J. Phys. Chem. B 2007, 111, 7331-7335
7331
One More Type of Extrathermodynamic Relationship Oleg A. Khakhel′, Tamila P. Romashko, and Yuriy E. Sakhno* PoltaVa Department of Ukrainian Academy of Technological Cybernetics Sciences, PoltaVa 36007, Ukraine ReceiVed: March 31, 2007
A new type of extrathermodynamic relationship is presented: ∆H ) Ti[∆S + R ∆(ln Ω)] + β, where Ω is the phase volume and β is a constant. This type of relationship holds for a series of systems in the case where δ∆(ln Ω) is changeable. The relation between this type of correlation and linear free-energy relationships is shown. An example of the specified correlation for energy parameters of excimers forming in different dipyrenylalkanes is demonstrated.
1. Introduction Linear free energy relationships (LFERs) correlating enthalpy (∆H) and entropy(∆S) have been found for a great variety of phenomena.1-9 LFER manifestations are also known as the compensation effect and isokinetic (or isoequilibrium) relationships. Because LFERs have been discovered in many unrelated fields, one can consider them as a manifestation of some common thermodynamic law. Additionally, because LFERs are relationships between quantities describing the thermodynamics of a series of systems, they are called extrathermodynamic equations. A series of systems might represent a series of processes in various mediums or those with different structures of reactants. LFERs are the empirical observations for which the physical origin has not been fully explained. Existing theories of LFERs, obviously, do not take into account some basic thermodynamic quantity. In this respect, it has been shown in the literature10 that it is necessary to consider phase volume as a significant quantity for LFERs. An invariability of phase volumes of a system’s states, for which a difference of free energy is determined, is the single factor generating LFERs. At present, as we know, LFERs are the sole type of extrathermodynamic relationship. In this work, we report a new type of extrathermodynamic relationship that follows from the consideration of a phase volume behavior in a series of systems. 2. Background In the general form, LFERs can be defined by the following equation,6 ∆G ) a ∆Gs + b
(1)
where ∆G is the free energy of a process such as a ratio or an equilibrium, ∆Gs is the free energy of some standard process, a is the similarity, and b is a constant. Equation 1 can be rewritten in other terms: log(K/K0) ) a log(Ks/Ks0)
(2)
where K and Ks are the rate or the equilibrium constants. Equation 2 represents a correlation equation of the Hammett type. Such equations can be divided into LFERs of two classes.6 * To whom correspondence should be addressed. E-mail: sakhno2001@ mail.ru; tel.: +380532677716.
Class I LFERs compare a rate constant with the equilibrium constant of the same process. The rate or equilibrium constant in Class II LFERs is related to the rate or equilibrium constant of an unconnected process. When passing from one system to another system of their series, the ∆H and ∆S values, which determine the magnitude of ∆G in eq 1, are changed proportionally. δ∆S/δ∆H ) 1/Ti
(3)
This is a precondition for LFERs. Equation 3, in which the δ operator reflects changes in magnitudes of a series, defines the compensation effect. From eq 3, one obtains eq 4: ∆H ) Ti∆S + R
(4)
where R is the constant, and Ti is the so-called isokinetic temperature. At this temperature, all systems of a series have the same value for ∆G, and hence, also have identical rate constants, for which ∆H and ∆S were calculated.2,4 It is generally considered that the R constant from eq 4 has no physical meaning. However, it was shown10 that R ) -TiR ∆(ln Ω), where R is the gas constant, and Ω is the phase volume. A linear relation between ∆H and ∆S for a series of systems is a consequence of Ω being constant for this series, for which there is a simple explanation based on two statements. The first statement is that, considering some rate or equilibrium for a series of systems, we must consider the behavior of its standard free energy, which is defined by eq 5. ∆G′ ) ∆G + TiR ∆(ln Ω)
(5)
Second, at any present temperature of Ti, the magnitudes of G′ of any states n and m, which are equal between themselves in one system of a series, have to be equal between themselves in any other system of a series: G′n ) Gn + TiR(ln Ωn) ) Gm + TiR(ln Ωm) ) G′m
(6) δG′n ) δGn + TiR δ(ln Ωn) ) δGm + TiR δ(ln Ωm) ) δG′m That is, under the same external effect on the n and m states, their G′ magnitudes have the same response. Because δ∆(ln Ωnm) ) 0, eq 6 leads to eqs 3 and 4. Below, analyze the case where δ∆(ln Ω) * 0.
10.1021/jp0725275 CCC: $37.00 © 2007 American Chemical Society Published on Web 05/25/2007
7332 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Khakhel′ et al.
3. Analysis well-known11
that the free energy defined by the Gibbs It is distribution, Ψ, is considered in thermodynamics as the free energy in a general sense. Under different selections of a system’s external parameters, differing quantities represent its free energy. Where a system is examined under a constant spatial volume, V, the Helmholtz energy, F, is its free energy:
( )
exp -
Ψ ) RT
∫ ∫ exp(-
)
H(qi,pi) dqi dpi RT
( )
Ψ ) RT
∫∫∫∫
(
exp -
)
H(qi,pi) + PV + p2V /2M dqi dpi dV dpV (7) RT
Here, Ψ is already identified with the Gibbs energy, G. In eq 7, M is the mass causing the pressure, PV and p2V /2M represent potential and kinetic energies of the extended system. The cases of constant and changeable phase volumes, in our opinion, are fully analogous with those for the Helmholtz and the Gibbs energies where spatial volume passes from the category of external parameters to the category of coordinates. So, it is true of G, such that dG ) d(H - TS) ) V dP - S dT. Here, dG is presented as the sum of differentials of the external parameters, namely, of P and T. -V acts as a generalized force corresponding to the external parameter of P. The corresponding potential energy is defined as U ) ∫V dP ) PV. Moreover, V becomes the coordinate (see eq 7). Analogically, it is true of G′ that dG′ ) d(H - T(S - R ln Ω)) ) V dP -(S - R ln Ω) dT + (RT/Ω) dΩ. It is visible that -(RT/Ω) acts as a generalized force corresponding to the external parameter of Ω. Consequently, 1/Ω is the coordinate. Thus, if δ∆(ln Ωnm) ) 0, then we consider the standard free energy under the condition of a constant 1/Ω (i.e., an extrathermodynamic equivalent of the Helmholtz energy). If δ∆(ln Ωnm) * 0, then we must consider the standard free energy under the condition of varying 1/Ω (an extrathermodynamic equivalent of the Gibbs energy). By analogy with eq 7, we can now express the following equation:
( )
Ψ exp ) RT
(
)
()
H(qi,pi) + TR ln Ω + p2Ω/2M 1 dqi dpi d dpΩ RT Ω
This equation can be transformed:
( )
exp -
Ψ ) RT
∫∫∫
∫
( )
∫
∫
Ψ E Ω(E) dΩ ) exp dE - 2 RT RT Ω Ω p2Ω/2M E Ω(E) 1 exp dpΩ ) exp dE x2πMRT RT RT Ω Ω
)
∫
( )
Ψ′′ ) Ψ - TR ln Ω + TR lnx2πMRT ) -TR ln
[∫ ( ) exp -
]
E Ω(E) dE (9) RT Ω
Equation 9 defines the standard free energy of a state in the case where we change the phase volume. We can compare it with the one obtained in the case of constant phase volume: Ψ′ ) Ψ + TR ln Ω ) -TR ln
[∫ ( ) exp -
]
E Ω(E) dE RT Ω
(10)
If Ψ is the Gibbs energy, then eq 10 leads to eq 5, and eq 9 can be rewritten as eq 11: ∆G′′ ) ∆G - TR ∆(ln Ω) + TR ∆(lnxM)
(11)
Like the Helmholtz and Gibbs energies, the ∆G′ and ∆G′′ extrathermodynamic potentials defined by eqs 5 and 11, respectively, are different quantities. Also, the Ti in the case of ∆G′′ is no longer the isokinetic temperature in the sense of invariability of rate constants, but the main sense of Ti remains in both cases. At this temperature, the G′ or G′′ magnitudes for states n and m have to be equal between themselves in all systems of a series. In accordance with eq 6, we can write down this statement for G′′: G′′n ) Gn - TiR ln Ωn + βn ) Gm - TiR ln Ωm + βm ) G′′m δG′′n ) δGn - TiR δ(ln Ωn) + δβn ) δGm - TiR δ(ln Ωm) + δβm ) δG′′m (12) Equation 12 leads to eq 13: δ∆Snm + R δ∆(ln Ωnm) δ∆Hnm
)
1 Ti
(13)
Formally, if δ∆(ln Ωnm) ) 0, then eq 13 changes into eq 3. From eq 13, one obtains eq 14:
∫∫∫∫
exp -
( ) (
exp -
or
Here, Ψ ≡ F, and H(qi,pi) is the Hamilton function of a system; qi and pi are phase variables (i.e., positions and momenta). Where pressure, P, is chosen as an external parameter, we have eq 7: exp -
not depend on Ω. Therefore, we can obtain the following relationships from eq 8.
( ) (
exp -
)
RT 1n Ω + p2Ω/2M E exp Ω RT RT 1 (E) dE d dpΩ (8) Ω
()
Here, E is the energy, Ω(E) dE is the phase volume of the layer lying between surfaces of H(qi,pi,Ω,pΩ) ) E and of H(qi,pi,Ω,pΩ) ) E + dE, pΩ is the moment corresponding to the coordinate of 1/Ω, and M is some mass characterizing a state. The Ω(E) magnitude depends on Ω. However, the Ω(E)/Ω magnitude does
∆H ) Ti[∆S + R ∆(ln Ω)] + β ) Ti[∆S + R ∆(ln Ω)] TiR ∆(lnxM) (14) The effect defined by eqs 11-14 can be named the extended compensation effect. It is the extrathermodynamic relationship however, that is unconnected with a linearity of free energy. Let us consider a behavior of free energies in connection with the extended compensation effect. LFERs compare the change in ∆G1 in some reaction or process with the change in ∆G2 in a similar standard reaction or process. Because of the similarity of the processes, these changes are supposed to be correlating. However, if eq 15 holds, then there is not a linear correlation. δ∆G′′1 ) δ∆G1 - Ti12R δ∆(ln Ω1) + Ti12 δ∆(1n xM1) )
δ∆G2 - Ti12R δ∆(1n Ω) + Ti12 δ∆(1n xM2) ) δ∆G′′2 (15)
One More Type of Extrathermodynamic Relationship
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7333
TABLE 1: Thermodynamic Parameters for Intramolecular Excimer Formation with Dipyrenylalkanes in Various Solvents and the Results of Correlation Analysisa D*1 solvent
D*2
∆H kJ/mol
∆S J/(K‚mol)
3.1 -1.9
42.9 26.7
∆H kJ/mol
∆S J/(K‚mol)
Bichromophore 1 methylcyclo-hexane toluene n-pentane n-hexane n-heptane n-decane n-dodecane n-hexadecane toluene liquid paraffin
-17.5 -15.1 -15.5 -16.7 -17.0 -17.5 -24.0
-21.3 -14.0 -15.0 -19.0 -21.1 -21.3 -38.2
methylcyclo-hexane toluene n-octane
-18.5 -22.0 -19.7
-24.1 -32.2 -25.2
n-pentane n-hexane n-heptane n-hexadecane n-nonane n-decane n-dodecane n-octane liquid paraffin toluene methylcyclo-hexane
-23.9 -25.4 -25.9 -25.9 -26.7 -26.7 -27.2 -26.4 -24.3 -27.5 -20.4
-39.8 -44.1 -46.0 -46.0 -48.0 -48.0 -49.2 -45.8 -40.1 -52.2 -30.9
Bichromophore 2 -19.2 -22.4 -23.8 -24.7 -26.7 -26.0 -25.0 -27.1
T i, K R, kJ/mol
-40.2 -51.4 -55.4 -56.2 -63.8 -60.8 -61.5 -63.5
D*1: Ti ) 372.4, R ) -9.642, R ∆(ln ΩDM) ) 25.89, ΩD/ΩM ) 22.5, r ) 0.9961, sd ) 0.7709, n ) 7 D*2: Ti ) 372.3, R ) -3.154, R ∆(ln ΩDM) ) 8.47, ΩD/ΩM ) 2.8; r ) 0.9526, sd ) 1.5586, nb ) 7 Ti ) 414.8, R ) -8.803, R ∆(ln ΩDM) ) 21.22, ΩD/ΩM ) 12.8, r ) 0.9762, sd ) 1.3474, n ) 3
Bichromophore 4
Ti ) 351.6, R ) -9.817, R ∆(ln ΩDM) ) 27.92, ΩD/ΩM ) 28.7, r ) 0.9887, sd ) 0.9210, n ) 11
r ) 0.9988, sd ) 1.1755, n ) 23 332.1 (330c) -10.698 (-11c)
analysis results: ∆H ) Ti∆S + R
Ti ) 308.6, R ) -10.140, R ∆(ln ΩDM ) ) 32.86, ΩD/ΩM ) 52.0
Bichromophore 3
Bichromophore 5 -34.1 -39.7 -40.5 -40.9 -40.0 -38.7 -40.0 -40.3
n-pentane n-hexane n-heptane n-decane n-hexadecane liquid paraffin toluene methylcyclo-hexane
analysis results: ∆H ) Ti∆S + R; Ti, K; R, kJ/mol; R ∆(ln Ω), J/(K mole)
-85.6 -105.3 -109.1 -110.2 -107.4 -102.4 -112.3 -108.5
Ti ) 265.9, R ) -11.328, R ∆(ln ΩDM) ) 42.46, ΩD/ΩM ) 167.8, r ) 0.9756, sd ) 2.001, n ) 8
r ) 0.9975, sd ) 1.9127, n ) 16 307.8 (304c) -6.924 (-7c)
a Bichromophores: 1 ) meso-2,4-di(2-pyrenyl)pentane, 2 ) 1,3-di(1-pyrenyl)propane, 3 ) rac-2,4-di(2-pyrenyl)pentane, 4 ) 1,3-di(2pyrenyl)propane, and 5 ) 1,16-di(1-pyrenyl)hexadecane. b Data for n-pentane was not taken into account. c Data from the ref 2.
Equation 15 corresponds to the condition of T ) Ti12, where Ti12 is the iso temperature for processes 1 and 2. Moreover, these processes are characterized by their own iso temperatures of Ti1 and Ti2 (see eqs 12 and 13). By using these parameters, we can obtain δ∆G′′1 ) F δ∆G′′2, where, at an arbitrary temperature, F is defined by eq 16. F)
Ti2 - Ti12 Ti1 - T Ti1 - Ti12 Ti2 - T
this case, however, the Ti1 and Ti2 temperatures are defined by eq 3. Because δ∆(ln Ω1,2) ) 0, it follows that eq 19 is the same as eqs 1 and 2. δ∆G1 ) F δ∆G2
(19)
4. Example (16)
From here we can derive eq 17. δ∆G1 ) F δ∆G2 - FTR δ∆(ln Ω2) + TR δ∆(ln Ω1) (17) Because Ω1,2 are changeable, there is no LFER in this case. δ∆G′1 ) δ∆G1 + Ti1,2R δ∆(ln Ω1) ) δ∆G2 + Ti1,2R δ∆(ln Ω2) ) δ∆G′′2 (18) On the contrary, if eq 18 is true, then at an arbitrary temperature we have δ∆G′1 ) F δ∆G′2. Here, F is also defined by eq 16. In
To illustrate the above statements, one can consider data from ref 2 that represent a large enough set of values of ∆H and ∆S for the intramolecular excimers of a series of dipyrenylalkanes in a variety of solvents. These data are listed in Table 1. They were obtained by scanning the graph in the paper.2 According to the conclusions of the authors,2 data listed in Table 1 are divided into two series. Each series refers the data to two types of excimers, D1* and D2*. In the authors’2 opinion, the D1* and D2* excimers have different equilibrium geometries. Each series of ∆H and ∆S values for the formation of excimers can be approximated by the straight line in the ∆H∆S coordinates. According to eq 4, Ti and R act as the parameters for such lines. The authors’2 evaluation gives the
7334 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Khakhel′ et al. 5. Discussion
Figure 1. Plots of ∆H vs ∆S and ∆H vs ∆S + R ∆(ln Ω) for intramolecular excimer formation with a number dipyrenyalkanes in toluene.
magnitudes Ti ) 330 and 304 K and R ) -11 and -7 kJ/mol for excimers of D1* and D2*, respectively. Using data from Table 1, we determined these magnitudes to be equal to 331.1 and 307.8 K and -10.698 and -6.924 kJ/mol. As is visible from here, the results of our correlation analysis coincide with the evaluation of the original work.2 The reasons for referring the ∆H-∆S data points for excimers to two correlation series have been discussed in the previous work.2 However, we wish to raise an objection against such correlations. Most of the bichromophores form one excimer according to the kinetic scheme M* + M a D*. 1,3-Di(1pyrenyl)propane forms two excimers according to D1* a M* + M a D2*. Data for its excimers of D1* and D2* are referred in two series, which have different values of Ti. However, as was previously shown in ref 10, the temperatures of Ti for two excimers in the case where they are formed in the same bichromophore have to be identical. The linear approximation performed separately with the data points for the D1* and D2* excimers of 1,3-di(1-pyrenyl)propane shows exactly such a result. These calculations, together with separate calculations for other excimers, are listed in the last right-hand column of Table 1.As visible from Table 1, each excimer is characterized by its own values of Ti and R. Because R ) -TiR ∆(ln ΩDM), we can also determine the value of R ∆(ln ΩDM) for each excimer. Thus, the compensation effect for excimers is displayed only for a series of solvents. One can consider that the phase volume of each excimer is a constant magnitude in various solvents. At the same time, changes of interchromophoric chains in bichromophoric structure change the magnitude of ΩDM. However, the values of R ∆(ln ΩDM) are calculated, and we can demonstrate the extended compensation effect by plotting the data points for excimers in the coordinates of ∆HDM vs ∆SDM + R ∆(ln ΩDM) (see eq 14). Among the data in Table 1, the data for toluene have the largest statistics. In Figure 1, these data points are plotted in both coordinates of ∆HDM vs ∆SDM and ∆HDM vs ∆SDM + R ∆(ln ΩDM). The results of a linear correlation analysis for these data are listed in Table 2. It should be noted that 1,3-di(1-pyrenyl)propane forms two excimers, D1* and D2*; however, they represent two parts of the DΣ* excimeric state, which is the equilibrium for the D1* and D2* excimers. By this reason, Figure 1 and Table 2 contain data for the DΣ* excimer that were calculated by exp(-∆GDΣ/RT) ) exp(-∆GD1/RT) + exp(-∆GD2/RT) with T ) 250 and 350 K. The magnitude of ΩDΣ was determined as ΩDΣ ) ΩD1 + ΩD2.
The example given in Figure 1 and Table 2 shows that the correlation is better in the case of the extended compensation effect. However, as was reviewed in the previous work,4 experimental errors in ∆H, ∆S, and ∆G can lead to an apparent correlation. In our viewpoint, to draw a strictly defined conclusion concerning any correlation, we should have a clear notion concerning its origin. We connect the origin of correlations with a behavior of phase volume. This is a common approach, which is unconnected to any concrete field. Therefore, in each case a discussion of the nature of the phase volume is necessary. Generally, phase volume, or, more precisely, the relation of Ωn/Ωm is the magnitude proportional to the relation of probabilities of wn and wm, which feature the probability for a system to be discovered in states of n or m. In fact, such probabilities are wn,m ∼ exp(-Ψn,m/RT) or wn,m/Ωn,m ∼ exp(-Ψ′n,m/RT). From here, it can be shown that wn/wm ) exp(-∆Ψ′n,m/RT)(Ωn/Ωm). If T ) Ti, then one obtains Ωn/Ωm ) wn/wm. On the other hand, in the previous work10 concerning excimers it was shown that ΩD/ΩM can be interpreted as magnitude proportional to the area of interaction between excited and unexcited chromophores in a space of their mutual positions. At this point, it is an expected result that, among the excimers analyzed in this work, the excimer of 1,16-di(1-pyrenyl) hexadecane, i.e., the excimer forming a bichromophore with the longest interchromophoric chains, has the greatest value of ΩD/ΩM (see Table 1). More short chains in other bichromophores, obviously, restrict the area of interchromophoric interaction, owing to both a shortening of chains and steric hindrances. The scheme describing excimer formation, M* + M a D*, is formally identical to schemes describing many bimolecular chemical reactions. Therefore, it is expedient to identify the phase volume of a chemical reaction with the volume of the reactionary region. The latter is a quantity that is similar to the area of interchromophoric interaction in the case of excimers. In regard to chemical reactions, it should be mentioned that at present there is a tendency to relate the isokinetic temperature with transition structure.6 To the point, this is in accordance with a relation of the isokinetic temperature with the excimeric equilibrium geometry.2 However, there are reasons to suppose that the equilibrium structure of the excimer is not the same in different solvents.12 At the same time, there is no reason to suppose that, for any given bichromophore, the area of interchromophoric interaction appreciably changes in different solvents. When discussing the nature of LFERs, Hammett especially emphasized the distance separating a reactionary center from the position at which structural changes take place is an important factor to LFERs.1 In this way he advanced seven suggestions to explain LFERs. One can consider that long distances are the same that as the condition of invariability of volume of the reactionary region. Then, all suggestions can be reduced to the stipulations expressed by eqs 18 and 15. By denoting δ∆G2 as σ and δ∆G1 as log(K/K0), from eq 19 we obtain the Hammett equation, log(K/K0) ) Fσ. Similarly, from eq 17 we can obtain eq 20: log(K/K0) ) Fσ + FsEs
(20)
where Fs ) RT(1 - F) and Es ) δ∆(ln Ω). Here, it is taken into account that by using identical substituents in two series of reaction we obtain the same steric effect, and hence, δ∆(ln Ω1) ) δ∆(ln Ω2) ) δ∆(ln Ω). Equation 20 is an equation of the Taft type.
One More Type of Extrathermodynamic Relationship
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7335
TABLE 2: Thermodynamic Parameters for Intramolecular Excimer Formation with Dipyrenylalkanes in Toluene and the Results of Correlation Analysis D*1 bichromophore meso-2,4-di(2-pyrenyl)pentane 1,3-di(1-pyrenyl)propane rac-2,4-di(2-pyrenyl)pentane 1,3-di(2-pyrenyl)propane 1,16-di(1-pyrenyl)hexadecane analysis results
D*2 ∆S J/(K‚mol)
-1.9
26.7
-1.9
-18.65
-23.75
-18.65
-22.0
-32.2
-22.0
-11.00
-27.5
-52.2
-27.5
-24.29
-40.0
-112.3
-40.0
-69.71
59.55 2.78
∆H ) Ti(∆S + R ∆(ln Ω)) + β, Ti ) 296.6 K, β ) -19.478 kJ/mol, r ) 0.9992, sd ) 2.1756, n ) 5
r ) 0.9893, sd ) 8.4876, n)5
Now, let us discuss some terminology questions. LFERs compare energies of ∆G′1,2 for two series of reactions or equilibriums (see eq 18). Linear correlations between the magnitudes of ∆H and ∆S, which define the same energy of ∆G, are named the compensation effect. An origin of the compensation effect is featured by eq 6. However, energy is not an absolutely defined quantity. There is always a gap between the energies of any two states. Therefore, one can consider that in eq 6 G′n ) ∆g′n and G′m ) ∆g′m. Then, by comparing eqs 6 and 18, we can conclude that they have the same physical meaning. From here, LFERs can be named “secondary compensation effect” or, vice versa, the compensation effect can be named “primary LFERs”. And some authors consider that LFERs and the compensation effect are synonymous notions. For example, in the previous work2 the ∆H-∆S relationships for pyrene excimer formation have been named as LFERs. Analogously, eq 15 features a secondary extended compensation effect (SECE). It was known earlier in a form of the Taft equation (see eq 20). We emphasize once again that the compensation and the extended compensation effects are different, which show up under different circumstances. One cannot consider that the extended compensation effect is a deviation of the compensation effect. Both effects take place for their own series of closely related reactions or equilibriums. The notion about “closerelation” is unclear enough. Here we can propose its stricter definition. It is obvious that in the case of the extended compensation effect, closely related reactions or equilibriums are ones that will have the same value of the ∆(lnxM) magnitude. An invariability of the ∆(ln Ω) magnitude gives such a criterion in the case of the compensation effect. In other words, an invariability of the ∆(ln Ω) and ∆(lnxM) magnitudes is a necessary condition for the compensation and extended compensation effects, respectively. In regard to LFERs and SECE, the necessary conditions for them are an invariability of the ∆(ln Ω1,2) and ∆(lnxM1,2) magnitudes, respectively (see eqs 18 and 15). However, even under these conditions, these relationships can be distorted because of the subjective selection of systems in series 1 and 2. In fact, to compare two reactions, in which we realized identical changes of conditions, it is not necessarily that these
∆H kJ/mol
∆S + R ∆(ln Ω), J/(K‚mol)
∆H kJ/mol
changes will satisfy eqs 18 or 15. As a rule, the total influence on the change of reaction conditions consists of several effects, each of which can have different efficiencies in different reactions. However, in some of these problematic cases it is possible to guess these efficiencies, an example of that for the case of LFERs is the Yukawa-Tsuno equation.1 6. Conclusions There are two types of correlations: (1) the compensation effect [∆H ) Ti∆S - TiR ∆(ln Ω)] and (2) the extended compensation effect [∆H ) Ti(∆S + R ∆(ln Ω)) - TiR ∆(ln xM)]. The first one holds for a series of systems in the cases where phase volumes, Ω, of the system’s states, for which ∆H and ∆S are determined, are constant. The extended compensation effect holds for a series of systems in the case where δ∆(ln Ω) is changeable. These two types of correlations follow from a consideration of the standard free energy, -TR ln[∫ exp(-E/RT)(Ω(E)/Ω) dE]. In the first case, this is ∆G′ ) ∆G + TR ∆(ln Ω). In the second case, this is ∆G′′ ) ∆G - TR ∆(ln Ω) + TR ∆(lnxM). Both types of correlation are characterized by their own special temperatures of Ti, Ti ) δ∆H/δ∆S, and Ti ) δ∆H/[δ∆S + R δ∆(ln Ω)]. These temperatures define the F value, which is used in correlation equations of the HammettTaft type. References and Notes (1) Hammett L. P. Physical Organic Chemistry; McGraw-Hill: New York, 1970. (2) Zachariasse, K. A.; Duveneck, G. J. Am. Chem. Soc. 1987, 109, 3790. (3) Harrison, A. G. J. Mass Spectrom. 1999, 34, 577. (4) Lui, L.; Guo, Q.-X. Chem. ReV. 2001; 101, 673. (5) Grosman, C. J. Biol. Phys. 2002, 28, 267. (6) Williams A. Free Energy Relationships in Organic and Bioorganic Chemistry; RSC: Cambridge, 2003. (7) Diao, G.; Chu, L. T. J. Phys. Chem. A 2005, 109, 1364. (8) Miyabe, K.; Guiochan, G. J. Phys. Chem. B 2005, 109, 12038. (9) Schultz, D.; Nitschke, J. R. J. Am. Chem. Soc. 2006, 128, 9887. (10) Khakhel’, O. A. Chem. Phys. Lett. 2006, 421, 464. (11) Leontovich, M. A. VVedenie V Termodinamiku. Statisticheskaya Fizika [Introduction in Thermodynamics. Statistical Physics, in Russian]; Nauka: Moscow, 1983. (12) Krikunova, V. E.; Serov, S. A.; Khakhel, O. A. Opt. Spektrosk. 1999, 86, 373.
