Studies of Solvent Effects. 1. Discrete, Continuum, and Discrete

Studies of Solvent Effects

The Journal of Physical Chemistry, Vol. 82, No. 4, 1978 405

structure, led Srivastavalg to assign a planar structure to anthrone. Therefore, the enthalpy of formation (Table 111) and resonance energy (Table IV) of anthrone were estimated and compared with those of xanthone. The resonance stabilization energy of xanthone falls between those of anthraquinone and anthrone, both of which are known to be planar molecules. These findings suggest that xanthone may also be planar. The somewhat greater resonance energy of xanthone, when compared with that of anthrone, may be due to the contribution of the unshared electrons of the ether oxygen atom to the resonance conjugation. This is consistent with the reported results20-22that the xanthone molecules, to which the y-pyrone ring is coupled, are aromatic.

References and Notes (1) Throughout this paper 1 calm = 4.184 J and 1 atm = 101.325 kPa. (2) R. Hayatsu, R. G. Scott, L. P. Moore, and M. H. Studier, Nature (London),257, 378 (1975). (3) R. Hayatsu, R. E. Winans, R. G. Scott, L. P. Moore, and M. H. Studier, Fuel, submitted for publication. (4) D. W. Van Krevelen, “Coal”, Elsevier, Amsterdam, 1961. (5) A. F. Holleman, “Organic Synthesis”, Collect Vol. I, Wiley, New York, N.Y., 1941, p 552.

(6) W. N. Hubbard, C . Katz, and G. Waddington, J. Phys. Chem., 58, 142 (1954). (7) W. N. Hubbard, D.W. Scott, and G. Waddington in “Experimental Thermochemistry”, F. D. Rossini, Ed., Interscience, New York, N.Y., 1956, Chapter 5, p 75. (8) CODATA Bulletin No. 17, ICSU CODATA Paris, 1976. (9) S. W. Benson, “Thermochemical Kinetics”, 2nd ed, Wiley, New York, N.Y., 1976. (10) S. E. Stein, D. M. Golden, and S. W. Benson, J. Phys. Chem., 81, 314 (1977). (1 1) S. Stein, D. M. Golden, and S. W. Benson, Predictive Scheme for Thermochemical Properties of Polycyclic Aromatic Hydrocarbons, Appendix C, Report No. FE-2202-2, Stanford Research Institute, May 1976. (12) W. C. Herndon, Thermochim. Acta, 8, 225 (1974). (13) J. D.Cox and G. Pilcher, “Thermochemistry of Organic and Organometallic Compounds”, Academic Press, London, 1970. (14) R. C. C a s , S. E. Fletcher, C. T. Mortimer, H. D. Springall, and T. R. White, J. Chem. Soc., 1406 (1958). (15) A. Magnus, 2. Phys. Chem. (Frankfurt am Main), 9, 141 (1956). (16) F. Klages, Chem. Ber., 82, 358 (1949). (17) G. W. Whehnd, “Resonance in Organic Chemistry”, Wiley, New York, N.Y., 1955. (18) S. C. Biswas and R. K. Sen, Indian J. Pure Appl. phys., 7,408 (1969). (19) S. N. Srivastava, Acta Crystallogr., 17, 851 (1964). (20) J. Gayoso, H. Bouanani, and A. Boucekkine, Bull. SOC.Chim. Fr., 3-4, 538 (1974). (21) H. Bouanani and J. Gayoso, Bull. SOC.Chim. Fr., 3-4, 545 (1974). (22) G. H. Stout, T. S. Lin, and I. Singh, Tetrahedron, 25, 1975 (1969).

Studies of Solvent Effects. 1. Discrete, Continuum, and Discrete-Continuum Models and Their Comparison for Some Simple Cases: NH,’, CH30H, and Substituted NH4’ P. Claverie,” J. P. Daudey, J.

Langlet, B. Pullman, D. Piazzola,

Institut de Biologie Physico-Chimique, Laboratoire de Biochimie Thgorique associh au CNRS, 75005 Paris, France

and M. J. Huron Institut Francais du P6tro/e, 92, Rueil-Malmaison, France (Received March 28, 1977)

In order to study solute-solvent interactions, three different models are considered: (1) a “discrete” model, according to which a finite number of solvent molecules are placed around the solute molecule, and the total interaction energy is calculated by simplifiedformulas; (2) a “continuum” model, according to which the solvent surrounding the solute is simulated by a continuum medium (the method of calculation takes into account the actual shape and charge distribution of the solute molecule and is an extension of the solute-solvent problem of the method previously used by Huron and Claverie for pure liquids); (3) a “discrete-continuum” combined model, according to which a small number of solvent molecules (corresponding to so-called solvation sites) interacting strongly with the solute are treated as discrete, while the remaining solvent is simulated by a continuous medium. Then these models are applied to the study of simple cases: solvation of NH4+in water and ammonia, solvation of methanol in water. It appears that some caution is necessary, because the various theoretical steps (into which the solvation process is decomposed) do not always lead to similar values in different models, but a satisfactory agreement (between the various methods and previous available results) may nevertheless be obtained concerning the total values corresponding to the complete solvation process. Since such a comparison of the various methods reveals their respective shortcomings, it may be expected that the simultaneous use of such different models will finally lead to an optimum methodology and thus allow us to deal sucessfully with various problems involving solvent effects.

Introduction In recent years there has been considerable interest in the interpretation of solvent effects on the properties of molecules in particular in relation to conformation^^-^^ and electronic ~pectra.l~-~O The first purpose of this paper is t o present simple methods for calculating the molecular interactions in liquids and to correlate the calculated energies with thermodynamic properties. In liquids, the molecules are close to each other and we have to consider the interactions of a given molecule with all the surrounding ones. The large number of molecules 0022-3654/78/2082-0405$01 .OO/O

is one of the major difficulties of the treatment of liquids which leads to the use of approximate models: (1)We may consider the molecules of the solvent around the solute as individual entities, in which case, we use a microscopic representation of the solvent, and the simplification concerns the number of molecules of the solvent around the solute taken into account. Generally in these representations, only the solvent molecules near the solute are retained.1J-26 (2) We may represent the solvent as a continuous medium, in that case we consider all the solvent molecules

0 1978 American

Chemical Society

406

The Journal of Physical Chemistry, Vol. 82, No. 4, 1978

surrounding the solute, but we do not give these molecules a discrete representation. They form a polarizable dielectric continuous medium characterized by macroscopic proper tie^.^^-^^ Whatever be the choice made, we have to calculate the interaction energies between the solvent and solute molecules. It seems possible nowadays to do so by quantum mechanical computations. For the discrete model A and B Pullman’ has obtained substantial results, using the “iupermolecule” approach, in the framework of the Roothaan ab-initio self-consistent field molecular orbital procedure38 with a Gaussian basis set: STO 3G basis set39or a split 4-31G basis.40 On the other hand, both microscopic and macroscopic models may be used for calculating the energy with semiempirical potentials. According to a second-order perturbation treatment, the interaction energy of a system (A-B) of two molecules may be split into the sum of long-range contributions (electrostatic, Eel“, dispersion, ED,and polarization, Ep,energies which vary as l / r n ) and short-range contributions (repulsion energy, ER,which decreases exponentially with distance). The electrostatic energy is calculated between atomic charge distributions of the molecules. The dispersion energy may be calculated using Kitaygorodsky’s formula41with the help of the van der Waals radii or the London approximation, involving the bond polarizability tensors,42or a Kihara type potential.43 The repulsion energy may be calculated using Kitaygorodsky’s exponential type f ~ r m u l a . ~ ’ The polarization energy of each molecule characterized by its bond polarizabilities is obtained in terms of the total electric field created by the charge distribution of the other molecule.45 In this paper we will present and compare three semiempirical models: a discrete model described by Daudey et al.,25,26 a continuum (the Huron and Claverie model for and a “discrete-continuum” (combining both previous models). Then we study the accuracy of these three models on some simple cases. We proceed in three steps: (1)In view of its use in the discrete-continuum combined model we verify that the discrete model correctly obtains the different preferential solvation sites of the solute and the binding energies between the solute and the solvent molecules. ( 2 ) In order to check the accuracy of the three models for the study of energies, we compute the solvation energy in water of the ammonium ion (NH4+)and of methanol. (3) We test the ability of three models to reproduce the well-known factz4 that the substitution of the hydrogens of NH4+by one, two, or three methyl groups decreases the value of the solvation energy with respect to the corresponding value for the unsubstituted ion. From the technical point of view, we mention that, in these calculations, we have used CNDO charges, and we have taken into account the atomic (hybridization) dipoles; but, instead of using these atomic dipoles directly we have replaced them by effective (atomic) charges that have been added to the usual CNDO charges in order to obtain effective total (atomic) charges.44

Method. Discrete, Continuum, and Discrete-Continuum Models I. Discrete Model. In this model, several solvent molecules are placed around the solute, and the total interaction energy of this complex system is calpulated according to

Claverie et al.

i< j

where i = 1 corresponds to molecule A, i (or j ) > 2 corresponds to molecules B, i.e., as the sum of four contributions: electrostatic, dispersion, repulsion, and polarization. The detailed formulas are given e l ~ e w h e r eand ~~~,~ need not be repeated here. As an important practical point, let us mention that, at very short interatomic distances, it is appropriate to modify the analytical expressions of the dispersion, repulsion, and polarization terms, in order to avoid spurious energy minima, or spurious divergencies in statistical mechanical integrals. The modification used in the present work is the one described previously by Caillet et In order to calculate the total interaction energy E of the system (one solute molecule + N solvent molecules), it is necessary (a) to choose some value of N , and (b) to choose some positions for the molecules. These two points of the method are described in ref 25 and 26. Then usually qome minimization of this energy is performed, as a substitute for a statistical mechanical treatment which would be markedly more difficult, The minimization processes have been performed using the .method of Chandler.46 In practice we have to calculate

(2) - Esolvent + solute Thus we have to perform two kinds of calculation: (a) a calculation of the minimum energy of the system (solute + N solvent molecules) and (b) a calculation of the minimum energy of the system ( N solvent molecules alone). In that way we take into account the reorganization of the solvent in the presence of the solute molecule. With this method it is also possible to try to evaluate the cavitation eliergy which is given by Esolv = Esolvent done

E

= Esolvent alone - &(solvent surrounding an empty cavity

(3)

cqvity of the size of the solute)

Thus this model may be satisfactory but the computation time may become very long due to the .minimization processes involved. We have indeed to perform several series pf minimizations: solvent alone, solvent + solute, and eventually solvent + empty cavity. Furthermore it may be dangerous to consider the first so1v;ation shell only, since the electrostatic energy (important for polar niolecules) decreases very slowly with the distance. As an example, Monte-Carlo calculations which give rather exact evaluations of the statistical averages deal with a rather large number of solvent molecules: 64 water molecules surrounding the methane molecule in the calculation by Dashevsky and Sarki~ov.~’ 11. Continuum When we want to proceed from the “microscopic” to the “macroscopic” representation using empirical potentials, we have to replace the discrete summations by integrals for calculating the dispersion and repulsion energies. Concerning the electrostatic energy, the solvent is now represented by an infinite homogeneous continuous medium characterized by its dielectric constant D. The solute is represented by an empty cavity, inside which some electrostatic charge distribution is placed. The dielectric solvent is polarized by this charge distribution, and therefore generates a “reaction field”, which interacts with the charge distribution. The problem thqn consists of calculating the reversible work of “charging” the solute molecule (i.e., varying continuously the charge distribution from zero.to its final physical value) in the presence of this reaction field, thus obtaining the variation AF of the



Helmholtz free energy corresponding to this process.47 Of course, the shape of the cavity and the charge distribution should correspond as closely as possible to the corresponding characteristics of the solute molecule. However, for reasons of mathematical simplicity, drastic simplifications of both the shape of the cavity and the charge distribution have been very common in earlier ~ o r k s . 4More ~ ~ recently, ~ Huron and C l a ~ e r i considered e~~ cavities of quite general shape (e.g., the van der Waals ,volume obtained by the reunion of the van der Waals spheres of all atoms of the solute molecule) containing an arbitrary set of point charges and eventually dipoles (in actual practice, these charges and dipoles are placed at the positions of the atoms of the solute molecule); of course, the solution is no more analytic, and a numerical procedure has to be devised. A basic procedure underlying the continuum model is the evaluation of the variation of some thermodynamic functions (essentially the Helmholtz free energy F , and sometimes the internal energy E ) between two thermodynamic states, by resorting to the “charging-parameter’’ method. More precisely, let us denote the total interaction energy of the system studied by U ( Q )(where Q denotes the set of all position and angular variables of the molecules of the system) and the corresponding canonical probability density for the variable Q by f(a)= e - u ( a ) / d T / j , - u ( a ) / k T dfi (4) Then, we may consider this interaction energy or rather some specific part of it (e.g., the solute-solvent interactions, or, even more specifically their electrostatic part on one hand, and dispersion-repulsion part on the other hand) to be multiplied by a so-called “charging” or “switching” parameter l, so that this interaction is suppressed when 4 = 0 and may be continuously “switched on” up to its real physical value by varying l from 0 to 1. The amount of work associated with such a reversible charging process gives the variation AF of the Helmholtz free energy between the “uncharged” and the “charged” states of the system (it must be emphasized that the word “charging” is a general way of speaking, since all these notions apply to any kind of interaction, and not to the electrostatic one only). Then, as recalled in Appendix C of ref 36a (where references to previous works are given), the variations AF and AE corresponding to such a charging process are given by

where

(7) The subscript E means that the average { ) E is to be taken with the distribution function f(4, SZ) corresponding with the interaction energy U ( l , SZ). If 4 applies to an electrostatic term (some charge q is multiplied by E), aU/al is essentially an electric potential (derivative of an energy with respect to an electric charge), and {aU/at)[ therefore appears as a mean potential. By analogy, this name is kept in the general case. Now, one of the basic ideas of continuum models is to replace the exact statistical-mechanical evaluation of this

mean potential by some simplified “macroscopic-type” calculation: thus, in the Debye-Huckel theory of electrolyte solutions, the Poisson-Boltzmann equation is used for that purpose, and if we restrict ourselves to a solvent of neutral molecules only, the Laplace equation will be sufficient, as described in previous work.36b,eThe essential problem then becomes choosing appropriate initial and final states, in order that the associated switched-on interaction corresponds with a mean potential which is easy enough to evaluate. Now, essentially two important classes of charging processes (named respectively “Debye-Huckel” and “Guntelberg-Muller” processes: see, e.g.,53and Appendix C of ref 36a) may be distinguished. (1) In the Debye-Huckel process, the switching parameter l acts on the interactions between all molecules of the system (solute and solvent molecules in our case). The initial state could be either the “perfect gas” state (all interactions switched off) or some “hard molecule” state (some suitable hard core is associated with each molecule, and the total interaction energy is 0 when these hard cores do not intersect and +a if there is a t least one hard core intersection). The attractive feature of the “hard molecule” initial state is the closeness of its distribution function f([ = 0, 9 ) to exact one f(4 = 1, Q), owing to the fact that the structure of a liquid is qualitatively determined by the harsh repulsive part of the intermolecular interactions: as a consequence, it may be possible to neglect terms involving a f l a l , such as the second term in the right-hand side of eq 6. However the price to pay is the requirement of evaluating the variations of thermodynamic functions between the “perfect gas” and “hard molecule” states. As shown in the Appendix, calculation of the electrostatic part of the energy according to eq A.1 requires some evaluation of the dielectric constant D ( [ ) for the partially “charged” liquid. Anyway, since the Debye-Huckel process treats the interactions between all molecules on an equal footing, it seems a priori more suited for the study of an homogeneous liquid than for the study of a solute-solvent problem, where the solute molecule is of special interest. Thus we now consider the second type of process. (2) In the Guntelberg-Muller process, the switching parameter 4 acts only on the solute molecule; the interaction terms between the solvent molecules themselves are already completely switched on in the initial state. However, as was pointed out for the Debye-Huckel process, two different initial states may be considered concerning the solute: the pointlike (perfect gas) solute (i.e., the solute-solvent interaction terms reduced to zero), and the hard-core solute (the solute-solvent interaction reduced to a hard-core one). As mentioned previously, the advantage of the second choice is that the turning on of the exact solute-solvent interaction is accompanied by a rather small change of the distribution function f(4, 0) (nevertheless, this change cannot be completely neglected in the electrostatic interaction, as we shall see below), In this second choice, it is necessary to evaluate first the variation of thermodynamic functions when going from the “pointlike” to the “hard-core” solute; this corresponds to the creation in the solvent of a cavity having the size and shape of the solute molecule. We have finally adopted the Guntelberg-Muller process with the hard core solute as an intermediate state for treating the solute-solvent problems. Thus the solvation process is dealt with in two steps. (A) Creation in the solvent of a cavity suitable for accomodating the solute molecule.

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(B) Introduction into the cavity of a solute molecule which interacts with the solvent. We have to determine (a) the internal energy change AE associated with the “turning on” of the nonpolar part of the intermolecular energy (dispersion and repulsion), and (b) the internal energy change AE associated with the “turning on” of the polar part of the intermolecular energy (electrostatic). As explained in the Appendix, this energy may be split in two parts: the solute-solvent energy and the variation of the solvent energy itself (due to reorganization of the solvent which accompanies the turning on of the charges of the solute molecule). The former term is negative (stabilization) while the latter is positive (destabilization) but smaller than the absolute value of the former. In the present work, the solute-solvent term only is explicitly evaluated; the other term is implicitly taken into account through the use of an empirical factor f by which the solute-solvent term is to be multiplied (see section B). (A) Eoaluation of t h e Cavitation Energy. Earlier attempts for evaluating the cavitation free energy (and internal energy) made use of the m a c r o s ~ o p i c or ~ ~mi-~~ croscopic surface tension;58this last method has been used by several authors in studying solute-solvent syst e m ~ . However ~ ~ , ~in ~the~present ~ ~ version ~ ~ ~ of the continuum model we have decided to use the Pierotti formulae61 as a starting point for evaluating the thermodynamic functions associated with the cavitation, because Pierotti obtained rather successful results in agreement (i) with the qualitative concepts proposed by Frank and Evans62or Nemethy and S ~ h e r a g aand ~ ~(ii) with the recent results obtained by Dashevsky and S a r k i ~ o vusing ~ ~ the Monte-Carlo method in order to evaluate the thermodynamic functions of solvation in water. However we have introduced in Pierotti’s formulae some change suggested by the comparison with the Monte-Carlo results for water; namely, we kept unchanged the Pierotti formula for AF,but we introduced the factor -0.5 before A E (thus converting 0.9 kcal/mol to -0.45 kcal/mol, which is now very close to the Monte-Carlo result of -0.4 kcal/mol). The need for such a change is not astonishing, since the Pierotti formulae were based upon a hard-sphere model of the liquid (scaled-particle theory), and therefore cannot be expected to describe directly the behavior of a liquid such as water, where electrostatic interactions are important, and result in rather specific reorganization of the molecule around a cavity (“iceberg effect” of Frank and Evans;62see also ref 63). The formulae that we used for the variations of F , E , and S corresponding to the creation of a spherical cavity with diameter o2in water are therefore (with cr2 expressed in Angstrom units) AF, = ( 1 0 0 0 0 ~~ 2 0~ 0 8 0 ~+~1141) cal/mol

(8)

A E , = -8.212

(9)

where 0 1 2 = (2.75 + 02)/2 Z = ( 1 0 . 6 8 0 ~-~2 3 . 9 1 0 ~+~14.30) cal/mol AS, = (A U, - AF,)/T cal/mol/K

(10) (11)

For a molecular cavity with a general shape made of pieces of the van der Waals spheres of the atoms, we evaluated the quantities AF,, AE,, and AS, as sums of increments from these pieces of sphere, each increment being proportional to the area A, of the piece of the sphere ( i ) (with diameter denoted g1 = 2 X, R,). Thus, denoting AFc(crk)the value given by eq 8 for oz = ob,the increment corresponding since to the area A , of the sphere i is AF,(crJ(A,/(m,*)),

ru12is the total area of the sphere, and the total AF, for the molecule is then

AF, = ZAi/(~~i2)AF,(~,) i

(12)

and we have similar formulae for AE, and AS,. (B) Evaluation of the Solute-Solvent Energy. We used the continuum model proposed previously by Huron and C l a ~ e r i e .Although ~~ the applications presented in this paper concerned pure liquids only, the method itself was described for the general case (a solute molecule eventually different from the solvent); thus, we will only mention the two following complements: (1) Concerning the dispersion and repulsion energies, (RLe, RJe)and KIJT (R,“,RJe) the calibration constants KLJD (ref 36a, section 11.3) actually depend on a single variable, namely, the ratio R,/R,, and are displayed for a number of values of this ratio (ranging from 0.565 to 1.77) in ref 36c, Table LA of Appendix A. In the present work, we used the values of this table, and in cases where R,”/R: = X,R,/(h,R,) did not coincide exactly with one of the entries of the table, we used an interpolation procedure for evaluating the needed “discrete summation” values (the specific program used was the subroutine ALI of the IBM Mathematical Subroutine Package, which performs an Aitken-Lagrange interpolation). (2) Concerning the electrostatic energy, since we have used the Guntelberg-Muller charging process, the factor 1 / 2 in eq 4 of ref 36c is rigorous while it would not be for the Debye-Huckel charging process (see the Appendix of this paper and Appendix B of ref 36c). Furthermore, the corrective factor which multiplies the electrostatic free energy AFelechas a twofold origin in solutesolvent systems: (1)Following B j e r r ~ m it, ~is~ possible that, in a continuum model, the macroscopic dielectric constant does not give a good enough representation of the statistical behavior of the solvent molecules in the immediate vicinity of the solute molecule (recent references are given by Beveridge and S ~ h n u e l l e ~This ~ ) . reason for introducing a corrective factor was already mentioned in ref 36c. A peculiar aspect of this problem concerns the choice of some suitable boundary surface Z; as was already pointed out for the dispersion and repulsion terms (see ref 36a, section 11.3), the use of the surface Z made from the atomic spheres with radii XJ?, may be expected to lead to overestimated values; thus, if this surface is actually used, some “calibration factor” smaller than unity must be introduced. (2) As emphasized in the Appendix, the complete AF corresponding to the “turning on” of the charge distribution of the solute involves a (negative, Le., stabilizing) solute-solvent part, and a (positive) solvent-solvent part (reorganization of the solvent around the solute during the charging process of the latter). Now, we have at the present time a method for evaluating the solute-solvent part only, hence the necessity of reducing this part by some suitable factor, in order to compensate for the lack of an explicit evaluation of the solvent reorganization effect. The numerical results obtained in the framework of the “discrete” model give some order of magnitude for this effect (see paper 2); for NH4+in water, the solute-solvent part amounts to about -80 kcal/mol, while the reorganization of the water molecules results in an energy loss of about +10 kcal/mol. On the whole, we were lead to use for the continuum model of water an (empirically fitted) factor f N 0.63 which takes into account the two above-mentioned effects (see the beginning of section I1 in the Applications part). Thus, if we denote the free energy change associated with the solute-solvent interaction only by AF,-,,, we



W State 1

State 2

State 3

1

Corresponding energy change (with respect to the previous state): -ilE (interaction between the nd solvent molecules and the surrounding solvent) (the nd molecules of solvent playing the role of a solute). State 3. Solute + nd solvent molecules + solvent with cavity suppressed. Change -AE (cavitation corresponding to nd molecules). State 4. Solvent without cavity ‘‘nd solvated solute” (i.e., solute surrounded by a discrete solvation shell made from nd solvent molecules). Change AE (interaction between the solute molecule and nd solvent molecules). State 5. nd solvated solute + solvent with a cavity having the size and shape of the nd solvated solute. Change AE (cavitation corresponding to the nd solvated solute). State 6. nd solvated solute surrounded by the solvent. Change AE (interaction between the nd solvated solute and the solvent). The complex (solute nd solvent molecules) plays the role of a big solute molecule. The total energy change associated with the solvation therefore appears as a sum of five energy changes, among which the steps 1-2, 2-3 and 4-5,5-6 are evaluated according to the continuum model, while step 3-4 is evaluated according to the discrete model.

+

+

State 6

State 5

State

4

Figure 1. The states describing the solvation process in the combined discrete-continuum model (Ste represents solute).

evaluate the total free energy change formula

AF according to the (13)

AF = fAF,-,,

as corresponding to the “turning on” of the charges of the solute. Since previously we have calculated the dispersion and repulsion interactions that give a variation of internal energy AE c1 AF (for discussion of this assumption, see the Appendix and also ref 36a, Appendix C), it is necessary to calculate the electrostatic internal energy E connected with the free energy AFby the Gibbs-Helmholtz relation

A E = AF

a

-(AF) aT 111. T h e Discrete-Continuum Combined Model. When the solute molecule is ionic, there exist specific solvation (or hydration) sites: i.e., some regions of the solute molecule which strongly attract solvent (or water) molecules, It seems that the continuum model alone is not very well adapted to represent these very specific interactions.’ Thus it may be useful to use a “combined model”, namely, the following two-step treatment: (i) search for specific solvation (or hydration) sites using the “discrete model” (in that case the “discrete model” will be used with a small number N of solvent molecules); (ii) study supermolecules (solute with the previously determined few solvent molecules) by a continuum model. The use of such a combined model (called “supermolecule-continuum’’ model) has also been considered by Beveridge and S ~ h n u e l l e . ~ ~ ~ ~ ~ Thus the calculation of the mean energy of the first shell of solvation (by the discrete model) will be completed by the calculation of the interaction energy between the continuum and the first shell. In the continuum model the initial state is the system: solvent surrounding an empty cavity of the size of the supermolecule (solute + some “solvent molecules”). We will also have to calculate the cavitation free energy and the cavitation internal energy. The theoretical thermodynamic process corresponding to solvation in the discrete-continuum model may be described as follows (Figure 1): State 1. Solute + solvent separated (nd solvent molecules, which are intended to represent the discrete solvation shell, are explicitly shown in Figure 1). State 2. Solute + nd solvent molecules + solvent with a cavity corresponding to the nd molecules remaining. -

Applications I. T h e First Solvation Shell in the Discrete Model. Stepwise Hydration and Ammoniation of NH4+. There are experimental data for the energies for stepwise addition of H20 and NH3 molecules to NH4+.6SActually the gas phase equilibria NH,I(H,O), = NH;(H,O),-, NH,+(NH,), = NH,+(NH,),-,

+ H,O + NH,

have been measured with a pulsed electron beam high pressure ion source mass spectrometer yielding the AGO, AHo, and ASo values for equilibria from n = 1to 5. It was found that AHfl,n-land AGfl,,l are larger for NH3 than the corresponding values for HzO. The difference is the largest for the first step ( 1 , O ) and decreases progressively until a reversal occurs at the ( 5 , 4) step. This problem has been studied theoretically by Pullman and ArmbrusterZ1using STO 3G and 4-31 G basis sets. Their computations have reproduced all the essential features of the experimental data. These authors have given reasons for the initial preference of NH4+for NH3 over H 2 0 and for the reversal of the relative affinities in the second Furthermore these computations indicate a preference for an arrangement of the solvent molecules (both NH3 and HzO) around the ion involving direct hydrogen bonds as in I, rather than inclusion be-

\

/

-0,

0-

H20

I

\

I

‘\

I

I

I1

tween two NH bonds as in 11. Thus, this problem seems to be convenient for testing our “discrete model”. We have studied successively (1) the formation of a complex between NH4+ and one molecule of water or ammonia (complex geometry and binding energy), and (2) the attachment of the next n molecules of HzO (or NH3) with n = 2, 3, 4.

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Claverie et al.

TABLE I: Total Energy of NH,+. . .OH, Complex for Various Arrangements of H,Oa Arrangement I I1 I11 IV

ED -3.29 -3.46 -3.30 -3.37

ER +7.62 + 7.25 +7.50 + 7.44

EP,

Eel -15.78 -15.56 -15.75 -15.78

-0.93 -0.73 -0.91 -0.88

EPw -5.56 -4.91 -5.50 -5.43

E+,+ -17.94 -17.41 -17.96 -18.02

a Total energy (in kcal/mol) Etot = Eel + E R t E D + EPa + E.‘, Eel is the electrostatic interaction e n e r p ; E R the repulsion interaction energy; E D the dispersion interaction energy; E‘, the solute polarization energy; and E sv the solvent polarization energy. Arrangement I along the direction of N-H a t the equilibrium distance, I1 along the bisector of HNH, and I11 and IV as shown in Figure 2.

( kca I /mole)

t

I I I

I

I

I

I

I I

-l4I

I

IH1/

I

I I

-

2.2

2.4

2.6

2.6 dNo(%)

Flgure 2. Variation of the binding energy of H28 upon approach toward NH4+: (I) along an NH bond; (11) along the bisector of HNH.

(1) Monosolvation. (a) Geometry. The approach of one single molecule of water toward the NH4+ion along the NH direction and along the bisector of HNH is illustrated in Figure 2. In agreement with Pullman and Armbruster,21 we find the most favorable path of approach along the NH direction. The calculated equilibrium distance, N.-H = 2.65 A, is larger than the STO 3G value (which is underestimated as underlined in ref 21) and in agreement with the more reasonable 4-31G value. For the bisecting approach, the minimum occurs for a distance N-0 = 2.40 A. The difference in energy between the two configurations (I and 11),although in favor of the direct NH approach, is much smaller than in the STO 3G computation. Furthermore, our minimization yields two other configurations with practically the same energy as configuration I (configurations I11 and IV in Figure 3). These positions have not been explored in SCF computations.21a Concerning the complex NH4+-NH3, the present model indicates, in agreement with ref 21a, that the approach along the NH direction is the only favorable one. (b) Energy. Table I gives the total interaction energy (in kcal/mol) for the complex NH4+--OH2and the decomposition of this energy into different terms (dispersion (ED),repulsion (ER),electrostatic (Ee1), solute polarization (E:), solvent polarization (EsVp)) for the different dispositions of H 2 0 with respect to NH4+ defined in (a). Table I shows that (i) the calculated energy difference between configurations I and I1 is very small (0.6 kcal/ mol), as compared to the value calculated by the STO-3G ab initio method2I (21.3 kcal/mol); (ii) the main difference between type I and I1 comes from the solvent polarization energy which is stronger (-5.56 kcal/mol) when H 2 0 is along an NH bond than when H 2 0 is along the bisector of HNH (-4.91 kcal/mol); (iii) the different energy terms are nearly the same for the three minima I, 111, IV.

xd

I I

Ill

IV

Figure 3. Favorable arrangement for the complex NH:-OH2: (I) along an NH bond, water in plane yOr, NO = 2.65 A; (111) oxygen of water in a plane bissecting H3NH4;plane of water perpendicular to X O Z ; (IV) oxygen directly above an hydrogen of NH4+; plane of water perpendicular to x O r ; OH, = 2.7 A.

TABLE 11: Total Interaction Energy of the Complex NH,‘. .

ED

ER

Eel

EP,

-4.65

t11.19

-18.10

-1.63

a

E’, -8.29

Etn+ -21.48

Energies in kcal/mol, notations as in Table I.

Table I1 gives the total interaction energy (in kcal/mol) and its decomposition in the same way as above for the complex NH4+-NH3. Comparison of Tables I and I1 shows the following: (i) The calculated binding energy of one molecule of water to NH4+ is in very good agreement with the experimental value (18.0 vs. 17.3 kcal/mol). (ii) The calculated binding energy of NH3 to NH4+is somewhat weaker than the experimental value (21.5 vs. 24.8 kcal/mol). (iii) In agreement with experiment@and with theoretical calculations,21the binding energy to NH4+is stronger for one molecule of NH3 than for a molecule of water. (iv) The electrostatic energy, the polarization energy of NH4+ by the “solvent molecule”, and the polarization energy of the “solvent” molecule by NH4+are stronger for NH3 as solvent than for H20. Thus, the reasons underlying the fact that the hydrogen bond between NH4+and NH3 is stronger than between NH4+and HzO are in agreement with the previous discussions.21@ (2) Attachment of the N e x t n Molecules of H 2 0 (or N H J . (a) Geometry. For the complex (NH4+)4Hz0)2we obtained three configurations with nearly the same energy: 32.12,34.27 and 34.30 kcal/mol. In the first, the two water molecules lie along the direction of the NH bonds; the N-0 distance is 2.67 A. In the second, one molecule of water lies above H3 and may interact with H1 and H3, and the second one lies along the NH2 direction. In the third, the first water molecule lies above H3 and the second one



TABLE 111: Stepwise Hydration or ’ Ammoniation of NH,‘ a Hydration

2 3 4

16.2 14.8 13.0

Ammoniation

14.7 13.4 12.2

19.1 16.9 14.9

17.5 13.8 12.5

a Solvation along NH bonds in all cases. -AEn-l,n are the computed binding energies (kcal/mol). -AHn-,,n are the corresponding experimental enthalpies7 (kcal/mol).

is slightly displaced from the NH2 direction. For the complex (NH4+)-(0H2),we have also found three configurations with nearly the same energy: 48.82, 48.91, and 48.97 kcal/mol. In the first, the three water molecules lie along NH directions with the N-0 distance = 2.70 A. In the second, one water molecule is slightly displaced away from the NH direction. In the third, one water molecule is slightly displaced from the NH direction and one occupies a position similar to I11 in Figure 3. For the complex NH4+.-(0H2),only one configuration exists, in which all water molecules are bisected by the NH directions, the N-0 distance being 2.70 A. Ammonium Ion with Ammonia Molecules. For the approach of two, three, and four NH, molecules toward NH4+we find that the approach along the NH directions is most favorable, the N.-N distances at equilibrium being 2.64, 2.68, and 2.70 A, respectively. Thus for both solvents the structure of the tetrasolvate is in agreement with the SCF results21and the deductions from the experimental data.68 (b) Energy. Table 111 gives the binding energies computed for polyhydrates and polyammoniates of NH4+ for n = 1-4 and Table IV gives the corresponding energy decomposition. In agreement with experimental data68and theoretical ab initio results,21it appears that with the increase of the number of ligands, the Nk4+ interactions with water become closer in magnitude to those with ammonia. Our calculated binding energies in both cases (hydratation and ammoniation) decrease less rapidly than the experimental values. Table IV shows the following. (i) In both cases the solute-solvent interaction energy per molecule is nearly constant. (ii) In both cases the solvent-solvent interaction energy per molecule decreases when n increases. This results from two facts: first, the total electrostatic repulsion between the solvent molecules increases when n increases, and second, the polarization of the solvent TABLE IV:

molecules by solute decreases when n increases. (iii) The polarization energy of the solvent by the solute is stronger for NH3 than for H 2 0 when n = 1; when n increases the polarization energies of the solvent by the solute tend to become similar for NH3 and H20. 11. Comparative Solvation Studies by Different Models. In order to test our three models and compare their results we have studied two simple cases: the solvation in water of the ion NH4+ and of methanol. (a) Using the discrete model we have performed calculations with an increasing number N of molecules of water surrounding the solute molecule and studied the influence of N upon the calculated value of the solvation energy. (b) Using the continuum model we compared its results with those obtained from the discrete model. A preliminary requirement was to evaluate the corrective “Bjerrum factor” f for the electrostatic energy corresponding to water as a solvent (see ref 64, 36c, and Method, section 1I.B). We have thus fitted this corrective factor f so as to reproduce the experimental vaporization energy of water (9.91 kcal/molg at 5” = 298.15 K). This yields f = 0.63. (c) In the case of NH4+,there exist strong hydration sites; four water molecules are strongly “bound” to NH4+.21968For that case we have calculated the influence of a continuum around the complex NH4+-.(OH2)4using the discrete-continuum combined model. (d) Finally, we have considered the problem of the influence upon the solvation energy of the substitution of the hydrogens of NH4+ by methyl groups. It is known e~perirnentallf~that the heat of hydration decreases when the number of substituting methyl groups increases. The ab-initio STO 3G calculations by Port and Pullman22have actually shown that the hydration energy of the substituted ammonium ion decreases when the size and number of alkyl groups increases. Thus we performed some calculations of solvation energy in water for NB4+successively substituted by one, two, and three methyl groups. Since the discrete model would require a very large number of calculations, we studied this problem by the continuum model only. (A) Solvation of NH4+ in Water. Table V gives the solvation energy in water of NH4+calculated with (i) the discrete model for NH4+surrounded successively by 4,18, 22, and 57 water molecules; (ii) the continuum model a t T = 298.15 K; and (iii) the discrete-continuum combined model. This table shows (i) that the solvation energy calculated with the discrete model depends on the number N of water molecules surrounding the solute. With regard to the

Energy Decomposition for Stepwise Hydration or Ammoniationa

-AE(solute-solvent energy) ( n - l), n

n

-A ED,

1 2 3

+3.4 +2.9 +2.6 +2.3

41 1

-A ERsqsv -A Ee’+,,

-A EPs-,V

-AE(solvent-solvent energy) ( n - l), n -AEs-sv

-A EDsv-,,

-A ERsv-sv

-A Ee’sv-sv

0.0 0.0 0.0

0.0 -0.5 -1.1 -1.2

-A EPsV-sV -A Esv-sv

Hydration

4

-4.5

-11.2 -8.8 -7.2 -5.3

+ 15.8 + 15.2 + 14.7 + 14.4 + 18.1 + 17.2 + 16.6 + 15.8

t0.9 + 0.3 + 0.0 -0.2

+ 12.6 + 12.2 + 12.4 + 11.9

+ 1.6

Ammoniation + 13.2 0.0

0.0 0.0 + 0.2 + 0.3

0.0

t 0.7

+ 13.1

+0.1 t0.3 + 0.4

0.0 0.0 0.0 0.0

t 5.4 t 4.5 t 3.3

+ 5.4

t 2.7

+1.1

t4.0 +2.3

0.0 +8.3 +8.3 -0.6 -1.6.5 + 6.0 t 0.1 + 13.1 -1.3 + 4.8 + 3.8 +3.1 -0.5 t 13.1 -1.9 + 3.3 + 1.8 a Dispersion, repulsion, electrostatic, and polarization terms as in Table I. The subscripts represents the solute and the subscript sv the solvent molecule; A EDs-sv represents solute-solvent dispersion energy and A ED,,-,, the solvent-solvent dispersion energy. -A E,-,, and -A E,,-,, represent the total solute-solvent interaction energy and the total solvent-solvent interaction energy, respectively (all values in kcal/rnol). 1

2 3 4

+4.7 t4.0 +3.6

-7.5 -6.2 -4.9

412


Claverie et al.

TABLE V: Solvation Energy of NH,+ in Water Calculated with Our Different Modelsa Solvation energy in water, kcalimol

Model Discrete model N= 4 18 22 51 Continuum model Combined discrete-continuum model Exptl valueb See text.

-48.0 -61.2 -71.0 -79.8 -82.9 -80.5 -83.05

Reference 1.

experimental values, the calculated energies are too weak for small values of N (4 or 18), they increase with N , and become practically stabilized for N = 57. This result is understandable since the electrostatic energy decreases slowly with the distance, so that a large enough number of water molecules is necessary for reaching the convergence of this energy to its limit. The last result (for 57 water molecules) is reasonably close to the experimental solvation energy. (ii) The continuum model gives a solvation energy in water which is in very good agreement with experiment. (iii) The discrete-continuum combined model also gives satisfactory results with regard to experiment. (1)Analysis o f t h e Discrete Model Calculations. One may consider the final results in two different ways. (a) I n terms o f t h e various parts o f t h e interaction energy: dispersion (ED),repulsion (ER),electrostatic (E"), solute polarization (EP,), and solvent polarization (EP,,). Table VI gives these parts of the energy calculated for (i) solvent alone and (ii) solvent + solute. Table VI concerns the results obtained with N = 18,22, and 57. This table shows the following: (i) In the presence of the solute, the solvent molecules lose electrostatic energy (line f-i,,,J; the solvent molecules are attracted by the solute so that a competition takes place between solvent-solvent attractions and solutesolvent interactions until an equilibrium is reached. The loss of electrostatic energy of the solvent molecules does

not vary much when the number N of water molecules increases. (ii) On the contrary, the number N of water molecules influences appreciably the electrostatic solute-solvent interaction energy (line f,-,J. (iii) The polarization of the solute molecule by the solvent molecules is very weak. This result justifies the neglect of this term in the continuum model. (iv) The gain of the solvent polarization energy due to polarization of the solvent by the solute (difference EPsv with and without the solute) does not depend very much on the number N of water molecules surrounding the solute (EP= -21.5, -27.3, and 28 kcal/mol for N = 18, 22, and 57). (b) Another way to analyze the solvation energy is to consider different intermediate steps. (i) S t e p I . Creation in the solvent of an empty cavity having the size of the solute. This step gives the cavitation energy Ec as the difference between the energy of N solvent molecules in their most favorable positions without solute and the energy of the same system including an empty cavity with the size and shape of the solute. (ii) S t e p 11. Introduction of the solute molecule into the cavity created in the solvent. The solute polarizes the solvent molecules, inducing some reorganization of the solvent molecules around the cavity. The solvent molecules are reoriented so that their mutual interactions are weakened. The solvent reorganization energy ESR which is positive may be obtained by neglecting the solvent polarization energy due to the solute. (iii) S t e p III. Turning on of the solute-solvent interaction yields the "charging" energy. The solvation energy is the sum of the energy changes associated with these three steps. Table VI1 gives the energy of different steps for NH4+ surrounded by N = 18, 22, and 57 water molecules. We shall comment on these values in the next paragraph. (2) Comparison between the Discrete and Continuum Models. Table VI11 gives the solvation energy of NH4+ in water calculated with the continuum model at T = 298.15 K, and the different parts which can be defined inside this energy (cavitation, dispersion, repulsion, and

TABLE VI: Decomposition of the Solvation Energy (kcal/mol) for NH,t Surrounded by 18, 22, or 57 Water Molecules Calculated with t h e Discrete Modeln Etot

ED

ER

-159.5 -148.6 -72.1 -220.7 + 10.9 -61.2

-90.9 -79.3 -28.8 -108.1 +11.6 -17.2

-198.0 -184.5 -84.5 -269.0 +13.5 -71.0

-113.1 -97.2 -27.3 -124.5 +15.9 -11.4

Eel

EP

-139.8 -112.9 -71.6 -184.5 + 26.9 -44.7

-43.2 -58.1 (-36.6) -1.2 -59.3 -14.9 -16.1

-172.1 -136.3 -94.6 -230.9 + 35.8 -58.8

-54.3 -10.8 (-43.5) -1.8 -72.6 -16.5 -18.3

N = 18 1sv-sv

fsv-sv fs-sv

ftqt f-1SV-S"

f-itot kv-sv fsv-sv fs-S"

ftqt f-1Sv-W

f-itot

+ 114.4 + 101.7 + 29.5

-131.2 -12.7 + 16.8 N = 22 +141.5 + 119.8 + 39.2 + 159.0 -21.7 + 17.5 N = 57 +418.5 +409.6 + 32.8 +442.4 -8.9 + 23.9

-565.8 -358.6 -473.8 -151.9 -556.7 -353.3 -441.5 -171.5 (-143.5) fS-,, -89.0 -32.8 -88.1 -0.9 -529.6 -172.4 -386.1 -645.7 ftot + 32.3 -19.6 + 5.3 +9.1 f-isv-sv -55.8 -20.5 -27.5 f-itot -79.9 a Subscripts s and s t as in Table IV. f and i represent final state (solvent and solute molecules) and initial state (solvent molecules alone). The solvation energy and its com onents are calculated as the difference between the final and initial states. When considering the polarization energy ( E ) of the solvent in the final state, the first value is the polarization energy of the solvent in presence of the solute and the value in parentheses indicates the polarization energy of the solvent in the absence of solute. 1sv-sv

fsv-sv

J:



TABLE VII: Solvation Process in Water for t h e NH; Ion in the Discrete Model for N = 18, 22, 57a Solvent alone (state 1)

\1

18

22

57

[-159.51

[-198.01

[-565.81

Step I cavitation

1

1

+27.7

i+7.0 [-152.51

Solvent surrounding an empty cavity of size and shape of the solute (state 2 )

1

Step solvent of the I1 reorganization solute in presence of the

it25.4

Solvent surrounding the solute (state 3)b

1

[-127.11

Step 111 turning o n of the interaction

L

-72.1 -21.5

[-72.1 -148.6 J E,-sv Esv-sv -61.2

Solvent t solute in interaction (state 4 )

[-170.31

t27.3

[-538.51

/,+9.8

itl.3.1 [-157.21

i

[ 528.71

-84.5 -2 7.3

[-84.5 -184.51 E,-sv Esv-sv -71.0

b

-89.0 -28.0

[-89.0 -556.71 Es-sv Esv-sv -79.0

Esolv = E1 + E11 t E111 Analysis in terms of the different intermediate steps. The energies are given in kcal/mol. The values in brackets are the energy values of the different states. The italicized values at the right of the arrows represent the energy difference between two states, i.e., the energy lost (or gained) at each step of solvation. In step I11 at the right of the arrows, there are two values, the first one indicates the solute-solvent interaction energy and the second the gain of solvent polarization energy due solventrepresents solute-solvent interaction energy and to the polarization of the solvent by the solute. The solute is transparent, i.e., we d o not take into account the solute-solvent interaction nor solvenb interaction energy. the solvent polarization by the solute, but the solvent molecules have the same configurations as in the final state. a

TABLE VIII: Solvation Energy (Etot) of the NH,‘ Ion in Water Calculated with the Continuum Modela -0.7

ED

ER

-,9.5

1.2.7 v -82.2

-75.$

-82.9

a This table gives also the decomposition of the total solvation energy in different interaction energy terms. All the values are in kcal/mol. Ec represents the cavitation energy.

electrostatic). The continuum model simultaneously calculates steps I1 and I11 of the solvation process (above) and in order to compare its results with the discrete results we have to compare Table VI11 with Table IX which gives (when using the discrete model with N = 57) the difference in energy of the systems (solvent + solute in interaction) and (solvent surrounding the empty cavity). A first comparison of the two models indicates the following: (i) The discrete model does not give a satisfactory value for the cavitation energy (Table VII), since this value is an order of magnitude too high with regard to the one calculated with the modified Pierotti formula6I fitted with respect t o Monte-Carlo calculation^^^ (Table VIII; Ec = -0.65 kcal/mol). This failure of the discrete model concerning the cavitation energy can be understood as follows: this method overestimates the energy of the system “solvent alone” with respect to the energy of the system

“solvent surrounding an empty cavity”. Actually the method, not being a statistical one, gives only the energy of a privileged configuration, Le., of a local minimum, and not a statistical average of the energy (as Monte-Carlo methods). Thus the energy minimization leads to some kind of crystal-like configuration. In the system (solvent surrounding a cavity) we tend to a crystal-like structure with a hole (the empty cavity), which is less favorable than the crystal-like structure of the solvent molecules alone, without a hole. In reality, due to their thermal motion, the water molecules do not have exactly an “average energy” of crystal type. When creating a cavity in the solvent one should take into account the loss of energy due to this creation and the gain of energy due to the structuration of water molecules around the empty cavity but, in the discrete model, the initial state (solvent without hole) is too structured (crystal-like), so that, in the intermediate state, the increased structuration around the cavity (“iceberg effect”62)cannot be properly represented. Accordingly, the compensation between the loss of energy (resulting from the increased intermolecular distances) and the gain of energy (due to the increased structuration) cannot be properly realized in the framework of the discrete model, the energy loss predominating too much. (ii) Aside from this point examination of Tables VIII and IX shows that it is difficult to compare the continuum and discrete models. Actually in the discrete model the “dispersion, repulsion, and electrostatic” energies are stronger than in the continuum model.

TABLE IX: Solute-Solvent Energy for NH,‘ in Water According to the Discrete Modela Solute-solvent ERs-sV t 32.8 -88.1

’

Solvent-solven t ERsv-sv Ee‘sv-sv t409.6 -441.5 +407.9 -444.7 +1.7 t 3.2

EDsmsv EPs EDsv-sv Final system ( 4 ) b -32.8 -0.9 -353.3 -359.7 Initial system ( 2)c Total E ( 4 ) - E(z) -32.8 +32.8 -88.1 -0.9 t 6.4 The labels ( 2 ) and (4)refer t o the states considered in Table VII. E(4) - E(z) = t *-*-3C+= --107.2 kcal/mol ED ER Eel E Solvent and solute in interaction. Solvent surrounding an empty cavity of size and shape of the solute.

-*

EPsv -171.5 -142.0 -29.5

414


TABLE X: Solvation Energy in Water (kcal/mol) of NH,’ Calculated with the Combined Discrete-Continuum Modela

EC

ED

ER

Claverie et al.

TABLE XI: Solvation Energy of Methanol in Water Calculated with both the Discrete and Continuum Models

State 1

- 22.6

N H ~ +t

t 7.5 - 19.5 - 34.6

/Step I State 2

- 1.3

1.3

NH‘,.

t

+o

Discrete modela N = 16 51 55 Continuum model

-4.9 -8.1 -8.8 -10.3

a N is the number of water molecules surrounding the solute.

Etot -7.9

t 9 . 6 -15.7 -14.0

State 4 -11.9 t 2 4 . 0 -74.1 -62.0

State 5 -1.7 -31.3

t 9 . 9 -45.3 -68.4

-0.4 -12.7 t 1 6 . 8 -84.2 -80.5

Energies of states 3 and 4 have been calculated with the discrete model and energies of states 1, 2, and 5 have been calculated with the continuum model. Esolv= (E, t E, - E, - E, - E l ) . a

This may be understood in the following way: as already pointed out, the discrete model neglects thermal motion so that the actual average energy is necessarily higher than the absolute minimum computed, especially when the solute is an ion which strongly attracts the water molecules (owing to the predominant electrostatic term). By contrast, in our continuum model, the real intermolecular distances are simulated by the use of the factor h which multiplies the van der Waals radii in order to reproduce (which are in principle the experimental molar temperature dependent). It is therefore likely that the difference between the system (solvent and solute in interaction) and the system (solvent surrounding an empty cavity corresponding to the solute) is overestimated in the framework of the discrete model. This does not preclude the solvation energy proper (difference between states 4 and 1 in Table VII) to be rather well obtained in the framework of both discrete and continuum models; the cavitation step (energy change between states 1 and 2 in Table VII) leads to different values in both models, but this difference tends to cancel the previous difference between the results of the two models (for the energy change between states 2 and 4). A practical consequence is the impossibility of “mixing” intermediate results of the two models, e.g., evaluating the cavitation energy according to the continuum model (modified Pierrotti’s formula61) and the solute-solvent interaction (state 4-state 2) according to the discrete model. We therefore conclude the following: (i) Owing to the approximations involved in our models, it is not possible to have a good agreement for all intermediate steps of the solvation process, (ii) nevertheless, a satisfactory agreement

ED

ER

Eel

EP

-113.6 -101.9 -8.1 -110.0 t11.7 t3.6

-33.3 -30.2 -1.2 -31.4 +3.1 t1.9

N = 16 isvmsv fsv-sv

/Step 111

Esolvation

Solvation energy, kcal/mol

TABLE XII: Solvation Energy of Methanol Surrounded by 1 6 and 55 Water Molecules Calculated with the Discrete Model

IStep 11 State 3

Model

Eel Etot

-127.2 -117.4 -14.7 -132.1 +9.8 -4.9

-69.8 -62.1 -22.6 fs-sv -84.7 ftot f-lsv-sv +7.7 -14.9 f-hot N= is,-,, -534.7 -341.3 -524.0 -334.8 fsv-sv -19.5 -30.2 fS-SV -543.5 -365.0 ftot f-isvmsv +10.7 +6.5 -8.8 -23.7 f-itot

t89.5 t76.8 t17.2 t 94.0 -12.7 t4.5 55 +388.6 t 372.9 +22.8 +405.7 -15.7 +7.1

-442.2 -139.8 -428.4 -133.7 -10.0 -2.1 -438.4 135.8 t13.8 t6.1 +3.8 t4.0

is obtained concerning the solvation energy itself, provided a large enough number of water molecules is chosen in the discrete model, in order to reach practically the limit of the electrostatic solute-solvent interaction. (3) Results Obtained with the Discrete-Continuum Combined Model. Since we know that strong hydration sites exist for NH4+,3,4,7 it is possible to study the solvation of “4’ using the discrete-continuum combined model (see Method, section 111). In that case we study the action of a continuum upon the “supermolecule” NH4+ 4 water molecules. Table X gives the results obtained. The calculated solvation energy seems in good agreement with the one calculated by the continuum model. We feel that in the case of an ion the combined model seems better adapted than the continuum model alone. Effectively in that case we take into account the water molecules strongly bound to NH4+;furthermore, when we look at the different terms of the interaction energy, we notice the phenomenon showed by the discrete model, i.e., the preponderant role of the electrostatic term. When using this model we have four water molecules strongly attracted by the solute and therefore lying quite close to it. Thus it appears that the interaction energy terms given in Table VI11 represent average values, whereas those given in Table X should be quite near to reality. (B) Soluation of Methanol i n Water. (a) Table XI gives the solvation energy of CH30H in water calculated using (i) the discrete model with 16, 51, and 55 water molecules surrounding CHBOHand (ii) the continuum model. Here again we note that in the case of the discrete model, only calculations performed with a large number of water molecules give satisfactory results. (b) Table XI1 gives the decomposition of the solvation energy (obtained with the discrete model) into different terms. One observes that, contrary to the case of NH4+ ion, the solute-solvent electrostatic interaction energy is very weak, and does not counterbalance the loss of electrostatic interaction energy of the solvent in the presence

+



TABLE XIII: Solvation Process in Water for CH,OH (in the Discrete Model) Taking into Account the Different Intermediate StatesQ Solvent alone (state 1)

1

16

51

55

[-127.21

[-500.41

[-534.71

1

1

+10.0

Step I cavitation

Solvent surrounding an empty cavity of size and shape of the solute (state 2 )

Lt

+10.6

k 5 . l [-495.31

[-117.21

1

Step of the I1 reorganization solvent in presence of the solute

Solvent surrounding the solute (state 3)b

1

[ -494.1 ]

[-521.51

1.I

J-1.3 -14.7

Solvent + solute interaction (state 4 )

1;

1.2

[-116.11

Step I11 turning on of the interaction

bt

[-524.11

2.6

$,,

4 - -13.1 1.3

-19.5

[-14.7 -117.41

[-13.1 -495.41

[-19.5 -524.01

E,-,"

Es-sv

E,-sv

E1 + E11 -+ E111

Esv-sv -4.9

Esv-sv -8.1

Esv-sv -8.8

CI For an explanation of this table see Table VII. The solute is transparent Le., we do not take into account the solutesolvent interaction nor solvent polarization by the solute.

TABLE XIV: Solvation Energy (E,t) of CH,OH Calculated with the Continuum ModelQ

EC

ED -8.8

-0.2

ER

Eel

Etot

t1.3

-2.6

-10.3

This table gives the decomposition of the total solvation energy into different interaction energy terms. All values are in kcallmol.

' of solute. Furthermore since the electrostatic attraction of the solvent molecules by the solute is weak, the distances between the solute and the solvent molecules remain larger than in the case of NH4+, so that the repulsion term is weak. Thus, in the case of the solvation of methanol in water the dispersion interaction energy is the preponderant term. (c) Table XI11 gives the decomposition of the solvation process of methanol in water into different intermediates ill not comment again on the cavitation energy, steps. We w but we will note that in the case of methanol the loss of energy due to reorganization of the solvent in the presence of solute is weak compared to the value obtained for NH4+; in the case of methanol, the polarization of the solvent by the solute is not important. Table XIV gives the solvation energy of methanol in water calculated by the continuum model; here again the

dispersion energy appears as the preponderant part of solvation energy. As for NH4+ it is difficult to compare the different parts of the solvation energy calculated with the discrete and continuum models; in the latter the calculated values represent average values. (C) Solvation in Water of Methylammonium Ions. Figure 4 gives the calculated net charges population and total charges71 of NH4+, (CH3)NH3+,(CH3)2NH2+, and (CH3)3NH+.In agreement with ref 22, it is seen that the positive charge population which was localized on the four hydrogens of NH4+ tends to be spread out on the hydrogens of the methyl groups in the substituted methylammoniums. Table XV gives the solvation energy (kcal/mol) in water of NH3+(CH3),NH2+(CH,),, and NH+(CH3)3and its decomposition into different interaction terms. This table shows the following: (1)In agreement with experimenP9 and with ab initio calculation22the solvation energy decreases when the number of alkyl groups increases. However our calculated difference between the solvation energies of NH4+ and NH+(CH3)3is not as large as the experimental value. This could be due to the fact that the empirical correction factor fused for the calculation of electrostatic energy was fitted on the system (water surrounding one water molecule). The results obtained for the solvation energy in water of

TABLE XV: Solvation Energy (in kcallmol) Calculated with the Continuum Model for the Different Methyl Substituted NH,' Ionsu Molecules EC ED ER Eel Etot CH,NH: -0.8 -13.7 t 3.9 -68.0 -78.6 -17.4 t 5.1 -62.3 -75.6 (CH,)'",' -1.0 -20.6 t 6.0 -57.2 -73.0 (CH,),",' -1.2 This table gives the experimental value' Eexpt and the decomposition of energy into different interaction

Eemt -75.7 -70.2 -62.8 energy terms.

TABLE XVI Charging process ( I ) Debye-Huckel (all molecules charged with E ) (11) Guntelberg-Muller

(solute charged with 5 , solvent totally charged)

Dispersion-repulsion interactions

Electrostatic interactions

caU/aot= N t i i

tau/at)t= N t Z q j q j l [ t ]

AF = 1I2NG

A F = NWjJit*j[

j

i

tI d t

caUJaDt= iis

causSV/at)t = t zqj*jl[l]

AF=G,

A FsmSv =

j

' I z Z q j * j l [ 11 j

416

The Journal of Physical Chemistry, Vol. 82, No. 4, 1978 0.237

n.271

-u

I

H

0.271

N

~4+

(0.236)

H

I

H 0.237 (0.236)

N H i ( CH31

0,192 10.145)

0.210 (0.209)

H

I

i

A\

H H H __c

0.220 10.168)

0.052 I 10.064c

/I\

H H H

VIY

0.192 10.145)

Figure 4. Net charges (populations) and total charges (taking into account the atomic dipoles) (values in parentheses) for different methyl-substituted ammonium ions.

NH4+are satisfactory because a NH4+molecule is of the size of a water molecule.70 However when methyl groups are added, the size of the molecule increases and probably the loss of energy due to the reorganization of the solvent in the presence of the solute increases. In order to obtain good quantitative results for the solvation energy one should probably modify the factor f according to the size of the solute molecule. (2) When increasing the number of methyl groups, both the total dispersion and repulsion energies increase (in absolute value), but the leading term responsible for the decrease of the solvation energy is the electrostatic component which decreases in absolute value when the number of methyl groups increases. This fact is due to the smearing out of the positive charge when the number of methyl groups increases.

Conclusion In this paper we have recalled the main features of two empirical models which may be used for solvation studies: the discrete and continuum models; then we have presented a combined discrete-continuum model. However in particular for the continuum models we have brought some precisions and improvements regarding the initial versions of the model:36 thorough discussion of the charging process in the case of solute-solvent problems, discussion and choice of formulae for evaluating the cavitation energy. Then we have checked the accuracy of these three models by studying some simple cases of solvation. The three models that we have presented seem to be good tools for the study of the solvation process. The discrete model gives a representation of the “structure” of the solvent molecules around the solute and provides satisfactorily the primary solvation sites. The continuum model gives the solvation energy with good accuracy, and

Claverie et al.

the combination of discrete and continuum models should be especially appropriate for studying ionic molecules.

Acknowledgment. The authors express their thanks to Dr. A. Pullman for several helpful discussions and comments concerning especially the comparison between the present discrete model (which uses simplified formulas) and previous discrete model calculations where a quantum mechanical ab initio method was used for evaluating the energy. Appendix. Evaluation of E and F Associated with the “Turning On” of the Interaction Energy According to the Debye-Huckel and Guntelberg-Muller Charging Processes This Appendix improves appendix B of ref 36c in making clearer the differences of the Debye-Huckel and Guntelberg-Muller processes; this point is very important when dealing with solutions instead of pure liquids (which were considered in ref 36c). (1)Debye-Hiickel Process. The reference state may be the hypothetical ideal gas in which atoms and molecules are forbidden to interact. Thus in the initial state U(0, Q) = 0. (a) W e m a y first consider t h e turning o n of t h e dispersion and repulsion energies. In the Debye process one simultaneously “charges” all the molecules. The total dispersion-repulsion interaction for the value [ of the switching parameter is Ud,r(t,

=

xt2upu(a)

‘12

(A.1)

/J#U

where u,, denotes the physical interaction between molecules p and v, and the factor F2 appears because [ multiplies the interaction parameters for each of the interacting molecules. Then we obtain

t Pz+ zV

upu(a2)

( A 3

(a u d , r / a t ) E = t xf i .r v+u z u,d.n) q t , .n) d a

(A.3)

aud,r/at

=

The integral is the average interaction energy between the molecule y and all other molecules (the distribution function f([, Q) corresponding to the partially switched interaction (A.1)). Now, for an isotropic liquid, this average is independent on the peculiar molecule considered, and we shall denote it a([).Then, C,,lN may be replaced by N , the total number of molecules, and (A.3) becomes (a u d 9 r / a t ) t

= NE;([)

(A.4)

In principle a([) depends on 4 since f([, Q) does. However in the continuum model that we it is assumed that f(t, Q) = g(Q) remains independent o f t as far as the switching of the dispersion-repulsion interaction is concerned. This is an acceptable assumption if we choose as our initial state a hard core fluid (made of “hard molecules”) and not a perfect gas (point-like molecules). Then a ( l ) no longer depends on 5, and (A.4) gives

AE

2 A F = SANtu

dt

=

l/zNi

(A.5)

(b) Now we deal with a n assembly of N molecules with dispersion and repulsion interactions. W e want t o evaluate the turning o n of the electrostatic energy. This energy will be the variable part of the total energy U. W e have to calculate the free energy variation AF, by varying all the electrostatic charges of solute and solvent molecules from the value 0 to q , corresponding to the transformation from the initial state, “all molecules without charges”, to the final state, “all molecules charged”.



If Vel is the electrostatic part of the total interaction energy of an assembly of N molecules, we obtain as shown in Appendix B of ref 36c

Uel(t;, a)=

‘/z Z Z

u ~ ~. . ( ~q j p ~. . .t;qjv. ( . . .)

(A.6)

P + V

where p and u label the molecules and 4;””(i = l...nJ and q; (with j = L n , ) are respectively the net charges of atoms i and j belonging to the molecules p and v. Then as shown in Appendix B of ref 36c the free energy related with the “turning on” of the electrostatic energy is given by

A F = J;(aU/at;), dt; = N ~ q j v J ~ ( \ k ’ [ r j v l D (d( ()] i

64.7) is the reaction potential created by the total

where 9 I charges of the solute in a medium with a dielectric constant D (4). Thus in the Debye charging process we must calculate exactly the integral of eq A.7. The value of this integral does not reduce to 1/29’[r,”lD],because in the integrand 9’[r ’ID(()]depends on 4. (23 Guntelberg-Muller Charging Process. W e consider t h a t only t h e solute molecule is subject to t h e action of t h e “switching parameter” 4; t h e solvent molecules are “charged”. (a) We have just to consider as a first step the “turning on” of the dispersion and repulsion interaction energy between the solute molecule and the solvent molecules. We obtain

cau/at;),= J. . . j u v ( n )f ( ( ; a ) da

(Am

where u represents the solute molecule and u, = Clrzvull,. f(4, Q) corresponds to the following situation: solute “charge” a t the value 4, solvent totally charged; but if according to ref 36a, appendix C, we neglect the variation of f with 4, we obtain

(aulae), = iiv

(A.9)

and

A F = JAG3 Ulat;), dt; = UvJkdt; = Uv

(A.10)

The factor 112 does not appear in that case. Neglecting the variation of the distribution function f(4, Q) is approximately valid when taking as the initial state the solvent with an empty cavity corresponding to the solute. (b) Now we have to evaluate the “turning on” of the electrostatic energy. I n the Guntelberg charging process, 4 acts only on the solute electrostatic charges. T h e solvent molecules are totally charged from t h e beginning. We have to consider

(A.ll) where 4; represents the charge of the solute molecule and

When the charging process (C; varies from 0 to 1)takes place, the third term of (A.11), namely, the solute-solvent electrostatic interaction, is not the single varying term; the electrostatic part of the first term (solvent-solvent interaction) varies too, because the solvent molecules undergo some reorganization in the immediate neighborhood of the solute molecule as a consequence of its charging. In this Appendix, we shall limit ourselves to the evaluation of the free energy associated with the solute-solvent

interaction; the evaluation of the AF,,,, associated with the solvent reorganization would be more difficult. As mentioned in the main text, discrete model calculations suggest that the second term is actually smaller than the first, and the (empirically fitted) factor f u s e d for multiplying must compensate for the-lack of an explicit AF,,,, term. Denoting the third term of (A.11) by U,s”,namely, the solute-solvent interaction energy, we may write

aussv/at; = z q j v q ( T va) ,

(A.12)

j

and

(AX) (a ussv”/(), = J . . .j(a ussv/at;) f(t;, a)d a = X q i v J .. . .Jp(?’, Q)f(t;, Q ) d a (A.14) i

However in this case, since 4 in f(4, Q) corresponds to the charge variation in the solute molecule only, we obtain

where, in the same way as previously, 9’corresponds to total charges (4 = 1)on the solute. The dielectric constant now has a physical value, since the solvent is fully charged from the beginning. From (A.15) we obtain

I -+v

AFS-,, = 1 / 2 Z q j v \ k [ rj D] i

(A.16) (A.17)

~

Thus eq A.17 shows that in the expression of the solute-solvent part of the free energy associated with the electrostatic interaction turned on by E , there appears exactly the factor 112 when dealing with the Guntelberg charging process. Table XVI summarizes the values of AF associated with the interactions turned on by 4 in both Debye and Guntelberg processes (we denote for brevity 9][4] instead of +‘[Fj”ID([)] ).

References and Notes A. Pullman and B. Pullman, Q. Rev. Biophys., 7, 505 (1975). M. Avignon, C. Garrigou-Lagrange, and P. Bothorel, Biopolymers, 12, 1651 (1973). M. M. Brewer and M. G. Kennerley, J . Colloid. Interface Sci., 37, 124 (1971). N. S.Ham, A. F. Casy, and R. R. Ison, J. Med. Chem., 16,470 (1973). I?. Ison, J . Pharm. Pharrnacol., 24, 82 (1972). R. R. Ison, P. Partington, and C. C. K. Roberts, Mol. Pharrnacol., 9, 958 (1973). I. D. Kuntz, J. Am. Chem. SOC.,93, 514 (1971). I. D. Kuntz and W. Kauzmann, Adv. Protein Chem., 28, 239 (1974). R. J. Abraham and T. M. Siverns, Tetrahedron, 28, 3015 (1972). 6. Sunners, L. H. Pietto, and W. G. Schneider, Can. J . Chem., 38, 631 (1960). H. Kamei, Bull. Chem. Soc. Jpn., 41, 2269 (1968). T. Drakenburg and S. Forsen, J . Phys. Chem., 74, 1 (1970). W. E. Stewart and T. H. Siddal, Chem. Rev., 70, 517 (1970). H. Kessler, Angew. Chem., 9, 219 (1970). N. C. Yang and S. L. Murov, J. Chem. Phys., 45, 4358 (1968). P. S.Wagner, M. J. May, and E. Haug, Chem. Phys. Lett., 13, 545 (1972). M. Koyanagi, R. J. Zwarich, and L. Goodman, J. Chem. Phys., 96, 3044 (1972). N. Kanamaru, M. E. Long, and E. C. Lim, Chem. Phys. Left., 26, 1 (1974). H. H. Jaff6 and M. Orchin, ”Theory and Applications of Ultraviolet Spectroscopy”, Wiley, New York, N.Y., 1962. L. Salem, “The Molecular Orbital Theory of Conjugated Systems”, W. A. Benjamin, Amsterdam, 1966. (a) A. Pullman and A. M. Armbruster, Inf. J. Quantum Chem. Symp., 8, 169 (1974); (b) Chem. Phys. Left., 36, 558 (1975). H. N. J. Port and A. Pullman, FEBS Lett., 31 (1973); Theor. Chim. Acta, 31, 231 (1973); Int. J. Quantum Chem. Biol. Symp., 1, 21 (1974); Mol. Pharmacol., 10, 360 (1974).

418


(23) 8. Pullman, Ph. CourriBre, and H. Berthod, J. M e d . Chem., 17, 439 (1974). (24) (a) B. Pullman, A. Pullman, H. Berthod, and N. Gresh, Theor. Chim. Acta, 40, 93 (1975); (b) B. Pullman, H. Berthod, and N. Gresh, FEBS Left., 53, 199 (1975). (25) M. J. Mantione and J. P. Daudey, Chem. Phys. Lett., 6, 93 (1970). (26) M. J. Huron and P. Claverie, Chem. Phys. Left., 9, 194 (1971). (27) V. G. Dashevsky and G. N. Sarkisov, Mol. Phys., 27, 1271 (1974). (28) N. S. Bayless and L. Hulme, Austr. J. Chem., 6, 257 (1953). (29) S. Basu, Adv. Q. Chem., 1, 145 (1964). (30) T. Abe, Bull. Chem. SOC.Jpn., 41, 1260 (1968). (31) A. Schweig, Chem. Phys. Lett., 3, 542 (1969). (32) (a) 0. Sinanoglu in “The World of Quantum Chemistry”, R. Daudel and B. Pullman, Ed., Reidel, Dordrecht, Holland, 1974, p 265; (b) Chem. Phys. Left., 1,340 (1967); (c) Adv. Chem. Phys., 12, 283 (1968); (d) Theor. Chim. Acta, 33, 279 (1974); (e) in “Molecular Associations in Biology”, Academic Press, New York, N.Y., 1968, p 472. (33) D. L. Beveridge, M. P. Kelly, and R. J. Radna, J . Am. Chem. Soc., 96, 3769 (1974). (34) A. J. Hopfinger in ”Molecular and Quantum Pharmacology”, E. D. Bergmann and B. Pullman, Ed., Reidel, Dordrecht, Holland, 1974, p 131. (35) J. Hylton, R. E. Christoffersen, and G. Hall, Chem. Phys. Lett., 24, 501 (1974). (36) (a) M. J. Huron and P. Claverie, J. Phys. Chem., 76, 2123 (1972); (b) 78, 1853 (1974); (c) 78, 1862 (1974). (37) F. Birnstock, H. J. Hoffman, and H. J. Kohler, Theor. Chim. Acfa, 42, 311 (1977). (38) C. C. J. Roothaan, Rev. Mod. Phys., 23, 69 (1951). (39) W. J. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys., 51, 2657 (1969). (40) R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys., 54, 724 (1971). (41) A. I. Kitaygorodsky, Tetrahedron, 14, 230 (1961). (42) (a) P. Claverie in “Localization and Delocalization in Quantum Chemistry”, Vol. 11, 0. Chahret, R. Daudel, S. Diner, and J. P. Malrieu, Ed., Reidel, Dordrecht, 1976, p 123. (b) P. Claverie, “Elaboration of Approximate Formulas for the Interactions between Large Molecules. Applications in Organic Chemistry”, in ”Intermolecular Interactions: from Diatomics to Biopolymers”, B. Pullman, Ed., Wiley, New York, N.Y., in press; (c) A. T. Amos and R. J. Crispin, Mol. Phys., 31, 147 (1976); (d) in “Theoretical Chemistry, Advances and Perspectives”, Vol. 2, H. Eyring and D. Henderson, Ed., Academic Press, New York, N.Y., 1976, p 1. (43) F. Danon and K. S. Pitzer, J. Chem. Phys., 36, 425 (1962). (44) P. Claverie, “Elaboration of Approximate Formulas for the Interactions between Large Molecules. Application in Organic Chemistry”, section V-B and Appendix F in “Intermolecular Interactions: from Diatomics to Biopolymers”, B. Pullman, Ed., Wiley, New York, N.Y., in press.

Lucas et al. (45) (a) J. Caillet and P. Claverie, Biopolymers, 13, 601 (1974); (b) Acta Crystallogr., Sect. A , 31, 448 (1975); (c) J. Caillet, P. Claverie, and B. Pullman, Acfa Crysfallogr., Sect. B , 32, 2740 (1976). (46) J. P. Chandler, minimization program STEPIT, Quantum Chemistry Program Exchange, Program No. 66, Indiana University, Bloomington, Ind. 47401. (47) F. M. Westheimer and J. G. Kitkwood, J. Chem. phys., 6, 513 (1938); J. G. Kirkwood, ”Theory of Solutions”, Gordon and Breach, New York, N.Y., 1968, p 208. (48) L. Onsager, J. Am. Chem. Soc., 58, 1486 (1936). (49) B. Linder, Adv. Chem. Phys., 12, 225 (1967). (50) J. A. Abbott and H. C. Bolton, Trans. Fararky Soc., 48, 422 (1952). (51) A. Wada, J. Chem. Phys., 22, 198 (1954). (52) F. Buckley and A. A. Maryott, J. Res. Nafl. Bur. Stand., 53, 229 (1954). (53) H. Eyring, D. Henderson, B. J. Stover, ana E. M. Eyring, “Statistical Mechanics and Dynamics,” Wiley, New York, N.Y., 1964, Chapter 14, Section 5, p 401. (54) H. H. Uhlig, J. Phys. Chem., 41, 1215 (1937). (55) D. D. Eley, Trans. Faraday Soc., 35, 1281 (1939). (56) H. L. Clever, J. Phys. Chem., 62, 375 (1958). (57) J. H. Saylor and R. Battino, J. Phys. Chem., 62, 1334 (1958). (58) (a) R. C. Tolman, J. Chem. Phys., 16, 758 (1948); 17, 118 (1949); (b) J. G. Kirkwood and F. P. Buff, J. Chem. Phys., 17, 338 (1948); 18, 991 (1950); (c) F. P. Buff, ibid., 19, 1591 (1951); 23,419 (1955); (d) S. Kondo, ibid., 25, 662 (1956). (59) D. S. Choi, M. S. Jhon, and M. Eyring, J . Chern. Phys., 53, 2608 (1970). (60) R. B. Herman, J. Phys. Chem., 75, 363 (1971); 76, 2754 (1972). (61) (a) R. A. pierotti, J. phys. Chem. 67, 1840 (1963); (b) 89,281 (1965). (62) H. S. Frank and M. W. Evans, J. Chern. Phys., 13, 507 (1945). (63) G. Nemethy and H. A. Scheraga, J. Chem. Phys., 36,3382, 3401 (1962). (64) N. Bjerrum, Trans. Faraday Soc., 24, 445 (1927). (65) D. L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 79, 2562 (1975). (66) D. L. Beveridge and G. W. Schnuelle, J. Phys. Chem., 78, 2064 (1974). (67) G. W. Schnuelle and D. L. Beveridge, J. Phys. Chem., 79, 2566 (1975). (68) J. D. Payzant, A. J. Cunningham, and P. Kebarle, Can. J. Chem., 51, 3242 (1973). (69) D. H. Aue, H. M. Webb, and M. T. Bowers, J . Am. Chem. Soc., 94, 4726 (1972). (70) P. Souchay in “Fondements th6oriques des red-ierches sur les actions intermolbculaires”, 278 collcque national du CNRS, &litions du Centre National de la Recherche Scientifique, Paris, 1966, p 221. (71) The total charges given do not represent only the electronic populations; besides these,they also take into account suitable “effective” charges which simulate atomic dipole moments.

Effects of Sodium and Lithium Halides and of Hydrochloric and Hydrobromic Acids on the Coacervation of Aqueous Solutions of Tetraalkylammonium Halides A. Mugnier de Trobriand, M. Lucas,* DGR, B.P. No. 6, 92260 Fontenay-aux-Roses, France

J. Steigman,’ and L. L-Y. Hwang2 Department of Chemistry, Polytechnic Institute of New York, Brooklyn, New York 11201 (Received May 9, 1977) Publication costs assisted by Commissariat 5 Energie Atomique

The coacervation of aqueous solutions of Pn4NBr, Hex4NBr,Hex4NC1,and HeptlNCl was studied at 35 “C in the absence and presence of LiC1, LiBr, NaC1, NaBr, HC1, and HBr. The “invasion” of the R4NX-richaqueous solution by the acids is very marked. A comparison is made between the anion-exchange resin Dowex I X-10 and the R4NX-rich aqueous phase. Several theories which had been advanced to explain the “acid effect” in resins and the corresponding phenomena in quaternary ammonium salt solutions are discussed.

Introduction Aqueous solutions of tetraalkylammonium halides of sufficient chain length separate into two aqueous phases a t room temperature upon the addition of inorganic electrolytes. Since one aqueous phase contains most of the 0022-3654/78/2082-0418$01 .OO/O

inorganic salt, with a small fraction of the quaternary ammonium salt, and the other is rich in the organic salt, these coacervates have been used for the extraction of anionic complexes of ions such as U022+,and have been found to be very similar in behavior and selectivity t o 0 1978 American Chemical Society

Studies of Solvent Effects. 1. Discrete, Continuum, and Discrete

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