2338
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979
William H. Bishop
Energy and Entropy Relations in Highly Polar Liquids. Electrostatic Approximations Using the Onsager Model William H. Bishop Laboratory of Biophysical Chemistry, Environmental Blosclences Department, Naval Medical Research Institute, National Naval Medical Center, Bethesda, Maryland 200 14 (Received October 18, 1978; Revised Manuscript Received Apri/30, 1979) Publication costs assisted by the Na Val Medical Research and Development Command
The simple model for a polar liquid proposed by Onsager consists of an electric moment located in a molecular sized cavity surrounded by a dielectric medium. The free energy, entropy, and internal energy changes within this system are examined when the interaction between the central electric moment and its reaction field from the surrounding medium is permitted to occur reversibly. Also of interest is the reverse procedure in which, while keeping the cavity intact, the central moment is reversibly uncoupled from interacting with the medium. The entropy change is initially obtained from the free energy by differentiation with respect to temperature. In polar liquids of large dielectric constant, the model entropy change for the uncoupling approaches the interestingly small limiting value of R / 2 , about 1 eu. This single theoretical process is thermodynamically equivalent to the sum of two real transfer processes. Except for processes involving water, the sum of entropy changes for each of several appropriate pairs of transfers is found to be small in absolute value and positive. The appropriate entropy sums in water are small and negative. A second method with direct application of the first and second laws is used to derive the theoretical entropy change. The same limiting expression and small value for the entropy change is obtained. In this second method, the absolutely small entropy is shown to be the sum of two larger entropy changes which have opposing signs. One of these components of entropy change is positive for the uncoupling and is derived from the change in interaction between the electric moment within the cavity and the electric moments within the medium; the other component of entropy change is negative for the uncoupling and is strictly attributable to the change in mutual interaction among the electric moments located within the dielectric medium.
I. Introduction There has been sustained interest in theoretical model systems which have simple enough properties to permit straightforward calculations of interesting physicochemical quantities and are at the same time, if even in a primitive manner, reasonably faithful physical representations of real systems. One of the most successful in these regards has been Onsager’sl model for polar liquids where the approach is “primitive” in the sense that it incorporates only the most basic notions of electrostatic polarity in a fluid of high molecular number density. In this theory the essential physical quantity treated is the average effect of electrostatic interaction between a central molecular dipole moment and the dipole moments of all the surrounding molecules. Originally proposed in order to interpret the dielectric properties of polar fluids, this model has been generalized and refined in several way^.^-^ The oversimplifications in the original form of the model led to quantitative deficiencies when dealing with highly polar associated liquids such as water or the lower alcohols. Nevertheless, even for these complex systems the model has endured as a very useful conceptual approach. More extended versions of the original theory have been successful in accounting for experimental data in such highly polar l i q ~ i d s . Moreover, ~,~ the Onsager theory is consistent with basic aspects of the forma! quantum theoretical treatment of the dielectric constant given by van Vleck.5,6 A sizeable structural change occurs within the aqueous phase when small nonpolar molecules dissolve in water; the motional freedom of the water molecules adjacent to the nonpolar core is less than that of those distant from the core. The significance of such change is unclear, particularly regarding its effect on the solubility of the nonpolar molecules or on the important related problem of hydrophobicity in biological systems. Briefly stated, there are three possible cases. The structural change can
(i) favor or promote solubility,’ (ii) have no free energy consequence for solubility8 or, (iii) be unfavorable for solubility.s11 There are proponents for each viewpoint in this controversial problem. The present article does not enter this controversy. The aim is rather to use the Onsager model to demonstrate that several additive electrostatic potential energy changes are generated if the interaction between the central moment and the surrounding polar molecules is reversibly eliminated, thereby converting the central molecule from a polar to a nonpolar core. One of these potential energy components, which is negative, results from changes in the interaction of the surrounding polar molecules with one another. Thus, a t least one contributing mechanism to the problematic structural change in aqueous solutions of nonpolar molecules can arise as a constitutive property of highly polar liquids in general. This is not to argue against the presence of unique properties in liquid water which might augment or reinforce such generic behavior; in particular, as is well known, no explicit provision is made in the Onsager theory for the short-ranged force and torque of hydrogen bonding.* However, as demonstrated by Coulson and Eisenberg,12about 70% of the total electrostatic interaction among water molecules can be catagorized as dipole-dipole electrostatic energy. That is, dipole-dipole electrostatic energy is a very significant contribution to the total energy when molecules interact through hydrogen bonds. The viewpoint intended in the present article is similar to that expressed by D e ~ t c h : ’fOn ~ ~ the other hand it seems unlikely that a successful theory of water can be constructed without taking into account the essential features that arise when molecules interact by dipole-dipole forces”. In the present article, derivations and calculations are presented for some simple processes employing the Onsager system in its original form. The analysis presented here has two objectives. These are (1) to indicate the
This article not subject to US. Copyright. Published 1979 by the American Chemical Society
Energy and Entropy Relations in Highly Polar Liquids
A t i A
1
0 --A
Flgure 1. Coupling and uncoupling of the central electric murnent in the Onsager system (i k). The permanent mument 6oin cavity i is reversibly coupled to, or uncoupled from, the reaction field arising from the medium in region k, in differentialsteps f6,dA. The cavity radius a , the total volume of system (i k), and the temperature T remain constant as system (i k) changes between the states (A = 0) and (A = 1).
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relationships between the various energy and entropy components in the model processes and, (2) to demonstrate some correlations between calculations on the model processes and experimental data. Though conceptually very simple, it is necessary to begin with a fairly systematic discussion of this model. 11. Description of the Model The system and the procedures of interest are shown in Figure 1. The polar molecule i, fixed for the present purpose at the center of the system, is represented by the permanent dipole moment rii,, in a cavity i of radius a. Molecule i is surrounded by a set of polar, k-type molecules which occupy the region k. The set of polar molecules in region k is represented by a dielectric medium, with dielectric constant e of the pure liquid of interest. Molecule i and the molecules of the set k are all of the same species; the only difference between i and type-k is that the extent of coupling of riiato the rest of the system may be varied. The system is in thermal contact with a heat bath of temperature T. The k-type molecules are assumed to be, at all times, in full, strong, polar interaction among themselves. That is, the forces among the moments of all the k-type molecules remain fully c0up1ed.l~ The molecules of the set k, which are represented in the system by the dielectric medium, are assumed to undergo such translational and rotational motions relative to one another and to molecule i as are determined by the temperature T and the intermolecular forces acting in the system. Full specifications of these forces and of their detailed structural effects within the system are not attempted here. The forces between i and the set k (ik forces) are represented in the system by the force of interaction between a dipole in a cavity and a dielectric medium surrounding the cavity. The forces between the moments of the molecules in set k (kk forces) are represented in the system by the forces of interaction between parts of the dielectric medium. Furthermore, a description of the total energy of interaction between i and set k and of the total energy of interaction among the molecules in set k is also not attempted. Only the changes in these energies resulting from specific procedures are examined. Molecule i and the set k together form the complete system of interest. This complete system will be abbreviated by the expression “system (i + k)”. It is clear that the conceptual model beimg used to represent system (i + k) is that due originally to Onsager.’ In this model the average tendency for the moments of the molecules in region k to be in energetically favorable orientations with respect to the field from the ceniral molecule i is represented by the polarization vector P in the dielectric medium. Th_epolarized state of this medium produces a reaction field R which interacts with the total moment f i in the cavity. This total rii is composed of the
The Journal uf Physical Chemistry, Vul. 83. No. 18, 1979 2339 permanent- riiaand an induced moment aR due to the action of R on the polarizability, a,of molecule i. The energy changes of main interest here are (i) those which result from changes-in interaction between the moment in the cavity and R, and (ii) those which result from changes in interaction among the moments of the molecules within set k, with set k being represented by the dielectric medium. When the permanent moment fi0of molecule i is affiied to the center of cavity i and is interacting with the reaction field from the medium, riio is electrostatically coupled to this medium. In this state of system (i k), the coupling parameter A = 1. When fiahas been reversibly lifted, in infinitesimal steps, out of the reaction field arising from the medium (leaving cavity i intact) by forces originating in the environment, + isiuncoupled o from the medium and, in this state of system (i + k), h = 0. In either of these states the mass and framework of molecule i are considered to remain in cavity i and to be coupled to the rest of the system. When fi0is coupled, molecule i acts in the system as a polar molecule; when &is uncoupled, molecule i acts as a nonpolar molecule. Wbetber riiais “felt” (A = 1)or “not felt” (A = 0) in the system, fiaalways belongs to the system and is never regarded as part of the environment. The quantities of main interest here are the energy, entropy, and free energy changes of the complete working system (i + k) as it passes from either of these thermodynamic coupling states to the other. In these processes, the self-energy of the permanent electrostatic charge distribution of molecule i is not of interest and following common practice3 will not be considered. During the (A 1)process, the electrostatic force acting between the moment riio and all the moments of the medium performs, in infinitesimal steps, reversible work on the environment against constraining forces originating in the environment. During the (A 0 ) process, these externally originating, constraining forces perform, in infinitesimal steps, reversible work on system (i k) against the electrostatic force acting between rii,, and all the moments of the medium. Therefore, during passage of system (i + k) from (A = 0) to (A = l), some of the change in the electrostatic potential energy of interaction between Cia and the moments in the medium is expended in reversible, forced displacements of bodies which are external to system (i + k). However, during (A 1)or (A O),it is not necessary or relevant to specify the precise mechanism by which the forces acting within system (i k) (between fiaand the medium surrounding fro) are reversibly balanced by these external constraining forces.
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111. Thermodynamic Parameters for t h e Coupling Process In this section for purpose of review and in order to establish notation, the free energy, entropy, and internal energy changes are derived for the reversible coupling procedure discussed in the previous section. Because in Onsager’s model only electrostatic interactions are considered explicitly, lower case letters are employed throughout to denote thermodynamic quantities. In the following both the total volume of system (i + k) and the cavity radius are assumed constant; therefore, the Helmholtz free energy is sought. The electrostatic Helmholtz free energy change in system (i k), denoted by f, for reversibly coupling the permanent moment riio to the reaction field from the medium is obtained by integrating the negative of the coupling work, dw, as the parameter his changed from zero to unity?
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2340
The Journal of Physical Chemistry, Vol. 83,
No. 18, 1979
William H. Bishop
where No is Avogadro’s number.
n is the refractive index of the liquid. In these equations r?io(A) is the permanent moment coupled in cavity i when the coupling level is A and fio(A) is given by Afro; R(A) is the reaction field when the coupling level is A; a is the polarizability of molecule i; g is defined by the equation
g = - -2(t - 1) 1 (2t + 1) a3
(3)
where a is the cavity radius and t is the dielectric constant of the pure liquid. Assuming no variation of mo or a with temperature, we derived the entropy change in system (i + k), denoted by s, for the coupling process from the Helmholtz free energy:
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During (A l),the thermodynamic internal energy change in system (i k), denoted by u, is obtained from the expression, u = f Ts:
=
+
-(&)( fi)($)
-
An expression for the derivative on the right side of eq
4 is required. Onsager’s final equation for a pure polar liquid is1
+
n2)(2t n2) 47rNm: =--t(n2 + 2)2 9k T
(t -
(6)
In this equation n is the refractive index of the liquid; N is the number of molecules per unit volume; k is the Boltzmann constant. Assuming no variation of n or mo with T i n this equation, we may obtain the derivative in eq 4 2 - c ( l + c) (7) aT v 2 + c2
)
(s)(=;
The quantity c = n2/t, the ratio of the high-frequency dielectric constant to the low-frequency dielectric constant. For the highly polar liquids of interest here c is very small and eq 4 becomes:
It is of interest to examine the limiting form of s, in eq 8, when t becomes large. For this, the following assumptions and expressions employed by Onsagerl are introduced. These expressions give the values for N, the polarizability a , the relationship between the permanent moment mo,and the total moment m when A = 1and the high t form of eq 6:
--)
N- = 3 = No( density 47ra3 mol w t
m =-
l
no 1 - a g (n2
n2
e-
+2 3
+ 2)2 47rNmO2
t =
2 9kT Using eq 9-12 in eq 8, we found that the limiting value, as t becomes large, for the entropy change in system (i + k) during the (A 1) process is given by
-
s = -k/2
(13)
This result is seen to be independent of mo,a3,or T. For 1mol of coupling, the entropy change is -R/2 or about -1 eu. The entropy change for 1mol of (A 0) would then be +1 eu. These entropy changes are interesting, because 0) or (A 1) they are so small. Presumably, the (A procedures in the Onsager model are processes in which the relative spatial and orientational distributions of the strongly polar molecules within system (i + k) are undergoing significant change. For example, during (A 0) in the Onsager model for liquid water, the electrostatic free energy change is of the order of +6 kcal/rn01,~resulting from the loss of the strong structural correlations between r?io and the moments in the medium. Some component of entropy change, more positive than the +1 eu, which might reflect such an increase in freedom, might have been anticipated. These points are examined further in the next section. In the present section the entropy change, s, for the (A 1)process was obtained from the free energy change by differentiation. This method will be referred to as the “thermodynamic” method only to distinguish it from the procedure used to derive s in the next section.
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IV. An Electrostatic-Isothermal Method An alternative approach to the coupling entropy is examined in this section. This approach will be called an “electrostatic-isothermal’’ method, again only to distinguish it from the “thermodynamic” method in the previous section. Both methods have the same goal which is to determine the thermodynamic parameters of the reversible coupling and uncoupling procedures. The purpose of this electrostatic-isothermal method is to include certain energy components which were not explicit in the method of the previous section. For a reversible, constant V,T coupling step the expression combining the first and second laws is T ds = dQ = du + dw (14) In the Onsager system, the only kind of energy treated is potential energy; and the only kind of potential energy treated is electrostatic. Therefore, the only contributions to du are electrostatic potential energy changes. The term dw is the same electrostatic work element which occurred in the previous section. During a step dA of (A l),the change, du, in electrostatic potential energy within system (i + k) resides additively in (1)the change in interaction between f i of molecule i and the reaction field arising from the medium in region k (ik interaction), (2) the change in interaction between the positive and negative particles in molecule i itself, resulting from distortion of i by the reaction field (intra-i interaction), and (3) the change in interaction
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Energy and Entropy Relations in Highly Polar Liquids
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979 2341
among the moments located within region k (kk interaction): T ds = dUkk(X) + dui(X) dUik(h) + dw(X) (15) The form of the potential energy change from both the ik and kk contributions is defined in electrostatics_:16 if from a given source an electric field of intensity F ac$s across a dipolar charge distribution of total moment M , the chang_e in elestrostatic potential energy of i_nt_eraction between F and M is given by the change in -(F.M). The intra-i contribution to du is obtained from the following expression for ui(X), which is defined as the potential energy increase within molecule i, at coupli_nglevel A, due to intra-i distortion by the reaction field R(X).12 In this equation m(X) is the total moment in the cavity (permanent plus induced) when the coupling level is A:
ui(X) = [m(h)- rno(X)l2/2a
(16)
It is shown in the Appendix that the contribution to du resulting from the change in interaction among the moments in region k during the coupling step dX is dukk(X)=
(
(1 -a3
(24)
)g2mozX dX
In eq 14 and 15 the net change, du, in electrostatic potential energy within system (i + k) during the step dh is the sum of eq 21,23, and 24. As already noted, the last term of duik in eq 21 cancels dw of eq 22. Further, the second term of duik in eq 21 cancels dui of eq 23. T ds for a differential step of the (A 1) procedure is then a sum of the surviving potential energy changes within system (i k). These are dub(X), which is positive, and the first term of duik(h),which is negative. Neither of these energy changes results in or is the result of reversibly forced displacements of objects external to system (i k) because during dX the complete work, dw, is accounted for by the -R dmo term. Using the definition of g, we find that the result is
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From the earlier definition of g and because
eq 16 becomes (1 - ag)2(L)m$Xdh 26 + 1 a3 For a coupling level A, the expression for Uik(h), the electrostatic potential energy of interaction between the total moment in the cavity and its reaction field, is Uik(X)
Integration between
X = 0 and A
(25)
= 1 results in
(19)
= -2(X).fi(h)
+
Since f i ( X ) = f i o ( X ) ad(X),’ the contribution from the ik interaction to du, the total electrostatic potential energy change during a reversible coupling step, is dUik(X) = -(fio(X)-d8(X) + 2&(X)*d&(X) + fi (X)-d&( A) ] (20) Because R(X) = g f i ( X ) , l then, in explicit terms of the coupling level X and the coupling step dX, eq 20 becomes, after combining one of the two aR dR terms with the mo dR term:
(1 -cfga d 2gmo2hdh
+
(
L)gmo2X 1 - ag
dX) (21)
In eq 21, it is to be noted that the last term on the right side, -[l/(l- ag)]gmo2XdX, is the -R dm, term of eq 20. As discussed in the previous section, it is this particular electrostatic potential energy change within system (i k) which is responsible for performing the useful work, dw, on the environment during the coupling step dh. In the present section, as well as in the previous one
+
dw(X) =
(
L)gmozX 1 - ag
dh
That is, the -R dmo term is the complete contribution to the free energy differential during dX, but it is only one of several contributions to the internal energy differential. From eq 18, the contribution to du from the distortion of molecule i by the reaction field is
or
s=
(A)’( &,( ?)(-y-) (27)
The s of eq 27, obtained from the electrostatic-isothermal method, may be compared with the s of eq 8 derived from the thermodynamic method. Both expressions give the entropy change within system (i k) during the (A 1) procedure. It is noted that both expressions have similar forms and, in particular, that both approach the same limit in liquids of high dielectric constant. The small, negative s in eq 27, remarked on at the end of section 111, results from the presence in eq 26 of two contributions which are of similar magnitude but opposite sign. The positive term is the kk contribution; the negative term is the ik contribution. For (A 0) the kk contribution will be negative and the ik contribution positive.
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V. Discussion and Comparisons with Experimental Data In the (A 1) process the negative ik contribution to the net entropy change in eq 26 might have been expected on qualitative grounds. One aspect of the coupling procedure is an increase in structural correlations between the molecules in region k and molecule i (ik correlations) occurring against the randomizing forces of thermal molecular agitation. Perhaps less anticipated in eq 26 is the positive kk contribution to the entropy change during (A 1). Then for (X 0) the kk contribution is negative. Therefore, during (A 0) when system (i + k) changes coupled) to (rii,, uncoupled), leaving cavity i from (+io, intact, the molecules within region k become more ordered with respect to one another and less ordered with respect to molecule i. For (A 0) in highly polar liquids, the end
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2342
The Journal of Physical Chemistry, Vol. 83, No. 18,
1979
William H. Bishop
TABLE I: Entropies of Transfer between the Polar Liquid Phase and Vapor Phase for Polar and Nonpolar Moleculesa ASo(l+v),beu
H20i
H20i CH,OH~
CH,COCH, CH,COCH, C,H,OH C,H,OH
Aso(v+l),c eu
12.1 12.1 9.3
A. Isoelectronic Nef - 12.6 CHhg - 15.0 C A h - 5.0
11.5 11.5 15.6 15.6
B. Approximately Isochoric' cs, - 8.0 C6H6 -9.7 CS, - 10.9 C6H6 - 13.5
s , eu ~
-0.5 - 2.9 t 4.3
deviation from model,e eu -1.5 -3.9 t3.3
t 3.5 t 1.8
t 2.5
t 4.7 t 2.1
t 3.7 t 1.1
t 0.8
a The entropies are local standard entropies of transfer discussed in section V and ref 19. From fixed position in pure polar liquid t o fixed position in ideal gas phase, at 25 "C and 1 atm. From fixed position in ideal gas phase t o fixed position in pure polar liquid of first column, at 25 "C and 1 atm. Sum of As,'(l-tv) and As"(v-+l). e Algebraic difference between s and 1.0 eu. f Reference 9. g Reference 16. Reference 17. Reference 18.
effect of the two contributions to the entropy change is such that a t any temperature a net amount of heat, equivalent to the kinetic energy content of one molecular degree of freedom, enters the system (negative of eq 13). This does not mean that the total kinetic energy of system (i + k) changes. It is assumed that the total kinetic energy of system (i k) remains constant if T remains constant. Equation 26 for (A l),or the negative of eq 26 for (A 01, are each a sum of potential energy changes only. The theoretical uncoupling (or coupling) procedure does 0) not correspond to any single real process. The (A process is, however, thermodynamically equivalent to the sum of two appropriate real transfer procedures which could be considered to occur simultaneously. The process (A 0) is equivalent to the sum of (1)transfer of a polar molecule from its pure polar liquid into the vapor phase, and (2) transfer of the corresponding nonpolar molecule from the vapor phase into the same polar liquid. The process (A 1) would then be equivalent to the sum of the reverses of the two transfers. The word "corresponding" means here that, ideally, the two molecules being transferred (1) have the same number of electrons and (2) occupy the same volume in the polar liquid. In some cases described below the two relevant molecules are isoelectronic; in other cases, referred to below as approximately isochoric, the transfers involve polar and nonpolar molecules whose average molecular domains in the polar liquid are roughly equal. Table I gives for several systems the entropies of transfer between polar liquid phase and vapor p h a ~ e . ~ l ~Pro~-l~ vided that consistent standard state conventions for liquid phase and vapor phase are observed in each transfer, the result of adding the entropies of the two transfers (in order to produce the equivalent of (A 0)) will be independent of standard states. The entropies in Table I are the "local standard entropies of transfer" discussed by Ben-Naim;lg AS"(1-v) is the standard entropy change, on a mole basis, when a polar molecule is transferred from a fixed position in its pure liquid to a fixed position in an ideal gas phase, the process occurring at 25 "C and 1 atm; AS"(v-1) is the standard entropy change, on the mole basis, for transferring a nonpolar molecule from a fixed position in an ideal gas phase to a fixed position in the appropriate pure polar liquid, the process occurring at 25 "C and 1 atm. This particular standard state convention has been shown8J9to yield quantities which reflect exclusively local intermolecular effects generated by the interphase transfers; these entropies are obtained experimentally from the correctly signed derivative, with respect to T , of the quantity -kT In y where y is the Ostwald absorption coefficient.8 In keeping with the level of approximation intended and, as is common in liquid phase thermody-
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namics, no allowance has been made in this table for the fact that the model is a constant T,V (Helmholtz) system and the experimental data refer to constant T,P (Gibbs) systems. In the upper three transfer pairs in Table I the solvent and solute are isoelectronic; in the lower four pairs the solvent and solute are considered approximately isochoric (the assumption employed in the Barclay-Butlerl8 paper from which these transfer entropies are derived). The complete list of pure liquid molar volumes (mL) for the pairs used follows:2o CH,COCH, (73.4)/cs2 (60.2); CH&OCHs (73.4)/C& (89.0); C2HSOH (58.4)/CS2 (60.2); CzHjOH (58.4)/C& (89.0). The average absolu-te value of all the transfer entropies in Table I is large; JASI = 11.6 eu. A favorable comparison of the theoretical (A 0) with the experimental transfer pairs would require in each case that two of these absolutely large entropies oppose one another in such a manner that only +1 eu remains. The Onsager system is very indiscriminate with regard to any special properties a given polar liquid might possess. In view of this underspecification, a first procedure for comparing theory and experiment would be simply to add the sum of all entropies in the (1 v) column to the sum of all entropies in the (v 1) column and divide by the number of pairs. The result is 1.85 eu, not far from the theoretical 1.0 eu. The deviation, 0.85 eu, is 7 70 of the average absolute transfer entropy, 11.6 eu. From this undiscriminating viewpoint, the agreement between theory and experiment would seem satisfactory. A more stringent overall procedure, which permits the individuality of the various systems to come into play, is to compare the average absolute deviation from theory to the average absolute transfer entropy; this ratio is 2.4 eu:11.6 eu or about 20%. The agreement between theory and experiment is then mediocre, a result not unexpected in view of the simplicity of the theoretical model and the complexities of the real systems. Possibly more revealing and realistic would be to note that (1) the entropies of the combined transfers corresponding to ( h 0) tend, consistent with the theoretical model, to be absolutely small in all the highly polar liquids, and (2) the experimental results bracket the theoretical prediction of 1eu in a nonrandom manner; the nonaqueous polar liquid transfers produce s values which are small and positive while the water transfers yield s values which are small and negative. This is an example, among many other,8 in which the behavior of water diverges from that of other polar liquids. The negative of eq 26 gives Ts for (A 0) as the sum of a positive ik component and a negative kk component. This theoretical procedure is the thermodynamic equivalent of the experimental process described by fl AS" (1-4
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Energy and Entropy Relations in Highly Polar Liquids
The Journal of Physical Chemistry, Vol. 83, No. 18, 1979 2343
+
ASo(v-l)]. It might then seem indicated to identify the positive and negative terms in the theoretical expression with the positive and negative terms, respectively, in the experimental equation. For example, at 25 "C, NOukk/T for (A 0) in the Onsager model for water is -16 eu and N O U k k / T for ( A 0) in the methanol model is -5 eu. The experimental values of ASo (v-1) for the transfers of methane into water and ethane into methanol are -15 and -5 eu, respectively. The correlation, in these cases, between experimental and theoretical quantities can be shown to hold rather well over the temperature range 5-25 "C. However, it is not the closeness of the correlation that is stressed here, but is rather the relative magnitudes of the theoretical effect in the two polar liquids. There are reasons why NOukk/Tcannot correctly be identified with AS"(v-1): (1)U k k / T is strictly an electrostatic quantity while ASo(v-+l) is a more complex entity which has components for cavity formation and dispersion intera c t i ~ n ;(2) ~ ~in ,the ~ ~theoretical (A 0), ukk/T is part of a net entropy change and there is no way (A 0) can be performed that would allow ukk/T to be obtained as an isolated, independent quantity. Nevertheless, Ukk is the complete, formally separable and additive component which is strictly assignable to changes in the interactions of the molecules in region k with one another as (A 0). As such, ukkdescribes a single primal effect, namely, what happens to the mutual interactions within a group of polar molecules when these molecules change from a state where they surround another polar molecule like themselves to a state where they surround a nonpolar molecule. This effect can contribute to the entropy lowering process of transferring a nonpolar core into a highly polar liquid, but it cannot by itself account for or be identified with the entire process. Similar comments could be made regarding the relationship of the ik component of (A 0) to ASo(1-v).
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changes when nonpolar molecules dissolve in water. In this regard, the simple Onsager system may be performing a function in polar fluid thermodynamics similar to its function in polar fluid dielectric behavior;l>*the model is a phenomenologically sound theoretical construct which furnishes an idealized baseline behavior from which real systems will deviate to different extents for different individual reasons.
Acknowledgment. The author acknowledges his appreciation for very helpful discussions with, and suggestions from, Professors S. Prager and E. E. Schrier on the contents of this paper. Comments on this work from Professors S. L. Friess, F. R. N. Gurd, and F. M. Richards and from Doctors D. B. Millar and V. A. Parsegian are also gratefully acknowledged. This work was supported by the Naval Medical Research and Development Command, Research Task No. MF58.524.015.0044 and NIH Grant HD09140.
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Appendix During the coupling step dA in the (A 1) procedure the change in interaction among the moments located in region k results in a contribution, dukk(A),to the total electrostatic potential energy change. The derivation of this contribution, eq 24, is done here. In the Onsager system of Figure 1, the center of the cavity is taken as origin for a spherical polar coordinate frame with the moment Sopointing in the positive 2 direction. When the coupling level is A and_the permanent moment coupled is Aliio, the electric field E(r,8,4,A),acting on a small positive test charge q, placed at point (r,8,4) in region k, is given by3J5t21 Z(r,0,$,A) = &A) =
VI. Summary and Conclusion The unit vector lris located at (r,8,4)and_ points in the Some properties of the Onsager system of Figure 1may direction of increasing r; the unit vector is is located at be summarized as follows: (r,0,4) and points in the direction of increasing 8; at any (1)During (A O), the contribution to the net entropy (r,0,4), = 0. In the following the vector function will change of system (i + k) arising from changes in structural be abbreviated correlations of the ik type is positive. (2) During (A 0), the contribution to the net entropy change of system (i + k) arising from changes in structural correlations of the kk type is negative. (3) The net entropy change for (A 0), which is the sum @(A) has two components; (i) &,(A) is the component of of the two contributions, is small in the systems of interest E(A) produced by 6, the total qoment in the cavity, and approaches a value of R/2 (-1 eu) in liquids of high coqidered as a "bare" dipole; (ii) E,(A) is the component dielectric constant. of E(A) produced by the moments of all the molecules in (4) The functional forms and limiting values of this net region k as a result of their average orientations relative entropy change are the same in both the thermodynamic to the field from 6. That is, Ep(A)results from the poand the electrostatic-isothermal methods. larization of the medium by 6: ( 5 ) The uncoupling procedure in this model provides a 2(A)= &,(A) + ZP(A) (30) theoretical link between two experimental transfer entropies which at first examination appear unrelated. An since approximate, first-order level of agreement is found between theoretical prediction and experimental results. &,(A) = (%)2(r,8,$) 1 - ag With good reason, comparisons of water with other polar liquids have usually emphasized its u n i q u e n e ~ s . ~ 'It~ ~ ~ ~ - ~ ~ appears that such uniqueness is also manifest in the then with eq 28-31: present study as demonstrated by Table I; in water the negative entropy overrides the positive entropy instead of the converse. This demonstrates the presence in liquid water of specific entropy lowering effects not provided for in Onsager's system. Nevertheless, the strongly negative Zp(A)itself has two components: (1)&,(A) is the field at Ukk for (A 0) in the Onsager model for water shows that (r,0,4) resulting from the polarization of the particular more general entropy lowering mechanisms can also make molecule k' _whichoccupies the region containing the point contributions to the large, negative standard entropy (I-$,$);(2) &(A) is the field at (r,8,$) resulting from the
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The Journal of Physical Chemistry, Vol. 83, No. 18, 7979
Gerald R. Van Hecke
polarization of all the other molecules in region k: = Ek'(X)
Ep(X)
+ @k(h)
(33)
The type of electrostatic potential energy contribution of interest here, Ukk(X), results from interaction of the moment of molecule :k occupying the region containing (r,O,$),with the field Ek(X) from all the other molecules in region k. For simplicity in notation, the prime on one k in Ukk(X) has been dropped. In terms of the volume element, du, of region k, the differential contribution, f%.&k(X), to this energy is t?Ukk(X)
=
-Ek(X)'P(X) do
(34)
where P(X) is the polarization, or the electric moment per unit volume. P(h) a t (r,6',$) is given byI5 t-1P(X) = ---E(X) 4n
P(X) = 4 3(t ~ ( 2-+ t') 1)(-)Z(r,O,$) 1 - ag The field &,(A)
(35) (36)
is15
With the definition of g from eq 3, the result given in eq 24 is obtained:
References and Notes (1) L. Onsager, J . Am. Chem. SOC., 58, 1486 (1936). (2)J. G. Kirkwood, J . Chem. Phys., 7, 911 (1939). (3) B. Linder, Adv. Chem. Phys., 12, 225 (1967). (4) S. Nir, Biophys. J., 16, 59 (1976). (5) J. H. van Vleck, J. Chem. Phys., 5, 556 (1937). (6) R. Fowler and E. A. Guggenheim, "Statistical Thermodynamics", Cambridge University Press, London, 1952,Chapter 14. (7) K. Shinoda, J. Phys. Chem., 81, 1300 (1977). (8) A. Ben-Naim, "Water and Aqueous Solutions. Introduction to a Molecular Theory", Plenum Press, New York, 1974,Chapters 4,6-8. (9) H. S. Frank and M. W. Evans, J . Chem. Phys., 13, 507 (1945). (10) W. Kauzmann, Adv. Protein Chem., 14, 1 (1959). (11) G. Nemethyand H. A. Scheraga, J . Chem. Phys., 36, 3401 (1962). (12) C. Coulson and D. Eisenberg, Proc. R. SOC.London, Ser. A , 291, 454 (1966). (13)J. M. Deutch. Annu. Rev. Phvs. Chem., 24, 301 (1973) (14i J. G. Kirkwood. Chem. Rev..-l9. 275 11936). (l5j L. Page, "Introduction to Theoretical Physics", Van Nostrand, Princeton, N.J., 1952,Chapter 10. (16) A. Ben-Naim, J. Wilf, and M. Yaacobi, J. Phys. Chem., 77, 95 (1973). (17) M. Yaacobi and A. Ben-Naim, J . Phvs. Chem., 78, 175 (1974). (18) I. M. Barclay and J. A. V. Butler, Trans. Faraday SOC.,34, 1445 (1938). (19) A. Ben-Naim, J . Phys. Chem., 82, 792 (1978). (20) R. C. Weast, Ed., "Handbook of Chemistry and Physics", 55th ed, CRC Press, Cleveland, 1974. (21)J. T. Edsall and J. Wyman, "Biophysical Chemistry", Academic Press, New York, 1958,Chapters 2 and 6. (22) 0. Sinanoglu and S. Abdulnur, Fed. Proc., 24, S12 (1965). (23) R. A. Pierotti, J. Phys. Chem., 69, 281 (1965). (24) A. Ben-Naim, J . Phys. Chem., 69, 1922 (1965). (25) A. Ben-Naim, J . Chem. Phys., 54, 1387 (1971). (26) R. 6. Hermann, J . Phys. Chem., 75, 363 (1971). (27) M. Lucas, J . Phys. Chem., 76, 4030 (1972). (28) M. H. Klapper, Prog. Bioorg. Chem., 11, 55 (1973). (29) H. DeVoe, J . Am. Chem. SOC.,98, 1724(1976). (30) C. Tanford, "The Hydrophobic Effect", Wiley, New York, 1973. ~
(37) Therefore, from eq 28 and eq 32-37
Then, from eq 34 Ukk(h) = JBUkk(h)
=
-J[Ek(r,6',4,x).P(r,e,$,X)]du (39)
Substituting from eq 36 and 38, with dv = r2 dr sin 0 d8 d$, one finds that the integration, over r from a to 03, over 6' from 0 to n, and over 4 from 0 to 27r, gives
Use of Regular Solution Theory for Calculating Binary Mesogenic Phase Diagrams Exhibiting Azeotrope-Like Behavior for Liquid Two-Phase Regions. 1. Simple Minimum Forming Systems Gerald R. Van Hecke Department of Chemistry, Harvey Mudd College, Claremont, California 9 1711 (Received September 29, 1978, Revised Manuscript Received June 11, 1979) Publication costs assisted by Petroleum Research Fund and Harvey Mudd College
Many binary phase diagrams of mesomorphic components exhibit minimum azeotrope-like phase behavior. Regular solution theory is successfully applied to calculate such phase diagrams. In addition, an empirical correlation between the regular solution parameters and molar volume ratio of the components comprising a mixture is proposed and shown to have considerable utility.
