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Topic 2400 Isopiestic; Aqueous Salt Solutions An extensive literature reports applications of the isopiestic technique to the determination of osmotic coefficients and ionic activity coefficients for salt solutions [1-11]. In effect the technique probes the role of ion-ion interactions in determining the properties of real salt solutions. Several approaches have been reported for analysing isopiestic results. A common method starts with the isopiestic ratio R iso . For solutions in dishes A and B at equilibrium, the isopiestic equilibrium conditions is given by equation (a). B i i i A j j j ) m ( ) m ( ν φ = ν φ (a) The isopiestic ratio, A j j B i i iso ) m /( ) m ( R ν ν = (b) An important task formulates an equation relating the osmotic coefficient for a given salt solution and the mean ionic coefficient ± γ If the salt solution contains a single salt, then according to the Gibbs-Duhem Equation, ) ( ) ( ) / ( aq d m aq d M 1 j j 1 1 µ = µ (c) Hence (where pressure p is close to the standard pressure) )] m / m ln( T R Q ( ) aq ( [ d m )] m M T R ( ) l ( [ d ) M / 1 ( 0 j 0 j j j 1 * 1 1 ± γ ν + µ = ν φ µ (d) Then, )] ln( ) m [(ln( d m )] m [ d j j j ± γ + = φ (e) Or, )] ln( d m / ) m ( d [ m )] [ d m dm j j j j j ± γ + = φ φ (f) Equation (f) is integrated between the limits ‘m j = 0’ amd m j [3,4]. Then, φ + φ = γ ± ) j ( m 0 j ) m ln( d ) 1 ( ) 1 ( ) ln( (g) And, ± γ + = φ ) j ( m 0 j j ) ln( d m m 1 1 (h) Hence the dependences of both ± γ and φ are obtained [1] for salt solutions and of both γ j and φ for solutions containing neutral solutes [5]. An important challenge at this stage is to express the experimentally determined dependence of φ on m j . Having expressed this dependence quantitively, the dependence of γ ± on m j is obtained using equation (g). The integration can be done graphically [6] or numerically using a computer-

based analysis. The Debye-Huckle Limiting Law plus ... · based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of φ on mj. ()∑

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Page 1: based analysis. The Debye-Huckle Limiting Law plus ... · based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of φ on mj. ()∑

Topic 2400

Isopiestic; Aqueous Salt Solutions An extensive literature reports applications of the isopiestic technique to the determination of

osmotic coefficients and ionic activity coefficients for salt solutions [1-11]. In effect the

technique probes the role of ion-ion interactions in determining the properties of real salt

solutions.

Several approaches have been reported for analysing isopiestic results. A common method

starts with the isopiestic ratio Riso. For solutions in dishes A and B at equilibrium, the isopiestic

equilibrium conditions is given by equation (a).

BiiiAjjj )m()m( ⋅ν⋅φ=⋅ν⋅φ (a)

The isopiestic ratio,

AjjBiiiso )m/()m(R ⋅ν⋅ν= (b) An important task formulates an equation relating the osmotic coefficient for a given salt

solution and the mean ionic coefficient ±γ

If the salt solution contains a single salt, then according to the Gibbs-Duhem Equation,

)()()/( aqdmaqdM1 jj11 µ⋅−=µ⋅ (c)

Hence (where pressure p is close to the standard pressure)

)]m/mln(TRQ()aq([dm

)]mMTR()l([d)M/1(0

j0jj

j1*11

±γ⋅⋅⋅⋅⋅ν+µ⋅−

=⋅⋅ν⋅⋅⋅φ−µ⋅ (d)

Then, )]ln()m[(ln(dm )]m[d jjj ±γ+⋅−=⋅φ (e)

Or, )]ln(dm/)m(d[m )][dmdm jjjjj ±γ+⋅−=φ⋅−⋅φ− (f)

Equation (f) is integrated between the limits ‘mj = 0’ amd mj [3,4]. Then,

∫ ⋅−φ+−φ=γ±

)j(m

0

j )mln(d)1()1()ln( (g)

And, ∫ ±γ⋅⋅+=φ)j(m

0

jj

)ln(dmm

11 (h)

Hence the dependences of both ±γ and φ are obtained [1] for salt solutions and of both γj and φ

for solutions containing neutral solutes [5].

An important challenge at this stage is to express the experimentally determined dependence of

φ on mj. Having expressed this dependence quantitively, the dependence of γ± on mj is obtained

using equation (g). The integration can be done graphically [6] or numerically using a computer-

Page 2: based analysis. The Debye-Huckle Limiting Law plus ... · based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of φ on mj. ()∑

based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the

dependence of φ on mj.

( ) ∑=

=γ ⋅+⋅−=φ

ji

1i

ir0ji

0j mmAmm3S1 )()/(/)/( (i)

The parameter r(i) increases in quarter powers. Then [7,8],

)(/ )/()/()ln( ir0j

ji

1i i

ii

210j mm

r

1rAmmS ⋅

+⋅+⋅−=γ ∑

=

=γ± (j)

In more recent accounts, Pitzer’s equations have been used to represent the dependence of φ on

ionic strength [9,10].

If the isopiestic experiments are repeated at several temperatures, the relative partial molar

enthalpy of the solvent L1(aq) is obtained [10].

In summary a large scientific literature reports thermodynamic data for aqueous solutions

containing salts [11] and mixed salt [12] systems. Footnotes

[1] G. Scatchard, W. J. Hamer and S. E. Wood, J.Am.Chem.Soc.,1938,60,3061.

[2] For reviews and further data compilations see

(a) R. N. Goldberg and R. L. Nuttall, J.Phys.Chem.Ref.Data, 1978,7,263.

(b) E. C. W. Clarke, J.Phys.Chem.Ref.Data, 1985,14,489.

[3] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn.

(revised), 1965, p. 34.

[4] A. K. Covington and R. A. Matheson, J. Solution Chem., 1977, 6, 263; NH4 CNS(aq).

[5]

(a) G . Barone, E. Rizzo and V. Volpe, J. Chem. Eng Data. 1976,21,59; alkyureas(aq)

(b) O. D. Bonner, C. F. Jordan, R. K. Arisman and J. Bednarek, J. Chem. Thermodyn.,

1976,8,1173; thioureas(aq)

(c) O. D. Bonner and W. H. Breazeale, J. Chem. Eng. Data,1965,10,325; dextrose(aq);

dimethylurea(aq).

(d) H. D. Ellerton and P. J. Dunlop, J. Phys.Chem.,1966,70,1831; sucrose(aq).

[6] J. A. Rard and D. J. Miller, J. Chem.Eng. Data 1982,27,169; CsCl(aq) and SrCl2(aq).

[7] J. A. Rard, J.Chem.Eng.Data,1987,32,92. La(NO3)3(aq) and Eu(NO3)3(aq).

Page 3: based analysis. The Debye-Huckle Limiting Law plus ... · based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of φ on mj. ()∑

[8] J. B. Maskill and R. G. Bates, J.Solution Chem.,1986,15,418 Tris(aq).

[9] L.M. Mukherjee and R. G. Bates, J.Solution Chem.,1985,14,255; R4N+Br-(D2O).

[10] S. Lindenbaum, L. Leifer, G. E. Boyd and J. W. Chase, J. Phys. Chem., 1970,74, 761;

R4NX(aq)

[11]

(a) KCl(aq) at 45 Celsius; T.M.Davis, L.M.Duckett, J.F.Owen, C.S.Patterson and R.Saleeby,

J.Chem.Eng. Data, 1985, 30,432.

(b) NH4Br(aq); A.K.Covington and D.Irish, J.Chem.Eng.Data, 1972,17,175.

(c) Sodium benzoate and hydroxybenzoates; J.E.Desnoyers, R.Page, G. Perron, J.-L.Fortier,

P.-A.Leduc and R.F.Platford, Can. J.Chem.,1973,51,2129.

(d) CaCl2(aq); L. M. Duckett, J. M. Hollifield and C. S. Patterson, J.Chem. Eng. Data, 1986,

31, 213.

(e) CaCl2(aq); J.A.Rard and F.H.Spedding, J.Chem.Eng.Data, 1977,22,56.

(f) Borates(aq); R. F. Platford, Can. J.Chem.,1969,47,2271.

(g) Pr(NO3)3(aq) and Lu(NO3)3; J.A.Rard, J. Chem..Eng.Data, 1987,32,334.

(h) Alkali metal trifluoroethanoates(aq); O.D.Bonner, J.Chem.Thermodyn.,1982,14,275.

[12]

(a) J.A Rard and D.G.Miller, J.Chem.Eng. Data, 1987,32,85; and references therein.

(b) G. E. Boyd, J. Solution Chem.,1977,6,95; NaCl+Na p-ethylbenzenesulfonate.

(c) A. K.Covington, T. H. Lilley and R. A. Robinson, J.Phys.Chem.,1968,72,2579; M+X-

pairs(aq).

(d) C. C. Briggs, R. Charlton and T. H. Lilley, J. Chem.Thermodyn., 1973, 5, 445; HClO4 +

NaClO4 + LiClO4(aq).

(e) C. P. Bezboruah, A. K. Covington and R. A. Robinson, J. Chem.Thermodyn., 1970, 2,

431; KCL + NaNO3(aq).

(f) S. Lindenbaum, R. M. Rush and R. A. Robinson, J. Chem.Thermodyn., 1972,4,381.

(g) D. Rosenzweig, J. Padova and Y. Marcus, J.Phys.Chem.,1976,80,601; NaBr+ R4NBr(aq).

(h) I.R.Lantzke, A.K.Covington and R.A.Robinson, J.Chem. Eng. Data, 1973,18,421;

Na2S2O6(aq), Na2SO3(aq).

Page 4: based analysis. The Debye-Huckle Limiting Law plus ... · based analysis. The Debye-Huckle Limiting Law plus extended form can be used to express the dependence of φ on mj. ()∑

(i) W.-Y. Wen, S.Saito and C-m. Lee, J. Phys. Chem.,1966,70,1244; R4NF(aq).

(j) A.K.Covington, R.A.Robinson and R.Thomson, J.Chem.Eng. Data, 1973,18,422;

methane sulfonic acid(aq).