THEORY OF‘ ~YDROl?HOBICBONDING
363
Theory of Hydrophobic Bonding. I. The Solubility of Hydrocarbons in Water, within the Context of the Significant Structure Theory of Liquids by Robert a. Hermmn Eli L d I y afnd Company, Ind.&znupolis, Indiana &YO6
(Received July 18,1970)
PubZicatwn costs assisted by Eli Lilly and. Company, Indianapolis, Indiana
The theory of the thermodynamic behavior of solutions of aliphatic hydrocarbons in water is examined. The water molecules in the layer of water next to the hydrocarbon molecule are assumed to be in an asymmetric electric field, similar to what has been proposed to account for the surface tension of pure ,water. This asymmetric field acts on the water dipole, restricting the motion of the water molecules. A. partition function which accounts for the thermodynamic properties of aqueous solutions of hydrocarbons can then be written in terms of the significant structure theory. The agreement between theory and experiment for the free energy, internal energy, and entropy are good but the calculated heat capacity is too low.
Introduction
It has been recognized for some time that the interaction between a protein and its aqueous environment is a most important factor in determining its conformation. Hydrocarbon side chains are present in all olypeptides and it is principally these d to cluster in aqueous solution. The tendency toward the aggregation of such hydrocarbon moieties in aqueous solution is an example of hydrophobic bonding and i s related to the insolubility of hydrocarbons in water. I n order to understand the origin of hydrophobic bonding, it is necessary to understand the molecular interactions involved in aqueous ons which determine the observed ennodynamic properties of hydrocarbon aqueous solutions has been developed by N6methy (and Sehteraga.lv2 This theory is based on the Frank and Wen flickering cluster model for water.3 The distribution of water molecules among species participating in different numbers of hydrogen bonds is formulated in terms of a partition function. The water molecules are distributed over five energy levels, eorresponding to four, three, two, one, and no hydrogen bonds per molewle. This model is made to account for iceberg formation around the hydrocarbon, as proposed by Frank and Evans,4 which in turn accounts for the reduction of unitary entropy5 and the large heat capacity of h y ~ r ~ c a r bsolutions. on I n the following treatment some of the thermodyn ~ i c p r o ~ rofthydrocarbon ~~s solutions are calculated, x t of the significant structure theory 8 theory applied to water by Jhon, Eyring’ accounts quite well for such l t z energy, the entropy properties as the ~ ~ e l ~ n h ofree of vaporization, and the minimum in the volume a t ’. The n~~~~~~ in the heat capacity C, is accounted
for, indicating the quality of the second derivative of the partition function. The theory has been used to account for the surface tensions and the dielectric constante of water. While the description of the interactions between hydrocarbon and wa,ter proposed here are independent of any particular current theory of liquids, we choose to formulate these interactions in terms of the significant structure theory because of the success and simplicity of this theorgr and because it has been applied to water.
Partition Function for Hydrocarbon Solutions It is assumed that the mole fraction of hydrocarbon in water solution is so low that there are always at least two layers of water molecules between hydrocarbon molecules. The partition function for the solution of a hydrocarbon in water may be written as a function of partition functions for the hydrocarbon fR, for the adjacent water layer f ~and , for the bulk water fw
N1
+ Nz
N
(2)
(1) G. NBmethy and H. A. Scheraga, J. Chem. Phys., 36, 3382 (1962). (2) G.NQmethyand H. A. Scheraga, ibid., 36,3401 (1962). (3) H. S. Frank and W. Y. Wen, D~scuss.Faraday h e . , 24, 133 (1967). (4) H. S. Frank and M. W. Evans, J. Chem. Phys., 13, 507 (1945). (5) W. Kauzmann, Advan. Protein Chem., 14, 1 (1959). (6) H. Eyring and T. Ree, Proc. Nut. Acad. Scd. U. S., 47, 526 (1961); H. Eyring and R. P. Marchi, J . Chem. Educ., 40,562 (1963). See also subsequent papers. For a complete summary of contributions to the theory, see H. Eyring and M. 5. &on, “Significant Liquid Structures,” Wiley, New York, N. Y., 1909. (7) M.S.Jhon, J. Grosh, T. Ree, and H. Eyring, J . Chem. Phgs., 44, 1466 (1966). (8) M. 8. Jhon, E. R. Van Arlsdalen, J. G ~ Q &and H. Eyring, ibid., 47, 2231 (1967). (9) M.E. Hobbs, M. 8, Jhon, and H. Eyring PTOC. Nut. Acud. Sei. U.S.,56, 31 (1966).
The Journal of Physical ChemGtry, Vol. 76, No. 9,1971
ROBERT €3. HERMANN
364 where M is a random mixing factor, N is Avogadro's number, N I and N z are the respective numbers of total water and hydrocarbon molecules. L is the number of molecules of"water in the surface layer of water surrounding each hydrooarbon molecule. This model assumes that the hydrocarbon adjacent water layer and remaining bulk water are all statistically independent. They will, however, influence each other through time average forces and this will manifest itself in She parameters which enter into each species. The partition function from which will be calculated the free energy and entropy change associated with dissolving a hydrocarbon in water is defined as fD
=z
M-lf~ol/gRNz'fWN1
(3)
where is the partition function for pure hydrocarbon and fw is the partition function for pure water. The random mixing factor is omitted to simplify comparison C ~it~ is the unitary free energy and with earlier W G I Y ~so unitary entropy which is calculated. Incorporation of the Hydrocarbon Partition Function. According to significant structure theory the partition function for the hydrocarbon is written
I n the case of E,,', the value of this parameter is determined in the following way. The hydrocarbon in solution is now bound to L water molecules so
EBZf = L * '/&RW
(7)
where ERW is the average interaction energy between the hydrocarbon molecule and a bordering water molecule. This replaces the E,, for the pure hydrocarbon which is one-half the average interaction energy between the hydrocarbon and its hydrocarbon neighbors. The actual value of this parameter is determined by fitting the data. The parameter Ba is assumed to remain unchanged. Changes in B have previously been taken to be proportional to the square root of changes in the sublimation energy"
-
ICE,'/a (8) The change in characteristic temperature for the hydrocarbon-water structure may be considered to be a function of the average increase in energy of the overall hydrocarbon-water layer structure and may be incorporated if necessary via a change in the corresponding characteristic temperature for the water layer only. The above approximations to the parameters allow fD to be written B
-
VLNa/ Vzfw ( N t LNd
where E,,, V,,, n2, and a2 are parameters characteristic of the hydrooarbonb6il0 The function fsz represents the solid-like degrees of freedom and is assumed to be (5)
The value for V zto be used is partially dependent on the nature of the experimental data. The experimental data given in Table I1 were compiled by Nemethy and Scheraga2 and refer to the process12of dissolving a pure liquid hydrocarbon RHI in water.
RHI ( X
= 1,
ftranszfrotlfvibz
(6)
where ftrans2, fvibz, frotl are the translational, vibrational, and rotational (OF librational) partition functions, respectively, and & is the Einstein characteristic temperature. For V z = V,,, the partition function reduces to that of the corresponding solid, while for appropriately larger values of Vz,it represents a liquid or a gas. The function ~'IR in eq I will have the same form as gR in eq 4,but the parameters E,,', &',nz', az' in fR will not necessrsrily be the same as the corresponding parameter in gIL. I n the theory of significant structures, for simple liquids, n2 and a2 are functions of V 2 and V s , only.6 If it is assumed that the available hydrocarbon volume V z remains approximately unchanged upon dissolving the hydrocarbon in Walter, the effects of differences between n2 and nz' and between a2 and az' may be neglected. The Journal of .Phg&ol Chemistry, VoL 76,No. 3,1871
T,p
=
1 atm) -+
RH,,
while for the gas-like degrees of freedom fg*
/
(Xstd,
T,p
= 1
atm)
(10)
Propane and butane are gasses at the pressures and temperatures of interest, so it is assumed that the following process may be substituted for (10)
RHI ( X = 1, T,p = Peg)--+
RH,,
(Xstd,
T9p
= 1
a h ) (11)
because of the small vapor pressure and compressibility of the liquid. I n the cage of methane and ethane the critical temperatures are below the temperatures of interest. It was assumed that data for the following process may be used to estimate parameters for the process (10)
+
(10) The formula L = 0.7 1.82%kcal waa used for the sublimation energy of aliphatio hydrocarbons. This waa given in ref 14, p 679.
(11) W.C. Lu,M.8.&on, T. Ree, end a.Eyring, J . Chem Phys., 46, 1076 (1967).
(12) Xstd refers to the standard state mole fraction which is unity here.
365
THEORY OF HTJDROPHOBICBONDING RH,,I ( X
=
1 , T , p = 1 atm)
----+
RH,, (Xetd,T,p = 1 atm) (12) where RH,,I represents a nonpolar solution of the hydrocarbon. The appropriate volume, V Z to , be used in eq 9 is the one corresponding to the state represented by the lefthand side of' (10). I n the case of methane and ethane the appropriate vohume would be the partial molal volume for the left-hand side of (12). This volume is arbitrarily taken to be the volume of the pure liquid hydrocarbon at the melting point. For simple liquids the volume inorease on melting is about 12%. This fact is used in si ificant structure theory to arrive at appropriate valu for n and a. Due to the difficulties with methane :md ethane noted above, it is assumed for simplicity thal for all hydrocarbons treated here, that V8JV2 =
l/1 .I20
I n c o r p o r a t i o n of the Partition F u n c t i o n for W a t e r a n d Preliminaqi C a l c u l a t i o n of jD. In this section the partition function j ' ~is calculated for hydrocarbon sohtions, incoqmating the partition function for water as given by Jhsn, Grosh, €be, and Eyring.' Parameters defining the first water layer around the hydroeneral, different from the bulk liquid. It is shown in &hissection that the only way to get a reasonable entropy is to adjust 0 for the first water layer. Due to the meaning of this parameter, any physical interpretation is d a c u l t , especially when an attempt is made to apply it to a single layer of molecules. function for water, according to Jhon, jWI f g ( V ~ - l ' d
'VI
{,rBe(-tEsi/RT)(l+ nhe-e/RT
)I ( V s , / V d (13)
where ,fa
J's = ftrifrotJvib1
(14)
jFE.fIibfvibl' K K A 1 + K ) q
(15)
% =-
n1(V,
-
~Sl)/V,,
(16)
For the first layer of water, the partition function f~ will have the same form as fw, but the parameters E,L,BL, nh,,, and EL are not necessarily the same as E,,, 81, nhl, and €1. The parameter E.L will. be different from E,, in the following manner. Assuming a single monolayer of ice which contains a maximum number of water molecules, three of the four bonds from a given water molecule will be bound to other water molecules in that layer. Half of the water molecules must be bound to the layer of water above and half to the layer below. Then in a planar surface layer only an average of '/s of the water-water bonds are broken. Assuming that each water molecule is in some fashion bound to the hydrocarbon molecule &L
=
('/dEal
+
(19)
('/@RW
represents the sublimation energy associated with a molecule in the first layer of water. The value of n h L should remain unchanged from nh according to the following argument. I n the case of simple liquids a molecule in the bulk is surrounded by six positions in its own layer, three in the layer above and three in the layer below. A molecule at a wall or other boundary is surrounded by six positions in its own layer and three in the second layer. Any hole traveling through the second layer to the boundary layer must be reflected at the boundary, otherwise holes would pile up at the boundary layer. When the hole is reflected it goes through the second Payer once again. Thus during the same time interval the probability of finding a hole next to a molecule is the same whether the molecule is at the boundary layer or not. Since the parameter n for water is similar to that for simple liquids, and water could be similarly divided into layers, the above argument, with some modification as to the number of neighbors, should apply to water. The "boundary" here is represented by a large hydrocarbon molecule. If the hydrocarbon molecule is about as small as a water molecule, then the problem of the boundary would not arise. Holes would then move
Table I: Butane Solution Quantities at 298.15"K
The! functions .,ftrlj frotl, fvlbl, f ~and , f l i b refer to the translational, rotational, vibrational, Einstein, and librational partition functions, respectively, K is the equilibrium constant between ice I and ice 111, and q is 46, the number of water molecules per cluster. For f l i b those 4l;uthors choose
(18) so that the three Einstein degrees of freedom for the crystal vibration anld the three degrees of libration are given the same functional form and the same characteristic temperature of 216.1'. Further details may be found in ref 7. flib
fE
Using Partition Function from Eq 18 -Caloulation no.-----------
eL, OK ERW, cal VL,@cc
AA, kcal AE, kcal AS, eu c, cal mol-'
I
I1
IIX
IV
216.1 1883 17.91 0.479 -0.634 -3.73 1.3
219.4 1883 17.91 0.84 -0.78 -5.44 0.6
255 1883 17.95 5.27 -0.85 -20.6 4.0
216.1 2879 17.71 -17.19 -20.74 -11.91 8.2
6 The calculated V s at 298.15'K is 17.6564 cc and Vl, the calculated bulk molar volume of water, is 18.009 cc. V L then indicates the degree of increased cluster formation.
The Journal of Physical Chemktry, Vol. 76,No,8, 1071
ROBERTB. HERMANN
366 Table II : iUiphatic 13ydrocarbon Solution Quantities a t 298.15'K Using Partition Function Containing Restricted Rotation Terms0 Hydrooarbon
Lb
Vr,c
BA
AE
AS
A@
CV
Methane 13 18.164 1.48 -2.848 -14.53 7.1 Ethane 16 18.164 2.93 -2.400 -17.88 8.3 Propane 18 18.164 4.44 -1.559 -20.12 9.8 Butane 20 18.164 5.59 -0.7183 -22.36 11.0 a Equation 23. From ref 6. c V s a t 298.15 is 17.6564 cc/mol and cates degree of decrerwed cluster formation.
freely through thc liquid and n h L would again be equal to nL. The value of E remains unchanged since the energy required to push aside another water molecule would not be expected to change for any reason other than a change in sublimation energy. The value of e does not affect the results greatly? because of the small value of the parameter a. With the above parameter changes, fD waa evaluated for butane (1; = '20) at 298.15"K. The results for several values of BL and ERWare given in Table I. In elmholtz free energy was minimized with respect to volume. Column 1 uses OL = 01 and ERW was adjusted so as t o obtain the experimental AE- In 1 column 2, BL is adjusted in accord with eq 8. ERW is again adjusted slightly to get the experimental AE (Table 11). Neither treatment gives an acceptable entropy. If degree of iceberg formation is defined as the per cent decrease in volume VL of the water layer next to the hydrocarhon fmm the calculated volume of bulk water at 298.15'K ( V , = 18.009) to the calculated cluster volume at the same temperature (V, = 1?.6564), then there is about 17% iceberg formation, since VL wm flound i o be 17.950 cc. In column 3, the value of OL is adjusted to give a reasonable eri tropy. Because of the oversimplified model of molecular motions represented by a characteristic temperature, the physical interpretation of the increased value of OL is not apparent. I n column 4 a value of ERWis assumed, which gives about 87% iceberg formation. The value of ERW needed is much too high so that the AA is of the wrong sign. The enicropy is still too low, although a contribution could come from an adjustment of Or,. Introduction of an Asymmetric Electric Field in the First Water Layer. I n order to more easily account for the negative entmp
