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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 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-
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).
[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).
(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).