J. Phys. Chem. 1995, 99, 12076-12080
12076
An Estimate of Shape-Related Contributions to Hydrophobic Gibbs Energies Jose M. Sanchez-Ruiz Departmento de Quimica Fisica, Facultad de Ciencias, e Instituto de Biotecnologia, 18071 -Granada, Spain Received: April 6, 1995; In Final Form: May 17, 1995@
It is shown that Ben-Naim’s quantities for the solvation of hydrocarbons in their own liquids may be calculated from Pitzer’s extension of the law of corresponding states. This calculation involves the dissection of solvation quantities into two terms, which may be interpreted as corresponding to (1) the value of the solvation quantity for a hypothetical fluid of simple spherical molecules, with the same critical temperature and pressure as the actual hydrocarbon under consideration, and (2) the contributions that arise as a result of the deviation of the intermolecular potential function from that of simple spherical molecules (“shape” contributions). The values obtained for the latter contributions support that shape effects may be significant with regard to hydrophobic Gibbs energies; for instance, the value for the strength of the hydrophobic effect derived from liquid alkane dissolution would increase from about 120 to about 170 J/mol A2 upon correcting for the calculated shapeentropic contributions (for comparison, we note that correcting for the whole solvation entropy of the liquid hydrocarbon would raise the value to about 200 J/mol A2).
Introduction Hydrophobic contributions to protein folding (and other biologically relevant processes) are often estimated on the basis of hydrophobic Gibbs energies derived from the analysis of liquid hydrocarbon dissolution. A longstanding (and repeatedly controversial) problem is related to the need of correcting these experimental Gibbs energies for “undesirable” entropic contributions (cratic mixing contribution, translational entropy, etc.). The recent theoretical work of Chan and Dill’ indicates that, for a pure liquid of nonspherical molecules, an entropic contribution arises form the coupling (Le,, nonseparability) of rotation (or other intemal degrees) and translational freedom. This suggests the interest in providing estimates for the contributions to hydrophobic Gibbs energies which may arise from effects associated with molecular shape in the pure liquid hydrocarbon phase. It will be shown in this work that Pitzer’s extension of the law of corresponding allows Ben-Naim’s quantities for hydrocarbon solvation in its own liquid to be expressed as a sum of two terms, which may be interpreted as corresponding to (1) the value of the solvation quantity for a hypothetical fluid of simple spherical molecules, with the same critical temperature and pressure as the actual hydrocarbon under consideration, and ( 2 ) the contributions that arise as a result of the deviation of the intermolecular potential function from that of simple spherical molecules (Le., “shape” contributions). Of course, no claim is made at this stage that the shape contributions calculated in this way are identical with the coupling effects discussed by Chan and Dill,’ although at least a qualitative relationship appears likely. Corresponding states behavior is e ~ p e c t e d ~for - ~ classical substances with effectively spherical molecules if the intermolecular interaction energy is painvise additive and can be derived from a pair potential of the form U(r)/c= F(r/o)
(1)
where E and u are characteristic energy and distance parameters and F is the same function for all the substances considered; for instance, the Lennard-Jones potential is consistent with eq @Abstractpublished in Advance ACS Abstracts, August 1, 1995.
0022-3654/95/2099- 12076$09.00/0
1: V ( r ) = 4~[(a/r)’~ When these conditions hold, certain properties may be expressed as universal f;nc_tions of “molecular” reduced parameters: P = kTk, v* =_ V/(Nu3)and p* = pa3/€, where is the molar volume, N stands for Avogadro’s number, and k is Boltzmann’s constant. Corresponding states behavior also implie_s that reduced critical parameters (P, = kT&, V*, = V,/(Nu3),and p*, = p , 0 3 / ~ ) should have universal values, from whence it follows that the corresponding states, universal dependencies, can be cast in terms of “practical” reduced parameters: 6 = TIT,, n = p/p,, and 4 = Thus, certain magnitudes (M) may be expressed as universal functions of reduced temperature and volume or, altematively, as universal functions of reduced temperature and pressure (since 6 , n, and 4 are not independent):
v
v/v,.
M =fie,n)
(2)
Substances such as Ar, Kr, and Xe (sometimes called “simple fluids”) accurately follow this behavior. For other substances deviations from the corresponding-states behavior of simple fluids arise as the result of the occurrence of one (or several) of the following factor^:^*^^'^ (1) quantum effects in light molecules (He, H2, and to a lesser extent Ne); ( 2 ) polar interactions in molecules with high permanent dipole, hydrogenbonding capability, etc.; (3) lack of effectively spherical symmetry in such a way that the angular dependence of the intermolecular interaction cannot be neglected and the translational partition function is not separable from r ~ t a t i o n . ’ -Here, ~$~~ we will be concemed with substances (such as alkanes) in which factors 1 and 2 are unimportant, the so-called “normal fluids”. Pitzer found2q3that the deviations observed for normal fluids could be accounted for on the basis of a single additional parameter, the acentric factor, defined as
where p:.’ is the gas pressure in equilibrium with the liquid (Le., the vapor pressure) at a reduced temperature of 8 = 0.7. The acentric factor is defined (eq 3) in such a way that w = 0 for the simple fluids Ar, Kr, and Xe (also, w =; 0 for C h , which possibly achieves effective spherical symmetry by rapid rotation). w values have been tabulated for many substances; 0 1995 American Chemical Society
Letters
J. Phys. Chem., Vol. 99, No. 32, 1995 12077
in any case, they can be easily calculated from vapor pressure data by using the definition (eq 3) or from the values of the boiling temperature and the critical temperature and pressure by using the equation given by Edmister.'* The-acentric factor was said by Pitzer3 to "measure the deviation of the intermolecular potential functions from that of simple spherical molecules". For the n-alkanes considered in this work, this deviation is expected to mostly consist of an angular (i.e., mutual orientation) dependence of the intermolecular potential,' an effect obviously associated with the molecular shape. Corresponding-states dependencies (eq 2 above) can now be extended to include normal fluids:
M =f(e,n,@)
(4)
wherefis the same function for both simple and normal fluids. Equation 4 is usually expanded as a power series in an acentric factor, although for most practical purposes the series may be truncated (with negligible error) in the linear term:
and, using eqs 8 and 9 and the equation for AG,, may be expressed as AG, = -AG,
+ RT ln(pVLIRT)
(10)
where ?L is the liquid molar volume (@L = R / ~ Land ) we have used the ideal value for the number density of the gas phase (ec = NpIRT). Expressions for the solvation entropy and enthalpy are easily found from eq 10:
AS, = -(aAG,/aT)?, = -AS,
- R ln(pVLIRT)+ R(l
AH, = -f(a[AG,/T]Ia7),
- Ta,) (1 1)
+
= -AH, RT(l - Ta,) (12)
where a L stands for the liquid thermal expansion coefficient. Under equilibrium conditions, we have that p = p , (the vapor pressure at the temperature T), pc = p ~AG, . = AH,, - TAS, = 0, AH, = TAS, and eqs 10-12 may be written as AG, = RT In(p,V,lRT)
where Mo and MI are universal functions of the reduced parameters. Clearly, IWgives the value of M for a hypothetical simple fluid with the same critical temperature and pressure as the actual fluid under consideration. On the other hand, wM' reflects the effect of deviations from spherical symmetry in the . intermolecular potential, which are associated to molecular shape. The logarithm of the reduced pressure corresponding to the vapor pressure (that is, p,,/pc),as well as the vaporization entropy along the liquid-vapor equilibrium line, can be expressed3.l3as in eq 5 :
AS, = -AS, - R ln@,V,/RT) AH, = -TAS,
(13)
+ R(l - Ta,)
+ RT( 1 - Ta,)
(14) (15)
Equations 13- 15 yield solvation magnitudes for temperature and pressure conditions which belong to the liquid-vapor equilibrium line; nevertheless, the calculated values may be assigned to 1 atm, since the pressure effect on solvation quantities is usually very small.I6 Upon substitution of eqs 6 and 7 into eqs 13-15, the latter can be cast in a the same form as eq 5 (Le., a truncated power series in acentric factor):
AG, = A@
+ UAG;
(16)
AS, = A$
-+ w AS:
(17)
where In nt,In n:,A$!, and AS: are universal functions of the reduced temperature (0) exclusively. These functions have been tabulated or given in analytical form in several place^.^,'^.^^ where
Solvation of n-Alkanes in Their Own Liquids Consider the liquid and gas phases of a substance at any given temperature ( T ) and pressure (p) conditions (not necessarily corresponding to the liquid-vapor equilibrium). The chemical potentials of the liquid and the gas may be expressedI5 as A$ = -A$
+ kT ln{@,A3}
(8)
+
(9)
,MG= , u * ~
p, = , M * ~ kT l n { ~ , A ~ }
where p * ~and p * ~are the pseudochemical potentials (work associated with the addition of a molecule to a fixed position in the system), @cand QL stand for the number densities, and A is the thermal de Broglie wavelength of the molecule at the temperature T. The vaporization Gibbs energy is given by AC, = N(UG - p ~ )where , Avogadro's number appears because AG,, is defined per mole, while chemical potentials (eqs 8 and 9) are defined per molecule. Consider now Ben-Naim's definitionI5 of the process of solvation of a substance in its own liquid: transfer of a molecule from a fixed center-of-mass position in an ideal-gas phase to a fixed center-of-mass position in the liquid. Thus, under the ideal-gas approximation for the gas phase of the-substance, the solvation Gibbs energy (per mole) is AG, = N@*L - p*c),
- R In(p,V,lRT)
- R In n t
+ R(l
@ = -TA$
+ RT( 1 - Ta,)
- Ta,)
(21)
(23)
Equations 16-24 provide a simple procedure to predict BenNaim's solvation quantities (of a substance in its own liquid) from Pitzer's extension of the law of corresponding states (that is, by using the tabulated values of the acentric factor and the universal functions In nt,In nb,A q , and AS:). I have used this procedure to calculate the solvation quantities for n-alkanes (n = 3-8). The several parameters involved in the c a l c ~ l a t i o n ~ ~ are given in Table 1 and the calculated values are compared with those given by Ben-Naim and MarcusI6 in Figure 1, where
12078 J. Phys. Chem., Vol. 99, No. 32, 1995
Letters
TABLE 1: Parameters Involved in the Calculation of Ben-Nab’s Solvation Quantities from Pitzer’s Extension of the Law of Corresponding State# Tc (K)
e25
pC(atm)
0 VL (mUmo1)
aL (K-’1 -In ne -in ni
A$ AS:,
propane
butane
pentane
hexane
heptane
octane
369.85 0.806 41.92 0.153 81 1.9 x 10-3 1.307 1.194 41.47 53.70
425.16 0.701 37.47 0.200 100 1.7 10-3 2.297 2.308 55.00 75.60
469.65 0.635 33.25 0.252 115 1.6 x 10-3 3.091 3.293 64.68 92.40
507.85 0.587 29.92 0.303 130 1.35 x 10-3 3.772 4.181 72.98 107.08
540.16 0.552 27.00 0.350 146 1.24 10-3 4.330 4.932 79.95 1 19.44
568.76 0.524 24.45 0.398 162 1.14 x 10-3 4.818 5.601 86.19 130.50
a 825 is the reduced temperature corresponding to 25 “C (825 = 298.15/Tc). The values of A$ and AS: are given in J K-’ mol-’. See note 17 for details on the sources and calculation of the magnitudes given below.
60
40 20 0
40
,’
I
I
I
I
, ,
30
20
2
10
-
20
0
15
E
10
0 d
I
k
3 4 5 6 7 8
n Figure 1. Calculation of Ben-Naim’s solvation quantities for the solvation of n-alkanes (n = 3-8) in their own liquids from Pitzer’s extension of the law of corresponding states. (0) Simple-fluid contributions calculated by using eqs 19, 21, and 23, and the parameters given in Table 1. (M) Shape-related contributions calculated by using eqs 20, 22, and 24 and the parameters given in Table 1. (A) Solvation quantities obtained by adding the simple-fluid and shape-related contributions (eqs 11- 13). (0)Solvation quantities directly calculated from experimental data and taken from Ben-Naim and Marcus.I6
values of WAG:, oAS;, and oAH;(shown in Figure 1) as our sought-for, shape-related contributions. According to the above proposal, shape-related contributions to solvation enthalpy and entropy are significant, as reflected by the comparatively large values of wAH; and WAS: (Figure 1). We explain qualitatively (and tentatively) these large values as follows: Lower n-alkanes are more or less ellipsoidal in shape;6 the interaction energy between apolar, ellipsoidal molecules in a relatively well-packed liquid is expected to be larger than that for spherical molecules, since a larger molecular surface area (for the same volume) is available for intermolecular “contacts” in the former case; hence the significant oAHi contibution. Also, steric interferences between the orientations of neighboring ellipsoidal molecules (mainly associated with the repulsive, “nearly-hard-core’’ part of the intermolecular potential) should bring about a reduction in the available volume in multidimensional configuration space;1.2.5hence the significant oASj contribution. It must be noted that the entropic and enthalpic shape contributions cancel each other out to some extent, to give a relatively small shape contribution to the solvation Gibbs energy: WAG: = wAH; - TwAS: (Figure 1).
Hydrophobic Gibbs Energies Derived from n-Alkane Dissolution We consider now Ben-Naim’s local Gibbs energiesI5 for n-alkane dissolution in water (AG*); that is, the Gibbs energy changes (expressed per mole) associated with the transfer of a molecule from a fixed center-of-mass position in the liquid hydrocarbon to a fixed center-of-mass position in the aqueous solution. Any given AG* value is obviously equal to the solvation Gibbs energy in the aqueous solution minus that in the pure liquid alkane; for the sake of clarity, we will refer to the former as hydration Gibbs energy: A G . Thus
AG* = AG, - AG,
a good agreement is apparent. In fact, a good agreement is
found not only for alkanes but also for cycloalkanes, alkenes, and benzene derivatives (results not shown). It must be noted that A@, A$, and (see eqs 16-18) are the values that would take the solvation quantities for w = 0 and that their expressions (eqs 19, 21, and 23) contain the simple-fluid, universal functions: In n: and A$; then, we interpret the values calculated for A G , A g , and A@ (given in Figure 1) as the solvation quantities of the hypothetical, simple spherical fluid with the same critical temperature and pressure as the alkane under c o n s i d e r a t i ~ n . ~Consequently, ~.~~ we interpret WAG:, WAS:, and wAHl in eqs 16-18 as representing the effect of deviations from spherical symmetry in the intermolecular potential. That is, we take the calculated
(25)
or, if we decide to expose the shape contributions in the liquid alkane phase
AG* = AG,, -
AH^ + TAS, = AG, - AI$
+ TA$
WAH;
-
+ TWAS; (26)
A value for the “strength” of the hydrophobic effect can be calculated as the slope of the plot of AG* versus accessible surface area (ASA). Using the AG* data for n-alkanes20 (n = 3-8), we obtain 119 J/mol A* at 25 “C (see Figure 2), which agrees in fact with the most often quoted value (usually given as “about 25 caVmol A2”); of course, this value includes the
Letters
J. Phys. Chem., Vol. 99, No. 32, 1995 12079 I
at least in principle, other criteria for choosing the reference simple fluid might be ~ o n s i d e r e d . ~It~ is also clear that additional theoretical work is required in order to establish a quantitative relation between the shape contributions calculated in this work and the rigorous classical statistical-mechanical analysis of Chan and Dill,’ These obvious caveats notwithstanding, the present work (1) does support that shape-associated contributions may be important in complex solvation processes and (2) suggests simple procedures to estimate these contributions from widely available data.23 In addition, and regardless of molecular interpretations, the fact that the solvation quantities of a normal fluid in its own liquid can be accurately predicted from Pitzer’s extension of the law of corresponding states should be useful by itself.
I
20 200
I
I
300
400
ASA
(A2)
Figure 2. Plot of the Gibbs energy change for n-alkane dissolution in water versus the water-accessible surface area of the alkane: ASA (see note 20 for the source of these data). (+) Ben-Naim’s, local transfer Gibbs energies (AG*). (m) Local transfer Gibbs energies corrected for the shape-entropic contribution (AG* - TwAS:). ( 0 ) Local transfer Gibbs energies corrected for the whole solvation entropy of the liquid hydrocarbon (AG* - TAS,). The numbers alongside the lines give their slo es in terms of the Gibbs energy associated with the transfer of 1 mol of accessible surface area.
x2
effect of the enthalpic and entropic shape contributions. There appears to be no reason why we should correct for the enthalpic shape contribution in eq 26, if, as suggested above, the -wAH: term is associated with the intermolecular interactions in the liquid hydrocarbon phase (which must be taken into account in the required hydrophobic Gibbs energies). On the other hand, we may consider correcting for the shape-entropic contribution, since the term ToJAS~presumably reflects the steric interferences between the orientations of neighbor alkane molecules, an effect characteristic of the liquid hydrocarbon phase, and, possibly, extraneous to protein folding and other biologically relevant processes. These “shape-entropy-corrected” transfer Gibbs energies are given by AG* - TwAS:, and they lead to a significantly higher value for the strength of the hydrophobic effect: 170 J/mol A* (see Figure 2). For the sake of comparison, we have also calculated transfer Gibbs energies corrected for the whole solvation entropy of the liquid hydrocarbon: AG* - TAS,; these lead to a value of 197 J/mol A* for the strength of the hydrophobic effect (see Figure 2). The above reasonings and calculations are meant only to illustrate the possible effect of shape-entropy corrections on hydrophobic Gibbs energies and should be considered with due caution. It is interesting to note, nevertheless, that the corrected values of the strength of the hydrophobic effect qualitatively agree with that calculated by using solid hydrocarbon models (and used in a recent interpretation of the protein folding thermodynamics),21 as well as with those suggested by some analyses of the effect of aliphatic mutations on protein stability.**
Concluding Remarks Shape contributions given in this work are, in fact, equal to the difference between the actual solvation properties and those corresponding to a hypothetical simple, spherical fluid with the same critical temperature and pressure as the hydrocarbon under considerati~n.~~ This procedure, although reasonable and convenient from the point of view of corresponding states theory, might perhaps be regarded as somewhat arbitrary, since,
Acknowledgment. This work was supported by Grant PB931087 from the DGICYT (Spanish Ministry of Science and Education). I thank Dr. H. S . Chan and Dr. K. A. Dill for bringing their work to my attention prior to publication. References and Notes (1) Chan, H. S.; Dill, K. A. J . Chem. Phys. 1994, 101, 7007. (2) Pitzer, K. S. J . Am. Chem. SOC. 1955, 77, 3427.
(3) Pitzer, K. S.; Lippmann, D. Z.; Curl, R. F.; Huggins, C. M.; Petersen, D. E. J. Am. Chem. SOC.1955, 77, 3433. Pitzer, K. S. In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T. S., Sandler, S. I., Eds.; ACS Symposium Series 60; American Chemical Society: Washington, DC, 1977; pp 1-10, (4) Pitzer, K. S. J. Chem. Phys. 1939, 7, 583. ( 5 ) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley: Reading, MA, 1960; Chapter 16. (6) Hirschfelder, J. 0.;Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964; Chapter 4. (7) Kestin, J.; Dorfman, J. R. A Course in Statistical Thermodynamrcs; Academic: New York, 1971; Chapter 7. (8) Maitland, G. C.; Rigsby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Clarendon: Oxford, 1981; Chapter 3. (9) Guggenheim, E. A. Thermodynamics, 5th ed.; North-Holland: Amsterdam, 1967; Chapter 4. (10) Hakala, R. W. J . Phys. Chem. 1967, 71, 1880. (11) In fact, deviations may also arise with complex molecules of effectively spherical symmetry, if they can be modeled as “spherical shells”. In this case, there is no angular dependence of the intermolecular potential, but the & - ) / E versus r/a curve differs from the Lennard-Jones form obeyed by the simple, spherical fluids (Ar, Kr, Xe). A dramatic example of this situation is buckminsterfullerene CCQ[Girifalco, L. A. J . Phys. Chem. 1992, 96, 8581; indeed, CCQis expected to deviate significantly from the corresponding states behavior of simple fluids [Hagen, M. H. J.; Meijer, E. J.; Mooij, G. C. A. M.; Frenkel, D.; Lekkerkerker, H. N. W. Nature 1993, 365, 4251. It must be noted, however, that this spherical-shell situation does not apply to the n-alkanes discussed in the present work. (12) Edmister, W. C. Petroleum Refiner. 1958, 37, 173. (13) Lewis, G. N.; Randall, M.; Pitzer, K. S.; Brewer, L. Thermodynamics; McGraw-Hill: New York, 1961; Appendix 1. (14) Lee, B. I.; Kesler, M. G. AIChE J . 1975, 21, 510. (15 ) Ben-Naim, A. Statistical Thermodynamics for Chemists and Biochemists; Plenum: New York, 1992. (16) Ben-Naim, A.; Marcus, Y. J . Chem. Phys. 1984, 81, 2016. (17) Values used for the acentric factor and the critical temperature and pressure were those given in Table 1 of : Xu, J.; Herschbach, D. R. J . Phys. Chem. 1992, 96, 2307. Liquid molar volumes were taken from: Sharp, K. A.; Nicholls, A,; Friedman, R.; Honig, B. Biochemistry 1991, 30, 9686. Thermal expansion coefficients of liquid pentane, hexane, heptane, and Octane were from: Raznjevic, K. Handbook of Thermodynamic Tables and Charts; Hemisphere Publishing Cop.: Washington, DC, 1976. Thermal expansion coefficients for liquid propane and butane were estimated by linear extrapolation from a plot of aL versus n (number of carbon atoms) constructed with the data for n = 5-8. Values of the universal functions In z:, In J C ~ ,A e , and AS: for the reduced temperatures corresponding to 25 “C were interpolated from the data given in Table Al-1 (appendix 1) of ref 13. (18) For this interpretation to be rigorous, eqs 19, 21, and 23 should use the molar volumes and thermal expansion coefficient of the hypothetical simple fluid; nevertheless, the use of the actual VL and aL values instead does not introduce significant errors (calculations not shown). (19) For a fluid of spherical, nonpolar molecules interacting through a Lennard-Jones potential (a simple fluid), the scale factors u and E (eq 1) can be related with the critical temperature and pressure: c/k = 0.77Tc and
Letters
12080 J. Phys. Chem., Vol. 99, No. 32, 1995 bo = */37&u3 = 18.4Tc/pc,where bo is the covolume which, for a simple fluid, turns out to be about twice the liquid molar volume (ref 6). These equations may be used to calculate the factors E and u corresponding to the hypothetical simple fluid taken as reference from the T, and p c values of the normal fluid under consideration. It is of interest that, for the n-alkanes analyzed in this work, the calculated “hypothetical covolumes” are still around twice the molar volume of the actual liquids. (20) AG* values for n-alkanes dissolution in water were taken from: Sharp, K. A,; Nicholls, A,; Friedman, R.; Honig, B. Biochemistry 1991, 30, 9686. Water-accessible surface areas for n-alkanes were taken from: Giesen, D. J.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. 1994, 98, 4141. (21) Privalov, P. L.; Makhatadze, G.I. J . Mol. Biol. 1993, 232, 660. Makhatadze, G. I.; Privalov, P. L. Adv. Prof. Chem., in the press.
(22) Sharp, K. A.; Nicholls, A.; Friedman, R.; Honig, B. Biochemistry 1991, 30, 9686. Pace, N. C. J . Mol. Biol. 1992, 226, 29.
(23) In principle, other criteria for choosing the reference simple fluid could be easily implemented in the general procedure employed in this work. Assume for instance that a given statistical-mechanical analysis provided such criteria in terms of the values of the Lennard-Jones, scale factors for the reference fluid; then, its Tc and pc values could be derived from the equations given above (note 19) and the u and E values. Once the critical constants have been calculated, the universal functions (In ny,A$) and the solvation quantities ( A e , AC, A@ eqs 19, 21, and 23) of this new reference fluid could be easily obtained. JP950984S
