527

T\DL& \" -3

The stat ic die lectric coosta nt of water in its various pbases PlAUJ8J:RC & )1-\RYOTT, 1956)

,,.,. -l . ll•!l'h prt'"llrt' poi) 11111rph '

e 't·cpt: it·t> Il. h·e VII(

Liquid . .. .. . . • .

\ ' opor

o

o

91 l i)U I O :!nO

l llll

20 1\1 100 :15 IOO 1.11 1

In hytlrogcu-IJoruleù s tructures , the orientation of thc clipolrs cnn be d1 nngcd either h y s imultaueous jumps •>f a serics of hydrog•' n" ulong tiJ<• hyd.rogen boruls ("imilarly to the Grotthus mcchanism of clcctrk ronduc­tnnre) or by a corn•la ted o:erÌt'S 11f molecular r ot a tio os (Fig. V-8).

l'il'. \ 8. - The corrrlutl'd rotation of ,e, rrul \>Uler molt'cute, in 11 h ) ùrOjtl' ll·houded t·bcùn

or ut twork ~lructure, rr­,ulting iu a cbange of thr direcrion• of t be dipole uto· menh (inthrated b} Juon") arro~<,).

,.,H

~ ' H

H f " ' o,., ..

H , , ,.,o

H

Tn it• t·- II nnd vnr. a s (ll (•ntioned befort', th l' hydrl)gl' n pOI' ItiOIIS ari' li..:t•d and. thl' rt•forP. o are t he dircctions of the clipol c momenlQ, 1'his

r•· ... ult~ in a lo'' clir lcctric cons tanl. ' l' lw diclN·I ric ('Oll8 t ant by it st' lf d oPs no t gin~ inf'orrna tion o n t h(• lcwul

I'Orr··lrt tions uf rnolccular motions . rri) WC\' Cf, KlllK" 0() l) 's tht•ory (1939) of clic•lt•CtrÌC bl'faav iour ran be HS I'cl to intcrprct t Ili' HIJ ~I'f\ (•d fu \ al.tP..,.

,\ lt hough tlw 1 hPtH), in a s trict s e n se, applies nn ly lo i~utropir .. uhsl ant·l's, il •·un b r ust·d f()r u .., ,,mi-qunntitati ve nnalysis . For po lur <n.J) tt\11 (' 1'~. llw

'llllit· di<'l l'clric· c·nn .. la nt is giq ·n b )' rq. (5.5):

., , .. m · m * _ ? "* ~ - - ·• ·' ~r--- - :t_, (3.5)

,du·n• ~· - uurnbcr of mnlccull'i- unit ,·olttmr; m = aH·ragc dipo l1· uw­mcnt of t hl' rnolt•t·ttle in the loca l field (.u - ( L F ) : m* = 't't' lur :.um

A rtn . h l ::ìuD<r .)a n r() 119701 S. 1\lt-,\ll

5!!8

of Lbt· tlipc\1 ~· mooteut• fo r a mol•·ct.J,. aml it-. neil!h iHtr~. l t tlc>pcutJ, U!il ouly u11 fLI hut also on the rdulìn• c,ritnlaliun" of llt'Ì~hiJorÌII I! Jllult·t·u lc·-. i. t·. 011 IIH: CC)rrt ·la tio ll ur .molt•culnr uri.·ntutÌIIIl '-~ m - m : l!; ... thc kirl..· '' uod curr..lation parnmdl'r '' hidt dqwn cl:- (111 1~ un 1 h e re l al h t• orit•IJLH· l Ìnn,; an d ÌS U lli i'USU'fC• of tht• (' I)Trt•lutÌIIll uf tJTÌt•Jtl Ul ÌOII": \ lllllllbt•r

or m oh·c·ult: iu the jrl• ('Oord in a l iun .. }wll ( t•. ~· in ;,.,. ~l l. .\ ~ L~ ):

;•, = nnglc hd\\CCU thc JipoJ,. ruumt•nl :- uf th•· t•t•ulral ruoll't'ttlt· ntu11ho:-t· in thl' jtlt c•oordinaLion sh•·IL.

g = m · m * m~ -= l co.s ;',

If thcr c h; no cor rc· lution al ali. [! l (rauclont orit·ntntum of mol­

('t nlr,:) . Ff1r "' rung currdalion .... g i~ l ur~t·. lu ic••. g::: 3. lu t la t • it•t• poi)­morph,.. , r!: pruLably is nul ver y tlifrcr•·u L fron t irt·· I. ll tJ\H'\l"r . E J ; .. ~rt·att• r

than in Ìt'e, bc·c~tust• of thc iuc-rt·u•c> in N and m . l n Lht• lictuid, thc calc ulntion of' g d ept-ncl" cm tltr t ht•orctiral m ode l

ai'Osu mcc.l , but in generai. valut•s bt•twct·u :2.3 :wcl 2.8 a r e obtain('d. Thf'!>l' 'aluè~ are lower tltan Lho .. e in ic~·. whieh i::. reu~>onnblt· lwcau,e 1·· ~:- corn ·la­

t io n is c•xprctcd hctwcen tbt.• molecu),·,.: in tlw liquitl. Th1· d~·t·rea ... f' of !! by aboul 0.2 unit s h t' tWt'l'll 0° all(l 100" i;. in agrc·t·mt•ut \d t h Lltt• idt>a of tht•

breakiug clo'm of ... trurture.

H ACC IS, IlAsTED &. DucnA:-:A'' (1952) han fl"~">\lnwd a<c mh. t ur(· mutlt·l» (~<PC St>e. \ "l.A) h uving a s m all frnt.·tion of non-bydro{!•·n-LonrJ,.cJ molt·t.· ul t>-:

"itb thc fraction of brol..cn h ydrogcn bonJ:. tnkt-n a:, (1.09 a t O • C, tlu.· ~· go t guud agr••ement with tht• CXJI!'TÌJtlt'nta l di t>leetrit· CQU'- Innls. ll"iug g = ~.Hl

a t 0° for the hydrogcn·LondNl mull•cul c·~ and t: = l fnr tlH' unbontled OIH'• ·

Tht' latter value is most likt-ly t oo Jo,, . .'\ calrulation by P oi'LF (1951). h tt-. t' d on an c·ntirf:'l) dit.lrrPII I theor~· tica l

concep t. a« continuum model» for wa tc•r. ,,ilJ ht• tf i,cu ... ,,.,) lat•·r ( et·.\ l. B).

F. S .UR C/l(m iral S!tifi.

Hydrogc>n bonding of an 0-11 group cau~<'l- a rlm••nli .. Jd !l hift of Llw ~:\IH signaJ, b l'cnuse th•· sh ie lcling of tltt• H nurleu ~- j ,. d t"t: rt•a ;.NI. Si nct· thr obs<"rvcù chl'mic·al sh ift for n rapidi~ intt·rchnngiug mixturc of variou:' forms is an nvN age over th e cbemicul ~hift of eal·L ft•rm. thc l t>lllfW· rature d cpendence of tlw ch e micnl ::.hift ough t to ~;ivc informntion on tlw

equi librium distributiou of protonf: io l crtn :> uf mole fra l· tiou;. r ·

6 (t) = Xua(t) 6RD T ( 1-Xttn(t)) ~Xli

whcrr t1l<' ubscripl!> HB an d ~ U r efer t o tht• h~·drug<'n-bomlt•tl and nul

hydrogrn-honded molecull'l!, respt'cli\ ely.

529

The chemica l shift of liquid \.-at~>r, a compared with the vapor . occurs at luwPr lìdds, in agreement with this <·oncept, and the downfie ld shift is Jargl'r at low· t t>mperaturcs . 1\h' LLF.R (1965) bas founcl for :!5 < t < 100°

6 = - 4.58 + 9.5 . lO 3 t (p p m) (5 .8)

HtN UMA :'i (1966) has propost>d a :,tructural interpretation. H e asFum t·d tha l ,,. ,H f• r cons is ts of a mhture of icc-like molec ulcs anù of monom rrs whid1 arP like a normal liquid , i. t ' . fr ec ly rntutin g, anù that Lhe ir dipo i ~·

momrots tlo nut contribui r at all t o tbc a ttraetivc en crgy b <'lW Cl' ll wal t• r monomers. This a""urnption i :-~ alml);; t t't·rtuinly wrong. Th e extrapolu­tion of tht• propt•rtie:- of the>s t' watrr moltcuks from thosc of If:S anJ H 2SP, M'l'm~ irlt'orret' t. (Ch. I ). Whilc Hitulman 's usults inclieated thc prcsrnrf' of f:~idy lur~c h ydr•lucn-bonùt•d d u!'t<•r s. they tlo not a lluw di, tirtdion b ct· ,,.('('11 , -ariuus t hrorl' l ical modcls.

H UTEUJA s & ScH t-: llAGA (1<>66) maolc imilar measurl'm ent.c:, u~ i ng difl't'r(·ut appro'\imation,; for l he analy."i,; .

G. Light "cau:-ri11g .

Tt hnd bee n thought that light scauc·ring could b e used to th•t crmine tiH' auwunt of :< tructured r l'gions in waler , but this did not tu ru out to be thr l'nsr. llJH.l now it is d car that no such information <'an b e obtaineù in this mann er.

Lig bt scattering arises: (a) l'rom the anisotropy of polar izability of mol­l'c ulcs; (b) from s pontancous fluctua tions in t h c d eus ity of the liqn id due to llrownian motinn . Thc Raylo·igh ra tio, Ru(90), thc rof'astuc of the light :-raLtt•ring at 90° to the inddent beam, is proportional Lo ('d n 'd p)2. MYsEr." (1961) att emptt tl to caleulatt' the :'ì tructural het ercogeni ty from the cJitT­(' r èncr bPtwern the observ,,d anrl calcuhtecl R u . H e as!'umt·J. tha t thc exp r ri­m ental R ,, is a " um of thc , -alu c for « isotropic» ~ t~attcring ns in othcr liquitls, i.e. due t() den ,.ity Auctuations, ancl of an addi t ional scaltcring d uP to thl' inhom ogeneit y. causcd L. the prest>nce of dusters .

E x.-pcrimentally, R,.(90) = l. 7h . lO - • cm- 1 • Gsing thc rxperimcn­talJy m en>'llr ed ,-a hw"' of ('d o ' 'd p ). ?II\'Sel$ calt'ubtetl Ru(90)1ootropic = l..)()·

· lO :; r·m - 1 • Thi' Jitfere·nce bc twt·cn the tw o numher s is mall , and ~oo

\fy-..·1-< N>nl·)u d•·•l th :H t11<'r f' r nn h l' no rln, ters of many wate r molerules '' ith a ol t•ns ity di{fe•rt•ut f.'rom thnt of Lhe a\'eragc d cnsi ty of the liqujd, Ilnwenr, t his ••e)u clusiou ÌJS •·rroneous. as was l'h own by CoA Eli' & EISENBERG ( 1965). Tlw ' a l ne• uf (d n 1 'd p). which :M)·s,·ls uscd, is thc obscrved Yalue ro·f'err ing lo ,, a ll'r as a \\ holt•. T hu-. il \'Ontain:- UOL only lhe cont rihu t ion to n Il ' d p) elnr Lo tht• lluctnutions . a~ in au isot ropi<' liquid , but the t• fl'ect s ()[ ;, tntcturnl

ch ungc.b a:.' wcll. So tb l.' calculatiou o f H .. (9t1)i ... tn•Jol r .~hould bnve r c·pro­

dtwf'd t ltt• uh ..,rn <'t i n:tlu<· R u(911). W'il h nHJrt· ac•t·uratt• dt•tcrmiuation·

of' H u (911). thi ,. j., intlec'd thf' ca:H' {COllE'- & L I"E"U FIH_.. l t}(•:J).

li. Dielertrir [{efu:catilln.

Mo"t of tbt' m cn"urr·m cnt " cli·cu~·ed •Il far Ùt·al "ith .. t a tic prop!'rt il··

nf wntrr . F or tlw uuderl>ta lldin~ of tlt e l> lrul'lttn· 11 j;, a l-.<• impurt ant tu

con-iJcr t b r rat<· aml m cchaniHu of m otiun of 1 h e m olt·t·nl<'-.. TI. t· tt •f'hni­

<fll !'lì to be di.,cu .. :-NI Il<''' polt>ntiaJI~ j!Ìn· iofurmulic•u of thi ... t~ pc .

. \ :- lll l'ntionr cl abo' <' ( · ,., •• \~.E), Lhf· .b latir (ur lnw frt•cpa·uc·y) dic lt·ct ri •·

t·on'-lanl ha .. ù h igh ,·ahw dut· to reoril'ntaUon .. of tlw \HIIer moll'ctÙe:-. If 1lw frequ<·n ry of l hr upplit•d lìt·ltl is-: Hr~ hi~b. th t- muh·rular di polt·:- t'anno i

folluw tlu· rapidi~ 'ar~ i n~ lit'ld. nud tut' dic·lt·c•t ric c·rm'-lunt d r·c rf'n~e" t o n

:.m nll valuc t14

(d'. ict•· ll ). 'l'hl' tl i•JII'r· ion of thc diclt>rlric· t 'OD!i lanl. i. e.

Il., tl t•p<' IIÙCll t'(' 011 th t• frf•IJll l' r'll'~· l' = t•) :! ;; j., {!i\' 1'11 by t'<J. (5.9):

(5 .9)

'l'h•• , •alttl:' of r dcen •a-,l·S r a pitlly in tht· r t>giun whl·n· (•J Td:::: l. r,1 j ..

the diel ('(·Lrit r ela xation tinlf'. charal't{'rizin~ tlw d rc·n~ 11f J• tl lur izntiou in a

"uh.;tatwc afta th•· applicd flcld j ., r f'mO\Ctl. Jt i., ~unw"bu t lorger than th l·

mul,•ruJnr rotutionnl correlation t inw r , . t h•· a n·ra~c t i m .- be t" t'"en ori t•nta­l iounl motion~" of t h<• mol,•rulc.

l n Ì<'<'- l nt O. T.t :::: 2.10 - 5 "•·r. In the liquicl iti ... mud1 .-:horter (COLI.I E., l l.AsTED &. RtTSO'\ , 19·1·8). LI~<> sbm, n in TaLlt Y-4. Only u t- in j!l C r;•Jaxution

t imt• i!' fo uml a t cq·r y tempr·rat ur t·. Thi .. i,. nn tmpvrt aut r c:-u lt. intlicn t iug

l ha t t b en · j .., only on<· rritit al rdaxntion prut'l'" in tlw Jic1uid. 1'br heat of nctiHttion .1 H = (·an bi' ohtaint·d from tb1· t empt·ratun•

tlPpPnÙeu cP of r d • lis i n t! a va n ' t H o A'· t ~ 1w calculnt io n (Tahlo• \ ·- 1 ) . Tbc

<> maller J fld (and corrcsponding ly high r 11 ) iu wntt•r. n~ c11ml'ured ''itL i<·e.

inclit·a te~ t ba t thc rotation of th r molecul t· ... j, m n rh t•a ... i•·r . O n t h c o t bt·r

haucl , rd Ì!1 s tili mur h h ighN Lhan for th<' « monouwr » in n dilutl' ~ulution in

Lcnwnc. nnd no s pread of thf• rdaxat ion t imcs j,. ~t't'n. nhll nugh thP t•nvirun­

ru <·nt of t hc mCJlecult', in th<' l iquid mu.-t lw quitt· varicd ( t·-~· in Ì('e·Hl and

it·e-\"I. whcr r tht>rt• an· cvcrnl n ont•quh·a lcnt oricntaliml'- of mol<'cul•·~, thr clis tributilln i :- mu1·h bruad c r tùau in Ì<'t:. and ,,, <' n ::! 3 ti m <·.., bruudcr

l han in thc liquid ). Thc h est rxplnnation sct•m « t o h <' t h a t t ben· j .... ~11111' c'<IOJierati,·c prue•'"!'.

irn ulving '>C\ cral ,,·ntcr molrf'ulrf', a s i1t Fn.\~t(~ pru po:-al uf du,.,teri' (1958).

1'he prnblc m with thi5 l'Xplunation i:- that tlu· l H aut! J "''t·m 111 b (• lc.o

531

Diclcctric relaxation pa ra meters

(Adnt>ted from E 1sEN8ERC &. KAt:z~IANN, 1969. p. ~ll7)

't,_ ___ _ t Il

u,n o, o 11,0

.. (", \.tr.l m ul ------------~

ltl' l o -::-;;:. ~. 1o-• 12 . 7

J.irp t i<l . . . . . o 1,7,8 . J() t~ l l 5 l :w. ~ . J o-t~

!!O 9 .:;:>. 10-t" 12 3 . Jn-• !

4 . . :;

3. CI 60 LO ~ · IO-" L9U . IO t!

- - ,

Rt r.

f:hE'iBERG & " AI' l·

'"'' ' (l 9fi9, p. 111); At•rv& Cot. E (I'Ii2).

Cut t.tE, IIA.,n :n & HrT.•O" (19~8).

-------~--~ ---------~~

1 . 0 . l o-t~ l Wart'r in tlilute hrn~:ene ~tllution

20 G \ltG & ~'n T rr t iCJh"l)

s rnall Lo corrrsporrd to thc complet e breaking and formatio n of la rgc chr"l tl· rs

( EtsE ,BERC & KAL'l:\JA~N', 1969) . On thc other band. if molccul~·;;; rcori ~·nt indcprndcntly, or if thc: rc exi~t s mall a~gregatc \\ruch persi--t for times

longr r t ha n lO- 11 Ree, a!' propo~ra in ;;omt• thcorrt ical modrl-., raus ing d idee· tric rc·laxation by rot a tion uf thc rn lirl' aggregate, Lhen wide dis tribution

of rdnxation time~ s boulcl twcur. Thn" it M't'ms that in this casr tlwrc

is arr unre•·onril<·tl ron tratl iction (l'f. al so thr X -ray !li lfrartion da t a, Sec. V.A, un llw dimensions t> f thr molcculcr nggrcgatcs) .

I. .V .\JR Spin - T~attice Relaxatio11.

From mcasurr menls of thc relax ution time TI' fo r thc proton, s~rrTn & Po\HFS (1966) ha w detPrmint•d a r ota t io n n l corrdatiotl t ime of 2.6 · lO ': st:c a t ::!5", closc to that secn in o thcr m t'n'turem c ntl:> . a nù '~i thin nn onlt·r of

mag nitude of the dielcct ri <: n •laxation ti mc.

K. Sdf-JJiffusion.

Tlu! M·lf-difl'u)>ion cocffi ciPnt, D , il! a measurl' of the ratr uf tli:,plact·ure nt

of molt'l'Wl'!\ in tlw Liquid frorn their momrntary po~itions . l t is ùctermincd

fn• m dilfw·itm ralt' mea-.urt•nwnt of i&otropic t raccr•. e. ~- l r:•~o in l 1!0. It wa,., fouml (\\ A~G, TioRt , SON & ErH' I.:\IAN, 1953) that llOO aud JlTO do nut d_j(J'u, ,· fa, t cr than 1 1 ~180, nlthou~;h it mi~ht ha\'(' b t·<·n t'~ Jtt't' tetl

532

tbat jumps along a chain of hydrogcn honds (Sec. V.E) would spced up th1• diffusion of D aud T. H owevcr. a more dctailed analysis &hows tbi:-. not to be the case; after a hydrogen·hond jump, th~ molecttles in tbc chain must reorient heforc thc ncxt jump cau takr piace. Tbus th<' rotational relaxation scem s to detcrmiuc the rate of difi'u.sion of protoru- as wdl ns of entir·c molecules . w ,\NG. Ronmso~ &- EDELMA.N (1953) :md uthcr autbors bave fow1d D::::: 2.5. lo-s cm- 2 sec- 1 at 25° in H~O with a s trong tempera· ttue dcpcndcnce: 1.4 · I O-; at s.o. 5.4 · l o-s a t 55°. D is uhoul 12 ° 0 lower io D 20, indicating lower m obility. WANG, RoniNSON & EDELMAN (1953) calculatcd thc m ean size for the indi viduai diffusion stcps anù obtainrd a value of tbc arder of tbc scparation of nearest neighbors. Thcy also notcd that the cuergics of activatiou of sclf·di.Ifusion, diclectrir r claxation. aod viscmJ s flow are all about tbc samc. ncar 4.6 kcal/mol at 15°. This makt•,. plausible the assumption that the (' ritical elemcntary stcp in ali tbcse pro· cesses is the same (FnA NK, 1958).

As mentioned b efore, thc time int.erval between successiv(' j umps of water molecules was fouud hy ncutron scattering to be l to 2 · 10- 12 sec. Within an order of magnitud e, this is tbe relaxation time found for many translational and rotational processcs.

VI. THEORETICAL MODELS OF WATER STRUCTUR E

A. Ce n era/ D i.scussion.

l. Survey of Various Theories. « Continuum» t iS. « J\.1 ix tu re» 1\-fodek

1\[any difl'crent modcls of water s tructurcs havc been proposcd, par · Licularly during the last lO years. .M:os t of thcm are based on t1<: interprc· tation of only ccrtain <>xperim cntal data, with attempts to cxplain othcr types of data as well . For example. a particular ruoùcl may deal mostly with spectroscopic data. Another may apply spcctroscopic information to el.-plain thcrmodynamie data, and then is used to interpr et other proper· ties as well. Jn othcr cases, the m ode] may bave becn developed to deal ""ith dynamic processes.

A generai classification of various l)'TJCS of models is givcn in Tablt­VJ -l.. Therc are two gener ai schooh of thought about water structure today, an d theoretical propo~als fall into two groups, usually t crmed « continuum» ancl « mi.xture» models.

The coucept of the continuum model is favored mainJy hy spectro­scopis ts. According to this concept, no water mol<•culcs of qualitati· vcly di iTerent h.-inds exist in the liquid. lnstead, aU moleculc:, ha' c qualitati vcly similar environmcnts, i. e. ali are hyJrogen·bonded. but the

A nn. ht. Supc.,.. Sonilcl !1970) 6, tl)l-!iU~

533

T.uu VI-l

Cluaifi.cation or theoretical mode.. or water strocture

TYPE EXAMPLE

(A) Continuum modela POPLE (1951)

(B) Mixture modela

l) Sm.all agp-e&ate Eu<XE!'f ( 1948)

2) Simple two-state HALL (1948) Dnu & Lrrovrrz (1965) Juo", Gaosa, RE& & Enmc (1966)

3) Detailed cluster . NhmTBY & SJBUo.a. (19624) Buus & Cllorr.m (1963) V .um & SENIO& ( 1965) RAGLD, SCBU.A.CA. & NbnmiY (1971)

4) lntentitial S.uroiLOv (1946; 1965) PA1JLJI'(G ( 1959) ~ & QmsT (1961) N,UTE!'(, D.uoo&D & LKVY (1967)

(C) Molecular dynamie• (Monte Carlo) ealculationa B.u.;u & WATTS (1969)

(D) Quantum mechanìcal cal-cula tio111 . . Diti. BENit & POPL:& (1969; 1970)

HA.l'um, Moscowrrz & STJumcu (1970 o; •>

bonds are distorted to various extenta. Thua, at any instant, moleculee may have slighùy dift'ezent local environ.menta, due to variatiozu in the extcnt of distortion, but these dift'e:rencea :repreaent a continu0111 range, without any qualitatively dift'erent clasaes present. Although the « conti· nuum mode!» has been diacuued much in recen't yeara (e. g. KBLL, 1971), the only detailed quantitative example developed ia that of PoPLB (1951). and its applications so far have been rather limited.

Most of the other propoaala fall into the clou of « mixture modelu. In generai, ali of them assume that there ate at leut two qualitatively diff. erent states of the water molecules in the liqui~ since the:re ia a definite difference between hydrogen bonds which are formed and whicla are brok&A.­There is a temperature·, pressu.re-, and solute·dependent equilibriam bet• ween the two forma. The hydrogen-bonded moleculee form an open, tetrahedrally bonded structure, « ice-like» in the senso of a high molar volume and the preaence of hydrogen bonds, although not necessarily

.~ ""· I1t. Supe'l'. Sanlt4 (1970) e, 4GI-5~

534

with tbe structurc of ice-I. In earlier proposal , i t was suggestt•d tbat tbcr e are small aggregatcs, dimers, t etramcn, eh·. in tbc liquiù. Best known is thc model of EucKEN (1948) wich was sa ccessfuJ for empirical corrcla tions of Yarious thermodynamic propcrtics. JioweYer. today, tbi., t yp e of model u,.,ually is discountcd, ruostly oo the basi;; of X-ra~·

and rclaxational studies wh.icL show no indication of small aggregatf's (cf. Sec. V. H ).

In sirupler mixture modcls . ìt is assumed that therc are only two kinJ., of m olecules. and b\ùk propcrtie:> are assigued to thcm . An older example is the model by HALL (1948) dealing with ultrasonic phenomena. Although containing some r e6.nements, the theory of Davis and Litovitz is essentia lly a two-state model, and so are the two models proposed by Eyring and hi;; coworker s ~!ARcm & EYlliNG, 1964; )HON ('t al., 1966), b ased on Eyring'!) « theory of significant stru ctures» {E"l'll!:'iG & Jno~, 1969).

Simplc two-state models usually contain a contradiction: sinc<· tbc spatiaJ extent of the two structures m ust be very small (Sec. Y.A) it i:. fal se to ascribe to them the propertics of bulk pha.ses. The propcrticl'l on thc surface of thc various s tructures are impor tant, since this s urfuce composes a largc fraction of these stru ctures. This problem is resolvcd in the models which consider molecules having various extents of hydrogen bonding: in thls case, the inside and the boun<lary of regioas (or « clustcrs»} can be treated differently. This approach was introduced by JiAGClS, H ASTED & B ucHNA:'i (195~) and by NtMETRY & ScHERACA (1962a), and has b ecn u sed in analyses of spectroscopic data (Bum; & CaoPPlN, 1963; LucK, 1965), and in theoretical discussions (V A.~ & SE:\"'OR, 1965).

A somewh at s imilar but conccptually quite diH'ercnt s tructur<' is pro­posed in the « interstitial» models: thc molecules in the hydrogcn-bonded s tate form an open network s tructure; the non-h ydrogen-bonded molccules occupy thc eavities of the network. Qualitative models of this typc bave been proposed by SA!tiOILOV (1946; 1965) and b y PAULlNG (1959). More dctailed treatmcnts were developed by F'ru.NK & QutsT (1961) and by NARTEN, DANFORO & LEVY (1967) .

Recenùy, direct numericaJ calculations (of tbc Monte Carlo type) haYC been carried out for water (B .. uuu:n & WATTS, 1969). Thcsc mcthods werc developed for siruple liquids, and dcal with radiai dis trihutions and with the dynamic of molecular motions, su ch as diffusion. 1t is t oo early yet to cvalunte how much specific information on water can be derived from these calculations, as they are very sensitive tu the assumptions made.

Quantum m echanicaJ calculatio ns ha\'C been discusscd earJier (5ec. III.D). So far, they gave useful information on small aggregates of water molecules, but no dctailcd studies of lnrger structurcs havc b een carried out.

A nn. l at. Supcr. Sanitd ll9'i0) l , ' 01-502

535

I n a sense, thc sharp diffcrence b etween the « continuum» an d the « mixture» modcls is artifìcial and results from limitations involved in t heir fon:nulation . Ali models are approximatc and contain many simplifìcations. Thc actual s tructurn of water prcsumahly is more complex than that proposed in any of the models. Thc corrcct evaluation may be that each modcl em­pbas izes only certain aspects of thc structure of water and of the nature of t be hyùrogen bond while neglecting other aspects. and so it gives a partial picture of the structure of water.

The hydrogen hond is not rigid, in the sense of h eing either pcrfectly formed or completcly brokcn. The bond can be distorted to various extcnts . At !cas t at small cxtents of distorti(m, the cnergy does not change very sbarply. (EISE:.VBF.RG & K.WZMANN, 1969, p. 248). This aspect is emphasizcd in the« continuum models». On tbe other hand, the re seems to b e sufficient C\'Ì1lcnce that t hcre a re wntor molccules in qualitatively different physical ~ tatcs, i. e. it makes physical sen se to talk abont hyclrogcn bonds which are formed or broken. Thcre ruay b e a degree of distortion aftcr which t he intcraction encrgy of the mo1ccules varies much more rapidly than during ,.;mali distortions of the hydrogcn bond. Howcvcr , this is only p art of the distinction betwecn thc various statcs, poss ibly a minor one. There is another dis tinctive feature. The possibiJity of f,rrcatly incr easing the numher of nearest ncighbors (to 8 or more ius teaù of 4·) when the bydrogen bonds are brokcn represents a definite qualitative dis tinction h etween the honded

and nonh ondcd moleculcs, cven though therc are s trong interaclions h etwecn the molccules of thc Iatter kind (cf. Sec. VI. A.2.). In the Light of prcscnt <'xpcricnce, perhaps the best qualitative picture rnlight be the foUowing: thcre are distinct kinds of molecules, diffcrentiated according to their dcgree of hydrogen bonding, but the properties , especiaUy thc potential energy, of cach kind, may vary over a certain range, due to the rnesence uf dis tortion or of local variations of intermolecular interactions. lt is cven possiblc that some of t hese ranges overlap for the various kinds of molccules.

2. Hy drogert B omlin.g and Noncovalent Tnteraction Energy in Liquid Wnter.

The cnergy of hydrogen bonds usuully lies in the range 5-7 kralfmol (PIME;>ITEL & McCLELLA~, 1960). For water, this ngrees well with the internai energy of st•blimation of ice at 0°, L.l E~uh = 11.65 kcalfmol (DonsEY, 1940), which corresponds to the breaking of two mo1es of hydrogon bonds per mole of water. Thus the total energy of breaking a hyù.rogeu hond in ice is 5.8 kcalfmol.

However , the corresponding encrgy in the liquid mus t be much lower , for two reasons. (a} E ven after the hydrogen bond is broken, tbere is strong

A nn. h t. Supcr. Sani t<\ (1970) 6, fUl--t;92

536

attractioo betwcen tbe molecu]es, due to dispersion intcrartions and, to some extent, dipolc interact ion!'. Thcreforc, tbc Lreaking or onc mole of hydrogen bonds in the liquid r eqnircs lcss th:m 5.8 kcal mol. (b) Molecules which ari' not hydrogen-bonded can have more than four near('St ncighbors. In simple liquida, thr coorJination nwnbcr i:; 8.5- 10.5. For << uon-hydrogen· bondcd» liquid water it bas be~n estimated as 8-9 ( NÉlltETUY & ScHERACA,

1962a). This increase in coordination number lowcrs the l'nerb'Y of the nonboodcd moleculc even more. due to the increao;c in tbc nwubcr of favo· rable noncovalent interactions.

Tbus, tbc energy of hrcaking hydrogen bondl3 in tbc liquid can be descrih· ed schematically as shown in Fig. VI- l. Tbc lowest energy state corre· sponds t o ice, i. e. to completcly bydrogeo-bonded moleculc!> wit h a coordina­tion numher :l = 4. 'l'he energy state or tbc va por lies 11.65 kcalfhigher. corresponding to tbc totru energy 2 E~ of two bydrogen boud:..

E

. ~w

ub l,., ZEH

Hypotht Uca l Uqvld antcerm•cha t.~ ( • • 4)

Uq-.id l • > 4)

Ice ( • • 4 )

Fig. Vl - J. - P otential energy o( water moleculei> in variouo strut turai state&, 8~ discussed in thc t ext. The energy of breaking 8 b)drogcn bond in lhe liquid with no rtructural change would be 2fl? 8 • T bc actual encrg)' (corresponding t o the transition struchued - unbtructurtd liquid) Ì~> 2 Ea (tiee tcxt).

lf one cotlld break tbc byà rogen bonds in tbc liquid without changiog z, a hypotbetienl « tetra-coordioated liquid » would b e ohtained, with an euergy lower tban tbat of the vapor. due to effect (a) mentioned ahovc. 2 <fin represcnts the difference betwecn 2 ETt = Ll ~ub and tbe interaction energica of tbc four-coordioated non-bydrogen-hondcd moler ules. If z is now rulowed to take its nctual value io tbc liquid, thc energy is lowered by

A nn . ltt. Super. Sanltd (1970) C, • ut-sD2

537

a value ~w, corresponding to the increased exten t of van der \Vaals interac­tions. The remaining difference, with respect to ice, r epresents 2 Ea , thc « energy of hreaking 2 hydrogen bonds in liquid water». Estimates of Ea vary over a wide range, but values ohteincd from many experimental techniques fall into the range of 2.2-2.7 kc:ùfmol, which seems to be a r easonable valut>.

T his argwnent is v:ùid for ali models of water , cven for interstitial modcls . In the lattcr, the situation is more complicatcd: thc coordination number of molecules in the hydrogen-honded nctwork also ch anges, due to the occupation of the ca,' ities by water molcctùcs.

3. Generai R equirements for Theoretical Models.

There are some principal requirements which theoretical models ought to sati&fy. (n) or course, various experimcntal obscr vations sh ould be cxplainablf', the more kinds, the b ettcr. (b) H ydrogen-bondcd s tructurcs must exiRt in thc liquid, but thc Jilfercncc bctwccn the solicl nnd the liquid must be explainable. (c) Disordering of the~e s tructures must be postulatcd, eithcr by breaking or by distortion of the hydrogen bonds. The model mns t also p ropose (d) a mechani'im of changes bctween thc s tructures postn­latcd , in order to explain fluiùity nnd dynamic properties, and (e) tlhe brcak­ing down of the structural ordcr with temperature. (f) The hydrogen­bonded-rcgions should not have the structure of ice-l, (at !cast not predom­inantly), s ince otherwise the case of supercooling were difficnlt to explaio; if there wcre ice-l clusters in the liquid . they would act as nuclei of crystalli­zation. (t!) On the othcr hand, tbc s tructnres hotùd not involve much strain. (h) Tbc extension of t he mode! to deal with structural changes in tbc pn~scncc of solutcs ought to b e possible. (i) The num.ber of adjustable parametcrs should b e reasonably small.

4. Problems of Consistency.

Some experimcntal obscrvations present problcms which are diflìcult to reconcile while sctting up theoretical models. The ambiguities in the interpretation of infrared data in terms of the nature of the water molecules prescnt in the liquid have already been discussed (Sec. V. C).

According to cxperimental data on moiecuiar motions, especially relaxa­tion processes (Sec. V. H- K) , it is very unlikely that srnall molecular aggre­gatcs (consisting of only a few molecules) cxis t in the liqnid, even at tem­peratures nt>ar the normal b oiling p oint. On the other hand, the raùial distrihution curve derived from X-ray scattering secm s to indicate that thcr e is no ordering h cyond 8-9 A, i. e. if clusters of hydrogen-bonded mol­ccnlcs ar e presen t , tbcy mnst have dimcnsions whicb are smaller. Th is

.4 n n. lat. Su ve'T. Sanitll (1970) 6, 491-;.9~

538

would exclude large clusters. Many experimental data bave been intcr­preted as indicating tbat thc fracti~>n of brokcn hydrogen bonds is small. However, if clus ters (ordered rcgions) are small, this means that they have a relatively largc surface, and many hydrogen bonds must be hrokcn.

B. Pople's Colltinuum (Bent Bond) Model.

I n a model proposcd in 1951, POPJ.E considercd as unlikely tbat many hydrogcn bonds are hrokcn, h ecause of tbc low beat of fusion. H c regardcd tl1 e hydrogen hond only ns an elcctrostatic intcraclion betwcen tbc Jlroton of one molccule and thc lon c elcctrou pair of th ~> othcr. Thcrcforc, he assuru­ed that bending can he treated in terms of classica! stati~tical m cchanic , h ascd on tbc gcomctricaJ dcscription of tl1c structure. Tbc cnergy is a fw1ction (eq. 6.1) of the two angles OA and 013 desrribing the cxtcnt uf hcnding. Tbey are tbe angles formed by an 0 1 • .• 0 2 direction and thc 0 1-H bond or tbc di.rection of a Jon c elt'ctron pair ou 0 2 , respectivcly .

(6.1)

Pople assumed f = k cos 8 wberc k is a force conslant. An average valul~ of O could be calculat<'d by :fitting the radiul dis tribution curve, us ing k and thc numhcr of second and third nearest neighhors as adjustabl1: parametcrs. Tbc caJcnlated aver age distortions are < L1 8 > = 26° a t O oc an d <LI () > = 30° a t 100 oC, an d a rcasonably good fit of MORGAN ami

W ARREN's (J 938) radiai distribution function is obtained. Due to bending of tbc bonds, some second and third neighhors com e closer tban in ice, cau~; ing

tbe coordination numhcr to be z > 4, and accol.Ulting for thc volume dccreasc on m<'lting. Tbc lattcr wus catimatcd from the radiai clistrihution function. A density d = 0.99 gfcm3 was calculated for tbe liquid.

Thc only otltcr property wbich Pople calculated wa the static diclcctrÌI' constant E0 • l ìsing a simplc geometrical calculation of nearest ncighbor efl'ects, based on Kirkwood's theory of dieJectrie behavior (Sec. V. E), hc obtained an e0 wbich is about 20 % too low, aod a temperature coeffi­cient d E0 / dT baving ah out 15 % error. ElSENBERC & KAUZMANN (1969, p. 192) state, witbout describing details of calcnlation, tbat a very good fìt of e0 vs. T can be obtained by taking into account the interactious bet.ween more molecules.

Pople did not discuss thermodynamir. propcrties. E ISENllEJtC & KAUZ·

ltfANN (1969, p. 178) calculatc tbc internai cnergy contribution due to bydro­gen bond distortion for a bent bond model (not nocessarily that of Pople) . They assume that a Jinear correlation exists between bydrogen bond s trcngth and the shifting of the OH stretching frequency in water, in a manner s imilar to the correlation used by WALL & HoRNIG (1965) (cf. Sec. V. C). They

.~ "'' · 1st. Suvcr. Sn11ità (1970) 6, f91-5Q2

539

show that with this correlation and with very simple assumptions ahout the vihration of the molecules, the dependence of the internai energy on t emperature is reproduced quite wcll, and that the heat capacity, although not shown by them in detail, probahly is r easonahle. As discussed hefore (Sec. V. C), it is dubious whether the frequency correlation can be applied to liquid water. The entropy was not calculated by EISENBERG & KAUZ­

MAN;'II, although this would be important, since it seems that the model is very solid-likc, so that it would have a low entropy. The model also seems to predict t oo much rigidity and viscosity for a liquid, and it is not clcar how it could be applied to solutions.

Sometimes it is argued that some bonds in the Pople model are broken anù rc-form<'d , to account for the flu.idity, but that the fraction of broken honds is small. It seems that this requ.ires the introduction of a di:fferent typ(' of watC'r molecule, contrary t o the basic assumption of the continuum model. It is true that only a relatively small fraction of molecules may be in this state, but ali structu.res whose properties make a significant contri­hution to the description of the liquid, according to the model chosen , must be t nken in t o considcration.

C. D etailed Cluster Model of Némethy and Sclleraga and Related 1\.fodels.

l. Description o.f the JYfodel and o.f its R esults.

In this model, i~ was taken into account thar watcr molecules in a mi..xtu.re mode! m ust h ave more that two kinds of local environments (cf. Sec. VI.A.l). The interpretation of the properties of solutions was one of the main aims. The model was based on the assumption that molecules exis t either in compact clusters with maximal hydrogen bonding (but that hydrogen bonds are broken on the surfaccs of the clust ers) or as non-hydro­gt>n-bonded but strongly interacting monomers, and thc two states are mixed throughout t he liquid.

The existence of clustcr s (insted of small aggregates) and the condition of compactness was bascd on FRANK's (1958) proposal of cooperativity of hydrogen bonds (Sec. III.D.3) . Thus the model representcd a quantitative devclopment of F rank's proposal, although the latter was not r epresented in the m o del in a formally cxact manner.

Cluster s ware nssumcd to be compact, in order to minimizc their free energy by nllowing the maximum number of hydrogcn bonds to occur. Partially hydrogen-bonded molecules already have lost most of their cntropy, so tbc requ.irement of compactness does not have a large influence on the OV\~rall entropy.

Since it was prcdicted and found that clusters are small ( < 100 mol­ecuJes), therc must be many molec\Ùes on Lheir su.rface which bave onc or

11 nn. I 1t. Su per. Scmit.à (1970) 6, 49Hi92

540

more of their bydrogen bonds broken. Therefore, tbc water molecules were divided into 6vc classes i = O to 5, each witb a diffcrent energy E1 , depcnding on the number of hydrogen bonds broken. The energy diffcrence hetween moleculcs with four hydrogen bonds and witb none is 2 En (Fig. VJ- 1).

1f FRA l""K'il proposal (1958) were followed rigorously, tbe fi vc energy levels E , ought to bave unequal spacings; it should be harder to break a hydrogen bond of a four-bonded or a two-bonded moleculc than of a three or a onc-bonded OD I' (cf. also Sec. III.D.4). lnstead, the energy levcls

werc spaced equally, at intervals of ! En for a practical r cason. Since

En is not known , it had t o be uscd as an adjustablc parameter. The usc of unequal spacings would have r equired three more adjustable parameters, rendering comparison witb experimental dala mcaninglcss. A cooperati\ e effect was introdueed by restricting the relative numbers of molecules in the varioUB levels, assuming a priori that only compact clusters exist. In this way, the modcl effectively corresponds t o Frank's proposal, but this is achicveù in a formally incxac l mauncr.

E vcn if 01Ùy relatively large clusters exist , a distribution of cluster sizcs and shapcs is expected in the liquid. Tbis was disregardcd in the mode!, to makc it Jl1athematically tractahle and to avoid more unknown parameter s. lt was assumcù tbat thc calculated cluster size rcprescnts thc most probablt> cluster size aod thcrefore is a good averagc. This corresponds to the method of the maximum t erm in tbc evaluation of partitioo functions. H owevcr, its use in this case is questionalJle becausc, for small cluster sizes, deviations from the mean are not negligiblc. The cluster size and shapc may stili b e considered nearly aver age, but an entropy contribution due to the exist ence distribution of finite width is ruissing (lUCLER, Scn ERACA & NÉ!\fE­THY , 1971 ).

While tbe clusters are assumed to bave optimal hydrogen bonding, tbey do not ha,·c to possess a rcgulnr structure such as that of ice· I or any otber solid form, but can consist of irregular arrangemcnts of molecules.

Witb all tbe assumptions mcntiooed, a partition function can be set up in t erms of thc five classes of molecules:

Z = (6.2)

wherc the summation is over tbc possihle distribution of thc mole fractions of the species (x1). Thcsc are not indcpendent: not only is it reqnir t'd that

4

\ X t =l, (6.3)

A nn. l et. SuDer. Samtct (1970) 6, 40 1 592

541

but there are two more equations arising from the assumption of compact, near-spherical clusters (NÉMETHY & ScHERAGA., 1962a, eqs. 3--4). Thus the five variables of summation can be replaced by three independent varia­bles: xu = fraction of non-hydrogen-bonded molecules, Do= cluster size, and y1 = another parameter describing one-bonded molecules. g is a combi­national factor, representing the interchangeability of molecules. Assum­ing that the molecules can be freely interchanged, it was written in the simple form

No g = - 4---'-- (6.4)

rr N, i - O

where N1 = x1 N0 • g is a function of fio, Yh Xu • The ft represent the contributions to the partition function of translational, stational, and vibra· tional degrees of freedom. The intramolecular vibrations are practically unexcited and therefore contribute a factor of l . For the four hydrogen· bonded species, the translational and rotational motions correspond tovibra· tions, and their contribntion to the partitions function were represented as products of Einstein functions:

6 fl = f1 (1- e- hVtt f kT )- l (6.5)

·-· The freqnencies v,, were specified by the assignment of observed IR freq· uencies in the 60-800 cm- 1 region, according to the best data obtainable at that time. Some arbitrariness could not be avoided. In most cases, some variàtion is not very eignificant, and the choices were made in a self· consistent manner. For the unbonded molecules, it was aseumed that the rotation is restricted, and that the translation can be represented in tenni of « free volume» V,. This and ER were used as temperature·independent adjustable parameters in the fitting of the thermodynamic parameters from O to 800. The maximal errore were ± 0.6 % in the free energy, ± 2.5 % in the enthalpy, and ± 1.5 % in the entropy, bnt Cp decreased much faster, from 21.5 at 00 to 13.7 calfdeg·mol at 70°, than the experimental value which is between 18.0 and 18.2 cal/deg·mol in the same interval. The dependence of the volume on the temperature could be calculated with few assumptions, but more parameters were required to describe the unlmown volume of the non-hyàrogen-honded liquid.

It could he shown that the model is consistent with the X -ray radiai distrihution function of MoRGAN & WARRE]."( (1938), by fitting the fìrst peak of the function at both low and high t emperatures. While this is not a proof of the model, it is a necessary requirement of consistency.

A nn. l at. S upe1'. Sanità (1970) l. 4Gl - 60Z

542

The model applics to D20 as well (NÉJ\IF:THY & ScnERACA, 1964), with only a single change in a paramct er: En = 1.56 kcalfmolc, instcad of l .32 kcalJmolc for H20. The direction of thc change corresponds lo that expectccl for thc isotope effcct. Thc volum e vs. temperature curve. including thc minimum at 11.2°, could be r eproduced without any new parametcrs. The errors iu thc thermodynamic paramcters are larger, howevcr. than for H20 : up to 15 % in Go, 6 % in H 0

, 2 % in S0• Cp is not murb diil'erent

from th at calculated for II20. Tbc fraction of rcmaining ltydrugen honds atul the mean cluster sizes are larger than in 1J20 . Tht' wodcl can ])e applir ò in tbc a nalysi of dilute aqueous solutions (Sccs. VIT.B, VIIT. E).

2. The Models of Bu ijs and Choppin, and of Vand and Senior.

B uu s & CHOPI'IN (1963) intcrpretcd oear infrared data in tcrms of scve­ral kinds of water molecolrs . Thcy assumcd only tluee spectroscopicaUy different spccies: with none, onc, or two hydrogcns of a molecule bydrogcn­bonded. They calculated thc fractions of these species from thc tempera· ture dependence of the absorption intcnsities, aod found the calculated fractions t o be consistent with thc results of Ù1e Némethy-Schcraga modcl. Since the spccies werc defined difl'erenùy in the two models, thc correlation bctween them is not weU unambiguous. Criticisms of the interprctations of the data were mentioned above (Sec. V. C).

VANO & SENlOR (1965) developed a mode!, based on some of the con­cepts of the Némethy-Scheraga model, using Buijs and Choppin's iufrared data. Thcy criticized the former for the use of single encrgy levels aod proposed instead five energy bands of finite width, r eprcsented by Gaussian curves. Thcy could fit the thermodynamic parameters well, includiog Cv·

This is the only mixture mode] so far in which cnergy bands werc uscd. Howcver, this introduced several more arhitrary adjustahle parameters. The numher of ways of comhining watcr moleculcs was overcounted. In adrlition, most of the criticisms of thc Némethy-Scheraga model, lis ted below, apply to this model as weU.

3. Criticisms of the N;methy ·Scheraga Mode/.

Over the past years, seYeral of the assumptions of the Némethy-Seheraga theory carne under criticism. Some were already mentioned aboYe in the discussion of the approx.imations of thc modcl.

(a) Equal spacing of thc energy levcls. A correction was made for this by the use of thc cluster equations. Siuce this is not a formally correct treatment, it may introduce some error.

.4 nn. l at. Su per. Sanità (1970) 6, 4Ql- 592

543

(b) The use of energy levels instead of bands, as would be required in a realistic representation of a liquid . The latter would correspond better to IR data, too (cf. VAND & SENIOR, 1965).

(c) The usc of a single cluster size and shape, instcad of a distribution, thus neglecting a mi.xing cntropy.

(d) AH the generai objcctions raiscd against the mixture theories apply to this model as weU, as discussed for example by EISENBERC & KAUZ)fANN (1969, chaps. 4 and 5). although some of them are matters of interpretation (Sec.VI.A).

R ecently, some mor e fundamental objcctions have been raised. (e) More data are aYailable now on IR and Raman spectra, and some

of the frequency assignments used to calculate the f!'s are dubious or even incorrect. The band near 175 cm- 1 which was as signed by N btETBY & ScnERACA (1962a) t o a rotational mode, has been shown by W ALRAFEN (1968) to be a translational mode.

(f) I n a detailcd analysis, diffcrcnces of the vibrational zero·point encrgies of the various species r;eem to make a signi6cant contribution, nccording to recent calculations (HAcLER, ScaERAGA & N ÉMETBY, 1971). Jlowever , the numerica! magnitudc of this error may depend o n the assump· tions of the model.

(g) The form of the combinational factor used (eq. 6.4) introduces an error since it counts too many con6gurations (PERRAM 8• LEVINE, 1967,

+ 0

;::.;/ o-L

6

o

0 + 0

Fig. Vl- 2. - Schematic illustralion of the non-independence of molecular iuterchanges in a part iaUy hydrogen-bonded oetwork. Varioo.ly bydrogen-booded

moleeules are indieated by different ~ymbols: e 4-bonded, ® = 3-bonded

O = 1-bonded, ® = non-bonded. An intercbange o( e and ® .without any olher ~imultaneous intercbange is not po&&ible, tince it would result io t wo unattacbtd c halC » hydrogen bonds (lower left). However, e and

® may be inlercbunged if ® and one o{ tbe O moleculea is interchanged

~imultaneously (lower r ight). Adapted from H"CLER, S cnERACA. & Xt­METHY (1971).

.! 11 n . h t . Supe r. Sanltd (1970) 6, 401-~02

541

Lt::n~F. & P ERRA:M. 1968; HACLEll, CBERACA & NÉMETIIY, 1971). Thc to impl<' expression for g in eq. (6.4) corrcsponds to tlw intlcpcndent intcr · chnnge of any m oleculcs belonging to difl'crent ~peci<··. Jfowever , nol ali sucb intcrchangcs ar e possiblc. F or example, a four-lwnd <'tl and a three· honùctl m olecule are not intcr chnngcable wi thout intt·rchanging at lenst ouc otber · pair of neighhoring moleculcs as well, iu orcler t o avoid « h nlf» bydrogcn bond , not connect t'd to another m olccule (Fif!. VI-2).

4 . Cluster l'rfodel by l/agler. S cherasa & N énuthy (19i J).

A n cw mode) Las becn dcYelopcd . with tbc twofold aam of correl't ing some of thc weaknesses mcntioned ahovc, and of annlyzin~ tbP. effect of , ·uriou!> approximation!o mnd!' io tbis and in otber s truc tural modcls.

A partition function, simill\r l o t ltat of ~i:M E1' 11Y &. Scu .ERACA (1962a) is used , with equa! spaciug of tb.e cnergy lcveh;, and corrections for changcc in zero point energies. En remains an adjustable p3rtunclf·r. Thc clus ter sizcs are rcprcsented hy a continuous distribution . Tbl' cntropy of clu&tt' rs j., included through the utoc of an adjustable parumcter . Tbc Yibrational frequcncics are assigned in u sirnplcr manner , an d are con -; i ... t cnt with ~·AL·

RAFE ' . cinta (19(»: 1967). Tbe configurational (trant- lational) entropy of t h <• liquid is calcula ted witb tbc aid of the Percu~· Y <' ' ick equution, de, ·e lopcd originally for treating simple liquids (LF:aoWlTZ, 19M). R estricted rotation introduccs a third adjustalllc paramcter. \\hiltJ tbt• furm11lation of tbe va· riablcs cntails some unccrlainti<·~, rcquired for tbc JH'ct•s:oa ry approximations tbe modcl can reproduce tbc main thermodynnmic paraan ctt•rs (free cnergy, internai encrgy, entropy) bet" ecn O and 60 °C with dt•\ ia tions of O to 5 % from <'xperimcntal data a od error.- up lo 20 °.1 in C p . A p roblema tic re -ult secms to be tbc predic tiou that th<' median values of cluster s izcs are vcry small , of tbc ordcr of ll or less, especially al higher tcmperaturt'l'. For small clus tcrs, tbc thcory contains some weakcr a,sumption-,. As discussed abovc, tbc prescnce of many small aggregates does not '><'t'm to be consis tcnt with various experimental d ata.

D. Ry ring's Model Dased on the Sign~fìcant Structurc Thenry.

l. Genera/ Description of t!IC Theory .

Eyring and his coworkers d evclopcd a simplc theor<'lical model for liquidi', bao;ed on physically reasonable intuitive asswnpti<•m• (re\ieweò by F>k·nrNt; & .Tuo!", 1969).

Based on tbc hypothcsis of molecular·sizcd vucaHCit·l'l, it is asswned tha t thcr e is an equilibrium b t' twecn molecules \\ itb solid-like an d gas-likc

A. nn. l at. Su va. Snnitd (1970) 6. 401-50!!

..

545

degrees of freedom in the liquid. Consequently, the partition function can be wr itten as the prod<1ct of partition functions for these two kinds of mol­eculcll, with the mole fraction of the solid-like m olecules taken equal to the ratio of the volumes of the solid and the liquid, V1 / V :

(6.6)

The partition function zg has the form of that for an ideal gas, and z8 that of an Einstein-type oscillator; the latter also contai.ns a degeneracy factor to account for the incr cascd numher of positions available for the liquid.

Tbe theory has been used successfully for many monatomic and molectùar liquids, fused salts, etc., in the calculation of tbermodynamic and transport propertics.

The thr.ory has the correct limiting behaviour. If V- Y, Z- (z.)l'', i . e. thc partition fun ction for an Einstein solid. lf V- 00 , i. e. (V ­- V,) f V - l, Z- ( zg)~'~, i. e. the partition funct ion for an ideai gas.

2. Application of the Theory to Water.

If the theory is t o be applied to watcr, the « solid-like» state cannot be taken as that of ice, since therc is a volume decrease on mclting, in spite of thc postulated appearance of vacancies. In an early attempt, 1\f.ARCHI & EYRING (1964) assumed two « solid-like » specics, one hydrogen-bonded, like ice-I, the other consisting of non-hydrogen-bonded , freely rotating mole­cules, in addition to the « gas-like» dcgrees of freedom. Due to criticism, on spectroscopic grounds, of the assumption of a freely rotating monomer, and because the minimum in the molar volume at 4° could not be calculated, a different model was proposed by ]HON et al. (1966) . They, too, assume the existence of two solid-like s tructures. As the first structure they propose a « cage·like clust er» of about 46 molecules with the structure of ice-l, ·without an explanation of the choice of this numerica! value. They assUIUe t hat the seconcl « solid-like» structure is that of ice-Hl, i. e. a more dense form and that ice-I-like clusters are dispersed in this s tructure. It is assumed further that ruost of the ice-1-like structure is destroyed by 40 °C.

A partition function is written as in tlte generai theory (eq. 6.6), but a product is used for the solid-like partition function:

(6.7)

where nr and nm are the numbers of molecules in tbe two s tates, correlated by an equ ilibrium constant

K = {ice-1-like) · (ice-III-like)

{ Zs T )q = \ Zs Hl

(6.8)

A nn. l!t. S ttlJeT. S~tnitcl (1970) 6, 4.91-502

546

Thi c:. scems to iruply that q molcr.ule~ alway!' cbang<' tngeth<'r froru ont• ~trurture to thc othcr. Jn application 10 rxpcr.inal'n tal propl'rlies. JnoN eta/. (1966) bave to usr 8 adjustnblc· paramctt•r~o~. altlaough some of tbe values choscn cnn b e ral ionaliud.

Tbc calculatcd valucs fur thc molar volum e Òl'lwct•n O and 100° have < 2 1' ~ error. Tbc fr ee cnerg) of thr liquid and tlH· cn t rOJ'Y of vaporization dc' ia te cvcn lcM• from cxpcrimcntal valu es, botlt for TI20 und D20. E rrors in Cv are up to 12 °6 a nel l n o:, respc·ctin·l~. T la e t•ulculntcd n tpor prcssure an d viscosity are satisfactory.

3. Criticisms of the Application to Tr'ater.

Severa} of tbe assumptions of thc modt' l seem arbi trar ) or ar e incon· s ist ent.

(a) Therc appears to be no pbysical rcason for tlw assumption that one of the solid-l.ike forms should b e ice-111, and tbc assumption is not justified in the paper. For examplc, the phase cliagram (Fig. JV-2) indicates that tbc first solid pbase reached upon compression of tbc liquid at O oc would he ice-V.

(b) Tbc assumption of cluslcrs witb th<• structurc and density of ice-I , with a s ize of 46 moleculcs, indr pendcntly of the temperature, appears arbitrary .

(c) In tbc treatment of tbc solid as an Einli tcin oscillator, an averagc frN[ltency was assumed for ali s ix intcrmolccular mod1·s. T bc vulue cbosen, 150 cm- 1 , does not correspond to tbc saml' vibrationul cncrgy and entropy as the cxperimental far infrared frcqu('ncics ~IlACLEJl, ~CnERACA & NÉ . 1\tETOY, 1971).

In tbc application of the s ignitìcan l structurc tbcory t o watcr, som e principal shortcomings occur, du e to tbc assurned cxistencc of two solid structures.

(d) As indicated earlier (Sec. VJ.A.l ), tbr difl'crencies in properties of tbc water moleeules on tbe houndarics of tbc solid-like rcgion ought to be taken into account. This problem does not occur in the signifieant s tructure tbeory of simple liqtùds, but causes nn error in thi s case.

(e) Whatever spatial arrangement is assumcd for the two solid-likc molecules , a confìgurational entropy PbouJd be part of tbc partition function to account for tbc differcnt numbers of ways of dil"trihuting tbc two compo-' nents in space (lù.cLER, ScoERACA & NÉMETDY , 1971).

(f) Problcms arise at the limit of V - V, (HACLER, ScoERACA & NÉMETTIY, 1971). In other liquids z- z. (crystal). In water this is not tbc case, since V, (ice-I ) > V (liquid). Thorefore, JnoN et al. (1966) assurned V, (ice-III) = 17.65 cm3/mol for the seeond component. Whil c this value

.4 nn. I lt. Suv cr. Sanità (1970) 8. 491-502

5<&7

is less than V (liquid), it differs from thc molar volume y o = 15.7 cm3/mol for ice-III. Furthermore, upon compression, a solid s tate ought to be reached, uccording to thc theory, whcn V- Va (icc-111). Actually, the observcd molar volume reaches a value 15. i cm3/mol a t 500 atm. wcll below the liqniù- solid line in thc phasc diagrarn (Fig. JV-2).

It sccms that thc signifìcant s tructural thcory in its prescnt form is not applicahle to watcr.

E. The T tco-S tate Model of Davis und Litovitz.

l . Descriptìon of the Nfodel and its R<•sults.

DAVlS & LITOVITZ (1965) wanted to develop a two-state model '~hic b uvoids the assumption of many non-bydrogen-bonded monomcr!'.

Fcllowing FRANK's (1958) s uggcstion of the cooperativity of hydrogc:n boncling, they proposcd t hat six-ruemhered puckercd rings play an important rolc in both s tatcs. For tbc open structme thcy assumed that t he s tructure is iclcntica l with that of ice-I. For thc close-packed structur~ (state Il), thcy proposcd that it consists nf six-mcrubered rings in the « chair» forro; thus each mole,·ulc has two hydrogcn bonds. In contrast to ice-I, the rings are orientcd parallcl above cach other, so that no hydrogcn bonds can be formcd betwcen them, bnt thc molecules are close-packcd instead. The volume of a tructurc entirely composed of s uch close-packed rings wm1ld b e ul>out 13.7 cm3f mol, as calculatcd by Davis and Litovitz, compared with 20.7 cm3/mol for thc open structure. They point out that the numher of tbc n ellfe t neighbors has h een incr eased in the second s tructure. Each molecule has two hydrogen-boudcd ncighbors and 5 nonhonded neighbors. TIH' nonbonùcd ncighbor dis tance is calculatecl to be 3.03 A. It i s obtaioed. as an adjus tablc parametcr , togethcr with five others (including the hydrogen bonù lcngt h with values of 2.80 to 2.82 A, and the fraction of molecules. in s tate II), from the fìtting of tbc volume and tbc radial distrihution curve. The latter is fittccl quite well, includ.ing the small peak at 3.5 A. The cal­cu1ated coemcient of thcrmal expansion is very dose to the expcrimental v aluc. r o ùircct comparison of cxperimcnt al and thcoretical values for­any othcr property .is shown. According to the calcnlat.ion, the enthalpy diffcrence hctwecn the two s tatcs is LI H 0 = 2.6 kcal/mol. This corresponds to thc breaking of onc hydrogen bond. sincc one bond per moleculc is rctaincd in structure II.

The main purpose of the paper is thc culculation of « relaxational thcrmodynamic propertics ». Davis and L.itovitz show that cach property relatcd to tbc volume orto the internai ene rgy, sucl1 as the compressibility ;<,.

coellìcicnt of thermal expansion a, or beat r apacity Cp can be divided

A nn. Jst. Su per. Sani t!\ (1970) 6, 401-6'1}2:

548

into two parts. One of thcm, tbc- «lattice » contrihution, corrcsponds to the normal changcs tsuch as thermal expansion) within the two structurcs . A « relaxational» (or « structural») component arises from thc s tructural rearrangement caused by ch anges in pressurr. or temperature. For examplc, for thc thermal expansion,

a = a<IO + ar (6.9)

where a«> is the «lattice» component. a, can be cxpressed m tcrms of the eq1ùlibrium h etween the two s tructures. Since

(6.1 O)

a =~(2'~ ) =~ lx1 { "(lV!.) + xu ( "dVu) l V d 1 p "\ \ "d T P· xJJ d T p . xr

l r ( d v ) ( "d xr ) ] + y o dXJ T 1T,Jl (6.11)

so that

a - - --_ LI V ( d Xr) r - Vo "d T l ' (~) dT P

(6.12)

Since the equilibrium con stant is defincd as

K e - A Gu / R T

' (6.13)

x1 can b e expressed in therrnodynamic t erms, and

(6.14)

with similar equations for the other parameters. Davis and Litovitz calculated the relaxational contributions to a, ~.

and cp. They showed that thc•se t erms make a large contribution . For example, the calculated values of the relaxational beat capacity are Cr•.r = 12.1 to 10.6 cal/deg·mol at 0°, 9.2 to 7.6 at 50°, 6.0 to 4.0 at 100°, with the rauge shown depending on tìhe cboice of some adj ustable parameters. Thus Cp,r , contributes about half of the observed value, in agreement with the arguments presented earlier (Sec. V. B). Unfortunately, there is no direct way to test the values of the relaxational contributions, so that the main conclusions of the model cannot be checked directly.

Ann. Ist. Suv eT. SClnità (1970) 6, 491-502

549

2. Critical Evaluation.

The merit of DAVTS & LITOVrTz's paper (J965) is t bc cmphasis of thc role of the « rela."<Cational» parameters and tbe calculation of their numerica! value (this couJd be done with most otber theories, too) . They obtained an inùication of the relative importancc of tbese terms, altbougb tbe actua numeri<:al values depcnd on tbe assumptions of tbe model and tberefore may not :~ 11 be correct.

The main weakness of the modcl is t hat of most two-state t beories: the attribution of h ulk phase properties Lo the structures. In structure I all hydrogen bonds are assumed to be fo rmed, and the molar volume of ice-I is assigned to it, altbough in small aggregates both quantities are rcduced due to the dis.ruptions of the structure at tbe surface (•). T he error is lcss for larger clusters hut the clusters cannot becomc very large. Nwnerical calcuJations were based on the fitting of thc radiai ùistribution function, altbough bere too, the distances between neighboring molecules in different states should not be disregarded. No physical r eason is given for the assumption of thc close-packed rings in structure Il, witb tbe exclu­sion of otber s tructures a nd other spatial arrangements.

In tbe discussion of their results, Davis and Litovitz conclude from tbe temper ature dependence of t bc relaxational iaotbermal comprcssibility, "r that more tban one close-packed s tate may exist. Tbey suggest the possi· bility of close·packed monomers. This would result in a more complicated system, more r ealistic s ince tber c must be actually many different structural arrangements in tbe liquid, but it would not avoid the main probJem of simple two-state tbeor ies., discusscd above.

F. Interstitt.al Models . l. Samoilov' s Proposal.

SAMOILOV (1946; 1965) proposed a mode] for liquid water in whicb be suggest ed tbat hydrogen-honded molecules form a framework with tbe structure of ice-I, and monomeric molecules move mto tbe interstitial cavities.

Later, Gurikov (Gunu:ov, 1963, 1965; VooVENKO, GuruKov & LEGIN, 1967) developed a quantitative t reatment of the Samoilov mode!, in terms

(•) A limitiog example ma y be cited : 12 molecules can form an« ice-I -likcn structure. wilh only one inner cavity (cf. Fig. IY- 1), while in ice there would be one such cavity for every two molecules. Thus t he volume per molecule is la.rger in an infinite network. Tbc total number of hydrogen bonds in th.is structure is 15, i.e. 1.25 per molecule, as com­pBred with 2 per moJecule in ice.

Ann. ht. SupeT. Sanità (1970) 6, 491-592

550

of a two-state equilibrium, in a ruanner similar to Frank and Quist tsee h clow). but using a simplc formulation of thc equilibrium h etwecn the two s tates. Ile gcts good ngreement with p-V-T propertics nn d also with cp .

Strongl'r hydrogen-bonding nnd a more ordercd s tructure werc found for 1>20. In one of t hc papcrs of GURIKOV (1964.\, the strangc proposal is madc that the inters titial molecules form hydrogcn bonds with tbc f ramework molecules, distorting tbe &tructurc.

2. Pauling's Mode/.

PAULING (1959) proposed a s tructure for watcr based on that of the gas hydrates, especially cWorioe hydratc. Tbc basic s tructurc is a dodeca· hedron, formed by 20 molcculcs which are hydro~en-bonded t o each other. with an unbonùed molccul c in th e center. The dodecabcdrn may be linkcù to each othcr in various ways. Paultog favorcd tbis modcl bccausc it givcs the corrcct density, and the numhcr of hydrogen bonds in each complex of 21 molecules is higher than in other s tructurcs of a s imuar s izc. However, DAI'ir'ORD & LEVY (1962) showed that this modcl is not cons ist ent with the radiai dis trihution curve.

3. Frank and Quist's Theory .

FRANK & QutST {1961) presented a statisticnl-thcrmodynamic treatment, based on Pauling' s modcl, but the mathematical d rvclopmcnt is applicahlc to any interstitial mode!. Thcy assume t hat some of the molecules fo rm a hydrogen-bonded framework with interstitial cavitics. Thc latter are partiaUy filled with non-hydrogen-bonded monomcric molcculcs, and thc extcot of occupancy variet. with temperature. They assum e that thc intcrstitial molecules are freely rotatiog and thcrcforc do not intera..: t s trongly with the framcwork. Thcrc is an cquilibrium hetwccn the two kinds of mol1•culcs.

Thcre art> N m oleculcs, a fraction f of which (or ·r moleculcs) are in the framcwork and a fraction (l - f) are interstitinl. Tbc numbcr of inter· stitial s itcs is Nfjv wbcrc v depends on the particular structure assumed . For ice-T, ,, = 2. For Pauling's s tructure, v = 3.83. Of coursc, Nfjv > N (l - f ) is a rt·quired condition for tbc exis tence of a stablc structure.

A mixing entrop y arisrs from tbc rus tribution of the inters titial mol­t•culct. O\'Cr tbl' availablc sitcs. The numher of arrangcm ent is

(6.15)

A nn. h t S upor . S anità (1970) S, 401-51)2

Using Stirling's approximation, this gives

Sm = R [_!_l n f j v -(1 - f) v f

-- {l - f) v

551

In f (l - f) l (6.16)

-- {l -f) v

If Gr, and G~on are tbe free energies of tbe empty framework and of tbe monomers, respectively, tben tbe free energy of tbe liquid can be written as

G = f l Gr. - a ( 1 -;:- f) J + (l - f) G~on- T 5"' (6.17)

Tbe term a (l - f)/f describes tbe cbanges in tbe energy of a framework molecule when a neigbhoring interstitial site is filled . Tbe parameter f is variable with temperature; tbc constants a, Gr, , and G~n are used to fit the p-V-T data. A good fit is obtained for tbese. On tbe otber band, tbc beat capacity is too small, apparently even less tban 12 calfdeg. m o l. Presumably, this bappens because tbe model is too solid-like. Tbe calculation predicts that ali interstices are filled at 300, i.e. N f/v = N (l - f), so t hat there must be a transition to other structurcs ahove 30 °C.

FRANK & QUIST (1961) state tbat tbere is a s ingle relaxation process, the forming and dissolution of clusters, b ut the reason for tbis assumption is not clarified. Tbey propose tbat a third class of water molecules may have to be postulatcd, probahly a non-interstitial monomer, t o account for tbe fiuidity of the liquid, for tbe beat capacity, and for tbe properties of tbe liquid above 30<>C. The model can describe the solution of nonpolar :;olutcs , only for small molecules such as m etbane. Tbus the Frank­Quist modd appear s to bave limited applicability at low temperatures and is no t a generai mode) for li qui d water.

4. The Model of Narten, Danford and Levy.

This model was proposed after NARTEN, DANFORD & LEVY (1967) r epeated MoRCAN & W ARRE N (1938) X -ray measurements witb higber precision and extended tbem to t emperatures up to 200°. Tbey propose tbat tbe framework has a structure like ice-l, but tbat it is expanded aniso­tropically. At low temperatures, ahout balf tbe interstitial sites are occupi­ed by monomers, centered in the cavities {l - f = 0.2). Unoccupied lattice sites may occur. Narten et al.'s main concern is tbe fìtting of tbe r adiai distribution function curves. This is done with six parameters, whicb include several intermolecular distances ( determining tbe expansion of the framework), tbeir fiuctuations, and tbe fractional occupancy of tbe sites. Tbe curves ar e fitted very well, but the parameters are adjusted independently at eacb temperature, so tbat the true numher of parameters is rather large.

A n11. lat. Supe1'. SanitA (1970) S, 491-51>2

l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

552

In order to compare 1'0 111 <' othr r plty.:;il'al tprantrt re~. :\arten et al. cal­t·ulatt• approximatc intrrmolecular frequencit•s from tht• paramclers of tht· radiai distrihutiou fuu clion. obtaining 200 rm- 1 and 9() cm- 1, Yaluc•ì< not loo far from the ouscrvrd \ alucs of 175 cm- l an ti (,Q (•m-l for 1lw transla­tional m od es (W A LUA l'El\ , ] 964: l 967). H owcY('r. t hl' ralr ulatcd iutcnsity dt•t·rcascs much lcì<s witb incrcasing t empcrntun• t han 1 hl' obc;t'n·Nl vnlut· .

Tbc free encrgy. cutbalpy and cntrop~ of framework moleeulcs. inter· s titial molecules nnd intr retitial ;, i1 f'5 wcn• calc:ulnted frorn thr dibtriliution of framcworl aud interstitial moleculcs. Tht•,.c uumher~> c·annot b e compared tlircctly ,,·ith any C~"]>Crimrn 1 al quantity, but dt·rivcll comparisous ca n b e made. Tbc calculatcd hcal of vaporizatiou h a,. a Ytthw of 9.7 kcal mol, and i-. indcpcndcnt of temperature, as compart:d w i t h thc experimcntal 'alucs 10.7 kcal 'mol at 4° and 9.7 k cal /mol a t 1000 (Dons EY, 19-lO).

On the other har1d, thc t·n thalpy and cntropy of fus ion can be calculated (JIAGLErt , S CHF.RACA & ÉMETH\' , '1971) from th <> paramctcrs of the modcl, rcsulting in L1 s~ ~ 1.23 e.u. nn d L1 n~::::: 0.2 kcal, in:.tend of 1 hc exp erim cntal values of L1 S~ = 5.~5 c.u. nntl L1 H~= L.4 l.callmul. Thcse rcsults indicate that thc model is too rigiJ nnd solid-tikc. Thi:- is l> hown also by the beat capacit y, computcd b) XARTE , . D...\ 'FORU & Ln'Y (1967). Tb.ey find a valuc ncar 9 cui dcg. mol. Thus it secm ., tha t thc entire s truc· turai conttihution to <·1, in tbc liquid ·state il! mil!sing from thc mode! (HAGLEJt, ScnERAGA &. 1\ f: 'dETIIY, 1971).

5. Generai Evaluation of lnterstitial Models.

Thcir grcat aJvaot.agc is thut Lher .rcpretlcnt a mixturc-model, i .e. nllow the cxistencc of Yarious kind:. of water molcculcb, with Yery difl'crcnt contributions to tbe Yolumc. wilhout requiriug tht' prcst•nce of large regions of ditfercnt structure~ ns in tbe clus te r models, a conccpt questioned by tbc X-ray s tudics (Sec. Y .A).

On the other haud, thcy face severa! unsoh·cd criticai problems. (a) It is not clcar how water molecules could exist in tbc interstices

of a bydrogcn-bonded framework without interacting strongly with the surrounding molccules nnd thercby dest.roying thc framcwork. For example, 11u crystalline gas hydratcs (the analogs of tbc interstitial models) a re known for polar solutes which can form hydrogcn bonds. F rank suggested (.FRAXK & QUIST, 1961; FRANK, 1970) that the intcrstitial water molecules are frecly rotat.ing and han• rotational s tates which effccth·ely make them inr rt and nonintcracting, but thc argumcnt has not b cen developed in de tail.

(b) Another problem is the necd for expans ion, i.c. strctching of thc lattice to accomodate tbe intcr slitial molcculcs. It is truc that som e of

A nn. hl. Suver So111lil (1970 > 6, 691-:.11~

553

the encrgy r ccp1ired for trus is r cgaincd through the new non-honding inter­actions formcd and that s uch strained s tructure;; cxist in som e of the hi~h pressurr, i. e. constrained ire polymorphs, hut why ~honld t hey furm in the liquid if stra in-frce s tructurcs are poss iblc as well ?

(c) The thcrmody uamic propertics indicate that the rnodels proposcd so far are rather solid-Iikc. This shortcoming coulcl b1} relievc·1l by the introduc tion of a third componcnt (FnANK & QCIST, 1961), but then tbc advantage claimed by thc proporwnts of the model is lost: t he mode! bccomes like thc othl' r mixture modl'!s in that longcr-rangc inhomogcncities a re introduccd. (d) Without such a third co.mponcnt, long-rangc order ought to be sccn.

VII. .\QUEOCS SOLUTlO.\"S OF NO.\'PO LAR :\lOLEC Uf.ES

\. ExpPrimrntal Data.

Thc t lwrmodynamic pararnNcr s of soluti tHl of nonpolar sub:stan ccs 111 watcr arr unu:sual, a;. imlicatcd ca rlicr (Tablc II-:~). The low solubility ( corrc•:,ponding to d G~ > O is eauscd by thc highly unfavourable cntropy (.1 s~ ·~ O) rather Lhan by an unfavotuablc cncrgy balance <IS is thc reason for luw solubility in most othcr solvcnts. Actually, tbc l'ntha lpy is v~:ry dose to zero, and may be cven nega t ive, such as fo r srnaU alipha tic hydroearhons at low templ'ratun·~. A compilation of t hermodynamic data from various sourCCH was givcn by :Nf:~rETHY & Sc u f.RAG.\. (1962b) . Frorn solubility data, us ing thc van' t TTofr f'~(llation. valu es ;,hown in T ablc YII-L

TAIILE VII-l

Thermodynamic 1mrarueters (a) for tbe solution of hydrocarhons in water !b) a l 25o

(Babed on various literature data ~ummari:ted by , Él'I ETIIY & SCIIERACA, 1962 b)

SOI.Un: l l •• n

l Il~· . --~ ----- - -~-- --

::\1ethunc .l f- :! . 5 1 l ti + 3 . J.) - 2.81i lO - •l ., .. -. _,., 18.-t. ICI- 16.8 Etbanc.

:l :\.3:! t o - 3 .86 - :!.3~ t o - l.:!i

l'ropanc 1..90 Lo .l.. -L!Jl - 2.09 IO - I . . ~.:;

Butunc. ,.. ,i. 82 t o 1\,00 - 11 .96 t o - fl.i2

19.5 to - 16.8 23.5 lO 21. 3

- 22.i t o - 21.9

Ben.t.rnc . . .

1

1.61 () .. i8 J:l. 5 T o lurnc ... :;_ ;n U , (l l 15.7 F.thylhrnzcnl' . t• .117 0.:\') IIJ .u

m-X~ l~nc h . ll Il . ~l 19 .l p- X) lrm) . ·l !> . l ::! Il. Hi 1\1 . 0

-------~--_l

(n) Un al • U"-rtl : J (;.: nnil .J H: ìn lt•lil mol. l 5 ~in q)J.\ )r·:;·mol.

(b) Vn1uf' .. <ih''"" are the (( f''(Ce~~ ,, fuoc1i0111 for the tran .. ft- r from the li•tuid 'i tale o r u di Iute iolution in a

nonpolar ••Hth·f>ot into wnttr.

. l tllt. l st . S11pu. :)rmità (1970) 6, ~\H-r.9z

55 1

arr obtaincd for 25°. LI H: b ccom c ... less m ·gati' c "ith iucrca ~:> i n~ dtniu lt•ng th and al~o with incn •as ing t cJntll'ratun•. Jt b('come>- pcJ~-> iti n· n c:ar ·IO.

Tht• t·ntrop y hccorut>!- more uegali' e with incrca~ing dtain le n~-:th .

On thr otbcr hand its ab!'olutc ' 'aluf· dccr ea '-<'"• i.e. i t bt·come~:> morc JW ' ith t '

al higlwr l r mperalurc" (•). Thu:. llw free t•nergy h m• u lti~W ) positiH 'ulut· u l low trmpcra lun brcau'-t' of tlw largt· uufa\·orablc· entrop\·. A l h i~ lt

t emperature .. , Lht· fr et· ent·rg~· t:. e' e n mori' unfavorabl<'. but a n inc• r ca'- Ì0!-(1~

more important contribu tion com i'"' fw rn 1lw cntha lpy. whilt• lh l· rolc• nl' t !t e• c n Lropy d eert'a 'lt'S.

Sim ilar lrcnd'- of tb c· 'ariou ... tlwrmod) nn mie paruml'lt•r"' ar<· St't' ll for

MOmaLic• b ytl roearbon,.. Tlacr r un· M'' t• ral d ill'en·rwt•;o, in tlH· llllllH'r ic·td 'uluc•~:> (Table V 11 · 1 ). Tlu• aroma l i c· h ydro('arbom- are· m uch rnore solublt·

in watc·r. u::. indicatctl b~ tlw lo\•t•r frt'l' t' ll t'Tb'). Tbc• dilll'rcrwe b c•L\\t'l'n

thc LI G~ for hPxan l' an<l fur bt·uzt·nc• j :. aboul ~ .7 J.. cul , mol. rurr f'"' J'UIId in~

t o a 100-fold int· rNts c·. i11 soluhility.'' With thc addi tion of alkyl groupfò ,

..1 G~ b t•come,. nwn· p o'liti n·, ju ~t liJ.. t• the ulipbatic ~~~ droc·arholl '-. I 111 rrt·~ · tingi) . tht• j H , ,·a lm·s are notit•('ahly roure• po~<i l j, c•. ami do noi 'ar~

much for tlifre re nt cleri,·a l j, r s of l~t~ uzene. For se· ' ('rul dc•ri\'11 1 i\ c" and for

Lcn z('D(' i t~.rU', . 1 H. = n at I8 °C. Com .. f•qucntl~. thl' !'Olubilit~ h c·h:l\' Ìtlur i ~ <>till d ominalt·d h ) lht• t•ntropy, althou~h llw aOHllut t· \alut• of thr la llrr

is les!'i tba n for t ht• aliphutit,.

Tht· pari i al m olai Ili' a l rnpat·i 1 i rs, , 1 t 1, ~ of tlw l'olulf• ha v c lur~c·

posi l h<' vulue:.: about aO·hU t • al d q~· mol fur nwthanc• u11d l'tha11c·. irwrt·a ... ing

to 80 fM hutanc. an d n r ar '70 caJ ti t·~· mol for th<' aroma l i t· h) dnwarhon:.: . For ali thc parawrlc'r'- .. r('gu lar rorn· latiurt-. t'Oli he l'~ lnhli :;h c·d \\il h

t b e eh ai11 lcng th . with Lht· cxcf'ption of m etharw. For 1 hc lall!'r . Loth 1 H .. aud LI SZ are mon· uegati \ e than expected from tht• t rene],. ~btJ\\' 11 lJ) t h!' highcr homolog..... 1)rt'l' Ullt Ci hly . thi s j ., dtH' tCI difl'erc·lll .- 1 rudural fra l un· ,_

in the \\ alt•r surruundiug thc• uwthanc m olc·cul!'. bt•t·au't' of iL>. ~>mali si:~.,·.

imilar rdationship;, an• St'C'll for olht•r organic· cumpouniJ ..., contni n ing h)rdroca rbon chains, a,Q a functiou of tlH' l'ltaiu lr ng th . Additi\ il~ uf tb('

thcrmodynamic l'fl'l•rt :-; of n pular group ancl of tb(• hydrocarbun dtnin

attached to it oftcn secm ~ t o bt· \'Cr ified closwl). Sonwtimt''- i t hoJr] , only after th!' firs t o ne or two CJ [2 - groups lll'!lrc.'ì l tlt t• polar grnup han· l.w t' n

subtract ed.

Tbc ' olume chang•· of mixin~ of h ) òrocarbou -. with "ut<·r. t J' ~ . i;; ncgativl~. Tht• absolute valucs of LI V~ nn· higb r r fu r tht· alipha ti c· h ) dro·

carbon~ t han for thr aro ma l ics (M A"TERTO'\, 195 1) .

(• ) The data ~ho" n are 11 1' \.CI'Ss cntropics» i.e. correctcd fur the cutrup~ ideai d ilu tion ( Chap. Il. C).

l nn l t ! SUJh r. 8<J IItlll l9ì01 6, 4\11 !.02