On the partitioning of benzene between water and n-alkanes

Chemical Physics Letters 486 (2010) 44–47

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On the partitioning of benzene between water and n-alkanes Giuseppe Graziano * Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy

a r t i c l e

i n f o

Article history: Received 23 November 2009 In final form 29 December 2009 Available online 4 January 2010

a b s t r a c t The magnitude of the work of cavity creation increases with the n-alkane chain-length due to the volume packing density increase, in line with expectations based on correlations with surface tension, cohesive energy density, and the inverse of isothermal compressibility. Also the magnitude of the Gibbs energy gain to turn on benzene–alkane attractive interactions increases with the n-alkane chain-length, but to a lesser extent than the work of cavity creation, thus benzene solubility, under Ben-Naim standard conditions, slightly decreases on lengthening the alkyl chain, in line with experimental data. It is unjustified to apply a Flory–Huggins correction to the Ben-Naim standard Gibbs energy of transfer. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction In 1991 Honig and colleagues strongly claimed that [1,2]: (a) the common estimates of the hydrophobic effect (i.e., transfer of a nonpolar molecule from an organic liquid into water) were not quantitatively correct because it was necessary to account for the difference in size between the solute and solvent molecules; (b) the correction term was rightly provided by the Flory–Huggins theory of polymer solutions [3]. Among the arguments advanced to support their claim, Honig and colleagues used the results obtained by De Young and Dill [4], by measuring the partitioning of benzene between water and n-alkanes of 8–16 carbon atoms over the 5–55 °C temperature range. Since the principal conclusions of De Young and Dill proved to be largely independent of temperature, my analysis is restricted to data at 25 °C. De Young and Dill expressed the benzene partition coefficient in terms of molar fraction Kx, molar concentration Kc, and Flory– Huggins corrected molar concentration Kc,FH [4]. The Gibbs energy changes corresponding to Kx and Kc, for the water ? n-alkane transfer, are given by:

DG ¼ RT  ln K x ¼ RT  ln½xB ðn-alkaneÞ=xB ðwaterÞ DG ¼ RT  ln K c ¼ RT  ln½C B ðn-alkaneÞ=C B ðwaterÞ

ð1Þ ð2Þ

where xB and CB are the molar fraction and the molar concentration of benzene, respectively. The Gibbs energy of transfer expressed in terms of molar concentration, DG, corresponds to the so-called Ben-Naim standard [5,6], and the relationship between the two Gibbs energy quantities is:

DG ¼ DG þ RT  ln½v ðn-alkaneÞ=v ðwaterÞ * Fax: +39 0824 23013. E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.12.086

ð3Þ

where v(n-alkane) and v(water) are the molar volume of n-alkane and water (i.e., the latter is 18.07 cm3 mol1 at 25 °C and 1 atm [7]), respectively; DG describes the transfer of a solute molecule from a fixed position in water to a fixed position in n-alkane, at constant temperature and pressure [5,6]. Adopting the Ben-Naim standard, the difference in the volume available per solute molecule in the two liquid phases (due to the difference in molar volume) is separated out, and the corresponding contribution to the transfer Gibbs energy is eliminated [5,6]. The relationship to pass from the Ben-Naim standard Gibbs energy of transfer to the Flory–Huggins corrected values used by De Young and Dill is the following:

DGFH ¼ DG þ RT  f½v B =v ðn-alkaneÞ  ½v B =v ðwaterÞg

ð4Þ

where vB is the molar volume of benzene, 89.41 cm3 mol1 at 25 °C and 1 atm [8]. It is clear that the Flory–Huggins correction term is large negative for the systems investigated by De Young and Dill because vB < v(n-alkane) and vB > v(water). It amounts to 10.9 kJ mol1 in the case of water ? n-octane transfer, and to 11.5 kJ mol1in the case of water ? n-hexadecane transfer. The De Young and Dill results were the following: (a) both Kx and Kc depend on the chain-length of the n-alkane (thus, also the DG and DG values depend on the chain-length of the n-alkane); (b) the Flory–Huggins corrected molar concentration partition coefficients Kc,FH and the DGFH estimates are independent of n-alkane chainlength. All the relevant numbers at 25 °C, extracted from Fig. 9 of the De Young and Dill article are listed in Table 1. For instance: (a) DG (in kJ mol1 units) = 18.2 for the water ? n-octane transfer, 18.6 for the water ? n-dodecane transfer, and 19.0 for the water ? n-hexadecane transfer; (b) DG (in kJ mol1 units) = 12.7 for the water ? n-octane transfer, 12.3 for the water ? n-dodecane transfer, and 12.1 for the water ? n-hexadecane transfer;

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G. Graziano / Chemical Physics Letters 486 (2010) 44–47 Table 1 Partition coefficients of benzene between water and n-alkanes, at 25 °C, expressed in terms of molar fraction Kx, molar concentration Kc, and Flory–Huggins corrected molar concentration Kc,FH, as determined by De Young and Dill [4]; the corresponding transfer Gibbs energy changes are listed in the last three columns.

H2O ? nC8H18 H2O ? nC10H22 H2O ? nC12H26 H2O ? nC14H30 H2O ? nC16H34

ln Kx

ln Kc

ln Kc,FH

DG (kJ mol1)

DG  (kJ mol1)

DGFH (kJ mol1)

7.35 7.43 7.50 7.60 7.66

5.14 5.05 4.95 4.93 4.88

9.54 9.54 9.51 9.53 9.52

18.2 18.4 18.6 18.8 19.0

12.7 12.5 12.3 12.2 12.1

23.6 23.6 23.6 23.6 23.6

(c) DGFH = 23.6 kJ mol1 for all the investigated water ? n-alkane transfer processes. On this basis De Young and Dill wrote [4]: ‘An independence of the partition coefficient on the alkane chain-length is anticipated since the density of the n-alkanes is dependent simply on the van der Waals volumes of the CH2 and CH3 groups. It is reasonable, therefore, to expect that the interaction energetics are also approximately independent of chain-length and that the dominant effect is that of molecular size, taken into account by the Flory–Huggins correction.’ In other words, De Young and Dill assumed that the independence of the transfer Gibbs energy of the chain-length should be the fundamental argument to establish the rightness of the calculated values because such an independence would reflect the essentially identical chemical environments provided by n-octane or n-hexadecane for a benzene molecule. In fact, they stated [4]: ‘Our principal purpose is to choose a standard state wherein the dependence of partitioning on the hydrocarbon chain-length is taken directly into account so that contact interactions (van der Waals, hydrophobic contributions, etc.) may be approximately separated from the effects of the chain configurational entropy.’ On this ground the authors concluded that the Flory–Huggins theory satisfactorily corrects for the entropic contributions due to molecular size differences in solute and solvent molecules, providing the right value of the Gibbs energy of transfer between water and n-alkane of a nonpolar unit such as a CH2 group (i.e., the right measure of the energetics of the hydrophobic effect). In order to make a real progress in understanding, it is necessary to find a transfer Gibbs energy that possesses a well-defined physical meaning. Ben-Naim demonstrated that, if the translational degrees of freedom can be treated classically, DG takes into account all the relevant changes in coupling work between the fixed solute molecule and the surrounding solvent molecules associated with the transfer process [6]. There is no need to correct the experimental DG values by a term that accounts for the difference in size between the solute and solvent molecules [9]. I have confirmed the validity of the arguments by Ben-Naim in the case of fluids obeying the van der Waals equation of state [10]. The DG values by De Young and Dill (see Table 1) indicate that the benzene transfer is slightly less favoured on increasing the chain-length of n-alkanes; this finding calls for a physico-chemical explanation that cannot be provided by the Flory–Huggins theory. In fact, even though the Flory–Huggins correction eliminates the chain-length dependence of the partition coefficient for benzene between water and n-alkanes, it does not provide any physical insight on the molecular mechanism of the hydrophobic effect and its features (i.e., its entropic nature at room temperature and the large and positive heat capacity change), because the Flory–Huggins theory is a lattice approach grounded on some unrealistic assumptions and approximations [11,12]. In this Letter, I would like to provide an analysis of benzene partitioning between water and n-alkanes that emphasizes how: (a) both the Gibbs energy cost of cavity creation and the Gibbs energy

gain due to benzene–alkane attractive interactions depend significantly on the chain-length of the n-alkane molecule; (b) both these Gibbs energy contributions account for the difference in size between the solute and solvent molecules, disproving the need to correct the DG values. 2. Calculation procedure According to a well-founded statistical mechanical theory of solvation [13–15], DG for the gas-to-liquid transfer is given by:

DG ðg ! lÞ ¼ DGc þ DGa

ð5Þ

where DGc is the Gibbs energy cost to create in the liquid phase a cavity suitable to host the solute molecule, and DGa is the Gibbs energy gain to turn on the attractive interactions between the solute molecule inserted in the cavity and the surrounding solvent molecules. For the transfer of benzene from water to n-alkane one has:

DG ðwater ! alkaneÞ ¼ DG ðg ! alkaneÞ  DG ðg ! waterÞ 

1

where DG (g ? water) = 3.6 kJ mol so Eq. (6) becomes:

ð6Þ

at 25 °C and 1 atm [16], and

DG ðwater ! alkaneÞ ¼ DGc ðalkaneÞ þ DGa ðbenzene—alkaneÞ þ 3:6

ð7Þ

Therefore, to shed light on benzene partitioning, it is necessary to obtain estimates for the reversible work necessary to create a cavity suitable to host a benzene molecule in the n-alkane liquid phase and for the Gibbs energy gain associated with turning on the attractive interactions between the benzene molecule inserted in the cavity and the surrounding n-alkane molecules. In this respect, it is worth noting that: (a) Jain and Ashbaugh calculated DGc in n-alkanes of 6–12 carbons by means of molecular dynamics, MD, simulations in the NPT ensemble at 300 K and 1 atm [17]; (b) they showed that a molecular-detailed version of scaled particle theory, MSPT, developed by Ashbaugh and Pratt [18] to account for the intra-molecular structure of hydrocarbon molecules, accurately reproduces the DGc values obtained by means of MD simulations. According to MSPT [18], the van der Waals volume of a n-alkane molecule, in a first approximation, is given by:

v vdW ðn-alkaneÞ ¼ ½4pðrCC =2Þ3  nC =3  ½4pðnC  1Þ=3 3

 ½ðrCC =2Þ3  ð3l=4Þ  ðrCC =2Þ2 þ ðl =16Þ

ð8Þ

where the first term accounts for the contribution of the nC carbon units (i.e., CH2 and CH3 groups) whose diameter is rCC, while the second term accounts for the volume reduction due to the (nC  1) overlaps between bonded carbon atoms, l being the bond-length [19]. The DGc expression derived by Ashbaugh and Pratt is the following (the pressure–volume contribution is neglected for its smallness when P = 1 atm):

DGc ¼ RT  f lnð1  gÞ þ 3B  n  ½ð2Rcav =rCC Þ  1 þ ð3=2ÞA  n  ½ð2Rcav =rCC Þ2  1g

ð9Þ

where g is the volume packing density of the liquid, given by g = (vvdW  NAv/v), where vvdW is the van der Waals volume per molecule, NAv is the Avogadro’s number, and v is the molar volume of the liquid; n is the volume packing density expressed in terms of carbon units, given by n = (p  nC  rCC3  NAv/6v); and Rcav = (rsolute + rCC)/2. The A and B parameters are derived by imposing the continuity of the cavity contact correlation function and its first and second derivatives for Rcav = rCC/2; their mathematical expressions, that depend on g and n, are involved and the readers are referred to the original article of Ashbaugh and Pratt [18]. I have used the expression in Eq. (9) to calculate the DGc values for creating a cavity suitable to host a benzene molecule in

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G. Graziano / Chemical Physics Letters 486 (2010) 44–47

Table 2 Thermodynamic properties, at 25 °C and 1 atm, of n-alkanes: molar volume, thermal expansion coefficient, isothermal compressibility, boiling temperature, vaporization enthalpy, cohesive energy density, c.e.d. = (DHvap  RT)/v, bulk surface tension, and Egelstaff–Widom length. Data come from Refs. [8,28].

nC8H18 nC10H22 nC12H26 nC14H30 nC16H34

v (cm3 mol1)

a  103

bT  1011 (cm2 dyne)

Tb (°C)

DHvap (kJ mol1)

c.e.d. (J cm3)

c

(K1)

(dyne cm1)

bT  c (Å)

163.54 195.94 228.60 261.31 (294.04)

1.16 1.02 0.93 0.87 (0.83)

12.82 10.94 9.88 9.10 8.57

125.6 174.1 216.3 253.5 286.8

41.5 51.4 61.5 71.3 81.4

239 250 258 263 268

21.14 23.37 (24.88) 26.13 27.05

0.271 0.256 0.246 0.238 0.232

Note: the numbers in parentheses have been obtained by interpolation–extrapolation.

n-alkanes of 8–16 carbons at 25 °C and 1 atm. To perform MSPT calculations, I have used: (a) the experimental density of n-alkanes at 25 °C and 1 atm, listed in the second column of Table 2 in terms of molar volumes [8]; (b) a hard-sphere diameter for the CH2 and CH3 groups rCC = 3.8 Å, that is a customary value for such groups in computer simulations [17,20], and a carbon–carbon bond-length l = 1.53 Å [18]; (c) a hard-sphere diameter for benzene rB = 5.26 Å [8]. The DGa values have been obtained in a hybrid manner, by inserting the calculated DGc numbers in Eq. (7). In addition, they have been calculated by means of the Pierotti’s relationship [21]:

DGa ¼ ð64=3Þ  g  e  ðRcav =rCC Þ3

ð10Þ

1/2

where e = [e(benzene)  e(CH2)] , e(benzene)/k = 531 K [8], and e(CH2)/k = 190 K are the Lennard-Jones potential parameters of the benzene molecule and of the carbon unit constituting the n-alkane molecules, respectively. Notwithstanding the several assumptions (i.e., both the size and energy parameter of n-alkanes refer to the carbon unit, but the volume packing density refers to the whole molecule), this formula should be sound from a physico-chemical point of view [14,22]. 3. Results The DGc(alkane) numbers, calculated by means of MSPT at 25 °C and 1 atm, are listed in the fourth column of Table 3. The DGc magnitude increases with the chain-length of the n-alkane molecule: DGc (in kJ mol1 units) = 34.1 in n-octane, 37.1 in n-dodecane, and 38.7 in n-hexadecane. This trend is readily explained by looking at the g values (see the second column of Table 3) that show the same dependence on the chain-length of the n-alkane molecule: g = 0.529 for n-octane, 0.551 for n-dodecane, and 0.563 for n-hexadecane. An g increase indicates a decrease in unoccupied volume, rendering more costly the creation of a molecular-sized cavity in the liquid [23]. Since the n-alkane molecules are made up of the same carbon units, the g increase reflects an increase in the number density of carbon units. In fact, qC102 (in atoms per

Table 3 Values, at 25 °C and 1 atm, of the volume packing density of n-alkanes, their number density in carbon units, the Gibbs energy cost to create in n-alkanes a cavity suitable to host a benzene molecule, and the Gibbs energy gain to turn on the attractive interactions between benzene and surrounding n-alkane molecules. The DGa numbers labeled 1 are obtained by means of Eq. (7), while those labeled 2 are calculated by means of Eq. (10).

g nC8H18 nC10H22 nC12H26 nC14H30 nC16H34

0.529 0.542 0.551 0.558 0.563

qC  102 (groups Å3)

DG c (kJ mol1)

DGa (1) (kJ mol1)

DGa (2) (kJ mol1)

2.95 3.07 3.16 3.23 3.28

34.1 35.9 37.1 38.0 38.7

50.4 52.0 53.0 53.8 54.4

50.5 51.7 52.6 53.3 53.7

Å3) = 2.95 for n-octane, 3.16 for n-dodecane, and 3.28 for n-hexadecane (see the third column in Table 3). I have verified that a small rCC change produces a significant effect in the g values that, in turn, becomes a marked effect in the DGc numbers (data not shown). The sensitivity to the diameter value assigned to solvent molecules of the DGc numbers calculated by means of SPT is well known [24], and does not cause problems for the present analysis because the goal is to show that the DGc magnitude depends on the chain-length of the n-alkane molecule. Moreover, the g values obtained by calculating the van der Waals volume of n-alkane molecules by means of Eq. (8) are more reliable from a physicochemical point of view than those obtained by considering that a n-alkane molecule can be described as a single sphere [8]. It is important to note that Jain and Ashbaugh, by means of MD simulations, obtained the same result: the magnitude of the reversible work to create a spherical cavity of a given diameter in n-alkanes of 6–12 carbons increases with the chain-length of the n-alkane molecule [17]. In addition, it has been proposed and supported that DGc should be: (a) inversely proportional to the isothermal compressibility of the liquid, DGc / 1/bT [25]; (b) proportional to the bulk surface tension of the liquid, DGc / c [26]; (c) proportional to the cohesive energy density of the liquid, DGc / c.e.d. [27]. Accepting the validity of these correlations, and on the basis of the experimental data for bT, c, and c.e.d. reported in Table 2 [28], the expectation is that DGc should increase with the chain-length of the n-alkane molecule, exactly in line with MSPT calculations and MD results. This is an important result because it shows that, notwithstanding the essentially identical chemical environment provided by noctane and n-hexadecane, the DGc magnitude distinguishes the nalkane liquids, feeling the number density increase in carbon units on lengthening the n-alkane molecule. By inserting the DGc numbers in Eq. (7), one obtains the DGa values listed in the fifth column of Table 3. Such values are large negative, due to the strongly attractive energetic interactions existing between benzene and n-alkane molecules, and lead to the measured negative numbers for DG(water ? alkane). For instance, DGa (in kJ mol1 units) = 50.4 in n-octane, 53.0 in n-dodecane, and 54.4 in n-hexadecane. These values, since only van der Waals interactions are operative, may appear too large in magnitude; however, they are close to those calculated by means of Eq. (10), as readily verified by looking at the sixth column of Table 3 (note that Pierotti derived Eq. (10) in the assumption of pairwise additive interactions described by the Lennard-Jones 6–12 potential). To gain perspective, it is useful to look at the values of the vaporization enthalpy at 25 °C and 1 atm for these hydrocarbons, listed in the fifth column of Table 2 [28]. For instance, DHvap (in kJ mol1 units) = 41.5 for n-octane, 61.5 for n-dodecane, and 81.4 for nhexadecane, indicating that they are large and strongly dependent on chain-length. The DGa magnitude increases with the chainlength to a lesser extent than the DGc one, so that benzene solubility, under Ben-Naim standard conditions, slightly decreases on lengthening the alkyl chain (see the sixth column of Table 1).

G. Graziano / Chemical Physics Letters 486 (2010) 44–47

4. Discussion The numbers reported in Tables 2 and 3 indicate unequivocally that, at 25 °C and 1 atm, n-alkanes of 8–16 carbon atoms have different values of the volume packing density, thermal expansion coefficient, isothermal compressibility, boiling temperature, vaporization enthalpy, cohesive energy density, bulk surface tension, and Egelstaff–Widom length [29]. It is therefore expected that the benzene partitioning between water and n-alkanes depends on the chain-length. This was the result correctly determined by De Young and Dill [4], and measured by the Ben-Naim standard DG(water ? alkane) values, which contain all the information and have a precise physical meaning. Moreover, for all liquids, both the Gibbs energy terms in Eq. (5), DGc and DGa, depend on (a) g that, in turn, depends on the size of solvent molecules, and (b) the size ratio between solute and solvent molecules, as emphasized by Eqs. (9) and (10), respectively. As already demonstrated by several authors using different arguments [9–12,30–35], there is no physical reason to perform a correction of the Ben-Naim standard DG(water ? alkane) values by means of the Flory–Huggins term. In particular, Lee showed that the Flory–Huggins correction term is the mixing entropy in the assumption of an ideal gas reference state, that is not realistic nor useful for crowded liquid phases [35]. Real liquids consist of finite size molecules which give rise to an excluded volume effect that reduces the entropy of the system; in addition, real molecules strongly interact among each other, further reducing the liquid entropy. The ideal gas reference state is evident in the fact that the molar volume of the solute is present in Eq. (4), and not the values of its partial molar volume in the two solvents. The latter, in general, are different from the molar volume and between each other exactly because, in each solvent, the excluded volume effect and intermolecular attractive interactions are operative [36,37]. Therefore, the use of the Flory–Huggins correction term, that represents the mixing entropy of ideal gases, is unjustified. In a review article of 1997 [38], Chan and Dill reanalyzed the benzene partitioning between water and n-alkanes, and concluded that a statistical mechanical theory more rigorous than Flory–Huggins is required to treat the alkane polymers and benzene. However, Chan and Dill still claimed that ‘some partitioning processes, particularly involving polymeric solvents such as octanol or hexadecane, are governed not only by translational entropies and contact interactions, but also by free energies resulting from changes in the conformations of the polymer chains upon solute insertion’ [38]. The idea was to extract from the measured transfer Gibbs energies solely the contribution of contact interactions by subtracting both translational and conformational entropies with the purpose to use such contact contributions in computational approaches for protein folding and protein–protein recognition. This idea, however, should be considered not useful because globular proteins are heteropolymers for which the additivity principle does not hold [39,40]. The chain connectivity cannot be neglected, and the energetics of a given contact between a

47

pair of amino acid side-chains is not a general quantity, but it depends on the context in which it occurs, as well demonstrated by a lot of point mutation experiments [41]. In conclusion, MSPT calculations show that the DGc magnitude increases with the n-alkane chain-length due to the g increase, in line with expectations based on correlations with surface tension, cohesive energy density, and the inverse of isothermal compressibility. Also the DGa(benzene–alkane) magnitude increases with the n-alkane chain-length, but to a lesser extent than DGc. Therefore, benzene solubility in n-alkanes, under Ben-Naim standard conditions, slightly decreases on lengthening the alkyl chain, in line with experimental data by De Young and Dill. The idea to apply a Flory–Huggins correction to the Ben-Naim standard Gibbs energy of transfer is unjustified, because the latter already accounts for the difference in size between the solute and solvent molecules. References [1] K.A. Sharp, A. Nicholls, R. Friedman, B. Honig, Biochemistry 30 (1991) 9686. [2] K.A. Sharp, A. Nicholls, R.F. Fine, B. Honig, Science 252 (1991) 106. [3] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953. [4] L.R. De Young, K.A. Dill, J. Phys. Chem. 94 (1990) 801. [5] A. Ben-Naim, Y. Marcus, J. Chem. Phys. 81 (1984) 2016. [6] A. Ben-Naim, Solvation Thermodynamics, Plenum Press, New York, 1987. [7] G.S. Kell, J. Chem. Eng. Data 20 (1975) 97. [8] E. Wilhelm, R. Battino, J. Chem. Phys. 55 (1971) 4012. [9] A. Ben-Naim, Biophys. Chem. 51 (1994) 203. [10] G. Graziano, Thermochim. Acta 339 (2003) 181. [11] A. Holtzer, Biopolymers 32 (1992) 711. [12] A. Holtzer, Biopolymers 34 (1994) 315. [13] B. Lee, Biopolymers 31 (1991) 993. [14] G. Graziano, Can. J. Chem. 80 (2002) 401. [15] G. Graziano, Chem. Phys. Lett. 429 (2006) 114. [16] G. Graziano, B. Lee, J. Phys. Chem. B 105 (2001) 10367. [17] A. Jain, H.S. Ashbaugh, J. Chem. Phys. 129 (2008) 174505. [18] H.S. Ashbaugh, L.R. Pratt, J. Phys. Chem. B 111 (2007) 9330. [19] A. Ben-Naim, Water and Aqueous Solutions. An Introduction to a Molecular Theory, Plenum, New York, 1974. p. 48. [20] W.L. Jorgensen, J.D. Madura, C.J. Swenson, J. Am. Chem. Soc. 106 (1984) 6638. [21] R.A. Pierotti, Chem. Rev. 76 (1976) 717. [22] G. Graziano, Chem. Phys. Lett. 460 (2008) 470. [23] G. Graziano, J. Phys. Chem. B 106 (2002) 7713. [24] K.E.S. Tang, V.A. Bloomfield, Biophys. J. 79 (2000) 2222. [25] G. Hummer, S. Garde, A.E. Garcia, M.E. Paulaitis, L.R. Pratt, J. Phys. Chem. B 102 (1998) 10469. [26] H.S. Ashbaugh, L.R. Pratt, Rev. Mod. Phys. 78 (2006) 159. [27] M. Kodaka, J. Phys. Chem. B 108 (2004) 1160. [28] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 77th edn., CRC Press, Boca Raton, 1996. [29] P.A. Egelstaff, B. Widom, J. Chem. Phys. 53 (1970) 2667. [30] A. Ben-Naim, R. Mazo, J. Phys. Chem. 97 (1993) 10829. [31] A. Ben-Naim, R. Mazo, J. Phys. Chem. B 101 (1997) 11221. [32] K. Soda, N. Tsuruta, J. Phys. Soc. Jpn. 63 (1994) 814. [33] S. Shimizu, M. Ikeguchi, K. Shimizu, Chem. Phys. Lett. 268 (1997) 93. [34] S. Shimizu, M. Ikeguchi, S. Nakamura, K. Shimizu, J. Chem. Phys. 110 (1999) 2971. [35] B. Lee, Biophys. Chem. 51 (1994) 263. [36] G. Graziano, J. Chem. Phys. 123 (2005) 167103. [37] G. Graziano, J. Chem. Phys. 124 (2006) 134507. [38] H.S. Chan, K.A. Dill, Annu. Rev. Biophys. Biomol. Struct. 26 (1997) 425. [39] K.A. Dill, D. Stigter, Adv. Protein Chem. 46 (1995) 59. [40] K.A. Dill, J. Biol. Chem. 272 (1997) 701. [41] B. Lee, Protein Sci. 2 (1993) 733.