J . Phys. Chem. 1994,98, 404-407
404
Estimation of Heterolytic Bond Dissociation Energies of Alkanes Michael L. Mavrovouniotis' and Leonidas Constanthou Chemical Engineering Department, Technological Institute, 2145 Sheridan Road, Northwestern Uniuersity, Evanston, Illinois 60208-31 20 Received: October 1 I , 1993; In Final Form: November 18, 1993'
Gas-phase enthalpies of ionic intermediate compounds can be estimated using a new technique which is based on the analysis of conjugate forms of the compounds. For ionic intermediates, conjugation determines the extent of charge dispersion which is directly related to the thermodynamic stability of an ion. In a simple application, the proposed method estimates the heterolytic bond dissociation energy of alkanes, which can be used to determine the enthalpy of formation of alkyl cations.
Introduction
HBDE of Alkanes
Gas-phase enthalpies of ionic hydrocarbon intermediates are important for the analysis of many chemical reaction systems. Since very few experimental data are actually reported in the literature, one would like to be able to estimate enthalpies of formation from information on the molecular structures of the compounds. A number of simple correlationshave been developed for this purpose for various classes of ions (e.g., Holmes et al., 1981; Holmes and Lossing, 1982; Lossing and Holmes, 1984; Bowenand Williams, 1977). Group-contribution methods (Reid et al., 1987) with broad validity permit the estimation of the enthalpies of formation of neutral hydrocarbons, but groupcontribution methods are difficult to apply to ions, because the dispersion of the charge (which is formally placed on one atom) to several atoms is a major factor affecting the stability and properties of an ion. A new method has been under development for the estimation of physical and thermodynamic properties of compounds from their molecular structure (Mavrovouniotis, 1990; Constantinou et al., 1993). The basis of this method is conjugation, which describesthe delocalizationof electrons that changes the character of individual bonds and affects the stability and properties of a compound. In conjugation, a compound is viewed as a hybrid of a number of forms. Each conjugate form is an idealized arrangement of atoms with localized integer-order bonds and charges. The dominant conjugate is represented by the standard structural formula of a compound, but there are also other forms, which can be called recessive conjugates (Mavrovouniotis,1990). The basic premise of the approach is that the properties of the hybrid compound can be estimated as a suitable combination of contributions from all conjugate forms of the compound. The method has been successfully employed for the estimation of properties of neutral compounds. In this Letter, we extend the technique for an approximate estimation of the enthalpies of alkyl ions. More precisely, we estimate the heterolytic bond dissociation energy (HBDE)of alkanes, the amount of energy consumed when a carbon-hydrogen bond dissociatesand the bond's electrons move to the hydrogen:
A recessive conjugate refers to a specific alternative form of a particular compound. A whole class of conjugates can be described by a conjugation mode or conjugation operator which is a recipe that yields a recessive conjugate when applied to the dominant form of any compound from a given class. Each operator affects only a chain of atoms and bonds and can be described fully by its effect on that chain (Table 1). Whenever the chain occurs in a specific compound, the operator can be applied to the compound to yield a recessive conjugate. For example, an operator which is described as H-C-C+ H+-.-C=C can be applied to the dominant conjugate form of the isopropyl ion:
-.
+
RH R+ H(1) where R+ is an alkyl cation, H- a hydrogen anion, and RH an alkane. The HBDE is obviously related to the enthalpies of formation, through the expression
+
HBDE = E(R+) E(H-) - E(RH) (2) where E(R+), E(H-), and E(RH) are the enthalpies of formation of the respective species. Abstract published in Aduance ACS Abstracts, January 1, 1994.
0022-3654/94/2098-0404~04.50/0
-
CH~-+CH-CH,
-
H+...cH,=cH-cH,
This ion has a total of six recessive conjugatesof this type, because there are six different (although overlapping) chains HCC+ that match the operator. For the enthalpies of formation of alkyl cations, conjugation operators that affect either two, four, or six bonds are used (Table 1). Each of the operators of Table 1 corresponds to a different type of recessive conjugate. The central idea in this Letter is that the occurrences of conjugationoperators determine the extent of charge dispersionand consequent stabilization-of the ion. To derive a suitable expression for the HBDE, based on the definition (2), we must express the enthalpies of formation of alkanes E(RH) and alkyl cations E(R+)through additive contributions of conjugate forms. The equation for the estimation of the enthalpy of formation of a neutral alkane can be written as (Mavrovouniotis, 1990) E(RH) = E, - A i n ( l +
EN,exp[(E, - E,)/AIJ
(3)
izo
where E, is the energy of each type of recessive conjugate i, Ni is the number of recessive conjugates with energy Ei (Le., the number of Occurrences of the appropriate operator), EO is the energy of the dominant conjugate, and A is a constant with molar energy units. The enthalpy of formation of an alkyl cation can be written similarly: E(R+) = K, - A l n ( l +
EM,
exp[x,//ll),
i#O
X,
= K,,- K, (4)
where xi is defined as shown, K,is the energy of the rcccssivc conjugate i, A4,is the number of Occurrences of the corresponding operator, and KO is the energy of the dominant conjugate. Equations 3 and 4 resemble formulas for isomers in equilibrium (Smith and Missen, 1982), but oneshould not interpret conjugates as isomers. As discussed by Mavrovouniotis (1990), conjugate 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 405
Letters
TABLE 1: Types of Operators of Alkyl Cations, along with Their ADDNW~OIIS result of application subchain on which index the otwator acts of the operator
-
~
II
forms represent idealized extreme structures; the real compound is not an equilibrium mixture but rather a single hybrid structure. To derive concrete models from (3) and (4),one must relate El and Kl to specific atom and bond contributions. Each of the operators in Table 1 modifies, by a constant amount, the energy of any dominant conjugate of which it acts. Thus, each operator corresponds to a fixed difference KO - Kl, Le., a fixed xl. The contributions of the simplest possible operators will be treated as adjustable parameters: x , for H-C-C'
(5)
The contributions of the longer operators are actually not extra adjustable parameters, because they can be derived from those of the two-bond operators. Consider as an example operator 3 from Table 1. Its effect on the dominant conjugate can be decomposed as follows:
H-C-C-C-C'
x1
+ H-C-C+...C=C
XI
I
63
6
0
0
0
0
2
1
3
0
0
1032.2
5
1
0
0
0
968.2
8
1
0
0
0
1103.3
2
1
2
1
3
961.1
8
1
3
0
0
958.5
7
2
0
0
0
954.8
1 7 1 2 1 0 1 0 1 0 1
Figure 1. Cations, experimental values of their HBDE, and the number of occurrences of the conjugation operators in each cation.
H+..C =C...C=C
(7) The effect of operator 3 on the energy is simply the sum of the effects of the operators in the decomposition. Therefore x j = x,
I &
1044.7
I 113.8
+ x2
(8) A similar decomposition can be carried out for each of the other operators (4,5, and 6):
et al., 1993)for the enumeration. Figure 1, which gives the data set in this application, provides the Occurrences of the operators for each of the ions used here. Expression 4 can therefore be related to adjustable parameters. We could attempt a similar treatment for expression 3, but we will show here that this is not necessary for the estimation of HBDE. To this end, we substitute (3) and (4)into the HBDE expression (2) to obtain
HBDE = E(H-)
+ K, - A ln{l+ xiuie x p [ x , / ~-~Ej ,
+
X#O
To simplify this model, we note that conjugation is more important in unstable intermediatesthan in stablecompounds. This assertion is based on comparing recessive conjugates to the dominant conjugatefor the alkyl cations and the alkanes. If a cation contains the operator
H-C-C" Their contributions would be
Thus, regardless of the length of the operator chains used, their contributions can be expressed as functions of the two-bond operators, without any new adjustable parameters. The Occurrences of the conjugation operator of type i ( M I )can be easily determined manually for simplealkyl cations. For more complex structures, a computer algorithm can be used (Prickett
u
...
H'+ C=C
(16)
then the corresponding neutral alkane, which has a hydrogen attached in place of the positive charge, will contain the operator:
C-C-C-H
u
C'+...C =C ...H'-
(17)
Note that KOC K Iand Eo C E l , by the definition of the dominant conjugate as the lowest-energy form. In (16), the energy difference between the recessive Hl+--C=C and the dominant H-C-Cl+ is x1 = K1-KO,and it is due to the creation of a double bond replacing two single bonds. In (17),however, the corresponding energy difference (El - EO)is due to a net loss of one bond (since three single bonds are replaced by one double bond) and separation of charges. Thus, the energy difference in (16) is drastically smaller than in (1 7). Thus
xI = K,-KO
