ARTICLE pubs.acs.org/JPCA
An Exact Stoichiometric Representation of the Resonance Energy Ilie Fishtik* Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, United States ABSTRACT: The energetic measure of aromaticity usually referred to as resonance energy (RE) is shown to possess a remarkable stoichiometric interpretation. Namely, the RE may be uniquely partitioned into a linear sum of contributions associated with group additivity (GA) response reactions (RERs). This new result is a powerful tool for critical analyses of various energetic approaches to RE. In particular, the single reaction scheme approach that is routinely used to evaluate RE is shown to be a particular case of the general GA method.
’ INTRODUCTION There is an increasing number of publications in which the evaluation of the resonance energy (RE) is performed on the basis of various types of isodesmic and homodesmotic reaction schemes.1 It is further considered that these “are commonly regarded as providing the most sophisticated measure” of RE.2 The rationales behind these reaction schemes, however, are often obscure and, at best, are based on qualitative arguments. Although there is an awareness and concern that none of these reaction schemes is ideal, there is a lack of understanding that a reaction scheme is not a fundamentally new method of evaluation of the RE. Rather, a single reaction scheme is a particular case of the general approach based on group additivity (GA) methods. We have already mentioned on several occasions35 that the evaluation of resonance and strain energies based on GA methods possesses a remarkable stoichiometric interpretation. Namely, the RE may be partitioned into a sum of contributions associated with a special class of reaction schemes referred to as GA response reactions (RERs). Usually, GA methods involve a large number of reference species. Respectively, the RE is a sum of contributions associated with a large number of GA RERs. According to this interpretation the single reaction scheme approach is just one reaction scheme from an enormous number of reaction schemes that emerge from the GA method. As a result, the single reaction scheme approach is always inferior to the GA method. The above stoichiometric interpretation of the RE has been formulated so far only in qualitative terms. We are now in a position to present an exact stoichiometric expression of the RE. ’ THE GROUP ADDITIVITY RESPONSE REACTIONS We consider n þ 1 chemical species B1, B2, ..., Bn, Bnþ1 and a GA model comprising l types of groups g1, g2, ..., gl. It is further assumed that the first n species B1, B2, ..., Bn are reference species while the last one, Bnþ1, is an aromatic test species whose RE is sought. Let gij (i = 1, 2, ...,n, n þ 1, j = 1,2, ..., l) be the number of r 2011 American Chemical Society
groups gj (j = 1,2, ..., l) in the species Bi (i = 1, 2, ..., n, n þ 1). We thus can define a group matrix as
Often rank g0 = q e l, which means that the groups are linearly dependent. In such cases we can arbitrarily drop the linearly dependent columns in g0 . Without loss of generality, we assume that the first q columns in g0 and the first q groups g1, g2, ..., gq are linearly independent and define a reduced group matrix 3 2 g11 g12 ::: g1q 7 6 6 g21 g22 ::: g2q 7 7 6 ::: ::: ::: 7 ð2Þ g¼6 7 6 ::: 6g gn2 ::: gnq 7 5 4 n1 gnþ1;1 gnþ1;2 ::: gnþ1;q such that rank g = q. Alternatively, the transposed original group matrix g0 can be brought to an echelon form by employing a Gaussian elimination and, then, removing the zero rows. According to the RERs formalism a GA RER involves no more than rank g þ 1 = q þ 1 species.5 Consider the following matrix 3 2 g11 g12 ::: g1q B1 7 6 6 g21 g22 ::: g2q B2 7 7 6 6 ::: ::: ::: ::: ::: 7 ð3Þ 7 6 7 6g g ::: g B n2 nq n 5 4 n1 gnþ1;1 gnþ1;2 ::: gnþ1;q Bnþ1 Received: March 21, 2011 Revised: May 10, 2011 Published: May 31, 2011 6657
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As shown by us,5 any determinant of order q þ 1 formed from this matrix is a GA RER. Thus, a complete list of GA RERs may be generated by considering all of the possible combinations of q þ 1 species from a total of n þ 1 species B1, B2, ..., Bn, Bnþ1. That is, the total number M of GA RERs is equal to ! nþ1 ðn þ 1Þ! ð4Þ M ¼ ¼ qþ1 ðq þ 1Þ!ðn qÞ! Next, the total number M of GA RERs may be formally partitioned into two parts. The first one involves q þ 1 reference species only and their number P is equal to the number of ways q þ 1 reference species may be selected from a total of n ! n n! P ¼ ¼ ð5Þ qþ1 ðq þ 1Þ!ðn q 1Þ! Let the q þ 1 reference species involved in a GA RER involving only reference species be Bi1, Bi2, ..., Biq, Biqþ1, where i1, i2, ..., iq, iqþ1 is an ordered (q þ 1)-tuple set of integers satisfying the condition 1 e i1 < i2 < ... < iq < iqþ1 e n þ 1. Such a GA RER may be denoted as F(Bi1,Bi2,...,Biq,Biqþ1) thus specifying the q þ 1 reference species that are involved in the reaction. The general equation of this GA RER is
g g Þ : ::: gg
i1 ;1 i2 ;1
FðBi1 ;Bi2 ;:::;Biq ;Biqþ1
i;q ;1 iqþ1 ;1
gi1 ;2 gi2 ;2 ::: giq ;2 giqþ1 ;2
::: ::: ::: ::: :::
gi1 ;q gi2 ;q ::: giq ;q gi1qþ1 ;q
B i1 B i2 ::: ¼0 B iq Biqþ1
ð6Þ Clearly, the GA RERs involving only the reference species are linearly dependent while their number P exceeds the number m of linearly independent GA RERs, which is equal to m = n rank g = n q. To simplify the notation, in what follows a complete set of arbitrarily ordered GA RERs involving only the reference species is denoted by Fj(j=1,2,...,P). The second set of GA RERs involves the test species Bnþ1 and, hence, only q reference species. As shown below, there are these GA RERs that represent the stoichiometric expression of the RE. Let the q reference species involved in such a GA RER be Bi1, Bi2, ..., Biq, where i1, i2, ..., iq is an ordered q-tuple set of integers satisfying the condition 1 e i1 < i2 < ... < iq e n. A GA RER involving the test species Bnþ1 is denoted as F(Bi1,Bi2,...,Biq,Bnþ1) and its general equation is
g g Þ : ::: gg
i1 ;1 i2 ;1
FðBi1 ;Bi2 ;:::;Biq ;Bnþ1
i;q ;1 nþ1;1
gi1 ;2 gi2 ;2 ::: giq ;2 gnþ1;2
::: ::: ::: ::: :::
gi1 ;q gi2 ;q ::: giq ;q gnþ1;q
Bi1 Bi2 ::: ¼0 Biq Bnþ1
ð7Þ A complete list of GA RERs involving the test species Bnþ1 may be generated by selecting all combinations of q reference species from a total of n. Hence, the total number of GA RERs involving the test species Bnþ1 is equal to ! n n! ð8Þ Q ¼ ¼ q q!ðn qÞ!
so that P þ Q = M. Again, to simplify the notation, in what follows a complete set of arbitrarily ordered GA RERs involving the test species Bnþ1 will be denoted by Fj(j=Pþ1, Pþ2,...,M). From the above definitions it is clear that when the number of reference species B1, B2, ..., Bn is equal to the rank of the group matrix, i.e., n = q, there is only one GA RER and this GA RER involves the test species Bnþ1. In other words, when n = q the number of reference species is minimal in the sense that there is no way to write a balanced GA reaction involving only reference species. This is the case which in the literature is usually known as a single reaction scheme. It is obvious that if a single reaction scheme is a GA RER, the reaction scheme is stoichiometrically unique.
’ STOICHIOMETRIC REPRESENTATION OF THE GA METHOD Consider a complete list of GA RERs involving only the reference species along with their enthalpy changes Fj :
n
∑ vjiBi ¼ 0 i¼1
ΔHj ¼
n
exp vji Δf Hi ∑ i¼1
j ¼ 1; 2; :::; P
j ¼ 1; 2; :::; P
ð9Þ
ð10Þ
where vji (i = 1, 2, ..., n; j = 1, 2, ..., P) are the stoichiometric (i = 1, 2, ..., n) are the experimental coefficients and ΔfHexp i enthalpies of formation of the reference species. Notice that in this, conventional, notation the sum over the number of species formally runs from 1 to n. It should be kept in mind, however, that a GA RERs involves no more than q þ 1 reference species and, consequently, the stoichiometric coefficients of the remaining n (q þ1) reference species are simply equal to zero. The enthalpy changes ΔHj (j = 1, 2, ..., P) of GA RERs involving only reference species and given by eq 10 are a measure of the performance of the GA method; e.g., the closer to zero the enthalpy changes, the better the performance of the GA method and the lower the error of the GA method. The exact interrelation between the enthalpy changes of GA RERs and the error of the GA method may be easily established by realizing that the enthalpy changes of the GA RERs in terms of calculated enthalpies of (i = 1, 2, ..., n) of the reference species are formation ΔfHcalc i equal to zero 0¼
n
∑ vji Δf Hicalc i¼1
j ¼ 1; 2; :::; P
ð11Þ
Subtracting eq 11 from eq 10 we have ΔHj ¼
n
n
∑ vji ðΔf Hiexp Δf HicalcÞ ¼ i∑¼ 1 vji ei i¼1
j ¼ 1; 2; :::; P
ð12Þ
where exp
ei ¼ Δf Hi
Δf Hicalc
i ¼ 1; 2; :::; n
ð13Þ
are the errors of the reference species. As shown by us, the species error may be obtained by minimizing the square of 6658
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the error vector subject to constraints given by eq 12. The result is4 P
∑ vji ΔHj j¼1
ei ¼
i ¼ 1; 2; :::; n
1 P n 2 vjs m j¼1 s¼1
∑∑
ð14Þ
It is seen that the smaller the errors of the GA method ei (i = 1, 2, ..., n) the closer to zero is the right-hand side of eq 21 and, hence, the closer RE to ΔHj (j = P þ 1, P þ 2, ..., M). Multiplying both sides of eq 21 by vj,nþ1 and summing over a complete set of GA RERs involving the test species Bnþ1 we have M
Equation 14 may be regarded as the stoichiometric representation of the GA performance.
’ THE MAIN RESULT Now, our main finding may be formulated as follows. Consider a complete set of GA RERs involving the test species Bnþ1 along with their enthalpy changes Fj :
¼
∑ vji Bi þ vj;nþ1Bnþ1 ¼ 0
i¼1
j ¼ P þ 1; P þ 2; :::; M
ð15Þ
M
∑
vj;nþ1 i ¼ 1
exp
vji Δf Hicalc þ Δf Hn þ 1
exp
REnþ1
¼ ¼
n
vji Δf Hicalc þ vj;nþ1 Δf Hncalc ∑ þ1 i¼1
ð19Þ
n
∑ vji Δf Hicalc ¼
vj;nþ1 i ¼ 1
Δf Hncalc þ1
ð20Þ
Substituting eq 20 into eq 17 gives eq 18. Assuming that the error of the GA is small, it is to be expected that the experimental enthalpy change of a GA RER, eq 16, is close to the RE. To prove this statement, we subtract eq 17 from eq 16 ΔHj vj;nþ1 REnþ1 ¼ ¼
n
exp vji ðΔf Hi Δf Hicalc Þ ∑ i¼1 n
∑ vji ei i¼1
exp
Δf Hn þ 1 Δf Hncalc þ1 1 n exp calc vji Δf Hi þ Δf Hn þ 1 vj;nþ1 i ¼ 1
∑
M
¼
ð18Þ
for any j = P þ 1, P þ 2, ..., M. From eq 19 we have 1
∑
ð24Þ
2 vj;nþ1
’ INTERPRETATION From the above development it follows that we have three identical expressions for the RE of species Bnþ1
ð17Þ
Indeed, the enthalpy change of any GA RER involving the test species Bnþ1 in terms of calculated enthalpies of formation is identically equal to zero 0¼
M
j¼P þ 1
This equation is independent of the choice of GA RER. Moreover, this RE is identical to the conventional one, i.e. REnþ1 ¼ Δf Hn þ 1 Δf Hncalc þ1
ð23Þ
M
∑ vj;nþ1ΔHj j¼P þ 1
ð16Þ
If the experimental enthalpies of formation of the reference (i = 1, 2, ..., n) in eq 16 are replaced by their species ΔfHexp i (i = 1, 2, ..., n) via the calculated enthalpies of formation ΔfHcalc i GA method, then the enthalpy change of an arbitrary GA RER involving the test species Bnþ1 is equal to the RE of the test species (normalized per one mole of Bnþ1)5 n
n
REnþ1 ¼
j ¼ P þ 1; P þ 2; :::; M
ð22Þ
and we, finally, arrive at our main finding
n
1
n
∑ vj;nþ1 ði∑¼ 1 vji ei Þ ¼ 0 j¼P þ 1
exp exp vji Δf Hi þ vj;nþ1 Δf Hn þ 1 ∑ i¼1
REnþ1 ¼
M
∑ vj;nþ1 ði∑¼ 1 vji ei Þ j¼P þ 1
Employing the RERs formalism it may be shown that the lefthand side of eq 22 is identically equal to zero
n
ΔHj ¼
M
vj;nþ1 ΔHj ð ∑ v2j;n þ 1 ÞREnþ1 ∑ j¼P þ 1 j¼P þ 1
∑
j¼P þ 1
vj;nþ1 ΔHj
ð25Þ
M
∑ v2j;nþ1 j¼P þ 1
As may be seen, the classical expression of the RE, eq 18, possesses a remarkable stoichiometric interpretation. Namely, according to eq 24, the RE may be partitioned into a linear sum of contributions associated with GA RERs. The significance of this new result is immense. It shows that from a thermodynamic point of view there is one and only one way to evaluate the relative stability of chemical species, e.g., the RE. Equation 18 is currently viewed in the literature as just one approach from a plethora of other approaches. The identities presented here explicitly indicate a common ground for all energetic approaches to RE. This is especially transparent for the single reaction schemes approach that is obviously a particular case of the general GA approach. More specifically, the general GA based method is reduced to a single reaction scheme when the number of reference species is minimal; i.e., the number of reference species n is equal to the rank of group matrix q. In this case, the RE takes the form
ð21Þ
REnþ1 ¼ 6659
ΔHF vF;nþ1
ð26Þ
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i.e., the RE is equal to the normalized (per one mole of Bnþ1) enthalpy change of a single reaction scheme that, concomitantly, is a GA RER. It should be also noticed that a single reaction scheme approach is “exact” in the sense that it completely lacks error estimation. Indeed, when the number of reference species is equal to q, the group values are the (exact) solutions of a linear system of q equations in q variables. In contrast, a valid GA method always involves more than q reference species and, hence, one has to employ the least-squares method to evaluate the group values. Respectively, a valid GA method is subject to a certain error and so is the evaluation of the RE based on this GA method. Alternatively, a GA method represents a special average over an enormous number of GA RERs, or reaction schemes, while a single reaction scheme is just one of these GA RERs. For these reasons, the single reaction scheme approach is always inferior to the general GA based method.
’ STOICHIOMETRIC FACTORS OF GA RERS Within the conventional thermodynamic approach to chemical reaction systems6 the absolute values of the stoichiometric coefficients in chemical reactions are irrelevant. What really counts is the ratio of stoichiometric coefficients. Thus, the reaction F : v1 B1 þ v2 B2 þ ::: þ vn Bn ¼ 0 is equivalent to γF : γv1 B1 þ γv2 B2 þ ::: þ γvn Bn ¼ 0 where γ is a different from zero constant. The stoichiometric coefficients in GA RERs are not always equal to the smallest integers. Furthermore, the stoichiometric coefficients of some species may be equal to zero. As a result, some GA RERs become stoichiometrically identical. For interpretational purposes it is desired to eliminate the repetitions of stoichiometrically identical GA RERs, thus, significantly reducing the complete list of GA RERs. The RERs formalism requires the contributions from stoichiometrically identical GA RERs to be summed up into one overall contribution. In doing this it is convenient to consider GA RERs in which the stoichiometric coefficients of the species are equal to the smallest integers. Every GA RER may be factorized as γFF where F is a GA RER in which the stoichiometric coefficients are the smallest integers and γF is a stoichiometric factor. Similarly, the enthalpy change of a GA RER may be presented as γFΔHF whereΔHF is the enthalpy change of F. Because in all applications (formulas) the stoichiometric factors γF are present in the squared form γF2, the overall stoichiometric factors of stoichiometrically distinct RERs are sums over the squares of the stoichiometric factors of stoichiometrically identical RERs. With this in mind the errors of the GA method take the form ei ¼
∑F vFi γF2 ΔHF 1 m
n
∑F s∑¼ 1
vFs2 γF2
i ¼ 1; 2; :::; n
’ EXAMPLE 1 The above theoretical considerations are next illustrated with the help of two examples. The first one illustrates the stoichiometric arbitrariness of a single reaction scheme. Consider a homodesmotic reaction scheme that has been recently proposed to evaluate the RE of pyridine (B7)7
This reaction scheme comprises six reference species B1, B2, B3, B4, B5, B6 and seven groups, namely, g1: CH2—CH2, g2: CH2—CH, g3: CH—CH, g4: CHdCH, g5: N—CH2, g6: N— CH, and g7: NdCH. It may be easily verified that the above homodesmotic reaction scheme is balanced; i.e., the number and type of groups on both sides of the reaction are preserved. At a first glance it may, therefore, seem that this reaction scheme is correct and may be used to evaluate the RE of pyridine. Nonetheless, the above homodesmotic reaction is wrong in that it is stoichiometrically arbitrary and so is its enthalpy change or the RE of pyridine. To prove it, consider the group matrix
Here and below the sign w means elementary row operations to the transposed group matrix. It is seen that the rank of the group matrix is equal to q = 5. Hence, a GA RER in this system involves no more than q þ 1 = 5 þ 1 = 6 species and, therefore, we have only one GA RER involving the reference species. This GA RER is
1 0 0 FðB1 ;B2 ;B3 ;B4 ;B5 ;B6 Þ : 0 0 0
ð27Þ
0 1 0 0 0 1
0 0 1 0 0 1
0 0 0 1 0 0
0 0 0 0 1 1
B1 B2 B3 B4 B5 B6
¼ 0B1 B2 þ B3 þ B4 B5 þ B6 ¼ 0 or
where the summation runs over a complete set of stoichiometrically distinct GA RERs involving only reference species. Similarly, the RE is given by
∑F vF;nþ1γF2ΔHF REnþ1 ¼ 2 γF2 ∑F vF;nþ1
Here the summation runs over a complete list of stoichiometrically distinct GA RERs involving the test species Bnþ1.
F1 : B2 þ B5 ¼ B3 þ B6
ΔH1
To better visualize the group balance, we present this reaction in terms of structural formulas too
ð28Þ
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According to eq 27 the species errors are e1 ¼ 0 e2 ¼ ΔH1 =4 e3 ¼ ΔH1 =4
1 0 0 FðB1 ;B3 ;B4 ;B5 ;B6 ;B7 Þ : 0 0 1
e4 ¼ 0 e5 ¼ ΔH1 =4 e6 ¼ ΔH1 =4
Now, let us consider the GA RERs involving the test species B7. Because a GA RER involves no more than 6 species and one of them should be always pyridine (B7), their number is equal to the number of ways five reference species may be selected from a total of six, i.e., 6!/5!/1! = 6. According to eq 7 these GA RERs are given by
1 0 0 FðB1 ;B2 ;B3 ;B4 ;B5 ;B7 Þ : 0 0 1
0 1 0 0 0 1
0 0 1 0 0 3
0 0 0 1 0 1
0 0 0 0 1 3
B1 B2 B3 B4 B5 B7
1 0 0 FðB1 ;B2 ;B3 ;B4 ;B6 ;B7 Þ : 0 0 1
0 1 0 0 1 1
0 0 1 0 1 3
0 0 0 1 0 1
0 0 0 0 1 3
B1 B2 B3 B4 B6 B7
¼ B1 2B2 þ 0B3 B4 þ 3B6 þ B7 ¼0
1 0 0 FðB1 ;B2 ;B3 ;B5 ;B6 ;B7 Þ : 0 0 1
0 1 0 0 1 1
0 0 1 0 1 3
0 0 0 0 0 1
0 0 0 1 1 3
B1 B2 B3 B5 B6 B7
0 1 0 0 1 3
0 0 1 0 0 1
0 0 0 1 1 3
B1 B3 B4 B5 B6 B7
¼ ð B1 2B3 B4 þ 2B5 þ 3B6 þ B7 Þ ¼0
0 0 0 FðB2 ;B3 ;B4 ;B5 ;B6 ;B7 Þ : 0 0 1
1 0 0 0 1 1
0 1 0 0 1 3
0 0 1 0 0 1
0 0 0 1 1 3
B2 B3 B4 B5 B6 B7
¼ B2 B3 þ 0B4 þ B5 B6 þ 0B7 ¼0
¼ B1 þ B2 3B3 B4 þ 3B5 þ B7 ¼0
0 0 0 0 1 1
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It is seen that from a total of six possible GA RERs involving test species B7 only three are stoichiometrically distinct, namely, F2 : B1 þ 3B3 þ B4 ¼ B2 þ 3B5 þ B7 F3 : B1 þ 2B2 þ B4 ¼ 3B6 þ B7 F4 : B1 þ 2B3 þ B4 ¼ 2B5 þ B6 þ B7
vF;7 1 1 1
γF2 1 2 1
ΔHF ΔH2 ΔH3 ΔH4
Observe that the GA RER F3 occurs twice and, therefore, has an overall stoichiometric factor equal to 2 (12 þ 12 = 2). Also notice that there is a stoichiometrically distinct GA RER in which the stoichiometric coefficient of B7 is equal to 0 B2 þ B5 ¼ B3 þ B4 This GA RER formally coincides with the GA RER involving only reference species. Because the stoichiometric coefficient of the test species B7 is equal to 0, this GA RER has no direct role in the evaluation of RE of B7 and can be disregarded. For visualization purposes, the list of stoichiometrically distinct GA RERs involving the test species (pyridine) is presented below in terms of the structural formulas
¼ 0B1 B2 þ B3 B5 þ B6 þ 0B7 ¼0
1 0 0 FðB1 ;B2 ;B4 ;B5 ;B6 ;B7 Þ : 0 0 1
0 1 0 0 1 1
0 0 0 0 1 3
0 0 1 0 0 1
0 0 0 1 1 3
B1 B2 B4 B5 B6 B7
¼ ð B1 2B2 B4 þ 0B5 þ 3B6 þ B7 Þ ¼0
Thus, the RE of B7 is equal to RE ¼
ðΔH2 þ 2ΔH3 þ ΔH4 Þ 1 ¼ ðΔH2 þ 2ΔH3 þ ΔH4 Þ 1þ2þ1 4
This (correct) result differs from the one predicted by the single homodesmotic reaction scheme. In fact, the single homodesmotic 6661
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The Journal of Physical Chemistry A reaction scheme may be decomposed in various ways into a linear combination of GA RERs. For instance, if the single homodesmotic reaction scheme is denoted as F then F = F1 þ F2 and ΔH = ΔH1 þ ΔH2. To what extent the enthalpy change of F coincides with the exact value of RE depends on the performance of the GA additivity method, i.e., the enthalpy change of F1. If ΔH1 ≈ 0, then the enthalpy changes of F2, F3, and F4 are close, i.e., ΔH2 ≈ ΔH3 ≈ ΔH4 ≈ ΔH. Only in this particular case may the enthalpy change of F be considered as a measure of the RE of pyridine.
’ EXAMPLE 2 The second example illustrates the evaluation of the RE of benzene. As is well-known, benzene is the etalon of aromaticity; therefore, accurate evaluation of its RE is crucial in understanding and rationalizing the concept of aromaticity from a thermodynamic point of view.8 The unacceptable scan of RE values for benzene in the literature is mainly due to a large variety of reference states and single reaction schemes. With the exception of the reference states that involve a single reference species, e.g., vertical2 or isomerization RE,7 the selection of reference state, however, is surprisingly straightforward. As a matter of fact, we have only two options, the cyclic and noncyclic reference states. Here we perform a detailed analysis of the RE of benzene employing the cyclic reference structure. From any organic chemistry textbook it follows that the cyclic reference state comprises cyclohexane, cyclohexene, and 1,3-cyclohexadiene. Thus, the classical, Kistiakowski “heat of hydrogenation” approach9 is essentially an isodesmic reaction scheme approach
More recently, a homodesmotic reaction scheme has been proposed7
It was further claimed7 that the homodesmotic reaction scheme is “superior” in that the energetic imbalances in the reference species are lower compared with the bond additivity or isodesmic reaction scheme. This is not entirely true for several reasons. First, there is no proof whatsoever that the energetic imbalances in the isodesmic reaction scheme are higher than in the homodesmotic reaction scheme. Second, it is methodologically wrong to compare the RE of two reaction schemes that involve different sets of reference species (two reference species in the isodesmic reaction vs three reference species in the homodesmotic reaction) as well as different GA methods (isodesmic vs homodesmotic). Third, as pointed out by Wiberg,10 there is one more reference species that has been omitted from the analysis, namely, 1,4-cyclohexadiene. Next, we employ the cyclic reference state comprising four reference species cyclohexane (ΔfHexp = 29.43 kcal/mol), cyclohexene (ΔfHexp = 1.03 kcal/mol), 1,3-cyclohexadiene (ΔfHexp = 25.0 kcal/mol), and 1,4-cyclohexadiene (ΔfHexp = 25.04 kcal/mol) to evaluate the RE of benzene (ΔfHexp = 19.8 kcal/mol). The procedure is separately performed for isodesmic and homodesmotic GA models.
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Isodesmic GA Model. This model includes three types of bonds (groups) C—H (g1), C—C (g2), and CdC (g3). Thus, the group matrix is
It is seen that that the rank of the group matrix is equal to q = 2. Respectively, we have a total of 4!/3!/1! = 4 GA RERs involving only the reference species. Elementary calculations show that only three of them are stoichiometrically distinct. These GA RERs, along with their stoichiometric factors and enthalpy changes, are presented below
On the basis of these data, the reference species errors are 0.43, þ0.86, 0.23, and 0.19 kcal/mol for cyclohexane, cyclohexene, 1,3-cyclohexadiene, and 1,4-cyclohexadiene, respectively. The standard deviation of the species errors is 0.58 kcal/mol. Interestingly, for a bond additivity model this is a pretty decent performance. Now, let us turn to GA RERs involving benzene. Because a GA RER in this system involves no more than q þ 1 = 3 species and one of them should always be benzene, the total number of GA RERs involving benzene is 4!/2!/2! = 6. From these only five are stoichiometrically distinct
According to eq 28 the RE of benzene is RE ¼ ð 35:97 2 64:83 2 64:95 31:23 31:31Þ=ð1 þ 4 þ 4 þ 1 þ 1Þ ¼ 32:55 kcal=mol For illustration purposes, let us show that the same result is obtained within the conventional analysis. Because only two groups from a total of three are linearly independent we can arbitrarily disregard one of them, e.g., g2. Employing the 6662
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The Journal of Physical Chemistry A least-squares method, the group contributions for the remaining two groups g1 and g3 are equal to H1 = 2.41689 kcal/mol and H3 = 22.2844 kcal/mol. The RE of benzene is thus RE ¼ Δf H exp Δf H calc ¼ 19:8 ð6H1 þ 3H3 Þ ¼ 32:55kcal=mol Homodesmotic GA model. This model includes four groups, namely, CH2—CH2 (g1), CH2—CH (g2), CH—CH (g3), CHdCH (g4). The rank of the group matrix is equal to q = 3
ARTICLE
’ CONCLUDING REMARKS The rigorous thermodynamic and stoichiometric analysis of the RE performed in this work shows that the classical definition of the RE may be uniquely partitioned into a sum of contributions associated with GA RERs involving the test species. This new result and interpretation of the RE reveals an intimate interrelation between the conventional GA and single reaction scheme methods. Namely, the single reaction schemes method is a particular case of the general, GA based method. As such, the evaluation of the RE based on single reaction schemes is inferior to the GA method and should be always avoided unless the number of reference species is limited and there is only one possible reaction involving the reference and test species. ’ AUTHOR INFORMATION Corresponding Author
Hence, there is one GA RER involving only the reference species and this is
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’ REFERENCES
Because the stoichiometric coefficient of 1,3-cyclohexadiene is equal to 0, this GA RER formally coincide with one of the GA RERs comprising the bond additivity model. The reference species errors are 0.39, þ0.78, 0, and 0.39 kcal/mol for cyclohexane, cyclohexene, 1,3-cyclohexadiene, and 1,4-cyclohexadiene, respectively. These errors are only slightly lower than the errors of the bond additivity model, e.g., a standard deviation of 0.55 kcal/mol. Next, from a total of four possible GA RERs involving benzene only three are stoichiometrically distinct and these are
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This gives a RE equal to RE ¼ ð 28:86 2 64:71 31:19Þ=ð1 þ 4 þ 1Þ ¼ 31:58kcal=mol As follows from the above rigorous analysis, two different GA models comprising the same set of four reference species result in surprisingly close values of RE of benzene. Thus, the bond additivity and homodesmotic models predict a RE of benzene that differ by only 1 kcal/mol! From these two GA models, the homodesmotic RE is preferred not only due to a slightly smaller error but because the homodesmotic model is more general and includes the bond additivity model. It means that the homodesmotic model is concomitantly a bond additivity model, i.e., preserves the type and number of bonds too. The opposite is not true; i.e., the bond additivity model does not preserve the type and number of homodesmotic groups. 6663
dx.doi.org/10.1021/jp202660z |J. Phys. Chem. A 2011, 115, 6657–6663
