Kalia, Anjan Kumar

Sharon, M.

English

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--" _ ..--- ...._- ---------~- -_ ..._,.~.-----_ ..._-~"-~-----_.__ .- - --~..~_.-. __ ..,.~NOTES1Fig. 1- Test of instability for quartz/methanol systemFig. 2 - Test of instability for quartz/methyl ethyl ketonesystemA GeneralIsothermEquationfor aMonovalentIonic CrystalContainingImpurityin +nValencyStateM. SHARON & AN]AN KUMAR KALlADepartment of Chemistry, University of Pune, Pune 411007Received7 June 1976;revised1 Novernber1976accepted22 January 19771. PRIGOGINE, 1. & GLANSDORFF, P., Structure,stabilityandfluctuations(John Wiley, New York), 1971.2. NICOLlS, G., Adv. chern.Phys., 19 (1971),209.3. DAS, S. S., Indian]. Chern.,12 (1974),897.4. RASTOGI, R. p., SINGH, KEHAR & SRIVASTAVA, M. L.,J. phys. Chern.,73 (1969),46-51.5. RASTOGI, R. P., SRIVASTAVA, M. L. & SINGH, S. N.,J. phys. Chern.,74 (1970),2960.6. SRIVASTAVA, R. C. & AWASTHI, P. K, J. Hydrology,20 (1973),37-47.7. RASTOGI, R. P., SINGH, KEHAR & SINGH, S. N., J. phys.Chern.,73 (1969),1593.8. SINGH, S. N., Ph.D. thesis, Gorakhpur University, 1969.9. LORENZ, P. B., J. phys. Chern.,57 (1953),430.10. GORBACHUK, I. T., DUSHCHENK, V. P. & IVANILSKII,B. G., Teplofiz.Teplolekh.,24(1973), 89-93.11. HADERMANN, A. F., WATERS, P. F. & Woo, J. W.,J. phys. Chern.,78 (1974),1.ReferencesPolarizationat electrodes9,10anddissociationfieldeffectsll,whichofferseriouslimitationsfor experi-mentalstudiesat highervoltagesmay modifytheactual form of transport[Eq. (3)]. However,thetheoreticalapproachfor predictinginstabilitywillremainthe same. Further, the conceptof localequilibriumhasbeenfoundto be universallyvalid,sincea thermodynamictheoryof stabilityof steadystatesin non-linearregionsor far fromequilibriumsituationshas not yet beendevelopedl,wheretheconceptof local equilibriumis not valid. Thusthe resultsreportedin the presentnote are fully3000 justified.The authorsare highly thankfulto Prof..R. P.Rastogi for suggestingthe problemand to DrK. D. S. Yadavafor valuablesuggestions.500 1000 1500A cp (Volts)-0-0- ,AP=-0·77 ~104dyne em2,*-*,AP=-3 87X 104dyne em-2*-*, 6P:7.73 Xl04dyne cm-24 -2...•....•., AP=ll'60 Xlj.O dyne emo-10·0020·00-20,00-10,0020·00•...oX~+'dr--o~ 10,00~'dFig. 3_ Test of instability for quartz/acetone system10·00t-oXc::! a+'d-10'00500. 1000ACP(Volts}3000A ~eneralisothermequationhasbeenderivedforan ionic monovalentcrystaldopedwith cationof+n valencystate.A generalequationis derivedforcalculationof enthalpyof migrationof point defectfromtheenthalpyof associationof [impurityof +nvalentstate- (n-1 ) vacancies]complexes.LIDIARDI has derivedan isothermequationtocalculatethe enthalpiesof migrationof pointdefects(Em), associationof impurity-vacancycom-plex (Ea) and formationof intrinsic defects(EF)for an ionic crystalcontainingknownamountof adivalentimpurity. For similarionicdopedcrystals,Dreyfus2has derivedEq. (1),EI =Em+O·SEa ... (1)whereEI is the observedenthalpyof conductionobtainedfrom the extrinsicregionof conductivityplot. Both thesederivationsneeda modificationfor a monovalentioniccrystalcontainingdopantofvalencygreaterthan two. The presentnoteis an549INDIAN J. CHEM., VOL. 15A, JUNE 1977... (3)... (9)attem t to derive a general isotherm equation forcrysta s doped with impurity of +nvalent state.Wh n a monovalent ionic crystal is doped with adivale t impurity, it is assumed that (a) eachaliova nt impurity atom introduces only one defectand ( ) due to electrostatic forces which existbetwe n an aliovalent cation and a cation vacancy,the e ergy of vacancy is lowest when it occupiesone 0 the nearest neighbour cation sites to theimpur ty. Such impurity-vacancy pairs are calledassoci ted pairs or complexes. At low temperaturethese complexes do not contribute to the con-ducta ce, but at higher temperaturesome of thesecompl xes get dissociatedto contribute towards thecondu tion. Therefore, it is assumed that eachcation with +n valency ",,"ouldproduce (n-l)numb r of vacancies,thus producing[impurity-(n-l)vacan iesJ complexes. If c be the total concentra-tion f a dopant of +n valency and p be thedegre of associationof [impurity-(n-l) vacancies]compl xes, and if there exists at any temperaturea dy amic equilibrium between the associatedvacan ies and unassociatedvacancies,then from thisequili rium the number of the vacancies can becalcul ted by applying the law of mass actionto th quasi-chemicalreaction:Un ssociated+n-valentimpurity ions+unassociat-ed vacancies~ [impurity-(n--1 ) vacancy] com-pIe esorc(1-PH-(n-1)c(1-P) ~ pcAcdording to the law of mass action,pcK"(T)~.=c(I-P){c(I---PW-1 ... (2)If . k represents the concentration of im.puritycompex (i.e. pc) then Eq. (2) reduces to Eq. (3).Xk-(C-Xk)"Th massaction constant K,,(T) can be expressed1as E . (4).K,,(T =Zl exp(-Sa(k) exp(Ea/kT) ... (4)Zl is a constant which depends upon the crystalstruc ure, and Sa is the entropy term for theassocation process. If Xl and X2 are the molarfract ns of two complementary intrinsic defects,then one can derive1the Eq. (5).X1X2 - x~=ki1=exp(Ss/k) exp(-E,/kT) ... (5)The resenceof unassociatednth valent impuritygives an additional defect (n-l)(c-xk) which issimil r in nature to that of first type (i.e., Xl type)of c mplementary defects in the intrinsic region(Xl x2). Hence the charge neutrality equationcan e written as:Xl = (n-l)(c-xk)+x2 ... (6)wher molar fraction X2 is written for the molarfract on of complementaryXl intrinsic defects (i.e.,X1X2 ;\:~). Substitution of Xk from Eq. (6) inEq. 3) givesc =K (T) X1-X2[ __1_ +(X1-X2)"-1] ... (7)" n-l K,,(T) (n-l)"l-550Dividing Eq. (7)throughoutby %0and substitutingXO/X1=%2/XO=E1_ [ 1] -1 [ k1--(,,-ll/2K,,(T)c/xo - --- (E1--E1) 1+~----(n-1) . (n-1)"-1(E1-Ei1),,-1] ... (8)If Ei1 is very small then,1 [ k -(,,-1)/2K (T) ]c/xo=--E1 1+_1--- --,,- et-1(n~l) n~l)"-lThe relative conductivity1 is shown by Eq. (10),...La -1~ =c:~~~ •••(10)0'0 1+awhere 0 =[1.2![1.1'[1.1 and [1.2 are the mobilities of two complementarydefects. O'c and 0'0 are respectively the specificconductance of doped crystal with concentration cand the specificconductanceof ideally p~re.crystal.Neglecting the value of E1] and substItutmg theE1 values from Eq. (10) into Eq. (9) gives Eq. (11).C/(O'~)= _1_ xo(1+6) +xo k1-(,,-1{t ~,,(T) (1+6)n0'0 n-l (n-l)( ),,-1X :: ... (11)This is a conductivity isotherm equation for acrystal containing +n valent dopant. This is annth order polynomial whose nature can only bepredicted if graph is plotted from a large numberof c versus O'c/O'ovalues. When n =2 the Eq. (11)reduces to Lidiard1 isotherm equation (Eq. 12).c/(O'c!O'o)=Xo(l+8)+Xok1-0.5K2(T)(H--8)2(::) .,,(12)Now a plot of c/(O'c/O'o)versus (O'c/0'0)"-1shouldbelinear with a slope ofXO"l-["-l/! K,,(T)-- ..------ (1+8)"(n-l)"and an intercept of (n~I)Xo(1+8).K,,(T) and Xo can be calculated for differenttemperaturesfrom the slopes and intercepts of theisotherms of different temperatures. The values ofenthalpy for the association of [impurity-(n-l)vacancies]complexes (Ea) and the enthalpy of theformation of point defects (E]o') can then be calculat-ed by using the Eqs. (4) and (5) respectively.Generalequationfor theenthalpyfor associationofcomPlexes-- In order to calculate the enthalpyfor the migration of point defectsfor an ionic crystalcontaining a cation dopant of +nvalency Eq. (1)can be modified as follows:In Eq. (3), (n-l)(c-xk) representsthe number ofunassociatedvacancies (the number of defects N),which are responsible for the conduction. Hence(n-l)(c-xk) can be replaced by N, provided thenumberof vacanciescreated{(n-l)(c-xk} are muchgreater than the number of intrinsic defects (xo)present in the association region.NOTES!:.HsolnAgX (s) ----------.-~~ Ag+(aq)+X'(aq)IIIAg+(g)+X- (g)Fig. 1- Modesof dissolutionof solid silver halide in excesssolvent(A)minedfromthe temperature-dependenceof the cellemf,in thepresenceof excessof AgX in the melt.The detailshavebeenreportedearlier3•4.In a saturatedsolutionof a salt, the freeenergychange(AG~oln) for the transferof ,one mole ofsalt from solid to the solution in standardstateis givenby~G~oln=- RT loge KsThecorrespondingenthalpychange,AH~oln is equalto theheatof solution. Table1givesthecalculatedvaluesof AG~olnand AH~oln for the dissolutionofsilverhalidesin variousionicmelts. For comparison,correspondingvaluesin waterS(at25°)areincluded.It is interestingto note that the thermodynamicdatain the two typesof mediaare of the sameorder,althoughthereexistsa fundamentaldifferencein the processof dissolutionin ionic meltsand inwater. The processof dissolutionof solid silverhalide in excessof solventmay be visualizedtotake placethrougheither of the two pathways,shownin Fig. 1. Path I, involvingthe dissolutionof solid salt as simplemolecularspecies,givestheheat of solution (tiHsoln). Path II consists offormationof gaseousions from the solid AgX andtheirsubsequentdissolution;aprocesswhichdependslargelyon the natureof solvent. Hence- UAgX =(tlHso1n)AgX-(AHso1v)AgX ••.(1)The lattice energiesat 350°for silverhalideswerecomputedthroughtheconsiderationsof Born-Haberthermochemicalcycle,which relateslattice energyto the measurablethermaldata,-UAgX =QAgX +SAg +hg +!Dx -Exwhere UAgX=lattice energyof the salt AgX;QAgX=heat of formationof the salt from itselementsin their standardstates; SAg =heat ofsublimationof silvermetal;JAg =ionizationpoten-tial of Ag; Ex =electronaffinityfor the halogen;iDx =heatof dissociationof gaseoushalogenintoatoms.Availabledata6at 250havebeencorrectedfor thetemperatureincreaseto 350°. The valuesfor theionizationpotentialof silverandelectronaffinityforhalogensat 25° have not beencorrectedfor theAgNOg(M)+MXin lAgmolten salt solvent(sat. with AgX)(+)(Indicator Half-cell)/// !:.Hsolv//AgNOg (M)Ag I inmoltensalt solvent(--)(ReferenceHalf-cell)Thefreeenergiesofsolutionandsolvationfor silverhalidesin moltennitratesandperchlorateshavebeencalculatedfrom potentiometricdata. The solvationprocessappearsto beinfluencedbytheanionof suchionic solvents.MOST of the thermodynamicpropertiesof thesoluteareverysensitiveto thesolvationeffects.In aqueoussolutions,the ion is consideredto besurroundedby threeconcentricregions1.Contraryto this, in ionic melts,the processof solvationissomewhatdifficult to visu.alize,becauseof theabsenceof polarsolventmoleculesandpurelyionicnatureof thesolvent. FlengasandRideal2calculat-ed the enthalpyand free energyof solvationforsilver halides in molten NaNOa-KNOa. In thiscommunicationan efforthasbeenmadeto correlatethe enthalpyandfreeenergyof solvation,with thenatureof the solvent.The solubilityproductconstants(Ks) for silverhalidesat 623K weredeterminedby measuringtheemf of the cell (A) at variousconcentrationsofhalideion. Heat of solution (AHsoln) was deter-*Present address: Hindn College, University of Delhi,Delhi 110007.fPresent address:Allen Clark Research Centre, Caswell,Towcester,Northants (England).S. K. JAIN* & R. S. SETHItDepartmentof Chemistry,University of Delhi, Delhi 110007Received30 September1976;accepted28 January 1977Free Energiesof Solvationof SilverHalidesin Ionic MeltsCombiningEqs. (3) and (4) and substitutingNfor (n-l)(c-xk)'Xk--- Z1exp(-Sa/k) exp(Ea/kT) ... (13)Substitutingthe value of N in the generalcon-ductivityequation,i.e. 11 =eN[L, onegets,11=e[L(n-l)(xk<1-1)1/n exp(Sa/nk) exp(-Ea/nkT)where[L = (a2ev6)/kTexp(Sm/k) exp(-Em/kT).HenceI1T = [(a2e2v~)/kJ(n-l)(xkZ'-1)1/nxexp(Sm+l/nSa) exp[-(Em +l/nEa)/TkJ ... (16)Eq. (16)showsthat the enthalpyvalueobtainedfromtheslopeof linearplot betweenlog crT versus(liT) of the extrinsicregionshouldbe givenby,E1=Em+ linEa ...(17)Whenn =2, Eq. (17) takesthe formof Eq. (1).The Eqs. (11) and (17) areof generalformandareapplicableto a dopant in any oxidationstate.Further work to confirm the validity of theseequationsshallbe publishedshortly.One of us (A.A.K.) wishesto thank the UGC,New Delhi, for financialassistance.References1. LIDIARD, A. B., Handbuchder Physik (Springer-Verlag,Berlin), 1957,20; 246.2. DREYFUS,R. W. & NOWICK,A. S., Phys. Rev.,126(1962),1367.551______c__--.-,-_.-- ---.J

A General Isotherm Equation for a Monovalent Ionic Crystal Containing Impurity in +n Valency State

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A General Isotherm Equation for a Monovalent Ionic Crystal Containing Impurity in +n Valency State

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