Datta, R

Sengupta, B K

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Indian J. Phy0. 62-68 (1978)On some correction on free energy of an assembly of charged paticles due to hard core potentialB . K .  8ENGUPTARaja Pmrif Mohan College, UttarjMra, Hooghly ANDR .  D a t t aCentre of Advanced Study in Applied Mathematics,92, Acharya Prafulla Chandra Road, Calculta-700 009{Received 27 April 1977, revised 22 Amjust 1977)Tlio pax)cr an invostigatiuii of tho phymeal proporfcios of an assombly of cHargod particlos in a continuous dioloctric modium. Diitta’s distri­bution forniuJa (Dutta, 1952, 1953) is usotl to rox>laco tho Boltzmann formula in order to tako into account tho effect of hard ct>re i)otoutial of tho i)articles. The resulting non-lineai* Poisson equation is solved by the luethod of Onsagor and Fuoss (1901) and tho free energy and other functions associated witli it are obtaiiuwl. For exx»orimental verilication, tho results are ax)pliod to tho strojig oloctrol>i;os in dilute solution.1. I n t r o d u c t i o nWo shall consider an assembly of charged particles in a continuous modium whore tho effect of continuous medium is taken into account, through tho dioloctric constant. From the Avork of Dutta (1952, 1953), Bogolubov (1949), Songupta (1954, 1950) and others it is well known that tho long ranges Coulomb forces as Avell as short range forces are important in tliis system. One simple type of potential tor tlm short-range forces is the hard core potential.Distribution fonnula of Dutta (1952, 1953) AAluch takes into account of tho hard core 2>otontial i.e. the exclusion volumes in configuration sx)aco of j)articles, can bo ax)j)Iiod to any system of charged particles. In 1961, Onsagor and Fuoss considered the above system in dielectric medium and obtained tlio free energy and other tliermod i^iamic quantities associated A\itli it using tho Boltzmann distribution formula which does not tako into account the hard core potential. We sl^ all take into account the effect of tho luvrd core potential in tho said Onsagor and Fuoss {I9(il) theory on replacing the Boltzmann distribution by Diitta’s distribution (J951, 1952) formula for chargtKl i)articies in dielectric modium. Tim results Jiave been apxdknl to the case of strong electrolytes in solution wliich is also an example of an assembly of chaigiKl pai-ticles in continuous modium. The agreement wil^ i experimental residt is mucli better.Mathematical CalculationsFree energy o f charged particle assembly 63Horo wo tako an aHSombly of ions in a continuous inoflium witli dioloctrio constant D. The ions arc supposod to bo produced by coinploto dissociation of an acid or a salt in solution. Tho distribution of ions in the solution is given by Dutta’s formula (1952, 1953)1b ±(1)whoro ™ number densities of positive and negative ions at distancefrom a reference ion and at that position potential is }/rr;=  exclusion volumes of two like ions;=  valejicy of positive or negatives ions;-  parametc^rs of distribution; fc, T have their usual significances.For simplicity, wo assunu  ^1 |  “  1 aed b^  -^b (say), so tliat Vj. —’ — v(say). When 0, the distribution is, on th(^  average an uniform one, so thatthat wlw>ro 7i average niunlxu’ of ions p>r unit volume. Tlum thenbmodified Poisson equation takes the form„ , 4716 1 1___XI { 1 I ^ 1 -fi1l-H V—e t^kT(2)Wo shall rostriot oursolvos to tlio case wlioro tlio avorago concontration of ions por unit volnmo is small so that wo (mn tako * „ < < ! .  Noglocting highor ordor torms o f bn, tho oq. (2) Ixicomos4npwhoro p — —2e«.(l—6w) singh | l—26» (cosh |(3)(3a)is the avorago charge density at the distance r.Trocooding along the lino of Onsager and Fuoss (1901) wo sot, for sliortdistances, i.e. for r <  “  where /? =(4)64 B. K. iSwgupta and B< DuttaIn oq. (4) ^  is tho potential at a distance r from an isolated ion and f(r) is theDrperturbation in tho Coulomb potential produced by neighbouring ions. From oqs. (4) and (S) one obtainsV**(r) =  - 5 ?  il-hn) ainh { ±  }x {l-2 6 « (o o sh ^  ( F r+ * )-!)}• (6)We approximate 1. Duo to spherical symmetry cq. (5) reduces toIntegrating eq. (6) and substituting tho result in oq. (4), we get=  i f  +S+{l+2bn)Fi{r)-bn  J*(r)| (7)whore Kj® =  n6) =  k*(1—ni),A and i? are constants andW  =  .> " ( t ; - + | - ? ) + ( ^ - A ) a (A )(8)where E{{x) is the positive exponential integralX piuE((x) =  J du./?... (9)For distance  ^ ^ , the long range potential is obtained by integrating theequationr72*r, ^2r . i^r^ n 1—76n - « ... (10)which is deduced from eq.(3) retaining up to cubic terms in }fri; this is justifiedTby the fact that 0 as —> oo, a =s ionic radius.Wo got with tho boundary condition | ~ ^Free ewrgy of charged particle, ataembly esfi .  =  c. , STre^ w. I-Urnr 'where E „ =  J du. ... (12)The integration constants A, B, C aro ovalnatofl by the following conditions;—c — /  inpiMr (13a)and/dyrg \ _ l z z k - \\ dr /i!_o  \ dr /^ - o ’From eqs. (3a) and (13a) wo have== D f  r^ TjhJrdr -T)a^wlionco wo obtainr-^ ) 7>»* •Differentiating oq. (7) and substituting in oq. (15),A =Pwhereandm )  =  (l+ 26») [ -^ 2  (2)+F+F*— 6 •[ 0 2  (H -3p-2i»*)— 11»U,(2))]e* tf^  “  aDhT ~  am(13o)(14)(15)(16)(17)66 B .  K .  S e n g u p t a  a n d  R .  D u t t aEiiminating C botwoon oqa. nsh) , mo i , , ,O'!. (16) wo got  ^ substituting tJio valuo of ^  from^ - ( I H  r) >[_  ,il+2bn)(l+T)Fi j+Hl+T)J’3^|.j _2t Jm )whoroandT~2Now tho potontial ,//• at tho rofrronco i(18)(19)(20)1011 duo to tho prosonco of othor i:ions jsi;* -  fr.(a )- " , Da(7).,16) .'  ~iD ( 4 ^  (l+2fc,){ 4 '.  . ( 3 + ? + ^ = ) - J*’'{sp= 1 —~ ,1  .(«5^+S3*,,2,,+ (l+ 2 bn)Fj(a)~6„j!' (^a)j.(21)Tho froo olootrio onorgy ISGe =  i il^ »lly  tho roJation JkTbi/y =  i ? -(22)(23)(24)Pree enefgy o f charged particle assenddy 6 7gives for the activity coofiiciont), ... ,25)The above oq. (25) is similar to tlu> oq. (34) of Fuoss ami OiiKagoi' (l»(iJ), viz.(r' is writtoii lor t in On»sagor’s cquaiioTi only to distingiuHli it Iroin our r)Mhoro r — f'\/]^l)u (27)Thus in our resiult, tUo voluiuo corrootion i.c. tlio offcct of hard coro poioniial in introduced by the uho of Dutta’s (1952, 1953) distribution formula aiid tlio result reduces to that o f Onsagor & Fuoss (1961) wl o^n the volume correction is neglecstod.3. N umkrjoal K ksultsFor Verification Avitli <ixporimental data, \u\ consider the (uisti of strong oltn;- trojytes in dilute solution. For low concentrations, the (Xpuition (2»5) and (2<i) for activity cocfllcicnts bwome (retaining up to the leading terms in r)(28 )(29)The following gives a comparative study of the mimerical values obtained from tho present work and that of Onsager and Fuoss (19(U) with, tb,o data of observed results (Harned and Owen, 1943).The result (29) of Onsagor and I^ 'uoks (1961) does not contain any parajiieter. Our result (28) contains tho parameter h whicJi is the exclusion volume of two like ions and introduced tlxrougli the distribution iormula of Dutta (1952, 1953). This is related to the mean distance of closest approach a by the relation hna’*In tho tablef  ^  activity coefficient as given by ilio work of Onsagor and Fuoss (1961). /  — activity coefficient as given by the present work.obs =  observed results.Hei-e we have taken D == 78.6, T =  ajid the standard values of k and c.m B. K. Sengnpia and R. BottaNaCl a 4 .4 l KCl a =  4.iA HCl a =  4.eA/ ' obs / ' obs.001 .96486 .96487.005 .92826 .92636 .9283.01 .90033 .90066 .9031.02 .86737 .86797 .8726.06 .81237 .81446 .8221.1 .76296 .76804 .778.2 .70890 .72097 .732.6 .63663 .67190 .678.96486 .96486.92626 .92632,90033 .90049,86737 .86780.81237 .81384.76295 .769.70890 .71743.63563 .66107.927.901.815.769.719.651.96486 .06487.92626 .92634,90033 .90033.86737 .8679.81237 .81419•76295 .76740.70890 .71946.63553 .66553.966.929.905.875.830.706.767.7574. COKOLTTSlOHVor coiiiparitioii, wo have given in tho above table the valued of activity coefficieniH of aovoral 1—1 electrolytes in very dilute solution as given by our present work, the corresponding results of Onsager ajul ¥uoss (19bl) and tho ob­served values (Harned and Owen, 1943). Better agrwmeiit as comparted to tlie results o f Onsager and Fuoss (1961) are obtained. The above result as well as tile preceoding analysis sliows that tho modified distribution formula coupled with tho above method of solving Poisson's aquation can bo fruitfully utilised for any system of charged particles.6. AOKlsrOWLBTOllBmsThe authors are very grateful to Professor M. Dutta, Professor-in-charge, 8. N. Bose Institute of Physical Sciences, a»nd Professor, Centre of Advanced Study in Applied Mathematics, C.U. for his kind help and valuable suggestions in preparing this paper and also wish to thank I>r. M. Sengupfta, Reader, Depart­ment of Chemistry, C.U. for useful discussion. They also wish to thank tho referee for his valuable suggestions.RiraBBNcnsBogoliubov N. N. 1943 Studies in Statistical Mechanics Vol. 1, North Holland Publishing Com­pany Amsterdam.Dutta M. 1952 Naturwiss, 39, 108.---------  1953 Natunciss 40, 61.Dutta M. Songupta M. 1964 iVoc. Nat* Inst, Sc, Ind, 20, 1.Fuosa M. ^  Onsager L, 1961 Proc. Nat, Acad, Sciences, 47, 818.Haniod H. S. & Owen B. B. 1943 T/mj Physical Chemistry qf Kkctrclytic sdutums Keinliold Publishing Corporation, U.S.A.Songupta M. 1966 Proc, Nat, Itist, Sci, Ind. 22, 13.

On some Correction on Free Energy of an Assembly of Charged Particles Due to Hard Core Potential

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On some Correction on Free Energy of an Assembly of Charged Particles Due to Hard Core Potential

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