, if ∆Sf is not available and scheme II cannot be used where recommended, i.e., for acids, ketones, and aldehydes, scheme Ib should be used. Predicted ∆Sf values from methods such as that Dannenfelser and Yalkowsky20 could be used, but this would result in added uncertainly in the predicted ∆cp values.
(1) Gracin, S.; Rasmuson, A. C. Solubility of Phenylacetic Acid, p-Hydroxyphenylacetic Acid, p-Aminophenylacetic Acid, p-Hydroxybenzoic Acid, and Ibuprofen in Pure Solvents. J. Chem. Eng. Data 2002, 47 (6), 1379. (2) Lyman, W. J.; Reehl, W. F.; Rosenblatt, D. H. Handbook of Chemical Property Estimation Methods; American Chemical Society: Washington, DC, 1990. (3) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall: Englewood Cliffs, NJ, 1999. (4) Abildskov, J.; O’Connell, J. P. Predicting the Solubilities of Complex Chemicals I. Solutes in Different Solvents. Ind. Eng. Chem. Res. 2003, 42 (22), 5622. (5) Yalkowsky, S. H. Solubility and Partitioning. 5. Dependence of Solubility on Melting-Point. J. Pharm. Sci. 1981, 70, 971. (6) Yalkowsky, S. H.; Valvani, S. C. Solubilities and partitioning. 2. Relationships between aqueous solubilities, partition coefficients, and molecular surface areas of rigid aromatic hydrocarbons. J. Chem. Eng. Data 1979, 24 (2), 127. (7) Mauger, J. W.; Paruta, A. N.; Gerraughty, R. J. Solubilities of Sulfadiazine, Sulfisomidine, and Sulfadimethoxine in Several Normal Alcohols. J. Pharm. Sci. 1972, 61, 94. (8) Grant, D. J. W.; Mendizadeh, M.; Chow, A. H.-L.; Fairbrother, J. E.; et al. Non-Linear van’t Hoff Solubility Temperature Plots and their Pharmaceutical Interpretation Int. J. Pharm. 1984, 18, 25. (9) Neau, S. H.; Flynn, G. L. Solid and Liquid Heat Capacities of n-Alkyl Para-aminobenzoates Near the Melting Point. Pharm. Res. 1990, 7 (11), 1157. (10) Neau, S. H.; Bhandarkar, S. V.; Hellmuth E. W. Differential Molar Heat Capacities to Test Ideal Solubility Estimations. Pharm. Res. 1997, 14 (5), 601. (11) Mishra, D. S.; Yalkowsky, S. H. Ideal Solubility of a Solid Solute: Effect of Heat Capacity Assumptions. Pharm. Res. 1992, 9 (7), 958. (12) Gracin, S.; Brinck, T.; Rasmuson, A. C. Prediction of Solubility of Solid Organic Compounds in Solvents by UNIFAC. Ind. Eng. Chem. Res. 2002, 41 (20), 5114. (13) Ruzicka, V.; Domalski, E. S. Estimation of the HeatCapacities of Organic Liquids as a Function of Temperature Using Group Additivity. 1. Hydrocarbon Compounds. J. Phys. Chem. Ref. Data 1993, 22 (3), 597. (14) Ruzicka, V.; Domalski, E. S. Estimation of the HeatCapacities of Organic Liquids as a Function of Temperature Using Group Additivity. 2. Compounds of Carbon, Hydrogen, Halogens, Nitrogen, Oxygen, and Sulfur. J. Phys. Chem. Ref. Data 1993, 22 (3), 619. (15) Goodman, B. T.; Wilding, W. V.; Oscarson, J. L.; Rowley, R. L. Use of the DIPPR Database for Development of Quantitative Structure-Property Relationship Correlations: Heat Capacity of Solid Organic Compounds. J. Chem. Eng. Data 2004, 49, 24. (16) Hildebrand, J. H.; Scott, R, L. Regular Solutions; Van Nostrand Reinhold: New York, 1970. (17) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Hemisphere: New York, 1989. (18) Hayden, J. G.; O′Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem. Proc. Des. Dev. 1975, 14 (3), 209. (19) Quayle, O. R. The Parachors of Organic Compounds. An Interpretation and Catalogue. Chem. Rev. 1953, 53, 439. (20) Dannenefelser, R.-M.; Yalkowsky, S. H. Estimation of Entropy of Melting from Molecular Structure: A Non-Group Contribution Methodol. Ind. Eng. Chem. Res. 1996, 35 (4), 1483.
Received for review November 9, 2004 Revised manuscript received January 5, 2005 Accepted January 11, 2005 IE048916S
