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Extrathermodynamics: Varieties of Compensation Effect Oleg A. Khakhel', and Tamila P. Romashko J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b00493 • Publication Date (Web): 07 Mar 2016 Downloaded from http://pubs.acs.org on March 9, 2016

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The Journal of Physical Chemistry

Extrathermodynamics: Varieties of Compensation Effect Oleg A. Khakhel’ and Tamila P. Romashko* Poltava Department of Ukrainian Academy of Sciences of Technological Cybernetics, Poltava, 36009, Ukraine

Received:

There are several types of the H compensation. Along with well-known phenomenon of the H-S compensation, two more types of the H-(S+Rln) compensation are observed in some series of systems. The nature of these phenomena is connected with the behavior of phase volume of systems, . The role of other thermodynamic parameters, which describe series, in manifestation of this or that types of the H compensation is shown in light of molecular statistical mechanics.

Introduction The most known phenomenon of compensation for enthalpy, H, is enthalpyentropy, S, compensation.1 It consists in that that in a series of some reactions (i.e., in different systems), the linear correlation between values of H and S is observed, H/S=Ti. Here, Ti is temperature, the parameter characterizing the H–S linear correlation. This phenomenon is a manifestation of regularity, which is also known as isokinetic effect, linear free energy relationship etc.1 Since H and S are defined in thermodynamics as a magnitude that are independent of each other and their correlation is observed for different systems, these manifestations are called extrathermodynamic relationships. According to conclusions of authors of the work,1 which is dedicated to the critical analysis of existing interpretations of extrathermodynamic relationships, their nature is unknown. Meanwhile, in the work2 it was shown that nature of the H–S

*

To whom correspondence should be addressed. E-mail: [email protected] ACS Paragon Plus Environment

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compensation can be associated with the behavior of the phase volume of systems, , namely, the H–S compensation is realized in the case if only, in a series of systems, the value of  is constant. However, the H–S compensation is not single type of extrathermodynamic dependencies for H. As it has been shown in the work,3 the H–(S+Rln) compensation take place in the case of variable phase volume in a series of systems, i.e., H/(S+Rln)=Ti. Here, R is the universal gas constant. The character of H compensation (H–S or H–(S+Rln)) is determined by the type of extrathermodynamic potential characterizing a series of systems. At present, there are two types of extrathermodynamic potentials. The first holds for a series of systems that show the H–S compensation.2 The second holds for a series of systems that show the H–(S+Rln) compensation.3 In present paper, we want to report the existence of third type of extrathermodynamic potential.

Background In general, the thermodynamic potential, , is defined by the Gibbs distribution,  H qi , p i      exp  dqi dp i ,    exp  RT   RT  

(1)

where qi are coordinates of the system, pi are corresponding momenta, H(qi, pi) is the Hamilton function of the system. If the system is considered at constant pressure, i.e., one of qi coordinates is spatial volume, the thermodynamic potential represents the Gibbs energy, =H-TS.    E     exp  E dE ,  RT   RT 

Eq 1 can be presented in the equivalent form, exp 

where  is the phase volume of the layer lying between energy surfaces of H(qi, pi)=E and H(qi, pi)=E+dE. The integral,  E dE , is equal to the total phase volume of the system, . If we consider any series of systems, values of  in different systems can be both identical and different. ACS Paragon Plus Environment

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The interpretation of the H–S compensation, which was proposed in the work,2 is based on two obvious points. The first, comparing different systems in a series, it is necessary to take into account the behavior of their standard free energies, I,  I     Tk ln    E  E    exp  exp  dE .    exp   RT    RT    RT 

(2)

The second is that that if any two system’s states, 1 and 2, at some temperature Ti have equal values of the standard free energy,  I1=I2, they will be equal between themselves in other systems, too, 12І  1I  2I  12  Ti R ln 12  0 .

(3)

H 12  Ti S12  Ti R ln  12 ,

(4)

ln12 is ln(. Hence,

and, since ln12 is a constant, H12/S12=Ti. In the case where ln12 is not a constant, eq 4 is no longer correct. In this case, however, there is other extrathermodynamic dependency. So, as it has been shown in the work,3 the magnitude of 1/ can be considered as the coordinate, which relates systems in a series. Then, the thermodynamic potential,  II, is defined from the following equation,  RT ln   p 2 2 M   II   E       exp    exp   E exp  RT  RT   RT  

 1 dEd  dp  .  

(5)

Here, p is the moment corresponding to the coordinate of 1/; RTln and p2/2М are corresponding potential and kinetic energies, respectively; M is some constant characterizing a state. The magnitude (Е)/ does not depend on  and we obtain   II  1  E   E     exp  exp  dE 2MRT .    RT    RT 

Considering again some states, 1 and 2, we have 12II  1II  2II  12  Ti R ln 12  Ti R ln M 12  0 .

(6)

H 12  Ti S12  R ln  12   Ti R ln M 12 .

(7)

From here,

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Since lnМ12 is a constant, H12/(S12+Rln12)=Ti. As is seems, third type of the H compensation exists, too, that will be discussed further. Now, consider examples.

Examples As examples, let us consider the data from the work,4 which have already been analyzed in the works.2,3 These data represent a large array of experimentally determined H and S values characterizing a formation of pyrene excimers. Excimer is 

electronically excited physical dimer formed by the kinetic scheme, M  M *  D * . Here, M is a chromophore in the ground state, M* is an electronically excited chromophore, D* is an excimer. The data for pyrene excimers are listed in Table 1 and plotted in Figure 1. These data are the data for the intramolecular excimers formation with a few bichromophores dissolved in a number of solvents. In addition, in the case of 1,3-Di(1-pyrenyl)propane, the data for its two intramolecular excimers, D*1 and D*2, formed by the kinetic scheme, 



D1*  M  M *  D *2 , are presented. Here, examples of the H–S compensation can be found

only in series of the H and S values for the same bichromophore in different solvents. Straight lines illustrating this correlation are plotted in Figure 1 and corresponding results of the correlation analysis are shown in Table 1. Indeed, in the work2 it was shown that the phase volume of excimer can be associated with a spatial volume of interchromophoric interaction. The structure of bichromophore determines the individual steric hindrance thereby forming unique volume of interchromophoric interaction for each bichromophore. Obviously, a solvent insignificantly influences on the spatial volume of interchromophoric interaction, i.e., the condition of ln=const will be realized in this case and, hence, the H–S compensation will be realized, too.


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Taking into account eq 4, lines plotted in the H–S coordinates allow calculate values of Rln. These values for different bichromophores are listed in Table 1 and pointed in Figure 2. It should be emphasized that in Figure 2, the data for 1,3-di(1pyrenyl)propane is not represented. As can be seen from Figure 2, the array of data for pyrene excimers is well fitted by the averaged straight line in the coordinates of H–(S+Rln). From eq 7, it is possible to determine the value of RlnM, which in our case is equal to 64.7 J/(K mol). Thus, the distribution defined by eq 5 specifies the H–(S+Rln) compensation. In the H-(S+Rln) coordinates, the linear correlation take place for all pyrene excimers formed in different solvents and bichromophores. Energy parameters for the excimeric states of D*1 and D*2 in 1,3-di(1-pyrenyl)propane, apparently, also must obey to the dependence described by eq 7. However, it is not so that can be seen from Figure 3, which shows data for the D*1 and D*2 excimers in seven solvents. In each solvent, we have formally the series of two systems with different values of , but these series are characterized by different slopes in the coordinates of H–(S+Rln). The dissonance of these data with data for other pyrene excimers is especially noticeable when we compare them with the straight line in Figure 2 (see Figure 4). The observed fact needs an explanation. Our explanation is resulted below.

Explanation In the case of nonconstant value of , for an ordering of magnitudes of H, S, and , it was proposed3 to involve additional coordinate, 1/, that defined new thermodynamic potential related with it, II, and new type of the H compensation, H–(S+Rln). However, the coordinate of 1/ is not an ordinary coordinate of a system. In particular, it fails the theorem on the uniform distribution of energy on degrees of freedom. Thus, in the case of spatial coordinates this theorem can be written


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in the form,5 qi

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H H   H     qi e dq1 ...dq n dp1 ...dp n   , where Θ=kT and k is the q i q i

Boltzmann constant. We can represent the thermodynamic potential of  II as  H q i , pi    ln   p 2 2M   II    exp     exp      

 1 dqi dp i d  dp  . As can be seen from here,   

the Hamilton function is H  H qi , pi    ln   p 2 2 M . From this, we obtain 1 H 1 H  e  1     1    ln 1   1       1   e

 H 

H 

1  ln  1 dq i dpi d   dp   e  1   

1 1 dqi dp i d  dp      e  

 H 

 H 

1 dqi dp i d  dp   

1 dqi dp i d  dp   . 

In order to underline the difference of the 1/ coordinate from spatial coordinates, it can be named as extracoordinate. In this regard, we propose to name the thermodynamic potentials of  I and II as the extrathermodynamic potentials (these terms we have already announced above). Also, the magnitude of S+Rln can be named as extraentropy. Further, as is easy to see, in contrast to the series of systems, data for which are pointed in Figure 2 and which are characterized by the single extracoordinate of 1/, the D*1 and D*2 excimers formed in the 1,3-di(1-pyrenyl)propane are characterized by two extracoordinates. Really, in addition to its own variable of 1/, these excimers have one more extracoordinate associated with their summary phase volume, 1/. In other words, extrasystem has a degree of freedom on the coordinate of 1/. Moreover, at each point of the 1/ coordinate, the phase volume of extrasystem has an opportunity to vary on the coordinate of 1/. In this case, the Hamilton function of extrasystem, 2 apparently, has the following form, H  H q i , pi   RT ln   p 2 2M  p  2 M  . Here,

denotations are the same as in eq 4. p2/2M and p2/2M are kinetic energy related to coordinates of 1/ and 1/, respectively; RTln is the potential energy related to the coordinate of 1/. The potential energy related to the coordinate of 1/ is accepted equal to zero. As a result, we have ACS Paragon Plus Environment

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  III    exp   RT  2  p 2 2 M  p  2M  1  E    exp  RT E   exp  RT

  1 1 dEd  dp  d     

(8)

 dp  . 

The magnitude of (Е)/ does not depend on  and , hence, eq 8 can be transformed,   III    exp  RT    E  E  1  1  exp  RT   dE  d    d   

 p 2 2M    exp   RT  

  p 2 2M  dp   exp   RT  

  dp  . 

In this case, we have two extracoordinates, therefore, the standard free energy is   III  1  E   E     exp  exp  dE 2MRT 2M  RT    RT     RT 

or  III    TR ln   TR ln M  TR ln M  . Again, considering two states of this extrasystem (in the case of excimers formation, the dimeric and monomeric states), we can write down,  III    TR ln   TR ln M  TR ln M   0 . At present, unfortunately, the physical

meaning of the M and M parameters is unclear. We can suppose that in our example with excimers these magnitudes are characteristic of excited pyrene only, so that M=M. Hence,  III    Ti R ln   Ti R ln M  0

(9)

and H  Ti S  R ln    Ti R ln M .

(10)

The difference between eqs 10 and 7 is only that in eq 10 the last term is twice more than the same in eq 7. The Ti and RlnM values calculated for lines in Figure 3 according to eq 10 are listed in Table 2. For an unclear reason, the data for the excimers of D*1 and D*2 in liquid paraffin significantly differ from the others and we will not take them into account. Averaging the values of lnM for six solvents, we obtain R ln M  126.7 J/(K mol). This ACS Paragon Plus Environment

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number is twice more than the RlnM value calculated for the line in Figure 2 that, in our view, confirms the correctness of the above reasoning.

Discussion Extrathermodynamic dependencies are known in many different fields. Usually, their interpretations not go beyond concrete series of reactions and are associated with specific moments for these series. The approach used in this work is not restricted only by excimers, examples with which have allowed us to show various types of the H compensation. We do not see the reasons why this approach can not be applied in other fields, though some questions remain unclear. First of all, it concerns the parameter of M (see eqs 5-10), the physical meaning of which remains to be seen. It is clear, however, that this parameter defines series of systems. I.e., closely related reactions are characterized by the same value of M. In a modern extrathermodynamic, this magnitude characterizes the highest level in the hierarchy of "closely related" systems. On the lower level of "close relation", a series of systems is defined by the phase volume. In the case where the phase volume is variable, it acts as extracoordinate As shown in the present work, there are series of systems that are characterized by two extracoordinates. In this case, the Gibbs distribution can be described by eq 8 and the corresponding extrathermodynamic potential of  III has the appearance, as it is shown in eq 9. This potential defines the H compensation in the form of eq 10. In the case of single extracoordinate, the Gibbs distribution for extrasystem is given by eq 5, extrathermodynamic potential of  II is expressed as eq 6 and the H compensation is defined by the dependence in the form of eq 7. The simplest case of extrathermodynamics is the case where systems in a series have the same value of phase volumes. Then, the Gibbs distribution for this extrasystem (see eq 2) defines extrathermodynamic potential  I in the form of eq 3. Hence, eq 4 describes well-known phenomenon of the H-S compensation or, in its more ACS Paragon Plus Environment

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traditional form,1 H=+S, where  and  are constants. At the present time, except for this work and our previous publication, 3 all literature concerning extrathermodynamics is devoted to this simplest case. The recent state of knowledge in field of the H-S compensation is expounded in the review.1 Since then, several works devoted to ascertaining of the origin of the compensation have been published (see, for instant, refs 6-11). However, in our opinion, significant progress in this question has not occurred, mainly, because formal (phenomenological) thermodynamics, unlike statistical physics, does not know such concept as volume of phase space of system. And the nature of the compensation effect is precisely related to this parameter. In statistical physics, a molecular structure of systems under study is explicitly introduced into consideration.5 Different systems differ in their structure, nature of occurring processes, etc.; therefore, in each case, an individual molecular statistical mechanical analysis is required. Such analysis carried out in relation to excimers showed2 that their phase volume is associated with a spatial area of intermolecular interaction that we mentioned above, and, in the case of constant value of phase volume, yielded the result for the H-S compensation in the form of eq 4. The analysis for a series of systems with variable value of phase volume leads3 us to the description of the compensation effect by eq 7. Series of systems studied in this work have their own 



specificity conditioned by kinetic scheme, D1*  M  M *  D *2 , that yields the special form of the compensation (see eq 10). Our conclusion about relation between nature of extrathermodynamic dependencies and behavior of the space surrounding investigated species (in more general formulation – behavior of their phase volume) is not breakthrough. Previously, in various ways (analysis of experiments, consideration of theoretical models, intuitively), many researchers arrived at a conclusion about the role of spatial area of interaction. For example, in the case of chemical reactions, the phase volume of systems could be related to the reactionary region and, in one's time, Hammett has emphasized12 ACS Paragon Plus Environment

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the importance of this region that we earlier discussed3 in relation to the reactions described by equations of Hammett and Taft. Authors of the work6 concluded that changes in properties of a solvate shell of species yield the legitimate H-S compensation effect. Concerning the data presented in Figure 1, the authors4 believed that two correlation lines are revealed there and these straight lines are related to structures of excimers. The last example shows that a subjective view on the array of H-S data can be deceptive. Apparently, in each case, it is necessary to work out an appropriate model of extrasystem. Next step is a determination of the integral of states and other properties of extrasystem.

Conclusions Thus, in different series of systems (in different extrasystems), the following types of compensation can be reveal, H  Ti S  Ti R ln    S   H  Ti S  R ln    Ti R ln M H  Ti S  R ln    Ti R ln M

These three types of compensation effect received analytical argumentation and have experimental confirmation.

References (1) Liu L.; Guo Q.-X. Isokinetic Relationship, Isoequilibrium Relationship, and Enthalpy-Entropy Compensation. Chem. Rev. 2001, 101, 673-695. DOI: 10.1021/cr990416z (2) Khakhel’ O.A. Linear Free-Energy Relationship. Chem. Phys. Lett. 2006, 421, 464468. DOI: 10.1016/j.cplett.2006.01.084


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(3) Khakhel’ O.A.; Romashko T.P.; Sakhno Yu.E. One More Type of Extrathermodynamic Relationship. J. Phys. Chem. B 2007, 111, 7331-7335. DOI: 10.1021/jp0725275 (4) Zachariasse K.A.; Duveneck G. Linear Free Energy Relationships for Excimers. J. Am. Chem. Soc. 1987, 109, 3790-3792. DOI: 10.1021/ja00246a053 (5) Leontovich M.A. Vvedenie v Termodinamiku. Statisticheskaya Fizika [Introduction in Thermodynamics. Statistical Physics, in Russian]; Nauka: Moscow, 1983. (6) Liu L.; Guo Q.-X. A General Theoretical Model of Enthalpy-Entropy Compensation. Chin. J. Chem. 2001, 19, 670-674. DOI: 10.1002/cjoc.20010190708 (7) Kirk W.R. Entropy-Enthalpy Compensation Behavior Revisited. J. Theor. Comput. Chem. 2004, 03, 511-520. DOI: 10.1142/S0219633604001161 (8) Starikov E.B.; Nordén B. Enthalpy−Entropy Compensation: A Phantom or Something Useful? J. Phys. Chem. B 2007, 111, 14431-14435. DOI: 10.1021/jp075784i (9) Freed, K. F. Entropy−Enthalpy Compensation in Chemical Reactions and Adsorption: An Exactly Solvable Model. J. Phys. Chem. B 2011, 115, 1689–1692. DOI: 10.1021/jp1105696 (10) Ryde U. A Fundamental View of Enthalpy–Entropy Compensation. Med. Chem. Commun. 2014, 5, 1324-1336. DOI: 10.1039/c4md00057a (11) Pan A.; Biswas T.; Rakshit A.K.; Satya P.; Moulik S.P. Enthalpy–Entropy Compensation (EEC) Effect: A Revisit. J. Phys. Chem. B 2015, 119, 15876–15884. DOI: 10.1021/acs.jpcb.5b09925 (12) Hammett L.P. Physical Organic Chemistry; McGraw-Hill: New-York, 1970.


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Page 12 of 17

Figure 1. Examples of the H-S compensation for pyrene excimers formation.

Figure 2. Example of the H-(S+Rln) compensation for pyrene excimers formation.


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Page 13 of 17

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Figure 3. Other examples of the H-(S+Rln) compensation for pyrene excimers formation.

Figure 4. Two types of the H-(S+Rln) compensation for pyrene excimers formation.


13


Table 1. Experimental values* of H and S for pyrene excimers and the results of the

Compound

analysis of their linear correlation.

1,3-di-(2-pyrenyl)propane

Analysis results: H=TiS-TiRln S+Rln, Ti, K J/(K mol) Rln, J/K mol

H, kJ

S, J/(K mol)

3.1

42.9

-1.9

26.7

-18.5

-24.1

-22.0

-32.2

-19.7

-25.2

-3.98

n-pentane

-23.9

-39.8

-11.88

n-hexane

-25.4

-44.1

-16.18

n-heptane

-25.9

-46.0

-18.08

n-hexadecane

-25.9

-46.0

-18.08

n-nonane

-26.7

-48.0

-20.08

n-decane

-26.7

-48.0

n-dodecane

-27.2

-49.2

n-octane

-26.4

-45.8

-17.88

liquid paraffin

-24.3

-40.1

-12.18

toluene

-27.5

-52.2

-24.28

methylcyclohexane n-pentane

-20.4

-30.9

-2.98

-34.1

-85.6

n-hexane

-39.7

-105.3

n-heptane

-40.5

-109.1

-66.64

n-decane

-40.9

-110.2

-67.74

Solvent

meso-2,4- methylcyclodi(2-pyre- hexane nyl)penta toluene ne methylcyclorac-2,4hexane di(2-pyretoluene nyl)penta ne n-octane

1,16-di(1pyrenyl)hexad ecane

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 17


Ti=308.6 Rln=32.86

75.76 59.56 -2.88

Ti=414.8 Rln=21.22

Ti=351.6 Rln=27.92

Ti=265.9 Rln=42.46

-10.98

-20.08 -21.28

-43.14 -62.84

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Page 15 of 17

n-hexadecane

-40.0

-107.4

-64.94

liquid paraffin

-38.7

-102.4

-59.94

toluene

-40.0

-112.3

-69.84

methylcyclohexane

-40.3

-108.5

-66.04

1,3-di-(1-pyrenyl)propane (D*1)

H=Ti(S+Rln)-TiRlnM Ti=308.16 K RlnM=64.7 J/(K mol)

1,3-di-(1-pyrenyl)propane (D*2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


*

n-hexane

-17.5

-21.3

4.59

n-heptane

-15.1

-14.0

11.89

n-decane

-15.5

-15.0

10.89

n-dodecane n-hexadecane

-16.7 -17.0

-19.0 -21.1

toluene

-17.5

-21.3

4.59

liquid paraffin

-24.0

-38.2

-12.93

n-hexane

-22.4

-51.4

-42.93

n-heptane

-23.8

-55.4

-46.93

n-decane

-24.7

-56.2

-47.73

n-dodecane n-hexadecane

-26.7 -26.0

-63.8 -60.8

toluene

-25.0

-61.5

-53.03

liquid paraffin

-27.1

-63.5

-55.03

Ti=372.4 Rln=25.89

Ti=372.3 Rln=8.47

6.89 4.79

-55.33 -52.33

Data from the ref. 4.


15


Table 2. Values H and S+Rln for pyrene excimers of D*1 and D*2 formed in

n-heptane

n-decane

n-dodecane

n-hexadecane

-17.5

-15.1

-15.5

-16.7

-17.0

-17.5

-24.0

S+Rln, J/(K mol)

4.59

11.89

10.89

6.89

4.79

4.59

-12.93

H, kJ

-22.4

-23.8

-24.7

-26.7

-26.0

-25.0

-27.1

S+Rln, J/(K mol)

-42.93

-46.93

-47.73

-55.33

-52.33

-53.03

-55.03

Ti, K

103.11

147.91

156.94

160.72

157.56

130.16

73.63

RlnM, J/(K mol)

174.3

114.0

110.0

110.8

112.7

139.0

313.0

R ln M ,

J/(K mol)

126.7


toluene

n-hexane

H, kJ

Solvent D *1 D *2

liquid paraffin

1,3-Di(1-pyrenyl)propane and the results of the analysis of their linear correlation.

Analysis results: H=Ti(S+Rln)-TiRlnM

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 17

-

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TOC graphic

H  T S  T R ln    S   i i H T S  R ln T R ln M i i H  T  S  R ln    T R ln M i i


17

Extrathermodynamics: Varieties of Compensation Effect - The Journal

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