ESTIMATION OF THERMODYNAMIC QUANTITIES

CHAPTER 22 ESTIMATION OF THERMODYNAMIC QUANTITIES

In this chapter we shall review some empirical and theoretical methods of estimation of thermodynamic quantities associated with chemical transformations.

22.1 EMPIRICAL METHODS Precise thermodynamic data are available for relatively few compounds. However,

in many situations, it is desirable to have some idea of the feasibility or impossibility of a given chemical transformation even though the necessary thermodynamic data are not available. Several groups of investigators [1] have proposed empirical methods of correlation that allow us to estimate the thermodynamic properties required to calculate Gibbs functions and equilibrium constants. All of these methods are based on the assumption that a given thermodynamic property, such as entropy, of an organic substance can be resolved into contributions from each of the constituent groups in the molecule. With tables of such group contributions assembled from available experimental data, we can estimate the thermodynamic properties of any molecule by adding the contributions of the constituent groups. Additional corrections can be made for the effect of neighboring groups.

Generally, several alternative methods of choosing the groups exist into which a specified molecule is resolved. In the Anderson – Beyer – Watson – Yoneda approach, the thermodynamic properties in the ideal gaseous state are estimated by considering

a given compound as built up from a base group (such as one of those listed in

Chemical Thermodynamics: Basic Concepts and Methods, Seventh Edition . By Irving M. Klotz and Robert M. Rosenberg Copyright # 2008 John Wiley & Sons, Inc.

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.1. Base-Group Properties Heat Capacity Constants Base Group

a b c Methane

D f H8 m,298.15K

S8 m,298.15K

Table 22.1), which has been modified by appropriate substitutions to yield the desired molecule. Thus, aliphatic hydrocarbons can be built up from methane by repeated substitutions of methyl groups for hydrogen atoms. Other compounds are formed by substitution of functional groups for CH n groups. The heat capacity constants are those for a cubic polynomial in the temperature, which are similar to those discussed in Chapter 4.

In the method of Cohen and Benson [2], a group is defined as a polyvalent atom of ligancy modynamic values for 37 hydrocarbon groups, 61 oxygen-containing groups, 59 nitrogen-containing groups, 46 halogen-containing groups, 53 sulfur-containing groups, 57 organometallic groups, and 65 organophosphorus and organoboron groups.

Whatever the method for dividing a compound into groups, the group values for thermodynamic properties must be obtained from a database of experimental values. 1 Each experimental value for a compound yields a linear equation in which the experi- mental value is the sum of group contributions. Preferably, many more experimental values should exist, the dependent variables in the linear equations, than the total number of group values. Then, we have an overdetermined set of values of the group contributions. The best values of the group contributions are obtained by multivariable least-squares fitting of the equations to the experimental data [3].

We will consider in some detail only one of these procedures—that of Andersen, Beyer, and Watson, as modified by Yoneda [4]—to illustrate the type of approach used in these approximation methods.

Group Contribution Method of Andersen, Beyer, Watson, and Yoneda

Like several other systems, this method is based on the assumption that a given ther- modynamic property, such as entropy, of an organic substance can be resolved into

1 Pedley provides group values and the experimental data from which they have been derived. J. Pedley, Thermochemical Data and Structure of Organic Compounds– Vol. I , TRC Data Series, CRC Press,

Boca Raton, FL, 1994.

513 contributions from each constituent group in the molecule. With tables of such

22.1 EMPIRICAL METHODS

group contributions assembled from available experimental data, we can estimate the thermodynamic properties of any molecule by adding the contributions of the constituent groups.

The base groups of the Anderson – Beyer – Watson – Yoneda method are listed in Table 22.1. They are modified by appropriate substitutions to yield the desired molecule. Thus, aliphatic hydrocarbons can be built up from methane by repeated substitutions of methyl groups for hydrogen atoms. Other compounds are formed by substitution of functional groups for CH n groups. All values in the

tables are in units of J mol 21 or J K 21 mol 21 as appropriate. The heat capacity constants are similar to those discussed in Chapter 4 but for a quadratic polynomial in T /1000.

The thermodynamic quantities for large, complex molecules are obtained by adding the contributions of the appropriate substitution group to the value for the base group. Table 22.2 gives the contributions for the primary substitution of a

CH 3 group on a single carbon atom in each of the five base groups listed in Table 22.1. For the cyclic base groups—cyclopentane, benzene, and naphthalene— several carbon atoms are available for successive primary substitutions (no more than one on each carbon atom), and the magnitude of the contribution depends on the number and position of the added methyl groups as well as on the type of base ring.

A second substitution of a methyl group for a hydrogen on a single carbon atom of a base group is called a secondary substitution. These secondary replacements have to be treated in more detail because the changes in thermodynamic properties depend on the nature of the carbon atom on which the replacement is being made and on the nature of the adjacent carbon atom. For this reason, these carbon atoms are characterized by “type numbers,” as shown in Table 22.3. The thermodyn- amic changes associated with secondary methyl substitutions then can be tabulated as in Table 22.4. The number in Column A is the type number of the carbon atom on which the second methyl substitution is made, and that in Column B is the highest type of an adjacent carbon atom, with each number referring to the status of the carbon atom before the substitution is made.

The effect of introducing multiple bonds in a molecule is treated separately. The appropriate corrections have been assembled in Table 22.5 and require no special comments, except perhaps to emphasize the additional contribution that must

be introduced every time a pair of conjugated double bonds is formed by any of the preceding substitutions in this table. The changes in properties accompanying the introduction of various functional groups in place of one or two of the methyl groups on a given carbon are listed in Table 22.6. Data from Table 22.1 and Table 22.2 give the contributions for the appropriate methyl-substituted base groups. One should observe particularly that the 55O structure requires replacement of two methyl groups, which must

be added before they can be substituted by 55O. The symbol @ preceding a func- tional group means that the values refer to the substitution of that group on an

aromatic ring.

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.2. Contributions of Primary CH 3 Substitution Groups Replacing Hydrogen on Carbon

Heat Capacity Constants Base Group

Da Db Dc Methane

D(DH m )

DS m

29.83 43.30 9.92 103.81 243.51 Cyclopentane

49.25 8.74 68.24 223.18 b) Second primary substitution to form 1,1

a) First primary substitution

46.32 11.59 81.21 239.58 b) Second primary substitution to form 1,1

a) First primary substitution

47.91 5.77 64.43 219.50 b) Second primary substitution to form 1,2

a) First primary substitution

a) First primary substitution to form

44.39 10.67 61.76 220.17 b) Second primary substitution to form 1,2

515 TABLE 22.3. Type Numbers of Different

22.1 EMPIRICAL METHODS

Carbon Atoms Type Number

Nature 1 2CH 3

2 .CH 2 3 .CH2 4 .C,

9 C in aromatic ring

The addition of a functional group requires more corrections for the type (as in Table 22.3) of the carbon to which the functional group is attached. For example,

a keto group, 2C55O, is attached to a carbon atom of type 2, because that carbon is also attached to two atoms other than hydrogen. An aldehyde group, 2C55O, however, would have a correction for a type 1 carbon. The corrections would then

be two multiples and one multiple, respectively, of the entry in Table 22.7 for the substituent. The determination of type number is made after the substitution. Also, when multiple substitutions are made on the same carbon atom, primarily by halogens, corrections must be made for the number of pairwise interactions among the atoms substituted on the same carbon atom. Thus, when three halogen atoms are substituted on a carbon atom, three pairwise interactions are possible and

TABLE 22.4. Contribution of Secondary Methyl Substitution Type

Number Heat Capacity Constants A B D(DH m )

Da Db Dc 1 1 221.09

DS m

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.5. Multiple Bond Contributions Replacing Single Bonds Heat Capacity Constants Type of Bond or Correction

Da Db Dc 1¼1

D(DH m )

12.84 2127.03 51.67 Adjacent double bonds

9.75 27.78 2.13 Conjugated double bonds

215.31 216.99 26.69 37.28 227.49 Double bond conjugated with

27.20 29.50 5.36 29.08 5.19 aromatic ring Triple bond conjugated with

23.8 4.6 0.4 aromatic ring Conjugated triple bonds

3.3 14.6 214.6 Conjugated double and triple bonds

the entry in Table 22.7 must be multiplied by three for that case in addition to the type correction. A % symbol preceding a functional group indicates that the substituent can be added in either of two orientations and that the correct choice needs to be made.

The procedure followed in the use of the tables of Andersen et al. [1], and Yoneda [4] is illustrated below for the estimation of standard entropies. These tables also include columns of base structure and group contributions for estimating

D f H8 m,298.15K , the standard enthalpy of formation of a compound, as well as columns for a, b, and c, the constants in the heat capacity equations that are quadratic in the temperature. Thus it is possible to estimate D f G8 m,298.15K by appropriate sum- mations of group contributions to D f H8 m,298.15K and to S8 m,298.15K . Then, if information is required at some other temperature, the constants of the heat capacity equations can

be inserted into the appropriate equations for DG8 m as a function of temperature and DG8 m can be evaluated at any desired temperature (see Equation 7.68 and the relation between DG8 m and ln K ).

Typical Examples of Estimating Entropies The use of Tables 22.1 through 22.7 will be illustrated by two examples.

517 TABLE 22.6. Contributions of Functional Groups

22.1 EMPIRICAL METHODS

Heat Capacity Constants Functional Group a

Da Db Dc Oxygen

D(DH m )

DS m

¼O(aldo) 210.13

17.11 2214.05 84.27 ¼O(keto)

25.27 124.64 269.25 2COO 2 CO2

Fluorine 2F 2154.18

4.23 276.57 24.56 @F

6.49 259.50 18.37 @F(ortho)

2Cl 2.05 25.90 7.45 264.85 14.94 @Cl

2Br 49.54 13.10 11.13 249.92 13.05 @Br

57.57 7.28 12.30 270.33 28.91 2COBr

2105.73 68.6 20.9 243.5 9.2 @COBr

298.49 Iodine

14.56 11.38 272.51 18.28 @I

2I 101.13

8.8 12.6 292.9 34.3 2COI

238.03 88.3 23.4 233.1 9.6 @COI

231.0 (Continued)

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.6. Continued Heat Capacity Constants

Functional Group a D(DH m )

Da Db Dc Sulfur

DS m

2SH 60.33 24.06 14.39 265.94 28.41 @SH

239.7 2SO 2 2 2279.91 @SO 2 2 2276.48

2SO 3 H 21182.8 2OSO 2 2 2379.53 2OSO 3 2 2583.17

Nitrogen 2NH 2 61.46 13.10 7.49 237.66

1.3 4.6 24.2 5 5N5 5(keto)

2N(NH 2 )2

@N(NH 2 )2

77.4 15.1 23.8 212.6 @CONH 2 2141.13 2CONH2

2NO 2 11.51 45.52 4.77 4.64 214.56 @NO 2 17.99 45.6 4.6 4.6 214.6

2ONO 20.67 54.81 10.33 6.32 216.07 2ONO 2 236.69

a The symbol @ preceding a functional group means that the values refer to the substitution of that group on an aromatic ring.

519 TABLE 22.7. Corrections for Type Number and Multiple Substitutions of

22.1 EMPIRICAL METHODS

Functional Groups Heat Capacity Constants

Functional Group a D(DH m )

Da Db Dc Oxygen

DS m

¼O(aldo) 222.68 18.83 23.60 6.74 24.81 ¼O(keto)

2OO2 210.46 2COOH

25.06 36.0 0.0 0.0 0.0 2COO 2 CO2

221.3 HCOO2

33.43 22.5 2.1 25.0 3.3 2CO 3 2 21.21

Fluorine 2F 26.15 4.14 1.59 20.54 1.59 2F,2F

215.36 23.81 22.01 20.75 21.76 2F,2Cl

11.00 20.67 7.20 13.97 18.33 2F,2Br

17.53 6.82 4.14 216.78 4.39 2F,2I

17.24 20.38 7.03 26.49 4.23 2COF

Chlorine 2Cl

22.59 5.19 3.77 212.55 8.03 2Cl,2Cl

17.78 26.23 22.59 6.49 23.77 2CL,2Br

21.51 6.19 7.24 229.08 12.64 2Cl,2I

20.5 5.2 7.03 227.57 18.91 2COCl

Bromine 2Br

27.24 25.23 1.63 226.57 9.67 2Br,2Br

17.61 9.92 4.69 235.94 19.66 2Br,2I

20.5 7.95 21.59 232.38 16.07 2COBr

Iodine 2I 24.31 3.93 2.76 210.13

7.28 2I,2I

23.39 23.05 0.50 0.75 21.51 2COI

(Continued)

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.7. Continued Heat Capacity Constants

Functional Group a D(DH m )

Da Db Dc Sulfur

DS m

2SH 21.13 1.59 1.46 21.21 21.59 2S2

@SO 2 2 25.86 2SO 3 H 211.7 2OSO 3 2 210.75

Nitrogen 2NH 2 25.44 21.42 0.67 1.97 22.55 2NH2

%2N ¼ 23.8 2N ¼ N2

2NHNH 2 25.4 21.3

2N(NH 2 )2

@N(NH 2 )2 25.4 2NHNH2

2CN 212.9 2.3 4.3 220.4 18.7 2NC

213.0 2.5 4.2 220.5 18.8 ¼NOH

2CONH 2 0.13 36.0 0.0 0.0 0.0 %2CONH 2 2 25.0 %2NHCO2

29.6 @NHCO2

25.0 2NO 2 29.46 0.0 0.0 0.0 0.0 2ONO

a The symbol @ preceding a functional group means that the values refer to the substitution of that group on an aromatic ring. A % symbol preceding a functional group indicates that the substituent can be added in

either of two orientations and that the correct choice needs to be made.

521 Example 1. Estimate the entropy S8 m,298.1K of trans-2-pentene(g).

22.1 EMPIRICAL METHODS

Contribution

H 186.3 j

Base group, H22C22H j

H Primary CH 3 substitution ! CH 3 2 2CH 3 43.3

Secondary CH 3 substitutions ! CH 3 2 2CH 2 2 2CH 2 2 2CH 2 2 2CH 3

Type numbers

Carbon A

Carbon B

DS8 m298.15K

1 2 38.9 Introduction of double bond at 2-position:

211.4 Summation of group contributions

2 2 trans

339.7 J K 21 mol 21 Experimental value [5]

342.29 J K 21 mol 21

Example 2. Estimate the entropy S8 m298.1K of acetaldehyde(g).

Contribution

H 186.3

Base group, H22C22H

Primary CH 3 substitution ! CH 3 2 2CH 3 43.3

CH 3 j Secondary CH 3 substitutions22CH22CH 3 j CH 3

ESTIMATION OF THERMODYNAMIC QUANTITIES

Type numbers

Carbon A

Carbon B

DS8 m,298.15K

2 1 21.5 Substitution of 55O replacing 2 22CH 3 groups

CH 3 2 2C5 5O Type correction 1

Summation of group contributions 244.9 J K 21 mol 21 Experimental value [5]

250.2 J K 21 mol 21

These examples illustrate the procedure used in the Andersen – Beyer – Watson – Yoneda method. The first example shows moderate agreement; the second shows poor agreement. Generally, it is preferable to consider the group substitutions in the same order as has been used in the presentation of the tables. The best agreement with experimental values, when they are known, has been obtained by using the minimum number of substitutions necessary to construct the molecule. For cases in which several alternative routes with the minimum number of substitutions are possible, the average of the different results should be used.

Other Methods Although the tables presented by Parks and Huffman [1] are based on older data, they

are often more convenient to use, because they are simpler and because they have been worked out for the liquid and the solid states as well as for the gaseous phase. A complete survey and analysis of methods of estimating thermodynamic properties is available in Janz’s monograph [5], and in the work by Reid et al. [6]. Thermodynamicists should have a general acquaintance with more than one method of estimating entropies so they can choose the best method for a particular application.

Poling et al. [6] also describe methods for estimation of additional properties, such as critical properties, P – V – T properties, and phase equilibria.

Accuracy of the Approximate Methods Free energy changes and equilibrium constants calculated from the enthalpy and

entropy values estimated by the group-contribution method generally are reliable only to the order of magnitude. For example, Andersen et al. [1] have found that their estimated enthalpies and entropies usually differ from experimental values [7]

523 by less than 16.7 kJ mol 21 and 8.4 J mol 21 K 21 , respectively. If errors of this mag-

EXERCISES

nitude occurred cumulatively, the free energy change would be incorrect by approxi- mately 19.2 kJ mol 21 near 258C. Such an error in the free energy corresponds to an uncertainty of several powers of 10 in an equilibrium constant. With few exceptions, such an error is an upper limit. Nevertheless, it must be emphasized that approximate methods of calculating these thermodynamic properties are reliable for estimating the feasibility of a projected reaction, but they are not adequate for calculating equili- brium compositions to better than the order of magnitude.

Equilibrium in Complex Systems The computation of chemical equilibria in complex systems has been developed

extensively [8]. The computation requires a database of the Gibbs functions of for- mation of all substances present in the system. The equilibrium is determined by minimizing the total Gibbs function for the system, subject to the material balance constraints for all elements in the system, using the thermodynamic database for the compounds present. With the development of the Internet, several web pages provide fee-based software for carrying out the minimization for complex systems [9]. Geological systems are treated on the Java MELTS web page [10].

EXERCISES

22.1. a. Estimate S8 m,298.15K for n-heptane (gas) by the group-contribution method of Andersen, Beyer, Watson, and Yoneda. Compare with the result obtain- able from the information in Exercise 12.15.

b. Estimate S8 m,298.15K for liquid n-heptane from the rules of Parks and Huffman. Compare with the result obtained in (a).

22.2. a. Using the group-contribution method of Andersen, Beyer, Watson, and Yoneda, estimate S8 m,298.15K for 1,2-dibromoethane(g).

b. Calculate the entropy change when gaseous 1,2-dibromoethane is expanded from 1 atm to its vapor pressure in equilibrium with the liquid phase at 298.15 K. Neglect any deviations of the gas from ideal behavior. Appropriate data for vapor pressures have been assembled conveniently by Boublı´k et al. [11], or in the NIST Chemistry Webbook [12].

c. Using the data given by Boublı´k et al. [11], or in the NIST Chemistry Webbook [12], calculate the enthalpy of vaporization of 1,2-dibro- moethane at 298.15 K.

d. Calculate the entropy S8 m,298.15K for liquid 1,2-dibromoethane.

e. Compare the estimate obtained in (d) with that obtainable from the rules of Parks and Huffman [1].

f. Compare the estimates of (d) and (e) with the value found by Pitzer [13].

ESTIMATION OF THERMODYNAMIC QUANTITIES

TABLE 22.8. Thermodynamic Data for Butadiene and Cyanogen Substance

D f H8 m,298.15K /J mol 21 S8 m,298.15K /J mol 21 K 21 Butadiene(g)

277.90 Cyanogen(g)

22.3. Precision measurements of enthalpies of formation and entropies are probably accurate to perhaps 250 J mol 21 and 0.8 J mol 21 K 21 , respectively. Show that either one of these uncertainties corresponds to a change of 10% in an equilibrium constant at 258C.

22.4. It has been suggested [14] that 1,4-dicyano-2-butene might be prepared in the vapor phase from the reaction of cyanogen with butadiene.

a. Estimate D f H m 8 and S m 8 for dicyanobutene at 258C by the group-contri- bution method.

b. With the aid of Table 22.8, calculate the equilibrium constant for the suggested reaction.

22.5. a. Estimate D f H m 8 and S m 8 for benzonitrile, C 6 H 5 CN(g), at 7508C by the

group-contribution method using benzene as the base compound.

b. Combining the result of (a) with published tables, estimate DG m 8 at 7508C for the reaction

C 6 H 6 (g) þ (CN) 2 (g) ¼C 6 H 5 CN(g) þ HCN(g)

An estimate of 277.0 kJ mol 21 has been reported by Janz [15].

REFERENCES 1. G. S. Parks and H. M. Huffman, The Free Energies of Some Organic Compounds,

Reinhold, New York, 1932; J. W. Andersen, G. H. Beyer, and K. M. Watson, Natl. Petrol. News, Tech. Sec . 36, R476 (1944); D. W. Van Krevelen and H. A. G. Chermin, Chem. Eng. Sci. 1 , 66 (1951); S. W. Benson, Thermochemical Kinetics, 2nd ed., John Wiley, New York, 1967; S. W. Benson and N. Cohen, in Computational Thermochemistry , K. K. Ikura and D. J. Frurip eds., American Chemical Society, Washington, DC, 1998, Chapter 2.

2. N. Cohen and S. W. Benson, in S. Patai and Z. Rappoport, eds. The Chemistry of Alkanes and Cycloalkanes , John Wiley & Sons, New York, 1992.

3. D. W. Rogers, Computational Chemistry for the PC, VCH Publishers, New York, 1994, Chapter 6.

4. Y. Yoneda, Bull. Chem. Soc. Japan 52, 1297 (1979). Reprinted with permission of the Chemical Society of Japan.

5. G. J. Janz, Estimation of Thermodynamic Properties of Organic Compounds, rev. ed., Academic Press, New York, 1967.

525 6. B. E. Poling, J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids,

REFERENCES

McGraw-Hill, New York, 2001. 7. D. D. Wagman et al., The NBS tables of chemical thermodynamic properties, J. Phys.

Chem. Ref. Data 11 , Supplement 2 (1982). 8. F. Van Zeggeren and S. H. Storey, The Computation of Chemical Equilibria, Cambridge

University Press, Cambridge, UK, 1970; W. R. Smith and R. W. Missen, Chemical Reaction Equilibrium Analysis: Theory and Algorithms , Wiley-Interscience, New York, 1982; W. J. Lyman, W. F. Reehl, and D. H. Rosenblatt, eds., Handbook of Chemical Property Estimation Methods , McGraw-Hill, New York, 1982; C. M. Wal and S. G. Hutchison, J. Chem. Educ. 66, 546 (1989); F. G. Helferich, Chemical Engineering Education , 1989.

9. Thermo Calc, http: //www.thermocalc.com/Products/Databases/TC_databas.html; Chemsage, http: //gttserv.lth.rwth-aachen.de/cg/Software/IndexFrame.htm; FACT, http:

10. http: //melts.ofm-research.org/applet.html. 11. T. Boublı´k, V. Fried, and E. Ha´la, The Vapor Pressures of Pure Substances, Elsevier,

Amsterdam, The Netherlands, 1973. 12. NIST Chemistry Web Book. http: //webbook.nist.gov/chemistry/. 13. K. S. Pitzer, J. Am. Chem. Soc. 62, 331 (1940). 14. G. J. Janz, Can. J. Res. 25B, 331 (1947). 15. G. J. Janz, J. Am. Chem. Soc. 74, 4529 (1952).