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5588 J . Phys . Chem. 1985 ,89 ,5588-5593

4 - H

I I I I I

q x.5

Figure 2. Correlation between the dissociation energies ofthe 0,-H andO,*-H groups and the charges locatedon the ion bondedto the O 3oxygen calculated for clusters 03A13 006 (0H),o H2XwhereX standsforH, Li, Na, and K atoms).

acidity of the skeletal 0 1- H groups decreases in the orderH >Li > N a > K.

These results thus indicate that the acidity of the skeletalhydroxyl groups is determined by th e actu al charge localized on

the zeolite skeleton, in addition to a numb er of other factors (Si:AIratio, structural characteristics, etc.); thus, this acidity dependson the amount of electron density transferred from the skeletonto the ions compensating its negative charge. It follows from thecalculation that this transfer is smaller for the cations than forH ions forming th e hydroxyl groups (in fa ct, ab initio calculationspredict very small transfer of electron density to the cation6).Together with the fact that the acidity of the skeletal hydroxylgroups decreases with increasingly negative charge on the zeoliteskeleton, this fact leads to the following general conclusions: (i)th e acidity of the skeletal hydroxyl groups of the purely H formof the zeolite is always higher tha t tha t of these O H groups in

the indentical (structurally a nd with the same Si:Al ratio) zeolitescontaining both theOH groups and cations compensating thenegative skeletal charge; (ii) for various types of cations localizedto the same degree (th e same degree of decationization) in thesam e zeolite, the acidity of t he skeletalOH groups decreases withdecreasing electronegativity of the cation.

The same conclusions may be drawn by using Sanderson'sm od el of e l e c t r ~ n e g a t i v i t y. ~ ~ - ~ ~f the acidity ofOH groups ischaracterized by the charge located on theirH atoms (acidityincreases with increasing charge ), then for zeolites with the overallformula HJ1-,$i9hA1960384 (whereX is a monovalent cation and0 < n < 96) the average partial ch argeon the H atoms is given27as

(S04S,1SsiSHxSx1- )1/7 S H

2 .08SH I2qH =

where is the atomic electronegativity of individual atoms andx = n/96 is the degree of decationiz ation. As for the cationsstudied the electronegativities decrease2' in the following order:SH 3.55) > SL1 0.74) > SNa 0.70) > SK 0.56), it is apparentth at this appro ach results in the sa me conclusions as mentionedabove.

Finally, it should be noted that the terminalOH groups of theclusters with individual cations exhibit similar behavior as theskeletal O H groups. For the ions studied , the acidity of theseterminal O H groups increases in the orderK < N a < Li < H ,as indicated by the charges calculated on their H atoms (whichincrease), as well as by th e values of the0 - H bond orders (whichdecrease).

(27) Sanderson,R. T. Chemical Bonds and Bond Energy ,2nd ed.;Ac-

(28) Jacobs,P. A.; Mortier, W. J. ; Uytterhoeven, J . B. J . Znorg. Nucl.

(29) Jacobs, P. A, ; Mortier, W . J. Zeolites 1984, 2, 227.(30) Mortier,W . J. J . Catal. 1978, 55, 138.(31) HoEevar,S . ; Driaj, B. J . Mol . Catal. 1982, 7 3 , 205.

ademic Press: New York, 1976.

Chem. 1978, 40, 1919.

Reevaluation of the Born Model of Io n Hydration

Alexander A. Rash in+ and Barry Honig *

Department of Biochemistry and Molecular Biophysics, C olumbia U niversity, New York, NewYork 10032(Received: June 5 1985; In Final Form: August 23 1985)

In this paper we dem onstrate that the B orn theory provides an accurate m eans of calculating the solvation energies of ioin water. The well-known equationAG,' = - ( q 2 / 2 r ) ( l 1/D) is rederived in a somewhat modified form( r is a radius andD is the dielectric constant), and it is found that the value ofr most consistent with the model is the radius of the cavityformed by the ion in a particular solvent. The failures of the Born theory are attributed to the use of ionic radii rather thacavity radii. The ionic radii of anions are shown to be a reasonable measure of the cavity size, but for cation s we argubasedon electron density profiles in ionic crystals, that covalent radii rather than ionic radii must beused. When these measuresof cavity size are introduced into the Born equation, experimental solvation energies are fairly well reproduced. Moreovethe additionof a single correction factor into the model, an increase of7 in all radii, leads to excellent agreement withexperim ent for over30 ions ranging in charge from1- to 4+. This need for corrected radii may be duein part to an increasein cavity size resulting from packing defects and toour neglect of dielectric saturation effects. However,it appears tha tdielectric saturation is not a dom inant facto r since the model works quite well for polyvalent ions where saturation effecshould be strongest. Applic ations of the Bo rn method to the transfer of ions between different solvents are discu ssed, anthe relation of our results to detailed sim ulation s of ion-solvent interaction sis considered.

IntroductionTh e Born theory ,' proposed Over 60 years ago, has provided

a useful and intuitively simple method of estimating the solvationenergy of ions, Interaction s between the ion and solvent are

'Present address: Departmentof Physiology andBiophysics,Mount SinaiSchool of Medicine One Gustave L. Levy Place, New York, N Y 10029.

assumed to be electrostatic in origin with the ion viewed as acharged sphere of radiusr and the solvent as a dielectric continuumOf dielectric constantD he elec trostatic work associated withcharging the ion immersed in the dielectric continuum is then givenby

( 1 ) Born, M . Z hys. 1920, I 45.

0022-3654/85/2089-5588 01.50/0 1985 American Chem ical Society

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Reevaluation of the Born Model of Ion Hydration The Journal of Physical Chemistry, Vol 89, No. 26, 1985 5589

W = q 2 / 2 D r (1)

where q is the net cha rge of the ion. Th e free energy of trans-ferring the ion from vacuum to a medium of dielectric constantD is just the difference in the charging energies, i.e.

AGso = - 1 i)It has been traditional to use ionic radii derived from crystalstructures to evaluate eq2 (Le., to setr = r lon lc ) ,nd under thisassumption qualitatively reasonable results can be obtained.2Forexample, as predicted by eq2 , the solvation energies of halideanions and alkali cations are inversely related to th e ionic radius.However, Latim er et aL3 found th at it is necessary to add0.1 8,to the ionic radii of anions and0.85 8, o the ionic radii of cationsto mak e the Born equation fit experimental data on a series ofalkali halide salts. They provided a qualitative justification forthese corrections in terms of the distance from the cen ter of theion to the center of the nearest water dipole. While this adjustingof parameters may be viewed as evidencefor the failure of thetheory, it is rather remarkable th at such a crude electrostatic modelworks a t all and yields rather good agreement with experiment.Furthermore, the correction required for halide ions is smallenough to suggest th at the theory has the potential to yield rea-sonably accurate results.

Wit h this possibility in m ind, we have in this paper reevaluatedthe assum ptions implicit in the Born model. W e find (in agree-ment with standard practice) that the use of ionic radii for anionsis consistent with the m odel but th at it is inappropriate to use theionic radii of cations ineq 2 . Rather, we demon strate that covalentradii provide a logically consistent choice ofr for cations. Sincethe covalent radii of cations are on the order of0.6-0.8 8, largerthan the corresponding ionic radii,our results provide astraightforward quantitative explanation for the correction appliedto cations by Latim er et aL3 Mo reover, we demonstrate tha t theBorn theo ry in its simplest form provides an ac cura te mean s ofestimating solvation energies in water.

Detailed descriptions of various attem pts to improve the Bornmodel ap pea r in a num ber of excellent reviews (see e.g. ref2 and4-7). In th e following we briefly discuss the physical basis ofthe various corrections that have been introduced. As pointedout by Born,l th e free energy, AGso, of bringing a charged spherefrom vacuum to a dielectric continuum is equal to thesum of thefree energies of th e following th ree processes:(1) stripping thesphere of its charg e in vacuum , AGIO;( 2 ) transferring the un-charged sphere from vacuum into the solvent, AG20; and(3)recharging the sphere in the solvent, AG30. In the original Borntheory, AG2' was assumed tobe zero, and thus, only the chargingand discharging terms contribute to the solvation energy. Theselead directly to eq2 .

Equation 2 has only one parameter, the radiusr , which isgenerally set equal to th e ionic radius. As pointed ou t above, theuse of ionic radii leads to a n overestimate of experimental solvationenergy of anions, and in particular, the calculated solvation energyof cations can be almost100 kcal/mol greater tha n the experi-menta l values.* If the Born model is to be retained , there are,in principle, three ways to reduce the calculated solvation energies:(1) Increase th e effective radius used in eq2 . ( 2 ) Decrease theeffective dielectric constant of water.(3) Add a correction term

(2) Bockris,J. O'M.; Reddy, A.K. N. ''Modern Electrochemistry ; PlenumPress: New York, 1977;Vol. 1.

(3) Latimer,W. .; Pitzer, K. S.; Slansky, C. M.J . Chem. Phys. 1939,7, 108.

(4) Conway, B. E.; Bockris, J. O'M. In Modern AspectsofElectrochemistry ; Tompkins,F. C., Ed.; Academic Press: New York, 1954;VOl. 1, p 47.

(5) Rosseinsky,D. R. Chem. Rev. 1965, 65, 67.(6) Conway, B. E. Ionic Hydration in Chem istry and Biophysics ;El-

sevier: Amsterdam, 198 .(7) Desnoyers, J. E.; Jolicoeur, C. In Comprehensive Treatiseof

Electrochemistry ; Conway, E. B., Bockris,J. O M ., Yeager, E., Eds.; PlenumPress: New York; 1983;Vol. 5.

to account for AGzo, he energy of transferring the neutral spherefrom vacuum to water.

Latimer et ale3were able to fit experimental solvation energiesto the Born equation by increasing the effective radius of the ion s.Stokes6 suggested the use of van d er W aals radiifor calculatingthe charging energy in vacuum while retaining ionic radii for theions in water. Since it is the vacuum term th at m akes the largestcontribution to the solvation energy (due to th e1 / D term in eq2 ) , he use of van der W aals radii which a re larger than ionic radii,particula rly for cations, produced improved ag reemen t with ex-periment. While Stokes was the first to question the use of ionicradii in the Born expression, we will demonstra te below th at th esame radius should be used in vacuum an d in water and that vander Waa ls radii are not the optimal choice.

Most atte mpts to improve the Born model have been based ona reduction of the effective dielectric constant of the s ~ l v e n t . ~ -Th e justification fo r this procedure, suggested by Noyes: is tha tdielectric satura tion occurs in the vicinity of the solvated ion, andthus, the effective dielectric con stant in this region is less than80. However, we will demonstrate that the Born model producessatisfactory agreement with experiment even if the effects ofdielectric satu ration are ignored. In fact, our results suggest thatthe effects of dielectric saturation are relatively small, even formultivalent ions.

The third approach th at has been used to refine the Born modelis to explicitly accoun t for AG20, the energy of transferring the

discharged ion from vacu um to ~ a t e r . ~ ~ ~ - ' ~ * ' ~hile this term islikely to mak e some contribution , the m agnitu de of the effect ismall,̂ .̂ and it cannot by itself account for th e large discrepanciesbetween predicted and experimental solvation energies.

It should be pointed out that there has been considerableprogress in the explicit simulation of ion-water interactions.14-However, there still remain significant discrepancies betweentheoretical and experimental solvation energies, due in part to thevarious approximations (e.g., cutoffs, periodic boundary conditions,potential functions) th at are used in the simulations. Thu s,continuum models retain their value both as a simple means ofcalculating solvation energies and as a source of insight in to theresults of detailed simulations.

A Revised Born Model

Rederivation of the Born Equation. In this section we re-consider the Born cycle by breaking up the discharging andchargin g processes into a numbe r of discrete steps. Thi s will allowus to arrive at an unambiguo us definition of the radius to be usedin eq 2 . W e will carrq thro ugh t he derivation for the case of acation; the derivation for anions is completely analogous. It shouldfirst be pointed out t ha t since discharging a cation involves addingan electron to the ion, the total electrostatic energy actuallyinvolves a nuclear-electronic attrac tion term in addition to thepositive self-energy of the electronic shell itself. Howev er, thespherical symmetry of the ion makes it possible to describe thesituation in terms of discharging a shell of positive cha rgeso thatthe nuclear attraction term need not be treated explicitly.

The classical expression for the electrostatic work involved indischarging a cation in vacuu m1 is just

AGIO = - q 2 / 2 R , , (3)

where R,, is the orbital radius of the neutral atom. (No te that

(8) Stokes, R.H. J . A m . Chem. SOC. 964, 86 979.(9) Noyes, R. M. . Am . Chem.SOC. 1962, 8 4 , 513.(10) Millen, W. .; Watts, D. W . J . A m . Chem. SOC. 967, 89, 6051.(11) Padova, J.J . Chem. Phys. 1972, 56, 1606.(12) Beveridge,D. .; chnuelle,G . W . J . Phys. Chem. 1975, 79, 2562.(13) Abraham,M. .; iszy, J. J . Chem. SOC., araday Trans. I 1978,

(14) Mezei, M.; Beveridge,D. L. J . Chem. Phys. 1981, 7 4 , 6902.(15) Szasz, I ; Heinzinger,K. Z . Naturforsch. A: Phys. , Phys. Chem.,

(16 ) Chandrasekhar, J.; Spellmeyer,D. C.; Jorgensen, W. L.J . Am. Chem.

(17) Clementi, E.; Barsotti, R.Chem. Phys. Lef t . 1980, 59, 21.

7 4 , 1604.

Kosmophys. 1982, 38A, 214.

SOC. 984, 106, 10.

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5590

it is the orbital radius rather tha n th e ionic radius that enters intothe expression because the electron is added to this orbit duringthis discharging process.)

Transferring the ne utral ato m into the solvent produces a cavityof radiusR,, (ac denotes atomic cavity).R, , will in genera l belarger thanRaw The energy of recharging t he cation in the solvent,AG,', may be divided into three separate contributions: (a)AG3,',the energy of dispersing th e electron fromR, , to R,, in vacuum;(b) i\G3bo,he en ergy of dispersing the electron fromR,, to infinityin water; (c)AG3,', the work associated with the shrinking of thecavity around the ion to the ionic cavity radius,R , , ( R l c< Rat).Thus

The Journal of Physical Chemistry, Vol 89, No. 26, 1985 Rashin and Honig

AG3,0 is just the difference in the electrostatic energy of a sphericalshell of charge betweenR,, and R i , when filled with vacuum orwith the dielectric mediu m. This can easily be shown to be

AG3,' = q 2 / 2 R , , q 2 / 2 R i ,- q 2 / 2 D R i , q 2 / 2 D R a C 5 )

Thus

AG3' = q 2 / 2 R , , q 2 / 2 R i c+ q 2 / 2 D R i cand

AGIO + AG3' = ( q 2 / 2 R i , ) ( l / D -1) ( 7 )Equation 7 is just the Born expression with the ionic cavity

radius appearing explicitly in the denom inator. It should bepointed out tha t du e to a cancellation of termsonly the cavityradius in the solvent appears in thefinal expression (eq 8 . Notethat this radius would in generalbe expected to change in differentsolvents.

The Ionic Cavity Radiu s. Equation 7 states that the appropriateradius tobe used in theBorn expression is the radius of th e cavityformed by an ion in a particula r solvent. This is also quite rea-sonable on intuitive grounds since it is at this distance from theion that the dielectric constant becomes different than that ofvacuum a nd the medium actually begins. We are left then withthe problem of arriving a t appro priate values of the cavity radiusfor both cations and anions.

It seems plausible to define th e ionic cavity as a sphere whichcontain s a negligible electron density con tribution from th e sur-rounding solvent. Analysis of electron density distributions in ioniccrystals1* see e.g. Figure 1) reveals that th e electron density du eto positive ions begins to become significant at a distance of abo utth e ionic radius from the center of the anion . It thus providesa reasonable measure of the cavity radius formed by an anion.The situation is quite different for cations. The ionic radius ofthe cation does not extend o ut to th e high electron density regionof the bound anion. In fact, due to qua ntu meffect^, ̂ ^^ t̂heelectron cloud of the anion is unable to significantly penetrate theempty valence orbital of the cation. As a result, the electrondensity of the anion begins to become significant at a distancefrom the nucleus of the cation corresponding approximately tothe orbita l radius of its valence electron. This radius provides afar more ac cura te measure of the cavity size formed by a cation

than does the ionic radius. Since orbital radii correspond closelyto covalent radii, which are experimentally determined quantities,we propose the use of covalent radii as a reasonable and convenientestimate of the radius to be used in the Born equation as appliedto solvated cations. As discussed above, the ionic radius constitutesa good estim ate for the cavity radius formed by solvated anions.

Analysis of electron density maps dem onstrates that the covalentradii of cations and the ionic radii of anions are closely relatedand provide a useful first approximation of the cavity radius

18) Gourary, B.S.; Adrian, F. J. Solid State Phys. 1960 10 127.(19) Slater,J. C. J . Chem. Phys. 1964, 41 3199.(20) Vainshtein, B. K.; Fridkin, V . M.; Indenbom, V . L. Modern

Crystallography ; Vainshtein,B. K., Ed.: Nauka: Moscow, 1979; Vol. 2.(21) Lang's Handbookof Chemistry , 11th ed.; Dean, J.A,, Ed.;

McGraw-Hill: New York, 1973.

lA

Figure 1. Electron density distribution in LiF crystal along the lineconnecting the centersof the fluoride ion (point A) and lithium ion (pointE).I8 Segments AD andDE orrespond to ionic radiiof F and Li . CD= 0.32A, * D = 2 C D , A E = 1.96 A and DE = 0.6

TABLE I: Internuclear Distances in Alkali Fluorides, the Radii ofCavities Formed by Alkali Ions, and Their C ovalent Radii

Li+ Na+ K+ Rb+ Cs+internuclear distance2' 1.96 2.31 2.69 2.84 3.05cavity radii 1.24 1.59 1.97 2.12 2.33covalent radii2' 1.23 1.57 2.03 2.16 2.35

'Calculated by setting the hard-core radiusof F equal to 0.72 8(see text).

formed by the respective ions in salt crystals. Figure1 plots theelectron density distribution in LiFI8 whichis clearly quite sym-metric abou t the minimum, defined by pointC. The ionic radiusof F orresponds to the segment AD. The electron density dueto Li' begins to rise steeply at point D, which as discussed aboveis the justification for using ionic radii to d efine the cavities formedby anions. In order to obtain an internally consistent definitionof the cavity formed by cations, we use the electron density atpoint D, which corresponds to the ionic radius ofF o define theelectron density at the boundary of any ionic cavity. PointB hasthe sam e electron density as point D, and thus, the segmentEB

defines the cavity formed by the Li' ion. This is found to be1.24

A which is close to the covalent radius of Li of1.23 A. Thesegment AB, equal to0.72 A , may be viewed as defining a hardcore of theF ion beyond which the cavity formed by the cationbegin s. If we use this valu e in all alkali fluoride salts, it is possibleto obtain a consistent measure of the cationic cavity radii bysubtracting0.72 A from the internuclear distance.

In Tab le I, the cavity radii obtained in this way are comparedto covalent radii. Th e remark able correspondence between thetwo values strongly supports our suggestion that covalent radiibe used to define the cavity radius of cations. Moreover, itdemonstrates that the ionic radii of anions and the covalent radiiof cations measure the same property, i.e., the distance from th enucleus at which the electron density of the surrounding mediumbegins to become significant. This then provides the underlyingjustification for the use of these values in the Born equation.

It should be pointed out that previous attem pts have been madeto use electron density maps as a basis for defining ionic radii.In particular, Gourary and Adrian'* equated ionic radii to thedistance between the point of minimum electron density (pointC in Figure 1) and the center of either ion. The radiiso definedremoved the asymmetry in the solvation energies of positive andnegative ions but did not produce accurate agreement with ex-perimen tal values.22 In contrast to the radii of Gou rary andAdrian,18the sum of the radii used in this work is not equal tothe internuclear distance. This is reasonable since we are in-terested in obtaining cavity radii an d both th e anion and cation

(22) Blandamer, M.J.; Symons,M . C. R. J Phys. Chem. 1963,67, 1304.(23) Gold,E. S. Inorganic Reactions and Structure : Holt, Reinhart and

Winston: NewYork, 1960.

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Reevaluation of the Born Mo del of Ion Hyd ration

-140

TABLE 11: Experimental and Theoretical Values of the Heats ofSolvation of Salts (in kcal/mol)'

salt AHcxptl AHalfd error AHEald(cor) errorLi F

-

-245.2CIBrIN a FCIBrIK FCIBrIR b FCIBrIC s FCIBrI

-211.2-204.7-194.9-217.8-183.8-177.3-197.5-197.8-163.8-157.3-147.5-192.7-158.7-152.2-142.4-186.9-152.9-146.4-136.6

-261 O-227.7-221.1-21 1.7-23 1.6-198.3-191.7-182.3-207.5-174.2-167.6-158.2-202.6-169.3-162.7-153.3-196.4-163.1-156.5-147.1

6.47.88.08.66.37.98.18.84.96.46.57.25.16.76.97.65.16.76.97.7

-243.9-212.8-206.6-197.9-216.4-185.4-179.2-170.4-194.0-162.9-156.7-147.9-189.3-158.3-152.4-143.3-183.5-152.4-146.2-137.5

0.5-0.8-0.9-1.5

0.6-0.9-1.1-1.7

1.90.50.4

-0.31.80.2

-0.1-0.6

1.80.30.1

-0.7

'The experimental values are from ref 2. Th e enthalpies of the saltsare sums of the solvation enthalpies of the individual ions calculatedaccording to eq8 with radii taken from ref 21 and 23 (for halogens).

share a commo n low electron density region (segme nt B-D) aspar t of their cavities. Thu s, cavity radii will always sum to valueslarger than the internuclear distance.

Results and DiscussionSolvation Enthalpies of Ions in Water. Since the free energies

of solvation of individual ions cannot be measured directly,eq 2cannot be tested for individual ions. However, the entha lpies ofsolvation of various salts are known and can be compared totheoretic al values obtained from t he Born expression for the en-thalpy2

where T ( a D / a T ) / D = -1.357 for water at 298 K.' Table I1

compares calculated an d experimental values of th e enthalpiesof solvation of alk ali halide salts.As can be seen by comparingcolumns two and three, the ag reement between theory an d ex-perime nt, even with uncorrecte d radii, is quite good. This levelof accuracy has notbeen obtained previously from the uncorrectedBorn expression. It ap pears then tha t th e Born expression worksquit e well, even if dielectric saturat ion is ignored.

As can be seen from Ta ble 11, the calculated results are con-sistently larger than the experimental ones by about10-15kcal/mol. This error might be attributed to the inherent limi-tations of the theory, but the fact th at it isso systematic suggeststhat it is du e to an identifiable factor. On e possibility, the neglectof dielectric saturation, willbe discussed below. However, perhapsthe most straightforward explanation is that anionic radii andcovalent radii for cations underestima te the cavity size. Indeed ,as is evident from Figure1 the electron density of the atomssurroundin g the central ion continues to increase at distances fromthe nucleus that are greater than these radii. Moreover, theproblem of packing bound solvent molecules would be expectedto expan d the cavity somewhat. Thu s, it seems quite reasonableto expect that the cavity radii th at best fit the Born model besomewhat larger than those defined above. In order to obtain anoptimal fit to th e experimental results of Table 11, we have definedcorrected Born radii by increasing each ionic and covalent radiusby 7 . The corrected enthalpies of solvation obtained from theseradii deviate from the experimental values by a maximum of only4 kcal/mol.

In order to te st the reliability of our modified Born model, itis necessary to com pare calculated a nd experimental solvationenergies for ions not included in our original samp le (Tabl e11).Since experimental values for isolated ions canno t be determined

The Journal of Physical Chemistry, Vol 89 , No. 26, 1985 5591

-60

- 90

I-110-

120

130

\-150 3 .4 5 6 a . .9r - 1

Figure 2. Compa rison of experimental (dots) and predicted (solid line)enthalpies of ion hydration. Th e enthalpies are given per unit charge.

directly, we base our comparisonon the relative heats of solvation,AH,,, (relative to H+ ions).

These are defined by2

Affrel(x-) = AHabs(X-) Mabs(H+) ( 9 )

= ( 1 0 )where X- and M + denote respectively anions and cations andAHah

denotes absolute heats of hydration.The fourth column in Table I11 contains experimentally de-termined relative heats of hydration fo r3 ions. By calculatingthe absolute heat of solvation for each ion from the Born ex-pression, we can obtain from eq9 and 10 a value of AHab,(H+)which is appro priate for each ion. Since the heat of solvation ofa proton must be a constant, the variation in the calculated valuesfor AHab,(H+) is a test of t he in ternal consistency of th e model.We find for alkali and halide ions that AHab (H+) varies between-260.00 and -264.3 kcal/mol with a mean value of-262.18kcal/mol. Thus, to within1 , AHab,(H+) s constant.

As another test of the model we use the valueof -262.18kcal/mol for the solvation energy of a proton and the experi-mentally determined values for relative solvation energies to obtaina series of solvation enthalpies of th e individual ions. These ar elisted in Tab le I11 and plotted as a function of the corrected radiusin Figure2 . Th ere appears to be excellent agreement betweenthe exp erimental values and the straigh t line derived from eq8 .Thus by introducing only one adjustable param eter into the Bornmodel, we have successfully reproduced experimental results for3 1 ions. Previous attem pts to improve the theory have involvedthe use of more param eters and have not in general consideredas large a number of ions.24

Som e of the ions for which the relative heats of hydration ar e

(24) Since the covalent radiusof ammonium is not a well-definedquantity, we have obtained a measureof the cavity rad ius from th e followingprocedure. The Na+-o xygen nternucle ar separation in water is approximately2.35A.5 Since the N a+ cavity radius is1.68A, the hard-core radiusof theoxygen is approximately0.67 A. Since a typicalNH++.-Ohydrogen bond hasa lengthof 2.8 A, the cavity size of ammonium is2.13 A as listed inTableIV.

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5592 The Journal of Physical Chemistry, Vol 89, No. 26, 1985 Rashin and Honig

TABLE I11 Cavity Radii and Hydration Enthalpiesion corr ected radius,’ , AHcalcdrb cal/mol kcal/mol AHabs,C cal/m ol errorf AH,,(H+),d kcal/m ol

Li+ 1.316 -126.7 13 6.34 -125.8 -0.7 -263.0N a +K+Rb+c s +FCI-Br-I-

c u +Ag+c u z +

Mg2+C aZ+Sr2+Ba2+Zn2+C d2+Hgz+

SC’+Y’+La3+Ce4+Ce3+Ga”

N H 4 + gO H -S2-S H-

AI^+

1n3+

1.6802.1722.3112.5141.4231.9372.0872.343

1.2521.4341.252

1.4551.8622.0542.1191.3381.5091.541

1.3381.5411.7331.8081.7611.7611.3381.605

2. I301.4981.9691.969

-99.3-76.8-72.2-66.3

-1 17.2-86.1-79.9-71.2

-133.2-1 16.3-532.8

-458.4-358.3-324.7-314.8-498.7-442.1-432.9

-1 122.7-974.1-865.8-830.0

1 515.0-852.2

-1122.1-935.1

-77.9-111.3-338.8

-84.7

163.68183.74188.8194.6

-381.5-347.5-341.0-331.2

118.7147.1

19.9

62.0140.8176.1210.1

32.889.885.0

-331.6 .-153.3

-82.1-2.5

-508.0-67.0

-337.6-199.9

185.0-37 1 o-849.4-341 O

-98.5-78.4-73.4-67.6

-1 19.3-85.3-78.8-69.0

-143.5-115.1-504.5

-462.4-383.6-348.3-3 14.3-49 1.6-434.6-439.4

-1118.1-939.8-868.6-789.0

-1556.7-853.5

-1124.1-986.4

-77.2-108.8-325.0

-78.8

-0.82.11.71.91.8

-0.9-1.4-3.1

7.2-1.1-5.6

0.96.66.8

-0.2-1.5-1.7

1.5

-0.4-3.6

0.3-5.2

2.70.20.25.2

-1 .o-2.3-4.2-7.5

-263.0-260.5-261 O-260.9-264.3-261.4-261.1-260.0

“T he radii ar e taken from ref 21 and 23 (f or halogens) increased by 7% (see text). bT he enthalpies of solvation ar e calculated from eq8 with theradii from column two. CT he elative enthalpies of solvation are from ref 5, except for the first nine which are from ref 2. dT he absolute heatssolvation of hydrogen ion ar e calculated from eq 9 and10. CT he bsolute enthalpies of solvation are calculated from eq 9 andI O and the calculatedmean value of the absolute heat of solvationof the hydrogen ion, AHab (H+), of 262.18 kcal/mol (see text). /The errors are in calculated values fromcolumn three comparedto the AH,,, in the fifth column. gSe e ref 24.

available5 are no t listed in Table11. These includeTI+,TI”, Pb2+,Co2+,Ni2+, Mn2+, Cr2+,Cr3+,Fe2+, and Fe3+. For these ions thedifferences between the calculated a nd ex perimental enthalpies

exceed 10%. However, these ions form coordination complexeswith water (ref6, p 335) and, therefore, it would not be expectedthat the cavity size for these ions follows the same rules th at holdfor other ions. Finally, we have also excluded Be2+ rom th e table.For Be2+, he difference between the experimental a nd calculatedvalue is 17%, a discrepancy which may be due to large s aturationeffects resulting from the fact t hat Be2+ is a p articularly sm allion.

The success of a continuum dielectric model in reproducingexperimental results even for multivalent ions demonstrate s tha tcorrections due to the effects of dielectric saturation are not largein the calculation of solvation enthalpies (a point first made byLatimer et aL3). Th is may not be the case in the calculation ofother thermodynamic q~ an ti t i es .~ ,~lthough conflicting estimatesof saturation effects have been reported in the literatu re,6 ~ 1 0 ~ 1 3 3 2 5it is interesting to note tha t th e saturation effects obtained byMillen and Watts’O are on th e order of the 7% correction factorin the cavity radius tha t we have introduced.

Solvation in Nonaqueous Media. It is of interest to considerhow the Born model mightbe applied to solvents other tha n wa ter.This question is of particula r importa nce for biological systemswhere problems concerning the free energies of transfer of ionsfrom water to proteins and membranes arise in a variety ofcontext^.^^.^

Th e Born equation a s developed above can in principlebe usedto calculate th e energies of transfer of ions from vacuum t o any

(25) Schellman, J. A . J . Chem. Phys. 1957, 26, 1225.(26) Parsegian,A . Nalure London) 969, 21, 884.(27) Honig,B. H.; Hubbell,W roc. Nul l . Acad. ci. U.S . .4 .1984, 81,

5412.

solvent. Howev er, since the cavity size produced by a partic ula rion will vary in different solvents, a general expression for transferenergies must take this into account. We obtain an expression

for the free energy of transfer of an ion between two solvents byconsidering th e process in which th e ion is first transferred fromone solvent to vacuum and then from vacuu m into the secondsolvent. Th e energies associated with eachof these processes aregiven by eq 7, but differen t cavity radii must be used. Thu s

where R,, an dR,, are the two cavity radii. Whe n R,, is equalto R,,, the sta nda rd Born expression is recovered.

The correction to the st and ard expression can in principle bequi te large. Consid er for examp le the transf er of a C1- ion fromwater to hexane (Dl= 80,D2 = 2). If we use the corrected radiusof 1.937A (Table 111 for the C1- ion in water and as sume th atthe cavity radius in hexane is equal to the van der Waals radiusof the C1- ion (Rc2= 2.252 A), the total transfer energy is cal-culated to be 47.8 kcal/mol or6 kcal/mol larger than ifR,, =R,, = 1.937 A. While it is not clear that the value we haveassumed for th e cavity radius in hexane is the correct one, themagnitude of the effect emphasizes the need to account forvariations in cavity radii in estimating transfer free energies.

It should be pointed out that the cavity radii we have usedshould constitute a good first approximation for any hydrogen-bonding solvents. This implies th at the correction term will besmall in going, say, from water to ethanol. Since the1 D termsar e small in both solvents, the free energies of transfer should alsobe small, as is observed e~p erim en ta1 ly. l~ imilarly, the Bornexpression using the corrected radii of Table I11 should be ap-propriate for many applications to proteins where ions and ion-izable groups appear to always be hydrogen bonded.28

8/9/2019 Borns Model1

6/6

J . Phys. Chem. 1985 ,89 , 5593-5599 5593

Relation to Simulations. The success of the Born model inreproducing th e observed solvation energies suggests that it m aybe capab le of providing useful insights regarding m olecula r dy-namics and Mo nte Carlo simulations of ions in water. On e strikingprediction of the Born model is that the transfer energy of an ionto a particular solvent is essentially independent of th e dielectricconstan t, once the dielectric constant is high, say above 30. Thisfollows from th e 1 1 / D dependence of the tran sfer energy.Forexample, the Born model predicts about a 3 kca l/mol differencein transfer energies between water an d meth anol for a univalention, a value which is in good agreem ent with exp erimental results.13It would appear then that any model for water which producesa high bulk dielectric constant shou ld be successful in reproducingsolvation energies, even if the calculated dielectric constant isincorrect.

On the other hand, the Born energy is highly sensitive to thechoice of the cavity radius. When translated into the parameters

(28) Rashin, A. A ,; Honig, B.J . Mol. Biol. 1984, 173, 515.

used in detailed sim ulations , this implies tha t it is necessary tohave accurate potential functions which successfully reproduceshort-range interactions and can account, for example, for bindingenergies in the gas ph ase.29 Finally, since the Born theory predictstha t ion-solvent interactio ns at fairly long distances a re stillsubstantial (Le., 10 kcal/mol for interactions above 10A , it wouldappear necessary to determine whether the use of periodicbound ary conditions is capable of accounting for this contributionto the total solvation energy. In any case, the app aren t successof the Born model in treating the solvation energies of ions suggeststh at the continuum model will continue to be useful in a varietyof applications.

Acknowledgment. We tha nk M ichael Gilson and Bruce Bernefor useful a nd stim ulating discussions. This work was supportedby Grants GM-30518 (N IH) and PCM82-07145NSF).

(29) Arshadi, M.; Yamdagui, R.; Kebarle,P. J. Phys. Chem. 1970, 74,1475.

Laser Multipho ton Dissociation ofAlkylCations: 1. Fragmentation Mechanism of

Homologous AlkylCations Produced from Their Iodid esH. Kiihlewind, H. J. Neusser,* and E. W. Schlag

Institut fur Physikalische und Theoretische Chemie der Technischen Uniuersitat Miinchen, 0 - 8 0 4 6 Garching,West Germany (Received: June 24, 1985)

A homologous series of alkyl cations and their corresponding multiphoton m ass spectra a re produced from alkyl iodidestest the degree of fragmentation pa ttern transferability from one ionCnH2,+l+ o the nex t larger one, etc., in multiphotonmass spectrometry. In electron impact ionization all ions in the sam e mass range, even when produced from different parentagproduce similar fragmentation. In direct contrast to this multiphoton mass spectra of homologous alkyl compounds displstrong differences. The wavelength of the exciting light influences the fragmentation patternin a direct way, thus leadingto two-dimensional control of the fragmentation pattern. These features can be exploited for mixture analysis using maspectrometry. Mechanistically this can be understood as being due to the ladder switching mechanism leading to opticselective photon absorption by fragmen t ions. Results are also presented that a classification in terms of a single averaenergy for decomposing ions is not adequa te.A parametrization in terms of differing internal energies for differing ionsis required.

I. IntroductionMultiphoton ionization (MPI) is known tobe an unique method

providing additional variables for the production of polyatomicmolecular ions in a mass spe~trometer. '-~One such possibilityis the use of high light inten sities in excess oflo7W/cm Z to leadto fragmentation of organic ions with a large amount of smallfragmen ts, evenC cations,2 these being energetically very highlying.

In order to exploit this new technique there is considerable

interest in the clarification of the complex mechanism of themultiphoton fragmentatio n processes of polyatomic molecular ions.Photon absorp tion exclus ively within the molecular paren t i ~ n , ~ , ~

(1) -1 U.; Neusser, H.J.; Schlag,E. W. 2 aturforsch., A 1978, 33A,

(2) Zandec, L.; Bernstein, R. B.; Lichtin,D. A. J . Chem. Phys. 1978, 69,

(3) Antonov, V . S.; Knyazev, I. N.; Letokhov, V. S.; Matiuk, V. M.;

(4) Rockwood,S.; Reilly, J. P.; Hohla, K.; Kompa, K. L.Opt. Commun.

5 ) Lubman,D. M.; Naaman, R.; Zare, R. N.J. Chem. Phys. 1980, 72,

( 6 ) Fisanick,G. .; EichelbergerIV, T. S . ; Heath, B. A,; Robin, M . B.J .

(7) Cooper, C.D.; Williamson, A.D.; Miller, J. C.; Compton, R. N .J .

1546.

3427.

Morshev, V. G.; Potapov, V. K. Opt. Let t . 1978, 3, 37.

1979, 28, 175.

3034.

Chem. Phys. 1980, 72, 5571.

Chem. Phys. 1980, 73, 1527.

competing light absorption and dissociation in the neutral ma-nifoldi0 and in the ionic manifold in a so-called ladder switchingmechanism, *I2 as well as ionization and dissociation of supe-rexcited ne utrals5 have been discussed. Two-laser experim entsof our groupI2J3 nd more recently photoelectron kinetic energymeasurements from other laboratories have shown tha t for aro-matic hydrocarbons, e.g., benzene, toluene, and chlorobenzene,' l6as well as for a series of small molecules17J8he ladder switching

(8) Rebentrost, F.; Kompa, K.L.; Ben-Shaul,A . Chem. Phys. Lett. 1981,

(9) Rebentrost,F.; Ben-Shaul, A .J . Chem. Phys. 1981, 74, 3255.(10) Yang, J. J.; Gobell, D. A,; Pandolfi, R.S ; l-Sayed, M. A .J . Phys.

77, 394.

Chem. 1983.87. 2255.(1 1) Diet;, W ; eusser, H . J. ; Boesl,U.; Schlag,E. W.; Lin, S. H. Chem.

(12) Boesl, U.; Neusser, H.J.; Schlag,E. W . J . Chem. Phys. 1980, 72,Phys. 1982, 66 105.

4771(13) Boesl, U.; Neusser, H.J.; Schlag, E.W . Chem. Phys. Lett. 1982.87,

(14) Long,S. .; Meek,J. T.; Reilly,J. P. J. Chem. Phys. 1983, 79, 3206.(15) Long, S. .; Meek, J.T.; Reilly,J. P. J . Phys. Chem. 1982.86,2809.(16) Anderson,S. L.; Rider, D. M.; Zare, R. N.Chem. Phys. Lett. 1982,

1.

93. 11.(17) Achiba, Y.; ato, K.; Shobatake, K.; Kimura, K.J . Chem. Phys.

( 1 8 ) Glownia, Y. H.; Riley,S. .; Colson,S. D.; Miller, J. C.; Compton,1982, 77, 2709.

R. N. J. Chem. Phys. 1982, 77, 6 8 .

0022-3654/85/2089-5593 01.50/0 1985 American C hemical Society