18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong electrolytes, tht is, those formed from strong cid neutrlized with strong bse, will dissocite lmost completely into ions, while wek electrolytes will dissocite only prtilly. In medi of lower dielectric constnt thn wter, such s furfurl, cetonitrile, lcohols, chlorocetic cid, dioxne, cetone, cetic cid, or in their mixtures with wter, conductivity mesurements show tht ll electrolytes re incresingly wek; tht is, they re prtilly ssocited, s the solvent moves down in the scle of dielectric constnts. Thus, the clssifiction of strong electrolytes s strong cids, bses, nd their slts (chlorides, fluorides, sulftes of sodium, potssium, mgnesium, copper, zinc, etc.) is only vlid in queous medi. On the other hnd, wek electrolytes such s cetic cid or chlorocetic cid in concentrted queous solutions cn ssocite to such high degree s to chnge the properties of wter s solvent. The dielectric constnt of ir is so low tht there re no ions present in the vpor phse over solution of voltile electrolyte. All molecules re fully ssocited. In mercury or sodium lmps, ions exist in the vpor phse under voltge difference nd in the bsence of ir. BASIC RELATIONS ith these considertions in mind, without specifying the extent of the ctul dissocition in queous solution, for 1 mole of n electrolyte E tht in totl dissocition would give + ctions C Z+ nd nions A Z, we write C A = C Z + A + Z + + For single electrolyte, the electroneutrlity condition reduces to v + Z + + v Z = 0 (18.1) In this eqution, the sign of the chrge is implicit in Z i. For clrity, it is better to hve the signs of the chrges explicit nd write v + Z + = v Z (18.1) 149
150 Clssicl Thermodynmics of Fluid Systems Vritions of this reltion tht re often used in the literture without further explntion my be confusing t first, so we write some of them in detil. 1 + = = + Z Z Z + + + (18.1b) 1 Z = Z + + (18.1c) One importnt form is obtined by multiplying Eqution 18.1 first by Z to obtin + Z + Z = (Z ) 2 nd then multiplying Eqution 18.1 by Z + to obtin + (Z + ) 2 = Z + Z Tking the difference of these two expressions nd rerrnging, we get ZZ + ( Z ) ( Z ) = + + + 2 2 (18.1d) with + + (18.2) According to Eqution 12.15, the equilibrium constnt for the ionic dissocition in terms of ctivities tkes the form K + (18.3) T = + E For the dissocition of the electrolyte E, ccording to Eqution 12.16, the vlue of the equilibrium constnt is obtined from the stndrd Gibbs energy chnge: K T θ µ E µ µ = exp RT + + θ θ (18.4) Some tretments of electrolyte solutions hve proposed to use mole frctions s mesure of composition. For ll prcticl purposes, the use of mollity is simpler nd gives better rnge of vlues. As n exmple, the solubility of common slt (NCl) in wter t 298 K is 360 g kg 1 of wter or 6.16 moles per 55.51 moles of wter. Thus, t sturtion, tht is, the mximum concentrtion of slt possible t this temperture, the mole frction of ech ion is 0.100, while the mollity is 6.16. Hving decided to
The Thermodynmics of Aqueous Electrolyte Solutions 151 use mollity s the mesure of concentrtion, the next step is to choose the stndrd sttes for the ctivity coefficients to be used in the evlution of the ctivities by Thus, for the ction, nd for the nion, i = iγ i =i γ i (18.5) = γ = γ (18.5) + + + + + = γ = γ (18.5b) In this expression, s the ctivities re dimensionless, i is the dimensionless mollity of the ion i nd is the dimensionless mollity of the electrolyte solute; tht is, the vlue of the mollity divided by 1[mole of i/1000g of solvent]. Similrly to the cse of the use of mollity for nonelectrolytes discussed in Chpter 15, the reference stte for the ctivity coefficient of the ions is their stte t infinite dilution, nd their stndrd stte is the idel solution in Henry s sense t 1[mole of i/1000g of solvent]. At the reference stte, the ctivity coefficient of n ion is normlized to unity. lim γ = 1 mi 0 i (18.6) MEAN IONIC ACTIVITY COEFFICIENT At the stndrd stte, the ctivity of n ion is equl to unity (dimensionless). This is so becuse in its stndrd stte the ion is in n idel solution t unit mollity. The normliztion of the ctivity coefficients of the ions to unity t their stte in n infinitely dilute solution is of gret importnce. At this stte, the presence of ny other ion is immteril, be it co-ion or counterion. Thus, the sme condition is vlid independently of the nture of the electrolyte generting the ion. ith this normliztion, lthough its vlue is not known, the stndrd stte potentil of n ion in solution is fixed nd well defined, nd it is independent where the ion cme from. The stndrd stte for the electrolyte is chosen so tht the constnt K T in Eqution 18.4 is equl to unity. µ = µ + µ θ E + + θ θ Agin here, the vlue of µ θ E for the electrolyte is not known, but we know tht for ech electrolyte it hs fixed nd well-defined vlue depending only on the temperture nd the pir of ions forming the electrolyte. Hence, from Eqution 18.3 we write E + ( + + ) ( ) ± ( + ) = + + + = γ γ = γ (18.7)
152 Clssicl Thermodynmics of Fluid Systems where the men ionic ctivity coefficient of the electrolyte γ ± is defined s γ± γ+ + γ (18.8) with v v + + v, s defined by Eqution 18.2. At infinite dilution, by normliztion of the ctivity coefficients of the ions, we hve limγ = 1 m 0 ± (18.9) OSMOTIC COEFFICIENT For single electrolyte queous solution t constnt temperture, neglecting pressure effects, the Gibbs Duhem eqution, which reltes the chnges in E with the chnges in the ctivity of wter, tkes the form n E d ln E + n d ln = 0 where ne = nd n = 1000/M re the number of moles of slt nd wter, respectively. 1000 md ln E + ln = 0 M d (18.10) The ctivity of wter is sometimes given in terms of the osmotic coefficient of the solution, defined s 1000 ϕ M ( ) ln (18.11) j For n queous solution of nonvoltile electrolyte, the ctivity of wter is obtined directly by mesuring the vpor pressure, P, of the solution t the temperture of interest. From Eqution 16.4, we write j P = x γ = P (18.12) S Here, P S is the vpor pressure of pure wter t the temperture of the system. For work t high pressure, the correction fctors included in Eqution 16.2 should be included in Eqution 18.12. It is of interest to obtin the reltions between the osmotic coefficient of single electrolyte solution, φ, nd the men ionic ctivity coefficient of the electrolyte. For single electrolyte in solution, Eqution 18.11 tkes the form ln M ϕ (18.11) 1000
The Thermodynmics of Aqueous Electrolyte Solutions 153 Then, M M d ϕ ϕ d 1000 1000 Combining this expression with the Gibbs Duhem eqution, Eqution 18.10, we get From Eqution 18.7, d E =dϕ+ϕ d E = + γ ± Thus, equting these two expressions nd rerrnging, the reltion between the osmotic coefficient of single electrolyte solution nd the men ionic ctivity coefficient tkes the form ( 1) γ ϕ ± = dϕ+ dm (18.13) This differentil reltion cn be used to obtin the men ionic ctivity coefficient in terms of the osmotic coefficient nd vice vers. Integrting between the limit t infinite dilution, where the men ionic ctivity coefficient nd the osmotic coefficient tend to unity, nd mollity m, ( ϕ1) ln γ ± = ( ϕ 1) + dm 0 (18.14) Becuse in Eqution 18.14 the mollity ppers s rtio, for simplicity, the tilde differentiting it from its dimensionless vlue is sometimes dropped. Rerrnging Eqution 18.13, we write or md ln γ = md ϕ+ϕd d = dm ( ϕ) d ± dm ( ϕ ) = md ln γ + d Integrting between the sme limits s before nd rerrnging, ± ϕ= 1 ln γ ± + 1 m md 0 (18.15) Agin, s the mollity ppers s rtio in this expression, sometimes the difference between the mollity nd its dimensionless form is ignored.