David L. Beveridge and Gary W. Schnuelle
2562
of benzene adsorption on both the aminated silica and the bound Magnus salt both indicate that only one process, physical adsorption is taking place. The adsorptions in both cases are therefore reversible, physical adsorption. The drop of the value of Qads below the value of QL for the benzene adsorption on the aminated silica may be caused by repulsions between the adsorbed species. The rapid return of the curve to the QL value may be caused by the onset of cooperative phenomena between the adsorbed phase and the forming multilayers. The Freundlich coefficients for the bound Magnus salt are given in Table VI1 and for the aminated silica in Table VIII. The average Qm value is 796 f 10 for the bound Magnus salt and for the aminated silica it is 860 f 5. There does appear to be a surface effect upon the value of Qm in this case. The ratios of CsiliJCbenzene range from 1.75 to 3.10. In all cases then fA(Silica) > f A~~~~~~~ and this ratio is greater than the one seen with the cyclohexane adsorption. We can conclude then that the benzene is more mobile on the aminated silica than on the bound Magnus salt. Conclusions The heats of adsorption derived from the isotherms of both cyclohexane and benzene on an aminated silica and a bound Magnus salt indicate that the interaction of the adsorbate with Magnus salt is greater than the interaction with an aminated silica. The adsorption interaction of both the cyclohexane and the benzene appears to be a one siteone adsorbate species interaction. Overall the adsorption of the benzene on the bound Magnus salt approaches chemisorption. However, all of the adsorptions were reversible physical adsorption. The system of bound Magnus salt is thus an ideal candidate for a model catalytic system and we are now studying the catalytic possibilities of this system.
Acknowledgments. We wish to thank the referees for their helpful comments and suggestions. We are indebted to the Robert A. Welch Foundation and the Research Corporation for supporting this research.
References and Notes (1) N. Kohler and F. Dawans, Rev. lnst. Fr. Pet. Ann. Combust. Liq.. 27, 105 (1972). (2) R. H. Grubbs and L. C. Kroll, J. Am. Chem. Soc., 93, 3062 (1971). (3) H. H. Weetall, Science, 166, 615 (1969). (4) W. Parr and K. Grohmann, Tetrahedron Len., 2633 (1971). ( 5 ) K. Schwetlick, J. Pelz, and K. Unverferth. Proc. lnt. Conf. Coord. Chem., 16th, 44 (1974). (6)D. C. Locke, J. T. Schmermund, and B. Banner, Anal. Chem., 44, 90 (1972). (7) M. Atoji, J. W. Richardson, and R. E. Rundle, J. Am. Chem. SOC., 79, 3017 (1957). ( 8 ) L. V. lnterrante and F. P. Bundy, lnorg. Chem., 10, 1169 (1971). (9) B. G. Anex, S.I. Foster, and A. F. Fucaloro, Chem. Phys. Lett., 18, 126 (1973). (10) D. 0. Hayward and B. M. W. Trapnell, “Chemisorption”, Butterworths, London, 1964. (1 1) It has been noted by the referees that the use of TIC14 to chlorinate the surface could lead to surface bound Ti species. They have suggested that SOClp would be a far better chlorinating agent. (12) H. C. Brown, W. R. Heydkamp, E. Breuer, and W. S. Murphy, J. Am. Chem. SOC.,86,3565 (1964). (13) R. N. Keller. “Inorganic Synthesis”, W. C. Fernelius, Ed. Vol. 2, Wiley, New York, N.Y., 1946, p 250. (14) This is the value of the surface area reported by the manufacturer (Waters Assoc.). (15) R. P. Messmer, U. Wahlgren, and K. H. Johnson, Chem. Phys. Lett., 18, 7 (1973). (16) R. J. H. Clark and C. S. Williams, J. Chem. SOC.A, 1425 (1966). (17) S. Brunaver, “The Adsorption of Gases and Vapors”, Vol I, Princeton University Press, Princeton, N.J., 1945. (18) T. Keii, T. Takagi, and S. Kanetaka, Anal. Chem., 33, 1967 (1961). (19) G. Halsey. J. Chem. Phys., 16, 931 (1948). (20) W. A. Steele. J. Chem. Phys., 25, 819 (1956). (21) American Petroleum Institute, Report on Research Project 44, Supp. VOl. A-62, 1972. (22) K.Unger, Agnew. Chem., lnt. Ed. Engl., 11, 267 (1972).
Free Energy of a Charge Distribution in Concentric Dielectric Continua David L. Beverldge, and Gary W. Schnuelle Department of Chemistry, Hunter College of The City university of New York, New York, New York 10021 (Received April 16, 1975) Publicatton costs asslsted by CUNY Faculty Research A ward Program
A theoretical treatment for dealing with the energetics of an arbitrary charge distribution imbedded in a central spherical cavity surrounded by two concentric dielectric continua is described. The results provide a general means for treating environmental effects using the continuum model. The form of the solution is particularly suited to identifying the contributions of the various dielectric regions.
I. Introduction The simplest model for the theoretical treatmentof ronmental effects on structure, properties, and chemical remodel.l actions in liquids and so~utionsis the the continuum model, the dissolved system is represented as discrete charges q k , k = 1, M or a charge distribution p(r) in a cavity imbedded in a structureless polarizable dielectric. The charges induce a reaction potential in the diThe Journal of Physical Chemistry, Vol. 79, No. 23, 1975
electric. The reaction potential acts back on the dissolved charges. The energy of interaction of the distribution with the environment is just the reversible work involved in charging the distribution in the presence of dielectric. This energy is thus a Helmholz free energy of polarization 1
A = - E qk@pR(rk)= 2 k
j’
p(r)+R(r)
dr
(1)
2563
Charge Distribution in Concentric Dielectric Continua
discrete distribution or points outside of a continuous where CPR is the reaction potential, d r is a volume element, charge distribution. The general solution of Laplace's and the integration extends over all space. equation at position (r,8,41 in polar coordinates is The development of the theoretical treatment of the continuum model began with the paper of Born2 on the revers+n ible work of charging an ion imbedded in a polarizable diCP = C (Bn,rn Pnm(cos8)eLm@(3) n=O m = - n electric. The case of a dissolved dipole was treated by Onsager3 in his classic treatment of the reaction field and the where the Pnm(cos8 ) are the associated Legendre polynodielectric constant of polar liquids. Kirkwood4concurrently mials and B,, and E,, are constants to be determined by treated the general case of an arbitrary charge distribution application of the boundary conditions appropriate for the in an extended Debye-Huckel treatment of chemical poparticular problem under consideration. tential. The theoretical methods developed in these papers The derivation is described in terms of a discrete distrihave found diverse application in theoretical c h e m i ~ t r y , ~ bution. For a collection of M point charges in a spherical physics,6 and bi01ogy.~ cavity of radius a imbedded in a polarizable dielectric conThe representation of solvent as a continuous homogetinuum (Figure l),the potential inside the cavity CPi and neous medium has some obvious shortcomings, since solthe potential outside the cavity a0are formulated indepenvent in the immediate vicinity of a dissolved molecule can dently, with boundary conditions applied so as to maintain be strongly influenced by the solute. Two extreme examthe continuity of potential across the cavity surface. Inside ples of this are dielectric saturation effects in aqueous ionic the cavity, the potential is taken as solutions6 where water molecules in the vicinity of an ion m +n are highly oriented, and hydrophobic bonding effectsg CPi = ti-l C (Bnmrn Pnm(cos@eim+(4) n=O m=-n where water molecules in the vicinity of a nonpolar solute or nonpolar region of a dissolved molecule form a clathrate. where t i is the dielectric constant inside the cavity, taken as Thus water in the vicinity of a dissolved molecule is not exunity in most applications but included here for generality. pected to behave like bulk water, and solvent is expected to The terms involving l/rn+lcan be identified with a multibe, to some extent, inhomogeneous. polar expansion of the central charge distribution Inhomgeneity in the vicinity of an ion was treated in the Frank and Wen model of ion hydration,1° where an ion in water is surrounded by three regions of solvent: (a) immobilized water molecules vicinal to the ion, (b) a disordered where zone of transition in structure as a result of opposing forces of the ion and more distant water molecules, and (c) structurally normal water or bulk solvent. Noye# has treated dielectric constant as a disposible parameter in a The terms involving rn define the reaction potential CPR inBorn charging model of ion hydration, and determined the side the cavity. local dielectric constant of water vicinal to a dissolved ion to be quite small. Gluekaufll and Booth12 have also contended with aspects of this problem. Friedman and Krishnan13 have detailed substantive aspects of the problem in a Outside the cavity, the potential is recent review article. There are two basic approaches to an improved theoretical treatment of environmental effects including dielectric saturation. The first solvation shell can be treated explicitsince the coefficients of terms in rn must be zero for the poly, with appropriate configurational degrees of freedom so tential to vanish properly at infinity. that saturation effects are manifest. The statistical thermoThe constants Bnm, C n m , and E n , are related by the dynamic supermolecule-continuum model currently being boundary conditions on the problem. The two boundary studied in this laboratory14J5is based on this idea. Alterconditions are imposed to assure (a) that the potential is natively, dielectric saturation effects can be represented in continuous across the cavity surface a purely continuum model if the continuum can be parti+i(r = a ) = CPJr = a ) (9) tioned into concentric shells characterized by different dielectric constants. We present herein the theoretical appaand (b) that the derivative of the potential, the normal ratus for dealing with the energetics of an arbitrary charge component of the displacement vector, is continuous across distribution imbedded in a central spherical cavity surthe cavity surface rounded by two concentric dielectric continua. (10) r=a 11. Background ar r=a The theoretical description of the free energy of a charge Since the Legendre polynomials constitute a linearly indedistribution in concentric dielectric continua is an extenpendent set of functions, the boundary conditions of eq 9 sion of Kirkwood's treatment of reaction p ~ t e n t i a l .We ~ and 10 can be applied term by term to ai and a0.Equation present in this section a brief review of relevant aspects of 9 leads to Kirkwood's derivation, sufficient for defining the notation and terms for the succeding development. The derivation is (11) based on Laplace's equation16 and eq 10 leads to v2* = 0 (2) En, ( n + 1)Cnm nBnman-l - ( n + 1)-= (12) an+2 valid for all points other than the sites of point charges in a an+2 01
+
-
The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
2564
David L. Beveridge and Gary W. Schnuelie 111. Theory
Consider now the charge distribution in a cavity imbedded in concentric dielectric continua, Figure 2. The central cavity contains the distribution and is characterized by radius a and dielectric constant ti. A first shell of polarizable dielectric extends from radius a to radius b with dielectric constant eloc, the "local" constant characteristic of the region vicinal to the central cavity. Extendihg beyond this from r = b to r = is the outside region, analogous to the region considered in the previous section and characterized by n dielectric constant to. The potential inside the cavity is +n
m
ti-'
@i
(Bnmr"
n=O m=-n
+
Figure 1. Definition of parameters for the simple continuum problem. where the origins of the terms in l/rn+l are the same as described in eq 5 and 6 of the preceeding section. The reaction potential inside the cavity becomes
Eliminating the Cnm,fromeq 11 and 12 we have
+n
w
@R
(13)
= ti-1
+
(Bnmrn Fnmrn)Pnm(cos %)eim$
n-0 m=-n
(21)
where t
= t0/ti
(14)
With Brim defined in terms of the basic characteristics of the charge distribution E,, and the cavity radius a , we have from eq 13 and 7 a general expression for the reaction potential of an arbitrary charge distribution. The polarization free energy of the system follows from eq 1:
where ei = 1 is assumed and
where the terms in Brim originate in the polarization of the local dielectric shell and the terms in Fnm originate in the polarization of the outside dielectric continuum. The potential in the local region is
=
k
1
qkqirknrlnpn(coS%kl)
(17)
where P,(cos & l ) is a simple Legendre polynomial. Each term in eq 15 represents polarization energy induced by the electric moment of order n of the distribution. For n = 0, QO= Z2 where Z is the net charge of the distribution, and 1 ( 1 - t) 22 Ao=--(18) 2 t a the ion polarization or Born charging2 energy. For n = 1, Q1 = y2, where p is the dipole moment of the distribution, and (19)
the dipole polarization or Onsager reaction field3 contribution. The n = 2, 3, . . . terms are the polarization energies induced by the quadrupole, octupole, . . . moments of the charge distribution. Kirkwood described the extension of this derivation to a classical and quantum mechanical continuous distribution in the original paper, ref 4. The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
C n=O m=-n
(Fnmr"
+ %)Pnm(cos%)eimm
(22) where the terms in l/rn+loriginate in the electrostatic potential of the central charge distribution and the terms in rn arise from the polarization of the outside continuum. Contrary to the situation described in the preceeding section, eq 8, the coefficients Fnmcan be nonzero since the local shell has finite bounds, Outside the central cavity and local shell, the potential is completely analogous to eq 8
pn (COS %k)pm" (cos %~)e-im('k-@') (16) Using the addition theorem of Legendre polynomials, eq 16 can be reduced to Qn
+ti
m
eloc = tioc-'
@O
= to-'
2
+n
n=O m=-n
Cnm
-Pnm(cos rn+l
%)eim@
(23)
AS above the coefficients Brim, Cnm, En,, Fnm, and Gnm are related by the boundary conditions on the problem and can be expressed in terms of the characteristics of the charge distribution Enm.The potential and its first derivative must be continuous across the boundary between the cavity and the local region and across the boundary between the cavity and the outside region. In the latter case, we have @loc(r= b ) = @& = b)
(24)
and (25)
Equation 24 leads to (26)
and eq 25 leads to Gnm nFnmbn-l - (n + 1) b n + 2 = -(n
Cnm + 1) bn+2
(27)
2565
Charge Distribution in Concentric Dielectric Continua
so that
and thus
With B,, in eq 34 and F,, in eq 39 expressed in terms of characteristics of the charge distribution E,, and the radii a and b , we have a general expression for the reaction potential of a discrete charge distribution imbedded in concentric dielectric continua. The polarization energy of the system follows from eq 1, using t i = 1 as before:
Figure 2. Definition of parameters for the concentric continua problem.
Eliminating the C,,
from eq 26 and 27 (28)
where cb
= fo/tloc
(29)
At the boundary between the cavity and the region of local dielectric, the boundary conditions are @i(r = a )
= @iOc(r= a )
(30)
and (31)
Application of eq 30 gives
+
Bnman Fnman +
and eq 31 leads to nBnman-l - (n
En, Giwn + 1) an+2 = -(n + I ) an+2
(33)
where a factor of nFnman appearing on both sides of the equation has been cancelled. Eliminating the Gn, from eq 32 and 33, and using eq 28, we have (34)
where tu’
[
= €a 1
1)(1- t Q ) ( l - tb) + ( n +((n+ l)cb + n ) b2,+l
and Ea
= tloclti
(36)
In the course of obtaining eq 34, we have as well
G,, = cQ’(Bn,a2n+1+E,,)
(37)
where the Q,’S are defined as in eq 15 and 16. We note in passing that the n = 0 term is implicit in eq 46 of ref 8. As a check on the derivation, eq 40 should reduce to eq 14 when eloc = eo and when eloc = t, and b = a. In the former case, t b = 1 and the second term in braces in eq 40 vanishes. Also tQ’ reduces to tQ, and eq 40 becomes identical with eq 14. In the latter case, tQ = 1 and t Q rreduces to tu, and the first term in braces in eq 40 vanishes. Equation 40 with ti identified with t and b with a reduces to eq 14.
IV. Discussion Equation 40 is thus a general expression for the free energy of an arbitrary charge distribution embedded in a spherical cavity surrounded by two concentric continua. The form of the solution is particularly useful for identifying the contributions of the local and the outer dielectric regions. The first term in brackets in eq 40 is due to the polarization charge in the local region, while the second term arises from the long-range polarization of the outer region. The parameter ea’ provides a measure of the combined effects of local region thickness, and of the increase in dielectric constant from local to outer regions. The parameters Q,’s which arise in eq 40 are identical with those given for the simpler Kirkwood problem. Indeed, the evaluation of the &, will be central to any similar discussion of environmental effects in which the charge distribution of interest is embedded in any type of spherical cavity. In this connection, note that the extension of this derivation to a continuous distribution involves merely redefinition of the Qn as detailed by K i r k ~ o o dThe . ~ use of the reaction potential as either a first order or finite perturbation in quantum mechanical problems follows along lines totally parallel to that described by Kirkwood4 and more recently by Hylton, Christoffersen, and HalL17 The results of this derivation provide a general means for treating environmental effects using the continuum model including dielectric saturation. The extension of this derivation to more than two concentric continua follows a development parallel to that presented in section 111, but is extremely lengthy. Acknowledgement. This study was supported by U.S. Public Health Service Research Career Development Award 6K04-GM21281-01A3 to D. L. Beveridge from the National Institute of General Medical Studies, a CUNY The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
2566
Gary W. Schnuelle and David L. Beveridge
Faculty Research Award, and the award of a CUNY Research Associate position to G . W. Schnuelle. References and Notes (1) C. J. F. Bottcher, "Theory of Electric Polarization", Vol. I, Elsevier, Amsterdam, 1973. (2) M. Born, Z.Phys., 1, 45 (1920). (3) L. Onsager, J. Am. Chem. SOC.,58, 1486 (1936). (4) J. G. Kirkwood, J. Chem. Phys., 1, 351 (1934). (5) T. Halicioglu and 0. Slnanoglu, Ann. N.Y. Acad. Sci., 158, 308 (1969); D. L Beveridge, M. M. Kelly, and R. J. Radna, J. Am. Chem. SOC.,06, 3769 (1974) (6) B. Linder, Adv. Chem. Phys., 12, 225 (1965). (7) 0.Sinanoglu in "Molecular Associations in Biology", B. Pullman, Ed.,
(8) (9) (IO) (11) (12) (13) (14) (15) (16) (17)
Academic Press, New York, N.Y.. 1968. p 427 ff; R. J. Kassner, J. Am. Chem. SOC.,95, 2674 (1973). R . M. Noyes, J. Am. Chem. SOC.,84,513 (1962). W. Kauzmann, Adv. Protein Chem., 14, 1 (1959). H. S.Frank and W.-Y. Wen, Discuss. Faraday. SOC.,24, 133 (1957). E. Gluekauf, Trans. Faraday. Soc.. 80, 572 (1964). F. Booth, J. Chem. Phys., 19, 391 (1951). H. L. Friedman and C. V. Krishnan in "Water-A Comprehensive Treatise", F. Franks, Ed., Plenum Press, New York, N.Y., 1973. D. L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 78, 2064 (1974). G. W. Schnuelle and D. L. Beveridge, J. Phys. Chem., following paper in this Issue. H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", Van Nostrand, Princeton, N.J., 1956. J. Hylton, R. Christoffersen, and G. G. Hall, Chem. Phys. Lett., 24, 501 (1974).
A Statistical Thermodynamic Supermolecule-Continuum Study of Ion Hydration. I. Site Method Gary W. Schnuelle and David L. Beveridge' Department of Chemistry, Hunter College ofthe City Univefsity of New York, New York, New York 10021 (Received April 16, 1974) Publication costs assisted by the National hstitutes of Health
The methodology and results of statistical thermodynamic supermolecule-continuum calculations on hydrated ions are described. Configurational averaging is carried out using the site method. Enthalpies, free energies, and entropies are calculated for the tetrahedral and octahedral coordination of the ions Li+, Na+, K+, F-, and C1-. The calculations utilize accurate quantum-mechanical representations of all interactions in the supermolecular assembly, and include no adjustable parameters. With the exception of the entropies for anions, all values agree satisfactorily with experimental results. The possibility of using extensions of this methodology for theoretical studies of polyatomic molecules in condensed phases is discussed.
I. Introduction
The statistical thermodynamic supermolecule-continuum model for the theoretical treatment of solvation energy and solvent effects on molecular structure and properties involves the calculation of a partition function for a dissolved molecule and its first solvation shell imbedded in a polarizable dielectric continuum.' The idea of treating solvent effects with a discrete representation of the solute and vicinal solvent molecules and a continuum representation for bulk solvent has been raised a number of times previously in the scientific literature, and dates back to the classic study of water and ionic solutions by Bernal and Fowler.2 Statistical aspects were first introduced with this model by Kirkwood3 in his early study of the dielectric constants of polar liquids. Contemporary use of this approach is found in theoretical studies of solvated electron systems, particularly in papers by Copeland, Kestner, and Jortner4 and by Fueki, Feng, and K e ~ a nBoth . ~ Newton6 and MOSkowitz, Boring, and Wood7 report quantum mechanical studies of hydrated electrons based on this model, but neglecting configurational averaging. Our recent interest in this approach arose in a consideration of the extension of current quantum theoretical calculations of solvent effects on biomolecular conformational stability as carried out in this laboratory using a continuum The Journal of Physical Chemistry, Vol. 79, No. 23, 1975
model8 and parallel studies elsewhere using the supermolecule modeLg When the problem is cast in the form of statistical thermodynamics, the relationship between the continuum model and supermolecule is clearly displayed and the union of these approaches into a supermolecle-continuum model is shown to accommodate, in principle, all the main factors which appear to contribute to solute-solvent interactions.' I t remains now to characterize the model in terms of (a) the methodology which renders supermolecule-continuum calculations of systems of chemical interest computationally tractable, and (b) the level of agreement to be expected between numerical supermolecule-continuum calculations and experiment. This series of two papers describes a statistical thermodynamic supermolecule continuum study of ion hydration, a prototypical solvation problem where both questions of numerical methodology and quantitative agreement between theory and experiment can be dealt with directly. Subsequent studies will deal with solvent effects on conformational stability.
11. Background We have previously described the supermolecule-continuum model in the context of solvent effects on conforma-
