Electrolyte Solutions: Thermodynamics, Crystallization, Separation methods

Transcription

1 Electrolyte Solutons: Therodynacs, Crystallzaton, Separaton ethods 9 Kaj Thosen, Assocate Professor, DTU Checal Engneerng, Techncal Unversty of Denark 1

2 Lst of contents 1 INTRODUCTION... 5 CONCENTRATION UNITS IDEAL SOLUTIONS DEFINITION 9 COLLIGATIVE PROPERTIES 13 4 CHEMICAL POTENTIAL AND ACTIVITY COEFFICIENTS CHEMICAL POTENTIAL EXCESS CHEMICAL POTENTIALS FOR REAL SOLUTIONS THE RATIONAL, UNSYMMETRICAL ACTIVITY COEFFICIENT THE MOLALITY ACTIVITY COEFFICIENT THE MOLARITY ACTIVITY COEFFICIENT THE ACTIVITY OF SPECIES 4.7 CHEMICAL POTENTIAL OF A SALT 5 MEASUREMENT OF CHEMICAL POTENTIALS IN SALT SOLUTIONS MEASUREMENT OF THE CHEMICAL POTENTIALS OF IONS 3 5. THE NERNST EQUATION THE HARNED CELL MEASUREMENT OF SOLVENT ACTIVITY FREEZING POINT DEPRESSION AND BOILING POINT ELEVATION MEASUREMENTS VAPOR PRESSURE METHODS ISOPIESTIC MEASUREMENTS OSMOTIC COEFFICIENT THE VALUE OF THE OSMOTIC COEFFICIENT AT INFINITE DILUTION MEAN ACTIVITY COEFFICIENT FROM OSMOTIC COEFFICIENT OSMOTIC PRESSURE 33 6 THERMODYNAMIC MODELS FOR ELECTROLYTE SOLUTIONS ELECTROSTATIC INTERACTIONS DEBYE-HÜCKEL THEORY DEBYE-HÜCKEL EXTENDED LAW DEBYE-HÜCKEL LIMITING LAW THE HÜCKEL EQUATION THE BORN EQUATION THE MEAN SPHERICAL APPROXIMATION EMPIRICAL MODELS FOR INTERMEDIATE/SHORT RANGE INTERACTIONS THE MEISSNER CORRELATION BROMLEY S METHOD THE PITZER METHOD... 5

3 6.3 INTERMEDIATE/SHORT RANGE INTERACTIONS FROM LOCAL COMPOSITION MODELS THE EXTENDED UNIQUAC MODEL THE ELECTROLYTE NRTL MODEL INTERMEDIATE/SHORT RANGE INTERACTIONS FROM EQUATIONS OF STATE FUGACITY COEFFICIENTS AND ACTIVITY COEFFICIENTS THE FÜRST AND RENON EQUATION OF STATE THE WU AND PRAUSNITZ EQUATION OF STATE THE MYERS-SANDLER-WOOD EQUATION OF STATE COMPARATIVE STUDY OF EQUATIONS OF STATE EQUILIBRIUM CALCULATIONS SPECIATION EQUILIBRIUM SOLID-LIQUID EQUILIBRIUM SATURATION INDEX VAPOR-LIQUID EQUILIBRIUM HENRY S CONSTANT LIQUID-LIQUID EQUILIBRIUM COMPOSITION DEPENDENCE OF EQUILIBRIUM CONSTANTS TEMPERATURE DEPENDENCE OF EQUILIBRIUM CONSTANTS PRESSURE DEPENDENCE OF EQUILIBRIUM CONSTANTS THE PRESSURE DEPENDENCE OF ACTIVITY COEFFICIENTS THERMAL AND VOLUMETRIC PROPERTIES PARTIAL AND APPARENT MOLAR PROPERTIES THERMAL PROPERTIES HEAT OF DILUTION HEAT OF SOLUTION MEASUREMENT OF HEATS OF DILUTION AND SOLUTION MEASUREMENT OF HEAT CAPACITY VOLUMETRIC PROPERTIES 84 9 PHASE DIAGRAMS PHASE RULE AND INVARIANT POINTS 9 9. BINARY PHASE DIAGRAM TERNARY PHASE DIAGRAM QUATERNARY SYSTEMS 94 1 CRYSTALLIZATION SUPERSATURATION THE KELVIN EQUATION FOR NUCLEATION ACTIVATION ENERGY FOR CRYSTAL FORMATION PRIMARY NUCLEATION RATE FRACTIONAL CRYSTALLIZATION

4 11.1 PRODUCTION OF KNO OPTIMIZATION OF FRACTIONAL CRYSTALLIZATION PROCESSES SIMULATION OF K SO 4 PRODUCTION PROCESS 18 4

5 1 Introducton Phase equlbra wth systes contanng electrolytes are of great portance. A few exaples ay llustrate ths: Producton of fertlzers and salts s often perfored by precptaton of pure solds fro ult coponent onc solutons. Scalng n heat exchangers s caused by soe salts for whch the solublty decrease wth ncreasng teperature. Scalng n ol producton and n geotheral heat producton s caused by soe salts for whch the solublty decrease wth decreasng pressure and decreasng teperature. Solublty of gases n electrolyte solutons s of portance n any polluton abateent processes. The nfluence of salts on the vapor pressure of aqueous solutons of organc ateral ay be portant for the proper choce of a separaton process. Salts ay even ntroduce a lqudlqud phase splttng n aqueous solutons of organc substances. Electrolytes dssocate nto ons when they are dssolved n polar solvents lke water or alcohols. A strong electrolyte wll dssocate copletely whle a weak electrolyte wll only dssocate partly. The presence of the charged ons causes the electrolyte soluton to devate uch ore fro deal soluton behavor than a non-electrolyte soluton does. Ths s the case even at very low electrolyte concentratons. The reason s that the ons nteract wth electrostatc forces whch are of uch longer range than those nvolved n the nteracton of neutral olecules. Ths effect s stronger the greater the charge on the ons. For a proper descrpton of electrolyte solutons not only the short range energetc nteractons but also the long range electrostatc nteractons have to be consdered. Another basc dfference between electrolyte and non-electrolyte solutons s the constrant of electroneutralty on electrolyte solutons. Because of ths constrant, a syste consstng of water and two ons s a bnary syste: The concentratons of the two ons cannot be chosen ndependently so the syste has two ndependent coponents. Consder salt S that dssocates nto ν C catons C and ν A anons A wth onc charges Z C and Z A. ZC Z A S = νcc + ν AA (1.1) Table 1.1 Salt dssocaton S C Z C v C A Z A v A NaCl Na Cl Na SO 4 Na SO 4-1 CaCl Ca + 1 Cl - -1 The electro-neutralty requreent gves for salt S: ν CZC + ν AZ A = (1.) In general the electro-neutralty of a ult coponent soluton contanng n oles of on wth the charge Z relatve to a hydrogen on can be expressed as: nz = (1.3) Sodu chlorde s often descrbed as a 1-1 salt, sodu sulfate as a 1- salt, calcu chlorde as a -1 salt, and calcu sulfate as a - salt, based on the values of the onc charges. 5

6 Concentraton unts In the followng, t s assued that the electrolytes are dssolved n water. Water s consdered to be the only solvent, electrolytes and non-electrolytes are consdered to be solutes. In a soluton contanng water, salts, and ethanol, water s consdered to be the solvent. The ons and ethanol are consdered to be solutes. As t wll be shown, ths approach has sgnfcant advantages when perforng sold-lqud equlbru calculatons and lqud-lqud equlbru calculatons for solutons contanng non-electrolytes n addton to electrolytes. In another approach often seen n the lterature, xtures of water and certan non-electrolytes such as organc solvents are consdered to be separate speces, xed solvents or pseudo solvents, whle other non-electrolytes and electrolytes are consdered to be solutes. For the descrpton of electrolyte solutons the ost coon concentraton unt s the olalty. The olalty unt s very often used n the presentaton of experental data, whle the ole fracton unt ost often s used n therodynac odels for electrolytes. The olarty unt s also often used, but s dependent on teperature and to a certan extent also on pressure. It s not a practcal unt because the densty needs to be known n order to convert olarty unts to olalty unts or ole fracton unts. The olalty of an on s the nuber of oles n, of the on per kg water n the lqud phase: n = ol/kg water (.1) nm w w The aount of water n the soluton s here calculated as the product of n w, the nuber of oles of water and M w, the olar ass of water n kg/ol. The olarty c of an on s the nuber of oles of the on per lter soluton: n c = ol/lter soluton (.) Vsoluton The volue of the soluton, V soluton, s related to ts ass and ts densty: nm w w + nm ons dsoluton = (.3) Vsoluton If the densty of the soluton d soluton s gven n kg/lter and the olar asses of water and ons are gven n kg/ol, the volue of the soluton wll be calculated n lter. Fro equaton (.1) t can be seen that the olalty concentraton unt s only dependent on the aount of the relevant solute and the aount of solvent. The ole fracton unt and the olarty unts on the other hand are also dependent on the aount of other solutes present. In addton, the olarty unt s also dependent on teperature and pressure because the densty of the soluton depends on teperature and pressure. Exaple.1: A soluton contanng 6 ol of sodu chlorde and one kg of water s a 6 olal soluton of sodu chlorde. The olalty of the sodu on n the soluton s 6. The olar ass of water s 18. gra/ol, and the nuber of ol water n one kg water can therefore be calculated as 1/18.. The olalty of the chlorde on n the soluton s also 6. The ole fracton of the sodu on s 6

7 x + Na n + Na 6 = = =.889 n + n + n /18. + Na Cl HO The ole fracton of the chlorde on s also.889. The ole fracton of water s 1/18. x w = = /18. Soetes the coposton of an electrolyte soluton s gven n ters of the aounts of water and salts nstead of water and ons. If ths approach s used, the coposton of the sae soluton can be descrbed by the ole fracton of sodu chlorde: x NaCl = nnacl n + n = /18. = NaCl HO The ole fracton of water s =.95 when ths approach s used. The densty of ths soluton s kg/lter at 5 C and 1 bar. The volue of the soluton can be calculated fro equaton (.3): V soluton nm w w + n + M + + n M Na Na Cl Cl = = = 1.131lter The olarty of the sodu on, the chlorde on and of sodu chlorde n the soluton s: c c c 6 + = = NaCl = = 5.35 ol/lter Na Cl At 1 C and 1 bar, the densty of the sae sodu chlorde soluton s kg/lter. The volue of the soluton s lter and the olarty of sodu chlorde s 5.14 ol/lter. For a salt S the olalty s: n S S = (.4) n w Mw If the salt dssocates nto ν C catons and ν A anons the olalty of the caton C s: ν CnS C = = ν CS (.5) nm w w The olalty of the anon A s: ν n A S A = = ν AS (.6) nm w w S of the salt can therefore be expressed ether n ters of the caton olalty or The olalty n ters of the anon olalty: C = A S or S ν = C ν (.7) A Most therodynac odels use the tradtonal ole fracton scale: 7

8 x n = n j The suaton n the denonator s over all solute and solvent speces. A relaton between the olalty unt and the ole fracton unt can be derved as follows: n n n M w w x = = = x wmw nj nj nwmw M w s the olar ass of water gven n kg/ol. The olarty unt s related to the ole fracton unt by: n n Vsoluton cv soluton x = = = (.1) nj nj Vsoluton nj A large nuber of dfferent concentraton unts are used to present experental data for electrolytes. These nclude: 1. Mass percent. olalty 3. Mole fracton (water, ons, and non electrolytes) 4. Mole fracton (water, salts, and non electrolytes) 5. Mass of salt per ass of H O 6. Mass of salt per volue soluton 7. Mole salt per volue of soluton 8. Mole salt per ass of soluton 9. Jänecke coordnates (Charge fracton + gra H O per ole salts) 1. Mass percent solvent (salt free) + olalty of salt 11. Mass percent solvent(salt free) + ass of salt per ass of xed solvent 1. Mole percent solvent(salt free) + olalty salt One reason why olalty s a popular unt for salt solutons s that the concentratons n olalty unts gve practcal nubers, often between and for ost salts, whle the concentratons n ole fracton unts are very sall as ndcated n fgure.1: The fgures show the phase dagra of the ternary syste NaCl-KCl-H O usng these two dfferent concentraton unts. The lnes n the phase dagras ark concentratons saturated wth ether NaCl or KCl. At 5 C, the solublty of NaCl s 6.15 ol/kg water,.997 salt ole fracton, or 6.4 ass percent. The correspondng nubers for KCl are 4.79 ol/kg water,.795 salt ole fracton, or.7 ass percent. (.8) (.9) 8

9 6 T= 5. C.1 T= 5. C 5 a.8 a KCl olalty 4 3 b KCl ole fracton.6.4 b NaCl olalty c NaCl ole fracton Fgure.1: Phase dagra for the NaCl-KCl-H O syste usng olalty as concentraton unt (left) and salt ole fracton (rght). The phase dagra conssts of two curves. On the curve a - b sold potassu chlorde s n equlbru wth saturated solutons. On b - c sold sodu chlorde s n equlbru wth saturated solutons. In the pont b both sold salts are n equlbru wth a saturated soluton. c 3 Ideal solutons 3.1 Defnton An deal soluton can be defned as a soluton n whch the olar Gbbs energy of speces s calculated as: G = G + RTln x d R s the gas constant, T s the absolute teperature n Kelvn, and x s the ole fracton of coponent. Based on the defnton of the deal soluton, the propertes of real solutons can be calculated by the su of two ters: an deal ter and an excess ter. Equaton (3.1) defnes the deal ter of the Gbbs energy and s therefore gven the superscrpt d. olar standard state Gbbs energy and s a functon of teperature and pressure. d, G = G + RTln (3.) The deal soluton Gbbs energy calculated fro equaton (3.1) s dfferent fro the one calculated fro equaton (3.). The latter s therefore arked wth superscrpt. s the G G (3.1) s the If there s only one coponent, the ole fracton x of coponent s equal to one. The ter RTlnx then becoes zero, and the Gbbs energy of s equal to the standard state Gbbs energy of at the teperature and pressure. Therefore ths standard state s often called the pure coponent standard state. An alternatve defnton of the deal soluton s based on the olalty scale: value of the olar standard state Gbbs energy on the olalty scale and s a functon of teperature and pressure. The entropy of coponent s related to the Gbbs energy through the fundaental therodynac relaton: 9

10 S G = T P, x The entropy of coponent n an deal soluton can be calculated fro equaton (3.1) usng the relaton n (3.): d d G G S = = Rln x = S Rln x (3.4) T T S s the olar standard state entropy of coponent and s a functon of teperature and pressure. Through the fundaental therodynac relaton G = H TS, the enthalpy of coponent n an deal soluton can be calculated fro equatons (3.1) and (3.3): H = G + TS = G + RT ln x + TS RT ln x = G + TS = H (3.5) d H s the olar standard state enthalpy of coponent and s a functon of teperature and pressure. The Gbbs energy and the entropy of a coponent n an deal soluton both depend on the coposton of the soluton accordng to equaton (3.1) and (3.3). Accordng to equaton (3.5), the enthalpy of a coponent n an deal soluton s not dependent on the coposton. Exaple 3.1 We wll calculate the phase dagra fro Fgure.1 assung deal soluton behavor. The phase dagra conssts of two equlbru curves. On one curve sold potassu chlorde s n equlbru wth a saturated soluton. On the other curve sold sodu chlorde s n equlbru wth a saturated soluton. The two equlbra can be expressed as: + K ( aq) + Cl ( aq) KCl( s) (3.6) + Na ( aq) + Cl ( aq) NaCl( s) The brackets (s) ndcate sold, crystallne phase, the brackets (aq) ndcate solutes n aqueous soluton. Equlbru s attaned when there s no Gbbs energy change for a par of ons that choose to go fro the crystallne phase to the aqueous phase or vce versa. The equlbra we consder n ths exaple are heterogeneous, nvolvng two sold phases and a lqud phase. The sold phases are pure, hoogeneous phases, not xtures. Even n pont b n Fgure.1 where two sold salts are n equlbru wth the sae lqud, the two sold salts wll for crystals of pure NaCl and of pure KCl. In other systes, for exaple xtures of aonu and potassu salts, there s a tendency to for xed crystals, sold solutons, due to the slarty of the two catons. The Gbbs energy of the pure sold salts can be found n therodynac tables. Also the Gbbs energy of the aqueous ons can be found n such tables. Table 5. contans the values necessary to calculate the phase dagra n Fgure.1. The Gbbs energy of each coponent n the olalty based deal soluton can be expressed usng equaton (3.): (3.3) 1

11 d, G =Δ G + RTln + ( ) f + K aq K ( aq) K+ d, =Δ ln Cl ( aq) f + Cl ( aq) Cl G G RT G G =Δ G =Δ G KCl( s) f KCl( s) NaCl( s) f NaCl( s) All the Gbbs energes used are Gbbs energes of foraton as ndcated by the subscrpt f. The Gbbs energes of foraton refer to the sae standard state: The natural state of the eleents at 5 C and 1 bar. The values therefore allow us to calculate the Gbbs energy change by reactons aong these copounds. The condton for equlbru of potassu and chlorde ons wth sold potassu chlorde s that the Gbbs energy s dentcal n the two phases, whch can be expressed as: G + G = G d, d, + K ( aq) Cl ( aq) KCl( s) or Δ G + RTln +Δ G + RTln =Δ G s f + K ( aq) K+ f Cl ( aq) Cl f KCl( ) A correspondng expresson for sodu chlorde could also be wrtten. By nsertng nubers fro Table 5., the followng expresson s obtaned for potassu chlorde: RT ln RT ln = 4914 K Cl The equaton can be odfed to (3.9) = exp exp K Cl = = RT By a slar ethod the correspondng equaton for the equlbru of sodu chlorde can be derved as: = exp Na Cl = RT The curve for KCl solublty can now be calculated at fxed concentratons of NaCl by usng equaton (3.9). Because the NaCl concentraton s fxed, the olalty of Cl - can be calculated as the olalty of NaCl plus the olalty of K +. The olalty of K + therefore s the only unknown. The curve for NaCl solublty can be calculated at fxed concentratons of KCl by usng equaton (3.1) n a slar anner. The pont b fro Fgure.1 can be calculated by solvng equaton (3.9) and (3.1) sultaneously. The result of the calculaton s shown n Fgure 3.1. The calculated solublty of sodu chlorde s very close to the KCl olalty T= 5. C Saturaton lne, experental KCl branch, deal soluton calculaton NaCl branch deal soluton calculaton Two-salt pont, deal soluton calculaton NaCl olalty Fgure 3.1: Phase dagra for the KCl-NaCl-H O syste calculated assung deal soluton behavor. (3.7) (3.8) (3.1)

12 Table 3.1: Calculaton of phase dagra for the KCl-NaCl-H O syste assung deal soluton behavor. Concentratons are n olalty (ol/kg water). (fxed) NaCl Cl K + actual solublty of sodu chlorde. For potassu chlorde on the other hand, the calculated solublty s uch lower than the actual solublty of potassu chlorde. Ths ndcates that one of these apparently slar salts has deal soluton behavor whle the other devates strongly fro deal behavor. Ths s a concdence caused by the fact that the ean olal actvty coeffcent of sodu chlorde has values near 1 at concentratons near saturaton of NaCl at roo teperature. Obvously, the deal soluton assupton gves results that devate sgnfcantly fro the experental value of the solublty n ths syste. In order to ake ths knd of calculatons correct, t s very portant to use a therodynac odel that takes the devaton fro dealty nto account. Exercse 3.1 In Exaple 3.1, the sold-lqud phase dagra of the ternary syste KCl NaCl H O syste at 5 C was calculated, assung deal soluton behavor and usng olalty as concentraton scale. Derve the correspondng equatons necessary for calculatng the sae phase dagra usng ole fracton as the concentraton scale, and calculate the phase dagra. The Gbbs energes of foraton for the ons based on the ole fracton scale are gven n table 3.. Na + K + Cl - Δ f G kj/ol Sold Phase KCl KCl KCl KCl KCl KCl KCl KCl (fxed) + Cl Na NaCl NaCl NaCl NaCl NaCl NaCl Na + + K Cl NaCl+KCl Table 3.: Standard state Gbbs energy of foraton of ons at 5 C. The values are based on the ole fracton concentraton scale. 1

13 Collgatve propertes Collgatve propertes are propertes that accordng to physcal chestry textbooks are ndependent of the type of speces n the soluton but are dependent on the aount of speces. As t wll be shown here, these collgatve propertes are not ndependent of the type of speces. On the contrary, they are strongly dependng on the type of speces. The collgatve propertes are freezng pont depresson and bolng pont elevaton. Vapor pressure lowerng and osotc pressure are usually entoned separately as collgatve propertes but these two latter propertes are so closely related to the bolng pont elevaton that they don t need to be dscussed separately. In a freezng pont depresson experent, the teperature at whch sold solvent (ce) s n equlbru wth a soluton s deterned. Ths teperature s lower than the freezng pont of pure water. In a bolng pont elevaton experent, the teperature at whch solvent vapor (stea) s n equlbru wth the soluton s deterned. At atospherc pressure, ths bolng pont s hgher than the noral bolng pont of pure water. These phenoena can be understood by consderng equaton (3.1), and (3.). Accordng to these equatons, the Gbbs energy of an deal soluton depends on the coposton of the soluton n addton to teperature and pressure. Ice and stea are two pure phases. The Gbbs energes of pure phases depend only on teperature and pressure. At equlbru between soluton and ce or soluton and stea, the Gbbs energy of water n soluton s dentcal to the Gbbs energy of ce or that of stea respectvely. By varyng the coposton, t wll therefore be possble to fnd a range of teperatures and pressures at whch there s equlbru between an deal soluton and the pure phases, ce and stea respectvely. The equlbra between solutons and pure phases can be expressed as: Δ G ( T, P) + RTln x =Δ G ( T, P) for freezng pont depresson f w w f ce and (3.11) Δ fgw( T, P) + RTln xw =Δ fgstea( T, P) for bolng pont elevaton The left hand sdes of the equatons (3.11) express the Gbbs energes of deal solutons as functons of coposton, teperature and pressure. The rght hand sdes of the expressons gve the Gbbs energes of the pure phases as functons of teperature and pressure. By varyng the water ole fractons t s possble to deterne a range of teperatures and pressures at whch these equlbra can be establshed. The expresson for the equlbru between aqueous soluton and ce, equaton (3.11) s plotted n fgure 3. together wth experentally easured freezng pont depressons. The pressure was held constant at 1 bar. The freezng pont teperatures were calculated at a nuber of water ole fractons, usng equaton (3.11). The water ole fractons were converted to ol solute per kg water. The experental data arked n Fgure 3. show a sgnfcant dfference between real electrolyte solutons and deal soluton behavor. The data also show a sgnfcant dfference between the freezng pont depressons caused by dfferent electrolytes. Apparently, sodu chlorde solutons are closer to deal soluton behavor than agnesu chlorde solutons are. A soluton contanng 5.8 oles NaCl per kg water freezes at C. Assung full dssocaton such a soluton contans 1.16 ol solutes (Na + and Cl - ons) per kg water. Ths soluton s called a eutectc soluton because t s the sodu chlorde soluton wth the lowest possble freezng pont. A eutectc soluton of agnesu chlorde 13

14 -5-1 Freezng pont of deal soluton Freezng pont of NaCl solutons Freezng pont of MgCl solutons Freezng pont of MgSO4 solutons Freezng pont C ol solutes/kg water Fgure 3.: The theoretcal freezng pont depresson of an deal soluton copared to easured freezng pont depressons of sodu chlorde, agnesu chlorde, and agnesu sulfate solutons. contans.73 oles agnesu chlorde and freezes at C. If full dssocaton s assued, ths soluton contans 8.19 ol of solutes (Mg + and Cl - ons). Ths eutectc soluton s therefore ore dlute than the eutectc sodu chlorde soluton. It would be expected that the ore concentrated soluton would have a lower freezng pont. Magnesu sulfate solutons have hgher freezng ponts than deal solutons have. The sae s the case for sodu sulfate solutons but the postve devaton for sodu sulfate solutons s lower than for agnesu sulfate solutons. The experental data for sodu sulfate solutons are not shown n Fgure 3.. The eutectc soluton of sodu sulfate only contans.84 ol solutes per kg water. 14

15 Exercse 3. Calculate the bolng pont elevaton of an deal soluton of solutes n water at 1 at pressure. Fnd the approprate Gbbs energes as a functon of teperature and pressure on the nternet or n stea tables. Alternatvely, you can use the relaton G = H-TS to calculate Gbbs energes n the relevant teperature range, based on table values at 1 C and 1 at. Copare the results wth the experental data fro Hakuta et al. [1] gven n Table 3.3. Plot the results n a graph slar to Fgure 3.. Table 3.3: Experental easureents of the bolng pont of electrolyte solutons at 1 at. NaCl Na SO 4 MgCl Bolng pont ass percent Bolng pont Mass percent Bolng pont Mass percent Hakuta T., Goto T., Ishzaka S., Bolng pont elevaton of Aqueous Solutons Contanng Inorganc Salts, Nppon Kasu Gakka-Sh, 8(1974)

16 4 Checal potental and actvty coeffcents 4.1 Checal potental μ The checal potental of a speces s the partal olar dervatve of the total Gbbs energy G, enthalpy H, Helholtz energy A, or nternal energy U of substance []: G H A U μ = = = n n n n T, P, n j S, P, n j T, V, n j S, V, n j In equaton (4.1), n s the aount of coponent, T s the teperature, P s the pressure, S s the entropy, and V s the volue. Matter flows spontaneously fro a regon of hgh checal potental to a regon of low checal potental just lke electrc current flows fro a regon of hgh electrc potental to a regon of low electrc potental and ass flows fro a poston of hgh gravtatonal potental to a poston of low gravtatonal potental. The checal potental can therefore be used to deterne whether or not a syste s n equlbru. When the syste s n equlbru, the checal potental of each substance wll be the sae n all phases of the syste. The checal potental of a speces n ts standard state s dentcal to ts olar standard state Gbbs energy. Equaton (3.1) can therefore be rewrtten: μ d μ RTln x = + (4.) 4. Excess checal potentals for real solutons It was shown n chapter 3 that aqueous salt solutons devate sgnfcantly fro deal soluton behavor. In order to descrbe phase equlbra of electrolyte solutons t s therefore necessary to ntroduce a correcton for the devaton fro deal soluton behavor. The dfference between the checal potental of a real soluton and that of an deal soluton s called the ex excess checal potental. The excess checal potental for coponent s μ = RT ln γ. γ s the actvty coeffcent of coponent. The actvty coeffcent s a functon of coposton, teperature and pressure. By ncludng ths excess ter, the checal potental of coponent n a real soluton s expressed as: μ = μ + μ d ex = μ + RT ln x + RT lnγ (4.3) = μ + RT ln ( xγ) The checal potental of water n an aqueous soluton can be calculated fro the defnton n equaton (4.1) and expressed through an equaton of the for gven n equaton (4.3): G μw= = μw+ RT ln ( xwγ w) (4.4) nw P, T, n w As shown n chapter 3, dlute solutons are exhbtng deal soluton behavor. In the lt of x w 1 t follows that γ w = 1 and the excess checal potental vanshes. The excess ter s (4.1) Prausntz J.M., Lchtentaler R.N., Azevedo E.G, Molecular Therodynacs of Flud-Phase Equlbra, thrd edton, Prentce Hall PTR, Upper Saddle Rver NJ 7458,

17 only relevant for xtures. The standard state checal potental of water, μ w s dentcal to the olar Gbbs energy of pure lqud water at teperature T and pressure P. 4.3 The ratonal, unsyetrcal actvty coeffcent The checal potental of an on ay be wrtten as: G μ = n P, T, n j The operaton pled n equaton (4.5) s physcally possble. Because of the electro neutralty constrant t s not possble to add ons of one type keepng the nuber of all other ons constant at the sae te. Therefore t s not possble to easure the propertes of a sngle on ndependent of other ons. What s easured s therefore always the su of the propertes of an anon and a caton. In order to get nuercal values of the standard state propertes of sngle ons, the propertes of one on are gven fxed values. By conventon, the standard state checal potental of the hydrogen on H + s J/ol. Slarly, by conventon the enthalpy of foraton, the standard state heat capacty and the standard state partal olar volue of the hydrogen on are gven the value. The propertes easured for HCl n dlute solutons then becoe the propertes of the chlorde on. It s of course possble atheatcally to wrte an expresson slar to equaton (4.4) for on : μ = μ + RTln( xγ) (4.6) In equaton (4.6), the sae standard state (pure on ) s used as n equaton (4.4). It s not possble to have pure on. Ths standard state s therefore not physcally possble but s a hypothetcal or atheatcal state used because t s convenent to use the sae defnton for all coponents. Tradtonally, the actvty coeffcents of solutes are noralzed so that the actvty coeffcent of a solute s 1 at nfnte dluton. Ths s acheved by defnng a new actvty coeffcent γ by the rato of the value of the actvty coeffcent at the relevant concentraton and the value of the actvty coeffcent at nfnte dluton, γ : γ γ = (4.7) * γ The nfnte dluton actvty coeffcent of a coponent n water s dependng on teperature * and pressure. It s clear fro equaton (4.7) that γ = 1 at nfnte dluton. γ s the ratonal, unsyetrcal, actvty coeffcent. The adjectve unsyetrcal refers to the fact that ths actvty coeffcent has a value of unty at nfnte dluton rather than n the pure coponent state. The defnton n equaton (4.7) s often referred to as a noralzaton of the syetrcal actvty coeffcent. The ratonal, unsyetrcal actvty coeffcent can be used n an expresson for the checal potental of on slar to equaton (4.6): μ = μ + RT ln( x γ ) * RT ln RT ln( x ) ln( ) = μ + γ + * * = μ + RT xγ * (4.5) γ (4.8) * 17

18 The ratonal, unsyetrcal standard state checal potental s defned by: * μ μ + RT ln γ (4.9) Ths standard state checal potental was ntroduced n equaton (4.8) n order to get a forulaton slar to the one n equaton (4.6). 4.4 The olalty actvty coeffcent The relaton between the ole fracton and the olalty of a solute speces was derved n equaton (.9): n n nwmw x = = = x wmw (.9) nj nj nwmw By usng ths relaton, the olalty concentraton unt can be ntroduced nto equaton (4.8): * * μ = μ + RT ln( xγ ) * * = μ + RT ln( xwm wγ ) (4.1) * * = μ + RTln ( Mw) + RTln( x wγ / ) In order to ake the ters densonless, = 1 ol/kg water has been ntroduced. A olalty actvty coeffcent s now defned: γ * ì w The olalty based standard state checal potental μ s defned: By ntroducng the ters for the olalty actvty coeffcent and the ter for the olalty based standard state n equaton (4.1) t can be wrtten: μ s the checal potental of solute n a hypothetcal, deal soluton ( γ = 1) at unt x γ (4.11) μ μ + RT ln ( M ) * w = μ + RTln ( γ Mw ) olalty. By nsertng = 1 ol/kg water and γ = 1 n equaton (4.13) the checal potental therefore becoes dentcal to μ. The ter = 1 ol/kg water s usually otted fro equaton (4.13). The olalty based standard state checal potental μ s often gven n tables of standard state propertes of electrolytes. One such table has been publshed by the US Natonal Insttute of Scence and Technology [3]. The reason why the unsyetrcal actvty coeffcent s used for ons s that the standard state propertes for ths standard state can easly be easured. The correspondng values for the pure coponent standard state wth syetrc actvty coeffcents on the other hand, cannot be easured for ons. (4.1) μ = μ + RTln( γ / ) (4.13) 3 NIST Checal Therodynacs Database Verson 1.1, U.S. Departent of Coerce, Natonal Insttute of Standards and Technology, 199, Gathesburg Maryland

19 4.5 The olarty actvty coeffcent The olarty concentraton unt can also be used for expressng the checal potental of a solute. In order to derve ths expresson, the olar volue Vˆsoluton s ntroduced. It s the volue per ol of soluton. The olar volue of the soluton s derved fro equaton (.3): Vˆ V soluton soluton = = nj j j d x M j soluton j (4.14) The relaton between the ole fracton and the olarty of a solute speces was derved n equaton (.1). By usng ths expresson, the olarty concentraton unt can be ntroduced nto equaton (4.8): * cv soluton * μ = μ + RT ln γ n j ˆ * ˆ V soluton * = μ + RT ln Vwc γ ˆ (4.15) Vw ˆ c cv soluton * = μ + RT ln γ c ˆ Vw Vˆw s the olar volue of water. In order to ake the expresson densonless, the factor c = 1 ol/lter has been ntroduced. A olarty actvty coeffcent s defned by: ˆ c V γ Vˆ soluton w γ * (4.16) Vˆ soluton 1 = γ Vˆ x w w It can be shown that the olarty actvty coeffcent can be calculated fro the olalty actvty coeffcent usng the expresson: c d w γ = γ (4.17) c It follows fro equaton (4.14) and equaton (4.15) that the olarty based standard state c checal potental of coponent, μ, s defned by: ( ˆ ) c M * ln * w μ = μ + RT Vwc = μ + RTln c dw RT c 1 = μ + ln dw γ Mwc = μ + RT ln dw Calculatons of the checal potental can then be perfored usng the sple expresson: (4.18) 19

20 c c RT ln ( c / c ) μ = μ + γ (4.19) Due to the teperature and pressure dependence of the olarty concentraton unt, the olar actvty coeffcent s seldo calculated drectly fro odels. If olarty actvty coeffcents are needed, actvty coeffcents are usually calculated on the ole fracton scale or olalty scale and then converted to the olarty scale usng equaton (4.16) or (4.17). Molarty actvty coeffcents wll therefore not be treated further n ths book. 4.6 The actvty of speces The actvty of water s defned as the ole fracton of water ultpled by the actvty coeffcent of water: aw xwγ w (4.) For solutes, a dfferent actvty s defned for each type of actvty coeffcent: a xγ when the syetrcal actvty coeffcent s used * * a xγ when the ratonal, unsyetrcal actvty coeffcent s used (4.1) a γ when the olalty actvty coeffcent s used 4.7 Checal potental of a salt The total Gbbs energy for a soluton of salts ay be wrtten as a su of the contrbutons fro water and ons: G = n μ + n μ (4.) w w ons The checal potental of a salt S can be calculated accordng to equaton (4.1): G μs = (4.3) ns PT,, n The checal potental of a salt, μ S s a physcal property of the soluton and can be evaluated drectly fro experental data. Salt S dssocates nto ν C catons C and ν A anons A accordng to equaton (1.1). For salt S n soluton, the total Gbbs energy can be expressed by: G = nwμw + n Sν CμC+nSν AμA (4.4) A partal olar dfferentaton s perfored accordng to equaton (4.3) at constant teperature, pressure, and ole nuber of water: G μw μc μ A μs = = n w + νcμc + n SνC + ν AμA + n Sν A (4.5) ns ns ns ns The Gbbs-Duhe equaton s at constant teperature and pressure expressed as: ndμ = (4.6) For an aqueous soluton of a sngle salt, S, the Gbbs-Duhe equaton gves: nd w μw + nsν CdμC + nsν AdμA = (4.7) The correspondng ters n equaton (4.5) therefore cancel. By rearrangng the reanng ters, they can be wrtten as:

21 ν C ( ) ( ) ( ) C ν ν A CμC νaμa C A C A A ln ( ) + RT ν ν μ S = + γ γ (4.8) A ean olalty and a ean olal actvty coeffcent are defned. Both are gven the subscrpt ± to ndcate that ths s an average (geoetrcal ean) of the value for the caton and the anon. The ean olalty s defned as: ν ( ) 1/ C ν A C A ν ± (4.9) The ean olal actvty coeffcent s defned n a slar way: ν (( ) ( ) ) 1/ C νa γc γ A ν + γ (4.3) The Greek letter ν s the su of the stochoetrc coeffcents: ν = νa + νc (4.31) If the standard state checal potental of the salt n soluton s wrtten as: + μ S = ν C μc ν Aμ A (4.3) Equaton (4.8) can now be wrtten as: μs = μs + νrt ln ( ± γ ± ) (4.33) If nstead ole fractons and ratonal actvty coeffcents were used, the expresson for the checal potental of the salt would be: * μs = μs + νrt ln ( x ± γ ± ) (4.34) * * * In equaton (4.34), μ S = ν μ ν μ C C + A A. The ean ole fracton s gven by: ν ( ) 1/ C ν A C A ν x x x The ean, ratonal, unsyetrcal actvty coeffcent s gven by: ± = (4.35) ν ( ( ) ( ) ) 1/ C νa γc γ A * * * ν + γ = (4.36) Equaton (4.8) can be derved n a spler way: The total Gbbs energy for a soluton of salts ay be wrtten as a su of the contrbutons fro water and salts: G n μ n μ = + (4.37) w w s s salts By equatng (4.) and (4.37) the followng equaton s obtaned: n μ = nμ (4.38) S S salts ons For salt S n soluton, the checal potental can therefore be expressed as: nsμ = nsν CμC + nsν Aμ S A (4.39) By cobnng wth equaton (4.13) for the ons C and A, equaton (4.8) s obtaned drectly. Exercse 4.1 Show how equaton (4.17) can be derved fro equaton (4.16). Exercse 4. Calculate the ratonal, unsyetrcal standard state checal potental and the pure coponent standard state checal potentals for the sodu on and the chlorde on at 5 C. 1

22 At ths teperature, the nfnte dluton actvty coeffcent of the sodu on s and the correspondng value for the chlorde on s standard state checal potentals of these ons can be found n Table 5.. γ Cl γ + Na = = The olalty based Soe values of the ean olal actvty coeffcents are gven n Table 5.3. Calculate the ean ratonal actvty coeffcents and the ean syetrcal actvty coeffcents of sodu chlorde fro the ean olal actvty coeffcents gven n that table. Plot the three types of actvty coeffcents n the sae coordnate syste. Notce that the values of the nfnte dluton actvty coeffcents gven above are not absolute values. These values were obtaned by defnng the nfnte dluton actvty coeffcent of the hydrogen on to be one at 5 C and 1 bar. Calculate the deal and the excess part of the checal potental of sodu chlorde n aqueous solutons as a functon of the coposton. Plot the results n graphs. Make also a plot of the total checal potental of NaCl n aqueous solutons. Use ths graph to explan why an aqueous NaCl soluton s saturated at a concentraton slghtly hgher than 6 olal.

23 5 Measureent of checal potentals n salt solutons 5.1 Measureent of the checal potentals of ons The nuercal values of the Gbbs energes of foraton of ons n ther standard state are apparently requred n order to perfor phase equlbru calculatons for solutons contanng electrolytes. The accurate easureent of these Gbbs energes s therefore portant. In voltac [4] or galvanc [5] cells checal energy can be converted to electrcal energy. The reacton between etallc znc and copper ons n a soluton of copper sulfate s one such exaple: + + Zn( s) + Cu ( aq) Zn ( aq) + Cu( s) (5.1) The reacton can be dvded nto two half-reactons: + Zn() s Zn ( aq) + e ( anode) and (5.) + Cu ( aq) + e Cu( s) ( cathode) In a Danell [6] cell, these two half cell reactons are takng place n two chabers separated by a porous ebrane that allow sulfate ons to pass fro the cathode to the anode and thus conductng electrcty. The anode and the cathode are connected wth a wre allowng electrons to be transported fro the anode to the cathode. The ebrane that separates the two chabers gves rse to a so-called lqud juncton potental. In order to avod the lqud juncton potental, the two chabers can nstead be connected wth a salt brdge. A salt brdge s a tube flled wth a concentrated soluton of a salt. The salt brdge conducts electrcty between the two chabers but prevents the two solutons fro beng xed. The soluton n the salt brdge can be prevented fro xng wth the solutons of the two chabers by addng agar. The ablty of a cell to create a dfference n electrcal potental s easured by ts electrootve force, whch s easured n volts (1V = 1 J/C). The electrootve force E of a voltac cell s dentcal to the electrcal potental at the cathode nus the electrcal potental at the anode. The aount of electrcal work perfored by the reacton on an nfntesal charge at constant teperature, pressure, and coposton s therefore dentcal to the product of the charge and the electrootve force. It follows that the Gbbs energy change of the su of the two cell reactons can be descrbed by the followng expresson: ν efe =Δ rg (5.3) The negatve sgn on the left hand sde sgnfes that the charge transported conssts of electrons and thus s negatve. F=96485 C/ol s Faradays constant and ν e s the stochoetrc coeffcent of the electrons n the half-reactons. The electrootve force s dependng on the teperature, pressure, and the coposton of the electrolyte solutons. When the cell reacton has reached equlbru, the electrootve force of the cell s zero. 4 Allesandro Volta ( ), Italan physcst whose nventon of the electrc battery n the year 18 provded the frst source of contnuous current 5 Lug Galvan ( ), Italan physologst who dscovered the effects of electrcty on anal nerves and uscles. 6 John Danell ( ), Brtsh chest, developed the Danell cell n

24 By usng equaton (4.13) ottng =1 ol/kg, equaton (5.3) for the cell reacton n equaton (5.1) can be converted to: ( ) μ ( ) FE = μ + RT ln a a RT ln a a + Zn + Zn SO4 + Cu + Cu SO4 a + a (5.4) Zn SO = μ 4 + μ + + RT ln Zn Cu a a + Cu SO 4 The sulfate on does not partcpate n the electrode reactons, but as the reactons take place n two dfferent chabers, the concentraton of sulfate on wll be dfferent n the two chabers. Therefore the actvty of the sulfate on s ncluded n equaton (5.4). In the lt of nfnte dluton the olal actvty coeffcents approach 1. Provded that the exact concentratons of the two ons are known, the dfference between ther olal standard state checal potentals can be calculated fro electrootve force easureents perfored n the vcnty of nfnte dluton. For such easureents to be relable, t s requred that no electrcal current s flowng. If there s an apprecable aount of electrcal current, concentraton and teperature wll change durng the experent. By usng a potentoeter, the exact electrootve force of the cell can be easured wthout the flow of electrons. 5. The Nernst equaton The general for of equaton (5.4) s: RT olal actvty product of products E = E ln ν ef olal actvty product of reactants wth the standard potental ΔrG E = (5.5) ν ef " products" and " reactants"refer to the total reacton ΔrG s the change n olal standard state gbbs energy by the total reacton Equaton (5.5) was frst forulated n 1889 by the Geran physcal chest Herann Walther Nernst and s known as the Nernst equaton. Fro the dervaton of the Nernst equaton (5.5) t can be seen that the ntroducton of the unsyetrcal actvty coeffcent provdes the necessary fraework for easurng standard state checal potentals. Obvously the ratonal, unsyetrcal reference syste could as well be used n the Nernst equaton (5.5). The use of the syetrcal reference syste would not work n the Nernst equaton, because t would requre that easureents were carred out n a concentraton range that cannot be realzed physcally. When a standard hydrogen electrode s used as ether anode or cathode, t s possble to easure the dfference between the standard state checal potental of the etal on n the other half reacton and the standard state checal potental of the hydrogen on. The latter s by conventon equal to zero. Metal ons havng a postve olal standard state checal potental can act as cathodes wth the standard hydrogen electrode as anode. Metal ons wth a negatve olal standard state checal potental act as anodes wth the standard hydrogen electrode as cathode. 4

25 5.3 The Harned cell A Harned cell [7] s a cell consstng of a standard hydrogen electrode and a slver chlorde electrode. For a standard hydrogen electrode the half cell reacton s 1 ( ) ( ) H g H + aq e + (5.6) The hydrogen electrode s a standard hydrogen electrode (SHE) when the partal pressure of hydrogen n the cell s 1 bar. If a slver chlorde electrode s used as cathode, the other half cell reacton s: AgCl() s e + Ag() s + Cl (5.7) The su of the two half reactons are: 1 + H( g) + AgCl( s) Ag( s) + H ( aq) + Cl ( aq) (5.8) The Nernst equaton for ths cell s: H H Cl Cl E = E RT ln 1/ F + γ + γ (5.9) PH The partal pressure of hydrogen s adjusted to one bar and the equaton reduces to: RT RT E = E ln ( + γ + γ ) = E ln ( HCl γ H H Cl Cl ± ) (5.1) F F The standard potental of ths cell s: μ μ μ μ + + H ( g) H Cl AgCl( s) E = (5.11) F By easurng the electrootve force of ths cell at hgh dluton and extrapolatng the value to nfnte dluton, the nuercal value of the standard potental of the cell can be deterned. When the standard potental of the cell s known, the cell can be used for deternng actvty coeffcents of aqueous HCl solutons. Experentally easured electrootve force values fro Harned and Ehlers [7], are gven n Table 5.1. The data are plotted n Fgure 5.1. The standard potental or standard electrootve force of the cell, E s calculated fro Equaton (5.1) as a functon of the olalty of HCl assung the ean olal actvty coeffcent to be 1. Ths assupton s true at zero olalty. By extrapolaton of the experental data to zero olalty the value of E can be found as the ntersecton wth the ordnate axs n Fgure 5.1. Because of the curvature of the lne along whch the data are to be extrapolated, ths extrapolaton cannot be done wthout usng an accurate actvty coeffcent odel. The extrapolaton ethod actually used for ths type of data was ntroduced by Htchcock n (198) [8]. The value of the standard electrootve force obtaned by a correct extrapolaton of the data n Table 5.1 s.41 Volt. Ths pont s arked n Fgure Harned H.S. and Ehlers R.W., The Dssocaton Constant Of Acetc Acd Fro To 35 Centgrade, J. A. Che. Soc., 54(193) Htchcock D.I, The Extrapolaton of Electrootve Force Measureents to Unt Ionc Actvty, J. A. Che. Soc. 5(198)

26 Table 5.1: Electrootve force easureents perfored n HCl solutons at 5 C [7]. Molalty of HCl, ol/kg EMF easured, Volt E+RT/F*ln() E olalty of HCl Fgure 5.1: Electrootve force easureents plotted as a functon of olalty. By assung deal behavor at nfnte dluton the standard electrootve force can be found by extrapolaton to zero olalty. An accurate extrapolaton requres the use of an actvty coeffcent odel known to be exact at nfnte dluton. The value of the standard electrootve force obtaned by a correct extrapolaton s E =.41 Volt Volt A varety of potentoetrc cells for deternaton of actvty coeffcents and standard state checal potentals were descrbed by Butler and Roy [9]. These nclude the glass ph electrode, aalga electrodes, caton-selectve glass electrodes, lqud and polyer based on-exchange electrodes, neutral, carrer-based electrodes, cells wth transference, sold-state ebrane electrodes, enzye and other bologcally based electrodes, and gas-senstve ebrane electrodes Standard state checal potentals for a few salts and ons are gven n Table 5.. Exercse 5.1 Use data fro Table 5. to deterne the standard electrootve force of the Harned cell. Calculate the ean olal actvty coeffcents of the HCl solutons for whch electrootve force easureents are avalable n Table Butler J.N. and Roy R.N. Experental Methods: Potentoetrc. Chapter n Actvty coeffcents n Electrolyte Solutons, nd Edton, Edtor K.S. Ptzer, CRC Press, Boca Raton, Florda,

27 Table 5.: Standard state propertes of ons and salts at 5 C. The values for ons are based on the olalty concentraton scale. Copound Δ f G kj/ol Δ f H kj/ol C p kj/ol/k KCl(s) NaCl(s) K SO 4 (s) AgCl(s) H + (aq) K + (aq) Na + (aq) Cl - (aq) SO 4 - (aq) Measureent of solvent actvty Instead of easurng the ean olal actvty coeffcents, checal potentals n an electrolyte soluton can be deterned by easurng the actvty of the solvent. By usng the Gbbs-Duhe equaton, experental solvent actvty coeffcents can be converted to ean olal actvtes and vce-versa. The solvent actvty n an electrolyte soluton can be easured n several dfferent ways: Freezng pont depresson and bolng pont elevaton easureents The prncple for freezng pont depresson easureents were descrbed n chapter 3. The actvty coeffcent of the solvent can be calculated fro the dfference between the easured freezng pont depresson and the freezng pont depresson expected fro deal soluton behavor. The sae apples to bolng pont elevatons. Equaton (3.11) s vald for deal soluton behavor: Δ G ( T, P) + RTln x =Δ G ( T, P) for freezng pont depresson f w w f ce and (3.11) Δ fgw( T, P) + RTln xw =Δ fgstea( T, P) for bolng pont elevaton By ntroducng the solvent actvty coeffcent, the equatons becoe vald for real solutons: ( γ ) Δ G ( T, P) + RTln x =Δ G ( T, P) for freezng pont depresson f w w w f ce and Δ G ( T, P) + RTln x =Δ G ( T, P) for bolng pont elevaton ( γ ) f w w w f stea (5.1) Assung that the Gbbs energy of foraton of water, ce and stea s known as a functon of teperature and pressure, the only unknown n equaton (5.1) s the water actvty. Fro easureents of freezng pont depresson or easureent of bolng pont elevaton t s therefore possble to deterne the water actvty n aqueous electrolyte solutons. Especally freezng pont depressons gve very accurate values of the water actvty. The easureent of freezng pont depresson s not very senstve to pressure varatons. The easureent of bolng pont elevatons on the other hand requres very accurate easureents of both 7

28 teperature and pressure Vapor pressure ethods Vapor pressures are usually easured by a statc ethod, a dynac ethod, or by the sopestc ethod. There are any varatons of the statc ethod for vapor pressure easureent. In soe applcatons of ths ethod, the dfference between the vapor pressure of the soluton and of the pure solvent s easured. In other applcatons, only the absolute vapor pressure of the soluton s easured. Chlled rror hygroeters easure the relatve hudty of a saple by a statc ethod. The relatve hudty s dentcal to the water actvty. In the btheral equlbraton ethod by Stokes [1], a steady state s establshed between a soluton at one teperature and pure water at a lower teperature. When no evaporaton/condensaton takes place through the tube connectng the gas phase of the saple and that of pure water, the vapor pressure of the soluton s dentcal to the vapor pressure of pure water at the easured teperature. Ths ethod requres a very accurate easureent of teperature but no easureent of pressure. The dynac ethod can also be appled n dfferent ways. Ha and Chan [11] used an electro-dynac balance to trap a sall saple of the soluton studed. The saple was kept floatng n the ar n a chaber wth a constant strea of hud ar flowng through. The ar hudty could be vared by xng a dry ar strea wth an ar strea of known hudty. If the ar strea had a vapor pressure of water exactly atchng the equlbru vapor pressure of the saple, the ass of the saple reaned constant. By easurng the relatve hudty of the ar leavng the chaber the water actvty of the saple was deterned. In equaton (5.1) the condton for equlbru between a soluton and a gas phase s gven. The equlbru between pure lqud water and pure stea can be expressed by a slar equaton: Δ G ( T, P) = Δ G ( T, Psat ) (5.13) f w f stea The Gbbs energy of foraton of the lqud phase s not very senstve to the pressure and can be assued ndependent of pressure wthn a large pressure range. P sat s the pressure of water vapor n equlbru wth pure water at the teperature T. The pressure dependence of the vapor phase s known fro fundaental therodynac relatons. At hgher pressures, P sat should be replaced wth the equlbru water fugacty: P Δ (, ) (, ) ln sat f Gstea T Psat =Δ f Gstea T P + RT (5.14) P By cobnng equaton (5.13) and (5.14) t can be concluded that: P Δ (, ) (, ) ln sat fgw T P Δ fgstea T P = RT (5.15) P The presence of ons causes a vapor pressure lowerng. For a soluton, the vapor pressure of water s therefore P w nstead of P sat. Equaton (5.1) at the sae teperature can therefore be wrtten: 1 Stokes, R.H., The Measureent of Vapor Pressures of Aqueous Solutons by B-theral Equlbraton Through the Vapour Phase, J. A. Che. Soc., 69(1947) Ha, Z.; Chan, C.K. The Water Actvtes of MgCl, Mg(NO 3 ), MgSO 4, and Ther Mxtures, Aerosol Scence and Technology 31(1999) (16) 8

29 P Δ (, ) ln ( ) (, ) ln w fgw T P + RT xwγ w =Δ fgstea T P + RT (5.16) P The result fro equaton (5.15) s nserted nto equaton (5.16) to gve: Psat Pw RT ln + RT ln ( xwγ w) = RT ln P P (5.17) Psat xwγ w = Pw After easurng the vapor pressure of water t s very sple to calculate the water actvty coeffcent fro equaton (5.17) Isopestc easureents The sopestc ethod for deternng the solvent actvty was ntroduced by Bousfeld [1]. In ths ethod the solvent actvty of a saple s deterned by brngng the saple n equlbru wth a reference solute of known water actvty. The ajorty of vapor pressure easureents on electrolyte solutons are ade by the sopestc ethod. Isopestc eans of equal pressure. A known aount of the salt under study s placed n a dsh. Wth xtures of salts, several saples can be put n separate dshes sultaneously. In another dsh a known aount of a reference solute wth known water actvty such as sodu chlorde s placed. Soetes ore than one reference solute s used n separate dshes. Water s added to the dshes, whch are placed n a contaner and brought nto good theral contact at a gven teperature. Ths s acheved by contanng the saples n etal dshes of hgh theral conductvty such as slver or platnu. The dshes rest on a thck copper block. Water wll dstll off the saples wth hgh water actvty and condense n saples wth low water actvty. After several days or weeks, the salt solutons n the dshes wll be n equlbru wth each other through the coon vapor phase. The asses of Fgure 5.: Isopestc easureent perfored n glass contaner the dshes are then easured n order to deterne the aount of water n each dsh. The water actvty of the reference salt soluton s a known functon of salt concentraton. At equlbru n the contaner, the water actvtes n all the dshes wll be equal and hence the water actvty of the soluton of the salt under study wll be known. One dsadvantage of the sopestc ethod s that by ths ethod water actvtes are deterned relatve to other water actvtes. In order to nterpret data obtaned by ths ethod, t s therefore necessary to use drect vapor pressure easureents or potentoetrc easureents n order to deterne the water actvty n the reference solutons. The sopestc ethod works best for solutons wth a olalty hgher than.1 olal. Below ths concentraton the ethod gves unsatsfactory results. Ths can be explaned by the fact 1 Bousfeld W.R. Vapour Pressure and Densty of Sodu Chlorde Solutons, Trans. Faraday Soc. 13(1918)41 9

30 that the water actvtes of all dlute solutons are very close to unty. The drvng force causng water to dstll off one dsh and condense n another dsh s therefore very low and requres a very long equlbraton te. 5.5 Osotc coeffcent The actvty of pure water s unty. In dlute solutons the actvty and the actvty coeffcents are very lttle dfferent fro unty. Ths eans that the water actvtes requre a large nuber of sgnfcant dgts to show varatons wth the salt concentraton. Ths led N. Bjerru to ntroduce the so-called osotc coeffcent n 1918 [13]. The osotc coeffcent ntroduced by Bjerru was later tered the ratonal osotc coeffcent. Bjerru gave ths ratonal osotc coeffcent the sybol f and defned t: f ln ( xwγ w) / ln x. Later, the practcal w osotc coeffcent Φ was ntroduced. The coputaton of practcal osotc coeffcent nvolved slghtly less trouble than the coputaton of ratonal osotc coeffcents as dscussed by Guggenhe [14]. Ths factor s not portant today, but the practcal osotc coeffcent s stll the only one used. The practcal osotc coeffcent or just the osotc coeffcent as t s known, s defned by: Φ - ln a w nw ln aw Mwν = S νn (5.18) S By sopestc easureents, the osotc coeffcent of a saple s deterned relatve to the osotc coeffcent of a known electrolyte. It s coon to use the osotc rato to present such data. The osotc rato R s defned as the rato between the osotc coeffcent of the saple and that of the reference: R Φ saple = Φref ( ν ) ( ν ) ref saple (5.19) The osotc coeffcent of the saple soluton s then related to the osotc coeffcent of the sopestc reference soluton wth the equaton: Φ saple = R Φ reference (5.) Solutons of two dfferent salts wth dentcal water actvty have dfferent osotc coeffcents f the stochoetrc coeffcents are dfferent. As entoned before, absolute values for osotc coeffcents can be easured by drect easureent of the partal pressures of water over salt solutons. It s coon to use a value of ν = 3 when reportng osotc coeffcents for a salt lke Na SO 4. For electrolytes lke H SO 4 and NaHSO 4, ν = s often used, soetes ν = 3 s used. It s portant to know whch stochoetrc coeffcents were used for a specfc set of osotc coeffcent data, n order to convert the to the correct water actvty. Table 5.3 shows soe values of a w, Φ and γ ± for aqueous solutons of NaCl at 5 C and 1 bar for dfferent olaltes. The water actvtes and the osotc coeffcents fro the table are llustrated n Fgure 5.3. The data shows that the water actvty s alost lnearly dependent of olalty. The osotc coeffcent on the other hand s uch ore senstve to concentraton changes and has a concentraton dependence that s far fro lnear. It s seen 13 N. Bjerru: De Dssozaton der starken Elektrolyte, Zetschrft für Elektrochee, 4(1918) Guggenhe E.A. The Specfc Therodynac Propertes of Aqueous solutons of strong electrolytes, Phl. Mag. 19(1935)588 3

31 fro Fgure 5.3 that a large osotc coeffcent eans a low water actvty. Table 5.3: Water actvtes, osotc coeffcents, and ean olal actvty coeffcents of aqueous NaCl solutons at 5 C, a w : water actvty, Φ: osotc coeffcent, γ ± : ean olal actvty coeffcent of NaCl NaCl ol/kg a w Φ γ ± water actvty Osotc coeffcent NaCl solutons at 5 C NaCl ol/(kg water) Fgure 5.3: The non-dealty of water n sodu chlorde solutons at 5 C, presented as water actvty and as osotc coeffcents. The data are dentcal wth the data n Table The value of the osotc coeffcent at nfnte dluton Fro the defnng equaton (5.18) the value of the osotc coeffcent at nfnte dluton s not obvous. The ole fracton of water can be wrtten as: x w = 1 x (5.1) ons By nsertng the relaton between ole fracton and olalty, equaton (.9) the ole fracton of water can be wrtten: 1 1 x w = 1 - xm w w x w = = 1 +M 1 +M ν ons w w ons S (5.) 31

32 The actvty of water a w = x w γ w. In the lt of nfnte dluton the actvty coeffcent of water s 1. ln a w can therefore be approxated as ( ) ln a w = ln x wγ w - ln (1 + M wν S ) - M wν (5.3) S By nsertng ths expresson n the defnng equaton (5.18) the ltng value of Φ at nfnte dluton can be deterned: Mwν S l Φ= = 1 (5.4) M ν S w S 5.6 Mean actvty coeffcent fro osotc coeffcent Based on the Gbbs-Duhe equaton, t s possble to calculate water actvtes (osotc coeffcents) fro electrootve force easureents (olal actvty coeffcents of ons). The relatonshp between osotc coeffcents and ean olal actvty coeffcents s based on the Gbbs-Duhe equaton at constant teperature and pressure: ndμ = (5.5) For an aqueous soluton of a sngle salt, S, ths can be expressed as: nd w μw + nd S μs = (5.6) Inserton of equatons (4.4) and (4.33) gves: ( ) ( ) nd w μw + RTln xwγ w + nd S μs + νrtln ± γ ± = (5.7) As the standard state checal potentals are constant at constant teperature and pressure, ths reduces to: ( ) nd w ln aw + nsνdln ± γ ± = (5.8) Usng the defnton of the olalty of a salt, equaton (.4), ths s transfored nto: ( ± γ ± ) dln a + M νdln = (5.9) w S w Replacng lna w wth the correspondng ter fro the defnton of the osotc coeffcent n equaton (5.18) gves: [ wν S] S wν ( ± γ ± ) d Φ M + M dln = (5.3) ± = S C A. Ths can be used to reduce equaton (5.3) further because the factor after S s a constant: d[ Φ S] = Sd ln ( Sγ ± ) (5.31) The dfferentals on both sdes are expanded: ν Fro equatons (.5), (.6), and (4.9) t s found that ( ) 1/ C ν A ν ν Φ d ln ln S + d S Φ= d S S + d S γ ± (5.3) The ters are rearranged to: d ln γ ± = d Φ+ ( Φ 1) d ln S (5.33) If ths equaton s ntegrated fro S = where Φ = 1 and ln γ ± =, the followng equaton s obtaned: ν 3

33 ln γ ± = Φ 1+ Φ 1 S ds ( ) (5.34) S If the osotc coeffcent has been deterned wth sall concentraton ntervals, equaton (5.34) can be ntegrated graphcally to gve the correspondng ean olal actvty coeffcents. Alternatvely, f the experental data for the osotc coeffcents can be gven an analytcal for n the range fro nfnte dluton to fnte olaltes, the ntegral n equaton (5.34) can be evaluated analytcally. It s also possble to go the opposte way and calculate Φ fro γ ±. Fro equaton (5.9) we get d ln a w+ Mwνd S +SMwνd ln γ ± = (5.35) Integraton fro S = gves S= S ln a + M ν + M ν d ln γ = (5.36) w w S w S S = ± Dvson wth M w ν S and usng the defnton of the osotc coeffcent n equaton (5.18) gves: S 1 Φ = 1 + S d ln γ + (5.37) S If ean olal actvty coeffcents have been deterned by a potentoetrc ethod (easureent of electrootve force), the correspondng osotc coeffcents can be obtaned by ntegraton of equaton (5.37). The ntegratons proposed n equatons (5.34) and (5.37) can be done graphcally, but that wll usually result n soe naccuracy. Osotc coeffcents easured wth the sopestc ethod usually do not extend to concentratons below.1 olal. By correlatng the experental data wth a therodynac odel that s accurate n the dlute regon, the ntegraton can be perfored analytcally nstead of graphcally. Norally the odel of Debye and Hückel (see chapter 6) s used for ths purpose. 5.7 Osotc Pressure If a contaner wth pure water s separated fro a contaner wth a soluton of salt by a se pereable ebrane that allows water but not ons to pass, water wll pass fro the contaner wth pure water (wth relatvely hgh water actvty) to the soluton contaner (wth relatvely low water actvty). Because water s attracted to the soluton contaner, the aount of soluton wll ncrease and the pressure n the soluton contaner wll ncrease untl equlbru s reached. If the teperature and pressure n the pure water contaner s T and P the checal potental of pure water s: μ = μ ( T, P w w ) (5.38) In the soluton contaner, the teperature and pressure at equlbru s T and P S, and the water actvty s a w. The checal potental of water n the soluton contaner s therefore: μ (, ) ln w = μw T PS + RT aw (5.39) At equlbru between the two contaners, the checal potental of water s the sae n the two contaners: 33

34 μw( T, P ) = μw( T, PS) + RTlnaw (5.4) The pressure dependence of the checal potental of speces s gven by: μ V P = (5.41) T V s the partal olar volue of coponent. The checal potental of a speces at a certan pressure can therefore be calculated by ntegraton of equaton (5.41) assung that the partal olar volue of the speces s ndependent of pressure: μw( T, PS) μw( T, P) = VwdP = Vw PS P P = VwΠ (5.4) In equaton (5.4) the sybol Π s used for the pressure dfference between the contaners. Π s called the osotc pressure. By ntroducng ths result nto equaton (5.4) ths equaton can be odfed to: RT Π= ln a w (5.43) Vw The osotc pressure s portant for a bologcal cell to antan ts structure. Two solutons wth the sae osotc pressures are sad to be sotonc. A hypotonc soluton s ore dlute and has lower osotc pressure. A hypertonc soluton s ore concentrated and has a hgher osotc pressure. If a bologcal cell s n a hypotonc envronent water wll flow nto t and cause t to expand. If a bologcal cell s n a hypertonc envronent, t wll loose water and shrnk, t becoes dehydrated. Fro the defnton of the osotc coeffcent n equaton (5.18) t can be seen that the osotc pressure and the osotc coeffcent are proportonal to each other: RT ν ns Π= Φ (5.44) Vw nw When pure water s produced fro sea water, a pressure correspondng to the osotc pressure has to be appled. Ths process s called reverse ososs. Exercse 5. P s ( ) Calculate the osotc pressure of sodu chlorde solutons at 5 C fro the data gven n Table 5.3. The densty of sodu chlorde solutons can be calculated fro the equaton 3 d = 44.5NaCl kg/. Assue that the partal olar volue of water s dentcal to the olar volue of the soluton. Textbooks on physcal chestry often gve the equaton for osotc pressure of dlute solutons as Π= crt, the van t Hoff equaton [ 15]. In ths equaton, the concentraton of solutes s c ol/ 3. Calculate the sae osotc pressures fro the van t Hoff equaton and show how ths equaton s related to equaton (5.44). 15 J.H. van `t Hoff, The Role Of Osotc Pressure In The Analogy Between Solutons And Gases, Zetschrft fur physkalsche Chee, 1(1887)

35 6 Therodynac odels for electrolyte solutons Therodynac odels for electrolyte solutons are developed n order to be able to atheatcally descrbe the propertes and the phase behavor of solutons. For the checal ndustry t s very valuable to be able to optze processes for the producton of checals. Electrolyte solutons are nvolved n any processes and t s therefore portant to have good odels for the descrpton of electrolyte propertes. In order to properly odel electrolyte systes, all dfferent types of nteractons: on-on, ondpole, dpole-dpole, olecule-olecule should be taken nto account. The potental energy caused by on-on nteractons s proportonal to the nverse separaton dstance, 1/r. Electrostatc on-on nteractons therefore have an effect over a relatvely long dstance and are called long range nteractons. The potental energy caused by olecule-olecule nteractons s proportonal to the sxth power of the nverse separaton dstance, 1/r 6. These nteractons are therefore called short-range nteractons. The potental energy of on-dpole nteractons s proportonal to 1/r and the potental energy of dpole-dpole nteractons s proportonal to 1/r 3. These nteractons could be called nteredate range nteractons. Most odels are structured wth ters representng only long range and nteredate/short range nteractons. 6.1 Electrostatc nteractons Debye-Hückel theory The frst really successful odel for the electrostatc nteractons between ons n aqueous electrolyte systes was developed n 193 by P. Debye and E. Hückel [16]. Debye and Hückel descrbed the therodynacs of deal solutons of charged ons. As entoned above, the electrostatc nteractons between charged ons only represent the long range nteractons n such solutons and not the short range nteractons. The nteractons between ons and water are not descrbed by the Debye-Hückel odel, whch has led people to descrbe ths odel as a delectrc contnuu odel. In ths odel, the solvent only plays a role due to ts relatve perttvty (delectrc constant) and ts densty. The Debye- Hückel odel can therefore not stand alone as a odel for electrolyte solutons. It only represents soe of the electrostatc nteractons and should be cobned wth a ter for short and nteredate range nteractons n order to fully descrbe the propertes of concentrated electrolyte solutons. In the Debye-Hückel theory, the electrostatc force that a postve on exerts on a negatve through the solvent edu s expressed through Coulobs law: 1 e F 4πε ε r r e s the electronc charge = C ε s the perttvty n vacuu = C J -1-1 = (6.1) ε r s the delectrc constant (relatve perttvty) of the solvent (untless). The value of the relatve perttvty of water s 78.4 at 98.15K 16 Debye P., Hückel E., Zur Theore der Elektrolyte. I. Gefrerpunktsernedrgung und verwandte Erschenungen, Physkalsche Zetschrft 4(193)

36 r s the dstance between the ons (eter). Posson s equaton gves a relatonshp between the charge densty (ρ C -3 ) around on and the electrcal potental (ψ J/C) for a sphere wth radus r around on : 1 d dψ ρ r r dr dr = (6.) εε r Due to the charges, the ons are not dstrbuted evenly or randoly n the soluton. Near a caton, anons tend to be n excess, near an anon, catons tend to be n excess. An on j has an electrcal potental energy of zeψ j f t s n the dstance r fro the on. Debye and Hückel assued the dstrbuton of the ons n the soluton to be a Boltzann dstrbuton. Ths assupton gves another relaton between the charge densty and the electrcal potental: ze j ψ nz j j kt ρ = en A e (6.3) all ons nv n j s the ol nuber of coponent j, z j s the charge of coponent j, N A s Avogadro s nuber = ol -1, k s the Boltzanns constant = JK -1, and T s the teperature n Kelvn, V s the olar volue of the soluton. Debye and Hückel cobned the Posson equaton and the Boltzann equaton thereby elnatng the charge densty. The resultng Posson-Boltzann equaton was solved for the electrcal potental ψ. In ther paper, Debye and Hückel fnally arrved at an excess Helholtz functon for an deal soluton of charged ons. It sounds lke a contradcton to have an excess Helholtz functon for an deal soluton of charged ons. Ideal solutons do not have excess ters. But as entoned before, ths excess Helholtz functon only takes the non-dealty caused by the electrostatc nteractons nto account and does not deal wth the tradtonal non-dealty, caused by short range forces. The olar excess Helholtz functon for the electrostatc nteractons can be expressed by the equaton: E A 1 = xzs κχ ( κ a) (6.4) RT 3 The ter s s defned by: e s = (6.5) 4πε ε rkt The dstance of closest approach to the on was gven the sybol a, (a for annäherungsabstand ). The dstance of closest approach s a paraeter for the radus of on, not ts daeter. It s expected that a s larger than the radus of the on, because the ons are thought to be surrounded by water that gves the ons a larger radus than the bare on. The product κa s densonless and κ s therefore a recprocal length. 1/κ s a characterstc length called the screenng length. The screenng length provdes a good frst estate of the dstance beyond whch Coulob nteractons can be essentally gnored, as well as the sze of the regon near a pont charge where opposte-charge counter-ons can be found. The expresson for κ s: 36

37 κ 1/ en nz A = εε rkt nv The functon χ s gven by: χ( x) = 3 + ln(1 + x) (1 + x) + (1 + x) x (6. 7) Apparently, the Helholtz functon of Debye and Hückel (6.4) has not been used by those who have developed odels for electrolyte solutons. Actually ths equaton s usually not even entoned. Instead, a nuber of splfcatons of the Debye-Hückel equaton have been used for odel developent and are often entoned n text books Debye-Hückel Extended law The so-called extended Debye-Hückel law represents a splfcaton of the orgnal Debye- Hückel equaton. The relaton between Gbbs energy and Helholtz energy s G = A + PV. No PV ter was added to the Helholtz functon. The PV ter was consdered nsgnfcant and was therefore dscarded. Checal potentals were derved fro the energy functon by olar dfferentaton at constant teperature and pressure, rather than olar dfferentaton at constant teperature and volue. Equaton (4.1) shows how the checal potental s derved fro the dfferent energy functons. G H A U μ = = = n n n n T, P, n j S, P, n j T, V, n j S, V, n j A Gbbs energy functon was created fro the Helholtz functon n equaton (6.4) by replacng the olarty concentraton unt wth olalty and splfyng the expresson for κ n equaton (6.6). The densty of an electrolyte soluton wth the olar volue V and the total volue nv can be wrtten as: nm w w + nm ons dsoluton = nv Ths expresson s converted to an expresson for nv, whch s nserted nto equaton (6.6): 1/ end nz A soluton κ = (6.9) εε rkt nwm w + nm ons Next, the approxaton s ade that the volue and the ass of the ons s zero. Ths approxaton represents a sall error for dlute solutons and a ore serous error for concentrated solutons. By akng ths approxaton, the densty of the soluton becoes equal to the densty of pure water, d kg/ 3. To ake the expressons spler, the onc strength, a concept frst ntroduced by Lews and Randall n 191 [17] s ntroduced: Z (6.1) I =.5 The expresson for κ can now be wrtten: (6.6) (4.1) (6.8) 17 Glbert N. Lews and Merle Randall, The actvty coeffcent of strong electrolytes, J. A. Che. Soc. 43(191)

38 1/ 1/ 1/ nz A A 1 z rkt nwm w rkt rkt end end end A 1/ κ = = I (6.11) εε εε εε Wth ths approxaton, the product sκ fro equaton (6.4) can be wrtten as: 1/ 3/ e end 1/ 1/ A e 1/ sκ I = ( πnad ) 4πε ε rkt ε ε rkt 4πε ε rkt Ths approxated value of sκ s expressed as AI ½ where A s the Debye-Hückel paraeter: 3/ I (6.1) 1/ e A= ( π NAd ) (6.13) 4πε ε rkt The value of the Debye-Hückel paraeter A s (kg/ol) ½ at 5 C. The ter κa fro equaton (6.4) was replaced by BaI ½ where a s a coon on sze paraeter replacng the ndvdual dstance of closest approach, a. The on sze paraeter a s often n the range B s derved fro the approxated value of κ n equaton (6.11) 1/ end A B = εε rkt The olar Gbbs excess functon arrved at by the splfcaton s: (6.14) E G Extended Debye-Hückel 4 3/ ½ = xwmw AI χ ( BaI ) RT 3 (6.15) 4A ½ ½ ( Ba) I = xm w w ln ( 1+ BaI 3 ) BaI + ( Ba) x w s the ole fracton of water, M w kg ol -1 s the olar ass of water. The functon χ(x) s gven n equaton (6.7). The extended Debye-Hückel law usually gves good results for actvty coeffcents up to an onc strength of about.1 olal. Above ths concentraton, short range nteractons apparently gve a sgnfcant contrbuton to the devaton fro dealty. The actvty coeffcents are derved fro the total Gbbs excess functon by olar dfferentaton: E ( / ) Extended Debye Hückel * ng RT A I = ln γ = Z n 1+ Ba I TPn,, j, j ì (6.16) The actvty coeffcents calculated wth the extended Debye-Hückel equaton are the ratonal actvty coeffcents and not the olal actvty coeffcents as soetes claed n text books. Accordng to the defnton of the olal actvty coeffcent n equaton (4.11), the olal actvty coeffcent accordng to the extended Debye-Hückel can be calculated fro: ( ) * A I ln γ = ln x wγ = ln xw Z (6.17) 1+ Ba I The extended Debye-Hückel ean olal actvty coeffcent of a salt wth caton C and anon 38

39 A calculated accordng to the defnton of the ean olal actvty coeffcent n equaton (4.3) s: 1 A I ln γ = ln x ± ν Z 1+ Ba I A I = ln xw ZCZA 1+ Ba I The last expresson n equaton (6.18) s obtaned fro the frst by usng the fact that ν Z + ν Z =. A A C C w ν (6.18) The actvty coeffcent of water calculated wth the extended Debye-Hückel equaton s gven by: E ( ngextended Debye Hückel RT ) / = ln γw = M wai nw 3 TPn,, 3 1 σ ( x) = 1 x ln(1 x) x 1+ x σ ( BaI ) 3/ ½ (6.19) Debye-Hückel ltng law The so-called Debye-Hückel ltng law s a further splfcaton of the orgnal Debye- Hückel theory. The olar excess Gbbs energy functon defnng the Debye-Hückel ltng law s gven by the expresson: G E Debye-Hückel ltng law RT 4 = x M AI 3 3/ w w (6.) Copared to the excess Gbbs energy of the Extended Debye-Hückel law n equaton (6.15), ths corresponds to consderng the ½ χ BaI to be one. value of the functon ( ) Typcal values of Ba s (kg/ol) ½..3 ½ Values of the functon χ ( 1.5I ) are shown..1 n Fgure 6.1 versus onc strength. Ths functon s obvously very senstve to changes n the onc strength. The functon has a ltng value of 1 at nfnte dluton. In a 1 olal soluton of a 1-1 salt, the value ½ of χ ( 1.5I ) s.48! The Debye-Hückel ltng law s therefore only correct n the lt of nfnte dluton. χ(1.5ι) Ionc strength, I, ol/kg χ(1.5ι) Fgure 6.1: The functon χ fro equaton (6.7) versus onc strength The actvty coeffcents for the Debye-Hückel ltng law are derved by olar dfferentaton of the excess Gbbs energy functon n equaton (6.): 39

40 E ( ngdebye Hückel ltng law / RT ) * = ln γ = - Z A I n T, P, nj, j ì (6.1) Equaton (6.1) s only applcable up to an onc strength of axu.1 olal,.e. only for extreely dlute solutons. The correspondng ean olal actvty coeffcent s: ln = ln x - Z Z A I γ ± w C A (6.) Experental.6 Debye-Hückel ltng law.5.55 Extended Debye-Hückel law Hückel, Molalty of HCl Molalty of HCl Fgure 6.: The ratonal ean onc actvty coeffcent of aqueous HCl at concentratons up to.3 olal, experental and calculated values Experental Debye-Hückel ltng law Extended Debye-Hückel law Hückel, 195 Fgure 6.3: The ratonal ean onc actvty coeffcent of aqueous HCl at concentratons up to 6 olal, experental and calculated values. Exercse 6.1 Show that the water actvty coeffcent accordng to the Debye-Hückel ltng law s: 3/ ln γ w = M wai. Hnt: Use the Gbbs-Duhe equaton (see secton 5.6) and use the 3 olalty of salt as ntegraton varable. Alternatvely the actvty coeffcent can be derved fro the excess Gbbs energy functon for the Debye-Hückel ltng law. Calculate the osotc coeffcent of water n sodu chlorde solutons usng water actvtes fro the Debye-Hückel ltng law and copare the results wth the experental values of these osotc coeffcents gven n Table 5.3. Use A = (kg/ol) ½ and Ba =1.5 (kg/ol) ½ The Hückel equaton In 195 t was found by E. Hückel [18] that the addton of an extra paraeter, C to the extended Debye-Hückel law ade t possble to calculate actvty coeffcents up to hgher concentratons wth good accuracy. Wth the C paraeter, a ter proportonal to the onc strength was added to the logarth of the ratonal, unsyetrcal actvty coeffcent: 18 Hückel E. Zur Theore konzentrerterer wässerger Lösungen starker Elektrolyte, Physkalsche Zetschrft 6(195)

41 * A I ln γ ± = - ZC Z A + CI (6.3) 1 + Ba I In hs paper, E. Hückel justfed the addton of the C paraeter by showng that accordng to theory, a ter proportonal to the onc strength would take the varaton of the delectrc constant wth the coposton nto account. The perforances of the Debye-Hückel ltng law, the Debye-Hückel extended law, and the Hückel equaton are llustrated n Fgure 6. and Fgure 6.3. In these fgures, the calculated actvty coeffcents of aqueous hydrochlorc acd solutons are plotted together wth experental values fro a nuber of sources. For the preparaton of Fgure 6. and Fgure 6.3, a value of 1.5 (kg/ol) ½ was used for Ba n equatons (6.16) and (6.3). The best value of the C paraeter n equaton (6.3) was deterned to be C =.31 kg/ol. Fgure 6.3 shows that the C paraeter n the Hückel equaton expands the concentraton range n whch the Debye-Hückel theory can be appled consderably. The C paraeter s a functon of teperature and of the ons n the soluton. In dlute solutons the actvty coeffcents of electrolytes decrease wth ncreasng concentraton, see Fgure 6.. For very dlute solutons the slope of the actvty coeffcent curve versus concentraton wll be nus nfnty. For ost electrolytes the actvty coeffcents wll pass through a nu and ncrease agan at hgh concentratons reachng values whch ay be uch hgher than unty. Soe systes however, show a behavor where the actvty coeffcent contnues to drop for ncreasng concentraton. Exercse 6. Show by usng the Gbbs-Duhe equaton (4.6), that the effect of the C paraeter n the Hückel equaton (6.3) on the natural logarth of the water actvty coeffcent s a MC w contrbuton of I to lnγ w. Z Z C A Deterne the C paraeter for NaCl at 5 C based on the experental values of the ean olal actvty coeffcents n Table 5.3. Use the Hückel equaton wth Ba=1.5 (kg/ol) ½ and A= (kg/ol) ½ to perfor the calculatons. Use the C paraeter to calculate osotc coeffcents for sodu chlorde solutons. Copare these calculated osotc coeffcents wth values calculated wth the extended Debye-Hückel law and wth experental values gven n Table 5.3. Exercse 6.3 The generally accepted correct ethod for extrapolatng electrootve force easureents [8] to nfnte dluton requres the use of the Hückel equaton. Use the Hückel equaton wth Ba=1.5 (kg/ol) ½ and A= (kg/ol) ½ to calculate the standard electrootve force of the Harned cell based on the experental easureents n Table 5.1. Note that a value of the Hückel C paraeter needs to be deterned at the sae te. Hnt: cobne the Hückel RT equaton wth the Nernst equaton to get an equaton of the for: y= E CI. F Deterne E and C fro the data for ths equaton. Plot the data. 41

42 6.1.5 The Born equaton Whle the Debye-Hückel theory deals wth the nteracton between charged ons, Born [19] derved an equaton for the nteracton between an on and the surroundng solvent. Around an electrcal charge there s an electrostatc feld. Polar olecules are affected by such felds and redrect theselves to have as low energy as possble n the feld. The postve part of a polar olecule s attracted to a negatve charge and vce versa for the negatve part. A orentaton polarzaton s takng place. A solvent consstng of polar olecules s polarzable and s also referred to as a delectrc edu. Hghly polarzable solvents have hgh relatve perttvtes (delectrc constants). As expressed by Coulob s law n equaton (6.1), electrostatc nteractons are lowered by solvents wth a hgh relatve perttvty. Salts do not dssocate spontaneously n a vacuu because the electrostatc nteractons between the ons are too strong. In water, the electrostatc nteractons between ons are lowered by a factor ε r = 78.4 at 5 C. The water olecules are sheldng the ons fro each other and allowng the to be separate. The relatve perttvty of a solvent s defned as the rato between the perttvty of the solvent and the perttvty of vacuu: ε ε r (6.4) ε Prevously, the relatve perttvty was called the delectrc constant and t was referred to as D. The relatve perttvty of a edu can be easured by exposng the edu to a perturbng external electroagnetc feld of sall feld strength. The edu s placed between the plates (conductors) of a capactor. The response of the edu to ths feld s easured, and fro ths response, the relatve perttvty of the edu can be deduced. Further nforaton on the easureent of reatve perttvty can be found n the book by Hll et al. [] and n the paper by Kaatze [1]. When ons are dssolved n a delectrc edu, the solvent olecules are polarzed by the electrcal charges. Ths on-solvent nteracton s called solvaton. For the specal case of water as solvent, the ter hydraton s used nstead. Here, hydraton s not the foraton of on-hydrates of a specfc stochoetrc coposton, but the redrecton of the polar water olecules around the charged ons. The energy change assocated wth solvaton s the solvaton energy or n the specal case of water, the hydraton energy. Consder a sphercal on wth charge Z and radus r n a edu wth the perttvty ε ε r. Born calculated the solvaton energy of ths on by ntegratng the energy of the electrcal feld fro the surface of the sphercal on to nfnty. Born found the electrostatc contrbuton to the Helholtz energy of the on to be: Z e A = 8πεεrr (6.5) The expresson was derved for a sngle on n a solvent wth the relatve perttvty ε r. When ore ons are present, the electrostatc felds generated by the ons nfluence each 19 Born M., Voluen und Hydratatonswäre der Ionen, Zetschrft für Physk, 1(19)45-49 Hll N.E., W.E. Vaughan, A.H. Prce, and M. Daves, Delectrc Propertes and Molecular Behavour, Van Nostrand Renhold, London (1969) 1 Kaatze, U., The Delectrc Propertes of Water n Its Dfferent States of Interacton, Journal of Soluton Chestry, 6(1997)

43 other, and these electrostatc nteractons nfluence the on-solvent nteractons. Equaton (6.5) shows that the Helholtz energy s lower n a edu of hgh relatve perttvty, ε r. The solvaton energy s therefore negatve f the on s transferred fro vacuu to a solvent. By ultplyng the energy n equaton (6.5) wth Avogadro s nuber, the Helholtz energy of one ol of ons wth charge Z s calculated. It s often of nterest to know the change n Helholtz energy when one ole of an on s oved fro a edu wth one perttvty to a edu wth another perttvty. Ths s the Helholtz energy of transfer. If one ole of ons s oved fro vacuu to a edu of relatve perttvty ε r, the energy change can be calculated as: ZeN A ZeN A ZeN 1 A Δ solv A = = 1 8πε ε rr 8πε r 8πε r ε (6.6) r It has been found that equaton (6.6) gves very accurate results copared to experental values of the solvaton energy. An effectve radus correspondng to the radus of the sphercal cavty n the solvent created by an on should be used n equaton (6.6) [, 3] rather than the onc radus of the on. Accordng to Rashn and Hong [3], the radus of the cavty produced by the sae on s dfferent n dfferent solvents. When the Helholtz energy of transfer between two solvents s calculated t s therefore necessary to use two dfferent rad for the sae on. Besdes, Rashn and Hong [3] found that the cavty radus of anons n water can be calculated by addng 7% to ther onc radus, whle the cavty radus of catons can be calculated by addng 7 % to ther covalent radus. The covalent radus s half the dstance between two dentcal atos bonded together by a sngle covalent bond. The covalent radus of a etal caton s usually larger than the onc radus but saller than the atoc radus of the correspondng etal. The atoc radus of sodu s 19 p, whle the covalent radus of sodu s 154 p and the onc radus of the sodu on s 1 p. The arguent for usng the covalent radus of the etal rather than the onc radus of the caton s that the covalent radus corresponds to the radus of the epty orbtal around the caton. Ths epty orbtal s assued to for a part of the cavty. For the transfer of one ole of on fro a edu wth relatve perttvty ε r,1 to a edu wth relatve perttvty ε r,, the Helholtz energy of transfer s accordng to Rashn and Hong [3]: ZeN 1 A ZeN 1 A Δ transfer A = 1 1 8πε r, ε r, 8πε r,1 ε r,1 ZeN A = 8πε + r, ε r, r, r,1 ε r,1 r,1 The rad of the solvent cavtes are arked wth subscrpt 1 and for the two solvents. (6.7) The Born ter n equaton (6.6) s often used wth equatons of state for electrolyte Later W.M, Ptzer K.S., and Slansky C.M., The Free Energy of Hydraton of Gaseous Ions, and the Absolute Potental of the Noral Caloel Electrode, J. A. Che. Soc., 7(1939) Rashn A.A. and Hong B., Reevaluaton of the Born odel of on hydraton, J. Physcal Chestry, 89(1985)

44 solutons. In ths case, the Born ter contrbutes to the actvty coeffcents because of the varaton of the relatve perttvty wth pressure. The Born ter s also used for calculatng the Gbbs energy of transfer. Ths s the dfference n standard state checal potental for a solute n two solvents wth dfferent relatve perttvtes. It s usually assued that the Helholtz energy dfference calculated wth equaton (6.6) s dentcal to the correspondng dfference n Gbbs energy. The Helholtz energy calculated fro equaton (6.5) s the dfference n Helholtz energy of a charged partcle n a delectrc edu copared to the Helholtz energy of an uncharged partcle of the sae sze, n the sae edu. The equaton thus represents the contrbuton to the excess Helholtz energy fro the nteracton between a sngle on and a nuber of solvent olecules. Helgeson et al. [4] used the equaton of Born (6.5) n cobnaton wth the extended Debye-Hückel law (6.16) to derve an actvty coeffcent odel. Ths actvty coeffcent odel s dentcal to the one derved by Hückel [18] n 195, but was derved on dfferent theoretcal assuptons. The Helgeson-Krkha-Flowers odel [4] can thus be seen as a ore odern valdaton of Hückel s equaton. Exercse 6.4 Accordng to Hefter et al. [5] the enthalpy of transfer of one ol of sodu chlorde fro pure water to pure ethanol s -13 J/ol and the correspondng entropy of transfer s -11 J/(ol K) both at K. The onc radus of the chlorde on s 181 pcoeter and the covalent radus of sodu s 154 pcoeter. The relatve perttvty of water s and the relatve perttvty of ethanol s 3.66 at K, both accordng to Albrght and Gostng [6]. Calculate the Gbbs energy of transfer for one ole of sodu chlorde beng transferred fro pure water to pure ethanol at 5 C. Assue that the cavty rad n water are the above rad plus 7 % and calculate the correspondng percentage that should be added to these rad n order to get the cavty rad n ethanol. Why are the cavty rad larger n ethanol than n water? The ean sphercal approxaton The Mean Sphercal Approxaton [7] uses a ore odern ethod of calculatng the excess Helholtz energy fro the electrostatc nteractons than Debye and Hückel used [16]. The resultng MSA ter sees to be ore coplcated than the correspondng Debye- Hückel ter, but gve slar nuercal results. It was stated by Zuckeran et al.[8] that At a purely theoretcal level, however, one cannot be content snce, a pror, there see no clear grounds for preferrng the DH-based theores apart fro ther ore drect and 4 Helgeson H.C., D.H. Krkha, G.C. Flowers Theoretcal Predcton of the Therodynac Behavor of Aqueous Electrolytes at Hgh Pressures and Teperatures: IV. Calculaton of Actvty Coeffcents, Osotc Coeffcents, and Apparent Molal and Standard and Relatve Partal Molal Propertes to 6 C and 5kb, Aercan Journal of Scence, 81(1981) Hefter G., Marcus Y., and Waghorne W.E., Enthalpes and Entropes of Transfer of Electrolytes and Ions fro Water to Mxed Aqueous Organc Solvents, Checal Revews, 8() Albrght P.S. and Gostng L.J, Delectrc Constants of the Methanol-Water Syste fro 5 to 55 C, J. A. Che. Soc. 68(1946) Lebowtz J.L. and Percus J.K., Mean Sphercal Model for Lattce Gases wth Extended Hard Cores and Contnuu Fluds, Physcal Revew, 144(1966) Zuckeran D.M., Fsher M.E., and Lee B.P., Crtque of prtve odel electrolyte theores, Physcal Revew E, 56(1997)

45 ntutve physcal nterpretaton rather than the ore odern (and fashonable) MSA-based theores whch snce they ental the par correlaton functons and the Ornsten-Zernke (OZ) relaton gve the presson of beng ore frly rooted n statstcal echancs. On the other hand, t has recently been shown that the DH theores yeld par correlatons satsfyng the OZ relaton n a very natural way. Furtherore, both theores have an essentally eanfeld character despte whch, n contrast to typcal ean feld theores for lattce systes, nether has any known Gbbs-Bogolubov varatonal forulaton or slar bass. How, then, ght the two approaches be dstngushed? The dervaton of the MSA ter for electrostatc nteractons s too coplex for these notes. The nterested reader s referred to papers lke Lebowtz and Percus, Mean Sphercal Model for Lattce Gases wth Extended Hard Cores and Contnuu Fluds [7]. The MSA expresson for the Helholtz energy of an electrolyte soluton s accordng to Harvey et al.[9]: E Ve ρz π VΓ A = Γ + Ω n + 4πε ε ons 1+Γσ Θ P 3π kt (6.8) The ters n ths equaton are: π ρσ 1 ρσ z Ω= 1 +, =, Θ 1+Γ Ω 1+Γ ons 3 1-, 6 ons 3 Pn σ ons σ e πσ P n Γ= ρ z ( 1+Γσ) εε kt Θ (6.9) π N and Θ= ρσ ρ = (nuber densty of speces ) V In ths for of the MSA ter the MSA screenng paraeter, Γ s gven wth an plct expresson. By usng an average daeter σ of the ons n the soluton, the expresson can be reduced to a spler, explct for: A E σ = 3 Γ RTV 3 = 1 + Γσ 3π N ons nσ n, 1 Γ= 1+ σκ 1 σ κ A ons 1/ ena = nz εε r RTV ons 1/ (6.3) The expresson for κ s dentcal to the expresson used n Debye-Hückel theory, equaton (6.6).The MSA ter for electrostatc nteractons s the preferred ter for any researchers 9 Harvey A.H., Copean T.W., and Prausntz J.M. Explct Approxatons to the Mean Sphercal Approxaton for Electrolyte Systes wth Unequal Ion Szes J. Phys. Che. 9(1988)

46 atteptng to develop equatons of state for electrolytes. It was shown by Y Ln [3] by Taylor expanson that there s vrtually no dfference between the MSA ter and the Debye-Hückel ter resultng fro slar splfcaton of the two theores. 6. Eprcal odels for nteredate/short range nteractons The theory presented by Debye and Hückel was only eant to take the electrostatc nteractons between sphercal ons nto account. The equatons presented n the prevous secton for the ean sphercal approxaton are also lted to the electrostatc nteractons between sphercal ons. These equatons therefore have to be cobned wth Fgure 6.4: The logarth of the reduced actvty coeffcent Γ versus onc strength μ over the onc strength range fro 1 to olal. odels or equatons that descrbe other types of nteractons such as on-dpole and dpoledpole nteractons and short range nteractons n order to be used for real solutons. Models for dpole-dpole nteractons wll not be consdered further n these notes The Messner correlaton In 197 H.P. Messner et al. started publcaton of a seres of papers on actvty coeffcents of strong electrolytes n aqueous solutons [31,3]. The bass for ther ethod was the observaton that curves of the reduced actvty coeffcent, Γ, versus the onc strength, I, for dfferent salts for a faly of curves that dd not cross each other. Ths only apples to curves of the reduced actvty coeffcent, not to curves of ean onc actvty coeffcents. One of the graphs presented n [31] s reproduced n Fgure 6.4. It was ponted out one year later by Broley [33] that Messner had panted an dealzed pcture of electrolyte behavor and that soe of the curves for coon salts actually do cross each other. The reduced actvty coeffcent, Γ, for a salt S was defned va the followng equaton: 1 Z C A CZ Z Z A ± or γ± ( γ ) Γ =Γ (6.31) Intally, the ethod of Messner and Tester [31] was a graphcal ethod for deternng actvty coeffcents n bnary solutons usng graphs lke the one shown n Fgure 6.4 vald 3 Ln Y. Developent of Equaton of State for Electrolytes, Ph.D. thess, Techncal Unversty of Denark, 8 31 Messner H.P and Tester J.W., Actvty Coeffcents of Strong Electrolytes n Aqueous Soluton, Ind. Eng. Che. Process Desgn and Developent, 11(197) Messner H.P., Kusk C.L. Actvty Coeffcents of Strong Electrolytes n Multcoponent Aqueous Soluton, AIChE Journal, 18(197) Broley, L. A, Therodynac propertes of strong electrolytes n aqueous solutons, AIChE Journal 19(1973)