Continuum reaction field calculation of

CONTINUUM REACTION FIELD CALCULATION OF

DIELECTRIC CONSTANT AND VAPOR PRESSURES

FOR WATER AND CARBON DISULFIDE

SHLOMONIR

From the Department of Experimental Pathology, Roswell Park Memorial Institute, Buffalo,New York 14263

ABSTRACT Continuum reaction field theory is applied to calculations of dielectric constant,contributionofintermolecular interactionstothe free energy ofaliquid,and heat of vaporization. Introduction of repulsive interactions and the use of one adjustable parameter, the freevolume, enables prediction of vapor pressures. The calculations areillustrated for a simple nonpolar liquid, carbon disulfide, and for liquid water. Itis shownthat when Onsager'sequationisrearranged to a quadratic equation, and arecently foundvalueofthepolarizabilityisemployed,its solutions for liquid water yield goodagreementwithexperimentalvaluesthroughoutthe whole tem-perature range. The decrease of the dielectricconstantwith temperature isessentially linearwith the inverse of absolute temperature, but there is additionalsignificant de-creaseduetothe decreaseofdensitywith temperature. Therelatively highvalue of the heat ofvaporization of liquid water is expressed in terms oflarge dipolar interactionof awater molecule with the environment, which is duetopolarizationeffects.

I. INTRODUCTION

Thisstudy presents a simple method for calculating the contribution of intermolecular interactions to the free energy of a liquid with the method of continuum reaction field

theory. The attractive interactions which will be considered are dispersion inter-actions and dipolar interinter-actions of a polarizable molecule embedded in a cavity with the entire environment. Previous attempts to obtain the heat of vaporization with the continuum reaction field approach (1) resulted in reasonable agreement with

experiment. This was an indication that in the case of simple liquids the model was applicable in accounting for intermolecular interactions, at least as a first order

ap-proximation. In the cases considered in ref. 1 dispersion interactions gave the domi-nantcontribution to intermolecular interactions. In the present work, further develop-ments intheapplication of the method are introduced, and the calculations on liquid waterillustrate the importance of the dipolar term.

It hasto be noted that in the calculation of both components of attractive inter-actionsno adjustable parameters are used. For the purpose of accounting for the incompressibility of liquids repulsive interactions will also be introduced. An intro-duction of repulsive interactions of exponential form, which is widely used, and the

use of anadjustable parameter, the free volume, enables an extension of the treatment for the predictionof vapor pressures at various temperatures.

Thecalculations will beillustrated for thecaseoftwo liquids, (a)carbon disulfide,

which stands for a simple nonpolar liquid, and (b) liquid water, which is a complex polar liquid. One reason for the choice ofliquidwater wasthe availabilityof extensive data. Another reasonwastheinterest in thestudyofliquidwateritself.

We will show that two of the anomalies attributed to liquidwater, itshighdielectric constant, E, and its highheat ofvaporization are explained in the framework of the reaction field theory, whose static part is the Onsagertheory(2), providedthat param-eters obtained from refractive index data, including theinfrared regionareemployed.

Incalculatingthedielectric constant, we employ a value ofthe static polarizability,

which accounts for atomic and molecular motions, and is 2.8 times larger (3, 4) than thepreviouslyusedvalue,whichonlytakes into account electronicmotions. The pre-viousapplication of the Onsager (2) equation to the case of liquid water resulted in an underestimate ofthe dielectric constant (30 instead of 80) when the vapor value for the permanent dipole moment was employed, or in an overestimate of the value of the permanentdipolemoment (5) (3 x

10-is

esuinstead of 1.84-1.85 x _10-

'8esu)

when the experimentalvalue of the dielectric constant was used.

Thisdiscrepancy led to a criticism of the Onsager theory and to the development

of a more general theory(6-8)for the dielectric constant. In parallel with this develop-ment various structural models (9-13) have been employed to calculate the dielectric constantofliquidwater. Bothapproacheshaveimprovedthe agreement with

experi-ment of calculated dielectric constants, but it was realized that the results were far from being satisfactory.

The main point of criticism of the Onsager theoryhas been its apparent neglect of effects of short range intermolecular forces (6, 8). The results of the present work would imply that perhaps the effects of short range intermolecular forces are partly included in the reaction field treatment through the values of the densities and

polar-izabilities. Theatomic and molecular contributions to thepolarizabilitymay depend

onthe molecularenvironment, which follows fromthefactthat several infraredbands

in theliquid state of water are absent in its vapor.

Itshould beacknowledgedthat the idea that the Onsager theory for the dielectric

constant may be applicableto the caseofliquidwater hasalreadybeen expressed by

Hill (14, 15) whoreplacedthe"internalrefractive index,"

n2,

in the visible byahigher

valueof its extrapolation in themicrowave region(vide

infra).

Our treatment is slightly different, makinguseofthevalue of the polarizability, which leads to a further im-provement of the agreement with experiment for the whole temperature range.

In section II we employOnsager'stheory to calculate the permanent dipole moment ofliquid water. In section III we will discuss the temperature dependence of the dielec-tric constant as given by the Onsager theory and will present results of calculation on water. In section IV we will present the calculations of free energy due to intermolecu-lar interactions and calculations of the heat of vaporization of carbon disulfide and

liquid waterwith thereaction field theory. In section V we will introduce repulsive interactions and willcalculate vapor pressures.

II. CALCULATION OF THE PERMANENT DIPOLE MOMENT OF LIQUID WATER WITH ONSAGER'S EQUATION

FollowingBottcher (5), Onsager'sequationfor the case ofapureliquidcanbewritten intheform

(e

-

1)(2e

+

1)/12re

=

[Na/(l -fa)]

+

[Ng2/(I

-

fa)23kT],

(1)

inwhichf is the factor of the reactionfield,

f

=

(l/a')[(2f

_-

2)/(2c

+

1)],

(2)

and a is given by

(4ir/3)Na3 = 1. (3)

Nis the number of molecules per unit volume, a is the polarizability, j is the permanentdipolemoment, k is Boltzmann's constant, andTis the absolute tempera-ture.

When Eqs. 1-3 are used for the case ofliquid water, using a = 1.45 x 10-24 cm3 (refs. 13, 16) (at t = 20C, e = 80.4) it turns out that ,u is approximately 3 x

10-"

esu.

Recentlywedevelopedasemiempirical procedure (3,4, 17) forobtainingthe repre-sentationoftherefractiveindex as afunction ofthe frequency (w)of the

electromag-neticfield. Thetheoryisbased on the model of bound anddampedoscillators and is described in ref. 18. For the purpose of the application in section IV we present a

descriptionof theprocedure (seeref. 4for moredetail). The basicequations we em-ploy are:

[p2(w)

-

lJ/[p2(w)

+ 2] =

(4,r/3)Na(w)

= _{,8(w), (4)}

and

P

(4r/3)Na(w) = , C,/[I _ (W/W,)2

-iyw/wf2],

(5)

1-1

in which wf is an absorption frequency and y, is a damping factor;

ni

is the complex refractive index given by

n(w) = _n(w) + ik(w), (6)

inwhich the real part, n(w), is the ordinary refractive index; C, is a coefficient pro-portional to the oscillator strength

f,;

C, =

4irNf,e2/3w2mi,

₍₇₎

where

e,

and

m,

stand for the charge and mass of the oscillator, respectively. For molecules possessing permanent dipole moments Eq. 4 is applicable only at sufficiently highfrequencies,i.e., when the permanent dipole moments cannot followthe rapidly

oscillatingelectromagnetic fields. For liquid water we used Eq. 4 at wavelengths shorter than10-2cm. In our procedure we used a least-squares fitforthe real part of Eqs. 4and 5. These equations represent a set of m linearequationsin the

C,'s

where m,the number of measurements of i(w), is much larger than P, the number of absorp-tionfrequencies (bands) employed. The

y,'s

mayberegarded as parameters, or,

alter-natively,aresetequal to thebandwidths. Whenthe

C,'s

areknownthe polarizability extrapolatedtoinfinite wavelength is alsoknownaccordingtothe relations

p

a(0)

-

a,(0),

(8)

and

al(O) =

3C,/4rN.

(9)

Theprocedurehasalready been applied for hydrogen,oxygen,andwater. inboth the gaseousand liquid states, and for carbondioxide and monoxide, carbondisulfide, ben-zene,glassand to solutions (19). In all cases agoodagreement betweencalculated and

observedn(w) wasobtained, in most casessignificantly betterthan according to pre-vious dispersionformulas. The calculations on liquid water, wherefour infraredbands and one ultraviolet band have been considered, yielded values of the polarizability a(0) significantly largerthanpreviouslyrecorded. When data on k(w) are included

weobtaina(0) = 4.04x 10-24 cm3,whereas neglecting k(w) leads to a(O) = 3.95 x

10-24 cm3. Both results areappreciably largerthan thepreviouslyused value(16),a = 1.45 x

10-1

cm3,which was determinedby ignoringtheinfrared bands in

extrapolat-ing from thevisibleandultraviolet regions towardsinfinite wavelengths. The extrapo-lation tow = 0 of the ultraviolet and visible contribution to a(w), leads to the same valuefor both the liquid and gaseous states of water, which alsocoincideswith the pre-viously obtained value (16). This part of a may be interpreted as due to electronic

motions. Thelargevalueof the staticpolarizability ofliquidwateris duetothe first twoinfrared bands which areinterpreted asarising from intermolecular motions (20)

andmightbe due to thehydrogenbonds in the structure ofliquidwater(Table I). Makinguseof Eqs. 3-5 we obtain: fora = 4.04 x 10-24 cm3,, = 1.70 x

10-18

esu

fora= 3.95 x

10-1

cm3,Mi

= 1.75 x

10-'8esu.

These values are closetothevaporvalueju = 1.84 x 10-18esuandarewell within the range of values obtained from dilute solutions ofwaterin nonpolar liquids,,u = 1.7

-1.9 x 10-18esu(see ref. 21, p. 31).

III. TEMPERATURE DEPENDENCE OF THE DIELECTRIC CONSTANT

Inorder tostudy thepredictions of the Onsager theory for the temperature dependence of the dielectric constant at different temperatures, we first make use of Eqs. 2 and 3

BIOPHYSICAL JOURNAL VOLUME 16 1976

TABLE I

ABSORPTION WAVELENGTHS AND BAND CONTRIBUTION TO THE STATIC POLARIZABILITY OF WATER

a%(0)

contribution Absorption tostaticpolarizability

Substance wavelength (XI024cm3)

Mm

Liquidwater(t-.20C,dataon 51.8 1.10

k(w) included in calculations), 14.3 1.33

a-4.04xl0-24cm3 6.08 0.03

2.86 0.14

0.099 1.44

Liquid water (t-20C, data on 51.8 1.11

k(w)notincluded in calculations) 14.3 1.23

a - 3.95x 10-24cm3 6.08 0.03

2.86 0.14

0.099 1.44

Gaseouswater(t-0C) 0.099 1.456

andrearrangeEq. 1 from an apparentlycubic equationto aquadraticequationin e,

Ae2- Be- C =

0,

(10)

inwhich A = 2(3 - 4irNa)2, B =

(36wrNs2/kT)

+ (3 + 8rNa) (3 - 4rNa), and

C=(3 + 8irNa)2.

Notethat inthe absence of a permanentdipole, or atoptical frequencies,when the permanentdipolemoments cannotfollow theoscillating fields, Eq. 10 is reduced to the Lorentz-Lorenzequation, Eq. 4,which is usedin ourprocedure for obtainingthe

polarizability from refractive index data. Although no approximation is needed for the solution of Eq. 10 itis instructivetoconsiderit forthe case A, C, << B,whereto first order ofapproximation,

e= B/A,

-l8rNjs2 +1 1 + 8rNa/3

kT(3

-

4rNa)2

2

(1

-

4irNa/3)(

Frominspection of Eq. 11itfollowsthatif changes indensitywith temperature may beignorededecreaseslinearly with I/T. However, the decrease of the

density

with

temperature introduces anothereffect whichmightbeof

importance.

Forinstance, in the caseof liquidwaterwhere thedensityat95°Cis

4%/cf

less thanits valueat20°C,this

effectintroduces anadditional decrease ofmorethan 10% in the value ofe at 95°C,

relative to its value at20C.

As waspointed outinsection I the ideathatthe Onsagertheory issatisfactoryfor the case ofliquidwater hasalready been expressed by Hill (14, 15) who replaced n2 by

ez. in thefollowing equation,

(-

-

n2)(2e

+

n2)/e(n2

+

2)2

=

4irNis2/9kT.

(12)

c. is thehigh-frequency limit ofewhich is usedto fit the experimental results in the microwaveregion. Hill (14) employedthetwo values: E. = 5 andE,0 = 4.5,resulting

in valuesfor,u between 1.68 and 1.61 and between 1.83 and 1.76, respectively, in the temperature range0-50°C. However, the temperaturedependenceofE. shouldnotbe neglected. Ifone assumes a tobe temperatureindependentandusesthe relation

E. = [1 +

(8ix/3)Na][1

- (47r/3)Na], (13) then duetochange indensity,E. in thecaseofliquidwater should decrease with tem-peratureby about 10%throughoutthe temperature range0-100°C. Asignificant im-provement inthe agreement withexperimentattemperatures between 0 and60°Cwas obtained by Hill (15) when using valuesofe. which decrease with temperature from 4.35at0°Cto4.05 at 60°C. In passing we will point out that this temperature de-pendence ofc. is compatible with Eq. 13 and with the assumption of temperature independence of the polarizabilityfor this temperature range.

Theneglectof the temperature dependenceof the polarizability is not an essential part of theprocedureoutlined in this section. Infact, theprocedureoutlined in Eqs.

4-6(see ref. 4) hastoberepeatedatvarious temperatures, thusyieldingthe

polarizabil-ityatanydesired temperaturedirectlyorbyaninterpolation.

CalculationsonLiquid Water

Itshouldbepointedoutthat inthis studythepolarizability was assumed to be

tem-peratureindependent, followingtheprocedurein previousstudies(2, 5-7, 13, 22) on

thedielectricconstant. Thereis ahost ofexperimentaldata(3-5, 21, 22)which indi-cates that the electronic contribution to the polarizability is essentially temperature independent.

In the case of liquid water the specific refraction, [(n2 - l)/(n2 +

2)](1

/d), at thesodium D line varies from 0.206254 at0°Cto0.205919 at60C(23)(d is the

den-sity).

The temperature dependence of the contribution of infrared bands to the polar-izability has not yet been studied. There have been studies which indicate temperature

dependenceof theinfrared absorption bands (24-26). Walrafen (25) has reported a substantial decrease with temperature in the integrated intensity of the far infrared bands ofliquidwater. Therefore it may beanticipatedthat thepolarizability ofliquid

wateris adecreasing function of temperature. However, the magnitude of this decrease should bedirectlyinvestigated byperforming refractiveindex measurements.

The calculated and observed valuesof e(T) aregiven in TableII. Incolumn 3 the valuesa = 3.95x 10-4cm3 andit = 1.75 x

10-18

esu are used in thecalculations. Incolumn4 wepresent resultswhichareobtained by using the vapor value for

.",

i.e., g = 1.84 x

10-1'

esu, where the value for a isobtained from Eq. 10 at 20C (a =

3.77 x 10-4 cm3), and is in agreement within the uncertainty (about

5%)

with the larger value.

Acomparison of calculated with experimental values ofE at temperatures between 0 and 100C indicates a good agreement. The differences between calculated and

TABLE II

CALCULATIONS OF DIELECTRIC CONSTANT OF LIQUID WATER AT DIFFERENT TEMPERATURES WITH ONSAGER'STHEORY

s(calc) e(calc)

e(obs)* p 1.75x 10

i8esu,

p 1.84

x,10-18esu,

a 3.95 xlo-24

cm3

a 3.77 X

10-24

cm3

oc

0 88 86.6 85.8

5 86 85.1 84.6

10 84.1 83.4 83.2

20 80.4 80.4t 80.4§

25 78.5 78.7 78.1

30 76.7 77.1 76.8

40 73.3 73.9 73.5

50 69.9 70.7 70.4

60 66.7 67.5 67.3

70 63.7 64.5 64.4

80 60.8 61.5 61.5

90 58 58.6 58.6

100 55.3 55.8 55.8

*Values of dielectric constants and densities of liquid water are taken from Handbook ofChemistry and Physics, The Chemical Rubber Co., 50th edition (1969-70).

fThe value of eat20 C wasused to determine p. §The value of eat20 Cwasusedtodeterminea.

observed values of eareabout 1%andarein most cases within theexperimental

un-certainty. The agreement withexperiment is somewhat better than that obtained in

ref. 15. This mightbeattributed to the higherprecisionof refractive index measure-mentscomparedwithmicrowave measurements of the real andimaginaryparts of the dielectric constant.

If thepolarizabilityofliquidwaterisaslightly decreasing function of temperature, thecalculated values of e will further decrease with temperature at temperatures above 20C and will further increase at temperatures below 20C so that the agreement of the calculated values with theexperimentalvaluesmight be found to be even closer.

IV. APPLICATION OF CONTINUUM REACTION FIELD TO CALCULATION OF FREE ENERGY OF INTERACTION

The continuum reactionfield approach considers a given molecule in a liquid to be embedded in acavity in a uniformsurroundingdielectric. This starting point is iden-tical with that of the Onsager (2) approach, whose equations for the dielectric constant andreaction field have been presented in previous sections. The method attempts to account for theinteraction of agiven molecule with the entire surrounding. The elec-trostatic part (orientation and induction effects) was developed by Martin (27) and Bell(28) and improved by Kirkwood (29) and Bonnor (30).

Thecontribution of the electrostatic (dipolar) interaction to the molar Helmholtz freeenergy in the liquid,

F.,

is(5)

e= _

[fAi2NA/(1

-_fa)],

in which NA is Avogadro's number and the other quantities appearing have been de-fined in Eqs. 1-3. The dispersion part was developed by Linder (1, 31-33) in a series apapers, in the first two of which (1) he also presented the calculation of energy of vaporization for several liquids and obtained a reasonably fair agreement with experi-mental values. In this section we will employ the version developed by Linder (33) who has also taken into account thermodynamical fluctuations according to the ap-proach of Callen and Welton (34). Following Linder (33) the expression we employ forcalculatingthe molar free energy of attractive intermolecular interactions accord-ing to the continuum reaction field theory is

Fat

NA

N j kT

E

f(i6n)a(it)].

(15)

inwhich the term n = 0 is given half weight, and

4,, = _2rnkT/A

(*iis Planck's constant divided by 2wr) has the meaning of circular frequency and is analogousto win Eqs. 4-7. Eq. 15 is obtained by decomposing the contribution toF into adipolar partasin Eq. 14 and addingto it thedispersion part, which is obtained from Linder (33) (Eq. 146) by identifying the generalized susceptibility

x(it,,),

with the polarizability

a(it.).

Alternatively, the same expression is obtained by con-sidering theabove-mentioned equation asrepresentingthe total interaction per mole-cule andreinterpretingthegeneralizedsusceptibilityat

t

= 0 according to the defini-tion ofCallenand Welton (34) andaccordingtoOnsager's(2)

equations.

ThetermsHelmholtz free energyorGibbs free energy can be usedinterchangeably

in aliquidat apressure below severalatmospheres, since theproductof the pressure withthe molar volume, Pv, which isoftheorder of calorie per mole is much smaller

than themagnitudeof

F,tt

in Eq. 15, which is of the order of several kilocalories per mole. In thecalculationswehaveemployedtherelation

f(ix,,)

=

f[2e(itj,)

-

2]/[2e(ij)

+

1])

-

(Il/a3).

(16)

Since forn . 1

E(it,,)

=

W2(it,,)

(4,

is acircularfrequency in theinfrared) Eqs. 4 and 5 wereemployed,sothat

X[1 _N + 4 kTfa + 3kT 2(j4) (17)

= -NA['

fa)

+ -Is

The firsttermin Eq. 17,whichvanishes unless permanent dipolesarepresent, has been calledadipolar term, whereas the lasttermisadispersionterm. Foranonpolar liquid the secondtermcorrespondsto n=0 in thedispersionsummation.

For a polarliquidthere may betwo interpretations howto calculate this term. If weobtain Eq. 15 by adding dispersion interactions to Eq. 14 then f(0) should be ob-tainedbyextrapolatingthe refractive indextozerofrequencyin Eq. 16. Wecalculated

BIOPHYSICAL JOURNAL VOLUME 16 1976

( 14)

TABLE III

CONTRIBUTION OF VARIOUS TERMSToFINT

FOR _LIQUIDWATERAT20-C*

Term Contributionto_Fijt

Kcal/mol Dispersion

n- 1,10,t1 -2.44x

1014

rad/s -0.615

n - 11,100{loo-2.44x

1016

rad/s -2.333

n -_101,1000, ₀₀₀-2.44 x1017_rad/s -0.392

n-_{1001,2500,t25} -6.11 x 1017rad/s -0.0008

Dispersion (total) -3.339

"Induction" -0.213

Dipolar(electrostatic) -6.513

Total_F,tt -10.065

*The values ofpand a used in the calculations were 1.84x 10-18 esuand 3.77 x10-24cm3, respectively, corresponding to column 4in Table II. The valueof thecoefficient Cuvat20'Cwas0.199,asin Table I, whereas the values usedfor the infraredcoefficientsaretakenas0.132, 0.146, 0.0019, and0.018,

whichyield, accordingto_Eqs.8 and 9 the value of 3.77 x 10-24 cm3 fora.

Thevalues ofthecoefficientsC,atother temperatures were obtained from the

abovecoefficients bymultiplications with

N(t)/N(20'C).

f(0)inthe secondtermaccordingtoEq. 2, bystarting from thegeneralized expression

of Linder (33) (Eq. 146), in whichweidentified thegeneralizedsusceptibility x(0)with

[(M2/3kT)/(l

-

fa)]

+ a. Since inthis casefdependsonpermanent dipole moments

wehave

designaied

it as an "induction" term. It should be recognized that (a) the

dipolarterm also includes induction effects and (b) the "induction" term does not vanish inthe absence of permanentdipoles, but becomesapartof thedispersionterm. Ourcalculation off according to Eq. 2 instead of from theextrapolated refractive index amounts, in the case of liquid water, toanincrease of

300%

in the magnitude of the second term. However, in anycasethemagnitude of the second term is a small fraction of that of the dispersion or dipolar terms, (see Table III) in analogy with the situation in gases (16), so that the distinction between thetwoversions isnotcritical.

Table III also illustrates the contribution of different spectralregions to dispersion interactions. Wehave included 2,500 terms although in practice the series has con-verged with 1,000 terms. Most of the contribution arises from the visible and ultra-violet regions. The practical implication of this analysis is that for the purpose of

calculating dispersioninteractions the absence of refractive index data in the infrared region is not crucial. Note that the value of

ft(itj

in the visible and ultraviolet regions is almost independent of infrared terms. Thismeansthat for nonpolar liquids neglect of consideration of the infrared regionwill not result in an appreciableerror in calculations of intermolecular interactions.

Theresults in TableIIIillustrate that in the case ofliquid water the dipolar term is about two times larger than the dispersion term. Thelargemagnitude of the dipolar

TABLE IV

CALCULATED VALUES OF FREE ENERGIES OF INTERACTION: COMPARISON OF CALCULATED WITHOBSERVED VAPOR PRESSURES

I _{Fatt Fjlt}(total) Pobs Pcalc

eC Kcal/mol mmHg

Liquidwater*

15 -10.08 -7.93 12.79 13.3

20 -10.06 -7.92 17.53 17.5t

40 -9.94 -7.84 55.32 50.3

60 -9.75 -7.72 149.4 135

80 -9.52 -7.57 355.1 339

100 -9.27 -7.41 760 790

Carbondisulfide§

-73.8 -9.30 -5.26 1 0.9

-44.7 -8.83 -5.04 10 9

-22.5 -8.47 -4.87 40 36

-5.1 -8.18 -4.74 100 92

25 -7.70 -4.51 353

28 -7.65 -4.49 400 400t

46.3 -7.38 -4.36 760 777

*The valuesof 1 andausedinthe calculationswere1.84x10-18esuand 3.77x 10-24 cm3,

respectively.

tTheabove valueofPob,wasused insolvingfor_Fp.l.

§The values of theparametersC1and Wiaretakenfromref.3 at25°C. Thedensity ofcarbon

disulfide is taken from International Critical Tables, 1928. McGraw-Hill, New York.

Vol. III: 23.

termarisesnotonly because of the permanentdipolemomentbut also because of the

polarizability. Aneglectof thecontribution of infrared bandstothestatic

polarizabil-itywillreduce thedipolartermby aboutafactor oftwo.

InTableIV wepresent results of

F.t,

(or Gatt) for liquid water and CS2 at various temperatures andatatmosphericpressure. In the calculations the temperature

depen-dence ofpolarizability wasignored. Thequantity

,8Q

T) was obtained from

#,

Tf)

at

Ti

from the relation

#,B

T) =

fQ(4, Ti)N(T)/N(Ti),

in which T, = 20°C for liquid

waterand

Ti

=25TCfor CS2.

Heatof Vaporization

Inthefollowingwewillemploythe"subscript "int,"todesignatethesumof attractive andrepulsive intermolecular interactions.

The purpose of thefollowing discussion is to illustrate how to obtain the heat of vaporization fromaknowledge of

Fi,

However, in the calculationswe will consider only

F.t,

which impliesthataroughestimate of the heatofvaporization can be ob-tained from theattractive interactionsalone,which have been obtained without any introduction ofadjustable parameters.

Thecalculations of Linder (1) and those presented here neglect the change in the contribution of internal degrees offreedom to the internal energy when going from

BIoPHYSICAL JOURNAL VOLUME 16 1976

the liquidtothe vapor. Hence thediscontinuous jumpin internal energy upon vapor-ization isattributed to the change in the magnitude of intermolecular interactions. At pressures of several atmospheres and lower, the intermolecular interactions in the vapor canbeneglected compared withthose in theliquidso that

AU

vaporization

-

Ui,(liq)

(18)

and the heat ofvaporization

AHvaporization = H gas - Hliquid,

= RT -

Ui,,(liq)

- Pv(liq), (19)

inwhich the term Pv(liq) may be neglected at pressures of several atmospheres or below.

The value of

Hi.,

(liq) is obtained by

Hint

(liq) =

Gint

+

Sint

T,

=

Gint

-

(d

Gint/d T)I

pT,

(20)

oralternatively from

Uint

=

Fint

-

(C'Fint/clT)JY

T.

(21)

Since thedensityanddielectricconstantsinour case areknown atatmospheric pres-sure wepreferredto useEq. 20 with theapproximation

Hint=

Hint(T, Po

= 1 atm) =

Gnt(T,

_PO)

- T lim [G(T+ A T,

PO)

- G(T, PO)]/AT,

AT-O

Gint

(T,

PO)

- _T(AG/AT)(T,

PO).

(22) Incalculating

Hint

itshould be realized that arelativelysmallerror in

AG/IAT

may result inalarge error inH.0. Forinstance,anerror of0.01 Kcal/moldegreewillresult in an error of 3.7 Kcal/mol in

Hi.,

(100C,

PO).

In view of this sensitivity and thefact that the parameters for liquidwater(in Table I)wereavailableonlyat 20°C,

the calculated valuesof

H10t

forliquidwater arereliable onlyat200C.

FromTable IVit isseen that

Gi.t

(or Gatt) isan increasingfunction of temperature (i.e.its magnitude decreases with temperature) and therefore

Sint

(or

Stt)

is a nega-tive quantity, which makes the absolute magnitude ofH larger than that of

Gi.t.

In the case ofliquid water the magnitude of

S5tt

(200, PF) turns out to be -4 e.u. from which the heat of vaporization, AH vaporization (20°C,

PO)

= RT -

Gatt

(20°C,

Po)

- Satt T turns out to be 11.8 Kcal/mol. iThe experimental heat of

va-porization,at200Cis(35) 10.5Kcal/mol. Hence, the abovecalculation overestimates the heatof vaporization by 11%,which may be regardedas areasonable agreement. If the value of

S,,t

is calculatedat100°C, thecalculated value of the heat ofvaporization

at 100°C becomes 14.5 Kcal/mol, whereas the experimental value is 9.7 Kcal/mol.

Thisdiscrepancy is attributedto thesensitivity of the calculated heat of vaporization to the temperature dependence of G, together with absence of sufficiently detailed informationon theparametersofliquidwater at 100°C. Ifthe value of Satt at 20°C is used, the result would be 11.5 Kcal/mol, i.e. less than the value at 20°C, in qualita-tive accord with the decrease of the heat of vaporization with temperature.

Inthe case of CS2 the absolute magnitude of

F,tt

at250C, +7.7 Kcal/mol, already exceeds the heat ofvaporization, which is (35) 6.7 Kcal/mol. Fromconsideration of attractive interactions only, the calculated heat of vaporization for CS2at250Cturns out tobeapproximately 12 Kcal/mol. Inthe two casespresented,thequantity

St,

is negative, corresponding to the ordering introduced by intermolecular interactions. Theabsolute values of

F,.

or Gatt are close to the values for heats ofvaporization, but the addition of theterm-Satt Tleadsto anoverestimate,whichinthecases treated varies between 10% and 80%. Inpart this overestimate isaresult of the fact that rela-tively small errors in the calculation of

Gat.

(T) can result in large errors in the

term-Stt

T. However, these results also illustrate the need to introduce repulsive intermolecular interactions.

V. REPULSIVE INTERACTIONS AND CALCULATION OF VAPOR PRESSURE

Aninadequacy of the expression of the free energy in section IV (Fatt) is its lack of accountingfor repulsive interactions, which prevent the liquid from being compressed.

Consequently,the relation

(cF/dv)T=

-P

(23)

isnotsatisfied. We have introduced repulsive interactions in such a way that Eq. 23 be satisfiedateach temperature.

Regardingthe functional form of the repulsive interactionswewill only report the results obtained with the useof an exponential potential, which stems from the form Aexp (-uR), which incidentally was also chosen in another recent study on water (36). Weexpress

F.,1

in the form

Freul

(T,v)

=

A(T)exp

(-C(VI/3

_{- V}

1/3))

(24)

in which vm is the volume per molecule, (vy =- 1/N), and

vor.

is the incompressible molecular volume. The coefficient C has been considered to be temperature indepen-dent. The expression for the total free energy function,

Fiiq,

isgiven by

Fljq

=

'-RTlng(int)

-

RTln(VFINAX3)

+

Fi0t,

(25) in which g (int) is the part of the partition function due to internal degrees of freedom, A =

h/(2irmkT)'/2,

m is the molecular mass, and VF is the molar free volume. (In ref.37the assumptions leadingtoEq.25 arediscussed.)

Fi.t

isgiven by

Fint(v,

T) =

Fatt(V,

_T) +

Ffepul(v,

T). (26)

Fromtheequilibrium relation for the chemical potentials

#(gas) = i(liq) (27)

andbyneglecting changes in the contribution of internal degrees of freedom to the free energy uponvaporization,itfollows that

P =

(RT/VF)

_exp

(Gi,n/R

T),

(28)

where P is the vapor pressure of the liquid. It is worthwhile to point out that Eq. 28 mayalso be derived from general thermodynamical considerations, without making useof Eq. 25, in which case the termR/

VF

willbe replaced by some parameter. Con-sider the relation

dlnF/dT = AH vaporization/RT2, (29)

which is an approximate form of Clapeyron's equation. FromEqs. (19) and (29) we obtain

dlnP/dT Z (I/T) -

(HintIRT2).

(30)

Fromtherelation

Hit

= -T2(O/d T)

(Gt

I/T)

it follows that

dlnP/dT = (1/T) + (a/8

T)(Gin/RRT).

(31)

Integration of Eq. 31 readily yields Eq. 28, inwhichR/VF is an integration constant. Hence, the form ofEq. 28 isquitegeneral and does not require the specific form of Eq. 25 for

Fijq.

Thefact that integration of Eq. 31 yieldsan apparent independenceof VF of temperature ismerely aresult of theneglectoftermsofrelatively small magni-tude, e.g., PVjjq. Since Vj,q itself varies only slightly with temperature, we do not anticipateappreciable changes of VF with temperature.

Since in Eq. 23Pis much smaller inmagnitude compared with theterms onthe left-hand side, Eq. 23wassolved bysettingto zerotheright-hand side. From Eqs. 23 and 15 we have determined the quantity

iv1/3CFP,uj

at each temperature. More details aregiven in theAppendix. Hence, anotherequationwas needed for the determination of

F..,.

Thus Eq. 28 for the vapor pressure wasusedat onetemperature. In orderto apply Eq. 28achoice hadtobe made for the valueof VF. In this work the free volume wasassumed to beproportional to the totalvolume. Once the valueof VF at one tem-perature was chosen, the vapor pressure was calculated for the whole temperature rangeand the results were compared with theexperimental values. Finally, the value of VF which gave the overall best agreement between calculated and experimental

values of vapor pressures was chosen. In Table IV we compare the experimental values of the vapor pressure with the calculated values. The good agreement obtained suggeststhat thetemperature dependence of

Gi.t

is in accord with experirnent. The prediction of vapor pressures provides another methodtopredict heats ofvaporization from Eq. 29.

From Table IV it follows that the magnitudes of repulsive interactions in liquid

water and carbon disulfide at atmospheric pressure are approximately 20% and

40%,

respectively, of the magnitudes of attractive interactions. This illustrates that a rough ideaabout the magnitude of intermolecular interactions may be obtained from a con-sideration of attractive interactions alone.

Thevalues of VF which gave the best agreement between calculated and observed vapor pressures are 1.3 cm3 for liquid water at 20°C, and 25 cm3 for carbon disulfide at25°C. As was shown in Eqs. 29-31 VF can be regarded merely as a parameter. Interpretation and discussion of the various aspects of the concept of the free volume aregiveninref.38. Ourinterpretationofthe free volume is accordingtothefollowing

relation

VF = Na

(Vm,

- _vcr)_'y, (32)

inwhich

'y

is some geometrical factor smaller than unity, which reflects the fact that part of the volume in a liquid is inaccessible because ofpacking of incompressible

units. The values of molar volumes inliquidwaterand carbon disulfide are - 18cm3

and 60 cm3 at 20°C. Hence, it might be that the difference vm -

vz,,

is several timessmaller in water than in carbon disulfide. However, more cases should be studied before drawing any conclusions. Inaddition,the determination of VFhas not yet been optimized. In the case of carbon disulfide any value of VFbetween 18cm3 and 30cm3

couldyieldanapproximate predictionof vapor pressures, whereas in waterthe range varies between 1 and 2cm3.

VI. DISCUSSION

Theresultsof this work present another illustration for the usefulness of the continuum reaction field theory (2, 31-33) for quantitative predictionsof dielectric constants (2)

and intermolecular interactions inliquids(1). It has to be realized that part of the com-plexity of intermolecular interactions hasalreadybeenimplicitlyabsorbed in the values of densities whichareemployedin thecalculations. In thecaseofliquidwaterit may be that in addition the value of the

polarizability arising

frominfrared bandsisaffected by theliquidenvironment. The basis for thispossibilitymay be the fact that thelarge

valueof the staticpolarizability ofliquidwater is due to the first two infrared bands (see Table I), whichareinterpretedasarisingfromintermolecular motions(20). These bandsareabsent in the vaporstateof water. However, refractive index data in the far infrared region are not available for the vapor, so that the difference between the

polarizabilityin theliquidand the vaporcanbespeculated.

Themagnitudes of the dielectric constant and intermolecular electrostatic interac-tionsinliquids of large dielectric constant depend not only on the magnitude of the permanentdipolemomentandon the numberof molecules per unit volume but also

criticallyonthepolarizability. Consequently, the atomic andmolecular contributions to thepolarizability should be taken into account in dielectric constant and in free energy calculations.

Itshould be stressed that acomparison of themagnitudeofdispersioninteractions

inliquid water with other liquids (1) does not indicate any anomaly for the case of liquid water. The magnitude of dispersioninteractions dependsonthe density andon the part of thepolarizabilityarising from visible and ultraviolet bands. The contribu-tion ofinfrared bands to dispersion interactions is relatively small (see Table III). Inthis regard it is noteworthy that the magnitude of thepolarizability arising from visible and ultraviolet bands is almost the same in the liquid and vapor states of water (3, 4) (see Table I).

The anomaloushigh value of the heat of vaporization ofliquidwater isdueto the

dipolarinteraction with thepolarizing medium. The treatment of the reaction field yields the high value of heat of vaporization ofliquidwater when thecontribution of

infraredbands tothe polarizability is taken into account.

Repulsive interactions have to be introduced in order to satisfythe thermodynamic relation (0Ff0V)T = -P and in order to account for the overestimate in the heat of vaporization when only attractive

interactjons

are taken into account. The use of the above relation provides a simpleprocedure for the introduction ofrepulsive

in-teractions,which employs a minimal number ofassumptionsandonlyone parameter,

VF,

whose

interpretation

isstilltobe

investigated

from calculations on more

liquids.

Theintroduction of repulsive interactions significantly improves the estimate of heat of vaporization and enables the prediction of vapor pressures, and therefore also boil-ing temperatures.

Inthe present treatment a molecule in a liquid is embedded in a spherical cavity whose volume is 1/N, and the permanent dipole moment resides at the center of the

cavity. The success of the model may stimulate reconsideration of a shift of the per-manentdipole moment from the center of the cavity, and a refinement of the concept ofthe cavity as well as varying its shape.

Whenthe contribution, (which may be negative or positive) of higher moments to intermolecular interactions in liquids becomes known, it can be simply added to the present expression for

F.tt

(or

G.).

Regarding the detailed structure ofliquid water, the present treatment does not exclude the possibility of a mixture (10, 39) model for liquid water. However, we avoided in thisframework the use of additional assumptions, e.g., the properties of thespecieswhich form the mixture in liquid water.

Thedegree of success of the application of the method for pure liquids may also provide an idea about the importance of various interactions in solutions. The

coup-ling of free-space quantum mechanical calculations with continuum reaction field treatmentof solvent effects may be a promising way of predicting conformations of molecules in various solvents (40, 41).

This work isanextendedversion ofaprevious articleandalecturegivenattheBiophysicalSociety sympo-sium_(42).

Iwould liketoexpress mygratitudetoDr. B.Linder for his usefulcomments.

NASA GrantNGR33-015-002is alsoacknowledged. Receivedforpublication 12May1975.

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APPENDIX

Wediscuss here the details of the determination of

Fp.j.

FromEqs. 23-25wehave

-RT(a

In

V1/a

v)

+

(cFatt/cv)

+

(clF.p,/cv)

O0.

(33)

Bymakinguseof the relations

vm = V/NA,

i.e.,

_a

/l

v =

(

I/NA)(c

/clv,,.),

and byneglectingthedependenceof

v..

onvm,weobtain

-RTv(cl

In

V./O8V)

+ _V,

(d

Ftt1V,,)

=

(C/3)

v

1/3

Ft.1p

(34)

Consider the left-hand side of Eq.34tobeagiven number, B, (T), i.e.,

BI(T)

=

(C/3)v1/3Ff,ul(T).

(35)

Since

BI

(T) isaknown numberateach temperature,aknowledgeofC,whichis assumed to

beindependent of temperature, suffices for the determination of

Fp.,

at any temperature. FromEq. 28wehave

RT

log(

VFP/R

T)

Gi

(36)

whichcan beapproximated by

RTlog(V,P/RT)

-

Fa

=

Ffpul(T).

(37)

Byusing Eq. 37 at one temperature,

TI,

andsubstitutingthevalue of

Fp,,,(T,)

in _Eq. 35 the unknown C is solved. In the following we will calculate the contribution to the term

v. (cl

Fatt

/dv").

Thederivativeofthedipolarterm is

-(C

/lv5,)([NAf

Jf2/(l

- = -[NA p2/2(1

-fa)2](C9f/Ov).

(38)

fcan be writtenin theform

f= (4w/3)[(2e - 2)/(2e + 1)J(l/v,,) =f1(e)/v,. (39) Hence,

Cif

fi

(e) I c_df (e)

_i(e)

f,

=_f

₍₀

__ V2 ,,v _ _ ~+(40)

inwhich the neglected term (I/v,,)[8f, (e)/cv,,,J is about two ordersofmagnitudesmaller than the firstterm,provided thate 2 10, whichcanbe assumed forapolarliquid. In view of the relation

d,O/dV, = -,B/v,,,, (41)

wefinallyobtain

cVl

NAA[2 (

fa)2

+

3fakT

+ 3kT 2

262((i{;)

+

62(itn)]

₍₄₂₎

VM

~

[2 (1_

fa)2

4 , (1 +

p(

J. (42

Theterm-vRTT( logVF/Ov)wasequated to -RT. If VFis interpretedaccordingto Eq. 32the term-RTshould bereplaced by -RT[V,,/(V -

vjwf)J,

butatthisstagewehavenotattempted

torefine thiscalculation, thus implicitly assumingv»>>v,. Inviewof the orders of magni-tudeof the terms RT and

v.m(cFatt/laV.),

even acompleteneglect ofthe termRTwould not

result inaserious error. Forinstance, atT - 300'K,RT= 0.6Kcal/mole,whereas

v.(a&tl

cv,,,) isabout 14Kcal/mol for carbon disulfideand 19Kcal/mol forwater.

BIOPHYSICAL JOURNAL VOLUME 16 1976 76