Hydrogen Atom Bond Increments for Calculation of Thermodynamic

J. Phys. Chem. 1995, 99, 14514-14527

14514

Hydrogen Atom Bond Increments for Calculation of Thermodynamic Properties of Hydrocarbon Radical Species Tsan H. Lay, Joseph W. Bozzelli,” Anthony M. Dean: and Edward R. Rittefl Department of Chemical Engineering, Chemistry and Environmental Science, New Jersey Institute of Technology, Newark, New Jersey 07102 Received: May 11, 1995; In Final Form: July 14, 1995@

Hydrogen atom bond increments (HBI) are defined and a data base is derived for accurately estimating AHfo298, s0298, and C,(Q (300 5 T/K I 1500) on generic classes of hydrocarbon (HC) radical species relevant to combustion and atmospheric chemistry, using these thermodynamic property increments. The HBI group technique is based on known thermodynamic properties of the parent molecule and calculated changes that occur upon formation of a radical via loss of a H atom. The HBI approach incorporates (i) evaluated literature bond energies, (ii) calculated entropy and heat capacity increments resulting from loss and/or change in vibrational frequencies including frequencies corresponding to inversion of the radical center, (iii) increments from changes in barriers to internal rotation, and (iv) spin degeneracy. Twenty five HBI groups corresponding to alkyl (primary, secondary, and tertiary), vinyl, allenic, allylic, benzyl, acetylenic, and other conjugated hydrocarbon radicals are defined, and their group values are calculated. The HBI groups, when coupled with thermodynamic properties of the appropriate “parent” molecule, yield accurate thermodynamic properties for the respective radicals.

Introduction Detailed kinetic reaction models using mechanisms based upon fundamental thermodynamic and kinetic principles are presently used and being developed by researchers attempting to optimize or more fully understand a number of systems comprised of many elementary chemical reactions. These include pyrolysis,’ combustion,2 igniti~n,~ atmospheric smog formation and transp0rt$3~ municipal and hazardous waste incinerati~n,~.~ chemical vapor deposition8 and semiconductor e t ~ h i n g ,rocket ~ propulsion,I0 and other related fields. One important requirement for reliable simulation of these systems is accurate thermodynamic property data for the molecular and radical species in the chemical mechanism. This data allow determination of the thermodynamic feasibility of reaction paths by allowing calculation of equilibrium constants (Keq). Also reverse rate constants ($) can be computed from the forward rate constant (kf) and Keq. The data also serve a vital role in estimating rate constants for endothermic reactions. Benson’s group additivity (GA) estimation technique is an accurate method for the estimation of ideal gas phase heat capacities, heats of formation, and entropies of stable molecules. This technique is discussed in Benson’s Thermochemical Kinetics” and other reference source^.^^,^^ The method assumes that the properties for a chemical substance are the sum of the contributions from each group or polyvalent atom (central atom) in that molecule. It is referred to as a second-order estimation technique since next-nearest-neighbor corrections, and to some extent, chemical structure, are accounted for. Estimations based upon chemical bond additivity alone are known as first-order estimation techniques, while those based upon atomic contributions alone are referred to as zero-order techniques.I2 The second-order estimates are obviously more accurate than lower order techniques because they include more parameters, but also require a larger data base. Several other estimation techniques

* E-mail:

[email protected].

Exxon Research and Engineering Co., Annandale, NJ 08801. 4 Present address: Department of Chemical Engineering, Villanova University, Villanova PA 19085-1681. Abstract published in Aduunce ACS Abstracts, September 1, 1995. +

@

0022-365419512099- 14514$09.0010

based upon group contribution principles are presented in Reid, F’rausnitz, and SherwoodI2 and Pedley, Naylor, and KirbyI4 (enthalpies only). Benson’s method is the most widely accepted one because of its ease of use and relative accuracy when compared with other techniques and has been incorporated in several computer program^.'^*'^ Group contributions (group values for GA) were derived by Benson and co-workers” and various other researcher^'^.'^^'^ by dividing similar molecules with known thermodynamic properties into their constituent groups and then performing multivariable linear regressions to find group contributions which gave the best fit to available experimental property data. The enthalpy group values for this GA approach were comprehensively reviewed and updated by Cohen and BensonI8in 1993 to make the prediction results more consistent with current experimental data. Benson and co-workers also derived several corresponding group values for estimating thermodynamic properties (hHfO298, s0298, and C,(TJ) for free radicals in the same manner as for stable Two classes of radical groups are used in Benson’s scheme: (i) groups for the radical-centered atoms and (ii) groups for radical-adjacent atoms. Here a radical will contain at least one of each type of radical group plus the normal groups for atoms not adjacent to radical center. Values of several of these radical groups for alkyl radicals were recently revised by Cohen.I3 In this work, we present details of an alternative approach, i.e. a single group, to estimate the thermodynamic properties (AiYf’298, s0298, and Cp(TJ,300 5 T/K 5 1500) for a series of hydrocarbon free radicals. Our method utilizes the thermodynamic properties of the parent molecules incorporated with a H atom bond increment (HBI) for each respective thermodynamic property of the parent which reflects the change due to loss of a H atom. We illustrate how HBI groups are derived on the basis of fundamental principles of statistical mechanics and thermochemistry. An advantage of this approach is use of a single HBI group to predict accurate properties of radical species. 0 1995 American Chemical Society

Thermodynamic Properties of Hydrocarbon Radical Species

J. Phys. Chem., Vol. 99, No. 39, 1995 14515

Consider the following homolytic reaction:

Vibrational Analysis CH3CH3

C.H2CH3

VS

The radical enthalpy of formation can be written as

One can calculate Mf0298(R')if one knows Mf0298(RH)and the bond strength for the R---H bond being broken to form the radical and H atom. Values of bond energies, D(R---H), in the HBI data base for the corresponding radicals are adopted from evaluation of the literature. To some extent, the molecular structure of a radical (R') is similar to that of the corresponding stable molecule (RH). The unpaired electron on the radical-centered atom is replaced by a bond to a H atom in the stable molecule, while most of the atom sequence and chemical bonds basically remain the same in the two species. If the differences in molecular structure and properties for R' and RH are properly taken into account, one can calculate SO298 and C,(T) values for R'from properties of the corresponding RH parent plus increment values for AC,(7) and AS0298 that account for these changes:

Frequencies (cm-1)

vibration

Frequencies

Mode

(cm-1)

3000

(x6)

C-H stretch

3000

(x5)

1400

(x6)

H-C-H bend (CH3 deform)

1400

(x4)

1150

(x2)

CH2twistIwag

1150

(x2)

993 822

(xl) (x2)

C-Cstretch H-C-C bend (C-CH3 rock) C-C.H2 inversion

975

(Xl) (xl) (Xl)

784 540

(3) Compare C.H2CH3 to CH3CH3 : lose one C-H stretching, two H-C-Hbending, one H-C-C bending, gain one C.H2 inversion, also, the bamer of internal rotation along C-C axis changes

where Sinto represents intrinsic entropy (excluding symmetry). These AS0298 and AC,(T) (300 IT K I1500) increments are group values for estimating the radical from the parent and are termed hydrogen atom bond increments (HBI) (AS0298(HBI) and AC,(T)(HBI)). They are used to calculate SO298 and C,(T) for a free radical formed via the elimination of H atom from its parent molecule. The values we report for AS0298 and AC,(T) are obtained by applying the principles of the rigid-rotorharmonic-oscillator (RRHO)I9model to account for the differences of molecular structures between R'and the corresponding RH (parent). Increment changes in the potential barriers of internal rotations about C-C bonds next to the radical center are also incorporated. Some increment values are obtained by difference using known properties of the parent molecule and the corresponding radical. The classification of HBI terms is based on generic types of R---H bonds. This work considers 25 types of hydrocarbon radicals. The development of HBI groups incorporates the following: (1) evaluated literature data corresponding to the bond energies, D(R---H) of the specific R---H bond; e.g. primary, secondary, tertiary, etc; (2) entropy and heat capacity increments accounting for loss and for the differences in vibrational frequencies of a parent molecule losing a H atom to form the radical; (3) gain of inversion frequencies at radical centers of carbon atoms; (4) entropy and heat capacity corrections accounting for the differences of rotational barriers of internal rotors in parent and radical; (5) entropy corrections for electron spin degeneracy. Entropy corrections accounting for changes in symmetry between the parent molecule and radical are not included in the HBI group values. These corrections need to be separately considered for each radical and parent molecule.

Methodology and Calculations Heats of Formation. The calculation of Mf0298(R') for a specific radical species uses literature values or group additivity

Conclusion :C.H2CH3

-

-

= C2H6 v/C-H/ 2v/H-C-H/

- v/H-C-C/ + v/inv-CHP/ - ir/CH3-CH3/ + ir/CH2-CH3/ Figure 1. Example of HBI approach.

(GA) for enthalpy of the parent molecule (Mfo298(RH))and a bond energy, D(R---H), for the specific H atom removed from the parent molecule to form the desired radical; see eq 2. Table 1 lists the bond energies and literature references. Change in Vibrational Frequencies and Rotational Barriers. SO298 and heat capacities C,(T) of free radicals are calculated by applying HBI values (AS0298 and AC,(T)) in eqs 3 and 4. Changes in both entropy and heat capacity values due to loss of a hydrogen atom are listed. We feel the specific vibrations that are lost or modified as a result of loss of a H atom are capable of being estimated more accurately than vibrations corresponding to loss of any other atom or group. The loss of one H atom from the parent molecule causes the daughter radical to lose 3 internal degrees of freedom (vibrations plus internal rotations) relative to the parent molecule (RH). As an example, we consider formation of C2H5 from C2H6. Figure 1 illustrates the calculation of ASo298 and AC,(T) values. Loss of a hydrogen atom results in the loss of one C-H stretch, two H-C-H bends, and one H-C-C bend. The radical center in ethyl is no longer the normal rigid tetrahedral structure, but is a more flexible, nonplanar configuration. Motion of the radical center is associated with the appearance of one low frequency20.2'in the vibration spectrum of ethyl, corresponding to inversion. This loss of four frequencies and gain of one are used to account for the differences in vibrational frequencies between the ethyl radical and ethane parent. The description of changes that occur in the loss of a H atom from ethane is still not complete. The rotational barriers about the C-C bond are very different for ethyl2' and This difference in rotational barriers, 2.922 and 0.1 kcal/mol,2' respectively, accounts for a significant fraction of AS0298 and AC,(T) contributions to HBI terms.

14516 J. Phys. Chem., Vol. 99, No. 39, 1995

Lay et al.

TABLE 1: Definition of HBI Group Term Name, Model Radicals, and Corresponding Bond Energy of the C-H Bond HBI"

BDEbD(R---H) (kcal/mol)

CCJ RCCJ ISOBUTYL NEOPENTYL CCJC RCCJC RCCJCC TERTALKYL VIN VINS C=C=CJ ALLYL2 ALLYL-S ALLYL-T BENZYLP BENZYL-S BENZYL-T CWJ CSCCJ CWCJC CSCCJC2 C=CJC=C C=CCJC=C C=CC=CCJ CJC=CC=C

101.1 101.1 101.1 101.1 98.45 98.45 98.45 96.3 111.2 109.0 89.0 88.2 85.6 83.4 88.5 85.9 83.8 133.2 89.4 87.0 84.5 99.8 76.0 80.0 81.0

definition

model radicals

primary alkyl radicals primary alkyl radicals primary alkyl radicals primary alkyl radicals secondary alkyl radicals secondary alkyl radicals secondary alkyl radicals tertiary alkyl radicals vinyl radicals 1-alkylvinyl radicals allenic allyl radicals 1-alkylallyl radicals 1,l-dialkylallyl radicals benzyl radicals 1-alkylbenzyl radicals 1,l-dialkylbenzyl radicals acetylenic ProPargY 1 secondary propargyl tertiary propargyl

a ' J in HBI group ID indicates the radical center located on carbon just before J; for example, CJC equals C'H2CH3. Bibliography for the bond energies in HBI data base. The bond energies are evaluated from the following literature data: 1. CCJ, RCCJ, ISOBUTYL, NEOPENTYL, CCJC, RCCJC, RCCJCC, TERTALKYL: (A) Seakins, P. W.; Pilling, M. J.; Nitranen, J. T.; Gutman, D.; Krasnoperov, L. N. J. Phys. Chem. 1992, 96, 9847; (B) Nicovich, J. M.; van Dijk, C. A.; Kreutter, K. D.; Wine, P. H. J. Phys. Chem. 1991, 95, 9890; (C) Russell, J. J.; Seetula, J. A.; Gutman, D. J. Am. Chem. SOC. 1988,110, 3092; (D) Gutman, D. Acc. Chem. Res. 1990,23, 375; (E) CH3C'HCHj: Chen, Y.; Rauk, A,; Tschuikow-Roux, E. J. Phys. Chem. 1990, 94, 2775. CH3CH?C'HCH3: Zbid. 1990, 94, 6250. 2. VINYL: (A) Ervin, K. M.; Gronert, S.; Barlow, S. E.; Gilles, M. K.; Harrison, A. G.; Bierbaum, V. M.; Depuy, C. H.; Lineberger, C.; Ellison, B. J. Am. Chem. SOC. 1990, 112, 5750; (B) Wu, C. J.; Carter, E. A. J. Phys. Chem. 1991, 95, 8352; (C) Curtiss, L. A.; Pople, J. A. J. Chem. Phys. 1988, 88, 7405; (D) Defrees, D. J.; McIver, R. T., Jr.; Hehre, W. J. J. Am. Chem. SOC. 1980, 102, 3334. 3. VINS, ajusted for difference between primary and secondary in alkyls. 4. C=C=CJ: Miller, J. A,; Melius, K. F. Symp. (Znt.) Combust. (Proc.) 1992, 93. 5. ALLYLT: Tsang, W. J. Phys. Chem. 1992, 96, 8378. 6. ALLYL-S: Roth, R. W.; Bauer, F.; Beitat, A.; Ebbecht, T.; Wustefeld, M. Chem. Ber. 1991, 124, 1453. 7. ALLYL-T, adjusted to have same difference from allyls as tertiary from secondary. Trends reported in: McMillen, D. F.; Golden, D. M. Ann. Rev. Phys. Chem. 1982,33,493. 8. BENZYL-P: (A) Gunion, R.; Gilles. M.; Polak, M.; Lineberger,W. C. Znt. J. Mass Spectrom. Zon Phys. 1993; (B) Defrees, D. J.; McIver, R. T., Jr.; Hehre, W. J. J. Am. Chem. SOC. 1980, 102, 3334; (C) Robaugh, D. A.; Stein, S . E. J. Am. Chem. SOC.1986, 108, 3224; (D) Tsang, W.; Walker, J. J. Phys. Chem. 1990, 94, 3324. 9. BENZYL-S, BENZYL-T: (A) Robaugh, D. A.; Stein, S. E. J. Am. Chem. SOC. 1986,108,3224; (B) Hippler, H.; Troe, J. J. Phys. Chem. 1990,94,3803. 10. C=CJ: (A) Ervin, K. M.; Gronert, S.; Barlow, S . E.; Gilles, M. K.; Harrison, A. G.; Bierbaum, V. M.; Depuy, C. H.; Ineberger, C.; Ellison, B. J. Am. Chem. SOC. 1990,112,5750; (B) Curtiss, L. A.; Pople, J. A. J. Chem. Phys. 1988,88,7405; (C) . Kiefer, J. H.; Sidhu, S. S.; Kern, R. D.; Xie, K.; Chen, H.; Harding, L. B. Combust. Sci. Technol. 1992,82, 101; (D) Wagner,A.; Keifer, J. Symp. (Znt.) Combust. (Proc.) 1992, 1107. 11. CGCCJ: (A) Tsang, W. J. Phys. Chem. Ret Data 1988,17, 109; (B) Orlov, Y. D.; Lebedev. Y. A. Russ. J. Phys. Chem. 1991,65, 153. 12. C=CCJC, CWCJC2: Orlov, Y. D.; Lebedev. Y. A. Russ. J. Phys. Chem. 1991, 65, 153. 13. C=CCJ-C: (A) Melius, K. F.; Miller, J. A. Symp. (Znt.) Combust. (Proc.) 1990; (B) Kuhnel, V. W.; Gey, E.; Ondruschka, B. Z. Phys. Chem. Liepzig 1987, 268, 23. 14. C=CCJC=C and C=CC=CCJ: (A) McMillen, D. F.; Golden, D. M. Ann. Rev. Phys. Chem. 1982,33,493; (B) Holmes, J. J. Phys. Org. Chem. 1992. 15. CJC=CC=C: Green, I. G.; Walton. J. Chem. SOC.,Perkin Trans 2 1984, 1253. BDE = bond dissociation energy of the corresponding C-H bond for the parent R' f H. molecule from which the radical is generated through the reaction RH 1'

1,

-

Vibrational Frequencies. The assignment of fundamental frequencies is adopted in part from the analysis of molecular vibrational frequencies by ShimanouchiZ3 and other studies on hydrocarbon radical structures and spectra. These studies include ethyl, n-propyl,20 i s o p r ~ p y l and , ~ ~ rert-butylZ5by Pac~ (22+) ansky and co-workers, sec-n-butyl by Chen et U Z . , ~ C2H by Kiefer e t al.,27and numerous others; see references in Table 2. The information of vibrational frequencies needed for some parenthadical couples in this study is unfortunately not always available. W e therefore perform MNDO/PM328 molecular orbital (MO) calculations using the MOPAC 6.029 program to obtain vibrational frequencies via normal mode analysis of nuclear coordinates for model radicals for which the information of vibrational frequencies is sparse. Example species include CHz=C'H, CHpC'CH3, and CH2=C=C'H. The vibrational frequencies of several free radicals and the corresponding stable molecules are listed in Table 2. These data help determine the differences and similarities for vibrations between a radical species and the parent molecule.

A data base of fundamental frequencies is assembled for relevant vibrational modes in Table 3. The hannonic vibrational frequencies for bond-stretching and bond-bending motions assigned and listed in Table 3 originate from Benson (Table A.13. in ref 11, page 300). Most assignments adopt an average value of characteristic frequencies identified in previous studies,' representing the standard stretching or bending motions. W e use the animation analysis in the Hyperchem software30 for assistance in the assignment of the approximate mode type for the PM3 vibrational frequencies. This is helpful for frequencies of hydrocarbon radicals not well characterized in the literature. Vibrations with low wavenumber, e.g. 5 1000 cm-I, have a larger contribution to the values of hS0298 and AC,(r) (300 I T/K I 1500) than high-frequency vibrations. Cohen also points out that the low-frequency out-of-plane methylene bend (541 cm-l) in the ethyl radical (C2H5) and (530 cm-I) in n-propyl (C'H2CH2CH3) is important in determining the values of AS0298 and AC,(r) for these two radi~a1s.I~W e assign this low-frequency vibration, 550 cm-I, to an inversion

Thermodynamic Properties of Hydrocarbon Radical Species

J. Phys. Chem., Vol. 99, No. 39, 1995 14517

TABLE 2: Fundamental Vibrational Frequencies (cm-') for Model Stable (Parent) Molecules and Model Free Radicals molecule

frequencies

CzHz

source

612(x2), 730(x2), 1974,3289,3374

ref 23

radical

frequencies

(2Z+) 371.6(~2), 1840,3328

C2H

source ref 27

(211)560( x 2), 1850,3460 C2H4 C2H6 CH2=C=CH2 C3H6 (1 intemal rotor)

826,943,949, 1023, 1236, 1342, 1444, 1623,2989,3103,3106,3206 822(x2), 995, 119O(x2), 1379, 1388, 1468( x2), 1469( x2), 2896,2954, 2969( x2), 2985( x2) 328( x2), 633( x2), 931,1053( x2), 1382, 1452(x2), 2142,2918, 3008(x2), 3334 428,575,912,914,945,990,1045, 1174, 1298, 1378, 1419, 1443, 1459, 1653,2932,2953,2973, 2991,3017,3091

ref 23

CH7=C'H

ref23

C2Hs

ref 23

CH2=C=C'H

U

CHFCHC'H~ (allyl)

CHz=C'CH3 C3HS (2 intemal rotors)

369,748,869,922,940, 1054, 1158, 1192, 1278, 1472, 1338, 1378, 1392, 1451, 1462, 1464, 1476, (2887 x 8)

ref 23

iso-C3H7 (1 internal rotor) tert-butyl

785.825.920. 1185. 1445. 1670,3115,3190,3265 541,713,948, 1123, 1206, 1370, 1427, 1445, 1462, 2842,2920,2987,3033,3112 378, 392, 708,747, 801, 946, 1151, 1342, 1996, 30N( x 3) 426, 503, 530,763, 781, 912,969, 1018, 1201, 1235, 1378, 1463, 1467, 3041,3062,3074,3167,3174 241,385,805, 899,912, 945, 1023, 1246, 1359, 1363, 1369, 1905,30OO(x5) 337,426, 836,917,932, 1029, 1089, 1159, 1342, 1408, 1411, 1460, 1467, 1470, 1475, (2900 x 7) 200, 541(x2), 733,992(~2), 1126, 1189(x2), 1252(x2), 1279, 1370(x3), 1455(x6), 2825( x2), 2931( x6)

ref 34 refs 20, 21 b

C

b ref 24

ref 25

a Dewar, M. J. S.; Ford, G . P. J. Am. Chem. SOC. 1977, 99, 1685. This work, PM3 MO calculations. Sim, F.; Salahub, D. R.; Chin, S.; Dupuis, M. J. Chem. Phys. 1991, 95, 4317.

TABLE 3: Assignment of Vibrational Frequencies symbol CT-H CD-H C-H H-C-H H -V -H,.W H-A-H H-C-C H-C-C,TR H-C-C,R H-A-C H-C#C,ub H -C#C .g H-C#C' H-C=C H-C'=C =C=C-H C=CC',OP CH,BEND

ccc,s

CCC,A cC'c,s CC'C,A

c-c=c c-C'=c C-A-C c=c C=C' c-c

C'-C C=C,TR H2C=CC,TR PH-CH2 PH-CHC PH-C2 INV-H3 INV-CH INV-H2 INV-C2

frequencya

approximate type of mode

3400 3100 3000 1400 950 500 1150 850 750 500 730 610 372 1050 785 840 770 500 370 440 540 730 420 3 10 5 40 1650 1900 1000 1350 1000 800 450 260 200 600 420 550 200

C-H stretch (next to C=C triple bond) C-H stretch (next to C-C double bond) C-H stretch (next to C C single bond) H-C-H bend (CH3 or CHI deform) H-C-H wag in C=CH2 vinyl group H-C-H wag in C=C-C'Hz allyl group H-C-C bend (CH3, CH2 twist & wag) CH3 rock in methyl group CH2 rock in chain -CHz- group C=C-C'HC wag in allyl group C-H bond bend (nu) in C W H in C W H C-H bond bend (ng) H-C#C' bend, ethynyl radical H-C=C bend, (CH2 wag & twist) H-C'=C bend, 2l/2 CC bond CHI wag in allene C=C-C' bend in allyl group C-H bend C-C-C deform, symmetric C-C-C deform, asymmetric C-C-'C deform in allyl group, symmetric C-C-'C deform in allyl group, asymmetric C-C=C deform, single bond & double bond C-C'sC deform, single bond & 2'12 bond C=C-C'C2 wag in allyl group C=C double bond stretch C=C' 2 & ]I2 bond stretch C-C single bond stretch C-'C 1 & '/z bond stretch torsion of double bond torsion of double bond torsion of Ph-C'H2 torsion of Ph-C'HC torsion of Ph-C'C:! inversion of C'H3 inversion of -C'HC inversion of -C'H2 inversion of -C'C2

a Frequency in cm-I; reference sources for those assignments: ref 11, refs 23-27, and MNDOIPM3 semiempirical molecular orbital calculation; see text. # represents triple bond.

mode. We also assign inversion modes for C-C'H-C (420 cm-l) and C'-(C)3 (200 cm-l) on the basis of the spectral data

determined by Pacansky and co-workers for isopropyl radical24 and for tert-butyl radical.25

Lay et al.

14518 J. Phys. Chem., Vol. 99, No. 39, 1995

TABLE 4: Moments of Inertia (Ir): Torsion Barriers (V): and Foldness Numbers (n)" to Hindered Rotation about Single Bonds rotor I, V n source commente molecule CH3-CH3 1.6 2.9 3 ref 11 large molecule 3.0 3.3 3 ref 13 RCH2-CH3 CH3-C2H5 RCH2-C2H5 (CH3)2CH-CH3 RCH(CH3)-CH3 R2CH-CH3 CH3-C(CH3)3 CH3 -VIN CH3-VINR CC-VINR CH3-PH C2H5-PH (CH3)2CH-PH CH3-BENZYL

2.8 20.8 3.0 3.0 3.0 3.0 2.8 3.0 20.8 3.0 20.2 40.5 3.0

3.3 3.3 3.8 3.8 3.8 4.7 2.1 2.1 2.1 0 0 0 1.65

3 3 3 3 3 3 3 3 3 6 2 2 3

ref 11 ref 11 ref 11

1.2 1.8 1.7 1.8 1.8 2.9 3.0 13.1

0.1 0.1 0.1 0.17 0.16 0.7 1.5 2.16

6 2 6 6 6 3 3 3

ref 13 assigned ref 13 ref 13 ref 13

radical CH3-CH2 RCH2-CH2 CH2-C2H5 CH2-CH(CH3)2 CH2-C(CH3)3 CH3-CHCH3 CH3-C'(CH3)2 CH3C0H-C2H5

Entropy corrections accounting for changes in symmetry are not included in the HBI group values. They need to be considered separately for each radical and parent molecule. Symmetry is not included in the values of AS0298(HBI). The calculation of SO298 for radicals is

=

+ AS"29,(HB1)

- ln(oradd/upparent) (5)

assignedd assigned ref 11 ref 37 assigned assigned ref 11 assigned assigned

large molecule large molecule

ref 11

BENZYL: -CHz(CsHs)

VIN: -CH=CH2

large molecule large molecule PH: phenyl, -c&

ref 13 ref 13 ref 13

a Unit: amu A*. Unit: kcal/mol. Number of maximum points in potential energy surface of internal rotation from torsion angle 0" to 360". Barriers are assigned in this work and are approximate. e Large molecules indicate molecules with more than four carbon atoms.

Hindered Internal Rotations. Barriers to hindered internal rotation(s) adjacent to a radical center are another important contribution to AS0298 and AC,(T). The rotational barrier about the C-C bond in the ethyl radical, C'H2-CH3, has a upper limit of 0.1 kcal/mol,2' while that in ethane is 2.9 kcal/mo1.22 The barriers of hindered internal rotations for stable molecules and free radicals considered in this work are listed in Table 4. The majority of data on rotational barriers in Table 4 are results of experimental determinations or ab initio MO calculations. When literature data are not available, the barriers are assigned by interpolation of the values from similar, but previously studied, internal rotor systems. The differences in the entropy of internal rotation between parent (molecule) and daughter (radical) resulting from changes in rotational barriers and moments of inertia are usually the major components of the entropy increment in HBI groups. The method and tables of Pitzer and G ~ i n n ~are ' - used ~ ~ to calculate the contribution of hindered internal rotations to the thermodynamic functions SO298 and Cp(T);see the Appendix for details. It is also suggested that this method be utilized to replace the determination of SO298 and C,(r) values using torsional frequencies calculated by various ab initio and semiempirical calculations, where the harmonic vibrational motions cannot correctly represent the hindered internal rotations, and incorrectly add a value of R/2 to the value of CJinfinity) above that of an ideal RRHO molecule. Spin Degeneracy and Symmetry. All free radicals estimated in this approach are considered to have only one unpaired electron (doublet) and assumed spin degeneracy equal to 2. The occupied orbitals of stable molecules in their ground state generally have paired electrons. The correction factor for spin degeneracy in the entropy increment is, therefore, R In 2 (R is the ideal gas constant), which has been added into all group values of AS"298.

where q a d l c a l and upwentare the symmetry numbers for radical and the corresponding parent molecule, respectively. No further correction is required for the calculation of C,(T) using HBI term values: C,(T)(R') =)"C ,

+ AC,(T)(HBI)

(6)

Calculation Details in Determining the Values of AS"298' (HBI) and AC,(T)(HBI). The values of AS"298 and AC,(T) that result from changes in the vibrational frequencies and intemal rotation barriers between a radical and its parent are presented in Table 5 for each HBI group. Contributions to AS"298 and ACJT) are listed for each vibrational frequency and intemal rotor, in addition to the sum of the respective contributions. Assignment of frequencies lost and gained and adjustment in potential barriers of internal rotation for the HBI As0298 and AC,(T) are discussed below. Primary Alkyl Radicals (CCJ, RCCJ, ISOBUTYL and NEOPENTYL). Four groups are derived to represent all primary alkyl radicals, C(CH3)xH3-xC'H2. The differences between the four groups are the number of substituted alkyl groups on the a-carbon (carbon next to the radical center): CCJ for none, RCCJ for one, ISOBUTYL for two, and NEOPENTYL for three alkyl substituents. The adjustments in frequencies accounting for loss of a H atom are based on the comparison of vibrational frequencies for ethane and ethyl, as illustrated in Figure 1 and discussed above. Frequencies lost and gained are assumed identical for all four. Rotational barriers account for the differences in the three HBI terms. The rotational barriers of the methyl group (C-CH3) vary in this series: ethane (2.9 kcaymol), propane (3.3 kcallmol), isobutane (3.8 kcallmol), and neopentane (4.7 kcaYmo1); also see Table 4. The barriers for the methylene group (C-C'H2) are similar for ethyl (